Solution of discrete ARMA-representations via MAPLE
Introduction
Consider the regular, discrete-time, autoregressive moving average (ARMA)-representation described by the matrix equationwherewhere at least one of Aq, Bq is non-zero, σ denotes the advance operator (i.e. σiy(k)=y(k+i)), defines the output and defines the input of the system , .
In the case where and the ARMA-representation , reduces to the generalized state space (GSS)-representationFurther if E is non-singular (i.e. |E|≠0) then (1.2) reduces to the state space (SS)-representation.
In [3] closed formulae for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) are presented. These formulae depend explicitly on the forward fundamental matrix Hk and the backward fundamental matrix Vk of A(z) both of which can be seen to be of fundamental importance in the analysis of discrete ARMA-representations. The closed formulae obtained can be seen to be natural extensions to the ones proposed by Lewis and Mertzios [4] for the GSS-representation (1.2) which correspondingly depend explicitly on the forward fundamental matrix φk and the backward fundamental matrix τk of (zE−A).
In this paper we implement the closed formulae as given in [3] for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) in the symbolic computational language MAPLE [1]. In Section 2 the definition and construction of Hk and Vk will be considered. In Section 3 the closed formulae for the solution of the ARMA-representation (1.1a) will be shown and these will be implemented via MAPLE in Section 4 where the corresponding MAPLE procedures will be presented. Finally in Section 5 these procedures will be illustrated via a numerical example.
Section snippets
Preliminaries
Consider the ARMA-representation (1.1a) and write it in the expanded formwhere we assume A(z) is regular (i.e. det[A(z)]≠0), k∈[0,N] and u(k) is non-zero for k=0,1,…,N−q. For such as case we can define the following Definition 2.1 Let the Laurent series expansion of A(z)−1 about z=∞ be given bywhere is the greatest order zero of A(z) at s=∞. Then the coefficient matrices , constitute the forward fundamental matrix Forward fundamental matrix sequence of A(z)
Solution of ARMA-representations
In [3] closed formulae for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) were obtained. In this section each of these three types of solution are considered and the corresponding closed formulae provided.
Implementation via MAPLE
In this section we implement the formulae for the solution of the ARMA-representation (1.1a) as presented in Section 3, in the symbolic computational language MAPLE [1]. One obvious advantage of using MAPLE is that it enables the user to implement any one of the “built in” procedures inherent in it predominantly, in this case, those in the linear algebra package linalg which contains a collection of procedures for matrix manipulation. This results in further simplification of any program code
Examples
In this section we will consider the solution of the ARMA-representationusing the MAPLE procedures presented in Section 4. Clearly from (5.1) we have q=2, r=3 and m=1.
The machine used is a SUN SPARC station10 (75 MHz SuperSPARC II). The last line of the output indicates the CPU time used in the computation. This is divided into three parts
- (i)
bytes used (integer) number of bytes of memory that have been requested up to that point in the
Conclusions
In this paper we have implemented the results presented in [3], concerned with the forward, backward and symmetric solution of an ARMA-representation, in the symbolic computational language MAPLE. We have also implemented the corresponding compatibility conditions for each such solution which define an admissible set of boundary conditions, in conjunction with the input sequence, for such a solution to exist. All the corresponding MAPLE code has been included and a concise illustrative example
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