Fuzzy clustering using multiple Gaussian kernels with optimized-parameters

In this paper, we propose a new kernel-based fuzzy clustering algorithm which tries to find the best clustering results using optimal parameters of each kernel in each cluster. It is known that data with nonlinear relationships can be separated using one of the kernel-based fuzzy clustering methods. Two common fuzzy clustering approaches are: clustering with a single kernel and clustering with multiple kernels. While clustering with a single kernel doesn’t work well with “multiple-density” clusters, multiple kernel-based fuzzy clustering tries to find an optimal linear weighted combination of kernels with initial fixed (not necessarily the best) parameters. Our algorithm is an extension of the single kernel-based fuzzy c-means and the multiple kernel-based fuzzy clustering algorithms. In this algorithm, there is no need to give “good” parameters of each kernel and no need to give an initial “good” number of kernels. Every cluster will be characterized by a Gaussian kernel with optimal parameters. In order to show its effective clustering performance, we have compared it to other similar clustering algorithms using different databases and different clustering validity measures.     

Fluctuation between Subsets of Evolutions Governed by Chaotic Systems

Some issues on fluctuation of solutions to Lorenz and Rossler systems, for instance, are related to viability kernels of subsets under continuous time systems, or in the case of Julia sets, for instance, under discrete time systems. It happens that viability kernels of subsets, capture basins of targets and the combination of the twos provide tools for the analysis of the local behavior around equilibria (local stable and unstable manifolds), the asymptotic behavior and the functuation of evolutions between two areas of a domain, etc. Since algorithms and softwares do exist for computing the viability kernels and the capture basins, as well as evolutions viable in the viability kernel until they converge to a target infinite time, we are able to localize fluctuation basins.     

Structure of Viability Kernels for Some Linear Differential Games

A convenient form of necessary and sufficient conditions of viability for differential games with linear dynamics is proposed. These conditions are utilized to construct maximal viable subsets of state constraints, viability kernels, in two illustrative two-dimensional examples. These examples demonstrate the relative simplicity of the structure of the viability kernels. It was found that the boundaries of the viability kernels consist of segments of the boundary of the state constraint and of lines defined by the first integrals of the governing equations as the players use extremal constant controls. It is conjectured that such a structure holds in high dimensional cases too.     

Local Linear Estimation of Jump-Diffusion Models by Using Asymmetric Kernels

This article addresses the problem of nonparametric estimation of the first and second infinitesimal moments by using the local linear method of the underlying jump-diffusion models. The motivation behind the study is to use the asymmetric kernels instead of standard kernel smoothing. The basic idea relies on replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the nonparametric estimation for jump-diffusion models. We prove that the estimators based on the local linear method for jump-diffusion models are consistent and asymptotically follow normal distribution under the condition of recurrence and stationarity.     

Optimal kernel estimation of spot volatility of stochastic differential equations

A unified framework to optimally select the bandwidth and kernel function of spot volatility kernel estimators is put forward. The proposed models include not only classical Brownian motion driven dynamics but also volatility processes that are driven by long-memory fractional Brownian motions or other Gaussian processes. We characterize the leading order terms of the mean squared error, which in turn enables us to determine an explicit formula for the leading term of the optimal bandwidth. Central limit theorems for the estimation error are also obtained. A feasible plug-in type bandwidth selection procedure is then proposed, for which, as a sub-problem, a new estimator of the volatility of volatility is developed. The optimal selection of the kernel function is also investigated. For Brownian Motion type volatilities, the optimal kernel turns out to be an exponential function, while, for fractional Brownian motion type volatilities, easily implementable numerical results to compute the optimal kernels are devised. Simulation studies further confirm the good performance of the proposed methods.     

Computing the viability kernel in large state dimension

T-viable states in a closed set K under a certain set-valued dynamic are states from which there exists at least one solution remaining in K until a given time horizon T. Minimizing the cost to constraints lets us determine whether a given state is T-viable or not, and this is implementable in large dimension for the state-space. Minimizing on the initial condition itself lets find viable states. Quincampoix's semi-permeability property helps find other states located close to the viability boundary, which is then gradually delineated. The algorithm is particularly suited to the identification of specific trajectories, such as the heavy viable solution, or to the computation of viability kernels associated with delayed dynamics. The volume of the viability kernel and its confidence interval can be estimated by randomly drawing states and checking their viability status. Examples are given.     

