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YULE WALKER METHOD

Presented By:Sarbjeet Singh

NITTTR- Chandigarh

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OVERVIEW OF MODELS

There are three types of model:

AR (auto regressive) model: a model which depends only on previous outputs of system.

MA model( moving average): model which depends only on inputs to system.

ARMA(autoregressive moving average): model based 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 as x(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

minimize the errors , e(n),for each sampled point, x(n), represented by an equation ‘1’.These errors are given by 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 n N ) ,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 to –Rxx(1).Equating the left and right sides gives

(7)

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CONTD.

For each value of k,1 ≤k≤p,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

that eqn(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 of solving eqn(8) for a(k) and the choice of the model order p, must first be described.

(12)

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SOLUTION OF THE YULE WALKER EQUATIONS

The autocorrelation method The covariance methodThe modified covariance methodThe 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 positive semi-definite .Equation (14) may be solved using the Cholensky decomposition method (Lawson & Hanson,1974 ).

Only N-p lagged components are summed , so for short 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 doesn’t 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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THANK YOU