## Periodic smoothing of time series.(English)Zbl 0628.65148

The periodic variation of a time series $$\{y_ i,t_ i\}$$ can be assessed by approximating the observed $$y_ i$$ with a general smooth periodic function $$f(\omega t_ i)$$ (where f($$\cdot)$$ is a periodic function with period 1). The basic idea is to choose the function f($$\cdot)$$ by smoothing $$y_ i$$ as a function of $$(\omega t_ i mod 1)$$. If the frequency $$\omega$$ is unknown, it may be determined so that it makes $$f(\omega t_ i)$$ a good approximation of $$y_ i.$$
If the periodic variation in the series is more complex (i.e. if several frequencies are involved), the approximation in the form of a sum of several smooth periodic functions can be found. The suggested method is repeatedly applied to the series of residuals, resulting from preceeding steps, until no improvement is possible. The method doesn’t require equally spaced data (in particular, it can handle missing data) and it is not necessarily sensitive to occasional gross errors in $$y_ i$$ (or $$t_ i)$$.
Reviewer: E.Rozmarova

### MSC:

 65C99 Probabilistic methods, stochastic differential equations 65D10 Numerical smoothing, curve fitting 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62M15 Inference from stochastic processes and spectral analysis 42A10 Trigonometric approximation 65T40 Numerical methods for trigonometric approximation and interpolation
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