Functional Series Time-dependent AutoRegressive Moving Average (TARMA) models form an important class of nonstationary stochastic models offering high parsimony, tracking of “fast” and “slow” variations, high accuracy and resolution, as well as accurate capturing of both resonances and antiresonances. This paper considers the estimation of Functional Series TARMA models with polynomial functional spaces based upon a novel matrix algebra that is isomorphic to that of the noncommutative ring of time-varying polynomial operators expressed in terms of the model’s functional spaces. The Generalized estimation method introduced offers important advantages, such as the use of properly contracted functional spaces that is necessary for the elimination of asymptotic bias errors, the ability to handle AR and MA functional spaces of different dimensionalities, as well as improved accuracy through a streamlined realization. The method’s excellent performance characteristics are confirmed via Monte Carlo experiments and comparisons with an earlier Polynomial-Algebraic approach and the adaptive Recursive Maximum Likelihood ARMA method.

Issue Section:
Technical Papers
Keywords:
signal processing, autoregressive moving average processes, matrix algebra, estimation theory, parameter estimation, stochastic processes
Topics:
Algebra, Matrix algebra, Polynomials, Space, Signals, Errors, Monte Carlo methods
1.
Newland, D. E., 1993, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, third edition, Addison Wesley Longman.
2.
Hammond
,
J. K.
, and
White
,
P. R.
,
1996
, “
The Analysis of Non-Stationary Signals Using Time-Frequency Methods
,”
J. Sound Vib.
,
190
, pp.
419
–
447
.
3.
Kitagawa, G., and Gersch, W., 1996, Smoothness Priors Analysis of Time Series, Springer-Verlag, New York.
4.
Grenier, Y., 1989, “Time and Frequency Representation of Signals and Systems,” Parametric Time-Frequency Representations, G. Longo and B. Picinbono, eds., Springer-Verlag.
5.
Tong, H., 1990, Non-linear Time Series: A Dynamical Systems Approach, Oxford University Press.
6.
Ben Mrad
,
R.
,
Fassois
,
S. D.
, and
Levitt
,
J. A.
,
1998
, “
A Polynomial-Algebraic Method for Non-Stationary TARMA Signal Analysis-Part I: The Method
,”
Signal Process.
,
65
, pp.
1
–
19
.
7.
Petsounis
,
K. A.
, and
Fassois
,
S. D.
,
2000
, “
Non-Stationary Functional Series TARMA Vibration Modelling and Analysis in a Planar Manipulator,”
J. Sound Vib.
,
231
, pp.
1355
–
1376
.
8.
Bellanger, M. G., 1987, Adaptive Filters and Signal Analysis, Marcel Dekker, New York.
9.
Stavropoulos
,
Ch. N.
, and
Fassois
,
S. D.
,
2000
, “
Non-Stationary Functional Series Modeling of Hardware Reliability: A Comparative Study Using Rail Vehicle Interfailure Times
,”
Reliability Eng. Sys. Safety
,
68
, pp.
169
–
183
.
10.
Fouskitakis, G. N., and Fassois, S. D., 2001, “Functional Series TARMA Modeling and Simulation of Earthquake Ground Motion,” Earthquake Eng. Struct. Dyn., in press.
11.
Conforto
,
S.
, and
D’Alessio
,
T.
,
1999
, “
Spectral Analysis for Non-Stationary Signals from Mechanical Measurements: A Parametric Approach
,”
Mech. Syst. Signal Process.
,
13
, pp.
395
–
411
.
12.
Rao
,
T.
,
1970
, “
The Fitting of Nonstationary Time-Series Models with Time-Dependent Parameters
,”
J. Royal Statist. Soc., Series B
,
32
, pp.
312
–
322
.
13.
Liporace
,
L. A.
,
1975
, “
Linear Estimation of Non-Stationary Signals
,”
J. Acoust. Soc. Am.
,
58
, pp.
1288
–
1295
.
14.
Charbonnier
,
R.
,
Barlaud
,
M.
,
Alengrin
,
G.
, and
Menez
,
J.
,
1987
, “
Results on AR-Modeling of Non-Stationary Signals
,”
Signal Process.
,
12
, pp.
143
–
151
.
15.
Aboutajdine, D., Alengrin, G., Barlaud, M., Grenier, Y., Menez, J., and Najim, M., 1988, “On AR Modeling of Nonstationary Signals,” Proceedings of the IFAC Symposium on Identification and System Parameter Estimation, Beijing, China, pp. 989–993.
16.
Tsatsanis
,
M. K.
, and
Giannakis
,
G. B.
,
1993
, “
Time-Varying System Identification and Model Validation Using Wavelets
,”
IEEE Trans. Acoust., Speech, Signal Process.
,
41
, pp.
3512
–
3523
.
17.
Kozin, F., 1977, “Estimation and Modeling of Non-Stationary Time Series,” Proceedings of the Symposium on Application of Computer Methods in Engineering, Los Angeles, Vol. 1, pp. 603–612.
18.
Ben Mrad
,
R.
,
Fassois
,
S. D.
, and
Levitt
,
J. A.
,
1998
, “
A Polynomial-Algebraic Method for Non-Stationary TARMA Signal Analysis-Part II: Application to Modeling and Prediction of Power Consumption in Automobile Active Suspension Systems
,”
Signal Process.
,
65
, pp.
21
–
38
.
19.
Stavropoulos, Ch. N., and Fassois, S. D., 2000, “The Impact of Scheduled Maintenance on the Failure Process of Electric Rail Vehicles,” Proceedings of the 8th Mediterranean Control Conference, Patras, Greece, pp. 6.
20.
Pandit, S. M., and Wu, S. M., 1983, Time Series and System Analysis with Applications, Wiley, NY.
21.
Abramowits, M., and Stegun, I. A., 1970, Handbook of Mathematical Functions, Dover Publications, New York.
22.
Fuhrmann, P. A., 1996, A Polynomial Approach to Linear Algebra, Springer-Verlag, New York.
23.
Bouthellier, P., and Ghosh, B. K., 1988, “Robust Stabilization of Discrete Time Single Input Single Output Linear Time-Varying Systems,” Proceedings of the 27th IEEE Conference on Decision and Control, Austin, Texas, pp. 39–44.
24.
Kamen
,
E. W.
,
1988
, “
The Poles and Zeros of a Linear Time-Varying System
,”
Linear Algebra Appl.
,
98
, pp.
263
–
289
.
25.
Ghosh
,
B. K.
, and
Bouthellier
,
P. R.
,
1993
, “
Simultaneous Coefficient Assignment of Discrete-Time Multi-Input Multi-Output Linear Time-Varying System: A New Approach for Compensator Design
,”
SIAM J. Control Optim.
,
31
, pp.
1438
–
1461
.
26.
Magnus, J. R., and Neudecker, H., 1988, Matrix Differential Calculus, Wiley, NY.
27.
Fraleigh, J. B., 1996, An Introduction to Abstract Algebra, fourth edition, Addison-Wesley, NY.
Copyright © 2001
 by ASME
You do not currently have access to this content.