Adaptive Estimation of Time-Varying Parameters in AR Models with Variable Forgetting Factor

Goran Kvaščev, Željko Đurović, Branko Kovačević Signals and Systems Department

University of Belgrade, School of Electrical Engineering Belgrade, Serbia

kvascev@etf.bg.ac.rs

Ivana Kostić Kovačević Department of Informatics and Computing

Singidunum University Belgrade, Serbia

Abstract— A new method for estimating time-varying parameters of nonstationary AR signal models, based on adaptive recursive least squares with variable forgetting factors, is described. The adaptive estimator differs from the conventional one by the simultaneously estimation of AR model parameters and scale factor of prediction residuals, while the variable forgetting factor values are adapted to the nonstationary signal via a new extended prediction error detection scheme. The method has good adaptability in the non-stationary situations, and gives low bias and low variance at the stationary situations. The feasibility of the approach is demonstrated with simulations.

Keywords— nonstationary signals, parameter estimation, adaptive estimation, variable forgetting factor.

I. INTRODUCTION The problem of modeling and processing the time-discrete

signals, described as quasi-stationary stochastic processes with time-varying or abruptly changing parameters, arises in various fields, such as speech and biomedical signal processing, image analysis, failure detection in measurement and control, etc [1-5]. In these cases, a general model structure can be used, in which parameters must be all estimated from measurements using an estimation procedure. The important step is data acquisition, the actual collection of the data, needed for parameter identification. Subsequently, the model representation must be chosen, and different model representations can be adopted. Depending on the model representation chosen, an identification method is selected. This method will yield the estimated parameter values of the model.

A variety of parameter identification algorithms have been developed, based primarily of one of the two aspects of underlying random variables: the distribution and the moments [6-9]. When the distribution of the random input signals is known, the maximum likelihood (ML) criterion can be applied to yield asymptotically optimal performance [7-9]. In general, the ML criterion is non-linear, and a set of nonlinear equations resulting from the optimality condition has to be solved iteratively. Furthermore, this procedure can be represented in a more suitable recursive form by a convenient approximations. This, in turn, results in the recursive ML parameter estimation algorithm of either a pseudo-linear type of a prediction error type [8,9]. In addition, for the quadratic ML criterion which is optimal for the normal distribution of the measurement noise,

the passed problem results in a recursive least-squares (RLS) parameter estimation algorithm [5-9]. Finally, more recently, a large number of parameter identification, has been developed using moment estimation, particularly second-order statistics (SOS) [10]. Usually, based on SOS estimation, the subspace method and moment matching are applied. However, the subspace technique is suitable only for stationary situations, since it requires the signal subspace to be time-invariant.

The essential common problem in time-varying parameter identification is localization of the boundaries of stationary parts of a measured signal, or equivalently detection of the instants of abrupt changes of the signal stationarity [8]. There are many results which theoretically treat the change detection problem in different uncertainty conditions, assuming different stochastic models of a signal [11-13]. Particularly, the classical auto-recursive (AR) modeling with variable forgetting factors (VFF) is commonly used [14-16]. Furthermore, in classical AR analysis the linear prediction (LP) parameters of the signal model are determined by RLS method. Thus, a strategy of choosing VFF has been recognized as one of the most important steps in a parameter estimation procedure for time-varying signal analysis. In addition, the concept of VFF also was introduced in adaptive control to avoid a 'blowing-up' of the covariance matrix of the estimates and subsequent unstable control [5]. Therefore, RLS algorithm with VFF represents the most interesting and powerful technique that implements adaptive parameter identification [5-9].

The most frequently, FF is chosen in a heuristic way, based on the fact that its small value corresponds to small memory of whole estimation scheme, and vice versa [8,9]. Therefore, for the stationary or near-stationary parts of signal, the FF should be near unity, allowing the adaptive algorithm to use most of the previous information in the signal, and for the non-stationary or transient part of signal, a small FF will shortening the effective memory length of the estimation process. Generally, we should estimate the degree of non-stationarity of signal to calculate the next value of FF. A different adaptation schemes have been proposed by varying the memory length of signal [11-16].

