112.215 Modern NavigationThomas Herring ([email protected]),MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/~tah/12.21510/28/2009 12.215 Modern Naviation L14 2Review of last class• Estimation methods– Restrict to basically linear estimation problems (alsonon-linear problems that are nearly linear)– Restrict to parametric, over determined estimation– Concepts in estimation:• Mathematical models• Statistical models• Least squares and Maximum likelihood estimation• Covariance matrix of estimated parameters• Statistical properties of post-fit residuals210/28/2009 12.215 Modern Naviation L14 3Todayʼs class• Finish up some aspects of estimation– Propagation of variances for derived quantities– Sequential estimation– Error ellipses• Discuss correlations: Basic technique used to makeGPS measurements.– Correlation of random signals with lag and noiseadded (varying amounts of noise)– Effects of length of series correlated– Effects of clipping (ex. 1-bit clipping)10/28/2009 12.215 Modern Naviation L14 4Covariance of derived quantities• Propagation of covariances can be used to determinethe covariance of derived quantities. Example latitude,longitude and radius. θ is co-latitude, λ is longitude, Ris radius. ΔN, ΔE and ΔU are north, east and radialchanges (all in distance units). € Geocentric Case : ΔNΔEΔU          =−cos(θ)cos(λ) −cos(θ)sin(λ) sin(θ)−sin(λ) cos(λ) 0X /R Y /R Z /R          A matrix for use in propagation from Vxx1 2 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 ΔXΔYΔZ         310/28/2009 12.215 Modern Naviation L14 5Covariance of derived quantities• Using the matrix on the previous page to find a linearrelationship (matrix A) between changes in XYZcoordinates and changes in the North (δφR), East(δλRcosφ) and height (Up), we can find the covariancematrix of NE and U from the XYZ covariance matrixusing propagation of variances• This is commonly done in GPS, and one thing whichstands out is that height is more determined than thehorizontal position (any thoughts why?).• This fact is common to all applications of GPS nomatter the accuracy.10/28/2009 12.215 Modern Naviation L14 6Estimation in parts/Sequentialestimation• A very powerful method for handling large data sets,takes advantage of the structure of the datacovariance matrix if parts of it are uncorrelated (orassumed to be uncorrelated).€ V10 00 V200 0 V3          −1=V1−10 00 V2−100 0 V3−1         410/28/2009 12.215 Modern Naviation L14 7Sequential estimation• Since the blocks of the data covariance matrix can beseparately inverted, the blocks of the estimation(ATV-1A) can be formed separately can combinedlater.• Also since the parameters to be estimated can beoften divided into those that effect all data (such asstation coordinates) and those that effect data a onetime or over a limited period of time (clocks andatmospheric delays) it is possible to separate theseestimations (shown next page).10/28/2009 12.215 Modern Naviation L14 8Sequential estimation• Sequential estimation with division of global and local parameters.V is covariance matrix of new data (uncorrelated with prioriparameter estimates), Vxg is covariance matrix of prior parameterestimates with estimates xg and xl are local parameter estimates,xg+ are new global parameter estimates.€ yxg      =AgAlI 0      xgxl      xg+xl      =AgTV−1Ag+ Vxg−1( )AgTV−1AlAlTV−1AgAlTV−1Al        −1AgTV−1y + Vxg−1xgAlTV−1y     510/28/2009 12.215 Modern Naviation L14 9Sequential estimation• As each block of data is processed, the localparameters, xl, can be dropped and the covariancematrix of the global parameters xg passed to the nextestimation stage.• Total size of adjustment is at maximum the number ofglobal parameters plus local parameters needed forthe data being processed at the moment, rather thanall of the local parameters.10/28/2009 12.215 Modern Naviation L14 10Eigenvectors and Eigenvalues• The eigenvectors and values of a square matrixsatisfy the equation Ax=λx• If A is symmetric and positive definite (covariancematrix) then all the eigenvectors are orthogonal andall the eigenvalues are positive.• Any covariance matrix can be broken down intoindependent 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 usinga matrix made up of columns of eigenvectors.610/28/2009 12.215 Modern Naviation L14 11Error ellipses• One special case is error ellipses. Normallycoordinates (say North and East) are correlated andwe find a linear combinations of North and East thatare uncorrelated. Given their covariance matrix wehave:€ σn2σneσneσe2      Covariance matrix;Eigenvalues satisfy λ2−(σn2+σe2)λ+ (σn2σe2−σne2) = 0Eigenvectors : σneλ1−σn2      and λ2−σe2σne      10/28/2009 12.215 Modern Naviation L14 12Error ellipses• These equations are often written explicitly as:• The size of the ellipse such that there is P (0-1) probability ofbeing inside is (area under 2-D Gaussian). ρ scales theeigenvalues€ λ1λ2   =12σn2+σe2±σn2+σe2( )2− 4σn2σe2−σne2( )      tan2φ=2σneσn2−σe2 angle ellipse make to N axis€ ρ= −2ln(1− P)710/28/2009 12.215 Modern Naviation L14 13Error ellipses• There is only 40% chance of being inside the 1-sigmaerror ellipse (compared to 68% of 1-sigma in onedimension)• Commonly you will see 95% confidence ellipse whichis 2.45-sigma (only 2-sigma in 1-D).• Commonly used for GPS position and velocity results• The specifications for GPS standard positioningaccuracy are given in this form and its extension to a3-D error ellipsoid (cigar shaped)10/28/2009 12.215 Modern Naviation L14 14Example of error ellipse-8-6-4-202468-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0Var2Var1Error Ellipses shown1-sigma 40%2.45-sigma 95%3.03-sigma 99%3.72-sigma 99.9%Covariance2+22+4Sqrt(Eigenvalues)0.87+and+2.29,Angle+‐58.3o810/28/2009 12.215 Modern Naviation L14 15Correlations• Stationary: Property that statistical