Equation differential, least squares data

Listing 11.31. Code for determining coefficients of differential equation by least squares fit to experimental data. 

In the work that follows, the experimental data were fitted by minimizing the sum of least squares and the differential equations were integrated numerically. 

Typically, process data are improved using spatial, or functional, redundancies in the process model. Measurements are spatially redundant if more than enough data exist to completely define the process model at any instant, that is, the system is overdetermined and requires a solution by least squares fitting. Similarly, data improvement can be performed using temporal redundancies. Measurements are temporally redundant if past measurement values are available and can be used for estimation purposes. Dynamic models composed of algebraic and differential equations provide both spatial and temporal redundancy. 

The described method allows identification of the model given by differential equation. Complexity of the problem is in the fact that possessing analytical solution (26) one may receive parameters of model determining its solution in accordance with least-squares method closely to experimental data. Such approach opens possibilities of reception of parameters estimation also for more complex dependences. 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

Data obtained under these conditions can be fitted using a least-squares procedure based upon the exact solution to the differential equations describing this mechanism [37, 44]. This yields values for the complex dissociation constant Ky, and the limiting first-order rate constant (a minimum value for the second-order rate constant for complex formation can also be obtained from this analysis). Note that AId refers to the interaction between reduced P and oxidized P, a situation that is observable only by kinetic methodology. 

For this case of study, it is supposed that the four states are directly measurable. The estimation problem is posed as a least squares objective function subject to the model nonlinear differential equations as constraints, restricting the mathematical program to the size of the moving window, and therefore ignoring the data outside such window. 

The linear least squares fitting of n experimental data points to the differential form of the equation of motion involves minimization of the summation... 

Determine the parameter values bj andZ>2 by using the data given in Example 9.1 and the nonlinear least squares method. Recall that in Example 9.1 we needed the elements of the Jacobian matrix 7 (see equation (9.142)). In this case, integrate simultaneously the time dependent sensitivity coefficients (i.e., the Jacobi matrix elements dyfdb and dyjdb2 ) and the differential equations. The needed three differential equations can be developed by taking the total derivative (as shown below) of the right hand side of equation (9.149) which we call h ... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

The differential method has also been used to determine deuterium isotope effects in the formation of the 2,4-dinitrophenylhydrazones of acetophenone-mef l-dg and other deuterated ketones (Baaen et ah, 1964), using carbon-14 as the tracer. The procedure requires that the isotope effects caused by the carbon-14 itself be known, and these were determined in separate experiments. Known mixtures of deuterated and undeuterated species were then prepared—one of which was always labeled with carbon-14— and the rate of change of carbon-14 content as a function of fraction of reaction was determined by removing small aliquots of reaction product at selected known intervals (or fractions f) of reaction. The data were then fitted to equations (35) or (36) by means of linear or non-linear least-squares codes, respectively, with an IBM 7090 computer. Some typical experimental results are given in Table 7. 

There are many proposals in the literature for curve fitting and digital differentiation, and the most important ones are listed in Table 3-9. Morrey [119], for example, fitted the digital data of a segment of a spectrum to a quartic equation and differentiated this convolute. A least-squares analysis of this equation gives rise to a set of linear equations. Peak positions were obtained without consideration of the type of distribution function representing the peak. 

The intersection of the straight line with the ordinate yields the rate constant k. The values for the rate Ta may be determined either directly in a differential reactor or via differentiation of integral data. In practice attempts will be made to fit a straight line to the experimental points as closely as possible. Adapting the equation by the least-squares method to the experimental points to fit the equation would of course be more exact. 

The following method uses the method of least squares. In this case, all data points are used to generate a second-order polynomial equation. This equation is then differentiated and evaluated at the point where the value of the derivative is required. For example, Microsoft Excel can be employed to generate the regression equation. Once all the coefficients are known, the equation has only to be analytically differentiated ... 

Statistical data of 2007-2012 was used for calculation of differential equations system (1) coefficients with least square method in cases of each country (8). 

When the mechanism is very complex or non-linear (as defined in section 2.4 p. 42) it is not possible to obtain a satisfactory analytical solution and (as discussed above) a set of differential equations, one for each state of the system, is the only available description of the mechanism. In such cases one can proceed either by trial and error (the above mentioned overlay method), using numerical simulation of the differential equations instead of the analytical equation. Alternatively Scientist, unique among the programs mentioned above, has facilities for non-linear least square fitting of data which can only be described by sets of differential equations. Kinetic instrument manufacturers are, increasingly, making such programs commercially available. 

Least squares multiple nonlinear regression using the Marquardt and Gauss-Newton methods. The program can fit simultaneous ordinary differential equations and/or algebraic equations to multiresponse data. 

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