Linear Filters

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Linear Filters
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Let

• denote a bivariate time series with zero mean.

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Suppose that the time series yt t ? T is
constructed as follows
The time series yt t ? T is said to be
constructed from xt t ? T by means of a
Linear Filter.
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The autocovariance function of the filtered series
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Thus the spectral density of the time series yt
t ? T is
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Comment A
is called the Transfer function of the linear
filter.
is called the Gain of the filter while
is called the Phase Shift of the filter.
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Also
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Thus cross spectrum of the bivariate time series
is

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• Definition

Squared Coherency function
Note
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Comment B
Squared Coherency function.
if yt t ? T is constructed from xt t ? T
by means of a linear filter
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Linear Filterswith additive noise at the output
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Let

• denote a bivariate time series with zero mean.

Suppose that the time series yt t ? T is
constructed as follows
t ..., -2, -1, 0, 1, 2, ...
The noise vt t ? T is independent of the
series xt t ? T (may be white)
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nt
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The autocovariance function of the filtered
series with added noise
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continuing
Thus the spectral density of the time series yt
t ? T is
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Also
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Thus cross spectrum of the bivariate time series
is

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Thus
Squared Coherency function.
Noise to Signal Ratio
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Box-Jenkins Parametric Modelling of a Linear
Filter
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• Consider the Linear Filter of the time series
  Xt t ? T

where
and
the Transfer function of the filter.
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• at t ? T is called the impulse response
  function of the filter since if X0 1and Xt 0
  for t ? 0, then

for t ? T
Xt
at
Linear Filter
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Also Note
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• Hence DYt and DXt are related by the same
  Linear Filter.

Definition The Linear Filter
is said to be stable if
converges for all B 1.
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Discrete Dynamic Models
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• Many physical systems whose output is represented
  by Y(t) are modeled by the following differential
  equation

Where X(t) is the forcing function.
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• If X and Y are measured at discrete times this
  equation can be replaced by

where D I-B denotes the differencing operator.
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• This equation can in turn be represented with the
  operator B.

or
where
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• This equation can also be written in the form as
  a Linear filter as

Stability It can easily be shown that this
filter is stable if the roots of d(x) 0 lie
outside the unit circle.
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Determining the Impulse Response function from
the Parameters of the Filter
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• Now

or
Hence
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• Equating coefficients results in the following
  conclusions
• aj 0 for j lt b.
• aj - d1aj-1 - d2aj-2-...- dr aj-r wj
• or aj d1aj-1 d2aj-2... dr aj-r wj
• for b j bs.
• and aj - d1aj-1 - d2aj-2-...- dr aj-r 0
• or aj d1aj-1 d2aj-2... dr aj-r for j gt bs.

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• Thus the coefficients of the transfer function,
• a0, a1, a2,... ,
• satisfy the following properties
• 1) b zeroes a0, a1, a2,..., ab-1
• 2) No pattern for the next s-r1 values
• ab, ab1, ab2,..., abs-r
• 3) The remaining values
• abs-r1, abs-r2, abs-r3,...
• follow the pattern of an rth order difference
  equation
• aj d1aj-1 d2aj-2... dr aj-r

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• Example r 1, s2, b3, d1 d
• a0 a1 a2 0
• a3 da2 w0 w0
• a4 da3 w1 dw0 w1
• a5 da4 w2
• ddw0 w1 w2
• d2w0 dw1 w2
• aj daj-1 for j 6.

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Transfer function at
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Identification of the Box-Jenkins Transfer Model
with r2
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• Recall the solution to the second order
  difference equation
• aj d1aj-1 d2aj-2
• follows the following patterns

• Mixture of exponentials if the roots of
• 1 - d1x - d2x2 0 are real.
• 2) Damped Cosine wave if the roots to
• 1 - d1x - d2x2 0 are complex.

These are the patterns of the Impulse Response
function one looks for when identifying b,r and s.
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Estimation of the Impulse Response function, aj

• (without pre-whitening).

