# 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