US 5,963,888 A · Fast method and apparatus for nonlinearly transforming mean and covariance estimates

Abstract

A system and a method for estimating the mean and covariance resulting from the application of a given nonlinear function or transformation when applied to a given mean and covariance derived from measurements of a physical system or process. The method comprises selecting a set of points having an arbitrary mean different from the given mean, but with covariance about the given mean equal to that of the given covariance; applying the given nonlinear function to each element of the selected set of points; computing an estimate of the mean of the nonlinearly transformed given mean and covariance; and computing the covariance of the nonlinearly transformed set of points about the estimated mean. The system and method provide an efficient procedure for accurately incorporating nonlinear models of quantities of interest in tracking systems, control mechanisms, and many other related signal processing systems.

Claims20

Dependent claims

A signal processing system for generating a mean vector and covariance matrix pair which estimates the result of applying a given nonlinear function f( ) to a given mean vector a and covariance matrix A, where said mean and covariance {a,A} are derived in part from measurements of a physical system or process, where said signal processing system comprises:

(1) means for measuring a mean and covariance estimate {a,A} of variables relating to a physical system or process;
(2) means for generating a first set of m vectors, {x.sub.1,x.sub.2, . . . ,x.sub.m,}, such that a minus 1/m times the sum of the elements of the set is nonzero, and a set of weights (w.sub.1,w.sub.2, . . . , w.sub.m }, the sum of the weights being W, so that a sum of 1 /W*w.sub.i *(x.sub.i -a)*(x.sub.i -a).sup.T, for index i ranging from 1 to m, is equal to A;
(3) means for determining a second set of vectors by applying a nonlinear function f( ) to each element of the first set of vectors to obtain a second set of vectors, {f(x.sub.1),f(x.sub.2), . . . , f(x.sub.m)};
(4) means for computing a transformed mean a.sup.+ ;
(5) means for computing a transformed covariance as a sum of 1/W*w.sub.i *(f(x.sub.i) -a.sup.+)*(f(x.sub.i)-a.sup.+).sup.T, for index i ranging from 1 to m;
and
(6) means for physically responding to a signal relating to the transformed mean and covariance.

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A computer readable memory storing instructions for configuring a signal processing system to perform a method according to the invention for generating a mean vector and covariance matrix pair which estimates the result of applying a given nonlinear function f( ) to a given mean vector a and covariance matrix A, where said mean and covariance {a,A} are derived in part from measurements of a physical system or process, where said signal processing system comprises:

(1) means for measuring a mean and covariance estimate {a,A} of variables relating to a physical system or process;
(2) means for generating a first set of m vectors, {x.sub.1,x.sub.2, . . . , x.sub.m }, such that a minus 1/m times the sum of the elements of the set is nonzero, and a set of weights {w.sub.1,w.sub.2, . . . , W.sub.m }, the sum of the weights being W, so that a sum of 1/W*w.sub.i *(x.sub.i -a)*(x.sub.i -a).sup.T, for index i ranging from 1 to m, is equal to A;
(3) means for determining a second set of vectors by applying a nonlinear function f( ) to each element of the first set of vectors to obtain a second set of vectors, {f(x.sub.1),f(x.sub.2), . . . , f(x.sub.m)};
(4) means for computing a transformed mean a.sup.+ ;
(5) means for computing a transformed covariance as a sum of 1/W*w.sub.i *(f(x.sub.i) -a.sub.+)*(f(x.sub.i)-a.sup.+).sup.T, for index i ranging from 1 to m;
and
(6) means for physically responding to a signal relating to the transformed mean and covariance.

4 more elements

Dependencies are inferred from the claim text.

Fast method and apparatus for nonlinearly transforming mean and covariance estimates

Life of this patent

Fast method and apparatus for nonlinearly transforming mean and covariance estimates

Life of this patent

Abstract

Claims20