Statistics 750   Multivariate Analysis

Prerequisite:    STAT 420 or STAT 700. Familiarity with some (any) statistical
software package would be very helpful.

Homework

Assignments, including any changes and hints, will continually be posted
here. The directory in which you can find old homework assignments and
selected problem solutions is Homework.


HW Set 7. This problem set will be due Monday, December 11.
R scripts for them have been provided (see below).

Problem 1. Simulate 1000 batches of normal data matrices of n=150
observations with 6 columns, say with mean 0 and covariance matrix
2*1^x2 + (1,-1,1,-1,1,-1)^x2 + diag(2,3,2,3,2,3). Perform on the dataset
for each such matrix a maximum likelihood factor analysis (with q=2),
and then use R function "promax" in package "mva" to do a Varmax
rotation (on the orthonormal-column matrix Lambda0 of factor loadings),
and tally for your 1000 datasets the total number of rotated
factor coordinates in the three ranges (0,.2), (.2,.8), and (.8,1),
and report the results. See R Log for a little R code and commentary on
how to do this. It seems to me that one cannot really know
what the rotated factors mean statistically without seeing
what they do typically in studies like these.

Problem 2. The second problem is about discriminant analysis with
cross-validated estimation of misclassification probability, for the
Swiss BankNote data. Here are the steps which I would like you to
follow. See R Script on Discrimination methods for R steps, implemented
on the Iris data, and the resulting picture. The progression of steps is:

(a) To find the estimated optimal quadratic discriminant region for
discriminating genuine from forged banknotes, and the (Fisher) linear
discriminant, and code them both.

(b) To estimate the probabilities of misclassification. This is done
both through a theoretical (or simulated) multivariate-normal probability
based on the estimated paramaters for the two banknote groups for the
the linear discrimination regions in (a), and then also by a cross-
validated estimation procedure. The cross-validated procedure would
successively leave out one or a few observations from one or both of
the two groups; re-estimate the discrimination regions using the
retained observations; and record whether the omitted observations are
properly discriminated (ie classified) or not, tallying the overall
relative frequencies of misclassification.

Note: for a small Log showing how to discriminate the BankNotes
essentially perfectly using two PC's, click here.

Problem 3. Use the R classification and clustering routines to see
(a) whether the grouping of US companies by industry section can be
reproduced by a formal classification algorithm, and
(b) whether the unsupervised clustering algorithms ("diana", "agnes", and
maybe "Kmeans") can provide any other meaningful grouping. Use the data
on US companies which can be found here. (It was downloaded from the
Hardle & Simar website: you can freely download html text (including
verbal description of data, i.e. descriptions of the variables which are
listed in Appendix B.5, pp. 455-6 of the Hardle & Simar book) and the
ASCII datasets themselves from the Hardle and Simar web-page.
In this dataset, you might want to explore the possibility of generating
new columns (e.g. ratios or interactions between columns and ratios,
formed from existing columns) before applying the clustering algorithms,
and you might want to reduce the dimension of the resulting explanatory-
variable sets via PC's to try some sort of visual clustering, as was
done in the small log referenced at the end of Problem 2 above. See
R Log on Clustering for some examples and details on using
the R clustering software. The "Dendrogram" pictures mentioned in the
Log in the small simulated data example are "agglomerative clustering"
and "divisive clustering".