R 入門

目  次

An Introduction to R

This is an introduction to R (“GNU S”), a language and environment for statistical computing and graphics. R is similar to the award-winning¹ S system, which was developed at Bell Laboratories by John Chambers et al. It provides a wide variety of statistical and graphical techniques (linear and nonlinear modelling, statistical tests, time series analysis, classification, clustering, ...).

This manual provides information on data types, programming elements, statistical modelling and graphics.

This manual is for R, version 3.4.1 (2017-06-30).

Copyright © 1990 W. N. Venables
Copyright © 1992 W. N. Venables & D. M. Smith
Copyright © 1997 R. Gentleman & R. Ihaka
Copyright © 1997, 1998 M. Maechler
Copyright © 1999–2016 R Core Team

Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.

序文

This introduction to R is derived from an original set of notes describing the S and S-PLUS environments written in 1990–2 by Bill Venables and David M. Smith when at the University of Adelaide. We have made a number of small changes to reflect differences between the R and S programs, and expanded some of the material.

We would like to extend warm thanks to Bill Venables (and David Smith) for granting permission to distribute this modified version of the notes in this way, and for being a supporter of R from way back.

Comments and corrections are always welcome. Please address email correspondence to R-core@R-project.org.

Suggestions to the reader

Most R novices will start with the introductory session in Appendix A. This should give some familiarity with the style of R sessions and more importantly some instant feedback on what actually happens.

Many users will come to R mainly for its graphical facilities. See グラフィクス, which can be read at almost any time and need not wait until all the preceding sections have been digested.

1 紹介と準備

1.1 R の環境

R はデータ操作、計算、グラフィックス表示といったソフトウェア機能の統合環境である。中でも 次のような機能を持つ

• 効率的なデータの操作と蓄積機能、
• 配列、特に行列に対する一連の操作、
• データ解析における中間的な操作のための、豊富で一貫した、そして統合された道具、
• データの解析や表示を直接計算機で表示したり、印刷するためのグラフィック機能、そして
• 条件実行、ループ、ユーザ定義の再帰的関数、そして入出力機能を持つ、周到に開発された単純 で効率的なプログラミング言語。(実際、システムに用意された関数のほとんどは、それ自身 S 言語によって書かれている。)

「環境」という言葉は、他のデータ解析ソフトにしばしば見られるような、次々と付け加えられた、 極めて局限され融通の効かない道具の蓄積ではなく、それが十分に計画され一貫したシステムである とことを特徴づけるために用いられている。

R は、対話的なデータ解析手法を新たに開発するための手段である。それは急速に発展し、パッケージの大規模なコレクションによって拡張されてきた。 しかし、R で書かれたほとんどのプログラムは、本質的に一時的なものであり、一部のデータ解析のために書かれている。 It has developed rapidly, and has been extended by a

1.2 関連するソフトウェアとドキュメント

R は、ベル研究所において Rick Becker、John Chambers、そして Allan Wilks によって開発され、S-Plus システムの基にもなっている S 言語を 実装したものと見なすことができる。

S 言語の発展は Jhon Chambers と共著者らによる 4 冊の本により特徴付けられる。R にとっての基本的文献は、Rick A. Becker、John M. Chambers、 そしてAllan R. Wilks によって書かれた『The New S Language: A Programming Environment for Data Analysis and Graphics』である。 S の 1991 年 8 月のリリース(S の第 3 版) における新しい機能は John M. Chambers と Trevor J. Hastie の編集による『Statistical Models in S』 に盛り込まれている。正確な文献は、Appendix F 参考文献を見よ。

そのほかにも、R をデータ解析や統計に使用する方法を説明する数多くの書籍があるとともに、S / S-PLUS 用のドキュメントはは通常、S の実装の違いを 念頭に置けば R でも使用できる。"The R statistical system FAQ" の中の R にはどんなドキュメントがあるの？ を参照。

1.3 R と統計

我々の R 環境への入門は「統計学」に就いて言及していなかったが、多くの人は R を統計システムとして使うであろう。 我々はそれを多くの古典的、現代的な統計学手法が実装された環境と考えたい。これらのいくつかは、基本的な R 環境に組み込まれているが、 多くは「パッケージ」として提供されている。R と共に提供されているパッケージは約 25 種類(“標準” や "推奨" パッケージと呼ばれる) ある。 もっと多くのパッケージが CRAN のインターネットサイト(https://CRAN.R-project.org 経由) で入手できる。

ほとんどの古典的統計学手法と最新の方法論の多くが R で使えるが、ユーザはそれを見つけ出すのに少々手間取ることを覚悟する必要があるだろう。

S（したがって R）と他の主要な統計システムの間には原理的に重要な違いがある。S では統計解析は普通何段階かのステップで行われ、 途中結果はオブジェクトに保管される。従って、SAS や SPSS は回帰や判別分析に対し、おびただしい結果を出力するが、R は最小限に出力を抑え、他の R 関数を 用いた次の疑問に応えるために、結果を適切なオブジェクトに保存する。

1.4 R とウィンドウシステム

R がもっとも便利に使えるのは、ウィンドウ・システムが動くグラフィクス・ワークステーションにおいてであろう。 このガイドもこの機能が使えるユーザを意図している。説明のほとんどすべては R 環境のあらゆる実装に対して適用されるが、 特に X-window システム上での R の用法にしばしば言及する。

多くのユーザは折にふれ、使用中の計算機の OS を直接操作する必要があるであろう。このガイドでは Unix マシンの OS との対話を中心に記述する。 Windows または macOS で R を動かしているなら、小さな調整が必要になるであろう。

R のカスタマイズ可能な特徴を、最大限に活用できるにワークステーションを設定することは、 少々退屈な手順であるため、ここでは今後もあまり触れないことにする。困難を感じたときは、周囲のエキスパートの助けを求めるべきである。

1.5 R の対話的な使用法

R プログラムを使うと、命令の入力が期待されるときはプロンプトを表示する。既定のプロンプトは ‘>’ で、 Unix ではシェルのプロンプトと同じかもしれないので、何も起きていないように見えるかも知れない。しかしながら、後で見るように、必要なら R のプロンプトを 別の物に変えるのは容易である。この文書では、今後 Unix のシェルプロンプトは ‘$’ であると仮定する。

Unix で R を使う際、推奨される最初に行う手順は次の通りである：

1. まずこの問題に対し R で作業するデータファイルを保持する独立したサブディレクトリ（例えば work）を作る。 これは、この特別な問題に対し R を使う際には、いつも作業ディレクトリになるであろう。

   $ mkdir work
   $ cd work
   

2. 次の命令で R を起動する

   $ R
   

3. ここでR の命令を入力できる(後を見よ)。
4. R を終了する命令は

   > q()
   

   ここで、R セッションのすべてのデータを保存するかどうかを聞かれるであろう。 終了する前に、データを保存するか、保存しないで終了するか、または R セッションに戻るために、それぞれ yes、no または cancel (頭の １ 文字 だけでよい) で答える。保存したデータは将来の R セッションで利用できる。

次回の R セッションは簡単である。

1. work を作業ディレクトリにし、前と同じようにプログラムを開始する：

   $ cd work
   $ R
   

2. R プログラムをを使い、セッションの最後で q() を用いて終了する。

Windows で R を使うときも、従う手順は基本的に同じである。作業ディレクトリとしてフォルダを作り、それを R のショートカット中の Start In 欄にセットすし、次にアイコンをダブルクリックし R を起動する。

1.6 入門セッション

先に進む前に、計算機における R の雰囲気を得たいと思う読者は、サンプルセッションにおける [入門セッション]を 実行してみることを強く勧める。

1.7 関数と機能に関するヘルプの利用

R には UNIX の man 機能に似た組み込みのヘルプ機能が備わっている。個々の名前のついた関数、例えば solve に ついて詳しいことが知りたいなら、命令は

> help(solve)

である。または

> ?solve

でも良い。特殊文字で指定される機能が知りたいなら、引数を二重引用符か一重引用符で囲み“文字列” にする必要がある： これは、if、for、function などの構文上の意味を持ついくつかの単語にも必要でである。

> help("[[")

2 種類の引用符の各々は、 "It's important" という例におけるように、他方をエスケープするのに使うことができる。 我々は便宜的に二重引用符を優先的に使うことにする。

R では、ほとんどの場合、ヘルプ文章を HTML 形式で得ることができ、命令

> help.start()

を実行するとウエッブブラウザを起動し、ヘルプ頁をハイパーリンクで閲覧できる。UNIX では後続するヘルプの要求は、HTML 形式の ヘルプシステムに送られる。help.start() によってロードされるページにある「Search Engine and Keywords」というリンクは、 高水準の概念の一覧表によって利用可能な関数から検索できるので特に有用である。手早く答えを得るのに重宝するし、R が提供するべきものの広がりをよく理解できる。

The help.search 命令 (または ??) よるヘルプの検索にはいろいろなやり方がある。 例えば、

> ??solve

?help.search によって詳細と使用例を見てほしい。

ヘルプのトピックに関する例は次のようにして実行できる

> example(topic)

R のWindows 版は他のオプションのヘルプシステムを持つ。詳細は命令、

> ?help

を使用せよ。

1.8 R コマンド、大文字と小文字の区別など

技術的に、R は非常に単純な構文を持つ「表現式言語」である。大抵の UNIX ベースのパッケージと同様に「大文字と小文字を区別」するので、 A と a は異なった記号であり、異なる変数として参照される。 R の名前に使えるシンボルのセットは、R が実行されているオペレーティングシステムと国に(技術的には使用している locale に) 依存する。 通常は、英数字すべて²(国によってはアクセントのついた文字を含む) に加え、 ‘.’ と ‘_’ が使えるが、‘.’ で始まる場合は、2 番目の文字は数字であってはならないと言う制約がある。名前の長さは、事実上制限がない。

基本的な命令は「式(expression)」または「代入(assignment)」からなる。もしある式が命令として与えられると、それは評価され、 （特に非表示にしない限り）表示され、そしてその値は失われる。代入も同様に、式を評価し、その値を変数に渡すが、結果は自動的には表示されない。

命令はセミコロン(‘;’) か、改行で区切られる。複数の基本的な命令は、 括弧 (‘{’ と ‘}’) で括ることにより、一つの式に束ねることができる。 注釈は、ほとんどどこにでも³ 置くことができ、ハッシュマーク(‘#’) からその行の末尾までは注釈と見倣される。

ある命令が行の最後で完結していなければ、R は別種のプロンプトを表示する、既定では

+

を 2 行目、そしてそれ以降の行に表示し、命令が構文的に完成するまで入力を読み続ける。このプロンプトはユーザが変更できる。 この文書では、この継続プロンプトは普通表示せず、単純な字下げで継続を示すことにする。

コンソールに入力するコマンドラインは、約 4095 バイト（文字ではない）に制限⁴されている。

1.9 前のコマンドの再呼び出しと修正

UNIX と Windows の多くのバージョンで、R は以前の命令の再呼び出しと再実行の機能を持っている。キーボードの垂直矢印キーを用いて 「命令履歴」を前後にスクロールすることができる。このようにして、一旦ある命令を選んだら、命令内を水平矢印キーを用いてカーソル移動でき、文 字を DEL キーで削除したり、他のキーで付け加えることができる。詳細は後で説明する、 コマンドラインエディタ を見よ。

再呼び出しや編集機能は大幅にカスタマイズできる。readline ライブラリに関するマニュアル項目を読めば、方法が分かるであろう。

あるいは Emacs テキストエディタが、R を用いた対話的な作業に関するより一般的な支援のメカニズム(ESS, Emacs Speaks Statistics を用いた) を提供してくれる。マニュアル The R statistical system FAQ の, R と Emacs を参照のこと。

1.10 ファイルからのコマンドの実行とファイルへの出力先変更

もし、命令⁵ が作業ディレクトリ work にある外部ファイル、 例えば commands.R に保存されているなら、それは R セッション中の任意の機会に、命令

> source("commands.R")

で実行できる。
Windows では、File メニューの Source が同じように使える。関数 sink

> sink("record.lis")

は、それ以降の端末からのすべての出力を外部ファイル record.lis に切替える。命令

> sink()

は、それをもう一度端末に戻す。

1.11 データの永続性とオブジェクトの削除

R が作ったり操作した実体は、オブジェクトと呼ばれる。これは変数、数の配列、文字列、関数、またはこれらの要素から作り上げられたより 一般的な構造であったりする。

一つの R のセッション中に、オブジェクトが作られ、名前を付けて保存される(このプロセスについては次のセッションで解説される)。 R 命令

> objects()

(または ls()) は、R 内に現在格納されている（ほとんどの）オブジェクトの名前を表示するために使用できる。 現在格納されているオブジェクトの全体を作業スペース(workspace) と呼ぶ。

オブジェクトを削除するには、rm 関数が利用できる：

> rm(x, y, z, ink, junk, temp, foo, bar)

一つの R セッション中に作られた全てのオブジェクトは、将来の R セッションで使えるようにファイルに永続的に保存できる。 各 R セッションを終了する際、現在利用できる全てのオブジェクトを保存する機会が与えられる。保存を選択すると、すべてのオブジェクトは 現在のディレクトリの .RData⁶ という名前のファイルに書き込まれ、 セッションで使用した命令行は .Rhistory というファイルに保存される。

後で R を起動したとき、このファイルから作業スペースを再読み込みする。と同時に関連する命令履歴も再読み込みされる。

R を用いた解析をするときは、別個の作業ディレクトリを使うべきである。ある解析において、x や y といった 名前のオブジェクトが作ることはよくあることである。このような名前は、単一の解析の文脈においては意味を持つことが多いが、 同一のディレクトリで複数の解析を行うならば、それが何を意味するかを判別するのは極めて困難になってしまう。

2 数とベクトルの簡単な操作

2.1 ベクトルと代入

R は、名前付き「データ構造(data structures)」を処理する。このような構造の中で最も単純なのは、順序付けられた数値の集合からなる単一の 実体である数値ベクトルです。5 つの数字（例えば、(たとえば 10.4, 5.6, 3.1, 6.4, 21.7）からなる x という名前のベクトルをつくるには、R の命令

> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)

を用いる。これは「関数(function)」c() を使った「代入(assignment)」であり、この文脈では、 任意の個数のベクトル「引数(arguments)」を取ることができ、その返り値はその引数を端から端まで連結して得られるベクトルである。 ⁷

ある式中に単独で現れた数値は、長さ 1 のベクトルと見なされる。

代入演算子（‘<-’）は、2つの文字 ‘<’ (“より小さい”) と ‘-’ (“マイナスs”) が厳密に並んだもので、式の値を受け取るオブジェクトを「ポイント」している点に 注意すること。ほとんどの場合、‘=’ 演算子を使用することもできる。

代入は、assign() 関数を使うことによっても可能である。上記の代入操作と同様のことを行うには：

> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))

とすればよい。通常の演算子 <- は、この構文の短縮形と考えることができる。

代入は、別向きで行うことも可能で、代入演算子の自明な変更を伴う。従って、同じことを

> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x

としてもよい。もし式が完全な命令として実行されるならば、その値が表示され、そして「失わ れる」⁸。そのため、もし命令

> 1/x

を実行すると、5 つの値の逆数がターミナルに表示される(そして、もちろん x の値は変化しない)。

更に次の代入

> y <- c(x, 0, x)

は、x の 2 つのコピーの真ん中に零がある、11 個の項目からなるベクトルr y を作るだろう。

2.2 ベクトル演算

ベクトルは算術式で使うことができ、その場合演算は要素毎に行われる。同じ式のベクトルは全て同じ長さである必要は無い。 長さが同じでなければ、式の返り値は、式内の最も長い配列と同じ長さの配列になる。式中のより短いベクトルは、それが最長のベクトルの長さに一致 するまで(おそらく部分的に) 必要なだけ「リサイクル」される。特に定数は単純に繰り返される。従って、上記の代入において、次の命令

> v <- 2*x + y + 1

は、要素毎にそれぞれ、2*x は 2.2 回、y は 1 回のみ、1 は 11 回 繰返すことで長さ 11 のベクトルを作成した後、v に演算結果を代入する。

基本的な演算子は、通常の +、-、*、/、^（冪乗）である。 更に普通の全ての算術関数を使うことができる。 log、exp、sin、cos、tan、sqrt 等は全てそれらの通常の意味を持つ。 max と min は、それぞれベクトル中の最も大きいものと最も小さいものを選択する。 range は、長さが 2 のベクトル、つまり c(min(x), max(x)) を値に持つ関数である。 length(x) は、x の要素数を返し、 sum(x) は、x の要素の総和を返し、 prod(x) は、全要素の積を返す。

統計関数として、標本平均を計算する mean(x) があり、これは sum(x)/length(x) と同じである。 var(x) は

sum((x-mean(x))^2)/(length(x)-1)

つまり標本分散を与える。もし var() の引数が n × p 行列なら、その値は列を独立な p-変量標本ベクトル とみなした p × p 標本共分散行列となる。

sort(x) は、x と同じ長さの、要素を昇順に並べ変えた配列を返す。しかしながら、その他にも もっと融通の効くソート機能がある（ソートのための順位を生成する order() や sort.list() をみよ)。

max と min は、引数が複数のベクトルでも、引数中の最大と最小の値を選択することに注意のこと。 「並列化(parellel)」最大・最小関数である pmax と pmin は、各入力ベクトルの位置にその最大(最小) 値を置い た(長さがその引数の長さの最大値である) ベクトルを作り出す。

ほとんどの目的にとっては、数値ベクトル中の“数” が整数とか、実数や複素数とかには関心が無いであろう。 内部的な計算は倍精度実数か、もし入力データが複素数なら倍精度複素数で行われる。

複素数を取扱うなら、虚数部を明示的に与える。したがって

sqrt(-17)

は NaN と警告を与えるが、

sqrt(-17+0i)

は、複素数として計算を行うであろう。

2.3 規則的な数列の生成

R はよく使われる数列を生成する幾つかの機能を持つ。例えば、1:30 は ベクトル c(1, 2,…, 29, 30) である。 コロン演算子は一つの式で最も高い優先度を持つので、例えば 2*1:15 は c(2, 4, …, 28, 30) というベクトルになる。 n <- 10 として、数列 1:n-1 と 1:(n-1) を比べてみよ。

構成 30:1 は、降順の数列を作るのに使うことができる。

関数 seq() は数列を生成するもっと一般的な関数である。これは 5 個の引数を持つが、特定の呼出しでは、その一部分だけを指定すればよい。 最初の 2 つの引数は。もし存在すれば。数列の最初と最後を指定し、もしこれだけが引数なら、結果はコロン演算子と同じになる。 つまり seq(2,10) は、2:10 と同じベクトルになる。

他の多くの R の関数と同様に、seq() へのパラメータは名前付き形式で与えることができ、その場合には現れる順序には無関係である。 最初の2 つのパラメータは、from=value and to=valuefrom=value とto=value と名前を付けることができる。 従って、seq(1,30), seq(from=1, to=30) と seq(to=30,from=1) はすべて 1:30 と同じ結果になる。 seq() の 2 つのパラメータは、by=value と length=value という名前を付けることができ、 それぞれ数列の増分と長さを指定する。もしどちらも指定しなければ、by=1 が既定として仮定される。

例えば

> seq(-5, 5, by=.2) -> s3

とすると、s3 は ベクトル c(-5.0, -4.8, -4.6, …,4.6, 4.8, 5.0) になる。同じく

> s4 <- seq(length=51, from=-5, by=.2)

とすると s4 も同じ配列になる。

5 番目のパラメータは、along=vector という名前を付けることができ、もし使うのなら、このパラメータだけを指定しなければならない。 これは、1, 2,…, length(vector) というベクトルを作るか、空のベクトルを指定(可能である) すると空の数列をつくる。

関連した関数は rep() で あり、さまざまな複雑なやり方でオブジェクトを複製するのに使うことができる。 最も簡単な例

> s5 <- rep(x, times=5)

は、x を端から端まで 5 つのコピーしたものを s5 にする。もう一つの便利なバージョンは

> s6 <- rep(x, each=5)

で、これは x の各要素を 5 回繰り返してから次の要素に移動する。

2.4 論理ベクトル

数値ベクトルと同様に、R は論理量を扱うことができる。論理ベクトルの要素はTRUE、FALSE、NA (欠損値のこと、後述) という値を持つことができる。初めの2 つは、それぞれ T と F に略される場合がある。 ただし、T と F は予約語ではなく、既定で TRUE や FALSEに設定されている変数にすぎないので、 ユーザが上書きすることが出来る。従って、常に TRUE や FALSE を使用すべきである。

論理ベクトルは「条件(conditions)」により生成される。例えば、

> temp <- x > 13

は、temp に x と同じ長さで、x の対応する要素が条件を「満足しなければ」FALSE を、 「満足すれば」TRUE にしたベクトルを設定する。

論理演算子には、<、<=、>、>=、完全な一致を表す == 、 不一致を表す != がある。 更に、もし c1 と c2 が論理式なら、c1 & c2 は論理積(“and”)、 c1 | c2 は論理和(“or”)、そして!c1 は c1 の否定である。

論理ベクトルは、通常の算術演算において使うことができ、そのときは数値ベクトルに「強制変換(coerced)」され、FALSE は 0 に、 TRUE は 1 は 1 になる。しかしながら、論理ベクトルとその強制変換された数値ベクトルが等価にならない場合がある。 例は、次の副節を参照のこと。