In this paper, a new VFF generating procedure is proposed, which is based on calculating of a suitable chosen discrimination function, representing the ratio between the extended prediction error on the data frame of a proper length

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and the noise variance. Since the noise variance is not exactly known in practice, a new adaptive RLS procedure for simultaneously estimation of AR parameters and scale factor of prediction residuals has been proposed. Finally, the value of VFF is calculating by passing the corresponding value of discrimination function through a suitably chosen polynomial approximation. To demonstrate the feasibility of the proposed approach, it has been applied to the test signal within the controlled experiment. The order and coefficients of the polynomial are determined by appropriate optimization procedure in the preprocessing phase.

II. PROBLEM FORMULATION A non-stationary stochastic process {𝑦(𝑘)} can be

represented by the following autoregressive (AR) signal model [1,7,8]

𝑦(𝑘) = −�Θ𝑖(𝑘)𝑦(𝑘 − 𝑖)𝑝

𝑖=1

+ 𝑢(𝑘) (1)

where the random input or excitation {𝑢(𝑘)} is a zero-mean stationary white noise process, and {Θ𝑖(𝑘), 𝑘 = 1, . . . , 𝑝} are time-varying parameters. It will be assumed that the constant p, the so-called order of AR process, is known a priori. Under local stationary conditions, the difference equation (1) can be rewritten in the form of ordinary linear regression [8,9]

𝑦(𝑘) = 𝑍𝑇(𝑘)Θ + 𝑢(𝑘) (2)

where regression vector

𝑍𝑇(𝑘) = [−𝑦(𝑘 − 1) . . .−𝑦(𝑘 − 𝑝)] (3) and the vector of AR parameters

Θ𝑇 = �Θ1 ⋯ Θ𝑝�. (4) The weighted recursive least-squares (WRLS) approach

requires finding the set of parameters {Θ𝑘} such that the cumulative squared error measure {𝐽𝑘(Θ)} is minimized, where [5-9]

𝐽𝑘(Θ) =1𝑘�𝜆𝑘−𝑖𝜀2(𝑖,Θ)

𝑘

𝑖=0 (5)

and the prediction error or measurement residual

𝜀(𝑘,Θ) = 𝑦(𝑘) − 𝑍𝑇(𝑘)Θ (6) The forgetting factor (FF), 𝜆 (0 < 𝜆 ≤ 1), is a data

weighting factor that may be used to weight recent data more heavily in the WRLS computation, thus permitting tracking of time-varying signal parameters. Although 𝜆 is very important in the application of WRLS to non-stationary data, it is often set to 1 if it is known beforehand that the data set is stationary. Progressively smaller values of 𝜆 result in the parameter estimates Θ�(𝑘) being computed with effectively smaller windows of data, that is beneficial in non-stationary situations. The update recursion for the WRLS prediction filter Θ�(𝑘) is given by [8,9]

Θ�(𝑘) = Θ�(𝑘 − 1) + 𝐾(𝑘)𝜀 �𝑘,Θ�(𝑘 − 1)� (7)

𝐾(𝑘) = 𝑃(𝑘 − 1)𝑍(𝑘)[𝜆 + 𝑍𝑇(𝑘)𝑃(𝑘 − 1)𝑍(𝑘)]−1 (8)

𝑃(𝑘) =1𝜆

[𝑃(𝑘 − 1) − 𝐾(𝑘)𝑍𝑇(𝑘)𝑃(𝑘 − 1)] (9)

where 𝑍(𝑘) and 𝜀(𝑘,Θ) are defined by (3) and (6), respectively. However, WRLS algorithm may have several sources of errors. The first of these comes from noise, e.g. measurement noise. This effect can be represented by adding a noise term to the output signal of AR process, yielding

[ ] [ ] [ ]z k y k s k= + (10)

where [ ]s k is zero-mean white noise with the corresponding variance, producing the desired signal-to-noise ratio (SNR).

