12.Estimation - Correlations

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MIT OpenCourseWarehttp://ocw.mit.edu

12.540Principles of Global Positioning SystemsSpring 2008

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12.540 Principles of the GlobalPositioning System

Lecture 12

Prof. Thomas Herring


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Estimation

Summary Examine correlations Process noise

White noise

Random walk First-order Gauss Markov Processes

Kalman filters Estimation in which the parametersto be estimated are changing with time


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Correlations

Statistical behavior in which random variables tend tobehave in related fashions

Correlations calculated from covariance matrix.Specifically, the parameter estimates from anestimation are typically correlated

Any correlated group of random variables can beexpressed as a linear combination of uncorrelated

random variables by finding the eigenvectors (linearcombinations) and eigenvalues (variances ofuncorrelated random variables).


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Eigenvectors and Eigenvalues

The eigenvectors and values of a square matrixsatisfy the equation Ax=x

If A is symmetric and positive definite (covariancematrix) then all the eigenvectors are orthogonal and

all the eigenvalues are positive. Any covariance matrix can be broken down into

independent components made up of theeigenvectors and variances given by eigenvalues.One method of generating samples of any randomprocess (ie., generate white noise samples withvariances given by eigenvalues, and transform using

a matrix made up of columns of eigenvectors.


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Error ellipses

One special case is error ellipses. Normally

coordinates (say North and East) are correlated andwe find a linear combinations of North and East thatare uncorrelated. Given their covariance matrix wehave:

n2 ne

ne e2

Covariance matrix;

Eigenvalues satisfy2(n2 +e2)+ (n2e2 ne2

) = 0

Eigenvectors:ne

1 n2

and2 e

2

ne


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Error ellipses

These equations are often written explicitly as:

The size of the ellipse such that there is P (0-1)probability of being inside is

1

2

=

1

2n

2 + e2 n

2 + e2( )2 4 n2e2 ne2( )

tan2= 2nen

2 e2

angle ellipse make to N axis

= 2ln(1 P)


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Error ellipses

There is only 40% chance of being in 1-sigmaerror (compared to 68% of 1-sigma in onedimension)

Commonly see 95% confidence ellipse whichis 2.45-sigma (only 2-sigma in 1-D).

Commonly used for GPS position and velocityresults


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Example of error ellipse

-8

-6

-4

-2

0

2

4

6

8

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0

Var2

Var1

Error Ellipses shown1-sigma 40%2.45-sigma 95%3.03-sigma 99%3.72-sigma 99.9%

Covariance2 22 4

Eigenvalues0.87 and3.66,

Angle -63o


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Process noise models

In many estimation problems there areparameters that need to be estimated butwhose values are not fixed (ie., theythemselves are random processes in someway)

Examples include for GPS Clock behavior in the receivers and satellites Atmospheric delay parameters Earth orientation parameters Station position behavior after earthquakes


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There are several ways to handle these types ofvariations: Often, new observables can be formed that eliminate the

random parameter (eg., clocks in GPS can be eliminated bydifferencing data)

A parametric model can be developed and the parameters ofthe model estimated (eg., piece-wise linear functions can beused to represent the variations in the atmospheric delays)

In some cases, the variations of the parameters are slow

enough that over certain intervals of time, they can beconsidered constant or linear functions of time (eg., EOP areestimated daily)

In some case, variations are fast enough that the process canbe treated as additional noise


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Characterization of noise processes

Firstly need samples of the process (often not easyto obtain)

Auto-correlation functions Power spectral density functions Allan variances (frequency standards) Structure functions (atmospheric delays) (see Herring, T. A., J. L. Davis, and I. I. Shapiro,

Geodesy by radio interferometry: The application ofKalman filtering to the analysis of VLBI data, J.Geophys. Res., 95, 1256112581, 1990.


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Characteristics of random processes

Stationary: Property that statistical propertiesdo not depend on time

Autocorrelation (t1,t2) = x1x2x1x2

f(x1,t1;x2,t2)dx1dx2

For stationary process only depends of = t1 t2

xx () = limT

1

2T x(t)x(t+ )dtPSD xx () = xx ()

eid


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Specific common processes

White-noise: Autocorrelation is Dirac-deltafunction; PSD is flat; integral of power underPSD is variance of process (true in general)

First-order Gauss-Markov process (one ofmost common in Kalman filtering)

xx () = 2e

xx () =22

2 +21

is correlation time


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Example of FOGM process


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Longer correlation time

Sh t l ti ti


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Shorter correlation time


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Summary

Examples show behavior of different correlationsequences (see fogm.m on class web page).

The standard deviation of the rate of changeestimates will be greatly effected by the correlations.

Next class we examine, how we can make anestimator that will account for these correlations.

Homework #2 in on the class web page (due April 5).

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12.Estimation - Correlations