WERKZEUGEDERMUSTERERKENNUNGUND Teil I … · 2020-02-14 · Waskann’R’?SprachumfangGraﬁkInstallation’R’-Sitzung ?plot Datenmatrizen,EinkanalbilderII WertedurchGrenzen WertedurchFarben

WERKZEUGE DER MUSTERERKENNUNG UNDDES MASCHINELLEN LERNENS

Vorlesung im Sommersemester 2020

Prof. E.G. Schukat-Talamazzini

Stand: 14. Februar 2020

Was kann ’R’? Sprachumfang Grafik Installation ’R’-Sitzung ?plot

Teil I

Einführung in die Sprache


Softwarewerkzeugeund ihre Benutzerschnittstelle

Graphikorientiert (GUI)MenüauswahlFormulareintragDrag & Drop

Sprachorientiert (PLI)Formale AnwendungsspracheInterpreterprogrammBS/PS-Interface

Befehlsorientiert (CLI)Fixiertes KommandoensembleAufrufparameter und -syntaxSkriptunterstützung

Programmorientiert (API)UnterprogrammbibliothekKlassenbibliothek(wie PLI) erweiterbar


Programmiersprachen für wissenschaftliches Rechnen


Inhaltlicher Vorlesungsaufbau

Die Programmiersprache ’R’

• Sprache ’R’ im ÜberblickHilfe, Installation, Sitzung

• Rechnen und DatentypenFelder, Listen, Funktionen,Operatoren

• Text und DatenZeichenketten,Eingabe/Ausgabe,Graphikausgabe

• Statistische ModelleRegression, Optimierung,Verteilungen

Anwendungsgebiete von ’R’

• ZeitreihenTrend/Saison, Faltung, ACF,Spektrum, AR/MA, GARCH

• TransformationenHauptachsen, MDS,Faktoren, ICA, NMF, LDA

• Klassifikatorenk-NN, MLP, SVM, SCT,OLS/GLS, LDA/QDA

• Clusteranalysedivisiv/agglomerativ,K -Means, Fuzzy,EM-Entmischung, spektral


Was kann ’R’?

Sprachumfang

Grafikausgabe

Installation des ’R’-Pakets

Ablauf einer ’R’-Sitzung

Elementare Grafikbefehle


Das System ist ...

• ein leistungsfähiger Taschenrechner• ein vektor/tensorbasierter Mathematikprozessor• eine funktional-objektorientierte Programmiersprache• ein Werkzeug zur statistischen Datenanalyse• ein mächtiges Kommandosystem für wiss. Grafiken• ein Interpreter für „die Sprache C ∪ Lisp“• ein offenes und kostenloses Softwareprodukt (S-Plus nicht!)• in den Bereichen Statistik, Datamining, Bioinf weit verbreitet• ein Algorithmen-Wiki für die internat. Datamining-Szene


Wissenswertes über

• Dokumente des ’R’-Pakets: file://usr/lib/R/doc/html

Tutorial, Language Definition, Installation, Erweiterungspakete, Datenim-/export, FAQ, ...

• Webseiten des ’R’-Projekts:international: http://www.r-project.org/ · national: http://www.r-project.de/

• Ausgewählte Handbücher und KursmaterialienContributed Documentation (≥ 50 Tutorials): http://cran.r-project.org/other-docs.html

• Interaktive Hilfe zur Programmlaufzeit(Funktionsaufruf) funcname (a,b,...)

1. Funktionsbeschreibung help (funcname)bzw. ?funcname

2. Beispieldemonstration example (funcname)3. Funktionsdeklaration funcname4. Suchfunktion help.search (topic)

bzw. ??topic


Was kann ’R’?