2.5 欠損値

ベクトルの要素が完全には分からないことがある。統計的な意味で要素や値が「利用不能（not available）」や「欠損値」であるとき、 ベクトル中のその位置に NA という特別な値を代入して保存しておくことができる。 一般的に、NA に対する演算は NA となる。この規則を設けた理由は単純で、もしある演算に対する指示が不完全ならば、 結果を知ることができず、したがって利用不可能だからである。

is.na(x) 関数は、x と同じ長さの（要素数が同じ）論理ベクトルで、x の中の対応する要素の値が NA の時のみ、 値 TRUE を返す。

> z <- c(1:3,NA);  ind <- is.na(z)

NA は実際は値でなく、利用不可能な量に対するマーカであるので、論理式 x == NA と is.na(x) は全く別物であることを注意しよう。そのため、x == NA とは x と同じ長さのベクトルで、論理式自体が不完全であり、 従って決定不能であるため、その「全て」の値が NA である。

数値計算によって生まれるもう一種の“欠損”値がある。いわゆる「非数(Not a Number)」で ある NaN 値である。例えば

> 0/0

または

> Inf - Inf

で、ともに意味のある結果が定義できないため NaN となる。

要約すると、is.na(xx) は、NA と NaN の双方に対し TRUE となる。 これらを区別するために、is.nan(xx) は NaN に対してだけ TRUE となる。

文字ベクトルが引用符なしで印刷されているときに、欠損値が <NA> と表示されることがある。

2.6 文字ベクトル

例えばプロットのラベル等、R では文字や文字ベクトルを頻繁に使う。必要に応じて、二重引用符で区切られた一連の文字、 例えば "x-values"、"New iteration results" によって示される。

文字列は二重引用符(") か一重引用符(') を使って入力されるが、出力するときは二重引用符である(引用符なしの場合もある)。 文字列は、エスケープ文字として \ を使った C スタイルのエスケープシーケンスを採用しているので、\\ と入力すると \ が出力され、二重引用符の中の " は \" と入力する。 その他の有用なエスケープシーケンスとしては、\n（改行）、\t（タブ）、code>\b（バックスペース）があるが、 全てのリストは ?Quotes で参照せよ。

複数の文字ベクトルを c()) 関数によって、1 つのベクトルに連結するすることができる。以下で多くの使用例が登場するであろう。

paste() 関数は任意の数の引数を取り、それらを 1 つの文字列に連結する。引数中に与えられた数値は明示的に文字列へ強制変換される。 つまり、それらが印字した時と同じような形になる。引数は、既定では 1 つの空白文字で区切られた形になるが、これは名前付きパラメータ sep=string で変更でき、区切り文字を string に変える。空文字(区切り文字無し) も可能である。

例えば、

> labs <- paste(c("X","Y"), 1:10, sep="")

は labs を文字ベクトル

c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")

にする。ここでも短いリストはリサイクルが行われるということを注意せよ；つまり、c("X", "Y") は 数列 1:10 にマッチするまで 5 回繰り返される。⁹

2.7 添字ベクトル：データセットのサブセットの選択と変更

ベクトルの要素の一部分は、ベクトル名に角括弧に入った「添字ベクトル(index vector)」を付加することにより選択することができる。 より一般的に、結果がベクトルとして評価される任意の式は、式の直後に角括弧に入った添字ベクトルを加えることにより、同じようにその要素の部分 集合を得ることができる。

このような添字ベクトルは、4 つの異なったタイプのいずれでも良い。

1. 論理ベクトル。この場合、添字ベクトルは、要素が選択されるベクトルと同じ長さまでリサイクルされる。 添字ベクトル中の TRUE に対応する値が選択され、FALSE に対応するものは無視される。例えば、

   > y <- x[!is.na(x)]
   

   は、x の欠損値でない値を、同じ順序でオブジェクト y を作る(または再作成する)。 もし x に欠損値があれば、y は x よりも短くなることを注意しよう。同様に

   > (x+1)[(!is.na(x)) & x>0] -> z
   

   は、オブジェクト z をつくり、その中の対応する x の値が欠損値でなく、且つ正の値の場合は ベクトル x+1 を設定する。

2. 正の整数ベクトル。この場合、添字ベクトル中の値は集合 {1, 2, …,length(x)} の中になければならない。 ベクトルの対応する要素が選択され、その順序で結果に連結される。添字ベクトルの長さは任意で、結果は添字ベクトルと同じ長さとなる。 例えば、x[6] は x の 6 番目の成分であり、

   > x[1:10]
   

   は、x の 最初の 10 個の要素を選択する（length(x) が 10 未満で無ければ)。同様に

   > c("x","y")[rep(c(1,2,2,1), times=4)]
   

   は、"x", "y", "y", "x" を 4 回繰り返す、長さ 16 の文字ベクトルを作る(余り行わないことだろうが)。

3. 負の整数ベクトル。この添字ベクトルは、選択するよりも「除外すべき」値を指定する。従って

   > y <- x[-(1:5)]
   

   は、x の最初の 5 つの要素を除いた残り全てからなる y が得られる。

4. 文字列ベクトル。これは、オブジェクトがその要素を特定するための names 属性を持つ場合のみに使われる。 このケースでは、名前ベクトルのサブベクトルを、2 つ上の項「正の整数ベクトル」と同様に使うことができる。

   > fruit <- c(5, 10, 1, 20)
   > names(fruit) <- c("orange", "banana", "apple", "peach")
   > lunch <- fruit[c("apple","orange")]
   

   文字と数字を組み合わせた「名前(names)」は、しばしば「数字の添字」よりも覚えやすいという利点がある。あとで分かるように、 このオプションはデータフレームとの関連でとりわけ役にたつ。

添字付き式は、代入される側にも現れることができる。この場合、代入演算は「ベクトルのそれらの要素に対してのみ」実行される。 式はベクトル vector[index_vector] の形式でなければならない。ベクトル名の代わりに任意の式にするのは ここではあまり意味が無いためである。

例えば

> x[is.na(x)] <- 0

は、x 中の欠損値を零に置き換え、そして

> y[y < 0] <- -y[y < 0]

は、次と同じ効果を持つ

> y <- abs(y)

2.8 他のタイプのオブジェクト

Vectors are the most important type of object in R, but there are several others which we will meet more formally in later sections.

• matrices or more generally 配列 are multi-dimensional generalizations of vectors. In fact, they are vectors that can be indexed by two or more indices and will be printed in special ways. See 配列と行列.
• 因子 provide compact ways to handle categorical data. See 因子.
• lists are a general form of vector in which the various elements need not be of the same type, and are often themselves vectors or lists. Lists provide a convenient way to return the results of a statistical computation. See Lists.
• data frames are matrix-like structures, in which the columns can be of different types. Think of data frames as ‘data matrices’ with one row per observational unit but with (possibly) both numerical and categorical variables. Many experiments are best described by data frames: the treatments are categorical but the response is numeric. See Data frames.
• functions are themselves objects in R which can be stored in the project’s workspace. This provides a simple and convenient way to extend R. See 自分自身の関数を書く.

3 オブジェクト、そのモードと属性

3.1 本質的属性：モードと長さ

The entities R operates on are technically known as オブジェクト. Examples are vectors of numeric (real) or complex values, vectors of logical values and vectors of character strings. These are known as “atomic” structures since their components are all of the same type, or mode, namely numeric¹⁰, complex, logical, character and raw.

Vectors must have their values all of the same mode. Thus any given vector must be unambiguously either logical, numeric, complex, character or raw. (The only apparent exception to this rule is the special “value” listed as NA for quantities not available, but in fact there are several types of NA). Note that a vector can be empty and still have a mode. For example the empty character string vector is listed as character(0) and the empty numeric vector as numeric(0).

R also operates on objects called lists, which are of mode list. These are ordered sequences of objects which individually can be of any mode. lists are known as “recursive” rather than atomic structures since their components can themselves be lists in their own right.

The other recursive structures are those of mode function and expression. Functions are the objects that form part of the R system along with similar user written functions, which we discuss in some detail later. Expressions as objects form an advanced part of R which will not be discussed in this guide, except indirectly when we discuss formulae used with modeling in R.

By the mode of an object we mean the basic type of its fundamental constituents. This is a special case of a “property” of an object. Another property of every object is its length. The functions mode(object) and length(object) can be used to find out the mode and length of any defined structure ¹¹.

Further properties of an object are usually provided by attributes(object), see 属性の取得と設定. Because of this, mode and length are also called “intrinsic attributes” of an object.

For example, if z is a complex vector of length 100, then in an expression mode(z) is the character string "complex" and length(z) is 100.

R caters for changes of mode almost anywhere it could be considered sensible to do so, (and a few where it might not be). For example with

> z <- 0:9

we could put

> digits <- as.character(z)

after which digits is the character vector c("0", "1", "2", …, "9"). A further coercion, or change of mode, reconstructs the numerical vector again:

> d <- as.integer(digits)

Now d and z are the same.¹² There is a large collection of functions of the form as.something() for either coercion from one mode to another, or for investing an object with some other attribute it may not already possess. The reader should consult the different help files to become familiar with them.

3.2 オブジェクト長の変更

An “empty” object may still have a mode. For example

> e <- numeric()

makes e an empty vector structure of mode numeric. Similarly character() is a empty character vector, and so on. Once an object of any size has been created, new components may be added to it simply by giving it an index value outside its previous range. Thus

> e[3] <- 17

now makes e a vector of length 3, (the first two components of which are at this point both NA). This applies to any structure at all, provided the mode of the additional component(s) agrees with the mode of the object in the first place.

This automatic adjustment of lengths of an object is used often, for example in scan() 関数 for input. (see The scan() function.)

Conversely to truncate the size of an object requires only an assignment to do so. Hence if alpha is an object of length 10, then

> alpha <- alpha[2 * 1:5]

makes it an object of length 5 consisting of just the former components with even index. (The old indices are not retained, of course.) We can then retain just the first three values by

> length(alpha) <- 3

and vectors can be extended (by 欠損値) in the same way.

3.3 属性の取得と設定

The function attributes(object) returns a list of all the non-intrinsic attributes currently defined for that object. The function attr(object, name) can be used to select a specific attribute. These functions are rarely used, except in rather special circumstances when some new attribute is being created for some particular purpose, for example to associate a creation date or an operator with an R object. The concept, however, is very important.

Some care should be exercised when assigning or deleting attributes since they are an integral part of the object system used in R.

When it is used on the left hand side of an assignment it can be used either to associate a new attribute with object or to change an existing one. For example

> attr(z, "dim") <- c(10,10)

allows R to treat z as if it were a 10-by-10 matrix.

3.4 オブジェクトのクラス

All objects in R have a class, reported by the function class. For simple vectors this is just the mode, for example "numeric", "logical", "character" or "list", but "matrix", "array", "factor" and "data.frame" are other possible values.

A special attribute known as the class of the object is used to allow for an object-oriented style¹³ of programming in R. For example if an object has class "data.frame", it will be printed in a certain way, the plot() function will display it graphically in a certain way, and other so-called generic functions such as summary() will react to it as an argument in a way sensitive to its class.

To remove temporarily the effects of class, use the function unclass(). For example if winter has the class "data.frame" then

> winter

will print it in data frame form, which is rather like a matrix, whereas

> unclass(winter)

will print it as an ordinary list. Only in rather special situations do you need to use this facility, but one is when you are learning to come to terms with the idea of class and generic functions.

Generic functions and classes will be discussed further in Object orientation, but only briefly.

4 順序付き因子と順序なし因子

A factor is a vector object used to specify a discrete classification (grouping) of the components of other vectors of the same length. R provides both ordered and unordered factors. While the “real” application of factors is with model formulae (see Contrasts), we here look at 具体例.

4.1 具体例

Suppose, for example, we have a sample of 30 tax accountants from all the states and territories of Australia¹⁴ and their individual state of origin is specified by a character vector of state mnemonics as

> state <- c("tas", "sa",  "qld", "nsw", "nsw", "nt",  "wa",  "wa",
             "qld", "vic", "nsw", "vic", "qld", "qld", "sa",  "tas",
             "sa",  "nt",  "wa",  "vic", "qld", "nsw", "nsw", "wa",
             "sa",  "act", "nsw", "vic", "vic", "act")

Notice that in the case of a character vector, “sorted” means sorted in alphabetical order.

A factor is similarly created using the factor() function:

> statef <- factor(state)

The print() function handles factors slightly differently from other objects:

> statef
 [1] tas sa  qld nsw nsw nt  wa  wa  qld vic nsw vic qld qld sa
[16] tas sa  nt  wa  vic qld nsw nsw wa  sa  act nsw vic vic act
Levels:  act nsw nt qld sa tas vic wa

To find out the levels of a factor the function levels() can be used.

> levels(statef)
[1] "act" "nsw" "nt"  "qld" "sa"  "tas" "vic" "wa"

4.2 関数 tapply() と不整配列

To continue the previous example, suppose we have the incomes of the same tax accountants in another vector (in suitably large units of money)

> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,
               61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,
               59, 46, 58, 43)

To calculate the sample mean income for each state we can now use the special function tapply():

> incmeans <- tapply(incomes, statef, mean)

giving a means vector with the components labelled by the levels

   act    nsw     nt    qld     sa    tas    vic     wa
44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250

The function tapply() is used to apply a function, here mean(), to each group of components of the first argument, here incomes, defined by the levels of the second component, here statef¹⁵, as if they were separate vector structures. The result is a structure of the same length as the levels attribute of the factor containing the results. The reader should consult the help document for more details.

Suppose further we needed to calculate the standard errors of the state income means. To do this we need to write an R function to calculate the standard error for any given vector. Since there is an builtin function var() to calculate the sample variance, such a function is a very simple one liner, specified by the assignment:

> stdError <- function(x) sqrt(var(x)/length(x))

(Writing functions will be considered later in 自分自身の関数を書く. Note that R’s a builtin function sd() is something different.) After this assignment, the standard errors are calculated by

> incster <- tapply(incomes, statef, stdError)

and the values calculated are then

> incster
act    nsw  nt    qld     sa tas   vic     wa
1.5 4.3102 4.5 4.1061 2.7386 0.5 5.244 2.6575

As an exercise you may care to find the usual 95% confidence limits for the state mean incomes. To do this you could use tapply() once more with the length() function to find the sample sizes, and the qt() function to find the percentage points of the appropriate t-distributions. (You could also investigate R’s facilities for t-tests.)

The function tapply() can also be used to handle more complicated indexing of a vector by multiple categories. For example, we might wish to split the tax accountants by both state and sex. However in this simple instance (just one factor) what happens can be thought of as follows. The values in the vector are collected into groups corresponding to the distinct entries in the factor. The function is then applied to each of these groups individually. The value is a vector of function results, labelled by the levels attribute of the factor.

The combination of a vector and a labelling factor is an example of what is sometimes called a ragged array, since the subclass sizes are possibly irregular. When the subclass sizes are all the same the indexing may be done implicitly and much more efficiently, as we see in the next section.

4.3 順序付き因子

The levels of factors are stored in alphabetical order, or in the order they were specified to factor if they were specified explicitly.

Sometimes the levels will have a natural ordering that we want to record and want our statistical analysis to make use of. The ordered() function creates such 順序付き因子 but is otherwise identical to factor. For most purposes the only difference between ordered and un順序付き因子 is that the former are printed showing the ordering of the levels, but the contrasts generated for them in fitting linear models are different.

5 配列と行列

5.1 配列

An array can be considered as a multiply subscripted collection of data entries, for example numeric. R allows simple facilities for creating and handling arrays, and in particular the special case of matrices.

A dimension vector is a vector of non-negative integers. If its length is k then the array is k-dimensional, e.g. a matrix is a 2-dimensional array. The dimensions are indexed from one up to the values given in the dimension vector.

A vector can be used by R as an array only if it has a dimension vector as its dim attribute. Suppose, for example, z is a vector of 1500 elements. The assignment

> dim(z) <- c(3,5,100)

gives it the dim attribute that allows it to be treated as a 3 by 5 by 100 array.

Other functions such as matrix() and array() are available for simpler and more natural looking assignments, as we shall see in array() 関数.

The values in the data vector give the values in the array in the same order as they would occur in FORTRAN, that is “column major order,” with the first subscript moving fastest and the last subscript slowest.

For example if the dimension vector for an array, say a, is c(3,4,2) then there are 3 * 4 * 2 = 24 entries in a and the data vector holds them in the order a[1,1,1], a[2,1,1], …, a[2,4,2], a[3,4,2].

Arrays can be one-dimensional: such arrays are usually treated in the same way as vectors (including when printing), but the exceptions can cause confusion.

5.2 配列の索引付け。配列のサブセクション

Individual elements of an array may be referenced by giving the name of the array followed by the subscripts in square brackets, separated by commas.

More generally, subsections of an array may be specified by giving a sequence of 添字ベクトル in place of subscripts; however if any index position is given an empty index vector, then the full range of that subscript is taken.

Continuing the previous example, a[2,,] is a 4 * 2 array with dimension vector c(4,2) and data vector containing the values

c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
  a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])

in that order. a[,,] stands for the entire array, which is the same as omitting the subscripts entirely and using a alone.

For any array, say Z, the dimension vector may be referenced explicitly as dim(Z) (on either side of an assignment).

Also, if an array name is given with just one subscript or index vector, then the corresponding values of the data vector only are used; in this case the dimension vector is ignored. This is not the case, however, if the single index is not a vector but itself an array, as we next discuss.

5.3 インデックス行列

As well as an index vector in any subscript position, a matrix may be used with a single index matrix in order either to assign a vector of quantities to an irregular collection of elements in the array, or to extract an irregular collection as a vector.

A matrix example makes the process clear. In the case of a doubly indexed array, an index matrix may be given consisting of two columns and as many rows as desired. The entries in the index matrix are the row and column indices for the doubly indexed array. Suppose for example we have a 4 by 5 array X and we wish to do the following:

• Extract elements X[1,3], X[2,2] and X[3,1] as a vector structure, and
• Replace these entries in the array X by zeroes.

In this case we need a 3 by 2 subscript array, as in the following example.

> x <- array(1:20, dim=c(4,5))   # Generate a 4 by 5 array.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    9   13   17
[2,]    2    6   10   14   18
[3,]    3    7   11   15   19
[4,]    4    8   12   16   20
> i <- array(c(1:3,3:1), dim=c(3,2))
> i                             # i is a 3 by 2 index array.
     [,1] [,2]
[1,]    1    3
[2,]    2    2
[3,]    3    1
> x[i]                          # Extract those elements
[1] 9 6 3
> x[i] <- 0                     # Replace those elements by zeros.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    0   13   17
[2,]    2    0   10   14   18
[3,]    0    7   11   15   19
[4,]    4    8   12   16   20
>

Negative indices are not allowed in インデックス行列. NA and zero values are allowed: rows in the index matrix containing a zero are ignored, and rows containing an NA produce an NA in the result.

As a less trivial example, suppose we wish to generate an (unreduced) design matrix for a block design defined by factors blocks (b levels) and varieties (v levels). Further suppose there are n plots in the experiment. We could proceed as follows:

> Xb <- matrix(0, n, b)
> Xv <- matrix(0, n, v)
> ib <- cbind(1:n, blocks)
> iv <- cbind(1:n, varieties)
> Xb[ib] <- 1
> Xv[iv] <- 1
> X <- cbind(Xb, Xv)

To construct the incidence matrix, N say, we could use

> N <- crossprod(Xb, Xv)

However a simpler direct way of producing this matrix is to use table():

> N <- table(blocks, varieties)

インデックス行列 must be numerical: any other form of matrix (e.g. a logical or character matrix) supplied as a matrix is treated as an indexing vector.

5.4 array() 関数

As well as giving a vector structure a dim attribute, arrays can be constructed from vectors by the array function, which has the form

> Z <- array(data_vector, dim_vector)

For example, if the vector h contains 24 or fewer, numbers then the command

> Z <- array(h, dim=c(3,4,2))

would use h to set up 3 by 4 by 2 array in Z. If the size of h is exactly 24 the result is the same as

> Z <- h ; dim(Z) <- c(3,4,2)

However if h is shorter than 24, its values are recycled from the beginning again to make it up to size 24 (see The recycling rule) but dim(h) <- c(3,4,2) would signal an error about mismatching length. As an extreme but common example

> Z <- array(0, c(3,4,2))

makes Z an array of all zeros.

At this point dim(Z) stands for the dimension vector c(3,4,2), and Z[1:24] stands for the data vector as it was in h, and Z[] with an empty subscript or Z with no subscript stands for the entire array as an array.

Arrays may be used in arithmetic expressions and the result is an array formed by element-by-element operations on the data vector. The dim attributes of operands generally need to be the same, and this becomes the dimension vector of the result. So if A, B and C are all similar arrays, then

> D <- 2*A*B + C + 1

makes D a similar array with its data vector being the result of the given element-by-element operations. However the precise rule concerning mixed array and vector calculations has to be considered a little more carefully.

5.4.1 混合ベクトルと配列の演算。 リサイクル規則

The precise rule affecting element by element mixed calculations with vectors and arrays is somewhat quirky and hard to find in the references. From experience we have found the following to be a reliable guide.

• The expression is scanned from left to right.
• Any short vector operands are extended by recycling their values until they match the size of any other operands.
• As long as short vectors and arrays only are encountered, the arrays must all have the same dim attribute or an error results.
• Any vector operand longer than a matrix or array operand generates an error.
• If array structures are present and no error or coercion to vector has been precipitated, the result is an array structure with the common dim attribute of its array operands.

