Optimization square curve-fitting

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

In this chapter you ll learn how to use the Solver, Excel s powerful optimization package, to perform non-linear least-squares curve fitting. 

These parameters are determined by non-linear least-squares optimization of the fit of the function to both the experimental storage and loss moduli curves. As emphasized, the two determiners of temperature-scan peak width referred to above (i.e., in terms of equation (2), activation energy AH of x0 and a ) have features that allow distinguishing... 

There are situations where a net atomic charge model does not give the desired accuracy of fit to the electric potential. Since the least-squares fitting procedure is a curve-fitting process, it is expected that addition of new variables of appropriate mathematical form will improve the fit. The addition of a new fixed site adds implicit variables, x, y, z, of the site location and an explicit variable, q, the site charge. The new site can be treated as a dummy atom with net charge q, and this charge can be optimized. 

Figure 7.17 compares the results of the two function fits to the data. In both cases all three coefficients are allowed to vary to optimize the data fit. As can be seen from the two curves and from the best fit values of the parameters, somewhat different curves and parameters are obtained depending upon whether the least squares technique minimizes the vertical distance between the theory and the data or minimizes the horizontal distance between the model and the data. This is a difference of nonlinear curve fitting as opposed to linear curve fitting. For a linear... 

Solver (or Optimizer in Quattro Pro) can also be used to do least-squares fitting of data with a theoretical function that cannot be cast in the form required for the usual linear regression. As an illustration, Fig. 4 shows a radiative decay curve measured by a student for a raby crystal as part of Exp. 44. After laser excitation the digitized emission signal I decreases exponentially and can be analyzed for k (the radiative rate constant, whose reciprocal is the lifetime) according to the relation... 

It is clear that the choice of the number as well as the place of the different sections necessary to obtain an approximate fit of the experimental curve is arbitrary. Obviously, accuracy increases with the number of sections. Once the number of sections is chosen, their locations can be optimized to obtain the minimal mean square deviation from the experimental results. However, as a general rule, the middle point of each section on the experimental curve is taken. The choice of the point where the first slope section starts determines the rest of the sections. The middle point of the last section can be determined once the values of the relaxed moduli are known and a careful choice has been made. In the regions of high and low frequencies, the horizontal sections will be longer than in the central zone of the relaxation, where the representative curve of the modulus has a greater slope. 

Multivariate curve resolution-alternating least squares (MCR-ALS) is an algorithm that fits the requirements for image resolution [71, 73-75]. MCR-ALS is an iterative method that performs the decomposition into the bilinear model D = CS by means of an alternating least squares optimization of the matrices C and according to the following steps ... 

Association constants were calculated by fitting the experimental titration curves to a 1 1 binding model based on (Equation (9.3)) and the mass balance (Equations (9.4) and (9.5)), by applying a least squares optimization routine ... 

Ultraviolet/visible (UV/vis) titrations were performed with a Varian Australia Pty Cary 3E UV/vis spectrophotometer. Titrations of Zn-porphyrin 1 were carried out by adding 5 pL aliquots (0.5 equiv.) of 10 2 M solutions of pyridine 2 or 4 to 1 mL of a 10-4 M Zn-porphyrin 1 solution, up to a maximum of 10 pyridine equivalents. Association constants were calculated by fitting the experimental titration curves to a 1 1 binding model based on (Equation (9.3)), on the mass balances (Equations (9.4) and (9.5)) and the Lambert-Beer law by applying a least squares optimization routine. 

Least-squared fit A fit of a line or other curve to a set of points that is optimized by minimizing the total distance of all points to the curve. Distances are squared to eliminate potential canceling with some (+) and others (—) relative to the curve. 

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