[ ]2

10 210log y

s

SNR dBσσ

= (11)

III. ADAPTIVE WRLS ALGORITHM WITH VFF The WRLS parameter estimate 𝜃� can be obtained from (3)

and (6-9), where the output signal of AR model in (2) is replaced by measurement signal in (10). One important part of the WRLS computation is an estimate of the scaling factor. Namely, the standard deviation 𝜎𝑦 in (11) is not exactly known in practice, and it has to be approximated by appropriate estimate, known as the scaling factor. A popular robust estimate of this factor is the median of absolute median of deviations [17]. However, WRLS approach using this estimate is found to be sensitive to abrupt noise level changes [18]. Therefore, we propose here the scaling factor to be estimated simultaneously with the signal model parameters 𝜃. Namely, if �̅�(𝑢) is the pdf of zero-mean Gaussian white noise 𝑢(·) in (2), with unit variance, then the pdf of the noise with some variance 𝜎2 is given by 𝑝(𝑢) = �̅�(𝑢/𝜎)/𝜎. Thus, one can define the conditional ML criterion

( ) ( ) ( ) ( ),/ / ; ln

kJ E F F u p u

εσ

σ

Θ Θ = Θ = − (12)

where 𝜀(·) is the measurement residual or prediction error in (6), as a generator for a class of Newton's stochastic gradient type algorithms [8,9]

𝜎�(𝑘) = 𝜎�(𝑘 − 1) − �𝑘𝜕2𝐽(𝜎�(𝑘 − 1)/𝜃�(𝑘 − 1)

𝜕𝜎2�−1

× �𝑘𝜕𝐽(𝜎�(𝑘 − 1)/𝜃�(𝑘 − 1)

𝜕𝜎�

(13)

where 𝜃�(𝑘) and 𝜎�(𝑘) are the corresponding estimates at time instance 𝑘 of 𝜃 and σ , respectively.

Let us consider, in addition to the average loss (12), the empirical average loss

( )( )1

1

ˆ,ˆ /

ˆ

k

ki

kJ k F

εσ

σ−

=

Θ Θ =

∑ (14)

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Under certain condition with 𝑘 growing 𝐽𝑘(⋅) in (14) converges to 𝐽(⋅) in (12) [9]. Moreover, since 𝑝(𝑢) =�̅�(𝑢/𝜎)/𝜎, one obtains from (12)

( ) ( )ln /F u uσ ϕ σ= + (15)

where 𝜑(𝑢/𝜎) = − ln �̅�(𝑢/𝜎). In addition, with large 𝑘 and by virtue approximate truth of the optimal conditions, yielding 𝐽𝑘−1(𝜎�/𝜃�) ≈ 0, one obtains from (14) and (15)

( ) ( ) ( ) ( )

( )

1

ˆ,ˆ ˆˆ ˆ ˆ/ ln 1 /

ˆ

ˆ,ˆln

ˆ

k k

kkJ k J

k

εσ σ σ ϕ

σ

εσ ϕ

σ

−

Θ Θ = + − Θ + ≈

Θ ≈ +

(16)

and

( ) ( ) ( )2

ˆ ˆˆ / ,,1ˆ ˆ ˆ ˆ

kJ kkk

σ εεψ

σ σ σ σ

∂ Θ ΘΘ ≈ − ∂

(17)

where 𝜓(⋅) = 𝜑′(⋅). Furthermore, one can also write from (12)

( ) ( ) ( )2

ˆ ˆ ˆ/ , ,1J k kE

σ ε εψ

σ σ σ σ

∂ Θ Θ Θ = − ∂

(18)

( ) ( ) ( )

( ) ( )

2 2

2 2 4

2

ˆ ˆ ˆ/ , ,1 '

ˆ ˆ, ,2

J k kE

k kE

σ ε εψ

σ σ σ σ

ε εψ

σ σ σ

∂ Θ Θ Θ = − + + ∂ Θ Θ +

(19)

