AS17: Kapitel 2 - Grundlagen aus der Mathematik
Jürgen Hedderich
2020-04-25
Hinweis: Die thematische Gliederung zu den aufgeführten Befehlen und Beispielen orientiert sich an dem Inhaltsverzeichnis der 17. Auflage der ‘Angewandten Statistik’. Nähere Hinweise zu den verwendeten Formeln sowie Erklärungen zu den Beispielen sind in dem Buch (Ebook) nachzulesen!
Aktuelle R-Umgebung zu den Beispielen
sessionInfo()
## R version 3.6.3 (2020-02-29)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 18363)
##
## Matrix products: default
##
## locale:
## [1] LC_COLLATE=German_Germany.1252 LC_CTYPE=German_Germany.1252
## [3] LC_MONETARY=German_Germany.1252 LC_NUMERIC=C
## [5] LC_TIME=German_Germany.1252
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## loaded via a namespace (and not attached):
## [1] compiler_3.6.3 magrittr_1.5 tools_3.6.3 htmltools_0.4.0
## [5] yaml_2.2.1 Rcpp_1.0.4 stringi_1.4.6 rmarkdown_2.1
## [9] knitr_1.28 stringr_1.4.0 xfun_0.12 digest_0.6.25
## [13] rlang_0.4.5 evaluate_0.142.3 (Grund-) Rechenarten
12 + 32 # Addition
## [1] 44
43 - 15 # Subtraktion
## [1] 28
Zahlen <- c(2, 5, 7, 8, 9, 6) # Werte in einem Vektor
sum(Zahlen) # Summe
## [1] 37
1:20
## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
sum(1:20) # Summe zu Rangzahlen
## [1] 210
4 * 17 # Multiplikation
## [1] 68
56 / 8 # Division
## [1] 7
Zahlen <- c(2, 3, 4,5)
prod(Zahlen) # Produkt
## [1] 120
1:10
## [1] 1 2 3 4 5 6 7 8 9 10
prod(1:10) # Fakultät
## [1] 3628800
# größter gemeinsamer Teiler (ggT)
ggT <- function(m, n) ifelse (n==0, m, ggT(n, m %% n))
ggT(21, 35)
## [1] 7
# kleinstes gemeinsames Vielfaches (kgV)
kgV <- function(m, n) abs(m*n) / ggT(m, n)
kgV(21, 35)
## [1] 105
ggT(3528, 3780); kgV(3528, 3780)
## [1] 252
## [1] 529202.3.2 Potenzen und Wurzeln
2^4 # Potenzieren
## [1] 16
12^4
## [1] 20736
sqrt(3) # Radizieren in R
## [1] 1.732051
sqrt(234)
## [1] 15.29706
35^(5/3)
## [1] 374.4956
pi # die Zahl Pi = 3.141593
## [1] 3.141593
exp(1) # die Zahl e = 2.718282
## [1] 2.718282
log(12, base = exp(1)) # Logarithmus zur Basis e
## [1] 2.484907
log10(16) # Logarithmus zur Basis 20
## [1] 1.20412
log2(20) # Logarithmus zur Basis 2
## [1] 4.3219282.3.4 Rundungen
# Rundungen
x <- c(1.347, 2.499, 3.501)
floor(x)
## [1] 1 2 3
ceiling(x)
## [1] 2 3 4
x <- c(1.347, 2.499, 3.501)
floor(x + 0.5)
## [1] 1 2 4
x <- c(-1.347, -2.499, -3.501)
ceiling(x - 0.5)
## [1] -1 -2 -4
# Rundung für x>=0
rundung <- function(x, m) floor((x/m + 0.5))*m
rundung(x=1.3567, m=c(0.1, 0.01, 0.001, 0.25, 0.5))
## [1] 1.400 1.360 1.357 1.250 1.500
x <- seq(1.1, 1.4, 0.02) # Tabelle
y <- rundung(x, m=0.1)
z <- rundung(x, m=0.25)
cbind(x,y,z)[1:10,]
## x y z
## [1,] 1.10 1.1 1.00
## [2,] 1.12 1.1 1.00
## [3,] 1.14 1.1 1.25
## [4,] 1.16 1.2 1.25
## [5,] 1.18 1.2 1.25
## [6,] 1.20 1.2 1.25
## [7,] 1.22 1.2 1.25
## [8,] 1.24 1.2 1.25
## [9,] 1.26 1.3 1.25
## [10,] 1.28 1.3 1.25
round(x=1.3567, digits=c(1,2,3))
