Least-square constraints polynomial

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

Prior to deconvolution, the background was subtracted and the data were smoothed with a 15-point quadratic least-squares polynomial followed by a 19-point quartic least-squares polynomial. The data were then scaled from 0 to 1. The S3 profile was deconvolved using a weight constraint of the form... 

Empirical polynomial descriptions of (approximately) how a response y depends upon proportions jcp jc2,. .., xp are popular mixture analysis tools. The process of fitting polynomials to mixture experiment data in principle uses the same least squares notion illustrated in the fitting of a parabola to the data of Table 5.2. However, the mechanics of using standard multiple regression analysis software in the mixture context is complicated somewhat by the basic constraint (5-15). For example, in view of (5-15) the basic (p + 1 parameter) linear relationship... 

As usual, the least squares procedure is employed to determine the values of the regression coefficients a, b, c, d, etc., giving the minimum deviation between the observed data and the model. Also, we impose the criterion that subsequent addition of higher-order terms to the polynomial will not change the value of the coefficients of lower-order terms. This extra constraint is used to evaluate the parameters etc. The coefficient a represents the average y value,... 

In this case, a linear least-squares problem with the linear equality constraints (7) needs to be solved (cf. [22, Chapter 21]). Finally, we remark that the entries of the system matrix M in (4) can be computed by using quadrature rules for triangles, that are exact for the desired polynomial degree n. A detailed list of quadrature rules for triangles is given in [9]. 

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