Continuous functions, linear least squares

T T. X, A continuous linear form, used in least squares method outline J i g) norm equivalent functional in least squares method outline ... 

SOLUTION. This is an example of linear least-squares analysis (LLSA), where the objective function is continuous. Typically, LLSA is performed on a discrete set of data points and one seeks to minimize the sum of squares of differences between the data and a continuous model function. In this case, we seek to minimize the square of the difference between two continuous functions over the complete range of reactant conversions that are possible (i.e., 0 < x < 1 for irreversible reactions). Hence, the sum of squares in the objective function to be... 

A modification of this procedure was proposed in the literature [389] and applied to determine the time constant distribution function [379]. This method is based on the predistribution of time constants uniformly on the logarithmic scale, and to improve the quality of the analysis, a Mmite Carlo technique was used to increase the number of analyzed time constants. Approximation was carried out using a constrained least-squares method and led to a continuous distribution function. This procedure converted the nonlinear problem to a linear one from which versus r , were obtained and produced positive values of the distribution function. The procedure was also applied to the distribution of the dielectric constants [379,389]. 

McCulloch (1971) proposes a more practical approach, using polynomial splines. This method produces a function that is both continuous and linear, so the ordinary least squares regression technique can be employed. A 1981 study by James Langetieg and Wilson Smoot, cited in Vasicek and Fong (1982), describes an extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount function and uses nonlinear methods of estimation. 

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