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SIGNAL

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Signal Processing 40 (1994) 269-285

On the singular value decomposition, applied in the analysis and prediction of almost periodic signals Sarbani Palit, P.P. Kanjilal* Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur 721302, India

Received 8 July 1993

Abstract Signals of a periodic nature are common in diverse fields, and it is often required to determine the one-cycle ahead prediction of such signals. ARIMA models have been used in the past for this job. This approach, however, has the problems of correct order determination and parameter estimation. The performance especially suffers degradation in the presence of noise, owing to lack of robustness. Numerical stability also poses a problem, to eliminate which, computationally intensive methods must be resorted to. Besides, being optimally designed for one-step prediction, such models consequently perform poorly for one-cycle ahead prediction. This paper analyses almost periodic signals using the singular value decomposition and proposes a new algorithm for the prediction of such signals, based on this analysis. The performance of the algorithm in the prediction of several real signals as well as ARIMA time series is found to be reasonably satisfactory. Its performance is compared with that of the Boxqenkins method for one-cycle ahead prediction, both in the absence and presence of additive white Gaussian noise, and found to be consistently favourable.

Zusammenfassung Signale von periodischer Natur sind auf verschiedenen Gebieten verbreitet, und es ist oftmals erforderlich, eine Pr~idiktion Rir einen darauffolgenden Zyklus durchzufiihren. In der Vergangenheit wurden fiir diese Aufgabe ARIMAModelle eingesetzt. Dieses Verfahren beinhaltet jedoch die Probleme der korrekten Ordnungs-Festlegung und der Parametersch~itzung. Die Eigenschaften verschlechtern sich unter RauscheinfluB wegen der mangelnden Robustheit. Numerische Stabilitfit stellt ein Problem dar, welches den Einsatz yon rechenintensiven Methoden erfordert. Daneben verhalten sich solche Modelle sehr ungfinstig ffir eine Ein-Zyklus Pr/idiktion, wenn sie optimal ffir eine Ein-SchrittPr~idiktion entworfen werden. In dieser Arbeit werden fast periodische Signale anhand einer Singul/irwertzerlegung analysiert und ein neuer Algorithmus zur Prfidiktion solcher Signale vorgeschlagen, der auf dieser Analyse basiert. Die Leistungsf'~ihigkeit des Algorithmus bei der Prfidiktion verschiedener realer Signale sowie yon ARIMA Zeitfolgen ist ziemlich zufriedenstellend. Seine Eigenschaften werden mit denen der Box-Jenkins-Methode zur Ein-Zyklus-Pr/idiktion mit und ohne additives weiBes gauBischen Rauschen verglichen und erweisen rich als fiul3erst gfinstig.

R~sum6 Les signaux de nature p~riodique sont communs fi divers domaines et il est souvent n~cessaire pour de tels signaux de disposer d'une pr6diction d'un cycle. Les mod61es ARIMA ont 6t~ utilis~ dans le pass~ pour ce travail. Cette approche a n~anmoins le probl~me de la d&ermination correcte de l'ordre et de l'estimation des param~tres. Les performances * Corresponding author. 0165-1684/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 1 6 8 4 [ 9 4 ) 0 0 0 8 1 - A

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souffrent en particulier de d6gradations en prSsence de bruit, entrainant un manque de robustesse. La stabilit6 num6rique pose 6galement un probl6me, qui peut 6tre 61iminer en utilisant des m6thodes de calcul intensif. De plus, 6tant conqu pour une pr6diction fi un pas, de tels mod61es ont en cons6quence de faibles performances pour la pr6diction ~ un cycle. Cet article analyse presque tout les signaux p6riodiques en utilisant la d6composition en valeur singuli6re et propose un nouvel algorithme, has6 sur cette analyse, pour la pr6diction de tels signaux. La performance de l'algorithme pour la pr6diction de nombreux signaux r6els ainsi que les s6ries teinporelles ARIMA est 8quitable satisfaisante. Sa performance est compar6e avec celle de la m6thode de Box-Jenkins pour la pr6diction fi un cycle avec et sans bruit blanc gaussien additif, et s'av6re 6tre favorable.

Keywords: Almost periodic signal; Singular value decomposition; Singular value; Data window matrix; Measure of aperiodicity; Prediction; Mean squared error

1. I n t r o d u c t i o n

Modelling and tracking of time series using ARIMA models is widespread. This approach has been used in [3, p. 300] to obtain the general seasonal multiplicative model for the prediction of seasonal time series. This approach has various numerical disadvantages. The degree of differencing needed to make the data stationary is determined by inspection of the autocorrelation which may not always lead to accurate results. Further, the non-linear estimation schemes used for the multiple parameters of this model often have convergence problems. Finally, these models are designed to yield one-step ahead predictions and hence cascading of single-stage predictions using predicted values has to be performed to obtain one-cycle ahead predictions. Consequently, the effect of the errors is cumulative, giving relatively poor quality forecasts. This paper presents analysis and prediction of almost periodic signals, i.e. signals with a dominant repeating pattern of fixed length but time-varying amplitude, using the singular value decomposition (SVD). Traditionally, almost periodic functions have been defined [2, 4] as those which can be uniformly approximated by a trigonometric polynomial (a function which consists of the sum of weighted sinusoids). These functions can be equivalently defined as those which can be uniformly approximated by time-translated versions of themselves. Almost periodic sequences can also be defined accordingly. It should be noted that these definitions impose a bound on the deviation of the function from periodicity unlike the description of

