Perform statistical normalization: zero mean $+$ unit variance

First, perform the MATLAB instruction:

$Z=$zscore$(x)$;

This command shifts and scales the column vectors to be zero$-$mean and unit$-$variance. Also, plot

the columns of the $Z$ matrix and discuss what you see. Is each column zero mean and unit variance?

Verify

Compute and plot the joint pdf

Recall that we can use histograms to "approximate" the true pdf$.$ However, there is no standard

MATLAB command to generate a $2$D histogram. Use the following code to compute the $2$D

histogram first and then to approximate the joint pdf$,$ say, between the $1$st and $2$nd stocks $($e$.$g$.,$

$x1=$A and $x2=B).$ Turn this into a MATLAB function so that you can use it to approximate

all other joint pdf$.$s $(1-3,$dots,$4-5).$ This code uses MATLAB's hist command to compute the

$1-$D histogram in an iterative way. The for loop segregates the stocks of X$1$ by the X$1-$value

bins. Then for each subset of X$1$ stocks, it creates a $1-$D histogram of the corresponding X$2$

stocks and puts them into the appropriate bins in the $2-$D histogram. It requires you to set bins,

which is the number of bins on each axis. You can use for instance: bins$=16$

Next, as for the marginal pdf approximation, we need to normalize the histogram so that the

joint pdf sums up to one:

pdf $=$ n$2$d$./($sum$($sum$($n$2$d$)))$;

Finally, plot the pdf using one of the following options, so that the pdf is clearly visible.

$1.$Image Plot: $h=$ imagesc $(x1,x2,pdf)$; colorbar;

Surface Plot: $,$ pdf Exercise # $1$:

Joint Distribution and Correlation

The weekly rates of return for five stocks listed on the New York Stock Exchange are given in the

file Stocks.dat. Call these stock column vectors: A$,$ B$,$ C$,$ D and $E.$ Let the data matrix be

B C D E$].$ In this assignment, first you will approximate the joint distribution of the pair of stocks.

Then you will find the covariance and correlation between each pair of stocks using the

approximated probability distribution functions $($pdfs$)$ and directly over the data $($sample

covariance$)$ too. The idea is to find out which two stocks have the highest correlation and how to

use this information for investment.

Here are the main steps:

Load the data file provided 'Stocks.dat' in MATLAB.

Visualize $($plot$)$ each of the $5$ stocks. OUTPUT: Stock plots.

Perform statistical normalization $($zero mean, unit variance$)$ of each stock data vector

OUTPUT: Plots of all normalized stocks

Approximate the joint pdf for your data. OUTPUT: Two different plots of your joint pdf$.$

Compute the covariance matrix and the correlation coefficient by a$)$ from the joint pdfs that

are estimated b$)$ directly from the sample covariance matrix OUTPUTS: the estimated

correlation from the joint pdfs and estimated correlation matrix from pdfs and the sample

correlation matrix.

You should create ONLY one single$-$file technical report $($in pdf$)$ containing comments,

discussions, MATLAB script and all outputs $($plots$,$ etc.$)$ of your assignment.