ARMA Modelling, Krylov Subspaces and ARMA Modelling via Krylov Subspaces

Steven Elsworth (The University of Manchester)

ATB Frank Adams 1,

ARMA models, first popularised by Box and Jenkins (1970), provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression and the second for the moving average. We discuss these models and explore how software (MATLAB and R) fit them.
Given a square matrix \(A\) and a nonzero vector \(\textbf{b}\), the \(r\)-order Krylov subspace is the linear subspace spanned by the image of \(\textbf{b}\) under the first \(r\) powers of \(A\). We discuss Krylov subspaces and the rational Arnoldi method. 
In 1984, George Cybenko published a paper outlining the connection between Krylov subspaces and Autoregressive models. We prove this connection and give a brief outline of the extension to ARMA models. 
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