5.5 2つの配列の外積

An important operation on arrays is the outer product. If a and b are two numeric arrays, their outer product is an array whose dimension vector is obtained by concatenating their two dimension vectors (order is important), and whose data vector is got by forming all possible products of elements of the data vector of a with those of b. The outer product is formed by the special operator %o%:

> ab <- a %o% b

An alternative is

> ab <- outer(a, b, "*")

The multiplication function can be replaced by an arbitrary function of two variables. For example if we wished to evaluate the function f(x; y) = cos(y)/(1 + x^2) over a regular grid of values with x- and y-coordinates defined by the R vectors x and y respectively, we could proceed as follows:

> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)

In particular the outer product of two ordinary vectors is a doubly subscripted array (that is a matrix, of rank at most 1). Notice that the outer product operator is of course non-commutative. Defining your own R functions will be considered further in 自分自身の関数を書く.

An example: Determinants of 2 by 2 single-digit matrices

As an artificial but cute example, consider the determinants of 2 by 2 matrices [a, b; c, d] where each entry is a non-negative integer in the range 0, 1, …, 9, that is a digit.

The problem is to find the determinants, ad - bc, of all possible matrices of this form and represent the frequency with which each value occurs as a high density plot. This amounts to finding the probability distribution of the determinant if each digit is chosen independently and uniformly at random.

A neat way of doing this uses the outer() function twice:

> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(fr, xlab="Determinant", ylab="Frequency")

Notice that plot() here uses a histogram like plot method, because it “sees” that fr is of class "table". The “obvious” way of doing this problem with for loops, to be discussed in ループと条件付き実行, is so inefficient as to be impractical.

It is also perhaps surprising that about 1 in 20 such matrices is singular.

5.6 配列の一般的な転置

The function aperm(a, perm) may be used to permute an array, a. The argument perm must be a permutation of the integers {1, …, k}, where k is the number of subscripts in a. The result of the function is an array of the same size as a but with old dimension given by perm[j] becoming the new j-th dimension. The easiest way to think of this operation is as a generalization of transposition for matrices. Indeed if A is a matrix, (that is, a doubly subscripted array) then B given by

> B <- aperm(A, c(2,1))

is just the transpose of A. For this special case a simpler function t() is available, so we could have used B <- t(A).

5.7 行列の機能

As noted above, a matrix is just an array with two subscripts. However it is such an important special case it needs a separate discussion. R contains many operators and functions that are available only for matrices. For example t(X) is the matrix transpose function, as noted above. The functions nrow(A) and ncol(A) give the number of rows and columns in the matrix A respectively.

5.7.1 行列の乗算

The operator %*% is used for 行列の乗算. An n by 1 or 1 by n matrix may of course be used as an n-vector if in the context such is appropriate. Conversely, vectors which occur in 行列の乗算 expressions are automatically promoted either to row or column vectors, whichever is multiplicatively coherent, if possible, (although this is not always unambiguously possible, as we see later).

If, for example, A and B are square matrices of the same size, then

> A * B

is the matrix of element by element products and

> A %*% B

is the matrix product. If x is a vector, then

> x %*% A %*% x

is a quadratic form.¹⁶

The function crossprod() forms “crossproducts”, meaning that crossprod(X, y) is the same as t(X) %*% y but the operation is more efficient. If the second argument to crossprod() is omitted it is taken to be the same as the first.

The meaning of diag() depends on its argument. diag(v), where v is a vector, gives a diagonal matrix with elements of the vector as the diagonal entries. On the other hand diag(M), where M is a matrix, gives the vector of main diagonal entries of M. This is the same convention as that used for diag() in MATLAB. Also, somewhat confusingly, if k is a single numeric value then diag(k) is the k by k identity matrix!

5.7.2 線形方程式と逆行列

Solving linear equations is the inverse of 行列の乗算. When after

> b <- A %*% x

only A and b are given, the vector x is the solution of that linear equation system. In R,

> solve(A,b)

solves the system, returning x (up to some accuracy loss). Note that in linear algebra, formally x = A^{-1} %*% b where A^{-1} denotes the inverse of A, which can be computed by

solve(A)

but rarely is needed. Numerically, it is both inefficient and potentially unstable to compute x <- solve(A) %*% b instead of solve(A,b).

The quadratic form  x %*% A^{-1} %*% x   which is used in multivariate computations, should be computed by something like¹⁷ x %*% solve(A,x), rather than computing the inverse of A.

5.7.3 固有値と固有ベクトル

The function eigen(Sm) calculates the eigenvalues and eigenvectors of a symmetric matrix Sm. The result of this function is a list of two components named values and vectors. The assignment

> ev <- eigen(Sm)

will assign this list to ev. Then ev$val is the vector of eigenvalues of Sm and ev$vec is the matrix of corresponding eigenvectors. Had we only needed the eigenvalues we could have used the assignment:

> evals <- eigen(Sm)$values

evals now holds the vector of eigenvalues and the second component is discarded. If the expression

> eigen(Sm)

is used by itself as a command the two components are printed, with their names. For large matrices it is better to avoid computing the eigenvectors if they are not needed by using the expression

> evals <- eigen(Sm, only.values = TRUE)$values

5.7.4 特異値分解と行列式

The function svd(M) takes an arbitrary matrix argument, M, and calculates the singular value decomposition of M. This consists of a matrix of orthonormal columns U with the same column space as M, a second matrix of orthonormal columns V whose column space is the row space of M and a diagonal matrix of positive entries D such that M = U %*% D %*% t(V). D is actually returned as a vector of the diagonal elements. The result of svd(M) is actually a list of three components named d, u and v, with evident meanings.

If M is in fact square, then, it is not hard to see that

> absdetM <- prod(svd(M)$d)

calculates the absolute value of the determinant of M. If this calculation were needed often with a variety of matrices it could be defined as an R function

> absdet <- function(M) prod(svd(M)$d)

after which we could use absdet() as just another R function. As a further trivial but potentially useful example, you might like to consider writing a function, say tr(), to calculate the trace of a square matrix. [Hint: You will not need to use an explicit loop. Look again at the diag() function.]

R has a builtin function det to calculate a determinant, including the sign, and another, determinant, to give the sign and modulus (optionally on log scale),

5.7.5 最小2乗とQR分解

The function lsfit() returns a list giving results of a least squares fitting procedure. An assignment such as

> ans <- lsfit(X, y)

gives the results of a 最小自乗法 fit where y is the vector of observations and X is the design matrix. See the help facility for more details, and also for the follow-up function ls.diag() for, among other things, regression diagnostics. Note that a grand mean term is automatically included and need not be included explicitly as a column of X. Further note that you almost always will prefer using lm(.) (see Linear models) to lsfit() for regression modelling.

Another closely related function is qr() and its allies. Consider the following assignments

> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)

These compute the orthogonal projection of y onto the range of X in fit, the projection onto the orthogonal complement in res and the coefficient vector for the projection in b, that is, b is essentially the result of the MATLAB ‘backslash’ operator.

It is not assumed that X has full column rank. Redundancies will be discovered and removed as they are found.

This alternative is the older, low-level way to perform 最小自乗法 calculations. Although still useful in some contexts, it would now generally be replaced by the statistical models features, as will be discussed in R における統計モデル.

5.8 cbind() と rbind() での区分行列のフォーマット化

As we have already seen informally, matrices can be built up from other vectors and matrices by the functions cbind() and rbind(). Roughly cbind() forms matrices by binding together matrices horizontally, or column-wise, and rbind() vertically, or row-wise.

In the assignment

> X <- cbind(arg_1, arg_2, arg_3, …)

the arguments to cbind() must be either vectors of any length, or matrices with the same column size, that is the same number of rows. The result is a matrix with the concatenated arguments arg_1, arg_2, … forming the columns.

If some of the arguments to cbind() are vectors they may be shorter than the column size of any matrices present, in which case they are cyclically extended to match the matrix column size (or the length of the longest vector if no matrices are given).

The function rbind() does the corresponding operation for rows. In this case any vector argument, possibly cyclically extended, are of course taken as row vectors.

Suppose X1 and X2 have the same number of rows. To combine these by columns into a matrix X, together with an initial column of 1s we can use

> X <- cbind(1, X1, X2)

The result of rbind() or cbind() always has matrix status. Hence cbind(x) and rbind(x) are possibly the simplest ways explicitly to allow the vector x to be treated as a column or row matrix respectively.

5.9 配列を使用した連結関数 c()

It should be noted that whereas cbind() and rbind() are concatenation functions that respect dim attributes, the basic c() function does not, but rather clears numeric objects of all dim and dimnames attributes. This is occasionally useful in its own right.

The official way to coerce an array back to a simple vector object is to use as.vector()

> vec <- as.vector(X)

However a similar result can be achieved by using c() with just one argument, simply for this side-effect:

> vec <- c(X)

There are slight differences between the two, but ultimately the choice between them is largely a matter of style (with the former being preferable).

5.10 因子による度数分布表

Recall that a factor defines a partition into groups. Similarly a pair of factors defines a two way cross classification, and so on. The function table() allows frequency tables to be calculated from equal length factors. If there are k factor arguments, the result is a k-way array of frequencies.

Suppose, for example, that statef is a factor giving the state code for each entry in a data vector. The assignment

> statefr <- table(statef)

gives in statefr a table of frequencies of each state in the sample. The frequencies are ordered and labelled by the levels attribute of the factor. This simple case is equivalent to, but more convenient than,

> statefr <- tapply(statef, statef, length)

Further suppose that incomef is a factor giving a suitably defined “income class” for each entry in the data vector, for example with the cut() function:

> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef

Then to calculate a two-way table of frequencies:

> table(incomef,statef)
         statef
incomef   act nsw nt qld sa tas vic wa
  (35,45]   1   1  0   1  0   0   1  0
  (45,55]   1   1  1   1  2   0   1  3
  (55,65]   0   3  1   3  2   2   2  1
  (65,75]   0   1  0   0  0   0   1  0

Extension to higher-way frequency tables is immediate.

6 リストとデータフレーム

6.1 リスト

An R list is an object consisting of an ordered collection of objects known as its components.

There is no particular need for the components to be of the same mode or type, and, for example, a list could consist of a numeric vector, a logical value, a matrix, a complex vector, a character array, a function, and so on. Here is a simple example of how to make a list:

> Lst <- list(name="Fred", wife="Mary", no.children=3,
              child.ages=c(4,7,9))

Components are always numbered and may always be referred to as such. Thus if Lst is the name of a list with four components, these may be individually referred to as Lst[[1]], Lst[[2]], Lst[[3]] and Lst[[4]]. If, further, Lst[[4]] is a vector subscripted array then Lst[[4]][1] is its first entry.

If Lst is a list, then the function length(Lst) gives the number of (top level) components it has.

Components of lists may also be named, and in this case the component may be referred to either by giving the component name as a character string in place of the number in double square brackets, or, more conveniently, by giving an expression of the form

> name$component_name

for the same thing.

This is a very useful convention as it makes it easier to get the right component if you forget the number.

So in the simple example given above:

Lst$name is the same as Lst[[1]] and is the string "Fred",

Lst$wife is the same as Lst[[2]] and is the string "Mary",

Lst$child.ages[1] is the same as Lst[[4]][1] and is the number 4.

Additionally, one can also use the names of the list components in double square brackets, i.e., Lst[["name"]] is the same as Lst$name. This is especially useful, when the name of the component to be extracted is stored in another variable as in

> x <- "name"; Lst[[x]]

It is very important to distinguish Lst[[1]] from Lst[1]. ‘[[…]]’ is the operator used to select a single element, whereas ‘[…]’ is a general subscripting operator. Thus the former is the first object in the list Lst, and if it is a named list the name is not included. The latter is a sublist of the list Lst consisting of the first entry only. If it is a named list, the names are transferred to the sublist.

The names of components may be abbreviated down to the minimum number of letters needed to identify them uniquely. Thus Lst$coefficients may be minimally specified as Lst$coe and Lst$covariance as Lst$cov.

The vector of names is in fact simply an attribute of the list like any other and may be handled as such. Other structures besides lists may, of course, similarly be given a names attribute also.

6.2 リストの作成と変更

New lists may be formed from existing objects by the function list(). An assignment of the form

> Lst <- list(name_1=object_1, …, name_m=object_m)

sets up a list Lst of m components using object_1, …, object_m for the components and giving them names as specified by the argument names, (which can be freely chosen). If these names are omitted, the components are numbered only. The components used to form the list are copied when forming the new list and the originals are not affected.

Lists, like any subscripted object, can be extended by specifying additional components. For example

> Lst[5] <- list(matrix=Mat)

6.2.1 リストの連結

When the concatenation function c() is given list arguments, the result is an object of mode list also, whose components are those of the argument lists joined together in sequence.

> list.ABC <- c(list.A, list.B, list.C)

Recall that with vector objects as arguments the concatenation function similarly joined together all arguments into a single vector structure. In this case all other attributes, such as dim attributes, are discarded.

6.3 データフレーム

A data frame is a list with class "data.frame". There are restrictions on lists that may be made into data frames, namely

• The components must be vectors (numeric, character, or logical), factors, numeric matrices, lists, or other data frames.
• Matrices, lists, and data frames provide as many variables to the new data frame as they have columns, elements, or variables, respectively.
• Numeric vectors, logicals and factors are included as is, and by default¹⁸ 文字ベクトル are coerced to be factors, whose levels are the unique values appearing in the vector.
• Vector structures appearing as variables of the data frame must all have the same length, and matrix structures must all have the same row size.

A data frame may for many purposes be regarded as a matrix with columns possibly of differing modes and attributes. It may be displayed in matrix form, and its rows and columns extracted using matrix indexing conventions.

6.3.1 データフレームの作成

Objects satisfying the restrictions placed on the columns (components) of a data frame may be used to form one using the function data.frame:

> accountants <- data.frame(home=statef, loot=incomes, shot=incomef)

A list whose components conform to the restrictions of a data frame may be coerced into a data frame using the function as.data.frame()

The simplest way to construct a data frame from scratch is to use the read.table() function to read an entire data frame from an external file. This is discussed further in ファイルからのデータ読み込み.

6.3.2 attach() と detach()

The $ notation, such as accountants$home, for list components is not always very convenient. A useful facility would be somehow to make the components of a list or data frame temporarily visible as variables under their component name, without the need to quote the list name explicitly each time.

The attach() function takes a ‘database’ such as a list or data frame as its argument. Thus suppose lentils is a data frame with three variables lentils$u, lentils$v, lentils$w. The attach

> attach(lentils)

places the data frame in the search path at position 2, and provided there are no variables u, v or w in position 1, u, v and w are available as variables from the data frame in their own right. At this point an assignment such as

> u <- v+w

does not replace the component u of the data frame, but rather masks it with another variable u in the working directory at position 1 on the search path. To make a permanent change to the data frame itself, the simplest way is to resort once again to the $ notation:

> lentils$u <- v+w

However the new value of component u is not visible until the data frame is detached and attached again.

To detach a data frame, use the function

> detach()

More precisely, this statement detaches from the search path the entity currently at position 2. Thus in the present context the variables u, v and w would be no longer visible, except under the list notation as lentils$u and so on. Entities at positions greater than 2 on the search path can be detached by giving their number to detach, but it is much safer to always use a name, for example by detach(lentils) or detach("lentils")

Note: In R リストとデータフレーム can only be attached at position 2 or above, and what is attached is a copy of the original object. You can alter the attached values via assign, but the original list or data frame is unchanged.

6.3.3 データフレームの使用

A useful convention that allows you to work with many different problems comfortably together in the same working directory is

• gather together all variables for any well defined and separate problem in a data frame under a suitably informative name;
• when working with a problem attach the appropriate data frame at position 2, and use the working directory at level 1 for operational quantities and temporary variables;
• before leaving a problem, add any variables you wish to keep for future reference to the data frame using the $ form of assignment, and then detach();
• finally remove all unwanted variables from the working directory and keep it as clean of left-over temporary variables as possible.

In this way it is quite simple to work with many problems in the same directory, all of which have variables named x, y and z, for example.

6.3.4 任意のリストのアタッチ

attach() is a generic function that allows not only directories and data frames to be attached to the search path, but other classes of object as well. In particular any object of mode "list" may be attached in the same way:

> attach(any.old.list)

Anything that has been attached can be detached by detach, by position number or, preferably, by name.

6.3.5 検索パスの管理

The function search shows the current search path and so is a very useful way to keep track of which data frames and lists (and packages) have been attached and detached. Initially it gives

> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"

where .GlobalEnv is the workspace.¹⁹

After lentils is attached we have

> search()
[1] ".GlobalEnv"   "lentils"      "Autoloads"    "package:base"
> ls(2)
[1] "u" "v" "w"

and as we see ls (or オブジェクト) can be used to examine the contents of any position on the search path.

Finally, we detach the data frame and confirm it has been removed from the search path.

> detach("lentils")
> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"

7 ファイルからのデータ読み込み

Large data objects will usually be read as values from external files rather than entered during an R session at the keyboard. R input facilities are simple and their requirements are fairly strict and even rather inflexible. There is a clear presumption by the designers of R that you will be able to modify your input files using other tools, such as file editors or Perl²⁰ to fit in with the requirements of R. Generally this is very simple.

If variables are to be held mainly in data frames, as we strongly suggest they should be, an entire data frame can be read directly with read.table() 関数. There is also a more primitive input function, scan(), that can be called directly.

For more details on importing data into R and also exporting data, see the R Data Import/Export manual.

7.1 read.table() 関数

To read an entire data frame directly, the external file will normally have a special form.

• The first line of the file should have a name for each variable in the data frame.
• Each additional line of the file has as its first item a row label and the values for each variable.

If the file has one fewer item in its first line than in its second, this arrangement is presumed to be in force. So the first few lines of a file to be read as a data frame might look as follows.

Input file form with names and row labels: Price Floor Area Rooms Age Cent.heat 01 52.00 111.0 830 5 6.2 no 02 54.75 128.0 710 5 7.5 no 03 57.50 101.0 1000 5 4.2 no 04 57.50 131.0 690 6 8.8 no 05 59.75 93.0 900 5 1.9 yes ...

By default numeric items (except row labels) are read as numeric variables and non-numeric variables, such as Cent.heat in the example, as factors. This can be changed if necessary.

The function read.table() can then be used to read the data frame directly

> HousePrice <- read.table("houses.data")

Often you will want to omit including the row labels directly and use the default labels. In this case the file may omit the row label column as in the following.

Input file form without row labels: Price Floor Area Rooms Age Cent.heat 52.00 111.0 830 5 6.2 no 54.75 128.0 710 5 7.5 no 57.50 101.0 1000 5 4.2 no 57.50 131.0 690 6 8.8 no 59.75 93.0 900 5 1.9 yes ...

The data frame may then be read as

> HousePrice <- read.table("houses.data", header=TRUE)

where the header=TRUE option specifies that the first line is a line of headings, and hence, by implication from the form of the file, that no explicit row labels are given.

7.2 scan() 関数

Suppose the data vectors are of equal length and are to be read in parallel. Further suppose that there are three vectors, the first of mode character and the remaining two of mode numeric, and the file is input.dat. The first step is to use scan() to read in the three vectors as a list, as follows

> inp <- scan("input.dat", list("",0,0))

The second argument is a dummy list structure that establishes the mode of the three vectors to be read. The result, held in inp, is a list whose components are the three vectors read in. To separate the data items into three separate vectors, use assignments like

> label <- inp[[1]]; x <- inp[[2]]; y <- inp[[3]]

More conveniently, the dummy list can have named components, in which case the names can be used to access the vectors read in. For example

> inp <- scan("input.dat", list(id="", x=0, y=0))

If you wish to access the variables separately they may either be re-assigned to variables in the working frame:

> label <- inp$id; x <- inp$x; y <- inp$y

or the list may be attached at position 2 of the search path (see 任意のリストのアタッチ).

If the second argument is a single value and not a list, a single vector is read in, all components of which must be of the same mode as the dummy value.

> X <- matrix(scan("light.dat", 0), ncol=5, byrow=TRUE)

There are more elaborate input facilities available and these are detailed in the manuals.

7.3 組み込みデータセットへのアクセス

Around 100 datasets are supplied with R (in package datasets), and others are available in packages (including the recommended packages supplied with R). To see the list of datasets currently available use

data()

All the datasets supplied with R are available directly by name. However, many packages still use the obsolete convention in which data was also used to load datasets into R, for example

data(infert)

and this can still be used with the 標準パッケージ (as in this example). In most cases this will load an R object of the same name. However, in a few cases it loads several objects, so see the on-line help for the object to see what to expect.

7.3.1 他の R パッケージからのデータのロード

To access data from a particular package, use the package argument, for example

data(package="rpart")
data(Puromycin, package="datasets")

If a package has been attached by library, its datasets are automatically included in the search.

User-contributed packages can be a rich source of datasets.

7.4 データの編集

When invoked on a data frame or matrix, edit brings up a separate spreadsheet-like environment for editing. This is useful for making small changes once a data set has been read. The command

> xnew <- edit(xold)

will allow you to edit your data set xold, and on completion the changed object is assigned to xnew. If you want to alter the original dataset xold, the simplest way is to use fix(xold), which is equivalent to xold <- edit(xold).

Use

> xnew <- edit(data.frame())

to enter new data via the spreadsheet interface.

8 確率分布

8.1 統計表の集合としての R

One convenient use of R is to provide a comprehensive set of statistical tables. Functions are provided to evaluate the cumulative distribution function P(X <= x), the probability density function and the quantile function (given q, the smallest x such that P(X <= x) > q), and to simulate from the distribution.

