《R语言与统计分析》-学习笔记4

探索性数据分析

常用分布的概率函数图

二项分布

n <- 20
p <- 0.2
k <- seq(0,n)
plot(k,dbinom(k,n,p),type = "h",main = "Binomial distribution,n=20,p=0.2",xlab = "k")

泊松分布

lambda <- 4.0
k <- seq(0,20)
plot(k,dpois(k,lambda),type = "h",main = "Poisson distribution,lambda=5.5",xlab = "k")

几何分布

p <- 0.5
k <- seq(0,10)
plot(k,dgeom(k,p),type = "h",main = "Geometric distribution,p=0.5",xlab = "k")

超几何分布

N <- 30
M <- 10
n <- 10
k <- seq(0,10)
plot(k,dhyper(k,N,M,n),type = "h",main = "Hypergeometric distribution,N=30,M=10,n=10",xlab="k")

负二项分布

n <- 10
p <- 0.5
k <- seq(0,40)
plot(k,dnbinom(k,n,p),type = "h",main = "Negative Binomial distribution,n=10,p=0.5",xlab="k")

正态分布

curve(dnorm(x,0,1),xlim = c(-5,5),ylim=c(0,0.8),col="red",lwd=2,lty=3)
curve(dnorm(x,0,2),add = T,col ="blue",lwd=2,lty=2)
curve(dnorm(x,0,1/2),add = T,lwd=2,lty=1)
title(main="Gaussian distributions")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("sigma=1","sigma=2","sigma=1/2"),lwd = c(2,2,2),lty = c(3,2,1),col = c("red","blue",par("fg")))

t分布

curve(dt(x,1),xlim=c(-3,3),ylim=c(0,0.4),col="red",lwd=2,lty=1)
curve(dt(x,2),add = T,col="green",lwd=2,lty=2)
curve(dt(x,10),add = T,col="orange",lwd=2,lty=3)
curve(dnorm(x),add = T,lwd=3,lty=4)
title(main = "Student T distributions")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("df=1","df=2","df=10","Gaussian distribution"),lwd = c(2,2,2,2),lty = c(1,2,3,4),col = c("red","blue","green",par("fg")))

卡方分布

curve(dchisq(x,1),xlim = c(0,10),ylim=c(0,0.6),col="red",lwd=2)
curve(dchisq(x,2),add = T,col="green",lwd=2)
curve(dchisq(x,3),add = T,col = "blue", lwd = 2)
curve(dchisq(x,5),add = T,col = "orange",lwd = 2)
abline(h=0,lty=3)
abline(v=0,lty=3)
title(main = "Chi square Distributions")
legend(par("usr")[2],par("usr")[4],xjust=1,c("df=1","df=2","df=3","df=5"),lwd=3,lty=1,col=c("red","green","blue","orange"))

F分布

curve(df(x,1,1), xlim = c(0,2),ylim=c(0,0.8),lty=1)
curve(df(x,3,1), add = T, lwd=2, lty=2)
curve(df(x,6,1), add = T, lwd=2, lty=3)
curve(df(x,3,3), add = T, col = "red", lwd=3,lty=4)
curve(df(x,3,6), add = T, col = "blue", lwd =3,lty=5)
title(main = "Fisher's F")
legend(par("usr")[2],par("usr")[4],xjust=1,c("df=(1,1)","df=(3,1)","df=(6,1)","df=(3,3)","df=(3,6)"),lwd = c(1,2,2,3,3),lty = c(1,2,3,4,5),col = c(par("fg"),par("fg"),par("fg"),"red","blue"))

对数正态分布

curve(dlnorm(x),xlim = c(-.2,5),ylim=c(0,1.0),lwd=2)
curve(dlnorm(x,0,3/2),add = T, col = "blue", lwd = 2,lty = 2)
curve(dlnorm(x,0,1/2), add = T, col = "orange", lwd = 2,lty = 3)
title(main = "Log normal distributions")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("sigma = 1","sigma = 2","sigma = 1/2"),lwd = c(2,2,2),lty = c(1,2,3),col = c(par("fg"),"blue","orange"))

柯西分布

curve(dcauchy(x),xlim = c(-5,5),ylim=c(0,.5),lwd=3)
curve(dnorm(x),add = T, col = "red", lty = 2)
title(main = "Cauchy distribution")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("Cauchy distribution","Gaussian distribution"),lwd = c(3,1),lty = c(1,2),col = c(par("fg"),"red"))

威布尔分布

curve(dexp(x),xlim = c(0,3),ylim=c(0,2))
curve(dweibull(x,1),lty=3,lwd=3,add = T)
curve(dweibull(x,2),col="red",add = T)
curve(dweibull(x,.8),col="blue",add = T)
title(main = "Weibull Probability Distribution Function")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("Exponential","Weibull,shape=1","Weibull,shape=2","Weibull,shape=0.8"),lwd = c(1,3,1,1),lty = c(1,3,1,1),col = c(par("fg"),par("fg"),"red","blue"))

