Solutions to HW 5
Exercise 5.1
X t +1 = Zt +1 + Zt ,
X t +h = Zt +h + Zt +h1
X t +1 = E ( X t +1 | X t , X t 1,.) = Zt
X t +h = E ( X t +h | X t , X t 1,.) = 0, h 2
et+1 = X t+1 X t+1 = Zt+1
Var(et+h ) = 2
(
)
et+h = X t+h X t+h = Zt+h + Zt+h1 Var(et+h )
STAT 420 Introduction to Time Series
Homework 1
1. Read Chapter 1 in the book, and answer the following question:
(a) Is the Marketing time series on page 3 continuous or discrete? give your reason.
Answer: Discrete. The observations are only taken every
STAT 350 (Fall 2016)
Homework 1
Name: xiaochuan chen
Professor: Rosario MONTER
Section:020
Time:11:30
1
There are two types of problems that I have included which you do not need to turn in.
1)
Before each problem (or set of problems) are listed some Prac
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 16: Birth-Death Processes
1
Birth-Death
1.1
Continuous-Time Stochastic Process
A Continuous-Time Markov Process
Two types of the state transitions are of only two types: births which increase
the sta
STAT 420 Advanced Statistical Methodology
Spring 2017
Lecture R
Introduction to R
0-0
STAT 420
Lecture R Introduction to R
Spring 2017
Outline
R and RStudio
The S Language
S Objects
Working Directory
Data Import/Export
Graphics
Dabao Zhang
1
Page 1
STAT 420
H10
X1, . , X t ,
predict the value it will assume at some specific future time point, X t + h .
The prediction problem: from the observed values of a time series at past points,
X t +h ,
We refer to X
In forecasting
t is called the forecast orig
STAT 420
H7
Given a set of observations
cfw_ X1,., X n from a stationary time series, the ACVF is estimated
by the sample autocovariance function defined as
h =
1 nh
( X X n )( X t X n ) ,
n t=1 t+h
1 n
where X n = X n .
n t =1
This also leads to estim
STAT 420
H8
HW3 (due Feb. 11)
Problem 1
Suppose that in a sample of size 100, you obtain
1 = 0.432
and
2 = 0.145 .
Assuming that
the data were generated from an MA(1) model, construct approximate 95% CIs for
Based on these two CIs, are the data consisten
STAT 420
H9
Trend stationary models
In practice, most time series are non-stationary. Real TS data often exhibit time trend and/or cyclic features
that are beyond the capacity of stationary ARMA models. Several examples we have considered came
from clearl
Solutions to HW 9
Problem 1
Xt
Let X t = X t1 , then the observation equation is X t = 1,0,0 X t , and the state equation is
X
t2
0.7 0.5 0.4
1
Xt = 1
0
0 X t + 0 Zt .
0
0
1
0
Problem 2
varve=read.table("data/varve.dat")
v=ts(varve)
lvarve =
STAT 350 (Fall 2016)
Homework 11 (20 points + 1 point BONUS)
Name:xiaochuan chen
Instructor: Rosario MONTER
Section:020
Practice Problems: 12.5 (p. 588), 12.9 (p.588)
(4 pts.) 1. For each of the following graphs, identify the form, direction (if possible)
STAT 350 (Fall 2016)
Homework 2 (20 points + 2 points BONUS)
1
Practice Problems: 3.1 (p.83), 3.33 (p.94), 3.37 (p.94), 3.71 (p.107), 3.73 (p.107), 3.99 (p.115)
(1.5 pts.) 1. The figures below display three density curves, each with three points marked on
STAT 420
Cheng Li
Homework 8
7.1
7.2
1
A cos
sin
T
N
T
A A 0
0
1
cos N
sin N
cos 2
sin 2
0
N
2
0
1
0
0
N
2
1
0
0
N
1
2
AT A 0
0
N
2
0
0
N
1
N
2
1
AT A AT cos
N
2
sin
N
1
N
2
cos 2
N
2
sin 2
N
1 N
Xt
N t 1
N
1
2
AT A AT X X t cos t
N
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 13
9
Linear Systems
9.1
Linear Systems in the Time Domain
Identify a model for the input and output.
