Predicting a continuous outcome using point process descriptors I have measured a series of times for discrete events along with a continuous variable.
So essentially I measure a point process $P: t_1, t_2, \dots, t_n$ and values $A_1(t=x_1), A_2(t=x_2), \dots, A_m(t=x_m)$ for the continuous variable $A$.
$A$ is sampled at regular intervals ($x_1, x_2, \dots, x_m$), which are always the same, independently of when the events in $P$ happen, so essentially in most cases $n \neq m$.
I have several of these measures, so at the end I find myself with a dataset like:
$t_{11}, t_{12}, \dots, t_{1n}; A_{11}, A_{12}, \dots, A_{1m}$
$t_{21}, t_{22}, \dots, t_{2n}; A_{21}, A_{22}, \dots, A_{2m}$
$\dots$
$t_{N1}, t_{N2}, \dots, t_{Nn}; A_{N1}, A_{N2}, \dots, A_{Nm}$
Now, I have reason to believe there is a relationship between the value of $A$ and the happening of an event in $P$. So, for instance, the higher $A$ the more likely $P$ is to happen. I should also add that the opposite is also true: that is, the happening of an event in $P$ may influence subsequent values of $A$.
With all this in mind, how would you proceed to model this? Ultimately what I would like to do is having a predictive model to determine $A$ given an arbitrary point process.
 A: I can think of many ways to models such a time series, for example, do you suspect that there is a temporal dependency between successive A values? I will present what I believe is a simple way to start. 
For each $A_{ij}$ estimate the rate of the point process $P$. This could of course be done in many ways where a simple method is to put a window around $A_{ij}$ and count the number $N_{ij}$ of occurrences from $P$ in that window. The width of the window could be decided from what you know about $P$ and could later be optimized by using, for example, cross validation. Now when you have a $N_{ij}$ for each $A_{ij}$ you can put this into a regression model: 
$$A_{ij} \sim \beta_0 + \beta_1 \cdot N_{ij}$$
This would just be a first step and there is much to play around with:


*

*Instead of using a rectangular window you could experiment with many other window functions.

*You can explore different regression models. If you believe there might be time dependencies in your data you could look into forecasting techniques such as vector autoregression.

*If prediction is your goal you could optimize your model(s) by evaluating different sets of parameters and models by cross validation.

