Prediction of the Poisson process

Question

I want to predict the time between traffic arrivals with a Poisson distribution. I am currently creating a (synthetic) arrival time with a Poisson process, so the time between arrivals has an exponential distribution.

Watching past data, I want to predict the next / future time between arrivals. For this, I want to implement a learning algorithm.

I used various approaches, for example, a Bayesian predictor (maximally posterior) and a multilayer neural network. In both of these methods, I use a moving window of a certain length n of input functions (time between arrivals).

In the Bayesian predictor, I use the mutual arrival times as binary functions (1-> long, 0-> short to predict that the next time between arrivals will be long or short), while for the neural network of n-neurons the input layer and the hidden layer m -neurons (n ​​= 13, m = 20), I entered n previous times between arrivals and generated the expected estimated time of arrival (threshold scales are updated by the backpropagation algorithm).

The problem with the Bayesian approach is that it becomes biased if the number of short intervals between receptions is greater than long. Thus, he never predicts a long period of inactivity (since the trailing edge of a short one always remains large. While in a multilayer neural predictor, the accuracy of the prediction is insufficient. Especially for higher times between arrivals, the accuracy of the prediction decreases sharply.

My question is "Can a random process (Poisson) be predicted with good accuracy? Or is my approach wrong?" . Any help would be appreciated.

Answer 1

If this really follows the distribution of the Poissons, you can predict the probability that the next traffic element will reach a given interval, and the probability curve is just a normalized integral (i.e., a curve with asymptote 1) of the Poisson distribution. Why all the riots with neural networks / Bayesian predictors?

Answer 2

Well, if the generation process is a homogeneous Poisson process, there is not much to predict, is it? There, a speed parameter that remains constant in time and can be estimated trivially, but past this, recent history should not affect the time between receptions. You use the binary functions of recent arrivals, but the whole point of homogeneous Poisson processes is that arrivals are exponential and exponential distributions without memory.

Now, if the homogeneous assumption is wrong, you need to think more about the details, and the answer depends on what middle measure you want to use for the process. Take a look at the Cox processes (double stochastic Poisson processes, where the average measure is also a random variable) or, possibly, the Hawks processes (where each arrival causes a surge in further activity).