Geometric derivative

One reason I believe that log-linear dynamical systems have so much potential as a modeling tool is that they measure the relative change in quantities rather than the absolute change. For example, suppose we’re tracking the heat of a system \(h_t\). The absolute change from time \(t\) to \(t+1\) is given by \(h_{t+1} - h_t\). The relative change is given by \(h_{t+1}/h_t\). I also call the relative change the geometric change, since it is based on ratios.

Looking at ratios rather than differences is a small change with big consequences. One consequence is shifting from the standard derivative to the geometric derivative, described in Appendix A of my paper on LLDS.

Differentiating differently

The standard derivative of \(f(t)\) is given by the limit

\[\lim_{\Delta \to 0}\frac{f(t + \Delta) - f(t)}{\Delta}.\]

By contrast, the geometric derivative of \(f(t)\) is given by the limit

\[\lim_{\Delta \to 0}\left(\frac{f(t + \Delta)}{f(t)}\right)^{1/\Delta}.\]

When we discretize the standard derivative in a simple way, we get a linear dynamical system. Suppose

\[\lim_{\Delta \to 0}\frac{f(t + \Delta) - f(t)}{\Delta} = C.\]

Let us fix \(\Delta\). Then we have

\[f(t + \Delta) - f(t) = \Delta C,\]

or if \(\Delta\) is our discrete time step

\[f(t+1) = \Delta C + f(t).\]

This is a linear dynamical system.

When we discretize the geometric derivative in the same way, we get a log-linear dynamical system. Suppose

\[\lim_{\Delta \to 0}\left(\frac{f(t + \Delta)}{f(t)}\right)^{1/\Delta} = C.\]

Let us fix \(\Delta\). Then we have

\[\frac{f(t + \Delta)}{f(t)} = C^{\Delta},\]

or equivalently

\[\log f(t + \Delta) - \log f(t) = \Delta \log C\]

If \(\Delta\) is our discrete time step, we get a linear dynamical system in \(\log f\):

\[\log f(t+1) = \Delta \log C + \log f(t).\]

In other words, a log-linear dynamical system.

Written on March 25, 2020