Zero mean unit variance normalization matlab

Matlab Code

WHO WANTS TO BE A MILLIONAIRE?

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

X = [A 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 pdf’s and directly over the data (sample covariance) too. The idea is to find

out which two stocks have the higher correlation and how to use this information for

investment.

Here are the main steps:

1. Load Stocks.DAT into MATLAB

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

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

OUTPUT: Plots of all normalized stocks

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

5. Compute the covariance matrix and the correlation coefficient by a) from the joint

pdf.s that are estimated, b) directly from the sample covariance matrix

OUTPUTS: the estimated correlation from the joint pdf.s and estimated correlation

matrix from pdf.s and the sample correlation matrix.

You should create ONLY one technical report (in pdf) containing: comments, discussion,

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

Assignment Details and Programming Hints

Load the data file, Stocks.dat into the data matrix, X, where the columns are the stock vectors.

X = [A B C D E]. Plot each column separetely in a single plot (different color) and observe.

Which stocks would you think that has the highest correlation?

1) Perform statistical normalization: zero mean + unit variance

First perform the MATLAB comment: Z = zscore(X);

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

Verify this by using mean() and var() commands. Also plot the columns of the Z matrix and

discuss what you see. Is each column really zero mean and unit variance? Verify.

2) 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 2D histogram. Use the following code to compute the 2D

histogram first and then to approximate the joint pdf, say between the 1st and 2nd stocks. Turn

this into a MATLAB function so that you can use it to approximate all other joint pdf.s (1-3, …

, 4-5). This code uses Matlab's hist command to perform most of the real work, and just

repeatedly applies it. The for loop segregates the stocks of X1 by the X1-value bins. Then for

each subset of X1 stocks, it creates a 1-D histogram of the corresponding X2 stocks and puts

them into the appropriate boxes in the 2-D histogram. It requires you to set bins, which is the

number of bins on each axis. I used: bins=16.

bins=16; % my choice.. feel free to change it..

stocks = Z’; % make them row vectors..

% The estimation and plotting of the 2D pdf

% 1. Use hist on all data to find bins and bin sizes.

[n, x1] = hist(stocks (1,:),bins);

[n, x2] = hist(stocks (2,:),bins);

delta_x1 = x1(2)-x1(1);

delta_x2 = x2(2)-x2(1);

% 2. Initialize a 2-D matrix for the 2-D histogram

n2d = zeros(length(x1), length(x2));

% 3. For each row, find the indices of the X_1 stocks

% which fall into that row.

% Compute a histogram for the X_2 stocks, and put it in the 2-D

% histogram for that row.

for i = 1:length(x1),

ind = find((stocks (1,:) > x1(i)-delta_x1/2) &

(stocks(1,:) <= x1(i)+delta_x1/2));

n2d(i,1:length(x2)) = hist(stocks(2,ind), x2);

end

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

up to one:

pdf = n2d./(sum(sum(n2d)));

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;

2. Surface Plot: h = surf(x1,x2, pdf); view(20,30);

3. Mesh Plot: h = mesh(x1,x2, pdf); view(20,30);

4. Contour Plot: h = contour(x1,x2,pdf); colorbar;

Do not forget to grid and label your plots. Insert and discuss two different joint pdf plots in

your report. Pick whichever two you think best show the features of your pdf.

3) Estimate the Covariance/Correlation from the joint pdf

As you have the approximated joint pdf, now you can now compute the Covariance for any

pair of stocks using the well-known formula:

You can use the following code to implement this summation:

CovXY = 0;

for x=1: bins,

for y=1: bins,

CovXY = CovXY + (x1(x)*x2(y)*pdf(x,y));

end

end

Now as you statistically normalize the data at the beginning, what do you say about

Covariance and Correlation? Repeat steps 2-3 for other stock pairs (1,3), …, (4,5). Which pair

of stocks is giving you the highest correlation? Put all the cross-correlation results into a 5x5

Cxy matrix where the (i,j)th and (j,i)th elements will both take the correlation value between

the ith and jth stock. The diagonal elements should be all 1 (no need for calculation) since the

correlation of a signal with itself is always 1.

4) Calculate sample Covariance/Correlation matrices {Bonus}

Each row of the data matrix S = stocks is a statistically normalized vector. Therefore, the

sample 5x5 covariance matrix Cov(S) can be computed as follows:

𝐶𝑜𝑣(𝐒) = 𝐒𝐒𝑇

𝑁 − 1

where N is the length of each stock data. Can you also find sample correlation matrix, Cor(S)?

Explain how?

Now examine the Cor(S) matrix. What do you see in diagonal elements? Why? Compare the estimated Correlation value between stocks 1 and 2 in the previous section with

the element at (1,2) or (2,1) of the Cor(S) matrix. Are they identical? Why not? Which one is a

better estimate for the cross correlation? Why?

According to the Cor(S) matrix, which pair of stocks have the highest correlation. Is Cor(S)

and Cxy matrices are identical? If not, which one do you think is a better estimate of the cross-

correlation?

Finally, how can you use the “stock correlation” information for investment (to reduce the risks

-OR- maximize the profits) and to become a millionaire? Briefly discuss.