C Simulation and visualization

Reading materials: Chapters 3, 5, and 7 in R for data science by Garrett Grolemund and Hadley Wickham (optional).

C.1 Simulation and visualization: univariate

## Random seed
set.seed(1) # set random number generator seed for reproducibility
runif(1)
runif(1) 

set.seed(2)
runif(1)

set.seed(1)
runif(1)
runif(1)

## Uniform distribution
### Draw ten uniform random variables:
runif(10)
# use ?runif to see what other options are available 

# To help understand this function, we can draw the density plot for this distribution
x.grid=seq(from=0,to=1,length.out=100)
unif.pdf=dunif(x.grid,min=0,max=1);
plot(y=unif.pdf,x=x.grid,type='l')

# Compare this with a histogram from a set of random variables from runif:

unif.rv.100=runif(100);
hist(unif.rv.100,freq=FALSE,main='Uniform',xlab='X')
lines(y=unif.pdf,x=x.grid,col='blue',lwd=3)
# Further compare it with a smooth density estimation based on the samples:
unif.pdf.est=density(unif.rv.100);
lines(unif.pdf.est,col='red',lwd=3)

# Wrap up the above plotting code as a function 

plot.density.empirical<-function(rvs,pdf.true,pdf.grid,main){
hist(rvs,freq=FALSE,main=main,xlab='X')
lines(y=pdf.true,x=pdf.grid,col='blue',lwd=3)
# Further compare it with a smooth density estimation based on the samples:
pdf.est=density(rvs);
lines(pdf.est,col='red',lwd=3)
}


plot.density.empirical(unif.rv.100,unif.pdf,x.grid,main='Uniform')

### Normal
### Draw ten normal random variables with mean 0 and variance 2
rnorm(10,mean=0,sd=sqrt(2))

### There are many other distributions in R
### try ?rchisq, ?rf, ?rt, ?rbeta, ?rpois, ...

## Other distributions
### Poisson  ?rpois

### Exponential ?rexp

### t ?rt


### Chisq ?rchisq


### F ?rf

## Sampling from a user-specified vector 

### Draw samples from any vectors using sample()
wrd = c("yet", "a","new","sentence")
sample(wrd,size=2)
# use ?sample to see other options

### Visulize the simulated data

y<-rnorm(100);
# We start with the very basic histogram 
hist(y)
# Use ?hist to modify the plot 

# And the boxplot:
boxplot(y)

# We can also draw visualize the data using ggplot2
library(ggplot2)
dat=data.frame(y=y)
ggplot(dat, aes(x=1,y=y)) + 
  geom_violin(trim=FALSE)

# Read more about the violin plots in the post here:
# http://www.sthda.com/english/wiki/ggplot2-violin-plot-quick-start-guide-r-software-and-data-visualization

C.2 Simulation and visualization: multivariate

# We can generate many independent univariate random variables using code in the previous section 

# Here we will generate dependent random variables, for instance, 
n=50;# sample size
x1=rnorm(n);
x2=x1*2+runif(n)*2;
x3=x1*x2*rpois(n,lambda=3);

# You can also generate a random variable using its probability density function using importance sampling
# We will be fine with the simple data generating method in this class 

# We put the three random variables into one data.frame for ease of plotting 
dat.multi<-data.frame(x1=x1,x2=x2,x3=x3)

### Visualization

### Pairwise scatter plot
pairs(dat.multi,pch='.',cex=4)


### Visualization in ggplot2
library(GGally)
ggpairs(dat.multi)


### For the pair x1 and x3

# A more informative scatterplot, using base R 
plot(x=dat.multi$x1,y=dat.multi$x3,data=dat.multi,pch=16,cex=2)
lines(lowess(dat.multi$x1, dat.multi$x3), col=2,lwd=3)


# Or using ggplot2
ggplot(data = dat.multi) + 
  geom_point(mapping = aes(x = x1, y = x3)) +
  geom_smooth(mapping = aes(x = x1, y = x3))