Optimally fuzzy scale-free memory

Any system with the ability to learn from a time series and predict the future must have a memory representing the information from the recent past. In cases where the external environment generating the time series has a fixed scale, the memory can be a simple shift register—a moving window of finite width extending into the past. The width of the window should be large enough to describe the largest scale relevant for predicting the signal. However, such a traditional buffer is inappropriate if the longest relevant scale is not known a priori, or if the signal has structure at many different time scales. It is well known that signals with scale-free long range correlations are found in many physical environments. Hence we argue in favor of a memory that is a scale-free fuzzy buffer which implicitly accounts for scale-free fluctuations in naturally generated signals. Based on a neuro-cognitive model of internal time, we construct a fuzzy buffer that optimally sacrifices the accuracy of information representation in order to represent exponentially long time scales without an explosion in capacity demands. Using several illustrative time series we demonstrate the advantage of the fuzzy buffer over the shift register in time series forecasting. We suggest that this method for representing time-varying signals may be of broad utility in a variety of applications.