The Algebraic Structure of Transformed Time Series

Abstract

Invertible transformations are often applied to time series data to generate a distribution closer to the Gaussian, which naturally has an additive group structure. Estimates of forecasts and signals are then typically transformed back to the original scale. It is demonstrated that this transformation must be a group homomorphism (i.e., a transformation that preserves certain arithmetical properties) in order to obtain coherence between estimates of quantities of interest in the original scale, and that this homomorphic structure is ensured by defining an induced group structure on the original space. This has consequences for the understanding of forecast errors, growth rates, and the relation of signal and noise to the data. The effect of the distortion to the additive algebra is illustrated numerically for several key examples.