Continuous Vector Functions Research Articles

Tests for equality of several mean vector functions for multivariate functional data with applications

In various scientific and technological fields, it is common to encounter multivariate functional data, where continuous vector functions for statistical units are observed over a specific interval. This paper aims to investigate the equality of mean vector functions in multi-sample multivariate functional data. To accomplish this, two novel global testing statistics are introduced by integrating or maximising the pointwise Lawley-Hotelling trace test statistic. The paper also provides asymptotic expressions for these proposed testing statistics under the null hypothesis and establishes their consistency with respect to the square root of the sample size. The performance of the two new testing procedures is demonstrated through simulation studies and the analysis of a real-world data example, highlighting their advantages over existing methods.

Iterative Methods with Self-Learning for Solving Nonlinear Equations

This paper is devoted to the problem of solving a system of nonlinear equations with an arbitrary but continuous vector function on the left-hand side. By assumption, the values of its components are the only a priori information available about this function. An approximate solution of the system is determined using some iterative method with parameters, and the qualitative properties of the method are assessed in terms of a quadratic residual functional. We propose a self-learning (reinforcement) procedure based on auxiliary Monte Carlo (MC) experiments, an exponential utility function, and a payoff function that implements Bellman’s optimality principle. A theorem on the strict monotonic decrease of the residual functional is proven.

STABILITY OF SOLUTIONS TO SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH INFINITE DISTRIBUTED DELAY

The paper considers a certain class of systems of nonlinear differential equations with infinite distributed delay. It is assumed that the coefficients in the linear terms are T-periodic, the nonlinear term is a continuous Lipschitz-like vector function, whose the degree of smallness greater than one. Such systems of differential equations arise when modeling various processes occurring in biology, chemistry, physics, economics. The Lyapunov - Krasovskii functional is proposed, on the basis of which sufficient conditions for exponential stability of the zero solution of the class of systems under consideration are established, estimates for the set of attraction of the zero solution and estimates for the norm of the solution to the initial value problem characterizing exponential decrease at infinity are indicated. All parameters involved in the estimates are specified explicitly. The conditions of exponential stability of the zero solution established in the paper are expressed in terms of integral inequality. The conditions of global exponential stability of the zero solution are also obtained.

Linear and nonlinear integral functional on the space of continuous vector functions

Linear and nonlinear integral functional on the space of continuous vector functions

Stability of spectral characteristics of boundary value problems for 2 × 2 Dirac type systems. Applications to the damped string

The paper is concerned with the stability property under perturbation Q→Q˜ of different spectral characteristics of a boundary value problem associated in L2([0,1];C2) with the following 2×2 Dirac type equation(0.1)LU(Q)y=−iB−1y′+Q(x)y=λy,B=(b100b2),b1<0<b2,y=col(y1,y2), with a potential matrix Q∈Lp([0,1];C2×2) and subject to the regular boundary conditions Uy:={U1,U2}y=0. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. Our approach to the spectral stability relies on the existence of the triangular transformation operators for system (0.1) with Q∈L1, which was established in our previous works. The starting point of our investigation is the Lipshitz property of the mapping Q→KQ±, where KQ± are the kernels of transformation operators for system (0.1). Namely, we prove the following uniform estimate:‖KQ±−KQ˜±‖X∞,p2+‖KQ±−KQ˜±‖X1,p2⩽C⋅‖Q−Q˜‖p,Q,Q˜∈Up,r2×2,p∈[1,∞], on balls Up,r2×2 in Lp([0,1];C2×2). It is new even for Q˜=0. Here X∞,p2, X1,p2 are the special Banach spaces naturally arising in such problems. We also obtained similar estimates for Fourier transforms of KQ±. Both of these estimates are of independent interest and play a crucial role in the proofs of all spectral stability results discussed in the paper. For instance, as an immediate consequence of these estimates we get the Lipshitz property of the mapping Q→ΦQ(⋅,λ), where ΦQ(x,λ) is the fundamental matrix of the system (0.1).Assuming the spectrum ΛQ={λQ,n}n∈Z of LU(Q) to be asymptotically simple, denote by FQ={fQ,n}|n|>N a sequence of corresponding normalized eigenvectors, LU(Q)fQ,n=λQ,nfQ,n. Assuming boundary conditions (BC) to be strictly regular, we show that the mapping Q→ΛQ−Λ0 sends Lp([0,1];C2×2) either into ℓp′ or into the weighted ℓp-space ℓp({(1+|n|)p−2}); we also establish its Lipshitz property on compact sets in Lp([0,1];C2×2), p∈[1,2]. The proof of the second estimate involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. It is also shown that the mapping Q→FQ−F0 sends Lp([0,1];C2×2) into the space ℓp′(Z;C([0,1];C2) of sequences of continuous vector-functions, and has the Lipshitz property on compacts sets in Lp([0,1];C2×2), p∈[1,2].Certain modifications of these spectral stability results are also proved for balls Up,r2×2 in Lp([0,1];C2×2), p∈[1,2].Note also that the proof of the Lipshitz property of the mapping Q→FQ−F0 involves the deep Carleson-Hunt theorem for maximal Fourier transform, while the proof of this property for the mapping Q→ΛQ−Λ0 relies on the estimates of the classical Fourier transform and is elementary in character.We apply our previous results to the damped string equation to establish the Riesz basis property and the asymptotic behavior of the eigenvalues of the corresponding dynamic generator under the assumptions d∈L1[0,ℓ], ρ∈W1,1[0,ℓ] on the damping coefficient and the density of the string, that are weaker than previously treated in the literature. We also establish Lipshitz dependence on d and ρ in ℓp-spaces of the remainders in the asymptotic formula for the eigenvalues.

