Statistics 783 Homework Assignments

Homework Assignments

Number	Due date	Assignment
1	January 21	Exercises 2.2, 2.6, 3.1 from the online notes. Data analysis: Find a time series with possible seasonal components (many can be found at the Federal Reserve's data site, but note that some are seasonally adjusted). Estimate the annual component. Check the residuals for a semi-annual or other periodic component. Extend your model as necessary.
2	February 4	Write a program that, given n (even or odd), constructs the orthogonal matrix F_n whose columns are the cosines and sines (the latter only when not all zero; normalize each column) of the frequencies f_j=2πj/n, 0≤f_j≤0.5. Use it to find F₅ and F₆, and verify that the rows of both are also mutually orthogonal and normalized. Use F₁₂ to carry out the harmonic analysis of the first 12 observations of the data series that you used in Assignment 1. Use a Fast Fourier Transform (FFT) to do the same analysis, and verify that the results are equivalent. Note that the results may differ by a scale factor. Use a FFT to analyze the entire data series that you used in Assignment 1, and comment on the frequencies that appear to be present in the series. Explore the impact of using a data window in the analysis of your data. Note that the R function `spectrum()` may be used to compute and graph the periodogram, by default with 10% of the series tapered at each end.
3	February 18	For the series that you used for earlier Assignments, or another series: construct the corresponding complex series. Use complex demodulation of the complex series to explore possible time variations in the seasonal structure. Choose a set of time series data with no obvious seasonal (or other periodic) component. You could use your earlier series with estimated periodic components removed. Compare the periodograms of the first and second halves of the series. Make periodograms of segments of the series of length ~n/9 (possibly overlapping) and average them. Make a spectrum estimate by smoothing the periodogram with 3 simple averages (modified Daniell) of length 5, and compare it with the estimate you found by segment-averaging. Does length 5 give the most similar estimate?
4	April 1	Find a collection of at least 5 related time series. Estimate the spectral density functions of all 5 series. For a few pairs of series, estimate the cross spectral density function, and graph the squared coherence and phase. Choose the amount of smoothing so that the null significance levels of squared coherence are low enough to be meaningful, while retaining enough resolution to distinguish frequency bands with different degrees of coherence. Carry out a conventional Principal Components Analysis of the 5 series, and comment on the interpretation, if any, of the components. Carry out a Complex Principal Components Analysis of the 5 series; you may want to limit the frequency band in the light of the coherency plots from the earlier part. Does the CPCA show non-trivial phase relationships? How does the interpretation of the first CPC differ from that of the first (or other) conventional PC?
More to come