Distribution	R name	additional arguments
beta	beta	shape1, shape2, ncp
binomial	binom	size, prob
Cauchy	cauchy	location, scale
chi-squared	chisq	df, ncp
exponential	exp	rate
F	f	df1, df2, ncp
gamma	gamma	shape, scale
geometric	geom	prob
hypergeometric	hyper	m, n, k
log-normal	lnorm	meanlog, sdlog
logistic	logis	location, scale
negative binomial	nbinom	size, prob
normal	norm	mean, sd
Poisson	pois	lambda
signed rank	signrank	n
Student’s t	t	df, ncp
uniform	unif	min, max
Weibull	weibull	shape, scale
Wilcoxon	wilcox	m, n

Prefix the name given here by ‘d’ for the density, ‘p’ for the CDF, ‘q’ for the quantile function and ‘r’ for simulation (random deviates). The first argument is x for dxxx, q for pxxx, p for qxxx and n for rxxx (except for rhyper, rsignrank and rwilcox, for which it is nn). In not quite all cases is the non-centrality parameter ncp currently available: see the on-line help for details.

The pxxx and qxxx functions all have logical arguments lower.tail and log.p and the dxxx ones have log. This allows, e.g., getting the cumulative (or “integrated”) hazard function, H(t) = - log(1 - F(t)), by

 - pxxx(t, ..., lower.tail = FALSE, log.p = TRUE)

or more accurate log-likelihoods (by dxxx(..., log = TRUE)), directly.

In addition there are functions ptukey and qtukey for the distribution of the studentized range of samples from a normal distribution, and dmultinom and rmultinom for the multinomial distribution. Further distributions are available in contributed packages, notably SuppDists.

Here are some examples

> ## 2-tailed p-value for t distribution
> 2*pt(-2.43, df = 13)
> ## upper 1% point for an F(2, 7) distribution
> qf(0.01, 2, 7, lower.tail = FALSE)

See the on-line help on RNG for how random-number generation is done in R.

8.2 データの集合の分布を調べる

Given a (univariate) set of data we can examine its distribution in a large number of ways. The simplest is to examine the numbers. Two slightly different summaries are given by summary and fivenum and a display of the numbers by stem (a “stem and leaf” plot).

> attach(faithful)
> summary(eruptions)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  1.600   2.163   4.000   3.488   4.454   5.100
> fivenum(eruptions)
[1] 1.6000 2.1585 4.0000 4.4585 5.1000
> stem(eruptions)

  The decimal point is 1 digit(s) to the left of the |

  16 | 070355555588
  18 | 000022233333335577777777888822335777888
  20 | 00002223378800035778
  22 | 0002335578023578
  24 | 00228
  26 | 23
  28 | 080
  30 | 7
  32 | 2337
  34 | 250077
  36 | 0000823577
  38 | 2333335582225577
  40 | 0000003357788888002233555577778
  42 | 03335555778800233333555577778
  44 | 02222335557780000000023333357778888
  46 | 0000233357700000023578
  48 | 00000022335800333
  50 | 0370

A stem-and-leaf plot is like a histogram, and R has a function hist to plot histograms.

> hist(eruptions)
## make the bins smaller, make a plot of density
> hist(eruptions, seq(1.6, 5.2, 0.2), prob=TRUE)
> lines(density(eruptions, bw=0.1))
> rug(eruptions) # show the actual data points

More elegant density plots can be made by density, and we added a line produced by density in this example. The bandwidth bw was chosen by trial-and-error as the default gives too much smoothing (it usually does for “interesting” densities). (Better automated methods of bandwidth choice are available, and in this example bw = "SJ" gives a good result.)

We can plot the empirical cumulative distribution function by using the function ecdf.

> plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)

This distribution is obviously far from any standard distribution. How about the right-hand mode, say eruptions of longer than 3 minutes? Let us fit a normal distribution and overlay the fitted CDF.

> long <- eruptions[eruptions > 3]
> plot(ecdf(long), do.points=FALSE, verticals=TRUE)
> x <- seq(3, 5.4, 0.01)
> lines(x, pnorm(x, mean=mean(long), sd=sqrt(var(long))), lty=3)

Quantile-quantile (Q-Q) plots can help us examine this more carefully.

par(pty="s")       # arrange for a square figure region
qqnorm(long); qqline(long)

which shows a reasonable fit but a shorter right tail than one would expect from a normal distribution. Let us compare this with some simulated data from a t distribution

x <- rt(250, df = 5)
qqnorm(x); qqline(x)

which will usually (if it is a random sample) show longer tails than expected for a normal. We can make a Q-Q plot against the generating distribution by

qqplot(qt(ppoints(250), df = 5), x, xlab = "Q-Q plot for t dsn")
qqline(x)

Finally, we might want a more formal test of agreement with normality (or not). R provides the Shapiro-Wilk test

> shapiro.test(long)

         Shapiro-Wilk normality test

data:  long
W = 0.9793, p-value = 0.01052

and the Kolmogorov-Smirnov test

> ks.test(long, "pnorm", mean = mean(long), sd = sqrt(var(long)))

         One-sample Kolmogorov-Smirnov test

data:  long
D = 0.0661, p-value = 0.4284
alternative hypothesis: two.sided

(Note that the distribution theory is not valid here as we have estimated the parameters of the normal distribution from the same sample.)

8.3 1 サンプルと 2 サンプルのテスト

So far we have compared a single sample to a normal distribution. A much more common operation is to compare aspects of two samples. Note that in R, all “classical” tests including the ones used below are in package stats which is normally loaded.

Consider the following sets of data on the latent heat of the fusion of ice (cal/gm) from Rice (1995, p.490)

Method A: 79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97
          80.05 80.03 80.02 80.00 80.02
Method B: 80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97

Boxplots provide a simple graphical comparison of the two samples.

A <- scan()
79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97
80.05 80.03 80.02 80.00 80.02

B <- scan()
80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97

boxplot(A, B)

which indicates that the first group tends to give higher results than the second.

To test for the equality of the means of the two examples, we can use an unpaired t-test by

> t.test(A, B)

         Welch Two Sample t-test

data:  A and B
t = 3.2499, df = 12.027, p-value = 0.00694
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.01385526 0.07018320
sample estimates:
mean of x mean of y
 80.02077  79.97875

which does indicate a significant difference, assuming normality. By default the R function does not assume equality of variances in the two samples (in contrast to the similar S-PLUS t.test function). We can use the F test to test for equality in the variances, provided that the two samples are from normal populations.

> var.test(A, B)

         F test to compare two variances

data:  A and B
F = 0.5837, num df = 12, denom df =  7, p-value = 0.3938
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.1251097 2.1052687
sample estimates:
ratio of variances
         0.5837405

which shows no evidence of a significant difference, and so we can use the classical t-test that assumes equality of the variances.

> t.test(A, B, var.equal=TRUE)

         Two Sample t-test

data:  A and B
t = 3.4722, df = 19, p-value = 0.002551
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.01669058 0.06734788
sample estimates:
mean of x mean of y
 80.02077  79.97875

All these tests assume normality of the two samples. The two-sample Wilcoxon (or Mann-Whitney) test only assumes a common continuous distribution under the null hypothesis.

> wilcox.test(A, B)

         Wilcoxon rank sum test with continuity correction

data:  A and B
W = 89, p-value = 0.007497
alternative hypothesis: true location shift is not equal to 0

Warning message:
Cannot compute exact p-value with ties in: wilcox.test(A, B)

Note the warning: there are several ties in each sample, which suggests strongly that these data are from a discrete distribution (probably due to rounding).

There are several ways to compare graphically the two samples. We have already seen a pair of boxplots. The following

> plot(ecdf(A), do.points=FALSE, verticals=TRUE, xlim=range(A, B))
> plot(ecdf(B), do.points=FALSE, verticals=TRUE, add=TRUE)

will show the two empirical CDFs, and qqplot will perform a Q-Q plot of the two samples. The Kolmogorov-Smirnov test is of the maximal vertical distance between the two ecdf’s, assuming a common continuous distribution:

> ks.test(A, B)

         Two-sample Kolmogorov-Smirnov test

data:  A and B
D = 0.5962, p-value = 0.05919
alternative hypothesis: two-sided

Warning message:
cannot compute correct p-values with ties in: ks.test(A, B)

9 グループ化、ループ、条件付き実行

9.1 グループ化された表現式

R is an expression language in the sense that its only command type is a function or expression which returns a result. Even an assignment is an expression whose result is the value assigned, and it may be used wherever any expression may be used; in particular multiple assignments are possible.

Commands may be grouped together in braces, {expr_1; …; expr_m}, in which case the value of the group is the result of the last expression in the group evaluated. Since such a group is also an expression it may, for example, be itself included in parentheses and used as part of an even larger expression, and so on.

9.2 制御文

9.2.1 条件付き実行：if 文

The language has available a conditional construction of the form

> if (expr_1) expr_2 else expr_3

where expr_1 must evaluate to a single logical value and the result of the entire expression is then evident.

The “short-circuit” operators && and || are often used as part of the condition in an if statement. Whereas & and | apply element-wise to vectors, && and || apply to vectors of length one, and only evaluate their second argument if necessary.

There is a vectorized version of the if/else construct, the ifelse function. This has the form ifelse(condition, a, b) and returns a vector of the same length as condition, with elements a[i] if condition[i] is true, otherwise b[i] (where a and b are recycled as necessary).

9.2.2 繰り返し実行：for ループ、repeat と while

There is also a for loop construction which has the form

> for (name in expr_1) expr_2

where name is the loop variable. expr_1 is a vector expression, (often a sequence like 1:20), and expr_2 is often a grouped expression with its sub-expressions written in terms of the dummy name. expr_2 is repeatedly evaluated as name ranges through the values in the vector result of expr_1.

As an example, suppose ind is a vector of class indicators and we wish to produce separate plots of y versus x within classes. One possibility here is to use coplot(),²¹ which will produce an array of plots corresponding to each level of the factor. Another way to do this, now putting all plots on the one display, is as follows:

> xc <- split(x, ind)
> yc <- split(y, ind)
> for (i in 1:length(yc)) {
    plot(xc[[i]], yc[[i]])
    abline(lsfit(xc[[i]], yc[[i]]))
  }

(Note the function split() which produces a list of vectors obtained by splitting a larger vector according to the classes specified by a factor. This is a useful function, mostly used in connection with boxplots. See the help facility for further details.)

Warning: for() loops are used in R code much less often than in compiled languages. Code that takes a ‘whole object’ view is likely to be both clearer and faster in R.

Other looping facilities include the

> repeat expr

statement and the

> while (condition) expr

statement.

The break statement can be used to terminate any loop, possibly abnormally. This is the only way to terminate repeat loops.

The next statement can be used to discontinue one particular cycle and skip to the “next”.

制御文 are most often used in connection with functions which are discussed in 自分自身の関数を書く, and where more examples will emerge.

10 自分自身の関数を書く

As we have seen informally along the way, the R language allows the user to create objects of mode function. These are true R functions that are stored in a special internal form and may be used in further expressions and so on. In the process, the language gains enormously in power, convenience and elegance, and learning to write useful functions is one of the main ways to make your use of R comfortable and productive.

It should be emphasized that most of the functions supplied as part of the R system, such as mean(), var(), postscript() and so on, are themselves written in R and thus do not differ materially from user written functions.

A function is defined by an assignment of the form

> name <- function(arg_1, arg_2, …) expression

The expression is an R expression, (usually a grouped expression), that uses the arguments, arg_i, to calculate a value. The value of the expression is the value returned for the function.

A call to the function then usually takes the form name(expr_1, expr_2, …) and may occur anywhere a function call is legitimate.

10.1 簡単な例

As a first example, consider a function to calculate the two sample t-statistic, showing “all the steps”. This is an artificial example, of course, since there are other, simpler ways of achieving the same end.

The function is defined as follows:

> twosam <- function(y1, y2) {
    n1  <- length(y1); n2  <- length(y2)
    yb1 <- mean(y1);   yb2 <- mean(y2)
    s1  <- var(y1);    s2  <- var(y2)
    s <- ((n1-1)*s1 + (n2-1)*s2)/(n1+n2-2)
    tst <- (yb1 - yb2)/sqrt(s*(1/n1 + 1/n2))
    tst
  }

With this function defined, you could perform two sample t-tests using a call such as

> tstat <- twosam(data$male, data$female); tstat

As a second example, consider a function to emulate directly the MATLAB backslash command, which returns the coefficients of the orthogonal projection of the vector y onto the column space of the matrix, X. (This is ordinarily called the 最小自乗法 estimate of the regression coefficients.) This would ordinarily be done with the qr() function; however this is sometimes a bit tricky to use directly and it pays to have a simple function such as the following to use it safely.

Thus given a n by 1 vector y and an n by p matrix X then X \ y is defined as (X’X)^{-}X’y, where (X’X)^{-} is a generalized inverse of X'X.

> bslash <- function(X, y) {
  X <- qr(X)
  qr.coef(X, y)
}

After this object is created it may be used in statements such as

> regcoeff <- bslash(Xmat, yvar)

and so on.

The classical R function lsfit() does this job quite well, and more²². It in turn uses the functions qr() and qr.coef() in the slightly counterintuitive way above to do this part of the calculation. Hence there is probably some value in having just this part isolated in a simple to use function if it is going to be in frequent use. If so, we may wish to make it a matrix binary operator for even more convenient use.

10.2 新しい2 項演算子を定義する

Had we given the bslash() function a different name, namely one of the form

%anything%

it could have been used as a binary operator in expressions rather than in function form. Suppose, for example, we choose ! for the internal character. The function definition would then start as

> "%!%" <- function(X, y) { … }

(Note the use of quote marks.) The function could then be used as X %!% y. (The backslash symbol itself is not a convenient choice as it presents special problems in this context.)

The 行列の乗算 operator, %*%, and the outer product matrix operator %o% are other examples of binary operators defined in this way.

10.3 新しい2 項演算子を定義する

As first noted in 規則的な数列の生成, if arguments to called functions are given in the “name=object” form, they may be given in any order. Furthermore the argument sequence may begin in the unnamed, positional form, and specify named arguments after the positional arguments.

Thus if there is a function fun1 defined by

> fun1 <- function(data, data.frame, graph, limit) {
    [function body omitted]
  }

then the function may be invoked in several ways, for example

> ans <- fun1(d, df, TRUE, 20)
> ans <- fun1(d, df, graph=TRUE, limit=20)
> ans <- fun1(data=d, limit=20, graph=TRUE, data.frame=df)

are all equivalent.

In many cases arguments can be given commonly appropriate default values, in which case they may be omitted altogether from the call when the defaults are appropriate. For example, if fun1 were defined as

> fun1 <- function(data, data.frame, graph=TRUE, limit=20) { … }

it could be called as

> ans <- fun1(d, df)

which is now equivalent to the three cases above, or as

> ans <- fun1(d, df, limit=10)

which changes one of the defaults.

It is important to note that defaults may be arbitrary expressions, even involving other arguments to the same function; they are not restricted to be constants as in our simple example here.

10.4 ‘…’ 引数

Another frequent requirement is to allow one function to pass on argument settings to another. For example many graphics functions use the function par() and functions like plot() allow the user to pass on graphical parameters to par() to control the graphical output. (See The par() function, for more details on the par() function.) This can be done by including an extra argument, literally ‘…’, of the function, which may then be passed on. An outline example is given below.

fun1 <- function(data, data.frame, graph=TRUE, limit=20, ...) {
  [omitted statements]
  if (graph)
    par(pch="*", ...)
  [more omissions]
}

Less frequently, a function will need to refer to components of ‘…’. The expression list(...) evaluates all such arguments and returns them in a named list, while ..1, ..2, etc. evaluate them one at a time, with ‘..n’ returning the n’th unmatched argument.

10.5 関数中からの代入

Note that any ordinary assignments done within the function are local and temporary and are lost after exit from the function. Thus the assignment X <- qr(X) does not affect the value of the argument in the calling program.

To understand completely the rules governing the scope of R assignments the reader needs to be familiar with the notion of an evaluation frame. This is a somewhat advanced, though hardly difficult, topic and is not covered further here.

If global and permanent assignments are intended within a function, then either the “superassignment” operator, <<- or the function assign() can be used. See the help document for details. S-PLUS users should be aware that <<- has different semantics in R. These are discussed further in Scope.

10.6 より高度な例

10.6.1 ブロックデザインに於ける効率因子

As a more complete, if a little pedestrian, example of a function, consider finding the efficiency factors for a block design. (Some aspects of this problem have already been discussed in インデックス行列.)

A block design is defined by two factors, say blocks (b levels) and varieties (v levels). If R and K are the v by v and b by b replications and block size matrices, respectively, and N is the b by v incidence matrix, then the efficiency factors are defined as the eigenvalues of the matrix E = I_v - R^{-1/2}N’K^{-1}NR^{-1/2} = I_v - A’A, where A = K^{-1/2}NR^{-1/2}. One way to write the function is given below.

> bdeff <- function(blocks, varieties) {
    blocks <- as.factor(blocks)             # minor safety move
    b <- length(levels(blocks))
    varieties <- as.factor(varieties)       # minor safety move
    v <- length(levels(varieties))
    K <- as.vector(table(blocks))           # remove dim attr
    R <- as.vector(table(varieties))        # remove dim attr
    N <- table(blocks, varieties)
    A <- 1/sqrt(K) * N * rep(1/sqrt(R), rep(b, v))
    sv <- svd(A)
    list(eff=1 - sv$d^2, blockcv=sv$u, varietycv=sv$v)
}

It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines.

The result of the function is a list giving not only the efficiency factors as the first component, but also the block and variety canonical contrasts, since sometimes these give additional useful qualitative information.

10.6.2 プリントされた配列から全ての名前を取り除く

For printing purposes with large matrices or arrays, it is often useful to print them in close block form without the array names or numbers. Removing the dimnames attribute will not achieve this effect, but rather the array must be given a dimnames attribute consisting of empty strings. For example to print a matrix, X

> temp <- X
> dimnames(temp) <- list(rep("", nrow(X)), rep("", ncol(X)))
> temp; rm(temp)

This can be much more conveniently done using a function, no.dimnames(), shown below, as a “wrap around” to achieve the same result. It also illustrates how some effective and useful user functions can be quite short.

no.dimnames <- function(a) {
  ## Remove all dimension names from an array for compact printing.
  d <- list()
  l <- 0
  for(i in dim(a)) {
    d[[l <- l + 1]] <- rep("", i)
  }
  dimnames(a) <- d
  a
}

With this function defined, an array may be printed in close format using

> no.dimnames(X)

This is particularly useful for large integer arrays, where patterns are the real interest rather than the values.

10.6.3 再帰的な数値積分

Functions may be recursive, and may themselves define functions within themselves. Note, however, that such functions, or indeed variables, are not inherited by called functions in higher evaluation frames as they would be if they were on the search path.

The example below shows a naive way of performing one-dimensional numerical integration. The integrand is evaluated at the end points of the range and in the middle. If the one-panel trapezium rule answer is close enough to the two panel, then the latter is returned as the value. Otherwise the same process is recursively applied to each panel. The result is an adaptive integration process that concentrates function evaluations in regions where the integrand is farthest from linear. There is, however, a heavy overhead, and the function is only competitive with other algorithms when the integrand is both smooth and very difficult to evaluate.

The example is also given partly as a little puzzle in R programming.

area <- function(f, a, b, eps = 1.0e-06, lim = 10) {
  fun1 <- function(f, a, b, fa, fb, a0, eps, lim, fun) {
    ## function ‘fun1’ is only visible inside ‘area’
    d <- (a + b)/2
    h <- (b - a)/4
    fd <- f(d)
    a1 <- h * (fa + fd)
    a2 <- h * (fd + fb)
    if(abs(a0 - a1 - a2) < eps || lim == 0)
      return(a1 + a2)
    else {
      return(fun(f, a, d, fa, fd, a1, eps, lim - 1, fun) +
             fun(f, d, b, fd, fb, a2, eps, lim - 1, fun))
    }
  }
  fa <- f(a)
  fb <- f(b)
  a0 <- ((fa + fb) * (b - a))/2
  fun1(f, a, b, fa, fb, a0, eps, lim, fun1)
}

10.7 スコープ

The discussion in this section is somewhat more technical than in other parts of this document. However, it details one of the major differences between S-PLUS and R.

The symbols which occur in the body of a function can be divided into three classes; formal parameters, local variables and free variables. The formal parameters of a function are those occurring in the argument list of the function. Their values are determined by the process of binding the actual function arguments to the formal parameters. Local variables are those whose values are determined by the evaluation of expressions in the body of the functions. Variables which are not formal parameters or local variables are called free variables. Free variables become local variables if they are assigned to. Consider the following function definition.

f <- function(x) {
  y <- 2*x
  print(x)
  print(y)
  print(z)
}

In this function, x is a formal parameter, y is a local variable and z is a free variable.

In R the free variable bindings are resolved by first looking in the environment in which the function was created. This is called lexical scope. First we define a function called cube.

cube <- function(n) {
  sq <- function() n*n
  n*sq()
}

The variable n in the function sq is not an argument to that function. Therefore it is a free variable and the scoping rules must be used to ascertain the value that is to be associated with it. Under static scope (S-PLUS) the value is that associated with a global variable named n. Under lexical scope (R) it is the parameter to the function cube since that is the active binding for the variable n at the time the function sq was defined. The difference between evaluation in R and evaluation in S-PLUS is that S-PLUS looks for a global variable called n while R first looks for a variable called n in the environment created when cube was invoked.

