After all, if you have a system whose future evolution is important to predict, why not try to infer a plausible model instead of a convenient one?
I am in the process of taxonomising here. Stuff which fits the particular (likelihood) model of recursive estimation and so on will be kept there. Miscellaneous other approaches here.
A compact overview is inserted incidentally in Cosma’s review of Fan and Yao —- FaYa03 —- (wherein he also recommends Bosq98, TaKa00 and BoBl07.)
To reconstruct the state, as opposed to the parameters, you do state filtering. There can be interplay between these steps, if you are doing simulation-based online parameter inference, as in recursive estimation.
Anyway, for what kind of systems can you infer parameters? Mutually exciting point processes? Yep, EFBS04 do that.
From an engineering/control perspective, we have BrPK16, who give a sparse regression version. Generally it seems it can be done by indirect inference, or recursive hierarchical generalised linear models, generalising the process for linear time series.
There are many highly general formulations; Kita96 gives a Bayesian “smooth” one.
See e.g. the HeDG15 paper:
We address […] these problems with a new view of predictive state methods for dynamical system learning. In this view, a dynamical system learning problem is reduced to a sequence of supervised learning problems. So, we can directly apply the rich literature on supervised learning methods to incorporate many types of prior knowledge about problem structure. We give a general convergence rate analysis that allows a high degree of flexibility in designing estimators. And finally, implementing a new estimator becomes as simple as rearranging our data and calling the appropriate supervised learning subroutines.
[…] More specifically, our contribution is to show that we can use much-more- general supervised learning algorithms in place of linear regression, and still get a meaningful theoretical analysis. In more detail:
we point out that we can equally well use any well-behaved supervised learning algorithm in place of linear regression in the first stage of instrumental-variable regression;
for the second stage of instrumental-variable regression, we generalize ordinary linear regression to its RKHS counterpart;
we analyze the resulting combination, and show that we get convergence to the correct answer, with a rate that depends on how quickly the individual supervised learners converge
Also, sparsely or unevenly observed series are tricky. I’m looking at those at the moment.
- Pereyra et al (PSCP16)
Modern signal processing (SP) methods rely very heavily on probability and statistics to solve challenging SP problems. Expectations and demands are constantly rising, and SP methods are now expected to deal with ever more complex models, requiring ever more sophisticated computational inference techniques. This has driven the development of statistical SP methods based on stochastic simulation and optimization. Stochastic simulation and optimization algorithms are computationally intensive tools for performing statistical inference in models that are analytically intractable and beyond the scope of deterministic inference methods. They have been recently successfully applied to many difficult problems involving complex statistical models and sophisticated (often Bayesian) statistical inference techniques. This paper presents a tutorial on stochastic simulation and optimization methods in signal and image processing and points to some interesting research problems. The paper addresses a variety of high-dimensional Markov chain Monte Carlo It also discusses a range of optimization methods that have been adopted to solve stochastic problems, as well as stochastic methods for deterministic optimization. Subsequently, areas of overlap between simulation and optimization, in particular optimization-within-MCMC and MCMC-driven optimization are discussed.
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