Pseudo-fractional ARMA modelling using a double Levinson recursion

Abstract

The modelling of fractional linear systems through ARMA models is addressed. To perform this study, a new recursive algorithm for impulse response ARMA modelling is presented. This is a general algorithm that allows the recursive construction of ARMA models from the impulse response sequence. This algorithm does not need an exact order specification, as it gives some insights into the correct orders. It is applied to modelling fractional linear systems described by fractional powers of the backward difference and the bilinear transformations. The analysis of the results leads to propose suitable models for those systems. IntroductionPseudo-fractional auto-regressive moving average (ARMA) modelling is a pole-zero modelling of fractional linear systems. These are described by fractional differential equations in the continuous-time case or auto-regressive integrated moving average (ARIMA) models in the discrete-time case. The first case is based on the definition of fractional differintegration, whereas the second deals with the fractional differencing that is a fractional version of the well-known finite differences. These systems are characterised by having a long memory that cannot be explained by the usual linear systems that have short memory (exponential). The desire of finding a theoretical base for such systems led to the fractional calculus that has recently received a great deal of attention in the scientific literature, through the publication of books, special issues of journals, review articles, as well as a very large number of research papers. The interest in fractional calculus comes from the fact that it provides foundations for the understanding of several natural phenomena and the basic theory for building models for the systems underlying them. However, adoption of the fractional calculus by the physicists and engineering community was inhibited historically by the lack of clear experimental evidence for its need and by the difficulty in constructing simple models for simulation or even implementation of simple fractional systems. Fractional calculus is almost as old as the common calculus, but only since 30 years ago it has been a subject of specialised publications and conferences.The basic building block of this kind of systems is the non-integer order derivative and integral that have been approximated by fractional powers of the backward difference or the bilinear transformations -the former is exactly the building block of the fractional differencing, as said earlier. However, these approximations are described infinite impulse response (IIR) systems with nonrational transfer functions. For these, ARMA models are only approximations. However, the usefulness of ARMA models makes them very interesting when constructing discrete-time approximating models for fractional systems. In the last few years, a lot of attempts to obtain such models have been done [1 -5]. However, it is not clear how to perform such modelling and how to choose the most suitable orders, although there are a l...

A new least-squares approach to differintegration modeling

“…5 and 6 depict, there is no substantial or significant difference between the algorithm proposed in this work, LS, and that presented in [3].…”

“…This leads us to consider fractional orders a 2 ½À0:5; 0:5 . For comparisons, we will use the results presented in [3,4] whenever feasible. 1 In order to get some insight into the order of the ARMA models, experiments with ARMA(n,m), n ¼ 1; .…”

“…However, these more correct models are difficult to implement and model in practice. For them, ARMA models are only approximations that we call pseudo-fractional ARMA models [3]. In the last few years, a lot of attempts to obtain such models have been done (see [4][5][6][7][8][9]).…”

“…In impulse response modeling the well-known Pade´algorithm is frequently used [2]. In [3], we presented a suitable recursive algorithm for this modeling. Here, we will propose a different algorithm based on a least-squares (LS) criterion different from [4].…”

“…In minimizing the error power we obtain suitable normal equations that allow us to obtain the ARMA parameters. The algorithm is described in Section 2 where we compare it with the algorithm described in [3] through some current modeling examples.…”