## first evaluation in S
S> cube(2)
Error in sq(): Object "n" not found
Dumped
S> n <- 3
S> cube(2)
[1] 18
## then the same function evaluated in R
R> cube(2)
[1] 8

Lexical scope can also be used to give functions mutable state. In the following example we show how R can be used to mimic a bank account. A functioning bank account needs to have a balance or total, a function for making withdrawals, a function for making deposits and a function for stating the current balance. We achieve this by creating the three functions within account and then returning a list containing them. When account is invoked it takes a numerical argument total and returns a list containing the three functions. Because these functions are defined in an environment which contains total, they will have access to its value.

The special assignment operator, <<-, is used to change the value associated with total. This operator looks back in enclosing environments for an environment that contains the symbol total and when it finds such an environment it replaces the value, in that environment, with the value of right hand side. If the global or top-level environment is reached without finding the symbol total then that variable is created and assigned to there. For most users <<- creates a global variable and assigns the value of the right hand side to it²³. Only when <<- has been used in a function that was returned as the value of another function will the special behavior described here occur.

open.account <- function(total) {
  list(
    deposit = function(amount) {
      if(amount <= 0)
        stop("Deposits must be positive!\n")
      total <<- total + amount
      cat(amount, "deposited.  Your balance is", total, "\n\n")
    },
    withdraw = function(amount) {
      if(amount > total)
        stop("You don't have that much money!\n")
      total <<- total - amount
      cat(amount, "withdrawn.  Your balance is", total, "\n\n")
    },
    balance = function() {
      cat("Your balance is", total, "\n\n")
    }
  )
}

ross <- open.account(100)
robert <- open.account(200)

ross$withdraw(30)
ross$balance()
robert$balance()

ross$deposit(50)
ross$balance()
ross$withdraw(500)

10.8 環境のカスタマイズ

Users can customize their environment in several different ways. There is a site initialization file and every directory can have its own special initialization file. Finally, the special functions .First and .Last can be used.

The location of the site initialization file is taken from the value of the R_PROFILE environment variable. If that variable is unset, the file Rprofile.site in the R home subdirectory etc is used. This file should contain the commands that you want to execute every time R is started under your system. A second, personal, profile file named .Rprofile²⁴ can be placed in any directory. If R is invoked in that directory then that file will be sourced. This file gives individual users control over their workspace and allows for different startup procedures in different working directories. If no .Rprofile file is found in the startup directory, then R looks for a .Rprofile file in the user’s home directory and uses that (if it exists). If the environment variable R_PROFILE_USER is set, the file it points to is used instead of the .Rprofile files.

Any function named .First() in either of the two profile files or in the .RData image has a special status. It is automatically performed at the beginning of an R session and may be used to initialize the environment. For example, the definition in the example below alters the prompt to $ and sets up various other useful things that can then be taken for granted in the rest of the session.

Thus, the sequence in which files are executed is, Rprofile.site, the user profile, .RData and then .First(). A definition in later files will mask definitions in earlier files.

> .First <- function() {
  options(prompt="$ ", continue="+\t")  # $ is the prompt
  options(digits=5, length=999)         # custom numbers and printout
  x11()                                 # for graphics
  par(pch = "+")                        # plotting character
  source(file.path(Sys.getenv("HOME"), "R", "mystuff.R"))
                                        # my personal functions
  library(MASS)                         # attach a package
}

Similarly a function .Last(), if defined, is (normally) executed at the very end of the session. An example is given below.

> .Last <- function() {
  graphics.off()                        # a small safety measure.
  cat(paste(date(),"\nAdios\n"))        # Is it time for lunch?
}

10.9 クラス、総称的関数とオブジェクト指向

オブジェクトのクラス determines how it will be treated by what are known as generic functions. Put the other way round, a generic function performs a task or action on its arguments specific to the class of the argument itself. If the argument lacks any class attribute, or has a class not catered for specifically by the generic function in question, there is always a default action provided.

An example makes things clearer. The class mechanism offers the user the facility of designing and writing generic functions for special purposes. Among the other generic functions are plot() for displaying objects graphically, summary() for summarizing analyses of various types, and anova() for comparing statistical models.

The number of generic functions that can treat a class in a specific way can be quite large. For example, the functions that can accommodate in some fashion objects of class "data.frame" include

[     [[<-    any    as.matrix
[<-   mean    plot   summary

A currently complete list can be got by using the methods() function:

> methods(class="data.frame")

Conversely the number of classes a generic function can handle can also be quite large. For example plot() 関数 has a default method and variants for objects of classes "data.frame", "density", "factor", and more. A complete list can be got again by using the methods() function:

> methods(plot)

For many generic functions the function body is quite short, for example

> coef
function (object, ...)
UseMethod("coef")

The presence of UseMethod indicates this is a generic function. To see what methods are available we can use methods()

> methods(coef)
[1] coef.aov*         coef.Arima*       coef.default*     coef.listof*
[5] coef.nls*         coef.summary.nls*

   Non-visible functions are asterisked

In this example there are six methods, none of which can be seen by typing its name. We can read these by either of

> getAnywhere("coef.aov")
A single object matching ‘coef.aov’ was found
It was found in the following places
  registered S3 method for coef from namespace stats
  namespace:stats
with value

function (object, ...)
{
    z <- object$coef
    z[!is.na(z)]
}

> getS3method("coef", "aov")
function (object, ...)
{
    z <- object$coef
    z[!is.na(z)]
}

A function named gen.cl will be invoked by the generic gen for class cl, so do not name functions in this style unless they are intended to be methods.

The reader is referred to the R Language Definition for a more complete discussion of this mechanism.

11 R における統計モデル

This section presumes the reader has some familiarity with statistical methodology, in particular with regression analysis and the analysis of variance. Later we make some rather more ambitious presumptions, namely that something is known about 一般化線形モデル and nonlinear regression.

The requirements for fitting statistical models are sufficiently well defined to make it possible to construct general tools that apply in a broad spectrum of problems.

R provides an interlocking suite of facilities that make fitting statistical models very simple. As we mention in the introduction, the basic output is minimal, and one needs to ask for the details by calling extractor functions.

11.1 統計公式を定義する；公式

The template for a statistical model is a linear regression model with independent, homoscedastic errors

y_i = sum_{j=0}^p beta_j x_{ij} + e_i,     i = 1, …, n,

where the e_i are NID(0, sigma^2). In matrix terms this would be written

y = X  beta + e

where the y is the response vector, X is the model matrix or design matrix and has columns x_0, x_1, …, x_p, the determining variables. Very often x_0 will be a column of ones defining an intercept term.

Examples

Before giving a formal specification, a few examples may usefully set the picture.

Suppose y, x, x0, x1, x2, … are numeric variables, X is a matrix and A, B, C, … are factors. The following formulae on the left side below specify statistical models as described on the right.

y ~ x

y ~ 1 + x

Both imply the same simple linear regression model of y on x. The first has an implicit intercept term, and the second an explicit one.

y ~ 0 + x

y ~ -1 + x

y ~ x - 1

Simple linear regression of y on x through the origin (that is, without an intercept term).

log(y) ~ x1 + x2

Multiple regression of the transformed variable, log(y), on x1 and x2 (with an implicit intercept term).

y ~ poly(x,2)

y ~ 1 + x + I(x^2)

Polynomial regression of y on x of degree 2. The first form uses orthogonal polynomials, and the second uses explicit powers, as basis.

y ~ X + poly(x,2)

Multiple regression y with model matrix consisting of the matrix X as well as polynomial terms in x to degree 2.

y ~ A

Single classification analysis of variance model of y, with classes determined by A.

y ~ A + x

Single classification analysis of covariance model of y, with classes determined by A, and with covariate x.

y ~ A*B

y ~ A + B + A:B

y ~ B %in% A

y ~ A/B

Two factor non-additive model of y on A and B. The first two specify the same crossed classification and the second two specify the same nested classification. In abstract terms all four specify the same model subspace.

y ~ (A + B + C)^2

y ~ A*B*C - A:B:C

Three factor experiment but with a model containing main effects and two factor interactions only. Both formulae specify the same model.

y ~ A * x

y ~ A/x

y ~ A/(1 + x) - 1

Separate simple linear regression models of y on x within the levels of A, with different codings. The last form produces explicit estimates of as many different intercepts and slopes as there are levels in A.

y ~ A*B + Error(C)

An experiment with two treatment factors, A and B, and error strata determined by factor C. For example a split plot experiment, with whole plots (and hence also subplots), determined by factor C.

The operator ~ is used to define a model formula in R. The form, for an ordinary linear model, is

response ~ op_1 term_1 op_2 term_2 op_3 term_3 …

where

response

is a vector or matrix, (or expression evaluating to a vector or matrix) defining the response variable(s).

op_i

is an operator, either + or -, implying the inclusion or exclusion of a term in the model, (the first is optional).

term_i

is either

• a vector or matrix expression, or 1,
• a factor, or
• a formula expression consisting of factors, vectors or matrices connected by formula operators.

In all cases each term defines a collection of columns either to be added to or removed from the model matrix. A 1 stands for an intercept column and is by default included in the model matrix unless explicitly removed.

The formula operators are similar in effect to the Wilkinson and Rogers notation used by such programs as Glim and Genstat. One inevitable change is that the operator ‘.’ becomes ‘:’ since the period is a valid name character in R.

The notation is summarized below (based on Chambers & Hastie, 1992, p.29):

Y ~ M

Y is modeled as M.

M_1 + M_2

Include M_1 and M_2.

M_1 - M_2

Include M_1 leaving out terms of M_2.

M_1 : M_2

The tensor product of M_1 and M_2. If both terms are factors, then the “subclasses” factor.

M_1 %in% M_2

Similar to M_1:M_2, but with a different coding.

M_1 * M_2

M_1 + M_2 + M_1:M_2.

M_1 / M_2

M_1 + M_2 %in% M_1.

M^n

All terms in M together with “interactions” up to order n

I(M)

Insulate M. Inside M all operators have their normal arithmetic meaning, and that term appears in the model matrix.

Note that inside the parentheses that usually enclose function arguments all operators have their normal arithmetic meaning. The function I() is an identity function used to allow terms in model formulae to be defined using arithmetic operators.

Note particularly that the model formulae specify the columns of the model matrix, the specification of the parameters being implicit. This is not the case in other contexts, for example in specifying nonlinear models.

11.1.1 コントラスト

We need at least some idea how the model formulae specify the columns of the model matrix. This is easy if we have continuous variables, as each provides one column of the model matrix (and the intercept will provide a column of ones if included in the model).

What about a k-level factor A? The answer differs for unordered and 順序付き因子. For unordered factors k - 1 columns are generated for the indicators of the second, …, kth levels of the factor. (Thus the implicit parameterization is to contrast the response at each level with that at the first.) For ordered factors the k - 1 columns are the orthogonal polynomials on 1, …, k, omitting the constant term.

Although the answer is already complicated, it is not the whole story. First, if the intercept is omitted in a model that contains a factor term, the first such term is encoded into k columns giving the indicators for all the levels. Second, the whole behavior can be changed by the options setting for contrasts. The default setting in R is

options(contrasts = c("contr.treatment", "contr.poly"))

The main reason for mentioning this is that R and S have different defaults for un順序付き因子, S using Helmert contrasts. So if you need to compare your results to those of a textbook or paper which used S-PLUS, you will need to set

options(contrasts = c("contr.helmert", "contr.poly"))

This is a deliberate difference, as treatment contrasts (R’s default) are thought easier for newcomers to interpret.

We have still not finished, as the contrast scheme to be used can be set for each term in the model using the functions contrasts and C.

We have not yet considered interaction terms: these generate the products of the columns introduced for their component terms.

Although the details are complicated, model formulae in R will normally generate the models that an expert statistician would expect, provided that marginality is preserved. Fitting, for example, a model with an interaction but not the corresponding main effects will in general lead to surprising results, and is for experts only.

11.2 線形モデル

The basic function for fitting ordinary multiple models is lm(), and a streamlined version of the call is as follows:

> fitted.model <- lm(formula, data = data.frame)

For example

> fm2 <- lm(y ~ x1 + x2, data = production)

would fit a multiple regression model of y on x1 and x2 (with implicit intercept term).

The important (but technically optional) parameter data = production specifies that any variables needed to construct the model should come first from the production data frame. This is the case regardless of whether data frame production has been attached on the search path or not.

11.3 モデル情報を取り出す総称的関数

The value of lm() is a fitted model object; technically a list of results of class "lm". Information about the fitted model can then be displayed, extracted, plotted and so on by using generic functions that orient themselves to objects of class "lm". These include

add1    deviance   formula      predict  step
alias   drop1      kappa        print    summary
anova   effects    labels       proj     vcov
coef    family     plot         residuals

A brief description of the most commonly used ones is given below.

anova(object_1, object_2)

Compare a submodel with an outer model and produce an analysis of variance table.

coef(object)

Extract the regression coefficient (matrix).

Long form: coefficients(object).

deviance(object)

Residual sum of squares, weighted if appropriate.

formula(object)

Extract the model formula.

plot(object)

Produce four plots, showing residuals, fitted values and some diagnostics.

predict(object, newdata=data.frame)

The data frame supplied must have variables specified with the same labels as the original. The value is a vector or matrix of predicted values corresponding to the determining variable values in data.frame.

print(object)

Print a concise version of the object. Most often used implicitly.

residuals(object)

Extract the (matrix of) residuals, weighted as appropriate.

Short form: resid(object).

step(object)

Select a suitable model by adding or dropping terms and preserving hierarchies. The model with the smallest value of AIC (Akaike’s An Information Criterion) discovered in the stepwise search is returned.

summary(object)

Print a comprehensive summary of the results of the regression analysis.

vcov(object)

Returns the variance-covariance matrix of the main parameters of a fitted model object.

11.4 分散分析とモデル比較

The model fitting function aov(formula, data=data.frame) operates at the simplest level in a very similar way to the function lm(), and most of the generic functions listed in the table in モデル情報を取り出す総称的関数 apply.

It should be noted that in addition aov() allows an analysis of models with multiple error strata such as split plot experiments, or balanced incomplete block designs with recovery of inter-block information. The model formula

response ~ mean.formula + Error(strata.formula)

specifies a multi-stratum experiment with error strata defined by the strata.formula. In the simplest case, strata.formula is simply a factor, when it defines a two strata experiment, namely between and within the levels of the factor.

For example, with all determining variables factors, a model formula such as that in:

> fm <- aov(yield ~ v + n*p*k + Error(farms/blocks), data=farm.data)

would typically be used to describe an experiment with mean model v + n*p*k and three error strata, namely “between farms”, “within farms, between blocks” and “within blocks”.

11.4.1 ANOVA 表

Note also that the analysis of variance table (or tables) are for a sequence of fitted models. The sums of squares shown are the decrease in the residual sums of squares resulting from an inclusion of that term in the model at that place in the sequence. Hence only for orthogonal experiments will the order of inclusion be inconsequential.

For multistratum experiments the procedure is first to project the response onto the error strata, again in sequence, and to fit the mean model to each projection. For further details, see Chambers & Hastie (1992).

A more flexible alternative to the default full ANOVA table is to compare two or more models directly using the anova() function.

> anova(fitted.model.1, fitted.model.2, …)

The display is then an ANOVA table showing the differences between the fitted models when fitted in sequence. The fitted models being compared would usually be an hierarchical sequence, of course. This does not give different information to the default, but rather makes it easier to comprehend and control.

11.5 フィルタ・モデルの更新

The update() function is largely a convenience function that allows a model to be fitted that differs from one previously fitted usually by just a few additional or removed terms. Its form is

> new.model <- update(old.model, new.formula)

In the new.formula the special name consisting of a period, ‘.’, only, can be used to stand for “the corresponding part of the old model formula”. For example,

> fm05 <- lm(y ~ x1 + x2 + x3 + x4 + x5, data = production)
> fm6  <- update(fm05, . ~ . + x6)
> smf6 <- update(fm6, sqrt(.) ~ .)

would fit a five variate multiple regression with variables (presumably) from the data frame production, fit an additional model including a sixth regressor variable, and fit a variant on the model where the response had a square root transform applied.

Note especially that if the data= argument is specified on the original call to the model fitting function, this information is passed on through the fitted model object to update() and its allies.

The name ‘.’ can also be used in other contexts, but with slightly different meaning. For example

> fmfull <- lm(y ~ . , data = production)

would fit a model with response y and regressor variables all other variables in the data frame production.

Other functions for exploring incremental sequences of models are add1(), drop1() and step(). The names of these give a good clue to their purpose, but for full details see the on-line help.

11.6 一般化線形モデル

Generalized linear modeling is a development of linear models to accommodate both non-normal response distributions and transformations to linearity in a clean and straightforward way. A generalized linear model may be described in terms of the following sequence of assumptions:

• There is a response, y, of interest and stimulus variables x_1, x_2, …, whose values influence the distribution of the response.
• The stimulus variables influence the distribution of y through a single linear function, only. This linear function is called the linear predictor, and is usually written

  eta = beta_1 x_1 + beta_2 x_2 + … + beta_p x_p,
  

  hence x_i has no influence on the distribution of y if and only if beta_i is zero.

• The distribution of y is of the form

  f_Y(y; mu, phi)
    = exp((A/phi) * (y lambda(mu) - gamma(lambda(mu))) + tau(y, phi))
  

  where phi is a scale parameter (possibly known), and is constant for all observations, A represents a prior weight, assumed known but possibly varying with the observations, and $\mu$ is the mean of y. So it is assumed that the distribution of y is determined by its mean and possibly a scale parameter as well.

• The mean, mu, is a smooth invertible function of the linear predictor:

  mu = m(eta),    eta = m^{-1}(mu) = ell(mu)
  

  and this inverse function, ell(), is called the link function.

These assumptions are loose enough to encompass a wide class of models useful in statistical practice, but tight enough to allow the development of a unified methodology of estimation and inference, at least approximately. The reader is referred to any of the current reference works on the subject for full details, such as McCullagh & Nelder (1989) or Dobson (1990).

11.6.1 ファミリ

The class of 一般化線形モデル handled by facilities supplied in R includes gaussian, binomial, poisson, inverse gaussian and gamma response distributions and also quasi-likelihood models where the response distribution is not explicitly specified. In the latter case the variance function must be specified as a function of the mean, but in other cases this function is implied by the response distribution.

Each response distribution admits a variety of link functions to connect the mean with the linear predictor. Those automatically available are shown in the following table:

Family name	Link functions
binomial	logit, probit, log, cloglog
gaussian	identity, log, inverse
Gamma	identity, inverse, log
inverse.gaussian	1/mu^2, identity, inverse, log
poisson	identity, log, sqrt
quasi	logit, probit, cloglog, identity, inverse, log, 1/mu^2, sqrt

The combination of a response distribution, a link function and various other pieces of information that are needed to carry out the modeling exercise is called the family of the generalized linear model.

11.6.2 glm() 関数

Since the distribution of the response depends on the stimulus variables through a single linear function only, the same mechanism as was used for linear models can still be used to specify the linear part of a generalized model. The family has to be specified in a different way.

The R function to fit a generalized linear model is glm() which uses the form

> fitted.model <- glm(formula, family=family.generator, data=data.frame)

The only new feature is the family.generator, which is the instrument by which the family is described. It is the name of a function that generates a list of functions and expressions that together define and control the model and estimation process. Although this may seem a little complicated at first sight, its use is quite simple.

The names of the standard, supplied family generators are given under “Family Name” in the table in Families. Where there is a choice of links, the name of the link may also be supplied with the family name, in parentheses as a parameter. In the case of the quasi family, the variance function may also be specified in this way.

Some examples make the process clear.

The gaussian family

A call such as

> fm <- glm(y ~ x1 + x2, family = gaussian, data = sales)

achieves the same result as

> fm <- lm(y ~ x1+x2, data=sales)

but much less efficiently. Note how the gaussian family is not automatically provided with a choice of links, so no parameter is allowed. If a problem requires a gaussian family with a nonstandard link, this can usually be achieved through the quasi family, as we shall see later.

The binomial family

Consider a small, artificial example, from Silvey (1970).

On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease, the effects of which become more marked with increasing age. Samples of islander males of various ages were tested for blindness and the results recorded. The data is shown below:

The problem we consider is to fit both logistic and probit models to this data, and to estimate for each model the LD50, that is the age at which the chance of blindness for a male inhabitant is 50%.

If y is the number of blind at age x and n the number tested, both models have the form y ~ B(n, F(beta_0 + beta_1 x)) where for the probit case, F(z) = Phi(z) is the standard normal distribution function, and in the logit case (the default), F(z) = e^z/(1+e^z). In both cases the LD50 is LD50 = - beta_0/beta_1 that is, the point at which the argument of the distribution function is zero.

The first step is to set the data up as a data frame

> kalythos <- data.frame(x = c(20,35,45,55,70), n = rep(50,5),
                         y = c(6,17,26,37,44))

To fit a binomial model using glm() there are three possibilities for the response:

• If the response is a vector it is assumed to hold binary data, and so must be a 0/1 vector.
• If the response is a two-column matrix it is assumed that the first column holds the number of successes for the trial and the second holds the number of failures.
• If the response is a factor, its first level is taken as failure (0) and all other levels as ‘success’ (1).

Here we need the second of these conventions, so we add a matrix to our data frame:

> kalythos$Ymat <- cbind(kalythos$y, kalythos$n - kalythos$y)

To fit the models we use

> fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos)
> fml <- glm(Ymat ~ x, family = binomial, data = kalythos)

Since the logit link is the default the parameter may be omitted on the second call. To see the results of each fit we could use

> summary(fmp)
> summary(fml)

Both models fit (all too) well. To find the LD50 estimate we can use a simple function:

> ld50 <- function(b) -b[1]/b[2]
> ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fml)); c(ldp, ldl)

The actual estimates from this data are 43.663 years and 43.601 years respectively.

