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 amin Total Posts: 279 Joined: Aug 2005
 Posted: 2018-06-18 21:23 Dear friends, I was able to find very interesting applications of my research to non-linear filtering that would make non-linear filtering computationally very simple and efficient. The new method would work equally well for discrete time state space models. I will try to explain the idea with Bayesian filtering applied to a very simple set of univariate discrete time state space equations giving observation process equation and the State process equation.Before we work on a complete algorithm, there are a few principles that we need for that purpose.1. Mapping an arbitrary density on a normal (or other) density.We can map any density on a normal density. Densities have probability mass normalized to one and this mapping requires that corresponding subdivisions on two densities have the same probability mass. Sometimes, we would like to map a density on a normal density so that subdivisions on the normal density are equally spaced.2. Non-linear transformation of a densityWe can easily take non-linear transformation of a density and this requires scaling the density variable by the appropriate transformation but the density changes everywhere and new density for the transformed variable can be calculated from the pre-transformation density by finding the change of variable derivative for the densities. 3. Adding locally non-linear innovations to a density.Just like we did for SDEs, we can easily add local possibly non-linear (in the SDE variable) normal-based innovations to a density and calculate the resulting density. For this we first map the density to a normal density and then add innovations linearly along the appropriate value of normal variable associated with the particular subdivision of the density.4. Calculating Transition probabilities from one density to the other density when both densities have been mapped on a normal density.This is far simpler than people would otherwise think. And it would only require Green's function that takes into account appropriate variance of the innovations and most local non-linearities can be easily neglected. For example, in the case of SDEs, we could map the two densities of SDE on Brownian motion densities(by equating the probability mass) and then transition probabilities between any two subdivisions would equal the transition probabilities between corresponding two subdivisions of the Brownian motion densities and this requires only appropriate calculation of time elapsed Δt. And any other non-linearities in the SDE evolution can be safely neglected. Other non-linearities show up in local scaling of the density but they can very easily be accounted for by a change of variable for densities. I will be explaining it more clearly with equations tomorrow.5. Calculation of Filtered density once an Observation has been made.Suppose we have an updated state density and we possibly take a transformation of it (as dictated by observation/measurement equation) and then we add observation noise (or observation innovation) to it to calculate what I call the observation/measurement density. Calculation of filtered density from this observation/measurement density simply requires calculation of all the transition probabilities from the updated state density that would result in the observed point estimate(on the measurement/observation density). Again this filtered density is the transition density from all the points on the updated state density to the observed point on the observation/measurement density. This would not in general be a normalized density and we have to normalize it and this normalized density is the filtered density. And then we update this filtered density according to state update equation to calculate the updated state density. This would possibly require a transformation of the filtered density(2), mapping the transformed density on a normal density(1) and then adding local update innovations(3) required to calculate the state update density. And then we will again map the state update density on a normal density(1) and add observation noise/innovations(3) to calculate the measurement density. And then again find filtered density(5) again.Sorry that when I wrote about mapping a density of SDE on a normal density, I forgot to mention another important relevant fact and that is when densities of SDEs are generated by using Ito-Taylor algorithm which gives us the SDE variable X,as a function of standard normal, Z, and in that caseF(X(Z))=F(Z)where F means CDF of the density. So when SDE variable X is found as a function of standard normal variable Z, the CDF of X is exactly the same as the CDF of corresponding Z. And SDE variable X is a local scaling of the standard normal variable.we can calculate the transition probabilities of SDEs by mapping the density of SDEs on relevant generating Brownian motion densities and then transition probability between various subdivisions on SDE density would be exactly the same as the transition probability between the corresponding subdivisions of two Brownian motion densities. Here is the link to the paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3119980But I believe that normal density and densities of SDEs share a far stronger result which says that when we divide the evolving normal density or density of SDEs along CDF subdivisions or probability mass subdivisions, then the cumulative transition probabilities between different CDF subdivisions of both normal densities at different time remain the same. We are used to looking at transition probabilities in terms of absolute grids. Greater variance simply expands the CDF subdivisions and on fixed grids, it seems that density is expanding which is, of course, true with respect to fixed grids . But when considered in terms of CDF subdivisions (fixed probability mass subdivisions i.e subdivisions that expand so as to keep the probability mass the same in them), the local subdivisions also expand in a synchronized fashion so that transition probabilities between various expanding subdivisions really remain the same though on fixed grids one would only notice that normal density/SDE density is expanding while it remains constant when considered in terms of locally expanding fixed probability mass density subdivisions while transition probability dynamics remain constant in terms of expanding subdivisions.
 amin Total Posts: 279 Joined: Aug 2005