This is to announce the FASS Regression Solver. It is a product of The Fuzzy Analytical & Statistical Software Co. Ltd. (FASS). It solves the general (non-linear) regression problem.

FASS, as the name implies, is offering software in which there is intimate mixing, yet careful separation, between uncertainty of the probabilistic kind, and that of the fuzzy kind.

Uncertainty in the fitted parameters of the regression model is fuzzy. This is in general the case for uncertainty regarding a constant known only to within a fuzzy term of description. Uncertainty in the measured data is also represented as fuzzy in
general. Thus the data inputs are fuzzy, and the estimated parameters reported as outputs are fuzzy also. Both are standardized as LABR fuzzy sets. The error residuals are assumed to be zero-mean Gaussian.

Consistent with the LABR fuzzy representation for each parameter, any function of the parameters may likewise have its derived fuzzy uncertainty also represented as an LABR fuzzy set. Thus the prediction function of the mean is represented as a fuzzy
LABR swath, grosser or more precise according as the degree of fuzziness inherent in the fitted parameters deriving from the model and the data. Likewise, 5% and 95% probability quantiles are LABR fuzzy swaths w.r.t to the independent variables.

The fuzzy mapping of the uncertainty in each parameter accomplishes in principle that which the Bayesian approach seeks among others to do, which is to map the marginals describing the uncertainty in each fitted parameter. But it is insisted that the
proper calculus that should always have applied to uncertain constants is that of the possibility calculus. The fundamental mathematical object of the possibility calculus is the absolute likelihood function (a.l.f). Under this calculus, the need for a
subjective, or any other "prior", is obviated.

This calculus is a special case of a reformulated fuzzy logic. Jaynes was correct in his assertion that a logic that did not uphold Aristotle, where Aristotle applies, could not be a proper basis for addressing the statistical inference problem. The
fuzzy logic here deployed is one in which Aristotle is upheld. The reformulated fuzzy logic allows for flexible connectives, and shows how, within the theory, these may be selected. The max/min rules apply in some situations, the product/product-sum
rules in others, and the bounded-sum rules in others, and in general, the reformulated fuzzy logic proposes a linear combination of these three basic rules, mediated through the semantic consistency coefficient. The determination of the latter is
internal to the theory under a fairly straight-forward rule easy to motivate and justify.

The probabiility of the data may be represented as a product-sum (p-s) integral over the a.l.f. This provides an obvious maximization criterion (max p-s) for the general regression problem. It is akin to maximum likelihood obviously, but uses a product-
sum rule of disjunction rather than a maximization rule, or bounded-sum rule. Both of the latter are rejected in the light of the reformulated fuzzy logic, which reserves the maximum and bounded-sum rules of disjunction respectively for cases where there
is strong positive, or strong negative semantic consistency. In the case of statistical inference, it is a rule of semantic independence that must apply, consistent with the i.i.d assumption for a statistical sample.

This approach obviates Bayesian priors, as already mentioned. It also sidesteps the known difficulties of the maximum likelihood estimate (MLE) of being sometimes misleading both as to location and precision of the true value sought. And it may be used
to give a full mapping of the uncertainty in fitted parameters, as the Bayesian approaches (e.g. MCMC) rightly seek to accomplish.

The FASS Regression Solver rests on the analytic extensions to ideas explored in an earlier work, "Fuzziness and Probability" (1995). Those extensions are forthcoming in a monograph, "The unified theory of fuzzy logic, the possibility calculus, and
statistical inference, with application to Gaussian regression analysis".

See the website: http://fuzzastat.com, for further details of the FASS Regression Solver.

For those interested, note in particular that a seminar/tutorial is proposed for Jan 26-Feb 2, 2016, in Port of Spain, Trinidad.

Sidney
--
Dr. Sidney Thomas
Executive Chairman and Principal
The Fuzzy Analytical & Statistical Software Co. Ltd.
http://fuzzastat.com

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