Sprachumfang

Grafikausgabe





’R’ als Taschenrechner

• ’R’ ist interaktiv: der Programmtext wird interpretiert• ’R’-Programm =̂ Folge von Ausdrücken• der Ausdruck wird ausgewertet; der Wert wird angezeigt:(17+4)*(35-25)+4501 ... [1] 4711

• nach Namensbindungen wird die Wertausgabe unterdrückt:xval <- (17+4)*(35-25)+4501 ... (nix)

• der Wert wurde aber errechnet und gemerkt:xval + 1111 ... [1] 5822

• Bezeichner werden nach Erstauftritt automatisch registriert:objects() ... [1] ’DNA.prom’ ’my_url’ ’xval’

• Bezeichner sind keine Variablen:xval <- "School’s out forever!" ... (kein Problem)


Vektoren, Matrizen, Tensoren

• Die elementaren Datentypen von ’R’ sind Vektoren:x <- c(1,4,9,16) ; x ... [1] 1 4 9 16

• Es gibt zahlreiche Konstruktoren für Vektorobjekte:y <- 1:4 ; y ... [1] 1 2 3 4

• Arithmet./logische Verknüpfungen operieren komponentenweise:x+y ... [1] 2 6 12 20

• Matrizen bestehen aus Datenvektor & Strukturinformation:matrix (data=1:12, nrow=3, ncol=4, byrow=FALSE)

[,1] [,2] [,3] [,4][1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

• Es gibt zahlreiche Werteselektionsmöglichkeiten:mat[3] mat[2,3] mat[2,1:3] mat[2:3,2][1] 3 [1] 8 [1] 2 5 8 [1] 5 6


Listen, Ausdrücke & Parsebäume

• Listen enthalten Komponenten von unterschiedlichem Typ:lis <- list (zahl=12, TRUE, ’zwei Wörter’)

• Selektion per Index oder per Name:lis[[2]] ... [1] TRUE oder lis$zahl ... [1] 12

• Programmtexte können geparst werden:ex <- parse (text = "a-b; (17+4)*45; x+y")

• Es wird eine Liste von Parsebäumen erzeugt:ex2 <- ex[[2]] ; ex2 ... (17 + 4) * 45

• Jeder Ausdruck liegt als Kantorovic-Ableitungsbaum vor:ex2[1] ex2[2] ex2[3] ex2[4]’*’() (17 + 4)() 45() NULL()

• Kantorovic-Bäume können (u.U.) ausgewertet werden:eval(ex2) ... [1] 945eval(ex) ... Error in eval(expr, envir, enclos) : Object ’a’ not found


Datensätze laden — auch über das Internet• Als Dateinamen werden auch URLs akzeptiert:myurl <- ’http://stat.ethz.ch/Teaching/Datasets/NDK/sport.dat’

d.sport <- read.table (myurl)• Datensätze mit benannten Mustern & Merkmalen:

weit kugel hoch disc stab speer punkteOBRIEN 7.57 15.66 207 48.78 500 66.90 8824BUSEMANN 8.07 13.60 204 45.04 480 66.86 8706DVORAK 7.60 15.82 198 46.28 470 70.16 8664FRITZ 7.77 15.31 204 49.84 510 65.70 8644HAMALAINEN 7.48 16.32 198 49.62 500 57.66 8613NOOL 7.88 14.01 201 42.98 540 65.48 8543ZMELIK 7.64 13.53 195 43.44 540 67.20 8422GANIYEV 7.61 14.71 213 44.86 520 53.70 8318PENALVER 7.27 16.91 207 48.92 470 57.08 8307HUFFINS 7.49 15.57 204 48.72 470 60.62 8300PLAZIAT 7.82 14.85 204 45.34 490 52.18 8282MAGNUSSON 7.28 15.52 195 43.78 480 61.10 8274SMITH 7.47 16.97 195 49.54 500 64.34 8271MUELLER 7.25 14.69 195 45.90 510 66.10 8253CHMARA 7.75 14.51 210 42.60 490 54.84 8249

• Auswahl von Zeilen und Spalten, z.B. d.sport[,’kugel’][1] 15.66 13.60 15.82 15.31 16.32 14.01 13.53 14.71 16.91 15.57 14.85 15.52

[13] 16.97 14.69 14.51


Aufrufen & Deklarieren von Funktionen

• Funktionsdeklaration mit formalen Parametern:funcname <- function (arglist) {body}

• Parameterbezeichner vname (ohne Typangabe!)

• Parameter mit Voreinstellung vname=default• Restparameterliste ...

