Warm-up Activity

The goal of discriminant analysis according to Johnson and Wichern is

"To describe, either graphically (in three or fewer dimensions) or algebraically, the differential features of objects (observations) from several known collections (populations). We try to find 'discriminants' whose numerical values are such that the collections are separated as much as possible".

1.Consider two samples of data: X_1=[(3&7 2&4 4&7)] X_2=[(6&9 5&7 4&8)]

Find the discriminant function and use it (according to page 543) to classify a new data point X new ' = [2&7].

Page 543:

15.2.2 Classification

Let xnew be a new vector of observation. The goal is to classify it into one of the groups by

using the discriminant function. The simple, and fairly obvious, the technique is to first obtain the

discriminant score by

znew = axnew = (x1 x2)S1

pl xnew.

Next, classify xnew to group 1 or 2 accordingly, as znew is closer to z1 or z2. A simple illustration

is done next.

Principle-component analysis is concerned with explaining the variance/covariance structure of any sample through the linear combinations such variables with the main goal in mind of dimension reduction. If data is multicollinear this is the main tactic to tackle that issue. A main theoretical underpinning is the following fact:

2.Prove that a symmetric matrix has real eigenvalues and that the eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.Use this fact to prove that any symmetric matrix can be spectrally decomposed into with a diagonal matrix containing the eigenvalues and the normalized eigenvectors arrangedcolumn-wise.

3.Suppose the random variables have the population variance/covariance matrix:

Calculate the principal components and interpret their meaning. What can be gained from knowing these values?

= [(1&-2&0 -2&5&0 0&0&1)]