Information criteria and harmonic models in time series analysis

In this paper we present a novel methodology for estimating harmonic models in time series. The amplitude density function is derived from an inversion of the spectral representation of the series and is found to have the property of strong consistency. It is used in conjunction with non-linear least squares regression in an iterative procedure to yield precise estimates of the frequencies in the model. Information criteria are then used in determining how many frequencies to include in the model. The optimal procedure was tested using Monte Carlo techniques, and proved successful at correctly estimating the underlying harmonic model. The methodology was applied to a series of magnitudes of a variable star, to a series of signed sunspot numbers, and to a series of temperature data. A class of iterative procedures was investigated, and an optimal procedure proposed, which involves the use of non-linear least squares and an update of the amplitude density function after each new frequency has been estimated. Some information criteria, among the class of BICs, are found to yield relatively parsimonious models, while the AlC does not. A model with six harmonic components is selected for one of the BICs for the series of signed subspot numbers. Only the yearly cycle is found to be worth including in the series of Budapest temperature data. However the information criteria indicate a relatively large number of harmonic components for the series of magnitudes of a variable star.