Estimating rate of occurrence of an event with exponential smoothing and irregular events

Question

Imagine that I have a set of measurements of x that are taken by many processes x₀ ... x_N at times t₀ ... t_N

Answer 1

You could try this:

Keep an estimator zn so that at each event:

z_n = (z_n-1+κ).e^{-κ.(t_n-t_n-1)}

This will converge towards the event rate in s^-1. A sligtly better estimator is then (as there is still an error/noise related if you compute the estimate just before or just after an event) :

w_n = z_n.e^-κ/(2.z_n)

In your example it will converge to 2s^-1 (the inverse of 500ms)

The constant κ is responsible for the smoothing and is in s^-1. Small values will smooth more. If your event rate is roughly of seconds, a value of 0.01s-1 for κ is a good start.

This method has a starting bias, and z₀ could be set to an estimate of the value for faster convergence. Small values of κ will keep the bias longer.

There are much more powerful ways of analyzing poisson-like distributions, but they often require large buffers. Frequency analysis such as Fourier transform is one.