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#期末考试专项复习#一、矩阵与数据框#1.生成特定的矩阵与数据框#矩阵#方法一a=array(1:10,dim=c(2,5))rownames(a)=1:2colnames(a)=c(one,two,three,four,five)adimnames(a)=list(1:2,c(one,two,three,four,five))nrow=nrow(a)ncol=ncol(a)dim(a)#方法二a=matrix(1:10,nrow=2,byrow=F)rownames(a)=1:2colnames(a)=c(one,two,three,four,five)a=matrix(1:10,nrow=2,byrow=F,dimnames=list(1:2,c(one,two,three,four,five)))#数据框的生成df=data.frame(Name=c(Alice,Becka,James,Jeffrey,John),Sex=c(F,F,M,M,M),Age=c(13,13,12,13,12),Height=c(56.5,65.3,57.3,62.5,59.0),Weight=c(84.0,98.0,83.0,84.0,99.5));dfLst=list(Name=c(Alice,Becka,James,Jeffrey,John),Sex=c(F,F,M,M,M),Age=c(13,13,12,13,12),Height=c(56.5,65.3,57.3,62.5,59.0),Weight=c(84.0,98.0,83.0,84.0,99.5))LstLst[[Name]]Lst[Name]Lst[1]Lst[[1]]Lst$Namedf=as.data.frame(Lst)dfx=array(1:6,dim=c(2,3))as.data.frame(x)#数据框的引用df[1:2,3:5]df[[Height]]df$Weightnames(df)#此属性一定非空rownames(df)=c(one,two,three,four,five)dfattach(df)r=Height/Weightrdf$r=rnames(df)detach()r=Height/Weight#2.矩阵的运算a=diag(1:3)a[2][1]=1a#1转置运算t(a)#2行列式det(a)#3向量内积x=1:5y=2*1:5x%*%yt(x)%*%ycrossprod(x,y)#4向量的外积x%*%t(y)tcrossprod(x,y)outer(x,y)x%o%y#矩阵的乘法a=array(1:9,dim=c(3,3))b=array(9:1,dim=c(3,3))x=1:3a*ba%*%bx%*%a%*%xcrossprod(a,b)#t(a)%*%btcrossprod(a,b)#a%*%t(b)#矩阵的逆solve(a)b=1:3solve(a,b)#ax=b的解#矩阵的特征值与特征向量sm=eigen(a)sme=diag(1:3)svde=svd(e)svdeattach(svde)u%*%diag(d)%*%t(v)#与矩阵运算有关的函数#取维数a=diag(1:4)nrow(a)ncol(a)#矩阵的合并x1=rbind(c(1,2),c(3,4))x2=x1+10x3=cbind(x1,x2)x3x4=rbind(x1,x2)x4cbind(1,x1)#矩阵的拉直a=matrix(1:6,ncol=2,dimnames=list(c(one,two,three),c(first,second)),byrow=T)as.vector(a)#apply函数apply(a,1,mean)apply(a,2,sum)tapply(1:5,factor(c(f,f,m,m,m)),mean)#第二题#产生随机数x=rnorm(100,0,1)x#画随机数的直方图hist(x,freq=F)#核密度曲线density(x)lines(density(x),col=blue)#添加正态分布分布函数y=seq(-4,3,0.2)lines(y,dnorm(y,mean(x),sd(x)),col=red)#画随机数的经验分布函数z=rnorm(50,0,1)plot(ecdf(z),do.p=F,verticals=T)d=seq(-3,2,0.2)lines(d,pnorm(d,mean(z),sd(z)),col=red)y=rpois(100,2)plot(ecdf(y),col=red,verticals=T,do.p=F)x=0:8lines(x,ppois(x,mean(y)),col=blue)w=c(75,64,47.4,66.9,62.2,62.2,58.7,63.5,66.6,64.0,57.0,69.0,56.9,50.0,72.0)hist(w,freq=F)lines(density(w),col=blue)x=44:76lines(x,dnorm(x,mean(w),sd(w)),col=red)plot(ecdf(w),do.p=F,verticals=T)lines(x,pnorm(x,mean(w),sd(w)),col=red)#编写函数求随机数的各种描述统计量data_outline=function(x){n=length(x)m=mean(x)v=var(x)s=sd(x)me=median(x)cv=100*s/mcss=sum((x-m)^2)uss=sum(x^2)R=max(x)-min(x)#样本极差R1=quantile(x,3/4)-quantile(x,1/4)#四分位差sm=s/sqrt(n)#样本标准误g1=n/(n-1)/(n-2)*sum((x-m)^3)/s^3g2=n*(n+1)/(n-1)/(