Letters to Technical Analysis
Dear Dr. Warren:
Recently I purchased a book, Digital Foundations of Time Series Analysis, by Robinson and Silvia, that
touches upon the Maximum Eutropy Method (MEM). As much as I want to understand I the basics, my
college training doesn't give me the advanced mathematical concepts needed to fully appreciate what the
authors try to explain. For this reason, I am grateful for your series on the MEM applied to spectral
analysis and forecasting. It has already clarified a few basic ideas, for which I thank you. The MEM
appears to be an elegant and powerful analytic tool, and I look forward to building the core programs that
can execute on my TRS 80, Model I microcomputer.
There is one equation that you introduce, the Final Predictive Error (FPE), that I am perplexed by. FPE is
given as FPE(M) = (N+M)*P(M)N-M). I believe the variance for the time series of order M is
P(M)/(N-M), which is related to the statistical standard deviation—a measure of error.
I don't understand why the FPE is equivalent to the variance multiplied by (M+N) as the selection
criterion. I would think that it should be sufficient to find the local minima for the variance as a function
of order or the number of lags used. Apparently, multiplication of the variance by (M+N) makes a
difference, but I don't understand why. Could you find the time to send me a brief explanation?
As an applied mathematics consultant, Dr. Warren, perhaps you've published books, handbooks, or notes
that can better explain this and other tools without getting much more technical than the notations in your
article. I can follow your terminology, and would like the opportunity to buy your published works. You
may be able to broaden my understanding of time series analysis and its application, for which I would be
Dear D. U.,
Thank you for your recent letter commenting on the MEM articles published in the January and March
1984 issues of Technical Analysis. It is nice to hear from a reader now and then, to gain some insight on
reader perceptions and comprehension of such technical analysis articles.
Regarding your question on the selection of the FPE criterion, I am enclosing a technical paper which
more fully describes the MEM methodology, and summarizes and development of the FPE order
selection criteria. In brief, the FPE criteria is based on minimizing the expected prediction error
variance. This quantity is the sum of an error variance due to forecasting, and an error variance due to
incorrect estimation of the prediction coefficients. The sample variance due to forecasting is
(1) V = P(M) / (N-M),
as noted in your letter. Akaike showed that the total error variance, including that due to errors in
estimating the preditor coefficients, is approximately given by