# Probability of the occurrence of an event over time

I need to answer the following question:

What is the probability that the event 1 will occur at some point in the time for a new sample? At which time point is this more likely to occur?

1: event occurred at a specific time point, 0: event did not occur, -1: the opposite of the requested event occurred (also not interesting).

For each sample, 6 time points are measured over time. Let us assume this is for example every week (1w, 2w, 3w, 4w, 5w, 6w).

sample 1 = (0, 0, 0, 0, 0, 1)
sample 2 = (0, 0, 1, 1, 1, 1)
sample 3 = (0, 0, 0, 0, 1, 1)
sample 4 = (0, 0, 0, 0, 0, 1)
sample 5 = (0, 0, 0, 1, 1, 1)
sample 6 = (0, 0, 0, 0, 0, 1)
sample 7 = (0, -1, -1, -1, -1, 0)
sample 8 = (0, 0, 0, 0, 0, 0)


What model would be appropriate for this?

Apparently, there are also "noisy" samples where we do not observe the event (samples 7, 8).

This can be as a typical time-series prediction and/or sequence modeling problem. The bad news is that 6 time-stamps is pretty short and you may find it more difficult to model compared to long sequences.

A simple statistical analysis (and not necessarily a ML model) is actually to calculate the probabilities. You have 6 time-stamps ($$T$$), 3 values ($$X$$) and $$N$$ samples. You can have an overall idea by calculating all $$P(X_{i}=x|T=t, X_{i-1}=y)$$ where $$x,y\in X$$ (i.e. $$x$$ and $$y$$ get values from -1, 0 and 1).

This is pretty Markovian as I took only the previous event responsible for predicting the current event. Sequence models can learn the history thus giving more holistically reliable results.

PS: Wait for better solutions. It is just a starter idea.

• Thanks Kasra, you are right that samples are too few. But I could collect more data if there is a method for approaching the problem. Also this data is more like an example to explain what type of data I have. Mar 16, 2022 at 10:04
• Of course if you can collect "longer sequences" you can simply feed it to any sequence model, ranging from classic HMM to RNNs like LSTM and ESN, and get a pretty descent result. But still the idea above is not that stupid. I would suggest to at least try it in case you have a blocking point. This is very close to how all Bayesian Models work. Good luck and let me know if I you need more info and upvote if you got a little help from answer :))) Mar 16, 2022 at 10:10
• So it seems that I need longer sequences to be able to perform HMM and probably more samples (sequences) to be able to compute the Markovian probabilities. But given the data I have I could still compute the probabilities based on Markovian but I should have in mind that the sequences are very few. Do I get it right? (seems I need a bit more reputation for the upvote :) ) Mar 16, 2022 at 10:27
• That is true! Sequence models "work" on even short sequences but it is about "how well they work". Longer sequences, better performance Mar 16, 2022 at 10:37
• Thanks Kasra. Would a simple correlation metric also work here? Like if we would observe a pattern that the 0s increase from 0 to 1 across samples (independently of the time point that the transition from 0 to 1 occurs). Mar 16, 2022 at 14:31