• Funktionsaufruf mit aktuellen Parametern:funcname(arglist)

• Positionelle Übergabe expr• Namentliche Übergabe vname=expr• Restparameterübergabe ...

• Beispiel:zeichne <- function (x, y=NULL, title=’Grafik’, ...) {

if (is.null (y)){ y <- x; x <- 1:length(x) }

plot (y ˜ x, main=title, ...)}

zeichne (sin(1:20), cos(1:20), title=’Kreis’, col=’red’)


Was kann ’R’?

Sprachumfang

Grafikausgabe





Wertesequenzen & WertemengenSkalare Datensammlung mit/ohne Reihenfolgeinformation

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Just a Whisper of a Label

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1000 Normal Random Variates

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Häufigkeiten & ProportionenWahrscheinlichkeitsverteilungen auf diskreten Skalen

Blueberry

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Vanilla Cream

January Pie Sales

(Don't try this at home kids)

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Wagenradgrafik


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Zweikanal-Zeitreihen (2D-Trajektorie)


Datenmatrizen, Einkanalbilder IGebirgsmetapher durch Gitterprojektion oder 3D-Rendering

Maunga WhauOne of 50 Volcanoes in the Auckland Region.

Drahtgitterlandschaft

row

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volcano

Schattierung


Datenmatrizen, Einkanalbilder IIWerte durch Grenzen · Werte durch Farben

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Zweikanalige DatenScatterplot ohne/mit Klassenkennzeichnung

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165 170 175 180 185

−35

−30

−25

−20

−15

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long

lat

100 200 300 400 500 600

Given : depth

x3-partitionierte (x1, x2)-Diagramme

Sepal.Length

2.0 2.5 3.0 3.5 4.0

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0.5 1.0 1.5 2.0 2.5

4.5

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2.0

2.5

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●Sepal.Width

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Petal.Length

12

34

56

7

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4.5 5.5 6.5 7.5

0.5

1.0

1.5

2.0

2.5

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1 2 3 4 5 6 7

●●●●●

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●

Petal.Width

Edgar Anderson's Iris Data

alle möglichen (xi , xj)-Diagramme


Darstellung von VerteilungsmodellenEmpirische/geschätzte Dichte · Text- und Formelsatz

Estimated Density

Height (inches)

Ker

nel D

ensi

ty w

ith N

orm

al F

it

0.000.050.100.150.200.25

60 65 70 75 80

● ●● ●●●● ●●● ●●● ● ●● ● ●● ●●● ●● ● ●

Bass 2

●● ●● ●● ●●● ●● ● ● ●●●● ● ●● ●● ●● ●●● ●●● ●● ● ●●● ●●●

Bass 1● ●● ●● ●●● ●● ● ●●●● ●●● ●● ●

Tenor 2

0.000.050.100.150.200.25

● ●●● ●●●● ●● ●● ●●●● ●● ●●●

Tenor 10.000.050.100.150.200.25

●●●●● ●● ●● ●●● ●●● ●●●● ●●●●● ●●●

Alto 2

●● ●●●● ●●● ●●● ●●● ● ●●●● ●●●● ● ●● ●●● ● ● ●● ●

Alto 1● ●● ●●● ●●● ● ●● ●●● ● ●●●● ●● ● ●● ● ●● ●●

Soprano 2

60 65 70 75 80

0.000.050.100.150.200.25

●● ●●● ● ● ●●● ●●● ● ●●● ●● ● ●● ●● ● ●●● ● ●●● ● ●●●

Soprano 1

Gruppenbezogene Dichtevisualisierung

π∑0

nx

demonstrating

expressions

σi

γ2

α

β

γ

− π

2

0 π

2

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

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●

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●

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●

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●

●

●

●

●

●

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●

● ●

●

●

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●

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●

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●

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●

●

●

●

●

●

●

●●

●

●

●

●●

●

● ●

●

●

●

: ε

2α

α

β

γ

− π

2

0 π

2

●

●

●

●

●

● ●

●●●

●

● ●

●●

●

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●

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●

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●

●

● ●

●

●

●

●

●●

●

●

●

● ●●

●

●●

●●

●

●

●

●●

●

●

●

: ε

2β

α

β

γ

− π

2

0 π

2

●

●

●

●

●

●

●

●●

●●

●

●

●●

● ●

●●●

●

●

●

●

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● ●

●

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●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

