Transcript Document
Statistics 153 Review - Sept 30, 2008
Notes re the 153 Midterm on October 2, 2008
1. The class will be split into two groups by first letter of surname.
Letters A to L will take the exam in 330 Evans.
Letters M to Z will take it in 340 Evans.
2. The exam will cover material through Chapter 4.
3. There will be 2 questions, answer both.
4. The questions will be like the Assignment's, but the exam will be closed book - no books or notes allowed.
5. No questions to the Proctors about the exam content please. If unsure, make an interpretation, state it and answer that.
6. You will have 60 minutes, exactly, to work.
7. The exam itself will be handed out. There will be space on it to answer the questions.
8. The solutions to Assignment 3 will be posted in the glass case, center corridor, third floor Evans, Tuesday after class, but
the papers won't have been graded yet.
9.
I will do a review in class September 30. Suggest some topics. 3
Name:________________________
MIDTERM EXAMINATION
Statistics 153
October 2, 2008
D. R. Brillinger
Answer both questions in the space provided. Show your work.
If you are not sure of the meaning of a question, set down an interpretation, and
provide a reasonable answer.
You have exactly 60 minutes for the exam.
Question 1. Let {Zt} be a purely random process.
What is a time series?
a sequence of numbers, x, indexed by t
{xt } t 0,1,2,...
Data t 1,..., N
Stat 153 - 11 Sept 2008 D. R. Brillinger
Simple descriptive techniques
Trend
Xt = + t + t
Filtering/filters
yt = r=-qs ar xt+r
yt = k hk xt-k
p. 189
This form carries stationary into stationary
MA(q) X t 0 Z t 1Z t 1 ... q Z t q
AR(p) X t 1 X t 1 ... p X t p Z t
Sm( xt )
q
t q
xt q /(2q 1)
Filters may be in series
stationarity preserved if filter time invariant
Differencing
yt = xt - xt-1 = xt
"removes" linear trend
Seasonal variation model
Xt = mt + St + t
St St-s
12 xt = xt - xt-12 , t in months
Stationary case, autocorrelation estimate at lag k, rk
t=1N-k (xt- x)(xt+k - x )
over
t=1N (xt - x
)2
autocovariance estimate at lag k, ck
t=1N-k (xt - x )(xt+k - x ) / N
Stat 153 - 16 Sept 2008 D. R. Brillinger
Chapter 3
mean function
(t ) E[ X (t )]
variance function
(t ) Var[ X (t )]
2
autocovariance
(t , t ) Cov( X (t ), X (t )
1
2
1
2
Strictly stationary
All joint distributions unaffected by simple time shift
Second-order stationary
(t ) (0)
(t )
2
2
( ) Cov[ X (t ), X (t )]
( ) ( ) / (0), : lag
Properties of autocovariance function
( ) ( )
| ( ) | (0)
Does not identify model uniquely
Useful models and acf's
Purely random
(k ) Cov( Z t , Z t k ) Z2 , k 0
0, k / 1, / 2,...
(k ) 1, k 0
0, k 0
Building block
Random walk
X t X t 1 Z t , X 0 0
t
X t Zi
i 1
E ( X t ) t
Var ( X t ) t Z2
not stationary
X t X t X t 1 Z t ,
purely random
E(aX bY ) aE( X ) bE(Y )
Var (aX bY ) a 2Var ( X ) 2abCov( X , Y ) b 2Var (Y )
Cov(aX bY , cU dV )
acCov( X ,U ) adCov( X ,V ) bcCov(Y ,U ) bdCov(Y ,V )
*
Moving average, MA(q)
X t 0 Z t t 1Z t 1 ... q Z t q
If E ( Z t ) 0,
E( X t ) 0
(k )
0,
2
Z
k q
q k
i ik ,
i 0
k 0,1,..., q
(k )
stationary
MA(1). (k ) 1, k 0
1 /(1 12 ), k / 1
0, otherwise
From *
Backward shift operator
B j X t X t j
MA(q )
X t ( B) Z t
( 0 1 B... q B q ) Z t
( B ) 0 1 B... q B q
Linear process.
MA()
X t i Z t i
i 0
Need convergence condition Stationary
autoregressive process, AR(p)
X t 1 X t 1 ... p X t p Z t
( B) X t Z t
first-order, AR(1)
Markov
X t X t 1 Z t
X t Z t ( Z t 1 X t 2 )
Z t Z t 1 2 Z t 2 ...
*
Linear process
For convergence/stationarity root of φ(z)=0 in |z|>1
| | 1
a.c.f.
From *
(k ) |k| Z2 /(1 2 ), k 0, / 1, / 2,...
(k ) |k|
p.a.c.f.
vanishes for k>p
In general case,
( B) X t Z t
need roots of (z) 0 in | z | 1 for stationari ty
Very useful for prediction
X t 1 X t 1 ... p X t p Z t
ARMA(p,q)
( B) X t ( Z t )
X t 1 X t 1 ... p X t p 0 Z t t 1Z t 1 ... q Z t q
Roots of (z)=0 in |z|>1 for stationarity
Roots of θ(z)=0 in |z|>1 for invertibility
ARIMA(p,d,q).
d X t is stationary
Random walk
X t X t 1 (1 B) X t
X t Z t
ARIMA(0,1, 0)
Yule-Walker equations for AR(p).
Correlate, with Xt-k , each side of
X t 1 X t 1 ... p X t p Z t
(k ) 1 (k 1) ... p (k p), k 0
For AR(1)
(k ) k (0), k 0
Stat 153 - 23 Sept 2008 D. R. Brillinger
Chapter 4 - Fitting t.s. models in the time domain
sample autocovariance coefficient.
ck
N k
t 1
( xt x )( xt k x ) / N , k 0,1,2,...
Under stationarity, ...
E (ck ) (k ) as N
Cov(ck , cm ) O(1 / N )
Estimated autocorrelation coefficient
rk ck / c0
If x1 ,..., xN i.i.d and N large
E (rk ) 1 / N
Var ( rk ) 1 / N
asymptotically normal
approximat e 95% CI around 0 : 2 / N
interpretation
Estimating the mean
N
X Xt / N
t 1
E( X ) , unbiased
r
Var ( X ) [1 2 (1 ) (r )] / N
r 1
N
2
N 1
Can be bigger or less than 2/N
1
For AR (1) approx (
)/ N
1
2
Fitting an autoregressive, AR(p)
Easy. Remember regression and least squares
Yi xi i , i 1,..., n
min ( , ) [ yi xi ]2
normal equations
xi ˆ yi
2
xiˆ xi ˆ xi yi
1ˆ
X t 1 ( X t 1 ) ... p ( X t p ) Z t , Z ' s i.i.d .
S
N
t p 1
[ xt 1 ( xt 1 ) ... p ( xt p )]2
AR(1)
ˆ
N 1
t 1
( xt x )( xt 1 x ) /
Cp.
r1 c1 / c0
N 1
t 1
( xt x ) 2
Seasonal ARIMA. seasonal parameter s
SARIMA(p,d,q)(P,D,Q)s
p ( B) P ( B s )Wt q ( B)Q ( B s ) Z t
Example
(1 B)(1 B12 ) X t (1 B12 ) Z t
X t X t 12 ( X t 1 X t 13 ) Z t Z t 12
Residual analysis.
Paradigm
observation = fitted value plus residual
The parametric models have contained Zt
The residual is zˆt
AR(1).
zˆ t xt ˆxt 1
Portmanteau lack-of-fit statistic
K
Q N rz2,k
k 1
ARMA(p,q) appropriate?
Q K2 pq