###################################
# QERM 598
# Homework 0
# by Mike Keim and Eli Gurarie)
# 1.2.2007
###################################

#########################
# An introduction to R	#
#########################



# The following examples and exercises should give you a first look
# at what R does and how it works. 
#
# R is a command-line program, which just means commands are entered
# line-by-line at the prompt. Being a programming language it is very
# finicky. Everything has to be entered just right - including case-
# sensitivity.  Get used to seeing the following succinct and redundant
# message in blue: "Error: syntax error"
#
# More typically, extended pieces of code are written in a text-
# editor and copy-pasted into R.  

# There is a script editor in R itself (under the file menu), which 
# allows users to run code by highlighting text and hitting ctrl-R.

# Tinn-R (http://www.sciviews.org/Tinn-R/) is a more specifically 
# R-oriented text editor which has a similar feature to the R-script 
# editor.  

# A very popular one among Windows users (especiallin in QERM) is 
# "CRIMSON EDITOR" - (downloadable at http://www.crimsoneditor.com), 
# which is very versatile, can color-code many programming languages,
# and has a bunches of neat features. 

# The comment operator is '#' (which is why they're all over the 
# place in this script).  You can (just for fun) highlight this entire 
# document and paste it into R and watch all sorts of stuff happen).  
# It is a good habit in general to write lots of comments in your 
# code describing what you're having R do for you.

# If you're brand new to R, your best bet is probably to print out 
# this lab and type things in by hand... just to get used to the 
# syntax. 

# The assignment operator '<-'
x<-5			# sets x equal to 5

# Notice that using the assignment operator sets the value of x
# but doesn't necessarily return anything.  To see what x is, you
# need to type:
x				# tell us what x is

# x can be many things more than just a single number.  We consider
# some more simple examples first.
x<-c(3,4,5)		# sets x equal to the vector (3,4,5)

# "c" is a function - a very very useful function that creates "vectors"
# or "lists". In all functions, arguments are passed within parentheses. 

x<-3:5			# is a shortcut for the same thing
x<-seq(3,5)		# does the same thing with a different function.

# You can learn more about how functions work by using "?"

?c
?seq 

# A help window appears with all sorts of information about the functions
# Note, in particular, the examples at the bottom of the window


#########################################################
# Some notes on vectors, matrices and matrix arithmetic	#
#########################################################
# You might be wondering after the last example whether x is
# a column vector or a row vector.  Usually in statistics, vectors
# are considered column vectors, but this isn't *always* the case.

# Usual operators '+', '-', '*' and '/' operate on vectors element by
# element, so we can see
x+x
x-x
x*x
x^2
x/x
# give the expected returns


# Surrounding the operator with '%' specifies that matrix operations apply
x%*%x			# returns the inner product
t(x)%*%x 		# again returns the inner product, note that this implies
				# R was interpreting x as both a column and row vector in
				# the previous line of code...
x%*%t(x)		# returns the outer product

# Some 'semi-intuitive' things that R does:
x+1
y<-1
x+y
y<-c(1)
x+y			# so we can add vectors of length 1 to other vectors, but
y<-c(1,2)
x+y			# this will make R choke.


# What about a matrix?
y<-matrix(c(1,2,3,4,5,6,7,8,9,10,11,12),nrow=4)
y
# Notice that we can do this:
y+x
# and this:
y+1
# and even this:
y+c(1,2)	
# but this may not be intuitive to everyone... e.g. me.

# Here's a tiresome calculation that R can do for you very quickly:
z<-matrix(rnorm(25),nrow=5)
z.inverse<-solve(z)
z%*%z.inverse
# Notice that the off-diagonals are 'zero' to within computer precision.
# Set your own precision like this:
round(z%*%z.inverse)

# What the heck is the "rnorm" function?  How does "matrix" make matrices?
# What can "round" do?   Find out all this and more yourself!
?matrix
?rnorm
?round


#############################################
# Selecting certain elements of an R object #
#############################################
# We can call out specific items in vectors and matrices using 
# square brackets (an important contrast from parentheses.  

x[1]		# returns the first element of x
x[-1]		# returns all but the first element of x
x[c(1,2)]	# returns the first two elements of x
# Square brackets can also use logical operators
x[x>=4]		# returns the elements of x that are >= 4
z[z>0]		# returns the positive elements of z.
z[1,]		# returns the first row of z
z[,1]		# returns the first column of z
z[1:2,3:5]	# returns the 2x3 submatrix in z's upper right corner






#####################################
# Homework for Wednesday 1.7.2007	#
#####################################

# 1.  Download R onto your computer from
# http://www.r-project.org/

# 2. Spend a little time poking around the website given above.

# 3. Create a vector of all the even numbers from 2 to 100 using the ':'
# operator described above in the definition of 'x'.

# 4. Create a vector of all the even numbers from 2 to 100 using the seq() function.

# 5. Create a vector of all the even numbers from 2 to 100 using
# the square brackets. (One way of doing this will seem very redundant
# after you do number 3.)

# 6. 
#	a) Create a 6x6 matrix of standard normal random variables.  
#	b) Invert it.
#	c) Extract the elements of the matrix that are <1
#	d) Use the mean() function to calculate the mean of your matrix.