INSTRUCTIONS. Failure to follow these instructions may result in points discounted.
Discussing this project with anyone besides your group partner (if any), the Instructor, or the TA is not permitted. By submitting a report, all its participants agree to comply with the CU Honor Code Policy.
Students registered for APPM 4540 may work in groups of up to 2 members, and submit one project report with all participant names on it. Submit a single report in CANVAS. Due to the COVID- 19 (coronavirus) pandemic, avoid meeting in person and instead collaborate remotely using some video conferencing such as Facetime, Skype, WhatsApp, or Zoom.
Students registered for APPM 5540 must work on the project on their own.
Your report is limited to 5 pages with a minimum font size of 11 points and 1-inch margins; in particular, please provide complete but brief answers. Appendices do not count toward the page limit!
To receive full credit, you must submit a professional report addressing all the instructions and questions in the same order as requested. Be sure to include all figures or tables and to label them (e.g., Figure 1, Table 2, etc). Also, be sure to cite any sources (textbooks, papers, websites, etc) you consult. Your write-up should include brief but complete answers to all the questions listed below with appropriate references to labeled figures or tables. At the end of your report, include an appendix with the R code you used to address the project. The code must include annotations.
In this final project you will partially analyze the following time-series (download the file PRO3DATA.TXT in Canvas for the dataset):
The data was generated as follows:k Xt = a + b • t + cj • cos(ωjt) + Zt, for t = 0, . . . , 250; j=1
where a, b, k, the cj’s, and the ωj’s are constants, and {Zt} ∼ IDD(0, σ2), with σ2 > 0. All these constants are fixed but unreported to you!
It is assumed that 0 < ωj < π for j = 1, . . . , k. The goal of the project is to estimate these frequencies as well as their total number. (Once we have estimates for these, the other constants may be estimated using standard regression techniques.) For this, it would be ideal if we could get rid of the noise {Zt}. Unfortunately, this is not possible! However, we may use a moving average filter to damp down the noise. So let 0 < l < 125, and consider:
XJ := 1 Σ X for t = l, . . . (250 − l).
1. Using that cos(α±β) = cos(α) cos(β)∓ sin(α) sin(β), show that:
k XtJ = a + b • t + uJj • cos(ωj t) + vjJ • sin(ωj t) + ZtJ,j=1
for suitable constants uJj and vjJ , and a weakly stationary mean-zero time series {ZtJ}.
2. Explain why for l large enough:
k XtJ ≈ a + b • t + uJj • cos(ωj t) + vjJ • sin(ωj t),j=1
From now on we will assume that l was chosen large enough so that the above ap- proximation is—for all practical purposes—an identity. Namely:
k XtJ = a + b • t + uJj • cos(ωj t) + vjJ • sin(ωj t) , for t = l, . . . , (250 − l). j=1
Next, to accomplish our goal, we would like to use the Discrete Fourier Transform (DFT). For this we first need to remove the linear trend in {XtJ}. To do so, consider the (causal) linear filter:
XtJJ := XtJ − XtJ 1, for t = (l + 1), . . . , (250 − l).
3.Using that cos(α − β) = cos(α) cos(β) + sin(α) sin(β) and sin(α − β) = sin(α) cos(β) −
cos(α) sin(β), show that:
k XtJJ = b + ujJJ • cos(ωj t) + vjJJ • sin(ωj t) , j=1
for suitable constants uJjJ and vjJJ. 1 2
4.If necessary do a literature search about the so-called Fast Fourier Transform (FFT). Explain— at a high level—what this algorithm does. Also, report what the following R-command re- turns: y = fft(x), when x is an n-dimensional vector.
1In practice, the precise relationship between the constants uj′′ and vj′′ and the original constants uj and vj is not relevant. What matters here is that {Xt′′} is a linear combination of sines and cosines with the same frequencies ωj’s as the original time series {Xt}.
The constant b may be thought of as associated with the frequency ω0 := 0
5.Explain how to use the fft command to determine the following version of the Discrete Fourier Transform (DFT) of x:
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