12.540 Principles of the Global Positioning System Lecture 11 Prof. Thomas Herring 03/1703 12540Lec11 Statistical approach to estimation Summar Look at estimation from statistical point of VIew Propagation of covariance matrices Sequential estimation 03/703 12540Lec11
03/17/03 12.540 Lec 11 1 12.540 Principles of the Global Positioning System Lecture 11 03/17/03 12.540 Lec 11 2 Statistical approach to estimation view Prof. Thomas Herring • Summary –Look at estimation from statistical point of –Propagation of covariance matrices –Sequential estimation 1
Statistical approach to estimation Examine the multivariate gaussian distribution Multivariant f(x) Minimize (x-u)V(x-w gives largest probability densi By minimizing the argument of the exponential in the probability density function, we maximize the likelihood of the estimates(MLE) This is just weighted least squares where the weight matrix is chosen to be the inverse of the covariance matrix of data noise 03/1703 12540Lec11 Data covariance matrix If we use the inverse of the covariance matrix of the noise in the data we obtain a mle if data noise is Gaussian distribution How do you obtain data covariance matrix? Difficult question to answer completely Issues to be considered Thermal noise in receiver gives on component Multipath could be treated as a noise-like quantity Signal-to-noise ratio of measurements allows an estimate of the noise(discussed later in course) thematical model of observables can sometimes be treated as noise-like Gain of GPS antenna will generate lower SNR at low elevation 03/703 12540Lec11
† 03/17/03 12.540 Lec 11 3 Statistical approach to estimation • Examine the multivariate Gaussian distribution: • probability density function, we maximize the likelihood of the estimates (MLE). • This is just weighted least squares where the weight matrix is chosen to be the inverse of the covariance matrix of data noise f (x) = 1 (2p) n V e -1 2 (x-m) T V -1 (x-m) (x - m) T V-1 (x - m) By minimizing the argument of the exponential in the Multivariant Minimize gives largest probability density 03/17/03 12.540 Lec 11 4 Data covariance matrix • If we use the inverse of the covariance matrix of the Gaussian distribution. • • Difficult question to answer completely • Issues to be considered: the noise (discussed later in course). sometimes be treated as noise-like. angles noise in the data, we obtain a MLE if data noise is How do you obtain data covariance matrix? – Thermal noise in receiver gives on component – Multipath could be treated as a noise-like quantity – Signal-to-noise ratio of measurements allows an estimate of – In-complete mathematical model of observables can – Gain of GPS antenna will generate lower SNR at low elevation 2
Data covariance matrix In practice in GPS (as well as many other fields), the data covariance matrix is somewhat arbitrarily chosen Largest problem is temporal correlations in the measurements. Typical gps data set size for 24- hours of data at 30 second sampling is 8X2880=23000 phase measurements. Since the inverse of the covariance matrix is required fully accounting for correlations requires the inverse of 23000X23000 matrix To store the matrix would require, 4Gbytes of memory Even if original covariance matrix is banded (ie correlations over a time short compared to 24-hours), 03/1703 12540Lec11 Data covariance matrix Methods on handling temporal correlations If measurements correlated over say 5-minute period, then Use full rate data, but artificially inflate the noise on each measurement so that equivalent to say 5-minute sampling(ie, sqrt(10)higher noise on the 30-second sampled values (GAMIT method) When looking a GPS results, always check the data noise assumptions(discussed more near end of course) Assuming a valid data noise model can be developed what can we say about noise in parameter estimates? 03/703 12540Lec11
