Goodness-of-fit tests for the error distribution in nonparametric regression

Suppose the random vector (X,Y) satisfies the regression model Y=m(X)+σ(X)ε, where m(⋅) is the conditional mean, σ²(⋅) is the conditional variance, and ε is independent of X. The covariate X is d-dimensional (d≥1), the response Y is one-dimensional, and m and σ are unknown but smooth functions. Goodness-of-fit tests for the parametric form of the error distribution are studied under this model, without assuming any parametric form for m or σ. The proposed tests are based on the difference between a nonparametric estimator of the error distribution and an estimator obtained under the null hypothesis of a parametric model. The large sample properties of the proposed test statistics are obtained, as well as those of the estimator of the parameter vector under the null hypothesis. Finally, the finite sample behavior of the proposed statistics, and the selection of the bandwidths for estimating m and σ are extensively studied via simulations.     

Unsupervised image segmentation using triplet Markov fields

Hidden Markov fields (HMF) models are widely applied to various problems arising in image processing. In these models, the hidden process of interest X is a Markov field and must be estimated from its observable noisy version Y. The success of HMF is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed one remains Markovian, which facilitates different processing strategies such as Bayesian restoration. HMF have been recently generalized to “pairwise” Markov fields (PMF), which offer similar processing advantages and superior modeling capabilities. In PMF one directly assumes the Markovianity of the pair (X, Y). Afterwards, “triplet” Markov fields (TMF), in which the distribution of the pair (X, Y) is the marginal distribution of a Markov field (X, U, Y), where U is an auxiliary process, have been proposed and still allow restoration processing. The aim of this paper is to propose a new parameter estimation method adapted to TMF, and to study the corresponding unsupervised image segmentation methods. The latter are validated via experiments and real image processing.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen–Cahn equation u_t − Δu+ ε⁻² f(u)=0. It is shown that the error depends on ε⁻¹ only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen–Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.     

An efficient estimator of pattern recognition system error probability

A new estimator of system error probability is proposed. The estimator combines the average conditional error method and the empirical error count method so that all the information available to the designer (test and reference data set samples and their labels) can be utilized most efficiently. It is shown that the proposed estimator is unbiased and has a lower variance than the average conditional error estimator proposed by Kittler and Devijver.⁽⁵⁾     

Fixed rank solutions of the matrix equation with statistical applications

By applying the canonical correlation decomposition of matrix pairs, the general fixed rank least square solutions of matrix equation Xβ=Y are derived. As statistical applications, an algorithm for computing the least square estimator of the multivariate reduced rank regression model Y=Xβ+?, r(β)=t is given.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

An interpolation problem associated with H-optimal design in systems with distributed input lags

A variety of H^∞ optimal design problems reduce to interpolation of compressed multiplication operators, f(s) → π_k(w(s)f(s)), where w(s) is a given rational function and the subspace K is of the form K=H² φ(s)H². Here we consider φ(s) = (1-e^α-5)/(s - α), which stands for a distributed delay in a system's input. The interpolation scheme we develop, adapts to a broader class of distributed lags, namely, those determined by transfer functions of the form B(e^−s)/b(s), where B(z) and b(s) are polynomials and b(s) = 0 implies B(e^−s) = 0.     

Two-dimensional diffusion processes as models in lifetime studies

Let X(t) denote the remaining useful lifetime of a machine, and Y(t) be a standard Brownian motion. Assume that the derivative ρ[X(t),?Y(t)] of X(t) is a deterministic function of (at least) Y(t). We consider the two-dimensional degenerate diffusion process (X(t),?Y(t)). We obtain explicit expressions for the expected value of the random variable T(x,?y) denoting the first time the machine must be replaced, or repaired, for various functions ρ[X(t),?Y(t)].     

Compact Recognizers of Episode Sequences

Given two strings X=a₁…a_n and P=b₁…b_m over an alphabet Σ, the problem of testing whether P occurs as a subsequence of X is trivially solved in linear time. It is also known that a simple O(n log |Σ|) time preprocessing of X makes it easy to decide subsequently, for any P and in at most |P| log |Σ| character comparisons, whether P is a subsequence of X. These problems become more complicated if one asks instead whether P occurs as a subsequence of some substring Y of X of bounded length. This paper presents an automaton built on the textstring X and capable of identifying all distinct minimal substrings Y of X having P as a subsequence. By a substring Y being minimal with respect to P, it is meant that P is not a subsequence of any proper substring of Y. For every minimal substring Y, the automaton recognizes the occurrence of P having the lexicographically smallest sequence of symbol positions in Y. It is not difficult to realize such an automaton in time and space O(n²) for a text of n characters. One result of this paper consists of bringing those bounds down to linear or O(n log n), respectively, depending on whether the alphabet is bounded or of arbitrary size, thereby matching the corresponding complexities of automata constructions for offline exact string searching. Having built the automaton, the search for all lexicographically earliest occurrences of P in X is carried out in time O(∑_i=1^mrocc_i·i) or O(n+∑_i=1^mrocc_i·i· log n), depending on whether the alphabet is fixed or arbitrary, where rocc_i is the number of distinct minimal substrings of X having b₁…b_i as a subsequence (note that each such substring may occur many times in X but is counted only once in the bound). All log factors appearing in the above bounds can be further reduced to log log by resorting to known integer-handling data structures.     