Optimal Stopping with a Probabilistic Constraint

We present an efficient method for solving optimal stopping problems with a probabilistic constraint. The goal is to optimize the expected cumulative cost, but constrained by an upper bound on the probability that the cost exceeds a specified threshold. This probabilistic constraint causes optimal policies to be time-dependent and randomized, however, we show that an optimal policy can always be selected with “piecewise-monotonic” time-dependence and “nearly-deterministic” randomization. We prove these properties using the Bellman optimality equations for a Lagrangian relaxation of the original problem. We present an algorithm that exploits these properties for computational efficiency. Its performance and the structure of optimal policies are illustrated on two numerical examples.     

Population Growth and Nash Equilibria Under Viability Constraints in the Commons

Population growth modifies the optimal equilibrium between a stationary population and its resource, producing instead a line of equilibria, characterized by fluctuating population size, resource quantity, harvest per head, and birth rates. The Pontryagin procedure allows the analytical expression of the Nash equilibria for two populations sharing a common resource and capable of growth. An alternative procedure, which avoids solving differential equations and inherently includes state constraints, involves building the capture-viability kernel of an auxiliary system. For two populations, all Nash equilibria under state constraints are obtained as the intersection of the boundaries of two capture-viability kernels. The two methods, Pontryagin and viability, yield concordant results. Viability is more flexible and avoids solving differential equations for each initial condition.     

Learning the optimal kernel for Fisher discriminant analysis via second order cone programming

Kernel Fisher discriminant analysis (KFDA) is a popular classification technique which requires the user to predefine an appropriate kernel. Since the performance of KFDA depends on the choice of the kernel, the problem of kernel selection becomes very important. In this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels, and formulate it as a second order cone programming (SOCP) problem. This formulation seems to be promising because the resulting SOCP can be efficiently solved by employing interior point methods. The efficacy of the optimal kernel, selected from a given convex set of basic kernels, is demonstrated on UCI machine learning benchmark datasets.     

Dilations of Some VH-Spaces Operator Valued Invariant Kernels

We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy??s and Stinesprings?? dilation type theorems.     

VIABILITY MODEL OF TROPHIC INTERACTIONS IN MARINE ECOSYSTEMS

ABSTRACT. An attempt to use viability models for studying marine ecosystems is proposed as a possible alternative to classical ecosystem modeling. Viability models do not consider optimal solutions but instead define all possible evolutions of a dynamical system under given constraints. Applied to marine ecosystems, a viability model is formulated based on the trophic coefficients of a mass‐balanced model. This requires relatively few assumptions about the processes involved and can integrate uncertainty associated with the required estimates of input parameters. An iterative algorithm is proposed to calculate the viability kernel, i.e., the envelope of all viable trajectories of the ecosystem. An application to the Benguela ecosystem is presented, considering interactions between detritus, phytoplankton, zooplankton, pelagic fish, demersal fish and fisheries. Results show how a viability kernel could be used to better define the healthy states of a marine ecosystem, by defining what states should be avoided. The paper discusses how viability models of trophic interactions could help to define a new ecosystem‐based indicator for fisheries management. It then discusses how this approach can potentially contribute to a paradigm shift that is emerging in the management of renewable resources.     

Stochastic control and compatible subsets of constraints

Given a stochastic differential control system and a closed set K in Rn, we study the that, with probability one, the associated solution of the control system remains for ever in the set K. This set is called the viability kernel of K. If N is equal to the whole set K, K is said to be viable. We prove that, in the general case, the viability kernel itself is viable and we characterize it through some partial differential equations. We prove that, under suitable assumptions, also the boundary of N is viable. As an application, we give a new characterization of the value function of some optimal control problem.     