Poisson models

With the Poisson family the default link is the log, and in practice the major use of this family is to fit surrogate Poisson log-linear models to frequency data, whose actual distribution is often multinomial. This is a large and important subject we will not discuss further here. It even forms a major part of the use of non-gaussian generalized models overall.

Occasionally genuinely Poisson data arises in practice and in the past it was often analyzed as gaussian data after either a log or a square-root transformation. As a graceful alternative to the latter, a Poisson generalized linear model may be fitted as in the following example:

> fmod <- glm(y ~ A + B + x, family = poisson(link=sqrt),
              data = worm.counts)

Quasi-likelihood models

For all families the variance of the response will depend on the mean and will have the scale parameter as a multiplier. The form of dependence of the variance on the mean is a characteristic of the response distribution; for example for the poisson distribution Var(y) = mu.

For quasi-likelihood estimation and inference the precise response distribution is not specified, but rather only a link function and the form of the variance function as it depends on the mean. Since quasi-likelihood estimation uses formally identical techniques to those for the gaussian distribution, this family provides a way of fitting gaussian models with non-standard link functions or variance functions, incidentally.

For example, consider fitting the non-linear regression y = theta_1 z_1 / (z_2 - theta_2) + e which may be written alternatively as y = 1 / (beta_1 x_1 + beta_2 x_2) + e where x_1 = z_2/z_1, x_2 = -1/z_1, beta_1 = 1/theta_1, and beta_2 = theta_2/theta_1. Supposing a suitable data frame to be set up we could fit this non-linear regression as

> nlfit <- glm(y ~ x1 + x2 - 1,
               family = quasi(link=inverse, variance=constant),
               data = biochem)

The reader is referred to the manual and the help document for further information, as needed.

11.7 非線形最小自乗法と最尤モデル

Certain forms of nonlinear model can be fitted by Generalized Linear Models (glm()). But in the majority of cases we have to approach the nonlinear curve fitting problem as one of nonlinear optimization. R’s nonlinear optimization routines are optim(), nlm() and nlminb(), which provide the functionality (and more) of S-PLUS’s ms() and nlminb(). We seek the parameter values that minimize some index of lack-of-fit, and they do this by trying out various parameter values iteratively. Unlike linear regression for example, there is no guarantee that the procedure will converge on satisfactory estimates. All the methods require initial guesses about what parameter values to try, and convergence may depend critically upon the quality of the starting values.

11.7.1 最小自乗法

One way to fit a nonlinear model is by minimizing the sum of the squared errors (SSE) or residuals. This method makes sense if the observed errors could have plausibly arisen from a normal distribution.

Here is an example from Bates & Watts (1988), page 51. The data are:

> x <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56,
         1.10, 1.10)
> y <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)

The fit criterion to be minimized is:

> fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2)

In order to do the fit we need initial estimates of the parameters. One way to find sensible starting values is to plot the data, guess some parameter values, and superimpose the model curve using those values.

> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 200 * xfit/(0.1 + xfit)
> lines(spline(xfit, yfit))

We could do better, but these starting values of 200 and 0.1 seem adequate. Now do the fit:

> out <- nlm(fn, p = c(200, 0.1), hessian = TRUE)

After the fitting, out$minimum is the SSE, and out$estimate are the 最小自乗法 estimates of the parameters. To obtain the approximate standard errors (SE) of the estimates we do:

> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))

The 2 which is subtracted in the line above represents the number of parameters. A 95% confidence interval would be the parameter estimate +/- 1.96 SE. We can superimpose the 最小自乗法 fit on a new plot:

> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 212.68384222 * xfit/(0.06412146 + xfit)
> lines(spline(xfit, yfit))

The standard package stats provides much more extensive facilities for fitting non-linear models by 最小自乗法. The model we have just fitted is the Michaelis-Menten model, so we can use

> df <- data.frame(x=x, y=y)
> fit <- nls(y ~ SSmicmen(x, Vm, K), df)
> fit
Nonlinear regression model
  model:  y ~ SSmicmen(x, Vm, K)
   data:  df
          Vm            K
212.68370711   0.06412123
 residual sum-of-squares:  1195.449
> summary(fit)

Formula: y ~ SSmicmen(x, Vm, K)

Parameters:
    Estimate Std. Error t value Pr(>|t|)
Vm 2.127e+02  6.947e+00  30.615 3.24e-11
K  6.412e-02  8.281e-03   7.743 1.57e-05

Residual standard error: 10.93 on 10 degrees of freedom

Correlation of Parameter Estimates:
      Vm
K 0.7651

11.7.2 最尤法

最尤法 is a method of nonlinear model fitting that applies even if the errors are not normal. The method finds the parameter values which maximize the log likelihood, or equivalently which minimize the negative log-likelihood. Here is an example from Dobson (1990), pp. 108–111. This example fits a logistic model to dose-response data, which clearly could also be fit by glm(). The data are:

> x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113,
         1.8369, 1.8610, 1.8839)
> y <- c( 6, 13, 18, 28, 52, 53, 61, 60)
> n <- c(59, 60, 62, 56, 63, 59, 62, 60)

The negative log-likelihood to minimize is:

> fn <- function(p)
   sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x))
           + log(choose(n, y)) ))

We pick sensible starting values and do the fit:

> out <- nlm(fn, p = c(-50,20), hessian = TRUE)

After the fitting, out$minimum is the negative log-likelihood, and out$estimate are the 最尤法 estimates of the parameters. To obtain the approximate SEs of the estimates we do:

> sqrt(diag(solve(out$hessian)))

A 95% confidence interval would be the parameter estimate +/- 1.96 SE.

11.8 幾つかの非標準モデル

We conclude this chapter with just a brief mention of some of the other facilities available in R for special regression and data analysis problems.

• Mixed models. The recommended nlme package provides functions lme() and nlme() for linear and non-linear mixed-effects models, that is linear and non-linear regressions in which some of the coefficients correspond to random effects. These functions make heavy use of formulae to specify the models.
• Local approximating regressions. The loess() function fits a nonparametric regression by using a locally weighted regression. Such regressions are useful for highlighting a trend in messy data or for data reduction to give some insight into a large data set.

  Function loess is in the standard package stats, together with code for projection pursuit regression.

• Robust regression. There are several functions available for fitting regression models in a way resistant to the influence of extreme outliers in the data. Function lqs in the recommended package MASS provides state-of-art algorithms for highly-resistant fits. Less resistant but statistically more efficient methods are available in packages, for example function rlm in package MASS.
• Additive models. This technique aims to construct a regression function from smooth additive functions of the determining variables, usually one for each determining variable. Functions avas and ace in package acepack and functions bruto and mars in package mda provide some examples of these techniques in user-contributed packages to R. An extension is Generalized Additive Models, implemented in user-contributed packages gam and mgcv.
• Tree-based models. Rather than seek an explicit global linear model for prediction or interpretation, tree-based models seek to bifurcate the data, recursively, at critical points of the determining variables in order to partition the data ultimately into groups that are as homogeneous as possible within, and as heterogeneous as possible between. The results often lead to insights that other data analysis methods tend not to yield.

  Models are again specified in the ordinary linear model form. The model fitting function is tree(), but many other generic functions such as plot() and text() are well adapted to displaying the results of a tree-based model fit in a graphical way.

  Tree models are available in R via the user-contributed packages rpart and tree.

12 グラフィックスの取扱い

Graphical facilities are an important and extremely versatile component of R の環境. It is possible to use the facilities to display a wide variety of statistical graphs and also to build entirely new types of graph.

The graphics facilities can be used in both interactive and batch modes, but in most cases, interactive use is more productive. Interactive use is also easy because at startup time R initiates a graphics device driver which opens a special graphics window for the display of interactive graphics. Although this is done automatically, it may useful to know that the command used is X11() under UNIX, windows() under Windows and quartz() under macOS. A new device can always be opened by dev.new().

Once the device driver is running, R plotting commands can be used to produce a variety of graphical displays and to create entirely new kinds of display.

Plotting commands are divided into three basic groups:

• High-level plotting functions create a new plot on the graphics device, possibly with axes, labels, titles and so on.
• Low-level plotting functions add more information to an existing plot, such as extra points, lines and labels.
• Interactive graphics functions allow you interactively add information to, or extract information from, an existing plot, using a pointing device such as a mouse.

In addition, R maintains a list of graphical parameters which can be manipulated to customize your plots.

This manual only describes what are known as ‘base’ graphics. A separate graphics sub-system in package grid coexists with base – it is more powerful but harder to use. There is a recommended package lattice which builds on grid and provides ways to produce multi-panel plots akin to those in the Trellis system in S.

12.1 高水準プロット命令

High-level plotting functions are designed to generate a complete plot of the data passed as arguments to the function. Where appropriate, axes, labels and titles are automatically generated (unless you request otherwise.) 高水準プロット命令 always start a new plot, erasing the current plot if necessary.

12.1.1 plot() 関数

One of the most frequently used plotting functions in R is the plot() function. This is a generic function: the type of plot produced is dependent on the type or class of the first argument.

plot(x, y)

plot(xy)

If x and y are vectors, plot(x, y) produces a scatterplot of y against x. The same effect can be produced by supplying one argument (second form) as either a list containing two elements x and y or a two-column matrix.

plot(x)

If x is a time series, this produces a time-series plot. If x is a numeric vector, it produces a plot of the values in the vector against their index in the vector. If x is a complex vector, it produces a plot of imaginary versus real parts of the vector elements.

plot(f)

plot(f, y)

f is a factor object, y is a numeric vector. The first form generates a bar plot of f; the second form produces boxplots of y for each level of f.

plot(df)

plot(~ expr)

plot(y ~ expr)

df is a data frame, y is any object, expr is a list of object names separated by ‘+’ (e.g., a + b + c). The first two forms produce distributional plots of the variables in a data frame (first form) or of a number of named objects (second form). The third form plots y against every object named in expr.

12.1.2 多変量データを表示する

R provides two very useful functions for representing multivariate data. If X is a numeric matrix or data frame, the command

> pairs(X)

produces a pairwise scatterplot matrix of the variables defined by the columns of X, that is, every column of X is plotted against every other column of X and the resulting n(n-1) plots are arranged in a matrix with plot scales constant over the rows and columns of the matrix.

When three or four variables are involved a coplot may be more enlightening. If a and b are numeric vectors and c is a numeric vector or factor object (all of the same length), then the command

> coplot(a ~ b | c)

produces a number of scatterplots of a against b for given values of c. If c is a factor, this simply means that a is plotted against b for every level of c. When c is numeric, it is divided into a number of conditioning intervals and for each interval a is plotted against b for values of c within the interval. The number and position of intervals can be controlled with given.values= argument to coplot()—the function co.intervals() is useful for selecting intervals. You can also use two given variables with a command like

> coplot(a ~ b | c + d)

which produces scatterplots of a against b for every joint conditioning interval of c and d.

The coplot() and pairs() function both take an argument panel= which can be used to customize the type of plot which appears in each panel. The default is points() to produce a scatterplot but by supplying some other low-level graphics function of two vectors x and y as the value of panel= you can produce any type of plot you wish. An example panel function useful for coplots is panel.smooth().

12.1.3グラフィックスの表示

Other high-level graphics functions produce different types of plots. Some examples are:

qqnorm(x)

qqline(x)

qqplot(x, y)

Distribution-comparison plots. The first form plots the numeric vector x against the expected Normal order scores (a normal scores plot) and the second adds a straight line to such a plot by drawing a line through the distribution and data quartiles. The third form plots the quantiles of x against those of y to compare their respective distributions.

hist(x)

hist(x, nclass=n)

hist(x, breaks=b, …)

Produces a histogram of the numeric vector x. A sensible number of classes is usually chosen, but a recommendation can be given with the nclass= argument. Alternatively, the breakpoints can be specified exactly with the breaks= argument. If the probability=TRUE argument is given, the bars represent relative frequencies divided by bin width instead of counts.

dotchart(x, …)

Constructs a dotchart of the data in x. In a dotchart the y-axis gives a labelling of the data in x and the x-axis gives its value. For example it allows easy visual selection of all data entries with values lying in specified ranges.

image(x, y, z, …)

contour(x, y, z, …)

persp(x, y, z, …)

Plots of three variables. The image plot draws a grid of rectangles using different colours to represent the value of z, the contour plot draws contour lines to represent the value of z, and the persp plot draws a 3D surface.

12.1.4 高水準プロット関数への引数

There are a number of arguments which may be passed to high-level graphics functions, as follows:

add=TRUE

Forces the function to act as a low-level graphics function, superimposing the plot on the current plot (some functions only).

axes=FALSE

Suppresses generation of axes—useful for adding your own custom axes with the axis() function. The default, axes=TRUE, means include axes.

log="x"

log="y"

log="xy"

Causes the x, y or both axes to be logarithmic. This will work for many, but not all, types of plot.

type=

The type= argument controls the type of plot produced, as follows:

type="p"

Plot individual points (the default)

type="l"

Plot lines

type="b"

Plot points connected by lines (both)

type="o"

Plot points overlaid by lines

type="h"

Plot vertical lines from points to the zero axis (high-density)

type="s"

type="S"

Step-function plots. In the first form, the top of the vertical defines the point; in the second, the bottom.

type="n"

No plotting at all. However axes are still drawn (by default) and the coordinate system is set up according to the data. Ideal for creating plots with subsequent low-level graphics functions.

xlab=string

ylab=string

Axis labels for the x and y axes. Use these arguments to change the default labels, usually the names of the objects used in the call to the high-level plotting function.

main=string

Figure title, placed at the top of the plot in a large font.

sub=string

Sub-title, placed just below the x-axis in a smaller font.

12.2 低水準プロット命令

Sometimes the high-level plotting functions don’t produce exactly the kind of plot you desire. In this case, 低水準プロット命令 can be used to add extra information (such as points, lines or text) to the current plot.

Some of the more useful low-level plotting functions are:

points(x, y)

lines(x, y)

Adds points or connected lines to the current plot. plot()’s type= argument can also be passed to these functions (and defaults to "p" for points() and "l" for lines().)

text(x, y, labels, …)

Add text to a plot at points given by x, y. Normally labels is an integer or character vector in which case labels[i] is plotted at point (x[i], y[i]). The default is 1:length(x).

Note: This function is often used in the sequence

> plot(x, y, type="n"); text(x, y, names)

The graphics parameter type="n" suppresses the points but sets up the axes, and the text() function supplies special characters, as specified by the character vector names for the points.

abline(a, b)

abline(h=y)

abline(v=x)

abline(lm.obj)

Adds a line of slope b and intercept a to the current plot. h=y may be used to specify y-coordinates for the heights of horizontal lines to go across a plot, and v=x similarly for the x-coordinates for vertical lines. Also lm.obj may be list with a coefficients component of length 2 (such as the result of model-fitting functions,) which are taken as an intercept and slope, in that order.

polygon(x, y, …)

Draws a polygon defined by the ordered vertices in (x, y) and (optionally) shade it in with hatch lines, or fill it if the graphics device allows the filling of figures.

legend(x, y, legend, …)

Adds a legend to the current plot at the specified position. Plotting characters, line styles, colors etc., are identified with the labels in the character vector legend. At least one other argument v (a vector the same length as legend) with the corresponding values of the plotting unit must also be given, as follows:

legend( , fill=v)

Colors for filled boxes

legend( , col=v)

Colors in which points or lines will be drawn

legend( , lty=v)

Line styles

legend( , lwd=v)

Line widths

legend( , pch=v)

Plotting characters (character vector)

title(main, sub)

Adds a title main to the top of the current plot in a large font and (optionally) a sub-title sub at the bottom in a smaller font.

axis(side, …)

Adds an axis to the current plot on the side given by the first argument (1 to 4, counting clockwise from the bottom.) Other arguments control the positioning of the axis within or beside the plot, and tick positions and labels. Useful for adding custom axes after calling plot() with the axes=FALSE argument.

Low-level plotting functions usually require some positioning information (e.g., x and y coordinates) to determine where to place the new plot elements. Coordinates are given in terms of user coordinates which are defined by the previous high-level graphics command and are chosen based on the supplied data.

Where x and y arguments are required, it is also sufficient to supply a single argument being a list with elements named x and y. Similarly a matrix with two columns is also valid input. In this way functions such as locator() (see below) may be used to specify positions on a plot interactively.

12.2.1 数式による注釈

In some cases, it is useful to add mathematical symbols and formulae to a plot. This can be achieved in R by specifying an expression rather than a character string in any one of text, mtext, axis, or title. For example, the following code draws the formula for the Binomial probability function:

> text(x, y, expression(paste(bgroup("(", atop(n, x), ")"), p^x, q^{n-x})))

More information, including a full listing of the features available can obtained from within R using the commands:

> help(plotmath)
> example(plotmath)
> demo(plotmath)

12.2.2 Hershey ベクトルフォント.

It is possible to specify Hershey ベクトルフォント. for rendering text when using the text and contour functions. There are three reasons for using the Hershey fonts:

• Hershey fonts can produce better output, especially on a computer screen, for rotated and/or small text.
• Hershey fonts provide certain symbols that may not be available in the standard fonts. In particular, there are zodiac signs, cartographic symbols and astronomical symbols.
• Hershey fonts provide cyrillic and japanese (Kana and Kanji) characters.

More information, including tables of Hershey characters can be obtained from within R using the commands:

> help(Hershey)
> demo(Hershey)
> help(Japanese)
> demo(Japanese)

12.3 グラフィックスの対話的操作

R also provides functions which allow users to extract or add information to a plot using a mouse. The simplest of these is the locator() function:

locator(n, type)

Waits for the user to select locations on the current plot using the left mouse button. This continues until n (default 512) points have been selected, or another mouse button is pressed. The type argument allows for plotting at the selected points and has the same effect as for high-level graphics commands; the default is no plotting. locator() returns the locations of the points selected as a list with two components x and y.

locator() is usually called with no arguments. It is particularly useful for interactively selecting positions for graphic elements such as legends or labels when it is difficult to calculate in advance where the graphic should be placed. For example, to place some informative text near an outlying point, the command

> text(locator(1), "Outlier", adj=0)

may be useful. (locator() will be ignored if the current device, such as postscript does not support interactive pointing.)

identify(x, y, labels)

Allow the user to highlight any of the points defined by x and y (using the left mouse button) by plotting the corresponding component of labels nearby (or the index number of the point if labels is absent). Returns the indices of the selected points when another button is pressed.

Sometimes we want to identify particular points on a plot, rather than their positions. For example, we may wish the user to select some observation of interest from a graphical display and then manipulate that observation in some way. Given a number of (x, y) coordinates in two numeric vectors x and y, we could use the identify() function as follows:

> plot(x, y)
> identify(x, y)

The identify() functions performs no plotting itself, but simply allows the user to move the mouse pointer and click the left mouse button near a point. If there is a point near the mouse pointer it will be marked with its index number (that is, its position in the x/y vectors) plotted nearby. Alternatively, you could use some informative string (such as a case name) as a highlight by using the labels argument to identify(), or disable marking altogether with the plot = FALSE argument. When the process is terminated (see above), identify() returns the indices of the selected points; you can use these indices to extract the selected points from the original vectors x and y.

12.4 グラフィックスパラメータを使う

When creating graphics, particularly for presentation or publication purposes, R’s defaults do not always produce exactly that which is required. You can, however, customize almost every aspect of the display using graphics parameters. R maintains a list of a large number of graphics parameters which control things such as line style, colors, figure arrangement and text justification among many others. Every graphics parameter has a name (such as ‘col’, which controls colors,) and a value (a color number, for example.)

A separate list of graphics parameters is maintained for each active device, and each device has a default set of parameters when initialized. Graphics parameters can be set in two ways: either permanently, affecting all graphics functions which access the current device; or temporarily, affecting only a single graphics function call.

12.4.1 永続的変更：par() 関数

The par() function is used to access and modify the list of graphics parameters for the current graphics device.

par()

Without arguments, returns a list of all graphics parameters and their values for the current device.

par(c("col", "lty"))

With a character vector argument, returns only the named graphics parameters (again, as a list.)

par(col=4, lty=2)

With named arguments (or a single list argument), sets the values of the named graphics parameters, and returns the original values of the parameters as a list.

Setting graphics parameters with the par() function changes the value of the parameters permanently, in the sense that all future calls to graphics functions (on the current device) will be affected by the new value. You can think of setting graphics parameters in this way as setting “default” values for the parameters, which will be used by all graphics functions unless an alternative value is given.

Note that calls to par() always affect the global values of graphics parameters, even when par() is called from within a function. This is often undesirable behavior—usually we want to set some graphics parameters, do some plotting, and then restore the original values so as not to affect the user’s R session. You can restore the initial values by saving the result of par() when making changes, and restoring the initial values when plotting is complete.

> oldpar <- par(col=4, lty=2)
  … plotting commands …
> par(oldpar)

To save and restore all settable²⁵ graphical parameters use

> oldpar <- par(no.readonly=TRUE)
  … plotting commands …
> par(oldpar)

12.4.2 一時的変更：グラフィックス関数への引数

Graphics parameters may also be passed to (almost) any graphics function as named arguments. This has the same effect as passing the arguments to the par() function, except that the changes only last for the duration of the function call. For example:

> plot(x, y, pch="+")

produces a scatterplot using a plus sign as the plotting character, without changing the default plotting character for future plots.

