Segment 18. The Correlation Matrix

From Computational Statistics (CSE383M and CS395T)
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Problems

To Calculate

1. Random points i are chosen uniformly on a circle of radius 1, and their <math>(x_i,y_i)</math> coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variables <math>X</math> and <math>Y</math>? (Hint: Transform probabilities from <math>\theta</math> to <math>x</math>. Second hint: Is there a symmetry argument that some components must be zero, or must be equal?)

2. Points are generated in 3 dimensions by this prescription: Choose <math>\lambda</math> uniformly random in <math>(0,1)</math>. Then a point's <math>(x,y,z)</math> coordinates are <math>(\alpha\lambda,\beta\lambda,\gamma\lambda)</math>. What is the covariance matrix of the random variables <math>(X,Y,Z)</math> in terms of <math>\alpha,\beta,\text{ and }\gamma</math>? What is the linear correlation matrix of the same random variables?

To Think About

1. Suppose you want to get a feel for what a linear correlation <math>r=0.3</math> (say) looks like. How would you generate a bunch of points in the plane with this value of <math>r</math>? Try it. Then try for different values of <math>r</math>. As <math>r</math> increases from zero, what is the smallest value where you would subjectively say "if I know one of the variables, I pretty much know the value of the other"?

2. Suppose that points in the <math>(x,y)</math> plane fall roughly on a 45-degree line between the points (0,0) and (10,10), but in a band of about width w (in these same units). What, roughly, is the linear correlation coefficient <math>r</math>?