Quantitative Lower Bounds for the Full Boltzmann Equation,Part I: Periodic Boundary Conditions

ABSTRACT

We prove the appearance of an explicit lower bound on the solution to the full Boltzmann equation in the torus for a broad family of collision kernels including in particular long-range interaction models, under the assumption of some uniform bounds on some hydrodynamic quantities. This lower bound is independent of time and space. When the collision kernel satisfies Grad's cutoff assumption, the lower bound is a global Maxwellian and its asymptotic behavior in velocity is optimal, whereas for noncutoff collision kernels the lower bound we obtain decreases exponentially but faster than the Maxwellian. Our results cover solutions constructed in a spatially homogeneous setting, as well as small-time or close-to-equilibrium solutions to the full Boltzmann equation in the torus. The constants are explicit and depend on the a priori bounds on the solution.     

On Lacunary Analogs of the Poincare Theta-Series and Their Applications

We consider Fuchsian groups of linear-fractional transformations such that each vertex of the fundamental polygon is common for an even or infinite number of fundamental congruent polygons meeting at this point. The whole collection of transformations splits into two disjoint sets. For these sets we introduce two lacunary kernels whose sum represents the well-known analog of Chibrikova and Sil'vestrov's kernel and study their properties. We introduce automorphic forms of dimension –4m that differ from the Poincare theta-series. We indicate an application of one of the constructed lacunary kernels which does not include the Cauchy kernel to solving some boundary value problem with a shift of the contour inside the domain.     

Sampling Theorems Associated with Singular q-Sturm Liouville Problems

In this paper we introduce three sampling theorems for transformations defined in terms of Jackson q-integration when the kernel of the transformation is a solution or the Green’s function of singular q-Sturm–Liouville problems. We consider the problem when the q-Sturm–Liouville problem is singular either at infinity or at zero with detailed investigations when the singular point is infinity. This approach allows the derivation of sampling representations for transforms whose kernels are linear combinations of q-Bessel functions, not just a single one as previously established.     

基于虚拟完整约束的欠驱动起重机控制方法

欠驱动系统的控制是非线性控制的一个重要领域,欠驱动系统指系统控制输入个数小于自由度个数的非线性系统.目前,欠驱动非线性系统动力学和控制研究的主要方法包括线性二次型最优控制方法和部分反馈线性化方法等,如何使系统持续的稳定在平衡位置一直是研究的难点.虚拟约束方法是指通过选择一个周期循环变化的变量作为自变量来设计系统的周期运动.该文以典型的欠驱动模型起重机为例,采用虚拟约束方法,使系统能够在平衡位置稳定或周期振荡运动.首先,通过建立虚拟约束,减少系统自由度变量;然后,通过部分反馈线性化理论推导出系统的状态方程;最后,通过线性二次调节器设计反馈控制器.仿真结果表明,重物在反馈控制下可以在竖直位置的附近达到稳定状态,反映了虚拟约束方法对欠驱动系统的有效性.     

Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc.     

Cutoff-type Boltzmann equations: Convergence of the solution

Using an approach similar to Tanaka's we prove the convergence toward equilibrium for general classes of models which correspond to Boltzmann equations of the cutoff type. A major step consists in showing a convex inequality involving the Kantorovich-Vasherstein metric. This requires assumptions on the interacting kernels. These assumptions are very natural from a physical point of view. In particular, our classes include models recently developed by physicists to study relaxation of closed oscillator systems.     

Equilibrium in semimonotone market games

This paper analyzes the existence of equilibrium for a class of market games in which agents are allowed to follow different patterns of behaviour, including cases where the strategy sets are neither compact nor convex. Agent’s behaviour is modelled in terms of “inverse reply correspondences” (mappings that associate to each agent’s strategy those outcomes that she finds acceptable). Sufficient conditions for an equilibrium to exist are provided     

A dynamic theory of the credit union

A topic of recent interest in the retail financial sector has been the growth of credit unions or “pure cooperatives”. Past credit union researchers built mathematical models of credit union operations. These models identified important operating characteristics but were modeled under assumptions of static operating environments. The model presented in this paper departs from the traditional static models and examines dynamic operation for a United States credit union. Its inter-temporal structure clarifies a number of issues—such as optimal equity retention and inter-temporal rate policy—not addressed by earlier studies. Given initial conditions, the model specifies equity retention and inter-temporal deposit and loan rate policies until an equilibrium state is reached.