Unfortunately, this is not implemented entirely consistently and it is sometimes necessary to set and reset graphics parameters using par().

12.5 グラフィックスパラメータのリスト

The following sections detail many of the commonly-used graphical parameters. The R help documentation for the par() function provides a more concise summary; this is provided as a somewhat more detailed alternative.

Graphics parameters will be presented in the following form:

name=value

A description of the parameter’s effect. name is the name of the parameter, that is, the argument name to use in calls to par() or a graphics function. value is a typical value you might use when setting the parameter.

Note that axes is not a graphics parameter but an argument to a few plot methods: see xaxt and yaxt.

12.5.1 グラフィックスの要素

R plots are made up of points, lines, text and polygons (filled regions.) Graphical parameters exist which control how these グラフィックスの要素 are drawn, as follows:

pch="+"

Character to be used for plotting points. The default varies with graphics drivers, but it is usually a circle. Plotted points tend to appear slightly above or below the appropriate position unless you use "." as the plotting character, which produces centered points.

pch=4

When pch is given as an integer between 0 and 25 inclusive, a specialized plotting symbol is produced. To see what the symbols are, use the command

> legend(locator(1), as.character(0:25), pch = 0:25)

Those from 21 to 25 may appear to duplicate earlier symbols, but can be coloured in different ways: see the help on points and its examples.

In addition, pch can be a character or a number in the range 32:255 representing a character in the current font.

lty=2

Line types. Alternative line styles are not supported on all graphics devices (and vary on those that do) but line type 1 is always a solid line, line type 0 is always invisible, and line types 2 and onwards are dotted or dashed lines, or some combination of both.

lwd=2

Line widths. Desired width of lines, in multiples of the “standard” line width. Affects axis lines as well as lines drawn with lines(), etc. Not all devices support this, and some have restrictions on the widths that can be used.

col=2

Colors to be used for points, lines, text, filled regions and images. A number from the current palette (see ?palette) or a named colour.

col.axis

col.lab

col.main

col.sub

The color to be used for axis annotation, x and y labels, main and sub-titles, respectively.

font=2

An integer which specifies which font to use for text. If possible, デバイスドライバ arrange so that 1 corresponds to plain text, 2 to bold face, 3 to italic, 4 to bold italic and 5 to a symbol font (which include Greek letters).

font.axis

font.lab

font.main

font.sub

The font to be used for axis annotation, x and y labels, main and sub-titles, respectively.

adj=-0.1

Justification of text relative to the plotting position. 0 means left justify, 1 means right justify and 0.5 means to center horizontally about the plotting position. The actual value is the proportion of text that appears to the left of the plotting position, so a value of -0.1 leaves a gap of 10% of the text width between the text and the plotting position.

cex=1.5

Character expansion. The value is the desired size of text characters (including plotting characters) relative to the default text size.

cex.axis

cex.lab

cex.main

cex.sub

The character expansion to be used for axis annotation, x and y labels, main and sub-titles, respectively.

12.5.2 軸と目盛マーク

Many of R’s high-level plots have axes, and you can construct axes yourself with the low-level axis() graphics function. Axes have three main components: the axis line (line style controlled by the lty graphics parameter), the tick marks (which mark off unit divisions along the axis line) and the tick labels (which mark the units.) These components can be customized with the following graphics parameters.

lab=c(5, 7, 12)

The first two numbers are the desired number of tick intervals on the x and y axes respectively. The third number is the desired length of axis labels, in characters (including the decimal point.) Choosing a too-small value for this parameter may result in all tick labels being rounded to the same number!

las=1

Orientation of axis labels. 0 means always parallel to axis, 1 means always horizontal, and 2 means always perpendicular to the axis.

mgp=c(3, 1, 0)

Positions of axis components. The first component is the distance from the axis label to the axis position, in text lines. The second component is the distance to the tick labels, and the final component is the distance from the axis position to the axis line (usually zero). Positive numbers measure outside the plot region, negative numbers inside.

tck=0.01

Length of tick marks, as a fraction of the size of the plotting region. When tck is small (less than 0.5) the tick marks on the x and y axes are forced to be the same size. A value of 1 gives grid lines. Negative values give tick marks outside the plotting region. Use tck=0.01 and mgp=c(1,-1.5,0) for internal tick marks.

xaxs="r"

yaxs="i"

Axis styles for the x and y axes, respectively. With styles "i" (internal) and "r" (the default) tick marks always fall within the range of the data, however style "r" leaves a small amount of space at the edges. (S has other styles not implemented in R.)

12.5.3 図の余白

A single plot in R is known as a figure and comprises a plot region surrounded by margins (possibly containing axis labels, titles, etc.) and (usually) bounded by the axes themselves.

A typical figure is

Graphics parameters controlling figure layout include:

mai=c(1, 0.5, 0.5, 0)

Widths of the bottom, left, top and right margins, respectively, measured in inches.

mar=c(4, 2, 2, 1)

Similar to mai, except the measurement unit is text lines.

mar and mai are equivalent in the sense that setting one changes the value of the other. The default values chosen for this parameter are often too large; the right-hand margin is rarely needed, and neither is the top margin if no title is being used. The bottom and left margins must be large enough to accommodate the axis and tick labels. Furthermore, the default is chosen without regard to the size of the device surface: for example, using the postscript() driver with the height=4 argument will result in a plot which is about 50% margin unless mar or mai are set explicitly. When multiple figures are in use (see below) the margins are reduced, however this may not be enough when many figures share the same page.

12.5.4 複数の図用の環境

R allows you to create an n by m array of figures on a single page. Each figure has its own margins, and the array of figures is optionally surrounded by an outer margin, as shown in the following figure.

The graphical parameters relating to multiple figures are as follows:

mfcol=c(3, 2)

mfrow=c(2, 4)

Set the size of a multiple figure array. The first value is the number of rows; the second is the number of columns. The only difference between these two parameters is that setting mfcol causes figures to be filled by column; mfrow fills by rows.

The layout in the Figure could have been created by setting mfrow=c(3,2); the figure shows the page after four plots have been drawn.

Setting either of these can reduce the base size of symbols and text (controlled by par("cex") and the pointsize of the device). In a layout with exactly two rows and columns the base size is reduced by a factor of 0.83: if there are three or more of either rows or columns, the reduction factor is 0.66.

mfg=c(2, 2, 3, 2)

Position of the current figure in a 複数の図用の環境. The first two numbers are the row and column of the current figure; the last two are the number of rows and columns in the multiple figure array. Set this parameter to jump between figures in the array. You can even use different values for the last two numbers than the true values for unequally-sized figures on the same page.

fig=c(4, 9, 1, 4)/10

Position of the current figure on the page. Values are the positions of the left, right, bottom and top edges respectively, as a percentage of the page measured from the bottom left corner. The example value would be for a figure in the bottom right of the page. Set this parameter for arbitrary positioning of figures within a page. If you want to add a figure to a current page, use new=TRUE as well (unlike S).

oma=c(2, 0, 3, 0)

omi=c(0, 0, 0.8, 0)

Size of outer margins. Like mar and mai, the first measures in text lines and the second in inches, starting with the bottom margin and working clockwise.

Outer margins are particularly useful for page-wise titles, etc. Text can be added to the outer margins with the mtext() function with argument outer=TRUE. There are no outer margins by default, however, so you must create them explicitly using oma or omi.

More complicated arrangements of multiple figures can be produced by the split.screen() and layout() functions, as well as by the grid and lattice packages.

12.6 デバイスドライバ

R can generate graphics (of varying levels of quality) on almost any type of display or printing device. Before this can begin, however, R needs to be informed what type of device it is dealing with. This is done by starting a device driver. The purpose of a device driver is to convert graphical instructions from R (“draw a line,” for example) into a form that the particular device can understand.

デバイスドライバ are started by calling a device driver function. There is one such function for every device driver: type help(Devices) for a list of them all. For example, issuing the command

> postscript()

causes all future graphics output to be sent to the printer in PostScript format. Some commonly-used デバイスドライバ are:

X11()

For use with the X11 window system on Unix-alikes

windows()

For use on Windows

quartz()

For use on macOS

postscript()

For printing on PostScript printers, or creating PostScript graphics files.

pdf()

Produces a PDF file, which can also be included into PDF files.

png()

Produces a bitmap PNG file. (Not always available: see its help page.)

jpeg()

Produces a bitmap JPEG file, best used for image plots. (Not always available: see its help page.)

When you have finished with a device, be sure to terminate the device driver by issuing the command

> dev.off()

This ensures that the device finishes cleanly; for example in the case of hardcopy devices this ensures that every page is completed and has been sent to the printer. (This will happen automatically at the normal end of a session.)

12.6.1 文章の製版用のPostscript 図式

By passing the file argument to the postscript() device driver function, you may store the graphics in PostScript format in a file of your choice. The plot will be in landscape orientation unless the horizontal=FALSE argument is given, and you can control the size of the graphic with the width and height arguments (the plot will be scaled as appropriate to fit these dimensions.) For example, the command

> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)

will produce a file containing PostScript code for a figure five inches high, perhaps for inclusion in a document. It is important to note that if the file named in the command already exists, it will be overwritten. This is the case even if the file was only created earlier in the same R session.

Many usages of PostScript output will be to incorporate the figure in another document. This works best when encapsulated PostScript is produced: R always produces conformant output, but only marks the output as such when the onefile=FALSE argument is supplied. This unusual notation stems from S-compatibility: it really means that the output will be a single page (which is part of the EPSF specification). Thus to produce a plot for inclusion use something like

> postscript("plot1.eps", horizontal=FALSE, onefile=FALSE,
             height=8, width=6, pointsize=10)

12.6.2 複数のグラフィックスドライバ

In advanced use of R it is often useful to have several graphics devices in use at the same time. Of course only one graphics device can accept graphics commands at any one time, and this is known as the current device. When multiple devices are open, they form a numbered sequence with names giving the kind of device at any position.

The main commands used for operating with multiple devices, and their meanings are as follows:

X11()

[UNIX]

windows()

win.printer()

win.metafile()

[Windows]

quartz()

[macOS]

postscript()

pdf()

png()

jpeg()

tiff()

bitmap()

…

Each new call to a device driver function opens a new graphics device, thus extending by one the device list. This device becomes the current device, to which graphics output will be sent.

dev.list()

Returns the number and name of all active devices. The device at position 1 on the list is always the null device which does not accept graphics commands at all.

dev.next()

dev.prev()

Returns the number and name of the graphics device next to, or previous to the current device, respectively.

dev.set(which=k)

Can be used to change the current graphics device to the one at position k of the device list. Returns the number and label of the device.

dev.off(k)

Terminate the graphics device at point k of the device list. For some devices, such as postscript devices, this will either print the file immediately or correctly complete the file for later printing, depending on how the device was initiated.

dev.copy(device, …, which=k)

dev.print(device, …, which=k)

Make a copy of the device k. Here device is a device function, such as postscript, with extra arguments, if needed, specified by ‘…’. dev.print is similar, but the copied device is immediately closed, so that end actions, such as printing hardcopies, are immediately performed.

graphics.off()

Terminate all graphics devices on the list, except the null device.

12.7 動的なグラフィックス

R does not have builtin capabilities for dynamic or interactive graphics, e.g. rotating point clouds or to “brushing” (interactively highlighting) points. However, extensive 動的なグラフィックス facilities are available in the system GGobi by Swayne, Cook and Buja available from

http://www.ggobi.org/

and these can be accessed from R via the package rggobi, described at http://www.ggobi.org/rggobi.

Also, package rgl provides ways to interact with 3D plots, for example of surfaces.

13 パッケージ

All R functions and datasets are stored in パッケージ. Only when a package is loaded are its contents available. This is done both for efficiency (the full list would take more memory and would take longer to search than a subset), and to aid package developers, who are protected from name clashes with other code. The process of developing packages is described in Creating R packages in Writing R Extensions. Here, we will describe them from a user’s point of view.

To see which packages are installed at your site, issue the command

> library()

with no arguments. To load a particular package (e.g., the boot package containing functions from Davison & Hinkley (1997)), use a command like

> library(boot)

Users connected to the Internet can use the install.packages() and update.packages() functions (available through the パッケージ menu in the Windows and macOS GUIs, see Installing packages in R Installation and Administration) to install and update packages.

To see which packages are currently loaded, use

> search()

to display the search list. Some packages may be loaded but not available on the search list (see Namespaces): these will be included in the list given by

> loadedNamespaces()

To see a list of all available help topics in an installed package, use

> help.start()

to start the HTML help system, and then navigate to the package listing in the Reference section.

13.1 標準パッケージ

The standard (or base) packages are considered part of the R source code. They contain the basic functions that allow R to work, and the datasets and standard statistical and graphical functions that are described in this manual. They should be automatically available in any R installation. See R packages in R FAQ, for a complete list.

13.2 提供パッケージと CRAN

There are thousands of contributed packages for R, written by many different authors. Some of these packages implement specialized statistical methods, others give access to data or hardware, and others are designed to complement textbooks. Some (the recommended packages) are distributed with every binary distribution of R. Most are available for download from CRAN (https://CRAN.R-project.org/ and its mirrors) and other repositories such as Bioconductor (https://www.bioconductor.org/). and Omegahat (http://www.omegahat.net/). The R FAQ contains a list of CRAN packages current at the time of release, but the collection of available packages changes very frequently.

13.3 名前空間

Packages have namespaces, which do three things: they allow the package writer to hide functions and data that are meant only for internal use, they prevent functions from breaking when a user (or other package writer) picks a name that clashes with one in the package, and they provide a way to refer to an object within a particular package.

For example, t() is the transpose function in R, but users might define their own function named t. Namespaces prevent the user’s definition from taking precedence, and breaking every function that tries to transpose a matrix.

There are two operators that work with namespaces. The double-colon operator :: selects definitions from a particular namespace. In the example above, the transpose function will always be available as base::t, because it is defined in the base package. Only functions that are exported from the package can be retrieved in this way.

The triple-colon operator ::: may be seen in a few places in R code: it acts like the double-colon operator but also allows access to hidden objects. Users are more likely to use the getAnywhere() function, which searches multiple packages.

Packages are often inter-dependent, and loading one may cause others to be automatically loaded. The colon operators described above will also cause automatic loading of the associated package. When packages with namespaces are loaded automatically they are not added to the search list.

14 OS の機能

R has quite extensive facilities to access the OS under which it is running: this allows it to be used as a scripting language and that ability is much used by R itself, for example to install packages.

Because R’s own scripts need to work across all platforms, considerable effort has gone into make the scripting facilities as platform-independent as is feasible.

14.1 ファイルとディレクトリ

There are many functions to manipulate ファイルとディレクトリ. Here are pointers to some of the more commonly used ones.

To create an (empty) file or directory, use file.create or create.dir. (These are the analogues of the POSIX utilities touch and mkdir.) For temporary files and directories in the R session directory see tempfile.

Files can be removed by either file.remove or unlink: the latter can remove directory trees.

For directory listings use list.files (also available as dir) or list.dirs. These can select files using a regular expression: to select by wildcards use Sys.glob.

Many types of information on a filepath (including for example if it is a file or directory) can be found by file.info.

There are several ways to find out if a file ‘exists’ (a file can exist on the filesystem and not be visible to the current user). There are functions file.exists, file.access and file_test with various versions of this test: file_test is a version of the POSIX test command for those familiar with shell scripting.

Function file.copy is the R analogue of the POSIX command cp.

Choosing files can be done interactively by file.choose: the Windows port has the more versatile functions choose.files and choose.dir and there are similar functions in the tcltk package: tk_choose.files and tk_choose.dir.

Functions file.show and file.edit will display and edit one or more files in a way appropriate to the R port, using the facilities of a console (such as RGui on Windows or R.app on macOS) if one is in use.

There is some support for links in the filesystem: see functions file.link and Sys.readlink.

14.2 ファイルパス

With a few exceptions, R relies on the underlying OS functions to manipulate ファイルパス. Some aspects of this are allowed to depend on the OS, and do, even down to the version of the OS. There are POSIX standards for how OSes should interpret ファイルパス and many R users assume POSIX compliance: but Windows does not claim to be compliant and other OSes may be less than completely compliant.

The following are some issues which have been encountered with ファイルパス.

• POSIX filesystems are case-sensitive, so foo.png and Foo.PNG are different files. However, the defaults on Windows and macOS are to be case-insensitive, and FAT filesystems (commonly used on removable storage) are not normally case-sensitive (and all ファイルパス may be mapped to lower case).
• Almost all the Windows’ OS services support the use of slash or backslash as the filepath separator, and R converts the known exceptions to the form required by Windows.
• The behaviour of ファイルパス with a trailing slash is OS-dependent. Such paths are not valid on Windows and should not be expected to work. POSIX-2008 requires such paths to match only directories, but earlier versions allowed them to also match files. So they are best avoided.
• Multiple slashes in ファイルパス such as /abc//def are valid on POSIX filesystems and treated as if there was only one slash. They are usually accepted by Windows’ OS functions. However, leading double slashes may have a different meaning.
• Windows’ UNC ファイルパス (such as \\server\dir1\dir2\file and \\?\UNC\server\dir1\dir2\file) are not supported, but they may work in some R functions. POSIX filesystems are allowed to treat a leading double slash specially.
• Windows allows ファイルパス containing drives and relative to the current directory on a drive, e.g. d:foo/bar refers to d:/a/b/c/foo/bar if the current directory on drive d: is /a/b/c. It is intended that these work, but the use of absolute paths is safer.

Functions basename and dirname select parts of a file path: the recommended way to assemble a file path from components is file.path. Function pathexpand does ‘tilde expansion’, substituting values for home directories (the current user’s, and perhaps those of other users).

On filesystems with links, a single file can be referred to by many ファイルパス. Function normalizePath will find a canonical filepath.

Windows has the concepts of short (‘8.3’) and long file names: normalizePath will return an absolute path using long file names and shortPathName will return a version using short names. The latter does not contain spaces and uses backslash as the separator, so is sometimes useful for exporting names from R.

File permissions are a related topic. R has support for the POSIX concepts of read/write/execute permission for owner/group/all but this may be only partially supported on the filesystem, so for example on Windows only read-only files (for the account running the R session) are recognized. Access Control Lists (ACLs) are employed on several filesystems, but do not have an agreed standard and R has no facilities to control them. Use Sys.chmod to change permissions.

14.3 システムコマンド

Functions system and system2 are used to invoke a system command and optionally collect its output. system2 is a little more general but its main advantage is that it is easier to write cross-platform code using it.

system behaves differently on Windows from other OSes (because the API C call of that name does). Elsewhere it invokes a shell to run the command: the Windows port of R has a function shell to do that.

To find out if the OS includes a command, use Sys.which, which attempts to do this in a cross-platform way (unfortunately it is not a standard OS service).

Function shQuote will quote ファイルパス as needed for commands in the current OS.

14.4 圧縮とアーカイブ

Recent versions of R have extensive facilities to read and write compressed files, often transparently. Reading of files in R is to a vey large extent done by connections, and the file function which is used to open a connection to a file (or a URL) and is able to identify the compression used from the ‘magic’ header of the file.

The type of compression which has been supported for longest is gzip compression, and that remains a good general compromise. Files compressed by the earlier Unix compress utility can also be read, but these are becoming rare. Two other forms of compression, those of the bzip2 and xz utilities are also available. These generally achieve higher rates of compression (depending on the file, much higher) at the expense of slower decompression and much slower compression.

There is some confusion between xz and lzma compression (see https://en.wikipedia.org/wiki/Xz and https://en.wikipedia.org/wiki/LZMA): R can read files compressed by most versions of either.

File archives are single files which contain a collection of files, the most common ones being ‘tarballs’ and zip files as used to distribute R packages. R can list and unpack both (see functions untar and unzip) and create both (for zip with the help of an external program).

付録 A サンプルセッション

以下のセッションには、R 環境のいくつかの機能を使用しながら紹介します。 システムの多くの機能は、最初は不慣れで困惑するでしょうが、 この難問はすぐに解消できます。

使用しているプラットフォームに合わせて R を適切に起動します（R の呼び出しを参照）。

The R program begins, with a banner.

(R コード内では、左側のプロンプトは混乱を避けるために表示しません。)

help.start()

オンラインヘルプへの HTML インターフェイスを立ち上げます(あなたの計算機で利用可能な wen ブブラウザを使用します)。 マウスを使ってこの機能の特徴を手短に調べてみてください。

ヘルプウィンドウをアイコン化し、次に進んでください。

x <- rnorm(50)

y <- rnorm(x)

x- と y-座標を表す 2 組の正規乱数を生成します。

plot(x, y)

点を平面にプロットします。グラフィックスウィンドウが自動的に表示されます。

ls()

R の作業スペースにどういう R オブジェクトがあるかを表示します。

rm(x, y)

不要なオブジェクトを削除(掃除)します。

x <- 1:20

x = (1, 2, …, 20) を作ります。.

w <- 1 + sqrt(x)/2

標準偏差の「重み」ベクトル。

dummy <- data.frame(x=x, y= x + rnorm(x)*w)

dummy

2 つの列ベクトル x と y の「データフレーム」を作成し、表示します、

fm <- lm(y ~ x, data=dummy)

summary(fm)

Fit a simple linear regression and look at the analysis. With y to the left of the tilde, we are modelling y dependent on x.

fm1 <- lm(y ~ x, data=dummy, weight=1/w^2)

summary(fm1)

Since we know the standard deviations, we can do a weighted regression.

attach(dummy)

Make the columns in the data frame visible as variables.

lrf <- lowess(x, y)

Make a nonparametric local regression function.

plot(x, y)

Standard point plot.

lines(x, lrf$y)

Add in the local regression.

abline(0, 1, lty=3)

The true regression line: (intercept 0, slope 1).

abline(coef(fm))

Unweighted regression line.

abline(coef(fm1), col = "red")

Weighted regression line.

detach()

Remove data frame from the search path.

plot(fitted(fm), resid(fm),

     xlab="Fitted values",

     ylab="Residuals",

     main="Residuals vs Fitted")

A standard regression diagnostic plot to check for heteroscedasticity. Can you see it?

qqnorm(resid(fm), main="Residuals Rankit Plot")

A normal scores plot to check for skewness, kurtosis and outliers. (Not very useful here.)

rm(fm, fm1, lrf, x, dummy)

Clean up again.

