Yule walker method

1

YULE WALKER METHOD

Presented By:Sarbjeet Singh

NITTTR- Chandigarh

2

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 .

3

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)

4

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

5

CONTD.

Rewriting equation

x(n)+ a(k) x(n) =[1+ a(k) ] x(n)=e(n)

x(n) =

= H(z)

H(f) =

6

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

7

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.

8

CALCULATIONS

Computation of model parameters-Yule Walker equations

9

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)

10

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)

11

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)

12

CONTD.

And so equation (4) simplifies to

Giving for kth p/m:

(5)

(6)

13

CONTD.

Writing out the LHS of equation (4) for the e.g. case of k=1,gives

14

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)

15

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)

16

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)

17

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

18

CONTD.

(11)

19

CONTD. From eqn (5),which is true for all k , it is seen

that eqn(11)

Hence eqn(11) simplifies to

20

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)

21

SOLUTION OF THE YULE WALKER EQUATIONS

The autocorrelation method The covariance methodThe modified covariance methodThe Burg method

22

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.

23

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)

24

CONTD.

The equivalent of eqn (8) is

where

(14)

(15)

25

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)

26

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)

27

THE BURG METHOD

This method relies upon aspects beyond the present scope . It produces accurate spectral estimates for AR data.

28

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.).

29

THANK YOU