1. Project order. x = scan("roster.txt",what="CHAR",sep="\n") sample(x) n = length(x) y = sample(x) for(i in 1:n){ if(i == 1) cat("\n\n Mon Nov21 \n") if(i == 11) cat("\n\n Wed Nov23 \n") if(i == 21) cat("\n\n Mon Nov28 \n") if(i == 31) cat("\n\n Mon Nov30 \n") cat(i,y[i],".\n") } Mon Nov21 1 WANG, ANDREA . 2 TEOH, KASEN . 3 SOUDA, NAVIN VARADARAJ . 4 PARAB, SHARDUL SHAILENDRA . 5 KUMAR, VAIBHAV . 6 ALSAFFAR, FADHEL . 7 ESCANDON VANEGAS, HECTOR . 8 FENG, JUSTIN ADAM . 9 COELHO, ABIGAIL . 10 MASHKARIA, SATVIK MEHUL . Wed Nov23 11 MOHAMMED, SHAKO AHMED MOHAMMED . 12 AGARWAL, AYUSHI . 13 NGUEN, CHUNG KYONG . -- not enrolled. 14 LAM, SHING HON . 15 SWOVELAND, JACOB MICHAEL . 16 TOLEDO LUNA, JOSE RODRIGO . 17 GOEL, SHASHANK . 18 KIM, NICKLAUS JUN . 19 WANG, MATTHEW XUEHAN . 20 YEHIA, NOUR . -- not enrolled. Mon Nov28 21 MCEVOY, KYLE ROBERTS . 22 SINGH, GAURAV . 23 KIM, DOEUN . 24 REN, ZHICHENG . 25 DUDEJA, GAUTAM . 26 ZHOU, CARTLAND . 27 WONG, EMILY FRANCES . 28 JAIN, PRANJAL . 29 STRAUS, GRAHAM PALEY . Wed Nov30 30 SAHU, DIPTI RANJAN . 31 ZHOU, ZHONGYI . 32 CHU, RAYMOND . 33 PATEL, PARTH . 34 SHAH, PARTH. 35 DEV, AISHWARYA . 36 WANG, JEREMY . 37 BAIERL, JOHN DUCHATEAU . 38 VO, DAVIS MINH . -- Attendance is mandatory at your own day plus at least 2 of the 3 other days. -- All oral projects will be on zoom! Join Zoom Meeting https://ucla.zoom.us/j/91509411456?pwd=aXNUMmhYRElBUzljeXdPMHExSkljZz09 Meeting ID: 915 0941 1456 Password: 235711. 2. Notes about projects. For your projects, you will find some time series data of your own, and will analyze the data using the methods we have talked about in class. You will write up your analysis in a written report, and will also make an oral presentation. The oral presentations will be 5-6 minutes each in total. You will receive one overall grade based on the oral report and written report, combined. Your dataset, which you will find yourselves, on the web, can be anything you choose, but it should be: a) univariate time series data. You should have at the very least n=100 observations. Between 100 and 5000 observations is best. b) something of genuine interest to you, and where you have more knowledge than an average person. Analyze the data using the methods we have talked about in class. If you have data problems, do whatever is convenient to clean up your data, but acknowledge them in conclusion. Estimate the sample acf and pacf. Remove the mean and any obvious trends and cycles. Then refit the sample acf and pacf, and fit an ARMA model. Show the residuals, and measures of how well it fits. Comment on your fitted model. Forecast with that ARMA? Interpret the coefficients of your fitted ARMA model. Are there outliers or features in your dataset, that are influencing these estimates? What do the ARMA fits and ACF and pacf telling you about your data? Smoothness? Predictability? Estimate the spectrum (non-parametrically and/or parametrically), and interpret the results. Are there significant cycles? Can you interpret the estimated spectrum in terms of how rough data are? Variation at different frequencies. Are there cycles you can remove and then re-estimate the spectrum? Comment on any clear cycles and the overall distribution of the variance by frequency, and its relationship to the smoothness of your time series. Try to forecast using your model. Comment on these forecasts. Your final project should be submitted to me in pdf by Dec9, 11:59pm, by email to frederic@stat.ucla.edu. They are all due the same date, regardless when your oral presentation is. Final project and oral report tips. For the oral projects. Rule 1: 5-6 minutes per oral report, plus questions at the end. I will cut you off if you go over 6 min. You can have someone in the audience help you with the time if you want. Rule 2: Everyone must be respectful and quiet during other people's talks. You can ask clarifying questions but keep deep questions until the end. Rule 3: Don't look at me! Rule 4: Speak very slowly in the beginning. Give us a sense of your data. Assume that the listener knows what the statistical methods you are using are, but knows nothing about the subject matter. Tell us what the methods say about your data. Emphasize the results more than the methods. Make sure to go slowly and clearly in the start so that the listener really understands what your data are. Rule 5: Speculate and generalize but use careful language. Say "It seems" or "appears" rather than "is" when it comes to speculative statements or models. For example, you might say "The data appear to be uncorrelated" or "an AR model seems to fit well" but not "The data are white noise" or "The data come from an AR model". Rule 6: Start with an introduction explaining what your data are, how you got them, and why they are interesting (roughly 1-2 minutes), then show your results as clearly as possible, with figures preferred (roughly 3 minutes), and then conclude (roughly 1 minute). In your conclusion, mention the limitations of your analysis and speculate about what might make a future analysis better, if you had infinite time. This might include collecting more data, or getting data on more variables, as well as more sophisticated statistical methods. For your written reports, rules 4-6 apply again. Have just the 4-6 pages of text in the beginning, and then the figures and your code at the end. Email your pdf document to me, at frederic@stat.ucla.edu . 3. Causal and invertible ARMAs. a. Xt − Xt−1 = Wt − 1/2 Wt−1 − 1/2 Wt−2. Identify the ARMA(p,q) model, and say if it is causal or invertible. The AR polynomial is φ(z) = 1 − z, which has root 1. The MA polynomial is θ(z) = 1 − z/2 − z^2/2, which has roots [-b+/-√(b^2-4ac)]/2a = [1/2 +/- √(1/4 + 2)]/-1 = -1/2 +/- 1.5, or −2 and 1. Since these polynomials share a common root, they have the common factor 1 − z. Factoring these out, the irredundant representation has AR polynomial φ(z) = 1, which has no roots, so it is causal, and MA polynomial θ(z) = 1 + z/2, which has root −2, so it is invertible. This is a causal and invertible ARMA(0, 1) process. The reduced process is Xt = Wt + 1/2 Wt-1. Xt - Xt-1 = Wt + 1/2 Wt-1 - Wt-1 - 1/2 Wt-2 = Wt - 1/2 Wt-1 - 1/2 Wt-2. b. Xt − 2Xt−1 + 2Xt−2 = Wt − 8/9 Wt−1. The AR polynomial is φ(z)=1−2z+2z^2, which has roots [2+/-√(4-8)]/4, or 1/2 ± i/2. These roots are inside the unit circle because 1/2^2 + 1/2^2 = 1/2 < 1. The MA polynomial is θ(z) = 1 − 8z/9, which has root 9/8. So this is an ARMA(2, 1) process which is invertible but not causal.