: ε

2γ

α

β

γ

− π

2

0 π

2

●

●

●

●

●●

●

●●

● ● ●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

● ●

●

●

●

●

●●

●

●

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●

●●

●

●●

●●

●

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●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

: ε

2δ

ei1α+2β

● small

BIGθζ

Formelsatz in LATEX-Syntax


Clusteranalyse (Gruppierung)Hierarchische Gruppierung · Unscharfe Gruppierung

DK L F I

UK D B

NL E

IRL

GR P

010

3050

Ward−Distanz

Lanc

e−W

illia

ms

scor

e

Dendrogramm (AGNES)

−30 −20 −10 0 10 20 30

−30

−20

−10

010

20

clusplot(fanny(x = USArrests[, 3:4], k = 7, memb.exp = 1.4))

Component 1

Com

pone

nt 2

●

●

●

●

●

●

●●

●

●

●

Partition (fuzzy K -means)


Was kann ’R’?

Sprachumfang

Grafikausgabe





am Institut für Informatik’R’ ist verfügbar an allen Rechnern im Linux-PC-Pool des FRZ

myself@mipool> R R version 3.3.1

R version 3.3.1 (2016-06-21) – "Bug in Your Hair"Copyright (C) 2016 The R Foundation for Statistical ComputingPlatform: x86_64-suse-linux-gnu (64-bit)

R ist freie Software und kommt OHNE JEGLICHE GARANTIE.Sie sind eingeladen, es unter bestimmten Bedingungen weiter zu verbreiten.Tippen Sie ’license()’ or ’licence()’ für Details dazu.

R ist ein Gemeinschaftsprojekt mit vielen Beitragenden.Tippen Sie ’contributors()’ für mehr Information und ’citation()’,um zu erfahren, wie R oder R packages in Publikationen zitiert werden können.

Tippen Sie ’demo()’ für einige Demos, ’help()’ für on-line Hilfe, oder’help.start()’ für eine HTML Browserschnittstelle zur Hilfe.Tippen Sie ’q()’, um R zu verlassen.


zu Hause unter Linux installieren

1. Rufen Sie die Webseite http://cran.r-project.org/CRAN = Comprehensive ’R’ Archive Network

2. Nutzen Sie die ONE-CLICK-InstallationR binaries linux suse SuSE 12.1 R-patched-devel-2.14.1

3. ... oder laden Sie sich eine Distribution herunter undinstallieren Sie das Paket als ’root’:rpm -i /home/schukat/hereis/R-base-2.4.0-1.i586.rpm

error: Failed dependencies:xorg-x11-fonts-100dpi is needed by R-base-2.4.0-1blas is needed by R-base-2.4.0-1libgfortran.so.0 is needed by R-base-2.4.0-1

4. Installieren Sie die fehlenden Pakete nach (YAST2), z.B.blas · xorg-x11-fonts-100dpi · fortran o.ä.


zu Hause unter Windows installierenWarum sollte das eigentlich nicht möglich sein? · Wer hat da eben gelacht?

〈Algorithmus〉

1 Laden Sie sich eine Binärversion für Windows XP herunter.2 Überzeugen Sie den Installationsassistenten Ihres Vertrauens

davon, daß es sich beim ’R’-Paket um einEgoshooterprogramm handelt.

3 Führen Sie einen Systemneustart durch.4 Entfernen Sie die verkohlten Kunststoffreste von der

Auslegeware.〈Algorithmus〉


Was kann ’R’?

