###Some Basic commands in R
###Course: APMA168-Nonparametric Statistics, Brown University
###Instructor: Konstantinos Spiliopoulos

###############################################################################################################
##############################Basics###########################################################################

x<-c(1,2,3,4,5)

y<-c(3,5,6,7,8,9)
y
#[1] 3 5 6 7 8 9

y1<-y[y != 7] #is the vector of elements of y excluding the element that has value 7
y1
#[1] 3 5 6 8 9

 sin(y) #it applies the sin function to each element of y-vector
#[1]  0.1411200 -0.9589243 -0.2794155  0.6569866  0.9893582  0.4121185


z<-c(x,y) #Combining the two vectors in one

x[2] #extractring the 2nd ellement of the x-vector
#[1] 2

mean(x) #The mean value of vector x
#[1] 3

sd(x) #The standard deviation of vector x
#[1] 1.581139

 median(y) #The median of the vector y
#[1] 6.5


objects() #In case you want to find out the objects that you have defined.
[1] "x" "y" "z"

hist(x) #ploting the histgram of vevtor x

sequence<-seq(1,3, by=0.2)#It creates a sequence of numbers from 1 to 3 with step 0.2
#[1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0




###############################################################################################################
#####Doing TESTS###############################################################################################
t.test(x,alternative="greater",mu=5) # will test the null hypothesis mu=5 vs. the alternate that mu > 5. 
#   One Sample t-test

#data:  x 
#t = -2.8284, df = 4, p-value = 0.9763
#alternative hypothesis: true mean is greater than 5 
#95 percent confidence interval:
# 1.492557      Inf 
#sample estimates:
#mean of x 
#        3 


#Besides just getting the result of a function, we can store the results somewhere. For example

tx<-t.test(x,alternative="greater",mu=5)

attributes(tx) # will show the various properties of tx. For example
#$names
#[1] "statistic"   "parameter"   "p.value"     "conf.int"    "estimate"   
#[6] "null.value"  "alternative" "method"      "data.name"  
#
#$class
#[1] "htest"

tx$p.value # returns the p-value.
#[1] 0.9762897


###############################################################################################################
##############Generating random variables######################################################################

#Examples of generating random numbers following various discrete and continuous distributions

normal<-rnorm(12,5,10)
# [1]  16.4273175  13.6126756  -2.7347741   8.6122029 -16.7411542   1.4438525
# [7]  -2.9433208   7.0977609  -3.1812454  26.7573497   8.5117967  -0.5879782

chisquare<-rchisq(200,4)


pnorm(30, mean=50, sd=sqrt(400)) #This is P(X < 30) when X is normal with mean 50 and standard deviation 20. 
pnorm(30, 50, 20)         #This is P(X < 30) when X is normal with mean 50 and standard deviation 20.
#Remark: R wants the s. d. as the parameter, not the variance. We'll need to take a square root!!!!!!!!!!!!!! 
#[1] 0.1586553   Namely P(X < 30)=0.1586553

qnorm(0.95, mean = 100, sd = 15) # This will give the 95th  quantile (or percentile) of the normal distribution
#[1] 82.89707


###A nice example is the following:

x <- rnorm(1000, mean = 50, sd = 20)  #generates 1000 i. i. d. normal random numbers
hist(x, freq = FALSE) #plots their histogram
curve(dnorm(x, mean = 50, sd = 20), add = TRUE) #graphs the p. d. f. of the same normal distribution


 
#######Similarly for Binomial, Poisson, Exponential, Gamma, Beta, etc....Check out R-manual and help!!!




###############################################################################################################
######Some Graphing commands################################################################################### 

plot(x)
plot(sin(x))

#One of the most useful commands is help("anything you want"). For example

help("plot") 

##Moreover, R supports for loops, if commands etc....


##A nice way of creating plots and saving them somewhere in your computer is the following. 
##In the example that follows we assume that we want to save our figure in 
##C:\Documents and Settings\Kostas\Desktop with name Figure1 and type .eps
##It also shows how one can put multiple graphs in one figure

x<-seq(-4,4,by=0.01)
postscript("C:\\Documents and Settings\\Kostas\\Desktop\\Figure1.eps", horizontal=F,paper="special", height=8, width=9, pointsize=12)
plot(x,(1/2)*x^2, xlim=c(-2,2), ylim=c(0,3), xlab="x",ylab=" V(x)", type="l", lty=1, col="black")
lines(x,(1/2)*x^2+0.1*(cos(x/0.01)+sin(x/0.01)), type="l", lty=1, col="black")
dev.off() 


###############################################################################################################
#######Defyning your own functions in R########################################################################


#The following command (function) defines a new function  called trial1 that takes as input a vector x and returns 
#a new vector with the elements squared

trial1 <- function(x) {
    x1 <- x^2 
    return(x1)
}

####An application:
  y
#[1] 3 5 6 7 8 9
 trial1(y)
#[1]  9 25 36 49 64 81

##However x1 is defined as a local variable. This means that it is recognized only inside the function trial1. 
#For example, if we run x1 we get an error message:

> x1
#Error: object 'x1' not found



##################################################################################################################
#####For loops and if statements in R#############################################################################

#A very useful command is the for loop:

for (i in 1:10000) {
    ### some R statements that do something here
}

#does the same thing (whatever is done by the R statements inside) 10000 times. 

#An example. Suppose we want to simulate the mean of a random sample of 
#size n from the normal distribution n times.

m <- 200
n <- 1000
MuHat <- double(n)
for (j in 1:n) {
    x <- rnorm(m)
    MuHat[j] <- mean(x)
}
hist(MuHat, freq = FALSE)
curve(dnorm(x, sd = sqrt(1 / m) ), add = TRUE)


###An example that combines a for loop and and if-elseif-else statement

w<-seq(1,20,by=0.2)
group<-NULL
for (i in 1:length(w)) {
  group[i] <- if (w[i] <= 5) 1 else if (w[i] <= 12) 2 else 3
}
group
#[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
#[39] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
#[77] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3



###############################################################################################################
#Practice yourself and use the R-manual and Help as much as possible!!!!!!!!!!!!!!!