n-2)/(n-3)*sum((x-m)^4)/s^4-3*(n-1)^2/(n-2)/(n-3)data.frame(N=n,Mean=m,Var=v,std_dev=s,Median=me,std_mean=sm,CV=cv,CSS=css,USS=uss,R=R,R1=R1,Skewness=g1,Kurtosis=g2,row.names=1)}x=rnorm(100)data_outline(x)#第三题#r,p,q,drnorm(100,0,1)pnorm(1:5,0,1)dnorm(-3:3,0,1)qnorm(seq(0,1,0.25),0,1)rbeta(100,2,2)rbinom(100,100,0.5)pbinom(1:100,100,0.5)dbinom(1:5,100,0.5)qbinom(seq(0,1,0.1),100,0.5)rchisq(100,1）qchisq(seq(0,1,0.2),10)pchisq(1:10,10)dchisq(1:10,10)rexp(100,0.5)rpois(100,2)ppois(1:1000,2)dpois(1:100,2)runif(100,0,1)qunif(c(0,0.2,0.8),0,1)punif(seq(0,1,0.2),0,1)dunif(seq(0,1,0.01),0,1)rt(100,2)qt(0.8,2)pt(-3:3,2)dt(-3:3,2)rf(100,1,2)qf(0.8,1,2)#四置信区间#1#（1）sigma已知interval_estimate1=function(x,side=0,sigma=1,alpha=0.05){xb=mean(x);n=length(x)if(side0){tmp=sigma/sqrt(n)*qnorm(1-alpha)a=-Inf;b=xb+tmp}elseif(side0){tmp=sigma/sqrt(n)*qnorm(1-alpha)a=xb-tmp;b=Inf}else{tmp=sigma/sqrt(n)*qnorm(1-alpha/2)a=xb-tmp;b=xb+tmp}data.frame(mean=xb,a=a,b=b)}x=rnorm(100,0,4)interval_estimate1(x,sigma=4,side=0)interval_estimate1(x,sigma=4,side=-1)interval_estimate1(x,sigma=4,side=1)#（2）sigma未知interval_estimate2=function(x,side=0,alpha=0.05){xb=mean(x);n=length(x)if(side0){tmp=sd(x)/sqrt(n)*qt(1-alpha,n-1)a=-Inf;b=xb+tmp}elseif(side0){tmp=sd(x)/sqrt(n)*qt(1-alpha,n-1)a=xb-tmp;b=Inf}else{tmp=sd(x)/sqrt(n)*qt(1-alpha/2,n-1)a=xb-tmp;b=xb+tmp}data.frame(mean=xb,a=a,b=b)}x=rnorm(100,0,1)interval_estimate2(x,side=-1)interval_estimate2(x,side=0)interval_estimate2(x,side=1)t.test(x,side=-1)t.test(x,side=0)t.test(x,side=1)#两个总体sigma1=sigma2但未知interval_estimate3=function(x,y,alpha=0.05){xb=mean(x);yb=mean(y)n1=length(x);n2=length(y)sw=((n1-1)*var(x)+(n2-1)*var(y))/(n1+n1-2)tmp=sqrt((1/n1+1/n2)*sw)*qt(1-alpha/2,n1+n2-2)a=xb-yb-tmp;b=xb-yb+tmpdata.frame(mean=xb-yb,a=a,b=b)}x=rnorm(100,0,1)y=rnorm(100,1,1)interval_estimate3(x,y)t.test(x,y)-0.03643479-0.98699097#第五题假设检验#（1）sigam已知，双侧，检验mu=mu0mean.test1=function(x,mu=0,sigma=1){xb=mean(x);n=length(x)z=(xb-mu)/sigma*sqrt(n)p=pnorm(z)if(p=1/2)P=2*pelseP=2*(1-p)data.frame(mean=xb,Z=z,p_value=P)}x=rnorm(100,0,2)mean.test1(x,mu=0,sigma=2)#(2)sigma未知，双侧，检验mu=mu0mean.test2=function(x,mu=0){xb=mean(x);n=length(x)z=(xb-mu)/sd(x)*sqrt(n)p=pt(z,n-1)if(p=1/2)P=2*pelseP=2*(1-p)data.frame(mean=xb,Z=z,p_value=P)}x=rnorm(100)mean.test2(x,mu=0)t.test(x,mu=0,alt=two.side)#两个总体sigma1=sigma2但未知，检验mu1=mu2mean.test3=function(x,y,mu=0){xb=mean(x);yb=mean(y)n1=length(x);n2=length(y)sw=((n1-1)*var(x)+(n2-1)*var(y))/(n1+n2-2)t=(xb-yb-mu)/sqrt(sw*(1/n1+1/n2))p=pt(t,n1+n2-1)if(p=1/2)P=2*pelseP=