Problem
Today we introduce multivariable sample spaces. We will learn more about covariance or how closely two sets of data are correlated. Using MatLab we will be able to construct some random sets of data and observe the covariance. Specifically we will utilize MatLab’s 3D visualization tool to observe the PDFs of our random vectors. Plotting in 3D will allow us to see the covariance and other general characteristics of our two random variables.
First we will start with generating two random arrays which are independently generated uniformly between zero and one. The task has us noting the general shape and meaning behind our 3D PDF or surf plot.
Second we will tell MatLab to construct our two random variables in accordance to four different covariance matrices (seen in Figure 2). Using these covariances and also a mean of [6 6] for both randomly generate variables, we are able to construct distributions which are not all normal. To observe the covariance we create a support region or plane which intersects the distribution as seen in Figure 1. We will see for covariances matrices which are not the identity matrix the variances of our random variables is not equal in all directions. Note how the diagonal corresponds to the variance of the values and the off diagonal tells us the correlation.
Approach and Results
We begin with our first task of creating two randomly generated uniform variables between 0 and 1. We can use MatLab’s random number generator to complete the task. To find the histogram of the 2 variables we use MatLab’s function ‘hist3’. In the previous assignment we emphasized and proved how the number of samples you take can influence whether your data reaches your expected output.Therefore we generate 100,000 vectors and sum all the results so to guarantee our output produces accurate results. Below we plot our overall histogram: