Imagine we are running a logistic regression to identify opportunities for car sale promotion, using previous promotion campaign's result. Each $y$ is the increase of car sale after the promotion.
However, there is a problem here: once we launch a campaign and boost the sales for a specific month, another campaign in the next month won't boost the sales anymore, because anyone who wants to buy a car would have bought it in the first promotion.
Therefore, although the samples are independent, the results of the regression are NOT independent, because we are dealing with a dynamic system.
Probably we can add some predictors into the regression model, such as 'duration since last promotion', to make the prediction more realistic. That being said, just from a theoretical perspective, without introducing any new predictors, is there a way to introduce a constraint into the regression model, so that the resulting positive predictions are 'sparse' in some sense?
There are several ways we can define sparsity here. In a simple sense, we can demand that, if one sample is classified as $1$, the next $k$ samples must be $0$, or cannot contain more than $n$ $1$'s.
Or, we can demand that the sum of the distance between consecutive $1$'s must be above a threshold.
I don't really have a concert idea here. I think this looks like a control problem, similar to LQR.
Can anyone share some ideas or provide some references?
