#basic plot  
boxplot(x)  
qqplot(x,y) 

#linear regression  
n = 10  
x1 = rnorm(n)#variable 1  
x2 = rnorm(n)#variable 2  
y = rnorm(n)*3  
mod = lm(y~x1+x2)  
model.matrix(mod) #erect the matrix of mod  
plot(mod) #plot residual and fitted of the solution, Q-Q plot and cook distance  
summary(mod) #get the statistic information of the model  
hatvalues(mod) #very important, for abnormal sample detection  

#logistic regression  
x <- c(0, 1, 2, 3, 4, 5)  
y <- c(0, 9, 21, 47, 60, 63) # the number of successes  
n <- 70 #the number of trails  
z <- n - y #the number of failures  
b <- cbind(y, z) # column bind  
fitx <- glm(b~x,family = binomial) # a particular type of generalized linear model  
print(fitx)  
  
plot(x,y,xlim=c(0,5),ylim=c(0,65)) #plot the points (x,y)  
  
beta0 <- fitx$coef[1]  
beta1 <- fitx$coef[2]  
fn <- function(x) n*exp(beta0+beta1*x)/(1+exp(beta0+beta1*x))  
par(new=T)  
curve(fn,0,5,ylim=c(0,60)) # plot the logistic regression curve  

# bootstrap  
# Application: 随机采样，获取最大eigenvalue占所有eigenvalue和之比，并画图显示distribution  
dat = matrix(rnorm(100*5),100,5)  
 no.samples = 200 #sample 200 times  
# theta = matrix(rep(0,no.samples*5),no.samples,5)  
 theta =rep(0,no.samples*5);  
 for (i in 1:no.samples)  
{  
    j = sample(1:100,100,replace = TRUE)#get 100 samples each time  
   datrnd = dat[j,]; #select one row each time  
   lambda = princomp(datrnd)$sdev^2; #get eigenvalues  
#   theta[i,] = lambda;  
   theta[i] = lambda[1]/sum(lambda); #plot the ratio of the biggest eigenvalue  
}  
  
# hist(theta[1,]) #plot the histogram of the first(biggest) eigenvalue  
hist(theta); #plot the percentage distribution of the biggest eigenvalue  
sd(theta)#standard deviation of theta  
  
#上面注释掉的语句，可以全部去掉注释并将其下一条语句注释掉，完成画最大eigenvalue分布的功能  

#Application：判断一个自变量是否有影响 (假设我们喂3种维他命给3头猪，想看喂维他命有没有用)  
#   
y = rnorm(9); #weight gain by pig(Yij, i is the treatment, j is the pig_id), 一般由用户自行输入  
#y = matrix(c(1,10,1,2,10,2,1,9,1),9,1)  
Treatment <- factor(c(1,2,3,1,2,3,1,2,3)) #each {1,2,3} is a group  
mod = lm(y~Treatment) #linear regression  
print(anova(mod))  
#解释：Df（degree of freedom）  
#Sum Sq: deviance (within groups, and residuals) 总偏差和  
# Mean Sq: variance (within groups, and residuals) 平均方差和  
# compare the contribution given by Treatment and Residual  
#F value: Mean Sq(Treatment)/Mean Sq(Residuals)  
#Pr(>F): p-value. 根据p-value决定是否接受Hypothesis H0：多个样本总体均数相等(检验水准为0.05)  
qqnorm(mod$residual) #plot the residual approximated by mod  
#如果qqnorm of residual像一条直线，说明residual符合正态分布，也就是说Treatment带来的contribution很小，也就是说Treatment无法带来收益（多喂维他命少喂维他命没区别）