Estimate measured vector

In many cases, one may measure spectra of solutions of the pure components directly, and the above estimation procedure is not needed. For the further development of the theory of multicomponent analysis we will therefore abandon the hat-notation in K. Given the pure spectra, i.e. given K (pxq), one may try and estimate the vector of concentrations (pxl) of a new sample from its measured... 

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. 

Since the concept of observability was primarily defined for dynamic systems, observability as a property of steady-state systems will be defined in this chapter. Instead of a measurement trajectory, only a measurement vector is available for steady-state systems. Estimability of the state process variables is the concept associated with the analysis of a steady-state situation. 

The residuals are the portion of the calibration measurement vectors that are not fit by die estimated pure spectra and the known concentrations. 

Prediction Prediction is the process of estimating the characteristics of unknown samples by applying a calibration model to an unknown measurement vector. (5ee also Calibration.)... 

Studentized concentration residuals are concentration residuals that have been divided by the concentration standard error of estimate and VI— leverage. Sample leverage is a measure of the influence a sample measurement vector has on the model. 

It is the linearized estimate of the measured vector Xj when the equation g(z) = 0 has been linearized at point z. It is different from the final xj (10.5.9), hence also Q2 computed as... 

Bayes estimators assume that the parameter vector is the realization of a random vector 0, the a priori probability distribution of which p0(0) is available, for example, from preliminary population studies. Starting from the knowledge of both the model of Equation 9.16 and the probability distribution of the noise vector v, one can calculate the Hkelihood function p 0(z 0) (i.e., the probability distribution of the measurement vector in dependence of the parameter vector). From p0(0) and p 0(z 0), the a posteriori probability distribution pe z (0 z) (i.e., the probability distribution of the parameter vector given the data vector) can be determined by exploiting the Bayes theorem ... 

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

If this is an explicit equation with respect to a the estimation of the vector k is mathematically identical to a differential analysis. The only difference is that values of ki are searched, for which the concentrations calculated from the above equation are as close as possible to the measured concentrations. Below, a simple example illustrating both techniques is given. 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h. ... 

Step 6. Based on the additional measurement of the response variables, estimate the parameter vector and its covariance matrix. 

We now formulate the associated identification problem by arranging all the velocity measurements in a vector Y and, in the vector F, the corresponding values are calculated from the solution to Eqs. (4.1.24 and 4.1.25) using an estimate of the permeability function. The performance index is ... 

The pole placement design predicates on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasureable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state X is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skip these measurable states. 

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