Phil Diamond

Metric spaces of fuzzy sets

Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.

Fuzzy least squares

Abstract Several models for simple least-squares fitting of fuzzy-valued data are developed. Criteria are given for when fuzzy data sets can be fitted to the models, and analogues of the normal equations are derived.

Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations

Traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations (DEs). A recent interpretation in terms of differential inclusions, expressed level setwise, overcomes this deficiency and opens up for profitable investigation such properties as stability, attraction, periodicity, and the like. This is especially important for investigating continuous systems which are uncertain or incompletely specified. This paper studies attractors of fuzzy DEs in terms of cocycles and encompasses both the time-dependent and autonomous cases.

Stability and periodicity in fuzzy differential equations

Formulations of fuzzy differential equations (DEs) in terms of the Hukuhara derivative do not reproduce the rich and varied behavior of crisp DEs. Another interpretation in terms of differential inclusions, expressed level setwise, overcomes much of this deficiency and opens up for profitable investigation such properties as stability, attraction, periodicity, and the like. This is especially important for investigating continuous systems, which are uncertain or incompletely specified. This paper studies studies Lyapunov stability of fuzzy DEs and the periodicity of the fuzzy solution set for both the time-dependent and autonomous cases.

Brief note on the variation of constants formula for fuzzy differential equations

This note gives a theory of state transition matrices for linear systems of fuzzy differential equations. This is used to give a fuzzy version of the classical variation of constants formula. A simple example of a time-independent control system is used to illustrate the methods. While similar problems to the crisp case arise for time-dependent systems, in time-independent cases the calculations are elementary solutions of eigenvalue-eigenvector problems. In particular, for nonnegative or nonpositive matrices, the problems at each level set, can easily be solved in MATLAB to give the level sets of the fuzzy solution.

Extended fuzzy linear models and least squares estimates

Least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads [1]. The idea is to introduce best fit difference models which exploit both the Hukuhara difference and the L2-metric distance. The fuzzy models use LR-fuzzy numbers. Fitted models are compared by using the coefficient of determination, in a similar way to its use in classical statistical least squares fitting. The non-LR-fuzzy case is also considered.

Characterization of compact subsets of fuzzy sets

A characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space Rn, the metric for which is the supremum over the Hausdorff distances between corresponding level sets. It is shown that a closed subset is compact if and only if it is uniformly support-bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable.

Anisotropy-based performance analysis of linear discrete time invariant control systems

The anisotropic norm of a linear discrete-time-invariant system measures system output sensitivity to stationary Gaussian input disturbances of bounded mean anisotropy. Mean anisotropy characterizes the degree of predictability (or colouredness) and spatial non-roundness of the noise. The anisotropic norm falls between the H

Regularity of solution sets for differential inclusions quasi-concave in a parameter

The concept of quasi-concavity is extended to multifunctions. It is then shown that if the velocity of a differential inclusion is regularly quasi-concave in a parameter, the solution set and attainability set are also dependent upon the parameter in like manner. The result is applied to give a vastly improved notion of fuzzy differential equations

Robustness of variograms and conditioning of kriging matrices

Current ideas of robustness in geostatistics concentrate upon estimation of the experimental variogram. However, predictive algorithms can be very sensitive to small perturbations in data or in the variogram model as well. To quantify this notion of robustness, nearness of variogram models is defined. Closeness of two variogram models is reflected in the sensitivity of their corresponding kriging estimators. The condition number of kriging matrices is shown to play a central role. Various examples are given. The ideas are used to analyze more complex universal kriging systems.