The Inner Structure of Time-Dependent Signals

This paper shows how a time series of measurements of an evolving system can be processed to create an ”inner” time series that is unaffected by any instantaneous invertible, possibly nonlinear transformation of the measurements. An inner time series contains information that does not depend on the nature of the sensors, which the observer chose to monitor the system. Instead, it encodes information that is intrinsic to the evolution of the observed system. Because of its sensorindependence, an inner time series may produce fewer false negatives when it is used to detect events in the presence of sensor drift. Furthermore, if the observed physical system is comprised of non-interacting subsystems, its inner time series is separable; i.e., it consists of a collection of time series, each one being the inner time series of an isolated subsystem. Because of this property, an inner time series can be used to detect a specific behavior of one of the independent subsystems without using blind source separation to disentangle that subsystem from the others. The method is illustrated by applying it to: 1) an analytic example; 2) the audio waveform of one speaker; 3) video images from a moving camera; 4) mixtures of audio waveforms of two speakers.