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YULE WALKERMETHOD
Presented By:
Sarb jeet Singh
NITTTR- Chand igarh
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OVERVIEW OF MODELS
There are three types of model:
AR (auto regressive) model: a model which dependsonly on previous outputs of system.
MA model( moving average): model which dependsonly on inputs to system.
ARMA(autoregressive moving average): modelbased on both inputs and outputs .
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AUTOREGRESSIVE MODEL & FILTER
In an AR model of a time series the current value of
the series ,x(n),is expressed as a linear function of
previous values plus an error term, e(n),thus:
x(n)=-a(1)x(n-1)-a(2)x(n-2)-. . .a(k)x(n-k)--a(p)x(n-
p)+e(n)
{p previous terms & represent a model of order p.}
Also written asx(n)=- a(k)x(n-k)+e(n)=- a(k) x(n)+e(n)
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x(n)=-a(1)x(n-1)-a(2)x(n-2)-. . .a(k)x(n-k)--
a(p)x(np)+e(n)
Fig-AR Filter
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CONTD.
Rewriting equation
x(n)+ a(k) x(n) =[1+ a(k) ] x(n)=e(n)
x(n) =
= H(z)
H(f) =
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POWER SPECTRUM DENSITY OF AR
SERIES
The power spectrum density, , of the AR series
x(n) is required. This is related to power spectrum
density of the white noise error signal , ,which
is its variance , ,by
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YULE-WALKER METHOD
The Yule-Walker Method estimates the power
spectral density (PSD) of the input using the Yule-
Walker AR method.
This method, also called the autocorrelation method,
fits an autoregressive (AR) model to the windowed
input data.
An autoregressive model depends on a limited
number of parameters, which are estimated from
measured noise data.
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CALCULATIONS
Computation of model parameters-Yule
Walker equations
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CALCULATIONS
In an AR model of a time series the current value of
the series ,x(n),is expressed as a linear function of
previous values plus an error term e(n), thus:
x(n) = -a(n)x(n-1)-a(2)x(n-2)- . . . -a(k)x(n-k)- . . .
-a(p)x(n-p)+e(n) (1)
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CONTD.
The optimum model p/ms will be those which minimizethe errors , e(n),for each sampled point, x(n),represented by an equation 1.These errors are givenby re-ordering equation 1 to
e(n) = x(n)+ a((k)x(n-k)
A measure of the total error over all samples , N(1 nN ) ,is required . The mean squared error is given by:
(3)
(2)
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CONTD.
The optimum value of each p/m is obtained by setting the partial derivative of
equation (3) w.r.t. the model p/m to zero, we have:
Now,
(4)
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CONTD.
And so equation (4) simplifies to
Giving for kth p/m:
(5)
(6)
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CONTD.
Writing out the LHS of equation (4) for the e.g. case of
k=1,gives
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CONTD.
Since in the case of autocorrelation functions Rxx(-j)
= Rxx(j), the expression may be written as
The RHS of equation (6) is equal toRxx(1).Equating
the left and right sides gives
(7)
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CONTD.
For each value of k,1 kp,a similar equation may
be written.These equations may be written in
matrix form as
(8)
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CONTD.
The model p/ms,a(k), may now be obtained from this
set of eqns which are known as Yule Walker (YW)
equations. In matrix notation eqn (8) may be writtten
Hence ,in principle,
Rxx(k-j)is symmetrical Toeplitz
(9)
(10)
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CONTD.
Equation (3) allows calculation of E , but another
expression another in terms of autocorrelation
functions and the a(k) may be found as follows.
Assuming the a(k) are real & expanding equation
(3) gives
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CONTD.
(11)
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CONTD.
From eqn (5),which is true for all k , it is seen thateqn(11)
Hence eqn(11) simplifies to
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CONTD.
So that finally
Equation (12) or (3) and the model p/ms from eqn(10)
may now be inserted in eqn of power spectrum
density Px(f) to obtain the autoregressive power
density spectrum.However , the possible ways ofsolving eqn(8) for a(k) and the choice of the model
order p, must first be described.
(12)
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SOLUTIONOFTHEYULEWALKEREQUATIONS
The autocorrelation method
The covariance method
The modified covariance method
The Burg method
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THE AUTOCORRELATION METHOD
The autocorrelation method is based upon the
mean squared error expression in eqn (3) .
The Levinson-urbin (kay,1988;Pardey ,Roberts, and
Tarassenko.1996) provides a computation efficient
way of solving the YW equations of (8) for the
model p/ms.
This method gives poorer frequency resolution
than the other to be described , and is therefore
less suitable for shorter data records.
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THE COVARIANCE METHOD
In this method the limits of summation in eqn (3)
are modified to run from n=p to n=N .
Also, the average is calculated over N-p terms
rather than N.Thus , eqn (3) becomes
(13)
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CONTD.
The equivalent of eqn (8) is
where
(14)
(15)
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CONTD.
E is given by
The p p matrix Cxx(j,k) is Hermitian and positivesemi-definite .Equation (14) may be solved using the
Cholensky decomposition method (Lawson &
Hanson,1974 ).
Only N-p lagged components are summed , so forshort data length there could be some end effects.
The covariance method results in better spectral
resolution than the autocorrelation method.
(16)
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THE MODIFIED COVARIANCE METHOD
In this method the average of the estimated forward
and backward prediction errors is minimized
.EQUATION (14) & (16) still apply, but eqn (15) is
modified to
The method doesnt guarantee a stable all pole filter
,but this usually results . It yields statistically stable
spectral estimates of high resolution.
(17)
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THE BURG METHOD
This method relies upon aspects beyond the
present scope . It produces accurate spectral
estimates for AR data.
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APPLICATIONS
A high-order Yule-Walker method for estimation of
the AR parameters of an ARMA model
Microwave multi-level band-pass filter using
discrete-time Yule-Walker method
In radar applications , the number of observations is
small (say 63 observations) and asymptotic
descriptions do not cover the estimates (better than
1st order Talyer approx.).
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THANKYOU