In the vicinity of the optimal solution, one concludes

( )ˆ/ / 0J σ σ∂ Θ ∂ = in (18), from which it follows

( ) ( )2

ˆ ˆ, , 1k kE

ε εψ

σ σ σ

Θ Θ =

(20)

and by substituting (20) into (19), one concludes further

( ) ( )2 2

2 2 2

ˆ ˆˆ / /

ˆkJ J ak kσ σ

σ σ σ

∂ Θ ∂ Θ≈ =

∂ ∂ (21)

where:

( ) ( )2

2

ˆ ˆ, ,1 ' 1

k ka E

ε εψ

σ σ

Θ Θ = + ≈

(22)

Finally, if one substitute (17), (21) and (22) into the Newton's stochastic algorithm (13), one obtains the recursive relation

𝑘𝜎�(𝑘) = (𝑘 − 1)𝜎�(𝑘 − 1) + 𝜀2 �𝑘,𝜃�(𝑘 − 1)� (23)

As mentioned before, the forgetting factor (FF) 𝜆 in (8) and (9), belonging to the interval (0,1), has to weight the recent data more heavily, and thus permit tracking of time-varying parameters. With a value of 𝜆 close to unity, it takes relatively long time to estimate the correct parameters, but eventually the parameters are estimated with high accuracy when the signal experiences stationary. The speed of adaptation is determined by the asymptotic memory length [11]

𝑁 =1

1 − 𝜆 (24)

which implies that the information dies with 𝑁. Thus, progressively smaller values of 𝜆 results in parameter estimation being computed with effectively smaller windows of data that are beneficial in non-stationary situations.

If a signal is composed of sub-signals with different memory length (24), varying between a minimum memory length 𝑁𝑚𝑖𝑛 and a maximum one 𝑁𝑚𝑎𝑥 the time-varying signal parameters can be estimated by using (3,6-9), assigning the corresponding 𝜆 varying between 𝜆𝑚𝑖𝑛 and 𝜆𝑚𝑎𝑥 to each sub-signal. However, it is not realistic in practice to know the memory lengths and starting points of the sub-signals beforehand. Thus, we should estimate the degree of non-stationarity of the signal to calculate the next value of 𝜆. Although different adaptation schemes has been proposed by varying the memory length of a signal, the method based on the extended prediction error (EPE) criterion is emphasized, since it involves rather easy computation, and has good adaptability in non-stationary situations, and a low variance in the stationary situation [11]. Particularly, the extended prediction error is defined by

𝐸(𝑘) =1𝑀� 𝜀2 �𝑘 − 𝑖,𝜃�(𝑘 − 𝑖 − 1)�𝑀−1

𝑖=0

(25)

where 𝜀(·) is the prediction error or residual in (6), and 𝑀 is a free parameter which has to be adapted.

The quantity 𝐸(𝑘) in (25) represents a measure of the local variance of prediction residual at the given data frame of size M, and it contains the information about the degree of data nonstationarity. The value of M represents the trade-off between the estimation accuracy and tracking capability of time-varying parameters. On the other hand, the total noise variance estimate in (23) is rather insensitive to the non-stationary effects in signal. Therefore, in order to make the estimation procedure invariant to noise level, one can define the normalized measure of nonstationarity, so called discrimination function

[ ] [ ][ ]2ˆ

E kQ k

kσ= (26)

A strategy for choosing the VFF at current time instant, k, may now be defined by using the polynomial approximation

[ ] [ ]0

pni

ii

k Q kλ λ=

= ∑ (27)

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The polynomial order, 𝑛𝑝, and coefficients , 1,...,i pi nλ = can be determined by using an appropriate optimization within the controlled experiment, where the true time-varying parameters are exactly know. This step is not a part of the adaptive WRLS algorithm with VFF. Namely, if one defines the desired time-varying parameter trajectory, [ ], 1,...,i i NΘ =then the measure of the parameter estimation quality can be represented by the average loss

