Statistical signatures of structural organization: The case of long memory in renewal processes

• We calculate the excess entropy and statistical complexity for fractal renewal processes.
• Processes with tails proportional to t−αt−α exhibit long memory via a phase transition at α=1α=1.
• Excess entropy diverges only there and statistical complexity diverges there and for all α<1α<1.
• Fractal renewal processes with power-law autocorrelations and Hurst exponent h>1/2h>1/2 do not have divergent excess entropy.
• This breaks the intuition that second-order statistics can identify complex systems with long-term memory.

Identifying and quantifying memory are often critical steps in developing a mechanistic understanding of stochastic processes. These are particularly challenging and necessary when exploring processes that exhibit long-range correlations. The most common signatures employed rely on second-order temporal statistics and lead, for example, to identifying long memory in processes with power-law autocorrelation function and Hurst exponent greater than 1/2. However, most stochastic processes hide their memory in higher-order temporal correlations. Information measures—specifically, divergences in the mutual information between a process' past and future (excess entropy) and minimal predictive memory stored in a process' causal states (statistical complexity)—provide a different way to identify long memory in processes with higher-order temporal correlations. However, there are no ergodic stationary processes with infinite excess entropy for which information measures have been compared to autocorrelation functions and Hurst exponents. Here, we show that fractal renewal processes—those with interevent distribution tails ∝t−α∝t−α—exhibit long memory via a phase transition at α=1α=1. Excess entropy diverges only there and statistical complexity diverges there and for all α<1α<1. When these processes do have power-law autocorrelation function and Hurst exponent greater than 1/2, they do not have divergent excess entropy. This analysis breaks the intuitive association between these different quantifications of memory. We hope that the methods used here, based on causal states, provide some guide as to how to construct and analyze other long memory processes.