One of the key factors which limits the use of neural networks in many industrial applications has been the difficulty of demonstrating that a trained network will continue to generate reliable outputs once it is in routine use. An important potential source of errors is novel input data; that is, input data which differ significantly from the data used to train the network. The author …

A numerical method, based on neural-network-based functions, for solving partial differential equations is reported in the paper. Using a ‘universal approximator’ based on a neural network and point collocation, the numerical problem of solving the partial differential equation is transformed to an unconstrained minimization problem. The method is extremely easy to implement and is suitable for …

This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is …

Let Wt; t ϵ [0, 1] be a standard k-dimensional Weiner process defined on a probability space (Ω, F, P), and let Ft denote its natural filtration. Given a F1 measurable d-dimensional random vector X, we look for an adapted pair of processes {x(t), y(t); t ϵ [0, 1]} with values in Rd and Rd×k respectively, which solves an equation of the form: x(t) + ∫t1f(s, x(s), y(s)) ds + ∫t1 [g(s, x(s)) + y(s)] …

A common reaction among applied statisticians is that the Bayesian statistician's energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying Bayes's theorem to calculate conditional distributions of unknowns given knowns. I feel, however, that the applied Bayesian …

Suppose that a forecaster sequentially assigns probabilities to events. He is well calibrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent. We prove a theorem to the effect that a coherent Bayesian expects to be well calibrated, and consider its destructive implications for the theory of …

Reverse-time stochastic diffusion equation models are defined and it is shown how most processes defined via a forward-time or conventional diffusion equation model have an associated reverse-time model.

Any applied statistician who has analysed a number of sets of real data is likely to have come across ‘outliers’. The intuitive definition of an outlier would be ‘an observation which deviates so much from other observations as to arouse suspicions that it was generated by a different mechanism’. An inspection of a sample containing outliers would show up such characteristics as large gaps between …

This paper treats essentially the first derivative of an estimator viewed as functional and the ways in which it can be used to study local robustness properties. A theory of robust estimation "near" strict parametric models is briefly sketched and applied to some classical situations. Relations between von Mises functionals, the jackknife and U-statistics are indicated. A number of classical and …

A FORTRAN subroutine is described for minimizing a sum of squares of functions of many variables. Such problems arise in non-linear data fitting, and in the solution of non-linear algebraic equations. The subroutine is based on an algorithn due to Marquardt, but with modifications which improve the performance of the method in certain circumstances, yet which require negligible extra computer time …

This paper contains a new approach toward a theory of robust estimation; it treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators--intermediaries between sample mean and sample median--that are asymptotically most robust (in a sense to be specified) among all translation invariant estimators. For the general …

Most algorithms for the least-squares estimation of non-linear parameters have centered about either of two approaches. On the one hand, the model may be expanded as a Taylor series and corrections to the several parameters calculated at each iteration on the assumption of local linearity. On the other hand, various modifications of the method of steepest-descent have been used. Both methods not …

Consider a situation in which it is desired to gain knowledge about the true value of some parameter (or about the true state of the world) by means of experimentation. Let Ω denote the set of all possible values of the parameter θ, and suppose that the experimenter's knowledge about the true value of θ can be expressed, at each stage of experimentation, in terms of a probability distribution ξ …

The Michigan Mathematical Journal