Derivatives and Internal Models pp 682-698 | Cite as

# Pre-Treatment of Time Series and Assessment of Models

## Abstract

The *pre-treatment* for the transformation of a given data set into a *stationary* time series has been mentioned several times in the preceding sections and will receive detailed treatment in this section. The basis for pre-treating a time series is its decomposition into a *trend* component *gt*, a *seasonal* component,S_{t}, and a random component^{1} *Z*_{ t }:

*Z*_{ f } then represents the stationary time series with E[Z_{ t }] = 0. The *trend g*_{ t } is a deterministic function of the time variable f, which represents a long-term development, for example a polynomial or an exponential function.^{2} A weaker trend can sometimes be more readily recognized after a compression of the time axis. The *season s*_{ t } represents a periodic component with a period *p*:

It follows that the sum \(\sum\nolimits_{i\, = \,1}^p {{s_{t\, + \,1}}}\) of *p* successive values is a constant. This constant can be incorporated into the trend *g*_{ t } so that, without loss of generality, the sum can be assumed to be equal to zero:

## Keywords

Time Series Move Average Internal Model Conditional Variance Time Series Model## Preview

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