12.215 Modern NavigationReview of last classToday’s classCovariance of derived quantitiesCovariance of derived quantitiesEstimation in parts/Sequential estimationSequential estimationSequential estimationSequential estimationEigenvectors and EigenvaluesError ellipsesError ellipsesError ellipsesExample of error ellipseCorrelations Cross CorrelationExampleTime series (infinite signal to noise)Auto and cross correlationsSignal plus noiseAuto and cross correlationsLow SNR caseLow SNR, longer time seriesCross Correlations comparisonEffects of clippingAuto and cross correlationsSummary of class12.215 Modern NavigationThomas Herring11/08/2006 12.215 Modern Naviation L14 2Review of last class• Estimation methods– Restrict to basically linear estimation problems (also non-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 residuals11/08/2006 12.215 Modern Naviation L14 3Today’s class• Finish up some aspects of estimation– Propagation of variances for derived quantities– Sequential estimation– Error ellipses• Discuss correlations: Basic technique used to make GPS measurements.– Correlation of random signals with lag and noise added (varying amounts of noise)– Effects of length of series correlated– Effects of clipping (ex. 1-bit clipping)11/08/2006 12.215 Modern Naviation L14 4Covariance of derived quantities• Propagation of covariances can be used to determine the covariance of derived quantities. Example latitude, longitude and radius. θ is co-latitude, λ is longitude, R is radius. ΔN, ΔE and ΔU are north, east and radial changes (all in distance units). Geocentric Case :ΔNΔEΔU⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ =−cos(θ)cos(λ) −cos(θ)sin(λ) sin(θ)−sin(λ) cos(λ)0X /RY/RZ/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⎡ ⎣ ⎢ ⎢ ⎢ ⎤⎦⎥⎥⎥11/08/2006 12.215 Modern Naviation L14 5Covariance of derived quantities• Using the matrix on the previous page to find a linear relationship (matrix A) between changes in XYZ coordinates and changes in the North (δφR), East (δλRcosφ) and height (Up), we can find the covariance matrix of NE and U from the XYZ covariance matrix using propagation of variances• This is commonly done in GPS, and one thing which stands out is that height is more determined than the horizontal position (any thoughts why?).• This fact is common to all applications of GPS no matter the accuracy.11/08/2006 12.215 Modern Naviation L14 6Estimation in parts/Sequential estimation• A very powerful method for handling large data sets, takes advantage of the structure of the data covariance matrix if parts of it are uncorrelated (or assumed to be uncorrelated).V1000V2000V3⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ −1=V1−1000V2−1000V3−1⎡ ⎣ ⎢ ⎢ ⎢ ⎤⎦⎥⎥⎥11/08/2006 12.215 Modern Naviation L14 7Sequential estimation• Since the blocks of the data covariance matrix can be separately inverted, the blocks of the estimation (ATV-1A) can be formed separately can combined later.• Also since the parameters to be estimated can be often divided into those that effect all data (such as station coordinates) and those that effect data a one time or over a limited period of time (clocks and atmospheric delays) it is possible to separate these estimations (shown next page).11/08/2006 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 priori parameter estimates), Vxgis covariance matrix of prior parameter estimates with estimates xgand xlare local parameter estimates, xg+are new global parameter estimates.yxg⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =AgAlI0⎡ ⎣ ⎢ ⎤ ⎦ ⎥ xgxl⎡ ⎣ ⎢ ⎤ ⎦ ⎥ xg+xl⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =AgTV−1Ag+ Vxg−1()AgTV−1AlAlTV−1AgAlTV−1Al⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −1AgTV−1y + Vxg−1xgAlTV−1y⎡ ⎣ ⎢ ⎤⎦⎥11/08/2006 12.215 Modern Naviation L14 9Sequential estimation• As each block of data is processed, the local parameters, xl, can be dropped and the covariance matrix of the global parameters xgpassed to the next estimation stage. • Total size of adjustment is at maximum the number of global parameters plus local parameters needed for the data being processed at the moment, rather than all of the local parameters.11/08/2006 12.215 Modern Naviation L14 10Eigenvectors and Eigenvalues• The eigenvectors and values of a square matrix satisfy the equation Ax=λx• 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.11/08/2006 12.215 Modern Naviation L14 11Error 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:σn2σneσneσe2⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Covariance matrix;Eigenvalues satisfy λ2−(σn2+σe2)λ+ (σn2σe2−σne2) = 0Eigenvectors: σneλ1−σn2⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and λ2−σe2σne⎡ ⎣ ⎢ ⎤ ⎦ ⎥11/08/2006 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 of being inside is (area under 2-D Gaussian). ρ scales the eigenvaluesλ1λ2⎫ ⎬ ⎭ =12σn2+σe2±σn2+σe2()2− 4σn2σe2−σne2()⎛ ⎝ ⎜ ⎞⎠⎟tan2φ=2σneσn2−σe2 angle ellipse make to N axisρ=−2ln(1−P)11/08/2006 12.215 Modern Naviation L14 13Error ellipses• There is only 40% chance of being inside the 1-sigma error ellipse (compared to 68% of 1-sigma in one dimension)• Commonly you will see 95% confidence ellipse which is 2.45-sigma (only 2-sigma in 1-D).• Commonly used for GPS position and velocity