Application of the transformation of variables technique for uncertainty mapping in nonlinear filtering

Abstract

This paper addresses the impact nonlinear observations of state variables have on uncertainty accuracy associated with state estimation algorithms. The transformation of variables technique is applied to exactly map probability density functions (PDFs) between domains completely spanned by different combinations of basis vectors. The technique allows for proper generation of the likelihood density when converting from measurement to state variable space and for association of a present state distribution with prior observation data. The exact mapping of probability distribution functions between domains and proper characterization of prior knowledge allows for Bayesian estimation to be appropriately carried out. A Bayes filter utilizing the technique is developed which uses the technique to map the uncertainty in time for generation of the prior density and in space for generation of the likelihood density. The filter is compared with conventional nonlinear filtering techniques in multiple scenarios to demonstrate the utility and insight offered for object tracking and parameter estimation applications.

Appendix

This appendix demonstrates the TOV technique for instances where the desired domain has the same number of base functions as the initial domain and when the desired domain as less base functions than the initial domain.

1.1 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m , m = n$

Consider the two-dimensional mapping of Cartesian coordinates, $\mathbf {x} \doteq \left( x, y\right)$, from a range and angle pair set, $\mathbf {z}\doteq \left( \rho ,\alpha \right)$, Eq. (46). Let the spherical domain be the initial domain with a given continuous joint PDF, $p\left( \mathbf {z}\right)$, the Cartesian domain PDF, $p\left( \mathbf {x}\right)$, is computed by applying Eq. (5) and the result shown in Eq. (47).

$$\begin{aligned} \begin{aligned} \begin{array}{ccl|ccl} \rho &{}=&{} \sqrt{x^2+y^2}&{}x&{}=&{}\rho \cos (\alpha )\\ \alpha &{}=&{} \arctan \left[ \frac{x}{y}\right] &{}y&{}=&{}\rho \sin (\alpha )\\ \end{array} \end{aligned} \end{aligned}$$

(46)

$$\begin{aligned} p\left( \mathbf {x}\right) =p\bigg (\mathbf {z}=\varvec{\psi }\left( \mathbf {x}\right) \bigg )\bigg \vert \rho \bigg \vert _{\mathbf {z}=\varvec{\psi }\left( \mathbf {x}\right) }^{-1} \end{aligned}$$

(47)

1.2 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m , m < n$ (Auxiliary Variable Method)

When concerned with a subset of variables related to the initial domain, the method of auxiliary variables (Papoulis 1991) can be applied to populate a square Jacobian. Let the auxiliary variable be the angle, $\alpha$, this results in a square Jacobian, Eq. (48), and joint and marginal PDFs shown by Eq. (49). With $\rho$ as the auxiliary variable, Eq. (50) would result. Eqs. (49) and (50) are derived differently based upon elimination of auxiliary variables but are equivalent due to the invariance of total probability.

$$\begin{aligned} \frac{\partial \left( x, \alpha \right) }{\partial \left( \rho ,\ \alpha \right) }=\underbrace{\left[ \begin{array}{cc} \cos (\alpha )&{}-\rho \sin (\alpha )\\ 0&{}1 \end{array}\right] }_{J}\left[ \begin{array}{c}\rho \\ \alpha \end{array}\right] \end{aligned}$$

(48)

$$\begin{aligned} \begin{aligned} p\left( x,\ \alpha \right)&=p\bigg (\rho = \frac{x}{\cos (\alpha )},\ \alpha = \alpha \bigg )\bigg \vert \cos (\alpha )\bigg \vert ^{-1}\\ p\left( x\right)&= \int \limits _{\alpha _0}^{\alpha _f}p\left( x,\ \alpha \right) d\alpha \end{aligned} \end{aligned}$$

(49)

$$\begin{aligned} \begin{aligned} p\left( x,\ \rho \right)&=p\bigg (\rho =\rho ,\ \alpha = \arccos (x/\rho )\bigg )\bigg \vert \ \rho \sin (\alpha )\bigg \vert _{\alpha =\arccos (x/\rho )}^{-1}\\ p\left( x\right)&= \int \limits _{\rho _0}^{\rho _f}p\left( x,\ \rho \right) d\rho \end{aligned} \end{aligned}$$

(50)

1.3 Transformation of variables for $\mathfrak {R}^n\rightarrow \mathfrak {R}^m\ ,\ m < n$ (Dirac delta Method)

The reduced state variable PDF representation for continuous or discrete random variables can be computed using the Dirac delta method (Au and Tam 1999). The method applies the Dirac generalized function in order to transform only the needed variables to the state(s) of interest, the continuous random variables the application is given by Theorem 1. The Kronecker delta scaling, sifting, and translation properties allow for the theorem to be carried out. Theorem 1 produces Eq. (49) without the need for evaluating the Jacobian, Eq. (52).

Theorem 1

Suppose that $z_i,\,i=\left[ 1,n\right]$, are continuous random variables with joint probability distribution $p\left( z_1,\ z_2, \ldots , \ z_n\right)$. Let $\fancyscript{D}$ be the $n$-dimensional set of every possible outcome of the $z_i$’s. Then the continuous random variable

$$\begin{aligned} x = \varvec{\psi }^{-1}\left( z_1,\ z_2, \ldots , \ z_n\right) \end{aligned}$$

has the probability distribution given by use of the Kronecker delta $\delta _{a,b}$ in the form

$$\begin{aligned} p\left( x\right) =\int \limits _\fancyscript{D^\mathbf {z}}p\left( z_1,\ z_2, \ldots , \ z_n\right) \delta \left[ \varvec{\psi }^{-1}\left( z_1,\ z_2, \ldots , \ z_n\right) -x\right] dz_1dz_2\ldots dz_n. \end{aligned}$$

(51)

$$\begin{aligned} p\left( x\right)&= \int \limits _{\fancyscript{D}^\rho }\int \limits _{\fancyscript{D}^{\alpha }}p\left( \rho ,\alpha \right) \delta \left[ \rho \cos (\alpha )-x\right] d\rho d\alpha \nonumber \\&= \int \limits _{\fancyscript{D}^\rho }\int \limits _{\fancyscript{D}^{\alpha }}\frac{p\left( \rho ,\alpha \right) }{\left| \cos (\alpha )\right| }\delta \left[ \rho -\frac{x}{\cos (\alpha )}\right] d\rho d\alpha \nonumber \\&= \int \limits _{\alpha _0}^{\alpha _f}\frac{1}{\cos (\alpha )}p\left( \rho =\frac{x}{\cos (\alpha )},\alpha \right) d\alpha \end{aligned}$$

(52)