03/17/03 12.540 Lec 11 5 Data covariance matrix • In practice in GPS (as well as many other fields), the data covariance matrix is somewhat arbitrarily chosen. • Largest problem is temporal correlations in the measurements. Typical GPS data set size for 24- hours of data at 30 second sampling is 8x2880=23000 phase measurements. Since the inverse of the correlations requires the inverse of 23000x23000 matrix. • To store the matrix would require, 4Gbytes of memory • correlations over a time short compared to 24-hours), covariance matrix is required, fully accounting for Even if original covariance matrix is banded (ie., the inverse of banded matrix is usually a full matrix 03/17/03 12.540 Lec 11 6 Data covariance matrix • use samples every 5-minutes (JPL method) measurement so that equivalent to say 5-minute sampling (ie., (GAMIT method) assumptions (discussed more near end of course). • what can we say about noise in parameter estimates? Methods on handling temporal correlations: – If measurements correlated over say 5-minute period, then – Use full rate data, but artificially inflate the noise on each sqrt(10) higher noise on the 30-second sampled values – When looking a GPS results, always check the data noise Assuming a valid data noise model can be developed, 3
Propagation of covariances Given a data noise covariance matrix. the characteristics of expected values can be used to determine the covariance matrix of any linear combination of the measurements Given linear operation: y=Ax with Vas covariance matrix of x yy =< yy >=< AXx'A>=A<XXAT V=AⅴAT Propagation of covariance Propagation of covariance can be used for any linear operator applied to random variables whose covariance matrix is already Specific examples: Covariance matrix of parameter estimates from least squa Covariance matrix for post-fit residuals from least Covariance matrix of derived quant such latitude, longitude and height from coordinate estimates 03/703 12540Lec11
† 03/17/03 12.540 Lec 11 7 Propagation of covariances y = Ax with Vxx x Vyy =< yyT >=< AxxT A T >= A < xxT > A T Vyy = AVxxA T • Given a data noise covariance matrix, the characteristics of expected values can be used to determine the covariance matrix of any linear combination of the measurements. Given linear operation : as covariance matrix of 03/17/03 12.540 Lec 11 8 Propagation of covariance known. – least squares – Covariance matrix for post-fit residuals from least squares – Covariance matrix of derived quantities such as estimates. • Propagation of covariance can be used for any linear operator applied to random variables whose covariance matrix is already • Specific examples: Covariance matrix of parameter estimates from latitude, longitude and height from XYZ coordinate 4
Covariance matrix of parameter estimates Propagation of covariance can be applied to the weighted least squares problem x=(AVYA)"A'VYy XX >=(A VA)AV<yy >Va(aVa Vx=(AVa) Notice that the covariance matrix of parameter estimates is a natural output of the estimator if av-1A is inverted ( does not need to be 03/1703 12540Lec11 Covariance matrix of estimated parameters Notice that for the rigorous estimation, the inverse of the data covariance is needed ( time consuming if non-diagonal To compute to parameter estimate covariance, only the covariance matrix of the data is needed (not the inverse) In some cases, a non-rigorous inverse can be done with say a diagonal covariance matrix, but the parameter covariance matrix is rigorously computed using the full covariance matrix. This is a non-MLE but the covariance matrix of the parameters should be orrect ust not the best estimates that can found) This techniques could be used if storage of the full covariance matrix is possible, but inversion of the matrix is not because it would take too long or inverse can not be performed in place 03/703 12540Lec11
† 03/17/03 12.540 Lec 11 9 • Propagation of covariance can be applied to the • Notice that the covariance matrix of parameter estimates is a natural output of the estimator if ATV-1A is inverted (does not need to be) x ˆ = (ATVyy -1 A)-1 ATVyy -1 y < xˆxˆ T >= (ATVyy -1 A)-1 ATVyy -1 < yyT > Vyy -1 TVyy -1 A)-1 Vxˆx ˆ = (ATVyy -1 A)-1 Covariance matrix of parameter estimates weighted least squares problem: A(A 03/17/03 12.540 Lec 11 10 Covariance matrix of estimated • Notice that for the rigorous estimation, the inverse of the data covariance is needed (time consuming if non-diagonal) • To compute to parameter estimate covariance, only the covariance matrix of the data is needed (not the inverse) • In some cases, a non-rigorous inverse can be done with say a diagonal covariance matrix, but the parameter covariance matrix is rigorously computed using the full covariance matrix. This is a correct (just not the best estimates that can found). • This techniques could be used if storage of the full covariance parameters non-MLE but the covariance matrix of the parameters should be matrix is possible, but inversion of the matrix is not because it would take too long or inverse can not be performed in place. 5