Sprachumfang

Grafikausgabe





Beginn und Ende einer ’R’-SitzungKonfiguration einer individuellen Umgebung

• Starten einer Sitzung:.../working-directory> R

• Beenden einer Sitzung:quit() oder q() oder q(’yes’) oder q(’no’)

• Der individuelle Zuschnitt Ihrer ’R’-Umgebung:.../working-directory/.Rprofile

• Prolog zu Sitzungsbeginn:... die Funktion .First() wird ausgeführt

• Epilog bei Sitzungsende:... die Funktion .Last() wird ausgeführt

• BEISPIEL:Die Startdatei .Rprofile enthält die Funktionsdefinitionen:

.First <- function() { source (’./lib/mydefs.R’) ; cat (paste (date(), ’Hola’)) }

.Last <- function() { graphics.off() ; cat (paste (date(), ’Adios’)) }


Kommandos zur ObjektverwaltungListen · Löschen · Ein/Ausgabe von Daten und Kommandos

• Auflisten aller definierten Objekte:objects() oder ls()

• Löschen definierter Objekte:remove (x,y,fun) oder rm (list=ls()) (?!?)

• ’R’-Kommandos von Datei lesen (Eingabeumlenkung)

source (file)• ’R’-Ausgaben/Meldungen in Datei schreiben (Ausgabeumlenkung)

sink(file=NULL, append=FALSE)• ’R’-Objekte in eine Datei speichernsave(..., list=character(0), file=)

• ’R’-Objekte aus einer Datei laden (char vector)

load(file) oder loadURL(url)• Alle Sitzungsobjekte werden nach .RData geschrieben


Stapelverarbeitung für -ProgrammeAufruf von und Parameterübergabe an ’R’-Skriptdateien

• Abarbeitung einer Datei mit ’R’-Programmcode:

Rscript progfile.R

(entspricht Ausführung von source ("progfile.R") im ’R’-Interpreter)

• Aufruf eines ’R’-Skripts mit Parameterübergabe:

Rscript progfile.R string1 string2 string3 ... ... ...

Nach der Zuweisung

arg <- commandArgs (trailingOnly=TRUE)

finden sich in den Komponenten arg[1], arg[2] usw. descharacter-Vektors arg die Aufrufparameter als Zeichenketten.

• Etwaige Grafikausgaben finden wir im aktuellen Arbeitskatalog unterdem Namen Rplots.pdf.(sofern „PDF“ voreingestellt und kein Dateiname angegeben)


Was kann ’R’?

Sprachumfang

Grafikausgabe





Zeichnen von Funktionsverläufen

• Generische Funktion delegiert an zuständige Methode:plot(func,...) plot.function(func,...)

• Methode zum Kurvenzeichnen:curve (expr, from, to, n=101, add=FALSE, type=’l’,ylab=NULL, log=NULL, xlim=NULL, ...)

• Der Funktionsname ist gegeben, z.B.:plot (sin)

• Eine Funktionsdefinition ist gegeben, z.B.:plot (function(z) {2*z+5})

• Das Intervall für die Argumentwerte ist spezifiziert, z.B.:plot (sin, -2, +4)


Beispiel: eine Sinuskurve

plot ( sin, from=0, to=6*pi, col="blue",main="Darstellung Sinuskurve",sub="drei Perioden" )

0 5 10 15

−1.0

−0.5

0.00.5

1.0

Darstellung Sinuskurve

drei Periodenx

sin (x

)


Beispiel: eine Funktionsdeklaration

plot ( function(z) {sin(1/z)}, col=’blue’, n=256,main=’Funktionsinstanz’,sub=’eigenhaendig deklariert’ )

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

Funktionsinstanz

eigenhaendig deklariertx

functio

n(z) {

(x)

sin(1/

z) (x)

} (x)


Bildübergreifende Grafikbefehle

• Überlagern mehrerer Verläufe in einem Achsenkreuzplot (x,y, add=TRUE)

• Umleiten der Grafikausgabe von Schirm auf Dateipdf (file=’Rplot.pdf’, ???)und entsprechend für postscript pictex png jpeg GTK GNOME xfig bitmap

zurück mit X11 (display, ???)• Keine neue Zeichnung ohne Bestätigung (Stapelbetrieb)

par (ask=TRUE)• Aufteilung der Leinwand in mehrere Grafikfelderpar (mfrow=c(n,m)) oderpar (mfcol=c(n,m))


