Predict with some probability the day of the month paycheck received through daily transactions

I am using R to do machine learning. I have daily shopping expenditure data of individuals over a couple of months and my goal is to be able to identify through their shopping pattern the day of the month they receive their paychecks. That is, to measure with some probability the day of the month they received their paycheck. Paycheck can be received on the 1st of the month or weekly or bi-weekly. Hence, my goal is to look at their daily shopping expenditure pattern to predict with some probability the day of of the month they receive their paycheck with some probability.

I guess this would be an unsupervised learning problem since my goal is to predict the day the salary was received and I don't have it recorded in the data.

I had the following steps in mind. But stuck on what to do next and if it is the most efficient way to go about it?

Step 1: detect local peaks of daily shopping pattern for each individual over a month.

Step 2: from the local peaks, detect the global maxima, i.e the highest peak among the local.

Step 3: record the associated day of the month the global peak was observed.

Please let me know if this seems logical. And if there are resources out there that I can use for help.

Thank you.