Fisher criterion

The following equation shows calculation of the Fisher criterion ... 

Based on the Fisher criterion, the hypothesis that the mathanatical model is adequate to the real process makes sense only if the residual dispersion S of the... 

As an example, the verification of adequacy of the malhanatical model of the YSZ-based sensor, by using the statistical Fisher criterion, is shown that the hypothesis about the adequacy of the presented model to the real YSZ-based sensor with the NiO-SE sintered at 1300°C for measuranent 200 ppm NOj at T = 800 C is not rejected by the uniformity of dispersions with a significance level ofq = 0.05. 

The entry in parentheses means that the table value of the Fisher criterion [ ] corresponds to the number of degrees of freedom, 4 and 29, at a significance level, a = 0.05. 

For each s pificant descriptor set, obtained in the previous step, an additional noncoUinear descriptor scale was added, and the appropriate (n + l)-parameter regression treatment was performed. When the Fisher criterion at the given probability level, F (or the cross-validated correlation coefficient for leave-one-out Rcv(Q). obtained for any of these correlations was smaller than that for the best correlation of the previous rank, the latter was designated as the final result and the search was terminated. Otherwise, the descriptor sets with the highest regression... 

The final result has therefore the maximum value of the Fisher criterion and the highest value of the cross-validated correlation coefficient. According to these statistical criteria, it was considered as the best representation of the property in the given (large) descriptor space. The BMLR approach has a variation that takes care of the noncoUinearity of descriptors pairs, called the Heuristics method (1996JPC10400). The advantages of such methods are that they are fast and limit the chance correlation to minimum. Both techniques were successfully used by ARK for model building for a tremendous amount of chemical properties of compounds and heterocycHcs, in particular. 

The Ellis equation (and consequently the Meter equation) adequately describe the experimental dependence of the emulsion viscosity on the shear stress over the entire range of dispersed phase contents because the values of the Fisher criterion do not exceedthe tabular value (Table I). Thus, the Ellis equation allows the variation in emulsion viscosity with in the indicated range for different values of shear stress to not only be described, but also predicted. 

Table I. Values of the Fisher Criterion for the Emulsion Viscosity Using the Ellis Equation...

Fisher criterion value for dispersed phase content of 0.30 0.50 0.60 0.65... 

The results of the experiments coincide well with the calculated curves. The model adequate has been tested with the help of the Fisher criterion from... 

In order to verify the model above, experimental data, obtained for 25x25x3 mm Raschig rings at different width of WFDR and different distances between them, are used. The Fisher criterion shows that the model is adequate. The model gives toe possibility to calculate toe distribution of the liquid phase in the packing volume of an industrial column. 

The criterion for best fit is based on the maximum likelihood principle (Fisher 1922) where the best estimates of the model parameters should maximise the likelihood function, L, for the observation of N different experimental observations. 

When feature selection is used to simplify, because of the large number of variables, methods must be simple. The univariate criterion of interclass variance/intraclass variance ratio (in the different variants called Fisher weights variance weights or Coomans weights is simple, but can lead to the elimination of variables with some discriminant power, either separately or, more important, in connection with other variables (Fig. 36). 

Actually an additional stability criterion is needed, see M.E. Fisher, Archives Rat. Mech. Anal. 17, 377 (1964) D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, New York 1969). A collection of point particles with mutual gravitational attraction is an example where this criterion is not satisfied, and for which therefore no statistical mechanics exists. 

If a linear model is inadequate it means that the response surface is not approximated to the plane. Apart from Fisher s criterion, which is there to judge the lack of fit of a regression model, inadequacy may also be recognized in this way ... 

Where the reproducibility variance is Sy=375.0, the arithmetic value of Fisher s criterion is Fr=1.21. 

The hypothesis on lack of fit of a regression model is checked by Fisher s criterion ... 

Knowledge of Sad and Sy facilitates determination of both calculating the value of Fisher s criterion and simultaneously of the tabular value by which we may compare and accept or reject the hypothesis of lack of fit of the regression model. Systematically given formulas for calculating Fisher s criterion for different designs of experiments are presented in Table 2.182. 

A more powerful criterion of goodness of fit is the F-test, pioneered by Fisher (1925), of the variance ratio... 

The value of P is used as a criterion to decide whether chance is a plausible explanation of the difference seen in the study data. If P is sufficiently low, chance may be thought to be an implausible explanation for the difference. The famous British statistician and geneticist Sir Ronald Fisher FRS proposed that the value 0-05 (or 5%) should be used as the criterion for judging if a value of P was sufficiently low to reject chance as the explanation of the difference - this is called the 0-05 or 5% level of statistical significance. So, the P = 0-017 would be sufficiently low to permit the conclusion that the difference was statistically significant - that is, that chance is an implausible explanation of the difference. On the other hand, P = 0-095 is not sufficiently small for chance to be thought an implausible explanation for the result put another way, the difference is not statistically significant. 

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