## [1] 1.400 1.360 1.3572.4 Einführung in die Matrixalgebra
library(Matrix)
library(MASS)
A <- matrix( c(1, 2, 3, 6, 5, 4), nrow=2, ncol=3, byrow=TRUE)
A.trans <- t(A); A; A.trans # Transponieren einer Matrix
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 6 5 4
## [,1] [,2]
## [1,] 1 6
## [2,] 2 5
## [3,] 3 4
# Addition zweier Matrizen
A <- matrix( c(1, 2, 3, 6, 5, 4), nrow=2, ncol=3, byrow=TRUE)
B <- matrix(c(4, 5, 6, 9, 8, 7), nrow=2, ncol=3, byrow=TRUE)
C <- A + B; A; B; C
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 6 5 4
## [,1] [,2] [,3]
## [1,] 4 5 6
## [2,] 9 8 7
## [,1] [,2] [,3]
## [1,] 5 7 9
## [2,] 15 13 11
# Multiplikation mit einem Skalar
A <- matrix( c(1, 2, 3, 6, 5, 4), nrow=2, ncol=3, byrow=TRUE);
A; 2 * A
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 6 5 4
## [,1] [,2] [,3]
## [1,] 2 4 6
## [2,] 12 10 8
# Multiplikation zweier Matrizen
A <- matrix( c(1, 2, 3, 6, 5, 4), nrow=2, ncol=3, byrow=TRUE);
B <- matrix(c(4, 5, 6, 9, 8, 7), nrow=3, ncol=2, byrow=TRUE);
C <- A %*% B; A; B; C
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 6 5 4
## [,1] [,2]
## [1,] 4 5
## [2,] 6 9
## [3,] 8 7
## [,1] [,2]
## [1,] 40 44
## [2,] 86 103
# Skalarprodukt zweier Vektoren
a <- 1:3
b <- 4:6
c <- t(a) %*% b; a; b; c
## [1] 1 2 3
## [1] 4 5 6
## [,1]
## [1,] 32
# Norm eines Vektors
a <- c(1, 2, 3, 4, 5, 6)
a.trans <- t(a)
a.norm <- sqrt(a.trans %*% a)
a; a.norm
## [1] 1 2 3 4 5 6
## [,1]
## [1,] 9.539392
# Bestimmung der Determinante
A <- matrix(c(3, 1, 2, 4, 5, 6, 9, 7, 8), nrow=3, ncol=3, byrow=TRUE)
A.det <- det(A); A; A.det
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 4 5 6
## [3,] 9 7 8
## [1] -18
# Berechnung der inversen Matrix
A <- matrix(c(3, 1, 2, 4, 5, 6, 9, 7, 8), nrow=3, ncol=3, byrow=TRUE)
A.inv <- solve(A)
A; round(A.inv, 2); round(A %*% A.inv, 2)
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 4 5 6
## [3,] 9 7 8
## [,1] [,2] [,3]
## [1,] 0.11 -0.33 0.22
## [2,] -1.22 -0.33 0.56
## [3,] 0.94 0.67 -0.61
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
library(MASS) # auch mit ginv() aus library(MASS)
A.inv <- ginv(A)
A; round(A.inv, 2); round(A %*% A.inv, 2)
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 4 5 6
## [3,] 9 7 8
## [,1] [,2] [,3]
## [1,] 0.11 -0.33 0.22
## [2,] -1.22 -0.33 0.56
## [3,] 0.94 0.67 -0.61
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
# Loesung lineares Gleichungssystem
A <- matrix(c(3, 1, 2, 4, 5, 6, 9, 7, 8), nrow=3, ncol=3, byrow=TRUE)
b <- c(2,4,8); A; b
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 4 5 6
## [3,] 9 7 8
## [1] 2 4 8
x <- solve(A, b); round(x, 2)
## [1] 0.67 0.67 -0.33
A %*% x
## [,1]
## [1,] 2
## [2,] 4
## [3,] 8
# Eigenwerte und Eigenvektoren in A
A <- matrix(c(3, 1, 2, 4), nrow = 2, ncol=2, byrow=TRUE);A
## [,1] [,2]
## [1,] 3 1
## [2,] 2 4
l <- eigen(A)$values; round(l, 2)
## [1] 5 2
x <- eigen(A)$vectors; round(x, 2)
## [,1] [,2]
## [1,] -0.45 -0.71
## [2,] -0.89 0.712.5 Funktionen
2.5.1 Lineare Funktionen