( ) ( )2

1

1 ˆN

iJ i i

N =

= Θ −Θ∑ (28)

where ( )ˆ iΘ is the WRLS estimate in (3), (6-9). Thus, for the given SNR in (11), one can find the optimal sequence

[ ], 1,...,opt i i Nλ = by minimizing the performance index (28). To perform this task, one has to apply an appropriate optimization procedure, such is Nelder-Mead gradient type algorithm [19]. Of course, this is time consuming and rather complex technical task for large N, since one has to optimize the criterion function in (28) with respect to huge number of N variables. Therefore, we propose to simplify the optimization process by dividing the whole data set of size N to R subsets of the equal size L. This means that at each data subset of size L one has to apply the Nelder-Mead optimization procedure to minimize the criterion function in (28), where N is replaced by L N<< . As a consequence, one generates the suboptimal sequence of FF [ ], 1,...,subopt i i Nλ = . Finally, one can approximate the obtained suboptimal FF sequence by the polynomial (27). The polynomial order pn has to be adopted in advance, while the coefficients of the polynomial should be calculated by using nonrecursive least-squares method [5-9]:

( )0

11 * * *

p

T Tsub

n

Q Q Q

λλ

λ

−

Λ = = Λ

(29)

where

[ ] [ ][ ] [ ]

[ ] [ ]

*

1 1 1

1 2 2

1

p

p

p

n

n

n

Q Q

Q QQ

Q N Q N

=

,

[ ][ ]

[ ]

1

2subopt

suboptsub

subopt N

λ

λ

λ

Λ =

(30)

Finally, the whole proposed adaptive WRLS estimation with VFF is defined by (3),(6)-(9) and (23), (25)-(27). The figure of merit of the proposed approach will be given on the basis of simulations in the next section.

IV. EXPERIMENTAL ANALYSIS Efficiency of the proposed WRLS method with VFF is

evaluated through the analysis of test signals. The test signal is generated by the second-order AR model in (2):

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ]( ) [ ]

1 2

1 2

1 2

2cos 2 ; 0.9

y k a k y k a k y k u k

a k f k a kπ

+ − + − =

= − = (31)

The time-varying parameter 𝑓[𝑘] in (31) is presented in Fig. 1. In the first experiment, we used the fixed forgetting factor 𝜆[𝑘] = 𝜆 from the interval [0.5, 1]. Then we generated measurement noise with different SNR values in (11) and applied WRLS algorithm (3), (6-9) to compute the performance criterion (28). The obtained results are presented in Fig.2.

These results have shown that small changes in FF produce significant change in the estimation quality. In addition, the optimal FF depends on SNR. Finally, the optimal value of the performance index is pretty large, indicating a moderate accuracy of the WRLS estimation procedure with fixed FF.

Fig. 1. Time-varying changes of variable 𝑓[𝑘] defining 𝑎1 parameter trajectory

Fig. 2. The performance index J (27) for different SNR and different 𝜆 values

Therefore, in the next experiment we used VFF in WRLS algorithm (3),(6)-(9). In this experiment, we applied a

0 500 1000 1500 2000 25000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

k

f[k]

0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

λ

J(λ)

SNR=10dbSNR=15dbSNR=20dbSNR=25dbSNR=30dbSNR=35db

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suboptimal sequence of FF obtained by dividing the whole data set of length 2500N = into 250R = subsets of the length

10L = . Measurement noise was generated so that 25dBSNR = . Then, we found the optimal value of FF for the

each of these data frames by optimizing the criterion (28) where N is replaced by L. The obtained sequence of suboptimal VFF is depicted in Fig. 3.

Fig. 3. Time sequence of suboptimal VFF

It has been seen that the so-generated suboptimal FF is not enough smooth and therefore we filtered this sequence by the moving-average filter of the fifth order. The filtered sequence is presented in Fig. 4, together with the estimated 1a parameter trajectory.

