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Eos, Transactions American Geophysical Union

Are autoregressive models sensitive to initial conditions?



The answer is, it depends. If the poles of the autoregressive (AR) model all lie inside the unit circle on the complex z-plane, then it is not sensitive to the initial condition. Otherwise, it will be.

The question in the title arises from a discussion between Gray and Woodward [1992] and Tsonis [1991, 1992] concerning the sensitivity of climate models to initial conditions. It is an interesting, well-posed mathematical question which, unfortunately, was not treated mathematically in their comments. The answer given above is closely related to whether the AR process is stationary or minimum-phase; here I would like to offer a simple, analytic, intuitively clear proof, the essence of which can be traced back to a work by Prony [1795], which has been known for nearly 200 years.

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