Time Series
Fall 2013
Lecture Notes
1
Lecture 10
6
Stationary Processes in the Frequency Domain
Derivation of Special Density Function
3 Autoregressive Process AR(1):
Xt = Xt1 + Zt ,
2
with X
=
t
z2
.
1 2
2
X
1 2
1 2 cos + 2
f ()
1 2
1
.
f () = 2 =
1 2
Time Series
Fall 2013
Lecture Notes
1
Lecture 2
2
2.4
Time Series Analysis and Forecasting
More Examples of Time Series
Los Angeles Annual Rainfall
W
.S
ha
r
ab
at
i
# The package TSA should be installed before running this code
# Annual Rainfall in Los A
Time Series
Fall 2013
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 :
Output Yt
Yiel
Time Series
Fall 2013
Lecture Notes
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 trend, qua
Time Series
Lecture Notes
Fall 2013
1
Lecture 14
10
State-Space Models and The Kalman Filter
10.1
State-Space Models
Observation = Signal + Noise.
In statistical language, this is equivalent to
Data = Fit + Residual.
Fit = Explained variation. Residual =
Time Series
Lecture Notes
Fall 2013
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 point Xt+h .
Time Series
Lecture Notes
Fall 2013
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 , Y1 ), (X2
Time Series
Fall 2013
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 write X(
Time Series
Fall 2013
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 ) + Zt .
Xt X
Time Series
Fall 2013
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, for a fu
Time Series
Fall 2013
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
ck =
.
N
Time Series
Fall 2013
Lecture Notes
1
Lecture 5
3.3
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 if the
Time Series
Fall 2013
Lecture Notes
1
Lecture 9
6
Stationary Processes in the Frequency Domain
6.1
Introduction
The autocovariance and autocorrelation functions describe the evolution of a process through time (time domain).
Spectral density function de
Time Series
Lecture Notes
Fall 2013
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, = .
= cov (Xt , X
about where current theory may fall short. The [How Bad Are
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mistakes are not found in alternative books. Evgeny Lyandres Boston
University We owe it to students to explain when shortcuts wi
world so that we can act in the world. Thus, one should prefer a theory
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When it is not, then the theoryas presently constitutedis not
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different implication). Thus, we tentatively accept the theory and
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