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