Input Xt
Example 1. Xt :
reactor.
Physical System
Temperature at which reactor is kept. Yt :
Outpu
STAT 420 Final Exam
Wed, May 4, 2011
Time: 120 minutes
Name:
Section:
Materials permitted:
Two pages of cheat sheets with double sides and calculators are allowed. BUT, books,
notes, laptop computers, phones, or any devices capable of wireless communicati
/" h ?
,2
ts, Make
able.
1- [10 points] Please indicate True (TV or False (F" for the following stateme
sure that your hand-writing Of Tl. and Fl! in the parentheses are differenti
( F 3-1'ainfeli time series over 0mm is a continuous time series.
(T ) Fil
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 3
3
Probability Models for Time Series
3.1
Stochastic Processes and Their Properties
Stochastic (Random) Process
For each t, Xt is treated as a value of the random variable Xt , 0 t T .
Sometimes we
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 6
4
Fitting Time Series Models In The Time Domain
4.1
Estimating Autocovariance and Autocorrelation Functions
Estimating Autocovariance and Autocorrelation Functions
Recall,
N
k
X
t )(Xt+k X
t)
(Xt X
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 5
4
Invertible and Stationary Processes
Stationarity of AR(p)
AR(p)
Xt = 1 Xt1 + + p Xtp + Zt .
Xt = (1 B + + p B p ) Xt + Zt .
Let (B) = 1 1 B p B p . Then,
(B) Xt = Zt .
AR(p) process is stationary
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 14
10
State-Space Models and The Kalman Filter
10.1
State-Space Models
State-Space Models
Observation = Signal + Noise.
In statistical language, this is equivalent to
Data = Fit + Residual.
Fit = Expl
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 4
3
Probability Models for Time Series
3.3
Properties of The Autocorrelation Function
For the stationary stochastic process X(t) or Xt we have
=
= 2.
0
1. 0 = 1.
2. Covariance is symmetric, = .
= co
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 8
5
Forecasting
5.1
Introduction
Introduction
Forecasting: The prediction problem from the observed values of a time series at
past points X1 , , Xt predict the value at some specific future time poi
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 11
6
Spectral Analysis
6.1
Methods for Estimating the Spectrum
Fourier Analysis
The approximation of a function by taking sum of sine and cosine terms, is called
the Fourier series representation and,
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 12
8
Bivariate Processes
8.1
Cross-Covariance and Cross-Correlation
Consider (Xt , Yt ), where Xt and Yt are maximum/minimum daily temperature
respectively.
Observe the bivariate time series (X1 , Y
HW1
Xiaochuan chen 00274000201
2.2
a.
code:
cfw_r 1
points <-ts(data = c(1.6,0.8,1.2,0.5,0.9,1.1,1.1,0.6,1.5,0.8,0.9,1.2,0.5,1.3,0.8,1.2),start = 1,end =16)
plot(points)
b.
from the graph we could get nothing reasonable guess on r1.
c.
xt<-c(points[seq(1,
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 1
1
Time Series Analysis and Forecasting
1.1
Introduction to Time Series
1.2
Learning Objectives
Learn about different time-series forecasting models: moving averages, exponential smoothing, linear t
Time Series Analysis
Fall 2016
Lecture Notes
1
Lecture 7
4
Fitting Time Series Models In The Time Domain
4.3
Fitting an Autoregressive Process
Estimating Parameters of an AR Process
1 Ordinary Regression Model. (
= X)
) + + p (Xtp X
= 1 (Xt1 X
t ) + Z
Time Series Analysis
Lecture Notes
Fall 2016
1
Lecture 2
2
2.4
Time Series Analysis and Forecasting
More Examples of Time Series
Los Angeles Annual Rainfall
# The package TSA should be installed before running this code
# Annual Rainfall in Los Angeles
#