Reliability of fitted parameters

For a specified mean and standard deviation the number of degrees of freedom for a one-dimensional distribution (see sections on the least squares method and least squares minimization) of n data is (n — 1). This is because, given p and a, for n 1 (say a half-dozen or more points), the first datum can have any value, the second datum can have any value, and so on, up to n — 1. When we come to find the 

The situation is similar for a linear curve fit, except that now the data set is two-dimensional and the number of degrees of freedom is reduced to (n — 2). The analogs of the one-dimensional variance )/( 1) the standard 

If the data set is Puly nomial and the enor in y is random about known values of a , residuals will be distr ibuted about the regression line according to a normal or Gaussian distribution. If the dishibution is anything else, one of the initial hypotheses has failed. Either the enor dishibution is not random about the shaight line or y =f x) is not linear. 

If the mahix form of the fitting procedure is used to solve for the intercept and slope of a shaight line, Eq. (3-13) 

The variance of the regression times the diagonal elements of the inverse coefficient mahix gives the variance of the intercept and slope. 

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We wish to cany out a proceduie that is the multivariate analog to the analysis in the section on reliability of fitted parameters. A vector multiplied into its hanspose gives a scalar that is the sum of squares of the elements in that vector. The y vector leads to a vector of residuals... 

Very briefly, this rather large subject in the general area of chemical kinetics [43-45] was carried into electrochemistry in the studies by Bieniasz et al. [46-48]. It asks the question, when fitting some parameter to a proposed mechanism by means of simulation using some simulation output (concentrations or current or some other result), how sensitive to the changes in the output is the value of the fitted parameter. This is expressed in the form of a sensitivity function s. If the simulation yields, for example, an array of concentrations c x,t,p), where x are positions in space, t the time (which may enter the problem) and the parameter(s) p, then the function is defined [46] as 5 = dc/dp, which is an expression of the sensitivity to changes in concentration. This can be useful in estimating the reliability of fitted parameters by a series of simulations. This subject will not be persued further here. 

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