Algorithms for predicting a couple points in the future I'm familiar with supervised learning algorithms like regression and neural networks which look at a bunch of input points and learn a function which outputs a value (the value varying depending on whether the algo is a classifier, logistic regression, or standard regression)..but I am facing a problem now where each observation I have is a couple of data points (TAQ data), and then after a fixed amount of these chronologically sorted events there is a spike and what appears to follow an exponential decay mean reversion. Every such instance described here is one of my observations and my goal is to predict/model this exponential decay mean reversion.
What type of algorithms would do something like this? time series models a-la ARCH/GARCH followed by a prediction look-ahead of N steps after training them or what else?
Thank you, any type of suggestions or advice/references are greatly appreciated.
 A: To predict "a couple of points in the future" summarizes the singular focus of my life's work. To make a prediction I believe you must have a trusty signal. To me a trusty signal is one that is the result of separating signal from noise. The "proof of noise" allows one to state the signal has neither too many or too few statistically significant parameters. Neural Networks in my opinion is "regression without ethics" as it fits rather than constructs valid likelihood ratio tests to augment/simplify a current candidate model.On the other hand the identification of Pulses , level Shifts , Seasonal Pulses and local time trends are conducted using formal likelihood ratio tests for detection and retention . There are a number of requirements for an error series to be "proven" to be white noise. Failure of one or more of these assumptions leads directly to model augmentation prior to usage. Some of those augmentation strategies include forming a minimally sufficient ARIMA model, validating that the mean of the errors does not differ significantly from zero for all intervals , validating that the error process is homogeneous and thus free of deterministic change points , dependence of the expected value and the variability of the process and finally a lack of ARIMA structure needed to coax the variance to stability. I would suggest that you post one of your actual time series data sets and let the list readers pursue this problem at "ground zero" . A comparison or bake-off between different approaches would I believe be of interest to all.
