# 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

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.