Estimation (Will return to statistical model shortly) Most common estimation method is"least-squares"in which the parameter estimates are the values that minimize the sum of the squares of the differences between the observations and modeled values based on parameter estimates For linear estimation problems, direct matrix formulation for solution For non-linear problems: Linearization or search technique where parameter space is searched for minimum value Care with search methods that local minimum is not found (will not treat in this course) 12540Lec10 Least squares estimation Originally formulated by Gauss Basic equations: Ay is vector of observations A is linear matrix relating parameters to observables; Ax is vector of parameters; v is esidual △y=AAx+v minimize(v v); superscript T means transpose △x=(AA)A△y03/12/03 12.540 Lec 10 9 • (Will return to statistical model shortly) • minimize the sum of the squares of the differences on parameter estimates. • For linear estimation problems, direct matrix formulation for solution • minimum value • found (will not treat in this course) Estimation Most common estimation method is “least-squares” in which the parameter estimates are the values that between the observations and modeled values based For non-linear problems: Linearization or search technique where parameter space is searched for Care with search methods that local minimum is not 5 03/12/03 12.540 Lec 10 10 Least squares estimation D observables; D residual Dy = ADx + v minimize vT ( ) v ; Dx = (AT A) -1 AT Dy • Originally formulated by Gauss. • Basic equations: y is vector of observations; A is linear matrix relating parameters to x is vector of parameters; v is superscript T means transpose