Segment 48. Principal Component Analysis (PCA)

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Problems

To Compute

1. Suppose that only one principal component is large (that is, there is a single dominant value Failed to parse (unknown error): s_i ). In terms of the matrix Failed to parse (unknown error): \mathbf V (and anything else relevant), what are the constants Failed to parse (unknown error): a_j and Failed to parse (unknown error): b_j that make a one-dimensional model of the data? This would be a model where Failed to parse (unknown error): x_{ij} \approx a_j \lambda_i + b_j with each of the data points (rows) having its own value of an independent variable Failed to parse (unknown error): \lambda_i and each of the responses (columns) having it's own constants Failed to parse (unknown error): a_j,b_j .

2. The file dataforpca.txt has 1000 data points (rows) each with 3 responses (columns). Make three scatter plots, each showing a pair of responses (in all 3 possible ways). Do the responses seem to be correlated?

3. Find the principal components of the data and make three new scatter plots, each showing a pair of principal coordinates of the data. What is the distribution (histogram) of the data along the largest principal component? What is a one-dimensional model of the data (as in problem 1 above)?

To Think About

1. Although PCA doesn't require that the data be multivariate normal, it is most meaningful in that case, because the data is then completely defined by its principal components (i.e., covariance matrix) and means. Can you design a test statistic that measures "quality of approximation of a data set by a multivariate normal" in some quantitative way? Try to make your statistic approximately independent of Failed to parse (unknown error): N , the number of data points.