STATISTICS 730: APPLIED TIME SERIES ANALYSIS, Fall 2004, Professor Dickey

HOMEWORK 5: Finding Hidden Periodicities

DIRECTIONS: Complete the following problems below. Show all of your work to receive credit. Make sure to include output from the SAS output window and graphs from the SAS graphics window WHEN YOU NEED TO SUPPORT YOUR ANSWERS!!! Here is a link to the SAS ONLINE DOCUMENTATION.

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A coating is applied to fabric as it passes through a series of 6 rollers. The coating is supposed to have a thickness of 10 thousandths of an inch, however as the figures below indicate, there is some irregularity in the thickness. The thicknesses are measured every quarter inch along the fabric. Your engineers feel that some of the irregularity may be due to some or all of the rollers being off center; thus, inducing some sort of periodicity into the coating thickness related to the circumference of one or more rollers. The roller circumferences are 5, 6, 8, 11, 16 and 20 inches. Note: the DATA are artificial.

[1] Plot the thickness by location. Is any pattern evident?

[2] Run PROC UNIVARIATE on the thickness. Does the distribution of thicknesses seem normal?

[3] Run PROC ARIMA to test that the thicknesses are white noise. What do you conclude based on the Chi-square test on the first 6 lags?

[4] Use PROC SPECTRA to identify which, if any, rollers need to be checked to see if they are off center. Express the shortest of the periodicities you found (highest frequency) in several ways:

There seems to be a repeating pattern every ____ inches.
There seems to be a repeating pattern every ____ observations.
There seems to be a repeating pattern that goes through ____ cycles per inch.
There seems to be a repeating pattern that goes through ____ cycles per observation.

[5] Using PROC REG guided by your PROC SPECTRA results, estimate how much variability (variance) would still remain in the thicknesses if all defective rollers were replaced. (run a regression on the sines and cosines indicated by PROC SPECTRA - what is the MSE?)

[6] From the PROC REG results, do the residuals seem to be normal?

Does there seem to be any autocorrelation in the data?
Back this up with some formal testing.
Plot the periodogram of the residuals. Does it look "flat?" Why do I want it to look flat?

[7] Turn in your COMPLETE SAS program from the enhanced editor window.