The next section will look at data from the classical experiment of Michelson to measure the speed of light. This dataset is available in the morley object, but we will read it to illustrate the read.table function.

filepath <- system.file("data", "morley.tab" , package="datasets")

filepath

Get the path to the data file.

file.show(filepath)

Optional. Look at the file.

mm <- read.table(filepath)

mm

Read in the Michelson data as a data frame, and look at it. There are five experiments (column Expt) and each has 20 runs (column Run) and sl is the recorded speed of light, suitably coded.

mm$Expt <- factor(mm$Expt)

mm$Run <- factor(mm$Run)

Change Expt and Run into factors.

attach(mm)

Make the data frame visible at position 3 (the default).

plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.")

Compare the five experiments with simple boxplots.

fm <- aov(Speed ~ Run + Expt, data=mm)

summary(fm)

Analyze as a randomized block, with ‘runs’ and ‘experiments’ as factors.

fm0 <- update(fm, . ~ . - Run)

anova(fm0, fm)

Fit the sub-model omitting ‘runs’, and compare using a formal analysis of variance.

detach()

rm(fm, fm0)

Clean up before moving on.

We now look at some more graphical features: contour and image plots.

x <- seq(-pi, pi, len=50)

y <- x

x is a vector of 50 equally spaced values in the interval [-pi\, pi]. y is the same.

f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))

f is a square matrix, with rows and columns indexed by x and y respectively, of values of the function cos(y)/(1 + x^2).

oldpar <- par(no.readonly = TRUE)

par(pty="s")

Save the plotting parameters and set the plotting region to “square”.

contour(x, y, f)

contour(x, y, f, nlevels=15, add=TRUE)

Make a contour map of f; add in more lines for more detail.

fa <- (f-t(f))/2

fa is the “asymmetric part” of f. (t() is transpose).

contour(x, y, fa, nlevels=15)

Make a contour plot, …

par(oldpar)

… and restore the old graphics parameters.

image(x, y, f)

image(x, y, fa)

Make some high density image plots, (of which you can get hardcopies if you wish), …

objects(); rm(x, y, f, fa)

… and clean up before moving on.

R can do complex arithmetic, also.

th <- seq(-pi, pi, len=100)

z <- exp(1i*th)

1i is used for the complex number i.

par(pty="s")

plot(z, type="l")

Plotting complex arguments means plot imaginary versus real parts. This should be a circle.

w <- rnorm(100) + rnorm(100)*1i

Suppose we want to sample points within the unit circle. One method would be to take complex numbers with standard normal real and imaginary parts …

w <- ifelse(Mod(w) > 1, 1/w, w)

… and to map any outside the circle onto their reciprocal.

plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y")

lines(z)

All points are inside the unit circle, but the distribution is not uniform.

w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i)

plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y")

lines(z)

The second method uses the uniform distribution. The points should now look more evenly spaced over the disc.

rm(th, w, z)

Clean up again.

q()

Quit the R program. You will be asked if you want to save the R workspace, and for an exploratory session like this, you probably do not want to save it.

付録 B R の呼び出し

Users of R on Windows or macOS should read the OS-specific section first, but command-line use is also supported.

B.1 コマンドラインからのRの呼び出し

When working at a command line on UNIX or Windows, the command ‘R’ can be used both for starting the main R program in the form

R [options] [<infile] [>outfile],

or, via the R CMD interface, as a wrapper to various R tools (e.g., for processing files in R documentation format or manipulating add-on packages) which are not intended to be called “directly”.

At the Windows command-line, Rterm.exe is preferred to R.

You need to ensure that either the environment variable TMPDIR is unset or it points to a valid place to create temporary files and directories.

Most options control what happens at the beginning and at the end of an R session. The startup mechanism is as follows (see also the on-line help for topic ‘Startup’ for more information, and the section below for some Windows-specific details).

• Unless --no-environ was given, R searches for user and site files to process for setting environment variables. The name of the site file is the one pointed to by the environment variable R_ENVIRON; if this is unset, R_HOME/etc/Renviron.site is used (if it exists). The user file is the one pointed to by the environment variable R_ENVIRON_USER if this is set; otherwise, files .Renviron in the current or in the user’s home directory (in that order) are searched for. These files should contain lines of the form ‘name=value’. (See help("Startup") for a precise description.) Variables you might want to set include R_PAPERSIZE (the default paper size), R_PRINTCMD (the default print command) and R_LIBS (specifies the list of R library trees searched for add-on packages).
• Then R searches for the site-wide startup profile unless the command line option --no-site-file was given. The name of this file is taken from the value of the R_PROFILE environment variable. If that variable is unset, the default R_HOME/etc/Rprofile.site is used if this exists.
• Then, unless --no-init-file was given, R searches for a user profile and sources it. The name of this file is taken from the environment variable R_PROFILE_USER; if unset, a file called .Rprofile in the current directory or in the user’s home directory (in that order) is searched for.
• It also loads a saved workspace from file .RData in the current directory if there is one (unless --no-restore or --no-restore-data was specified).
• Finally, if a function .First() exists, it is executed. This function (as well as .Last() which is executed at the end of the R session) can be defined in the appropriate startup profiles, or reside in .RData.

In addition, there are options for controlling the memory available to the R process (see the on-line help for topic ‘Memory’ for more information). Users will not normally need to use these unless they are trying to limit the amount of memory used by R.

R accepts the following command-line options.

--help

-h

Print short help message to standard output and exit successfully.

--version

Print version information to standard output and exit successfully.

--encoding=enc

Specify the encoding to be assumed for input from the console or stdin. This needs to be an encoding known to iconv: see its help page. (--encoding enc is also accepted.) The input is re-encoded to the locale R is running in and needs to be representable in the latter’s encoding (so e.g. you cannot re-encode Greek text in a French locale unless that locale uses the UTF-8 encoding).

RHOME

Print the path to the R “home directory” to standard output and exit successfully. Apart from the front-end shell script and the man page, R installation puts everything (executables, packages, etc.) into this directory.

--save

--no-save

Control whether data sets should be saved or not at the end of the R session. If neither is given in an interactive session, the user is asked for the desired behavior when ending the session with q(); in non-interactive use one of these must be specified or implied by some other option (see below).

--no-environ

Do not read any user file to set environment variables.

--no-site-file

Do not read the site-wide profile at startup.

--no-init-file

Do not read the user’s profile at startup.

--restore

--no-restore

--no-restore-data

Control whether saved images (file .RData in the directory where R was started) should be restored at startup or not. The default is to restore. (--no-restore implies all the specific --no-restore-* options.)

--no-restore-history

Control whether the history file (normally file .Rhistory in the directory where R was started, but can be set by the environment variable R_HISTFILE) should be restored at startup or not. The default is to restore.

--no-Rconsole

(Windows only) Prevent loading the Rconsole file at startup.

--vanilla

Combine --no-save, --no-environ, --no-site-file, --no-init-file and --no-restore. Under Windows, this also includes --no-Rconsole.

-f file

--file=file

(not Rgui.exe) Take input from file: ‘-’ means stdin. Implies --no-save unless --save has been set. On a Unix-alike, shell metacharacters should be avoided in file (but spaces are allowed).

-e expression

(not Rgui.exe) Use expression as an input line. One or more -e options can be used, but not together with -f or --file. Implies --no-save unless --save has been set. (There is a limit of 10,000 bytes on the total length of expressions used in this way. Expressions containing spaces or shell metacharacters will need to be quoted.)

--no-readline

(UNIX only) Turn off command-line editing via readline. This is useful when running R from within Emacs using the ESS (“Emacs Speaks Statistics”) package. See コマンドラインエディタ, for more information. Command-line editing is enabled for default interactive use (see --interactive). This option also affects tilde-expansion: see the help for path.expand.

--min-vsize=N

--min-nsize=N

For expert use only: set the initial trigger sizes for garbage collection of vector heap (in bytes) and cons cells (number) respectively. Suffix ‘M’ specifies megabytes or millions of cells respectively. The defaults are 6Mb and 350k respectively and can also be set by environment variables R_NSIZE and R_VSIZE.

--max-ppsize=N

Specify the maximum size of the pointer protection stack as N locations. This defaults to 10000, but can be increased to allow large and complicated calculations to be done. Currently the maximum value accepted is 100000.

--max-mem-size=N

(Windows only) Specify a limit for the amount of memory to be used both for R objects and working areas. This is set by default to the smaller of the amount of physical RAM in the machine and for 32-bit R, 1.5Gb²⁶, and must be between 32Mb and the maximum allowed on that version of Windows.

--quiet

--silent

-q

Do not print out the initial copyright and welcome messages.

--slave

Make R run as quietly as possible. This option is intended to support programs which use R to compute results for them. It implies --quiet and --no-save.

--interactive

(UNIX only) Assert that R really is being run interactively even if input has been redirected: use if input is from a FIFO or pipe and fed from an interactive program. (The default is to deduce that R is being run interactively if and only if stdin is connected to a terminal or pty.) Using -e, -f or --file asserts non-interactive use even if --interactive is given.

Note that this does not turn on command-line editing.

--ess

(Windows only) Set Rterm up for use by R-inferior-mode in ESS, including asserting interactive use (without the command-line editor) and no buffering of stdout.

--verbose

Print more information about progress, and in particular set R’s option verbose to TRUE. R code uses this option to control the printing of diagnostic messages.

--debugger=name

-d name

(UNIX only) Run R through debugger name. For most debuggers (the exceptions are valgrind and recent versions of gdb), further command line options are disregarded, and should instead be given when starting the R executable from inside the debugger.

--gui=type

-g type

(UNIX only) Use type as graphical user interface (note that this also includes interactive graphics). Currently, possible values for type are ‘X11’ (the default) and, provided that ‘Tcl/Tk’ support is available, ‘Tk’. (For back-compatibility, ‘x11’ and ‘tk’ are accepted.)

--arch=name

(UNIX only) Run the specified sub-architecture.

--args

This flag does nothing except cause the rest of the command line to be skipped: this can be useful to retrieve values from it with commandArgs(TRUE).

Note that input and output can be redirected in the usual way (using ‘<’ and ‘>’), but the line length limit of 4095 bytes still applies. Warning and error messages are sent to the error channel (stderr).

The command R CMD allows the invocation of various tools which are useful in conjunction with R, but not intended to be called “directly”. The general form is

R CMD command args

where command is the name of the tool and args the arguments passed on to it.

Currently, the following tools are available.

BATCH

Run R in batch mode. Runs R --restore --save with possibly further options (see ?BATCH).

COMPILE

(UNIX only) Compile C, C++, Fortran … files for use with R.

SHLIB

Build shared library for dynamic loading.

INSTALL

Install add-on packages.

REMOVE

Remove add-on packages.

build

Build (that is, package) add-on packages.

check

Check add-on packages.

LINK

(UNIX only) Front-end for creating executable programs.

Rprof

Post-process R profiling files.

Rdconv

Rd2txt

Convert Rd format to various other formats, including HTML, LaTeX, plain text, and extracting the examples. Rd2txt can be used as shorthand for Rd2conv -t txt.

Rd2pdf

Convert Rd format to PDF.

Stangle

Extract S/R code from Sweave or other vignette documentation

Sweave

Process Sweave or other vignette documentation

Rdiff

Diff R output ignoring headers etc

config

Obtain configuration information

javareconf

(Unix only) Update the Java configuration variables

rtags

(Unix only) Create Emacs-style tag files from C, R, and Rd files

open

(Windows only) Open a file via Windows’ file associations

texify

(Windows only) Process (La)TeX files with R’s style files

Use

R CMD command --help

to obtain usage information for each of the tools accessible via the R CMD interface.

In addition, you can use options --arch=, --no-environ, --no-init-file, --no-site-file and --vanilla between R and CMD: these affect any R processes run by the tools. (Here --vanilla is equivalent to --no-environ --no-site-file --no-init-file.) However, note that R CMD does not of itself use any R startup files (in particular, neither user nor site Renviron files), and all of the R processes run by these tools (except BATCH) use --no-restore. Most use --vanilla and so invoke no R startup files: the current exceptions are INSTALL, REMOVE, Sweave and SHLIB (which uses --no-site-file --no-init-file).

R CMD cmd args

for any other executable cmd on the path or given by an absolute filepath: this is useful to have the same environment as R or the specific commands run under, for example to run ldd or pdflatex. Under Windows cmd can be an executable or a batch file, or if it has extension .sh or .pl the appropriate interpreter (if available) is called to run it.

B.2 windows からの R の呼び出し

There are two ways to run R under Windows. Within a terminal window (e.g. cmd.exe or a more capable shell), the methods described in the previous section may be used, invoking by R.exe or more directly by Rterm.exe. For interactive use, there is a console-based GUI (Rgui.exe).

The startup procedure under Windows is very similar to that under UNIX, but references to the ‘home directory’ need to be clarified, as this is not always defined on Windows. If the environment variable R_USER is defined, that gives the home directory. Next, if the environment variable HOME is defined, that gives the home directory. After those two user-controllable settings, R tries to find system defined home directories. It first tries to use the Windows "personal" directory (typically My Documents in recent versions of Windows). If that fails, and environment variables HOMEDRIVE and HOMEPATH are defined (and they normally are) these define the home directory. Failing all those, the home directory is taken to be the starting directory.

You need to ensure that either the environment variables TMPDIR, TMP and TEMP are either unset or one of them points to a valid place to create temporary ファイルとディレクトリ.

Environment variables can be supplied as ‘name=value’ pairs on the command line.

If there is an argument ending .RData (in any case) it is interpreted as the path to the workspace to be restored: it implies --restore and sets the working directory to the parent of the named file. (This mechanism is used for drag-and-drop and file association with RGui.exe, but also works for Rterm.exe. If the named file does not exist it sets the working directory if the parent directory exists.)

The following additional command-line options are available when invoking RGui.exe.

--mdi

--sdi

--no-mdi

Control whether Rgui will operate as an MDI program (with multiple child windows within one main window) or an SDI application (with multiple top-level windows for the console, graphics and pager). The command-line setting overrides the setting in the user’s Rconsole file.

--debug

Enable the “Break to debugger” menu item in Rgui, and trigger a break to the debugger during command line processing.

Under Windows with R CMD you may also specify your own .bat, .exe, .sh or .pl file. It will be run under the appropriate interpreter (Perl for .pl) with several environment variables set appropriately, including R_HOME, R_OSTYPE, PATH, BSTINPUTS and TEXINPUTS. For example, if you already have latex.exe on your path, then

R CMD latex.exe mydoc

will run LaTeX on mydoc.tex, with the path to R’s share/texmf macros appended to TEXINPUTS. (Unfortunately, this does not help with the MiKTeX build of LaTeX, but R CMD texify mydoc will work in that case.)

B.3 macOS からの R の呼び出し

There are two ways to run R under macOS. Within a Terminal.app window by invoking R, the methods described in the first subsection apply. There is also console-based GUI (R.app) that by default is installed in the Applications folder on your system. It is a standard double-clickable macOS application.

The startup procedure under macOS is very similar to that under UNIX, but R.app does not make use of command-line arguments. The ‘home directory’ is the one inside the R.framework, but the startup and current working directory are set as the user’s home directory unless a different startup directory is given in the Preferences window accessible from within the GUI.

B.4 R によるスクリプト

If you just want to run a file foo.R of R commands, the recommended way is to use R CMD BATCH foo.R. If you want to run this in the background or as a batch job use OS-specific facilities to do so: for example in most shells on Unix-alike OSes R CMD BATCH foo.R & runs a background job.

You can pass parameters to scripts via additional arguments on the command line: for example (where the exact quoting needed will depend on the shell in use)

R CMD BATCH "--args arg1 arg2" foo.R &

will pass arguments to a script which can be retrieved as a character vector by

args <- commandArgs(TRUE)

This is made simpler by the alternative front-end Rscript, which can be invoked by

Rscript foo.R arg1 arg2

and this can also be used to write executable script files like (at least on Unix-alikes, and in some Windows shells)

#! /path/to/Rscript
args <- commandArgs(TRUE)
...
q(status=<exit status code>)

If this is entered into a text file runfoo and this is made executable (by chmod 755 runfoo), it can be invoked for different arguments by

runfoo arg1 arg2

For further options see help("Rscript"). This writes R output to stdout and stderr, and this can be redirected in the usual way for the shell running the command.

If you do not wish to hardcode the path to Rscript but have it in your path (which is normally the case for an installed R except on Windows, but e.g. macOS users may need to add /usr/local/bin to their path), use

#! /usr/bin/env Rscript
...

At least in Bourne and bash shells, the #! mechanism does not allow extra arguments like #! /usr/bin/env Rscript --vanilla.

One thing to consider is what stdin() refers to. It is commonplace to write R scripts with segments like

chem <- scan(n=24)
2.90 3.10 3.40 3.40 3.70 3.70 2.80 2.50 2.40 2.40 2.70 2.20
5.28 3.37 3.03 3.03 28.95 3.77 3.40 2.20 3.50 3.60 3.70 3.70

and stdin() refers to the script file to allow such traditional usage. If you want to refer to the process’s stdin, use "stdin" as a file connection, e.g. scan("stdin", ...).

Another way to write executable script files (suggested by François Pinard) is to use a here document like

#!/bin/sh
[environment variables can be set here]
R --slave [other options] <<EOF

   R program goes here...

EOF

but here stdin() refers to the program source and "stdin" will not be usable.

Short scripts can be passed to Rscript on the command-line via the -e flag. (Empty scripts are not accepted.)

Note that on a Unix-alike the input filename (such as foo.R) should not contain spaces nor shell metacharacters.

付録 C コマンドラインエディタ

C.1 準備

When the GNU readline library is available at the time R is configured for compilation under UNIX, an inbuilt command line editor allowing recall, editing and re-submission of prior commands is used. Note that other versions of readline exist and may be used by the inbuilt command line editor: this used to happen on macOS.

It can be disabled (useful for usage with ESS ²⁷) using the startup option --no-readline.

Windows versions of R have somewhat simpler command-line editing: see ‘Console’ under the ‘Help’ menu of the GUI, and the file README.Rterm for command-line editing under Rterm.exe.

When using R with GNU²⁸ readline capabilities, the functions described below are available, as well as others (probably) documented in man readline or info readline on your system.

Many of these use either Control or Meta characters. Control characters, such as Control-m, are obtained by holding the CTRL down while you press the m key, and are written as C-m below. Meta characters, such as Meta-b, are typed by holding down META²⁹ and pressing b, and written as M-b in the following. If your terminal does not have a META key enabled, you can still type Meta characters using two-character sequences starting with ESC. Thus, to enter M-b, you could type ESCb. The ESC character sequences are also allowed on terminals with real Meta keys. Note that case is significant for Meta characters.

Some but not all versions³⁰ of readline will recognize resizing of the terminal window so this is best avoided.

C.2 編集動作

The R program keeps a history of the command lines you type, including the erroneous lines, and commands in your history may be recalled, changed if necessary, and re-submitted as new commands. In Emacs-style command-line editing any straight typing you do while in this editing phase causes the characters to be inserted in the command you are editing, displacing any characters to the right of the cursor. In vi mode character insertion mode is started by M-i or M-a, characters are typed and insertion mode is finished by typing a further ESC. (The default is Emacs-style, and only that is described here: for vi mode see the readline documentation.)

Pressing the RET command at any time causes the command to be re-submitted.

Other 編集動作 are summarized in the following table.

C.3 コマンドライン・エディタの要約

Command recall and vertical motion

C-p

Go to the previous command (backwards in the history).

C-n

Go to the next command (forwards in the history).

C-r text

Find the last command with the text string in it. This can be cancelled by C-g (and on some versions of R by C-c).

On most terminals, you can also use the up and down arrow keys instead of C-p and C-n, respectively.

Horizontal motion of the cursor

C-a

Go to the beginning of the command.

C-e

Go to the end of the line.

M-b

Go back one word.

M-f

Go forward one word.

C-b

Go back one character.

C-f

Go forward one character.

On most terminals, you can also use the left and right arrow keys instead of C-b and C-f, respectively.

Editing and re-submission

text

Insert text at the cursor.

C-f text

Append text after the cursor.

DEL

Delete the previous character (left of the cursor).

C-d

Delete the character under the cursor.

M-d

Delete the rest of the word under the cursor, and “save” it.

C-k

Delete from cursor to end of command, and “save” it.

C-y

Insert (yank) the last “saved” text here.

C-t

Transpose the character under the cursor with the next.

M-l

Change the rest of the word to lower case.

M-c

Change the rest of the word to upper case.

RET

Re-submit the command to R.

The final RET terminates the command line editing sequence.

The readline key bindings can be customized in the usual way via a ~/.inputrc file. These customizations can be conditioned on application R, that is by including a section like

$if R
  "\C-xd": "q('no')\n"
$endif

付録 D 関数と変数の索引