In a strict sense parameter estimation is the procedure of computing the estimates by localizing the extremum point of an objective function. A further advantage of the least squares method is that this step is well supported by efficient numerical techniques. Its use is particularly simple if the response function (3.1) is linear in the parameters, since then the estimates are found by linear regression without the inherent iteration in nonlinear optimization problems. [Pg.143]

In Section 3.4, traditional methods of obtaining values of rate parameters from experimental data are described. These mostly involve identification of linear forms of the rate expressions (combinations of material balances and rate laws). Such methods are often useful for relatively easy identification of reaction order and Arrhenius parameters, but may not provide the best parameter estimates. In this section, we note methods that do not require linearization. [Pg.57]

The lecist squares method for parameter estimation is a central technique in the area of process identification. The method itself is particularly simple to apply if the selected model structure has the property of being linear-in-the-parameters. In this case, the least squau es parameter estimates can be found suicilytically. For example, consider a model of the following form [Pg.60]

The previously described methods for parameter estimation are based on linear transformations, that is, the model expression is written in such a way that the rate constant ( a) can be solved from a linear expression. This is convenient from the numerical viewpoint, but the drawback is that the structure of the transformed data becomes biased, as discussed previously (Section 10.2). In effect, the most rational standpoint is to utilize the original experimental data as such, typically the experimentally recorded component concentrations (ca )> and to compare them with the predicted concentrations ic i). The objective function (Q) thus becomes [Pg.596]

The methods concerned with differential equation parameter estimation are, of course, the ones of most concern in this book. Generally reactor models are non-linear in their parameters and therefore we are concerned mostly with nonlinear systems. [Pg.113]

Below, some useful simple methods for the estimation of kinetic parameters by linearization are presented. The problem is treated in more detail in Appendix B. [Pg.315]

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax > 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x" X)x = (Xx) (Xx) > 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. [Pg.35]

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. [Pg.2]

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier [Pg.140]

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