Linearity, in the parameters

The first is the relational model. Examples are hnear (i.e., models linear in the parameters and neural network models). The model output is related to the input and specifications using empirical relations bearing no physical relation to the actual chemical process. These models give trends in the output as the input and specifications change. Actual unit performance and model predictions may not be very close. Relational models are usebil as interpolating tools. 

Several methods are used to fit rate models, the two most common of which often give erroneous results. The first is the transformation of a proposed rate model to achieve a model form that is linear in the parameters. An example is the nonlinear model ... 

Now if the function is linear in the parameters, the derivative dyidaj does not contain the parameters, and the resulting set of equations can be solved for the parameters. If, however, the function is nonlinear in the parameters, the derivative contains the parameters, and the equations cannot in general be solved for the parameters. This is the basic problem in nonlinear least-squares regression. 

Equation (7-29) is linear in the parameters and can be regarded as a multiple LEER. Many other equations of this form have been proposed. 

The method of least squares provides the most powerful and useful procedure for fitting data. Among other applications in kinetics, least squares is used to calculate rate constants from concentration-time data and to calculate other rate constants from the set of -concentration values, such as those depicted in Fig. 2-8. If the function is linear in the parameters, the application is called linear least-squares regression. The more general but more complicated method is nonlinear least-squares regression. These are examples of linear and nonlinear equations ... 

These functions are easily implemented on a computer, and even on many calculators. The procedure can be extended to cover other equations that are linear in the parameters. One can readily show, for example, that the least-squares rate constant for second-order kinetics from Eq. (2-13) is... 

A model described by this differential equation is linear in the parameters ki. .. k , if... 

Given the following equilibrium data for the distribution of S03 in hexane, determine a suitable linear (in the parameters) empirical model to represent the data. 

This method of least squares is not only intuitively desirable, but also provides estimates having desirable properties, if certain assumptions are met (D4). For models that are linear in the parameters, that is, models of the form... 

Many of the models encountered in reaction modeling are not linear in the parameters, as was assumed previously through Eq. (20). Although the principles involved are very similar to those of the previous subsections, the parameter-estimation procedure must now be iteratively applied to a nonlinear surface. This brings up numerous complications, such as initial estimates of parameters, efficiency and effectiveness of convergence algorithms, multiple minima in the least-squares surface, and poor surface conditioning. 

In this formula, f is a function of the independent variables x to x and the unknown parameters b. to b which is linear in the parameters. The function ... 

Non-linear curves may be treated using Equation 9 directly, using the techniques of non-linear least squares, when appropriate. (Note that a non-linear calibration curve does not necessarily imply non-linear least squares. The latter is necessary only if the problem is non-linear in the estimated parameters (16). For example, y = a+bx+cx and y = a+bx are both non-linear functions, but only the latter is non-linear in the parameters.)... 

We shall not treat the methods of fitting nonlinear equations, those that are not linear in the parameters, in detail, but we shall remind the reader that nonlinear least squares does not lead to a closed solution for the parameters, as in linear least squares. The method of nonlinear least squares requires a set of tentative values of the parameters, followed by an iterative process that is stopped when successive results are close... 

This book is limited to models that are linear in the parameters. 

This is the general matrix solution for the set of parameter estimates that gives the minimum sum of squares of residuals. Again, the solution is valid for all models that are linear in the parameters. 

Notice that the methods presented in Sections 1.0.2 and 1.0.3 can be extended to estimate the parameters in multivariable functions that are linear in the parameters. 

Eiy > = f(x, p) for the true value p of the parameters. The role of other assumptions will be clarified later. At this moment the most important message, coming from mathematical statistics, is as follows. If assumptions (i) through (iii) are satisfied, the model (3.2) is linear in the parameters, and we select the weighting coefficients according to w =, where a is a (possibly... 

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. 

Equations SQ(a,b)/aa = and SQ(a,b)/ab = 0 are linear in the parameters. Solving them simultaneously we obtain the least squares estimates... 

In linear calibrations (I use the term in its mathematical sense—linear in the parameters of x, allowing a quadratic calibration model = a + + b2xz... 

It should be pointed out that when one speaks of a linear model in regression the term linear means linear in the parameters po, p1 ..., Pp and not in the independent variable X. Other examples of linear models (linear in the parameters) are ... 

Each fault function in TA is assumed to have a linear-in-the-parameters structure, i.e.,... 

This model is called simple since there is only one independent variable (X), and linear because it is linear in the parameters. This means that no parameters appear as exponents or are multiplied or divided by another parameter (i.e., Xp). The term linear can cause some confusion in nonstatistical literature since second or higher order polynomial models are also called linear models (but in the terminology of linear in the parameters and not as a straight line). To avoid any confusion, here the term straight-line model is used for the above-described simple linear model. For an in-depth discussion of linear models, the reader is referred to an appropriate statistical manual [8]. 

If the model for the response variable is linear in the parameters, for each observation at chosen settings of the independent variables x then for the set of n observations and p parameters one can write [8] ... 

Most kinetic expressions, however, are not linear in the parameters and two approaches can be followed. The first is to rewrite the expression in a linear form and apply the linear least-squares minimization to obtain parameter values. Expression 14 can be reformulated into eq 47. 

We observe that this metamodel is linear in the parameters fij and Pyj but nonlinear in the variables Xj. At the end of the case study in Section 3, we try to validate the assumption that the model is adequate. 

If the function /is linear in the parameters aj, the fitting procedure is now complete and the values a) are the best values in a least-squares sense. If /is nonhnear in terms of any of the a, values, it is usually necessary to improve the a, values by carrying out another cycle of minimization, in which a] plays the role previously played by a°. The iteration process is repeated as many times as necessary to obtain convergence of the a, values to some predetermined level of accuracy. 

Carry out the least-squares minimization of the quantity in Eq. (7) according to an appropriate algorithm (presumably normal equations if the observational equations are linear in the parameters to be determined otherwise some other such as Marquardf s ). The linear regression and Solver operations in spreadsheets are especially useful (see Chapter HI). Convergence should not be assumed in the nonlinear case until successive cycles produce no significant change in any of the parameters. 

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