statistical orbit determination smoothing monte carlo simulation

Colorado Center for Astrodynamics Research The University of Colorado 1

STATISTICALORBIT DETERMINATION

SmoothingMonte Carlo Simulation

ASEN 5070

LECTURE 33

11/18/09

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Smoothing

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After some algebra it can be shown that

Smoothing

After someafter some algebra algebra

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Smoothing

Finally

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Smoothing

The equation for the smoothed covariance is given byThe equation for the smoothed covariance is given by

the

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Smoothing computational algorithm

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Filtering - Example (Problem 41 of Ch. 4 Exercises)

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where

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Computational Algorithm for Smoother

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Computational Algorithm for Smoother

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Smoothing - Example (Problem 42 of Ch. 4 Exercises)

Solve for the smoothed history of using the computational algorithm for smoother. Plot the true values of η, the filter values (determined by the algorithm below), and the smoothed values. Compute the RMS of the smoothed fit (data is noised single cycle of a sine wave). The sequential algorithm is given by

where

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Smoothing - Example (Problem 42 of Ch. 4 Exercises)

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Filter – Smoother Consistency Check

The smoother covariance at stage k is given by

where

is the filter covariance at stage k

and

is the filter covariance at stage k + 1 based on k observations

kkP

1k

kP

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Calculate the n x n difference matrix difference between the filter and smoother covariance at stage k

for k {0, 1, 2, …, }.

• should be non negative definite, i.e. have no negative eigenvalues

• Denote the square root of the ith main diagonal element of as

• Calculate the n x 1 difference vector between the filtered and smoothed state estimate

kk k kP P P

kP

kP ik

k x

ˆ ˆkk k kx x x

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Calculate the ratio

for each i {0, 1, 2, …, n} and k {0, 1, 2, …, }

• If for each i and k, we have

Then the filter-smoother solutions are said to be globally consistent.

• If for each i and k, we have

The filter-smoother solutions have globally failed the consistency test.

/i ik k kR x

3ikR

3ikR

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Monte Carlo Simulation

George H. BornASEN 5070 Statistical Orbit Determination

The University of Colorado, Boulder

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Monte Carlo Simulations

Assume we have a state vector X with associated error covariance TP E xx where x

is a vector of zero mean error realizations of the state vector X. Hence

TP E xx

factor P into

TP S S

where S is upper triangular and can be computed via Cholesky decomposition or orthogonal transformations. Note that S is not unique.

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Pre-multiply P by TS and post-multiply by 1S

1 1T T TS P S S S S S I

using

1 1

T

T T T

P E

S P S E S S I

xx

xx

let TS e x

so TE I ee

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Therefore e can be realized as an ,N O I vector of random numbers, and x

calculated from

TSx e

Therefore x is a realization of errors of the vector X for which P is the error covariance.

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Implementation procedure using Matlab

Given an n-vector X and P, compute S

cholS P

Generate an n-vector of Gaussian random numbers with ,N O I

randn ,1ne

If desired, an n-vector of Gaussian random number, b, with mean M and variance 2 can be computed from

2sqrt randn ,1M n b

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Compute a realization of error in x from

TSx e

Generate a new realization of X

n e w X X x

n e wX will have P as its error covariance

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Another realization may be computed by generating a new vector of random numbers, e.

Unless you specify the seed, Matlab will generate a different random vector each time randn n,1 is used

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We could also use A P in place of S

1 1

P A A

A P A I

Let

1Ae x , i s ,N O Ie

then

Ax e

Note that TA Se e

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Hence this will be a different realization of x given the same random vector, e.

However, it can be shown that

S Q A

where Q is an orthogonal transformation matrix.

Therefore,

1

2

T TS A Q

A

x e e

x e

and 1x and 2x have the same Euclidean norm.

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Proof that S Q A : from TP S S A A

using an orthogonal transformation matrix

T TP S S A Q Q A

let B Q A

then T TS S B B

or S B

S Q A

Because S is not unique i.e. there is more than one value of S for which

TS S P

We must choose Q so that S is upper triangular. If we do not require S to be triangular there are an infinite number of solutions for S, i.e. Q may be any orthogonal matrix.

statistical orbit determination smoothing monte carlo simulation

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