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• Suppose that Yt t ? T and Xt t ? Tare
  weakly stationary time series satisfying the
  following equation

Also assume that Nt t ? T is a weakly
stationary "noise" time series, uncorrelated
with Xt t ? T. Then
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Suppose that for s gt M, as 0. Then a0, a1, ...
,aM can be found solving the following equations
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If the Cross autocovariance function, sXY(h), and
the Autocovariance function, sXX(h), are unknown
they can be replaced by their sample estimates
CXY(h) and CXX(h), yeilding estimates of the
impluse response function
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In matrix notation this set of linear equations
can be written
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If the Cross autocovariance function, sXY(h), and
the Autocovariance function, sXX(h), are unknown
they can be replaced by their sample estimates
CXY(h) and CXX(h), yeilding estimates of the
impluse response function
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Estimation of the Impulse Response function, aj

• (with pre-whitening).

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• Suppose that Yt t ? T and Xt t ? Tare
  weakly stationary time series satisfying the
  following equation

Also assume that Nt t ? T is a weakly
stationary "noise" time series, uncorrelated
with Xt t ? T.
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• In addition assume that Xt t ? T, the weakly
  stationary time series has been identified as an
  ARMA(p,q) series, estimated and found to satisfy
  the following equation
• b(B)Xt a(B)ut
• where ut t ? T is a white noise time series.
  Then
• a(B)-1b(B)Xt ut
• transforms the Time series Xt t ? T into the
  white noise time seriesut t ? T.

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• This process is called Pre-whitening the Input
  series.
• Applying this transformation to the Output series
  Yt t ? T yeilds

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or
where
and
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In this case the equations for the impulse
response function - a0, a1, ... ,aM - become
(assuming that for s gt M, as 0)
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Summary

• Identification and Estimation of
• Box-Jenkins transfer model

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• To identify the series we need to determine
• b, r and s.
• The first step is to compute
• the ACFs and the cross CFs
• Cxx(h) and Cxy(h)
• Estimate the impulse response function using

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The Impulse response function at

1. Determine the value of b, r and s from the
   pattern of the impulse response function

Pattern of an rth order difference equation
b
s- r 1
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• Determine preliminary estimates of the
  Box-Jenkins transfer function parameters using
• for j gt bs. .
• aj d1aj-1 d2aj-2... dr aj-r
• for b j bs
• aj d1aj-1 d2aj-2... dr aj-r wj

• Determine preliminary estimates of the ARMA
  parameters of the input time series xt

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• Determine preliminary estimates of the ARIMA
  parameters of the noise time series nt

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Maximum Likelihood estimation of the parameters
of the Box-Jenkins Transfer function model
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• The Box- Jenkins model is written

The parameters of the model are
In addition

1. the ARMA parameters of the input series xt
2. The ARIMA parameters of the noise series nt

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The model for the noise ntseries can be written
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Given starting values for yt, xt, and
and the parameters of the transfer function model
and the noise model
We can calculate successively
The maximum likelihood estimates are the values
that minimize
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Fitting a transfer function model

• Example Monthly Sales (Y) and Monthly
  Advertising expenditures

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The Data
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Using SAS

• Available in the Arts computer lab

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The Start up window for SAS
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To import data - Choose File -gt Import data
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The following window appears
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Browse for the file to be imported
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Identify the file in SAS
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The next screen (not important) click Finish
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The finishing screen
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• You can now run analysis by typing code into the
  Edit window or selecting the analysis form the
  menu
• To fit a transfer function model we need to
  identify the model
• Determine the order of differencing to achieve
  Stationarity
• Determine the value of b, r and s.

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• To determine the degree of differencing we look
  at ACFs and PACFs for various order of
  differencing

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To produce the ACF, PACF type the following
commands into the Editor window- Press Run button
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• To identify the transfer function model we need
  to estimate the impulse response function using

• For this we need the ACF of the input series and
  the cross ACF of the input with the output

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To produce the Cross correlation function type
the following commands into the Editor window
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• the impulse response function using can be
  determined using some other package (i.e. Excel)

r,s 1
b 4
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To Estimate the transfer function model type
the following commands into the Editor window
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To estimate the following model
Use input( b (w -lags ) / (d -lags) x)
In SAS
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The Output
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The Output