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  Tue Mar2 \n")
if(i == 6) cat("\n\n  Thu Mar4 \n")
if(i == 11) cat("\n\n  Tue Mar9 \n")
if(i == 16) cat("\n\n  Thu Mar11 \n")
cat(i,y[i],".\n") 
}

  Tue Mar2 
 MCGOVERN, IAN RICHARD .

  Thu Mar4 
 WANG, TIANYI .
 XU. JING .
 LI, SIXUAN .
 LIU, NAN .

  Tue Mar9 
 HE, JIE .
 WANG, KAIXIN .
 PENG, KAIYAN .
 LI, JIAYI .
 PLITHIDES, MAX .
GENG, JIALIANG .

  Thu Mar11 
 LI, XIA .
 PAN, WENLAN .
 PIAO, JUAN .
 LU, QIUJING .
SRIRAMOJU, RAJA VYSHNAVI .

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, 
including the methods used for the homeworks and also other methods we 
have discussed in class. You will write up your analysis in a 
written report, and will also make an oral presentation. 
The oral presentations will be 
8-10 minutes each in total. You will receive one 
overall grade based on the oral report and written report, combined. 
2% extra credit will be based on class participation and attendance from Thu 1/28/21 on. 

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.
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. 
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 ARIMA model. Show the residuals, 
and measures of how well it fits. 
Comment on your fitted model. 
Estimate the spectral density using the periodogram and smoothed periodogram. 
Comment on any clear cycles and the overall distribution of the variance by frequency, and its relationship to the smoothness of your time series. 

Your final project should be submitted to me in pdf 
by Mon, Mar 15, 11:59pm, by email to frederic@stat.ucla.edu.

Do not send them via ccle.
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: 8-10 minutes per oral report, plus questions at the end. 
I will cut you off if you go over 10 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: I will make everyone speaking that day a co-host, so you can just share your screen when you give your presentation. 

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 2 minutes), then show your 
results as clearly as possible, with figures preferred (roughly 5 minutes), and 
then conclude (roughly 2 minutes). 
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 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.