almost periodic signals given in this paper. On the other hand, the traditional definition allows local variation in the length of the repeating pattern - a feature absent in the present definition. Such variations are considered only briefly here. Almost periodic signals are common in various natural as well as man-made processes and the one-cycle ahead prediction of these are often of importance and interest. For example, the prediction of the daily electrical load of an electrical substation is of great importance since prior knowledge of it enables the adequate and efficient generation and distribution of power. A novel approach of analysis and one-cycle ahead prediction of such signals using the SVD has been proposed. The SVD, which is an orthogonal decomposition, can separate a system into its structural modes and indicate their relative strengths. Owing to the availability of numerically robust techniques for SVD [5], it has found wide applications in control and system identification [I0], detection and estimation of signals in the presence of noise [1, 6, 15, 16] and order detection in AR models [11]. Fortran implementations of the SVD are available in [13]. A quantified measure of the degree of aperiodicity of a signal in terms of the singular values of its data matrix is presented. The degree of aperiodicity is used to study the effectiveness of the largest singular value as a trend indicator. Its applicability in detecting periodicity and determining a working period length for repetitive signals is also indicated. The principal components of SVD of almost periodic signals are studied theoretically and through simulations. On the basis of the obtained results, a robust yet simple algorithm is proposed

s. Palit, P.P. Kanjilal / Signal Processing40 (1994) 269-285 for one-cycle ahead prediction. The performance of the proposed method is compared with that of the Box-Jenkins seasonal multiplicative model for a real-life data series. The performance is also compared for prediction of time series generated by models satisfying the implicit model assumptions of each method separately.

size ml x n such that for the endpoint kn (k >~ml and refers to the kth cycle), the corresponding data window is given by

X(k)=] a~-ml+i.

The SVD of an m × n matrix A is defined [5] as (1)

where U = J i l l , . . . , Um'] ~ ~mxm, V = IV1 . . . . . Vn] R "×" and u r u = L VVV = / a n d , ~ s R m×". Z is equal to [diag{aa ..... a,}:0] or its transpose, depending on whether m < n or m ~> n; where p = min(m, n), al >1 ... >~a, >1 O, al, ..., ap are the singular values of A. The column vectors ui and vi, which correspond to the singular value a~, are called the ith left singular vector and the ith right singular vector, respectively. The right and left singular vectors form a basis for the row- and column-spaces of A, respectively. A large singular value represents a dominant mode of the matrix. If a singular value has multiplicity 1, the corresponding singular vector is also unique, up to sign. These properties of the SVD will be used in the following sections. Consider samples from a strictly periodic signal {x(.)}, with a period of length n, i.e. x(k + n) = x(k). Observations from m periods are assumed to be available. Then, the corresponding data matrix A ~ R m×" is formed by partitioning the signal into periods and placing each period (in phase) as a row of A, as shown below:

LaTJ

.

(3)

L aZ

2. Signal characterization using SVD

A = U~,V T,

271

This is henceforth referred to as the kth data window. The matrix X(k) has ml repeated rows and is of rank 1. Consequently it has only 1 non-zero singular value and m~ - 1 zero singular values. Since only a l ¢ 0, it is enough to consider simply Vl(k). The vector v~(k) represents one period of the signal normalized to a unit vector. Since the signal is perfectly periodic, it has the same repeating pattern and, consequently, vx(k) does not change with k, i.e. over successive data windows. The elements of the vector u~(k)a~(k) represent the amplitude scaling factors of each row within a data window. Since the amplitude of each period of the signal is the same, so are these factors. Consider the two following cases of signals with marginal deviation from periodicity. Case 1: Consider a signal {x(.)} which has the same repeating pattern of length n but with different scaling factors over different periods. It can be immediately inferred that the rows of X(k) are multiples of each other; hence the rank of X(k) still remains 1. The vector v~(k) does not change with k but the elements of u~(k)a~(k) are no longer identical; they now vary according to the amplitudes of the rows of X(k).

E x,1, x,2, . .' x,n1, :((m -- 1)n + 1) x((m -- 1)n + 2)

In order to analyse the signal as it evolves with time, the data matrix is divided into overlapping rectangular matrices referred to as data windows of

(2)

x(mn)

Case 2: Consider next a periodic waveform with amplitude varying from one period to another, plus noise. This includes, as a special case, a periodic

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signal plus noise. The matrix X(k) is now of full rank but a~(k) would be very large compared to the rest of the singular values, assuming that the SNR is high. The condition number will hence be high. The vector v~(k) will approximately represent one period (normalized to a unit vector) of the periodic part of the signal while its amplitude expansion coefficients will be given approximately by the elements of ul(k)at(k).

P

I]EIIF=

3.1. Measure of aperiodicity In practice, the periodicity of a signal is often buried under additive noise. Hence it is necessary to provide a measure of aperiodicity that would indicate the strength or weakness of the periodic content of a signal. Such a measure is suggested below. It is assumed that the period length is known and the data matrix is formed as indicated in the previous section by aligning the rows, period by period. Strictly speaking, the waveform is aperiodic due to the presence of noise. The data matrix Aaper of any general aperiodic waveform can be decomposed as Aapcr = Ape r + E,

where the SVD of Aaper is as in (1), Aper is the dominant periodic part (ulaa~ T) and E is the residual, which may be written as P

UI~lVT = ~ uiaivT.

P

~ a,u~vT i=2

P