ECE 5615/4615 Computer Project

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1 Set #1p Due Friday March 17, 017 ECE 5615/4615 Computer Project The details of this first computer project are described below. This being a form of take-home exam means that each person is to do his/her own work. Bring questions to me. A hints page will be provided as needed. 1. The random variable w is defined in terms of the random variables x, y, and z, to be w = xy + z The input rv are assumed to be mutually independent, with x ~ U(-1,1), y ~ U(- 1,1), and z ~ U(0,1). a.) Find the theoretical pdf f w w. Start by first finding the pdf on v = xy using the fact that for independent x and y 1, f v v 1 v = f w x wf y --- w d w = 1 v f w y wf x --- w d w Then find the pdf on the sum w = v + z from a convolution. Note that the rv v and z are also independent. Why? b.) Simulate the rv w using Python and the uniform random number generator rand(). Estimate the theoretical pdf using a scaled histogram, as described in the Chapter 3 IPython notebook pdf or the posted notebook itself. The value of N should be at least 100,000. Plot both the pdf estimate and theoretical pdf from part (a) on the same graph to see how well they compare.. In this problem you will generate correlated Gaussian random number pairs using a technique that is valid for random vectors of any dimension and finite length. This problem will be restricted to the -D case, so the jointly normal random variate pairs will be of the form Nm 1 m 1 which corresponds to a jointly normal pdf of the form Example histogram calculation using 50 bins (1) () 1. A. Papoulis and S. Pillai, Probability, Random Variable, and Stochastic Processes, McGraw Hill, New York, 00, pp Alberto Leon-Garcia, Probability and Random Processes for Electrical Engineering, third edition, Prentice Hall, New Jersey, Section 6.6, pp

2 f y y where y = y 1. Assume we are given an n -dimensional random vector X that has zero mean and covariance matrix C x = I, where I is the n n identity matrix. Our goal is to generate random variates corresponding to random vector Y such that they have a prescribed mean, m y', and prescribed covariance matrix, C y. If we transform X with matrix A and add a constant vector m y, i.e. Y = AX+ m y (4) we obtain a new random vector with mean m y and covariance matrix C y = AIA t = AA t. Clearly getting the prescribed mean is easy, and simply requires adding a constant vector to the random number pairs resulting from the transformation AX. How do we choose A to get the desired covariance matrix C y? Assuming that the desired C y is positive definite (determinant > 0 or all eigenvalues are positive), we can first expand C y as the following matrix product where P is a unitary matrix whose columns are the orthonormal eigenvectors of C y and D is an n n diagonal matrix of the eigenvalues of C y. We know that the eigenvalues are just the variances, i, of the individual components of Y, and they must all be positive, so we can expand D as D = D 1 D 1 (6) where D 1 is an n n diagonal matrix with entries being the square root of the entries of D, or simply the standard deviations,. The desired transformation we are seeking is finally As a check note that y 1 m y 1 m m m exp = t i C y = PDP t A = PD 1 (3) (5) (7) AA t = PD 1 PD 1 t = PD 1 D 1 P t = PDP t = C y (8) For the n = case the covariance matrix is simply C 1 1 y = 1 (9) PD 1 In Python obtaining is easy since eigenvalue decomposition can be performed to obtain both P and D using D,P = linalg.eig(c), where C is a square matrix and P and D are as defined above in Python terms. The see the IPython notebook example for Chapter. Page

3 a.) Write a Python function that generates N Gaussian (normal) random number pairs using the technique described above. Zero mean unit variance iid normal random variates can be obtained in IPython (assuming %pylab is loaded) from x = randn(,n), which produces matrix x having rows and N columns. The function should have an interface that takes as input the desired means, m 1 and m, the variances 1 and, and the correlation coefficient. The output should be a matrix, y, corresponding to the input variates x, e.g., def corrgauss(mean,var,p,n): Y = corrgauss(mean,var,p,n) Generates Correlated Gaussian RV's mean = mean vector [m1, m] var = variance vector [var1, var] p = correlation coefficient N = number of random pairs to generate X = randn(,n) # Your code goes in here to compute Y... return Y In my version I print some helpful information about the construction of Y, e.g., b.) Test the function of part (a) using mean = [1, 1]; var = [1, 1]; p = 0; N = 1000 by plotting scatter plots of the data for When doing a scatter plot in Python s matplotlib, keep the same x-y axis units so the symmetry is obvious, e.g., Comment on your results. Is this what you expected? c.) Repeat part (b) using N d.) Using the parameters part (c), but increasing the sample size to N = , compute the Page 3

4 sample statistics for the mean vector, and covariance matrix. Python supports the calculation of the mean vector and correlation matrix naturally, e.) Using a sample size of N = find experimentally the probability that the random vector Y lies a square with unit length sides centered at the origin. Use the parameters of part (c). To check your answer numerically integrate the theoretical joint pdf, i.e., calculate 1 1 P square = f y y 1 dy 1 d 1 1 See the notes below on how to do this using scipy.integrate. Mathematica works well also. f.) Repeat part (e) except now shift the square center to the point (,). (10) The scipy.integrate module contains functions for performing numerical integration. In particular quad() for 1-D integrals and dblquad() for -D integrations. The Chapter 3 notebook contains information on how to use dblquad() via an example. The example is repeated here as well. Page 4

5 Estimate of the error Answer Page 5