Abstract
This work investigates a spectral decomposition of the AR metric proposed as a measure of structural dissimilarity among ARIMA processes. Specifically, the metric will be related to the variance of a stationary process so that its behaviour in the frequency domain will help to detect how unobserved components generated by the parameters of both phenomena concur in specifying the obtained distance. Foundations for the metric are briefly reminded and the main consequences of the proposed decomposition are discussed with special reference to some specific stochastic processes in order to improve the interpretative content of the AR metric.
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Notes
- 1.
In order to simplify notation, in this chapter we are not formally considering multiplicative seasonal ARIMA processes; however, all the subsequent results straightforwardly follow. In addition, we are strictly using classical notation as in [5].
- 2.
In fact, to be correct we should define as δ 1 the first non-zero absolute difference between the AR expansions of X t and Y t processes. This consideration is important when comparing pure AR seasonal processes, for instance; however, we are omitting this point for simplicity of notation.
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The research has been supported by Department of Statistical Sciences, University of Naples Federico II.
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Iannario, M., Piccolo, D. (2013). Spectral Decomposition of the AR Metric. In: Torelli, N., Pesarin, F., Bar-Hen, A. (eds) Advances in Theoretical and Applied Statistics. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35588-2_11
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