$r(t)$ during a single time interval of length 100 is 0.016. We later demonstrate that the ratio of the period of stable oscillations to memory length obeys $$\frac{T}{m} \propto \sqrt{m},$$ at the point where oscillations begin to form, so that in systems where agents have longer memories, these memories cover a relatively shorter fraction of one oscillation.

To determine the values of $\beta_0, \beta_1$ which maximize the likelihood $\mathcal{L}_i$ we express the log likelihood as a function of these parameters: $$\begin{aligned}
\ln \mathcal{L}_i  &= \sum_{k = 1}^{m} I_R(i,k)[\beta_0 + \beta_1 t_{ik}] \\ &- \sum_{k = 1}^{m}\ln(1+e^{\beta_0 + \beta_1 t_{ik}}).\end{aligned}$$ This expression is then numerically maximized to find the most likely values of $\beta_0$ and $\beta_1$ which can then be used in equation (\[rlogit\]) to predict the current value of $r(t)$.

![ Dashed line shows the rock weight in a group of size $L=100$ with $\alpha=2, \beta=\gamma=1$. All agents have memory $m=100$ and update rate $\epsilon = 10^{-3}$. Open circles show binomial logistic regression estimates of the rock fraction made using the memory of a single randomly selected agent at a sequence of regularly spaced intervals. Black dots show similar estimates made by simply computing the fraction of rock interactions in 