Beispiel: zwei Kurven in einem Bild

plot ( sin, from=0, to=6*pi, col="blue",main="Darstellung Sinus- und Kosinuskurve",sub="drei beziehungsweise zwei Perioden")

plot ( cos, from=0, to=4*pi, add=TRUE, col="green")

0 5 10 15

−1.0

−0.5

0.00.5

1.0

Darstellung Sinus− und Kosinuskurve

drei beziehungsweise zwei Periodenx

sin (x

)


Beispiel: sechs Bilder auf einem Blatt

par (mfrow = c(2,3))curve (sin(10*x), col="blue"); curve (sin(20*x), col="green")curve (sin(30*x), col="red"); curve (sin(40*x), col="cyan")curve (sin(50*x), col="yellow");curve (sin(60*x), col="magenta")

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(10

* x)

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(20

* x)

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(30

* x)

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(40

* x)

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(50

* x)

0.0 0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.00.5

1.0

x

sin(60

* x)


Wertesequenzen und Punktesequenzen

• Wertesequenzen zeichnen: plot(x) , z.B.:plot (c (2,3,5,7,11,13,17))plot (1:20)plot (seq (0,8,0.1))plot (sin (seq (0,8,0.1)))

• Punktesequenzen zeichnen: plot(x,y)z.B. x <- seq (0,2*pi,0.1) nebstplot (x, sin(x)) oderplot (sin(x), x) oderplot (cos(x), sin(x))

• Kurvendarstellung durch type=’?’ gesteuert:points lines bothoverplotted contour-onlyhigh-density steps Steps ...


Beispiel: eindimensionale Wertefolgen

par (mfrow = c(1,3))plot (c(2,3,5,7,11,13,17,19,23), type=’l’, col=’blue’)plot (1:20, type=’p’, col=’blue’)plot (seq (0,8,1), type=’b’, col=’blue’)

2 4 6 8

510

1520

Index

c(2, 3,

5, 7, 1

1, 13, 1

7, 19, 2

3)

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5 10 15 20

510

1520

Index1:2

0

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2 4 6 8

02

46

8

Index

seq(0,

8, 1)


Beispiel: zweidimensionale Wertefolgen

par (mfrow = c(1,3))x <- seq (0, 6*pi, 0.1)plot (sin(x), cos(x), type=’b’, col=’blue’)plot (x, sin(x), type=’b’, col=’blue’)plot (sin(x), x, type=’b’, col=’blue’)

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15sin(x)

x


Leinwandaufteilung mit variablen Feldmaßen

layout (matrix (c(1,1,2,3), ncol=2), width=c(2,3), height=c(3,2))x <- c (runif (123), rnorm (50, mean=2, sd=0.5))boxplot (x, col="green")plot (sort(x), col="blue", type="h")hist (x, col="red", lab=TRUE, main="")layout (1) # nicht vergessen!!

0.00.5

1.01.5

2.02.5

3.0

0 50 100 150

0.00.5

1.01.5

2.02.5

3.0

Index

sort(x)

x

Freque

ncy

0.0 0.5 1.0 1.5 2.0 2.5 3.0

020

4060 58

65

6

24

10 10


Zusammenfassung (1)

1. ’R’ ist eine funktionale Programmiersprache mit rudimentärerObjektorientierung (polymorphe Aufrufe).

2. ’R’ arbeitet interaktiv (Interpreter statt Compiler), unterstützt aberStapelverarbeitung (Skripten).

3. ’R’ gilt als spektakulär für wissenschaftliches Rechnen, Statistik,Datenmodellierung und -visualisierung; für Bild- und Textverarbeitunggibt es performantere Wettbewerber.

4. Der elementare Datentyp von ’R’ ist der Vektor.

5. In ’R’ sind auch Funktionen und Ausdrücke Sprachobjekte, können alsozur Laufzeit manipuliert werden.

6. ’R’ unterstützt die E/A beliebiger ’R’-Objekte durch automatischeSerialisierung.

7. Grafiken werden kumulativ erzeugt; zuerst das Koordinatensystem, dannsukzessive weitere Zeichnungskomponenten.

8. Komplexere Grafiken werden durch Leinwandaufteilung und sequentielleKomposition einfacher Grafiken erzeugt.

Documents

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