12. 540 Principles of the Global Positioning System Lecture 12 Prof. Thomas Herring 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 1 12.540 Principles of the Global Positioning System Lecture 12 Prof. Thomas Herring
Estimation Summary Examine correlations Process noise White noise · Random walk First-order Gauss Markov Processes Kalman filters- Estimation in which the parameters to be estimated are changing with time 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 2 Estimation • Summary – Examine correlations – Process noise • W hit e n ois e • Random walk • First-order Gauss Markov Processes – Kalman filters – Estimation in which the parameters to be estimated are changing with time
Correlations Statistical behavior in which random variables tend to behave in related fashions Correlations calculated from covariance matrix Specifically, the parameter estimates from an estimation are typically correlated Any correlated group of random variables can be expressed as a linear combination of uncorrelated random variables by finding the eigenvectors(inear combinations)and eigenvalues(variances of uncorrelated random variables) 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 3 Correlations • Statistical behavior in which random variables tend to behave in related fashions • Correlations calculated from covariance matrix. Specifically, the parameter estimates from an estimation are typically correlated • Any correlated group of random variables can be expressed as a linear combination of uncorrelated random variables by finding the eigenvectors (linear combinations) and eigenvalues (variances of uncorrelated random variables)
Eigenvectors and Eigenvalues The eigenvectors and values of a square matrix satisfy the equation AXEnX If a is symmetric and positive definite(covariance matrix) then all the eigenvectors are orthogonal and all the eigenvalues are positive independent components made up of the into Any covariance matrix can be broken doy eigenvectors and variances given by eigenvalues One method of generating samples of any random process(ie, generate white noise samples with variances given by eigenvalues, and transform using a matrix made up of columns of eigenvectors 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 4 Eigenvectors and Eigenvalues • The eigenvectors and values of a square matrix satisfy the equation Ax = λ x • If A is symmetric and positive definite (covariance matrix) 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 the eigenvectors and variances given by eigenvalues. One method of generating samples of any random process (ie., generate white noise samples with variances given by eigenvalues, and transform using a matrix made up of columns of eigenvectors
Error ellipses One special case is error ellipses. Normally coordinates(say North and East) are correlated and we find a linear combinations of north and east that are uncorrelated. Given their covariance matrix we have: Covariance matrIX; Eigenvalues satisfy(o+o)n+(ofdf-0n)=0 Eigenvectors 12 n-OF 2/ and 03/18/02 12.540Lec12
03/18/02 12.540 Lec 12 5 Error ellipses • One special case is error ellipses. Normally coordinates (say North and East) are correlated and we find a linear combinations of North and East that are uncorrelated. Given their covariance matrix we have: σn 2 σne σne σe 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Covariance matrix; Eigenvalues satisfy λ 2 − (σn 2 + σe 2 ) λ + (σn 2 σe 2 − σne 2 ) = 0 Eigenvectors: σne λ1 − σn 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and λ2 − σe 2 σne ⎡ ⎣ ⎢ ⎤ ⎦ ⎥
Error ellipses These equations are often written explicitly as tang 26 w angle ellipse make to n axis The size of the ellipse such that there is P( 0-1) probability of being inside is p=√-2ln(1-P 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 6 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 λ1 λ2 ⎫⎬ ⎭ = 1 2 σ n 2 + σ e 2 ± σ n 2 + σ e 2 ( )2 − 4 σ n 2 σ e 2 − σ ne 2 ( ) ⎛ ⎝ ⎜ ⎞ ⎠ tan 2φ = 2 σ ne σ n 2 − σ e 2 angle ellipse make to N axis ρ = − 2ln( 1 − P )
Error ellipses There is only 40% chance of being in 1-sigma error(compared to 68% of 1-sigma in one dimension Commonly see 95% confidence ellipse which is 2.45-sigma(only 2-sigma in 1-D) Commonly used for GPS position and velocity results 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 7 Error ellipses • There is only 40% chance of being in 1-sigma error (compared to 68% of 1-sigma in one dimension) • Commonly see 95% confidence ellipse which is 2.45-sigma (only 2-sigma in 1-D). • Commonly used for GPS position and velocity results
Example of error ellipse TTTTTTTTTTTTTTTTT Error Ellipses shown variance 1-sigma 40% 2.45-sigma 95% 303-igma99% 4}3.72sgma99.9% 24 Eigenvalues 0.87and3.66 Angle-63° -6 -8 -8.0-60-4.0-2.00.02.04.06.080 Var1 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 8 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 shown 1-sigma 40% 2.45-sigma 95% 3.03-sigma 99% 3.72-sigma 99.9% Covariance 2 2 2 4 Eigenvalues 0.87 and 3.66, Angle -63 o
Process noise models In many estimation problems there are parameters that need to be estimated but whose values are not fixed (ie. they themselves are random processes in some way) EXamples include for GPS Clock behavior in the receivers and satellites Atmospheric delay parameters Earth orientation parameters Station position behavior after earthquakes 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 9 Process noise models • In many estimation problems there are parameters that need to be estimated but whose values are not fixed (ie., they themselves are random processes in some way) • Examples include for GPS – Clock behavior in the receivers and satellites – Atmospheric delay parameters – Earth orientation parameters – Station position behavior after earthquakes
Process noise models There are several ways to handle these types of variations Often, new observables can be formed that eliminate the random parameter(eg, clocks in GPS can be eliminated by differencing data) A parametric model can be developed and the parameters of the model estimated(eg. piece-wise linear functions can be used 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 be considered constant or linear functions of time(eg, EOP are estimated daily In some case, variations are fast enough that the process can be treated as additional noise 03/1802 12.540Lec12
03/18/02 12.540 Lec 12 10 Process noise models • There are several ways to handle these types of variations: – Often, new observables can be formed that eliminate the random parameter (eg., clocks in GPS can be eliminated by differencing data) – A parametric model can be developed and the parameters of the model estimated (eg., piece-wis e linear functions can be used 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 be considered constant or linear functions of time (eg., EOP are estimated daily) – In some case, variations are fast enough that the process can be treated as additional nois e