伽马分布

curve(dgamma(x,1,1),xlim = c(0,5),lwd=2,lty=1)
curve(dgamma(x,2,1),add = T,col="red",lwd=2,lty=2)
curve(dgamma(x,3,1),add = T,col="green",lwd=2,lty=3)
curve(dgamma(x,4,1),add = T,col ="blue",lwd=2,lty=4)
curve(dgamma(x,5,1),add = T,col ="orange",lwd=2,lty=5)
title(main = "Gamma distributions")
legend(par("usr")[2],par("usr")[4],xjust = 1,c("k=1 (Exponential distribution)","k=2","k=3","k=4","k=5"),lwd = c(2,2,2,2,2),lty = c(1,2,3,4,5),col = c(par("fg"),"red","green","blue","orange"))

贝塔分布

curve(dbeta(x,1,1),xlim = c(0,1),ylim=c(0,4))
curve(dbeta(x,3,1),add = T, col ="green")
curve(dbeta(x,3,2),add = T,lty=2,lwd=2)
curve(dbeta(x,4,2),add = T,lty=2,lwd=2,col="blue")
curve(dbeta(x,2,3),add = T,lty=3,lwd=3,col="red")
curve(dbeta(x,4,3),add = T,lty=3,lwd=3,col="orange")
title(main="Beta distributions")
legend(par("usr")[1],par("usr")[4],xjust = 0,c("1,1","3,1","3,2","(4,2)","(2,3)","4,3"),lwd = c(1,1,2,2,3,3),lty = c(1,1,2,2,3,3),col = c(par("fg"),"green",par("fg"),"blue","red","orange"))

直方图与密度函数的估计

直方图

它给出了数据的频率分布图形。
hist( )的调用格式

hist(x, breaks = “Sturges”, freq = NULL, probability = !freq,
col = NULL,
main = paste(“Histogram of” , xname),
xlim = range(breaks), ylim = NULL,
xlab = xname, ylab,
axes = TRUE, nclass = NULL)

说明: 若选项breaks取向量，则用于指明直方图区间的分割位置; 若取正整数,则用于指定直方图的小区间数. freq取T表示使用频数画直方图, 取F则使用频率画直方图. probability与freq恰好相反. col用于指明小矩形的颜色.

核密度估计

样本的直方图粗略地描述了样本的分布, 我们还可以用函数density()得到样本的核密度估计值, 并用lines( )得到密度估计的曲线.
density( )的调用格式

density(x, bw = “nrd0”,
kernel = c(“gaussian”, “epanechnikov”, “rectangular”,
“triangular”, “biweight”, “cosine”, “optcosine”),
n = 512, from, to)

说明: 选项bw指定核密度估计的窗宽, 也用字符串表示窗宽选择规则, 具体可参考函数bw.nrd( ). kernel为核密度估计所使用的光滑化函数, 缺省为正态核函数. n给出等间隔的核密度估计点。from与to分别给出需要计算核密度估计的左右端点。

例4.2.1从二项分布binom(100,0.9)中抽取容量为N=100000的样本.试作出它的直方图及核密度估计曲线.

N <- 100000 #重复N组
n <- 100    #每组100次试验
p <- .9     #每次试验的成功概率
x <- rbinom(N,n,p)  #每组成功的次数
hist(x,xlim = c(min(x),max(x)),probability = T,nclass = max(x)-min(x) +1,col = "lightblue",main="Binomial distribution,n=100,p=0.9")
lines(density(x,bw=1),col="red",lwd=3)

例4.2.2从负二项分布nbinom(10,0.25)中抽取容量为N=100000的样本. 试作出它的直方图及核密度估计曲线.

N <- 100000 #重复N组
n <- 10     #每组需要成功10次
p <- 0.25   #每次成功的概率是0.25
x <- rnbinom(N,n,p) #在成功10次前，要失败多少次
hist(x,xlim = c(min(x),max(x)),probability = T,nclass = max(x)-min(x) +1,col = "lightblue",main="Nbinomial distribution,n=10,p=0.25")
lines(density(x,bw=1),col="red",lwd=3)

单组数据的描述性统计分析

单组数据的图形描述

** 例4.3.1** 程序包DAAG中有内嵌数据集“possum”，它包括了从维多利亚南部到皇后区的七个地区的104只负鼠(possum)的年龄、尾巴的长度、总长度等9个特征值，我们仅考虑43只雌性负鼠的特征值，我们建立子集fpossum,考查雌性负鼠(fpossum)的总长度的频率分布.