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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

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