I am working with time-series data in Python to see if variables like the time of day and the day of the month and the month of the year affect attendance at a gym. I have read up on encoding the time series data cyclicly using sine and cosine. I was wondering if you can do the same thing for the day of the month. The reason I ask is that, unlike the number of months in a year or the number of days in a week, the number of days in a month is variable (for example, February has 28, whereas March has 31). Is there any way to deal with that?
Here is a link describing what I mean by cyclic encoding: https://ianlondon.github.io/blog/encoding-cyclical-features-24hour-time/
Essentially, what this is saying is that you can't just convert the hour into a series of values like 1, 2, 3, ..., 24 when you are doing machine learning because that implies that the 24th hour is further away (from a euclidean geometric perspective) from the 1st hour than the 1st hour is from the 2nd hour, which is not true. Cyclical encoding (assigning sine and cosine values to each hour) allows you to represent the fact that the 24th hour and the 2nd hour are equidistant from the 1st hour.
My question is that I do not know if this cyclical conversion will work for days in a month, seeing as different months can have different numbers of days.
You can implement this by dividing each month into 2π radians; then in a 28-day month, a day is 0.2234 while in a 31-day month, a day is 0.2026.
This obviously introduces a skew where a shorter month will appear to take up as much time as a longer one; but it will satisfy your requirement. If you only use this metric for normalizing a single feature, that should be inconsequential, and let you achieve the stated goal.
If you have points in time with a finer granularity than a day, you obviously can and probably should normalize those into the same projection.