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Yule walker method

Yule walker method






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  • October 31, 2012
  • October 31, 2012
  • October 31, 2012
  • October 31, 2012

Yule walker method Yule walker method Presentation Transcript

  • YULE WALKER METHOD Presented By: Sarbjeet Singh NITTTR- Chandigarh1
  • OVERVIEW OF MODELSThere 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 . 2
  • 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) 3
  • 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 4
  • CONTD. Rewriting equationx(n)+ a(k) x(n) =[1+ a(k) ] x(n)=e(n)x(n) = = H(z) 5 H(f) =
  • POWER SPECTRUM DENSITY OFAR 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 6
  • 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. 7
  • CALCULATIONS Computation of model parameters- Yule Walker equations8
  • 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) 9
  • 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) (2) A measure of the total error over all samples , N(1 n N ) ,is required . The mean squared error is given by: (3) 10
  • CONTD.The optimum value of each p/m is obtained by setting the partialderivative of equation (3) w.r.t. the model p/m to zero, we have: (4) Now, 11
  • CONTD. And so equation (4) simplifies to (5) Giving for kth p/m: (6) 12
  • CONTD. Writing out the LHS of equation (4) for the e.g. case of k=1,gives 13
  • 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) 14
  • 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) 15
  • 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 (9) Hence ,in principle, (10) Rxx(k-j)is symmetrical → Toeplitz 16
  • 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 17
  • CONTD. (11) 18
  • CONTD. From eqn (5),which is true for all k , it is seen that eqn(11) Hence eqn(11) simplifies to 19
  • CONTD. So that finally (12) 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. 20
  • SOLUTION OF THE YULE WALKER EQUATIONS The autocorrelation method The covariance method The modified covariance method The Burg method21
  • THE AUTOCORRELATIONMETHOD 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. 22
  • 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) 23
  • CONTD. The equivalent of eqn (8) is (14) where (15) 24
  • CONTD. E is given by (16) 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. 25
  • THE MODIFIED COVARIANCEMETHOD 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 (17) The method doesn’t guarantee a stable all – pole filter ,but this usually results . It yields statistically stable spectral estimates of high resolution. 26
  • THE BURG METHOD This method relies upon aspects beyond the present scope . It produces accurate spectral estimates for AR data. 27
  • 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.). 28
  • THANK YOU 29