Information-theoretic Underpinnings of the Effort-to-Compress Complexity Measure

Effort-to-Compress (ETC) is a measure of complexity based on a lossless data-compression algorithm that has been used extensively in characterization and analysis of time-series. ETC has been shown to give good performance for short and noisy time series data and has found applications in the study of cardiovascular dynamics, cognitive research and regulating the feedback of musical instruments. It has also been used to develop causal inference methods for time series data. In this work, a theoretical analysis helps us to demonstrate the links of ETC measure to the total self-information contained in the joint occurrence of most dominant (shortest) patterns occurring at different scales (of time) in a time-series. This formulation helps us to visualize ETC as a dimension like quantity that computes the effective dimension at which patterns in a time-series (translated to a symbolic sequence) appear. We also show that the algorithm that computes ETC can be used as a means for an analysis akin to ‘multifractal analysis’ using which the power contained in patterns appearing at different scales of the sequence/ series can be estimated. Multifractal analysis has been used widely in analysis of biomedical signals, financial and geophysical data. Our work provides a theoretical understanding of the ETC complexity measure that links it to information theory and opens up more avenues for its meaningful usage and application.