How to model multi-modal distribution or cyclical time series data?


I am looking for help on best practice to model this particular distribution. This is hourly sales volume of a particular item.


As you can see, the sales have peaks around 6:00, 9:00, 13:00. Otherwise, it has slow declines throughout the day.

What’s the best way to model this kind of behaviour?

I have tried:
a = pm.Normal(‘a’, mu=0.0, sd = 0.1, shape=2)
pm.math.exp(a[0]*hour_of_day + a[1]*hour_of_day**2)

Is there a better way of doing this?

Many thanks for suggestions.


If the data is indeed cyclical (night time is missing in the plot?) then you might want to try out the Von Mises distribution. It’s the equivalent of a Normal distribution for periodic data.

You might need a mixture of multiple components to account for the multiple peaks.

Another option might be to transform the data into frequency space using a Fourier series.

However, I’ve never put that to practice and just wanted to give you a direction. Surely someone else has more experience!