Both fuzzy logic, as the basis of many inference systems, and Neural Networks, as a powerful computational model for classification and estimation, have been used in many application fields since their birth. These two techniques are somewhat supplementary to each other in a way that what one is lacking of the other can provide. This led to the creation of Neuro-Fuzzy systems which utilize fuzzy logic to construct a complex model by extending the capabilities of Artificial Neural Networks. Generally speaking all type of systems that integrate these two techniques can be called Neuro-Fuzzy systems. Key feature of these systems is that they use inputoutput patterns to adjust the fuzzy sets and rules inside the model. The paper reviews the principles of a NeuroFuzzy system and the key methods presented in this field, furthermore provides survey on their applications for technical diagnostics and measurement.
We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from finite-sample system identification, that by perturbing the residuals in the gradient of the objective function, information can be extracted about the amount of uncertainty our model has. Particularly, we provide an algorithm to build exact, non-asymptotically guaranteed, distribution-free confidence regions for ideal, noise-free representations of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efficient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, ε-SVR and kernelized LASSO.
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