Fig. 4. True and estimated 𝑎1 parameter trajectory (a), Filtered suboptimal VFF sequence (b)

Finally, in the third experiment we applied the adaptive VFF in (23), (25)-(27). The obtained results are presented in Figs. 5-8. Here, 60, 2,pM n= = and 25dBSNR = is used. Obviously, the normalized indicator of nonstationarity in (26) detects nonstationary effects within the signal properly. Moreover, the increase of this discrimination function leads to

the decrease of VFF. Whenever the signal experiences abrupt changes, the VFF drops to the low values. As soon as the estimated parameter value approaches the neighborhood of the true one, the reduced value of the discrimination function changes the VFF close to unity. These experimental results also show that the proposed algorithm produces pretty small estimate bias and reasonable estimated variability over time. Moreover, the time delay between the true value and the estimated value comes from the fact that the algorithm needs a few steps to reflect the effect of the changed VFF.

Fig. 5. Discrimination function (26)

Fig. 6. Adaptive VFF in (27)

V. CONCLUSION The AR nonstationary signal, or system parameter

estimation problem, in the presence of additive measurement noise has been considered in the paper. A new algorithm for simultaneously estimation of AR parameters and unknown noise variance has been considered. The parameter estimation procedure uses the weighted recursive least-squares algorithm with variable forgetting factor. The variable forgetting factor is adapted to a non-stationary signal through the suitably defined

0 500 1000 1500 2000 25000

0.2

0.4

0.6

0.8

1

n

λ[n]

0 500 1000 1500 2000 2500-4

-2

0

2

4

a 1[k]

TRUEESTIMATED

0 500 1000 1500 2000 25000.8

0.9

1

VFF[

k]

k

0 500 1000 1500 2000 25000

1

2

3

4

5

6

Q[k

]

k

0 500 1000 1500 2000 25000.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

VFF[

k]

k

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discrimination function that is mapped by polynomial scheme into the VFF value. In this way, we obtain an adaptive recursive estimation procedure that exhibits good tracking performance of time-varying parameters together with satisfactory parameter estimation quality. The feasibility of the approach is demonstrated with simulation results within the controlled experiments.

The proposed adaptive method for generating VFF is general, while the parameter estimation procedure depends on adopted model structure. However, it can be easily applied to other commonly used signal or system models such as ARX, ARMA or ARMAX, or even non-linear models.

Fig. 7. The second order polynomial approximation in (27)

Fig. 8. True and estimated 𝑎1 parameter trajectory

ACKNOWLEDGMENT This research was supported by Serbian Ministry of Education and Science (Projects TR32038 and III42007)

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[12] M. Basseville and A. Benveniste, Detection of Abrupt Changes in Signals and Dynamical Systems, Springer Verlag, Berlin, 1986.

[13] M. Milosavljevic and I. Kovalinka, The Modified Generalized Likelihood Ratio for Automatic Detection of Abrupt Changes in Stationarity of Signals, Proc. 22nd Annual Conf. Inform. Sci. Systems, Princeton, NJ, 1988.

[14] Y.S. Cho, S. B. Kim and E.J. Powers, Time-Varying Spectral Estimation Using AR Models with Variable Forgetting Factors, IEEE Trans. Signal Proc., 39 (6), 1991, pp. 1422-1426.

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[16] B.D. Kovacevic, M.M. Milosavljevic and M.D. Veinovic, Robust Recursive AR Speech Analysis, Signal Processing, 44 , 1995, pp. 125-138.

[17] W. N. Venables and B.D. Ripeley, Modern Applied Statistics with S, Springer Verlag, Berlin, 2002.

[18] Z. Banjac and B.D.Kovacevic, Robust Parameters and Scale Factor Estimation in Nonstationary and Impulsive Noise Environment, Signal Processing, 1(1), 2004, pp. 1-12.

[19] L.C.W. Dixon, Nonlinear Optimization, English University Press, London, 1972.

1 2 3 4 50.65

0.7

0.75

0.8

0.85

0.9

0.95

1

λ[Q

]

Q

0 500 1000 1500 2000 2500-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

a 1[k]

k

TRUEVFF

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