Learning Tapestries: predictive statistical inference for possible futures in a noisy chaotic system

In several scenarios, events with significant impact arise from a combination of less impactful events happening in a sequence resulting in a cumulative effect surpassing a threshold. For example, floods can arise because of a sequence of non disastrous events soaking the ground and filling reservoirs, so one final event results in runoff that can not be soaked into the ground and reservoirs are overcome resulting in a flood. A simpler case is when an optical fiber is impacted with a sequence of radiation events, where each event is quickly recovered from but a build up of less annealable defects occur and the fiber has significantly higher long term darkening. The underlying chaotic systems such as earth’s weather or solar weather provide significant challenge for longer range prediction. A key characteristic of such systems is that two points measured arbitrarily close to one another will depart from each other exponentially fast. When the measurement itself is uncertain the future trajectories are better expressed in terms of probabilities. Delay maps, sequences of measurements with specific relationships between variables and time delays, provide an empirical method of reconstructing chaotic dynamics, hence can be used as a basis for empirical prediction. Here we use multiple delay maps to construct multiple sequences of predictions where predictions within a sequence are in a very strong sense connected, but the separate sequences are not. The sequences can be made to construct predictive densities which preserve the associations inherent in the chaotic system through time. In an inference scenario the predictive densities can be created from a training sample to construct likelihoods of a test sample which can be used for both testing of variable sets for better predictive learning and to test for learning across time by weighting each sequence of predictions by how close it is to the test data as it appears. If learning across time can be demonstrated, the approach allows exploration of multiple possible paths several steps into the future, conditioning on the next steps. The existence of learning behavior will be examined both with artificial chaotic noisy data and with real solar data using Ap, sunspots, and F10.7 solar measurements. This demonstration of learning in a tapestry of the solar data provides a necessary first step toward the implementation of a spreadsheet structure where different strategies could be evaluated against an unfolding sequence of events through time, for example providing a way to evaluate different paths of inner solar system space craft against the distribution of probable solar storms in space and time.