Estimating vector

Equations 6,71, 6.75 and 6.76 can be solved simultaneously to yield g(t) and G(t) when the initial state vector x0 and the parameter estimate vector k(J) are given. In order to determine ktl+l) the output vector (given by Equation 6.2) is inserted into the objective function (Equation 6.4) and the stationary condition yields,... 

If we compare Equations 6.79 and 6.11 we notice that the only difference between the quasilinearization method and the Gauss-Newton method is the nature of the equation that yields the parameter estimate vector k(l+l). If one substitutes Equation 6.81 into Equation 6.79 obtains the following equation... 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

The solution of this linear least-squares problem is exact and the estimate vector b for the parameter values 0, is given by the matrix notation [8]... 

It makes an initial choice of the estimated vector parameter (indeed, for i = 0 it chooses Po)-... 

For an estimated vector of the three parameters, curves of Za and Z as a function of time can be calculated by the integration (if necessary, numerical) of Eqs. (22) and (23) and compared with the experimental results. An objective function can be minimized by suitable-least-squares calculations as the vector of parameters is varied according to the proper procedures (27,106,107). 

In each step of the univariate search method, p — 1 parameters of the previous estimate vector remain fixed, while a minimum of the objective function is searched for by varying only the remaining parameter over a predetermined range. Starting from the initial parameter vector bo, in p steps all parameters are optimized in a fixed order, after which this process is repeated until each additional decrease of the objective function becomes negligible. 

The properties of the parameter estimates of linear models are no longer valid exactly for the estimates of nonlinear parameters, for the optimal estimates cannot be expressed as linear combinations of the observation outcomes. The estimating vector is not multidimensional normally distributed. Its distribution is not even known The true covariance matrix and the joint confidence region of the parameter estimates are approximated by the expression obtained in the case of linear model regression ... 

The Kalman filter algorithm is initialized using an initial state estimate vector X o i and its error covariance matrix P[oi i], both assumed known. Hereafter, it propagates by computing state estimates X i +in recursively from the following equation ... 

These residuals are the differences between the experimental observations F and the calculated values of Y using the estimated vector b. A common way for evaluation of the unknown vector b is the least squares method, which minimizes the sum of the squared residuals O ... 

Linear and nonlinear regression analyses, including least squares, estimated vector of parameters, method of steepest descent, Gauss-Newton method, Marquardt Method, Newton Method,... 

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