par(mfrow=c(1,1), font.axis=2, bty="l", ps=12, cex.axis=1.0)
x <- seq(-1.9, +1.9, by=0.02)
y.1 <- 2 + 3 * x
y.2 <- 5 - 2 * x
plot(x, y.1, type="l", axes=F, xlab="", ylab="", las=1, lwd=1.9,
xlim=c(-2, +2), ylim=c(-4,9))
axis(side = 1, pos = 0, xaxp=c(-2, 2, 8))
axis(side = 2, pos = 0, las=1)
lines(x, y.2, lwd=1.9)
points(0.6, 3.8, pch=16, cex=1.3)
text(1.2, 7.5, expression("f(x)=2+3x"), cex=1.2)
text(-1.3, 6, expression("g(x)=5-2x"), cex=1.2)
text(0.7, 1.1, "Schnittpunkt: (0.6, 3.8)", cex=1.2)
2.5.2 Nichtlineare Funktionen
par(mfrow=c(1,1), font.axis=2, bty="l", ps=12, cex.axis=1.0)
x <- seq(-2, +2, by=0.02)
y.1 <- 1 - 2*x + 3*x^2
y.2 <- 4 + 5*x - 3*x^2
plot(x, y.1, type="l", axes=F, xlab="", ylab="", las=1, lwd=1.9,
xlim=c(-2, +2), ylim=c(-6, +6))
axis(side = 1, pos = 0, xaxp=c(-2, 2, 8))
axis(side = 2, pos = 0, las=1)
lines(x, y.2, lwd=1.9)
text(-1.3, 4.0, expression(f(x) == 1 - 2*x + 3*x^2), cex=1.2)
text(-1.5, -4.0, expression(g(x)==4 + 5*x - 3*x^2), cex=1.2)
points(-(-2/(2*3)), 1-((-2)^2 / (4*3)), pch=19, cex=1.5)
points(-(5 /(2*(-3))), 4-( 5^2 /(4*(-3))), pch=19, cex=1.5)
2.5.3 Periodische Funktionen
par(mfrow=c(1,2), font.axis=2, bty="l", ps=12, cex.axis=1.0)
x <- seq(-1.0, + 1.0, by=0.0002)
y <- sqrt(1-x^2)
plot(c(x, x, 1), c(y, -y, 0), type="l", axes=FALSE, xlab="", ylab="",
las=1, xlim=c(-1.2, +1.2), ylim=c(-1.2, +1.2))
text(0.2,0.091, expression(x), cex=1.0)
lines(c(0.0, 0.0), c(-1,+1)); lines(c(-1,+1), c(0,0))
lines(c(0.0, 1.2), c(0.0, 1.0)); lines(c(0.0, 1.2), c(1.0, 1.0))
lines(c(1.0, 1.0), c(0.0, 0.822)); lines(c(0.77, 0.77), c(0.0, 0.65))
text(-0.06, -0.08, "0", cex=1.2, font=2)
text(1.06, -0.08, "A", cex=1.2, font=2)
text(0.78, -0.08, "B", cex=1.2, font=2)
text(0.78, +0.75, "C", cex=1.2, font=2)
text(1.08, +0.78, "D", cex=1.2, font=2)
text(0.06, +1.1, "E", cex=1.2, font=2)
text(1.21, +1.1, "F", cex=1.2, font=2)
x <- seq(0, 2*pi, by=0.01)
y.1 <- cos(x); y.2 <- sin(x)
plot(x, y.1, type="l", axes=F, xlab="", ylab="", lwd=2.0,
xlim=c(0, 2*pi), xaxp=c(0, 2*pi, 5), ylim=c(-1, +1))
axis(side=2, pos = 0, yaxp=c(-1, 1, 10), las=1, cex=1.0)
axis(side=1, pos = -1.0, at=c(0, pi/2, pi, 3*pi/2, 2*pi),
labels=expression(0, pi/2, pi, 3*pi/2, 2*pi), cex=2.0)
lines(x, y.2, lwd=2.0)
text(3.4, 0.8, expression(sin(x)), cex=1.2)
text(1.2, -0.8, expression(cos(x)), cex=1.2)
abline(h=0)
2.5.4 Expontialfunktion / Wachstumsfunktionen
# Exponentialfunktionen
par(mfrow=c(1,3), lwd=2.0, font.axis=2, bty="l", ps=12, cex.axis=1.5)
x <- seq(-2, 3, by=0.001)
y.1 <- exp(x)
y.2 <- (1/5)^x
plot(x, y.1, type="l", axes=TRUE, xlab="", ylab="", las=1,
xlim=c(-3, 3), ylim=c(0, 20))
lines(x, y.2)
text(2, 19, expression(y==e^x), cex=2.5)
text(-2.2, 4, expression(y==0.2^x) , cex=2.0)
# Logarithmusfunktion
x <- seq(0, 10, by=0.001)
y.1 <- log(x)
y.2 <- log10(x)
plot(x, y.1, type="l", axes=TRUE, xlab="", ylab="", las=1,
xlim=c(0, 10), ylim=c(-4,4))
lines(x, y.2)
text(4, 2, expression(y==ln(x)), cex=2.0)
