# Defining churn for customers with seasonal purchase patterns

I want to define customer churn accurately for the data showing seasonal patterns of not-purchasing.

Our customers purchase on the regular basis most time of the year, with approx. 97% of all orders having less than 31 days gap between them. However, about 75% of our customers will have at least 1 (max 9) long breaks between two orders: ranging from 32 to 200 days each. This pattern is characteristic for festive seasons: Christmas, Easter, summer holidays etc. Even though those orders with long gaps account for about 3% of all orders, less than 1% of them will become a "final" or last order, so they are not good at predicting churn.

How could I approach this problem to be able to better define and then model churn? Any suggestions are welcome. Feel free to ask for more info if needed

thanks, Kasia

I guess one way to model that problem is to assume that on normal days the waiting times between visits are distributed as some reasonably "compact" distribution, for example $\tau \sim Exp(1/2)$ however in festive season the waiting times are distributed as a mixture of original distribution and some much more extended distribution, for example N(20,5$^2$). You can then fit for those parameters (you may want to replace the normal and exponential distribution by something else). Here is an example likelihood for the waiting time between visits in a festive season. $$P(\tau|t_0,t_1,s_1,f) = f \exp(-\tau/t_0) + \frac{1-f}{\sqrt{2\pi} s_1} \exp (-\frac {(\tau-t_1)^2}{2s_1^2})$$