Time Series of Non-Additive Metrics: Identification and Interpretation of Contributing Factors of Variance by Linear Decomposition

Abstract

The research paper addresses linear decomposition of time series of non-additive metrics that allows for the identification and interpretation of contributing factors (input features) of variance. Non-additive metrics, such as ratios, are widely used in a variety of domains. It commonly requires preceding aggregations of underlying variables that are used to calculate the metric of interest. The latest poses a dimensionality challenge when the input features and underlying variables are formed as two-dimensional arrays along elements, such as account or customer identifications, and time points. It rules out direct modeling of the time series of a non-additive metric as a function of input features. The article discusses a five-step approach: (1) segmentations of input features and the underlying variables of the metric that are supported by unsupervised autoencoders, (2) univariate or joint fittings of the metric by the aggregated input features on the segmented domains, (3) transformations of pre-screened input features according to the fitted models, (4) aggregation of the transformed features as time series, and (5) modelling of the metric time series as a sum of constrained linear effects of the aggregated features. Alternatively, approximation by numerical differentiation has been considered to linearize the metric. It allows for element level univariate or joint modeling of step (2). The process of these analytical steps allows for a backward-looking explanatory decomposition of the metric as a sum of time series of the survived input features. The paper includes a synthetic example that studies loss-to-balance monthly rates of a hypothetical retail credit portfolio. To validate that no latent factors other than the survived input features have significant impacts on the metric, Statistical Process Control has been introduced for the residual time series.