text(8, 1.3, expression(y==lg(x)), cex=2.0)
# Glockenkurve
x <- seq(-3, 3, by=0.001)
y <- exp(-0.5*x^2)
plot(x, y, type="l", axes=TRUE, xlab="", ylab="", las=1,
xlim=c(-3, 3), ylim=c(0, 1))
text(2.0, 0.9, expression(y == e^-(frac(1,2)*x)^2), cex=2.5)
# Wachstumsfunktionen
par(mfrow=c(1,3), lwd=2, font.axis=2, bty="l", ps=15, cex.axis=1.5)
# exponentielles Wachstum
e <- exp(1); t <- seq(0, 20, by=0.1); lambda <- 0.1; N.0 <- 10
N <- N.0 * e^(lambda*t)
plot(t, N, ylim=c(0,100), type="l", las=1, cex.lab=1.8,
xlab="Zeit [t]", ylab="Bestand [N]")
abline(h=10, lty=2); text(10, 100, "A", cex=2)
# exponentielles wachstum mit Sättigung
N.max <- 90; lambda <- -0.2; N.0 <- 10
N <- N.max - (N.max-N.0)*e^(lambda*t)
plot(t, N, ylim=c(0,100), type="l", las=1, cex.lab=1.8,
xlab="Zeit [t]", ylab="Bestand [N]")
abline(h=10, lty=2); abline(h=90, lty=2); text(10, 100, "B", cex=2)
# logististische Wachstumsfunktion
N.max <- 90; a <- 5; b <- 0.5
N <- N.max / (1+e^(a - b*t))
plot(t, N, ylim=c(0,100), type="l", las=1, cex.lab=1.8,
xlab="Zeit [t]", ylab="Bestand [N]")
N.0 <- N.max / (1+e^a)
abline(h=N.max, lty=2); abline(h=N.0, lty=2); text(10, 100, "C", cex=2)
2.5.5 Fläche unter einer Funktion
par(lwd=1.5, font.axis=2, bty="n", ps=10, cex.axis=1.07, mfrow=c(1,1))
x <- seq(0.5, 4.5, by=0.001)
f <- function(x) - 0.5*x^2 + 2*x + 2
y <- f(x)
plot(x,y, type="l", axes=TRUE, lwd=1.2, xlab=" ", ylab=" ",
xlim=c(0,5), ylim=c(-1,5))
x1 <- seq(1.5, 3, by=0.1)
y1 <- f(x1)
polygon(c(1.5, x1, 3), c(0, y1, 0), density=5, lwd=0.7)
polygon(c(3, 3, 3.25, 3.25), c(0, f(3), f(3), 0))
polygon(c(3, 3, 3.25, 3.25), c(0, f(3.25), f(3.25), 0),angle=135,
lwd=0.7, density=10)
abline(h=0)
text(2.3, 2, "F(x)", cex=2, font=4)
text(1.5, -0.28, "a", cex=1.5, font=4)
text(3.25, -0.28, "b", cex=1.5, font=4)
text(3.7, 3.8, expression((b-3)*(f(3)-f(b))), cex=1.3, font=4)
2.6 Kombinatorik
Funktion permn() in library(combinat)
library(combinat) # Permutationen
x <- c("a","b","c")
permn(x)
## [[1]]
## [1] "a" "b" "c"
##
## [[2]]
## [1] "a" "c" "b"
##
## [[3]]
## [1] "c" "a" "b"
##
## [[4]]
## [1] "c" "b" "a"
##
## [[5]]
## [1] "b" "c" "a"
##
## [[6]]
## [1] "b" "a" "c"
n <- 8 # Produkt 1:n
prod(1:n)
## [1] 40320
n <- 20
prod(1:(2*n -2)) / (2^(n-1)*prod(1:(n-1)))
## [1] 8.200795e+21
n <- 9; k <- 7 # Binomialkoeffizient
choose(n, k)
## [1] 36
combn(letters[1:5], 3) # Kombinationen
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] "a" "a" "a" "a" "a" "a" "b" "b" "b" "c"
## [2,] "b" "b" "b" "c" "c" "d" "c" "c" "d" "d"
## [3,] "c" "d" "e" "d" "e" "e" "d" "e" "e" "e"
choose(10, 3) * prod(1:3) # Verteilung von Preisen
## [1] 720
choose(5+12-1, 12) # Bonbonsorten
## [1] 1820
26^3 + 26^2 + 26 # Worte
## [1] 18278
sum(choose(10, 1:10)) # Ausstattungsvarianten
## [1] 1023
# Multinomialkoeffizient
karten <- 52 # Anzahl der Karten
spieler <- 4 # Anzahl der Spieler
k.spiel <- karten/spieler # Anzahl Karten pro Spieler
prod(1:karten)/(prod(1:k.spiel)^spieler)
## [1] 5.364474e+28