Small area estimation using a nonparametric model-based direct estimator

Nonparametric regression is widely used as a method of characterizing a non-linear relationship between a variable of interest and a set of covariates. Practical application of nonparametric regression methods in the field of small area estimation is fairly recent, and has so far focussed on the use of empirical best linear unbiased prediction under a model that combines a penalized spline (p-spline) fit and random area effects. The concept of model-based direct estimation is used to develop an alternative nonparametric approach to estimation of a small area mean. The suggested estimator is a weighted average of the sample values from the area, with weights derived from a linear regression model with random area effects extended to incorporate a smooth, nonparametrically specified trend. Estimation of the mean squared error of the proposed small area estimator is also discussed. Monte Carlo simulations based on both simulated and real datasets show that the proposed model-based direct estimator and its associated mean squared error estimator perform well. They are worth considering in small area estimation applications where the underlying population regression relationships are non-linear or have a complicated functional form.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Optimal diagonal scaling for infinity-norm optimization

A solution is presented for the previously unsolved diagonally scaled multivariable infinity-norm optimization problem of minimizing D(s)(A(s) + Ψ(s) X(s))D⁻¹(s)_∞ over the set of stable minimum-phase diagonal D(s) and stable X(s). This problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of ‘structured’ plant uncertainty. The result facilitates the design of feedback controllers which optimize the ‘excess stability margin’ [3] (or, equivalently, the ‘structured singular value μ’ [4]) of diagonally perturbed feedback systems.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Monoid-labeled transition systems

Given a -complete (semi)lattice , we consider -labeled transition systems as coalgebras of a functor ⁽⁻⁾, associating with a set X the set ^X of all -fuzzy subsets. We describe simulations and bisimulations of -coalgebras to show that L⁽⁻⁾ weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID).Exchanging for a commutative monoid , we consider the functor ⁽⁻⁾_ω which associates with a set X all finite multisets containing elements of X with multiplicities m M. The corresponding functor weakly preserves nonempty pullbacks along injectives iff 0 is the only invertible element of , and it preserves nonempty kernel pairs iff is refinable, in the sense that two sum representations of the same value, r₁ + … + r_m = c₁ + … + c_n, have a common refinement matrix (m_{(i, j)}) whose k-th row sums to r_k and whose l-th column sums to c_l for any 1≤ k ≤ m and 1 ≤ l ≤ n.     

A note on two-sided stochastic control problems

We consider a class of two-sided stochastic control problems. For each continuous process π_t = π_t⁺ − π_t⁻ with bounded variation, the state process (x_t) is defined by x_t = B_t + f₀^t I(x_s - a)^d_πs+ − f₀^t I(x_s a)^dπ_s−, where a is a positive constant and (B_t) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function J(π) = E [f₀^∞ e^−αsX_s² ds], with discount rate α > 0, associated with π.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

Recursive Data Types in Algebraically ω-Complete Categories

A category (of data types) is called algebraically ω-complete provided that for each endofunctor T the data-type equation T(X) X has a solution constructed as a colimit of the ω-chain → T() → T²()..., where is the initial data-type. Examples include the categories of (1) countable sets and (total, partial, or nondeterministic) functions, (2) countably dimensional vector spaces and linear functions, (3) countable well-ordered sets and join-preserving functions. In the case of categories enriched over CPO (the category of complete partial orders and strict, continuous functions) a stronger property holds for all locally continuous functors T: the data-type equation is both a limit and a colimit of the finite iterations of T over the initial data-type.     

Fuzzy statistics: regression and prediction

Our method of estimation of parameters in statistics uses a set of confidence intervals producing a triangular shaped fuzzy number for the estimator. In crisp linear regression we use this to obtain fuzzy number estimators for τ and λ. This is then employed in fuzzy prediction and fuzzy hypothesis testing about the values of τ and λ.