Retrospective Change-Points Detection for Multidimensional Time Series of Arbitrary Nature: Model-Free Technology Based on the ϵ-Complexity Theory.

We consider a retrospective change-point detection problem for multidimensional time series of arbitrary nature (in particular, panel data). Change-points are the moments at which the changes in generating mechanism occur. Our method is based on the new theory of -complexity of individual continuous vector functions and is model-free. We present simulation results confirming the effectiveness of the method.

Model-free classification of panel data via the -complexity theory

In this paper, we propose a novel method for the classification of so-called panel data. Our approach is based on a notion of the -complexity of continuous vector-functions. The proposed methodology does not use any prior information on data-generating mechanisms, therefore, can be utilized for multidimensional time series of arbitrary nature. We demonstrate the performance of our methodology on simulations.

Analytic vector-functions in the unit ball having bounded $\mathbf{L}$-index in joint variables

In this paper, we consider a class of vector-functions, which are analytic in the unit ball. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ We present necessary and sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables. They describe the local behavior of the maximum modulus of every component of the vector-function or its partial derivatives.

Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables.

In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables for vector-functions analytic in the unit ball, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc. The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii, for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii. In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.

Operators on Spaces of Functions and Measures. Vector Invariant (Fractal) Measures

We consider a general schema involving measure spaces, contractions and linear and continuous operators. Within the framework of this schema we use our sesquilinear uniform integral and introduce some integral operators on continuous vector function spaces, which lead us to operators on spaces of vector measures. Using these last operators, we generalize the Markov operators, obtaining via contractions vector invariant (fractal) measures. Concrete examples are provided.

Special Issue on Recent Trends in Stochastic Models, Statistics and Their Applications

This special issue was organized on the occasion of the 13th Workshop on Stochastic Models, Statistics and Their Applications (SMSA 2017) held at Humboldt University of Berlin on February 20-24, 2017. Key areas of the workshop focused on models and methods for problems arising in industry, finance, natural sciences, engineering, and technology. In these areas, solutions are needed, which are able to deal with complex and possibly high-dimensional data, make use of innovative methodologies, and have proven applicability to real-world data. In this special issue, we have 13 papers. The papers consider stochastic models for diverse data sets and streams such as bridge vehicle loads, watermark identification data, text documents, images of digitized handwritten digits, financial asset returns, option prices, and equity data. The methods and algorithms required to handle come from machine learning, stochastic processes, sequential (on-line) change-point detection, regression analysis, image analysis, and robust divergence measures. The work of Barbulescu analyzes the properties of Generalized Regression Neural Network (GRNN) model applied to the prediction of the selected Romanian stock market prices. In the work of Luo et al, a nonlinear PDE model to estimate the American call options and put options under financial crisis scenario is implemented with additional penalty introduced to Black-Scholes formula. The work of Liu et al proposes a compound Poisson bivariate process model of vehicle load with Levy copula function and validates it in a reliability analysis of bridges in China. Migalska proposes in her paper a density-free symmetry verification test for arbitrary symmetries in images. In their work, Darkhovsky et al propose a novel method based on the notion of ϵ-complexity of continuous vector functions for an off-line detection of multiple change-points in multidimensional time series. The work of Rafajlowicz et al proposes a method of spike detection and suppression (SDS method), which extends the jump detector proposed by the Authors in their earlier work and applies it for a laser power control in a 3D additive manufacturing process. Sliwinski et al propose two algorithms for nonlinearity recovery in Hammerstein systems, one based on orthogonal series expansions and the other using aggregation and validate them in numerical simulations. The work of Sousa et al provides an insight into the behavior of GARCH process describing financial markets affected by rumours and other random factors through an analysis of process mean and variance by estimating run length. Kisslinger et al describe a new toolkit based on scaled Bregman divergences for detecting distributional changes in random data, including streams and clouds. The work of Sarnowski et al derives an optimal decision function for a sudden change detection in political business cycle model based on the Markovian process. The work of Walkowiak et al proposes a two-step procedure for open-set classification of text documents and validates it on simulated data and on the collection of Wikipedia documents. The work by Xie et al combines two-tier stochastic frontier model with the traditional Richardson Model and applies it to the Chinese stock market analysis. Finally, the work of Yuan et al derives the Hamilton-Jacobi-Bellman equation for an optimal consumption problem of maximizing the expected discounted value of utility in an economic growth model with random technological shocks. We thank numerous anonymous reviewers for helping in evaluating all submitted manuscripts.

Model‐free offline change‐point detection in multidimensional time series of arbitrary nature via ϵ‐complexity: Simulations and applications

A novel method for offline detection of multiple change points in multidimensional time series is proposed. It is based on the notion of ε‐complexity of continuous vector functions. The proposed methodology does not use any prior information on data‐generating mechanisms; therefore, it can be applied to multidimensional time series of arbitrary nature. Its performance is demonstrated in simulations and an application to high‐frequency financial data.

On p-convergent Operators on Banach Lattices

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

Binary classification of multichannel-EEG records based on the ϵ-complexity of continuous vector functions

Background and objectiveA crucial step in a classification of electroencephalogram (EEG) records is the feature selection. The feature selection problem is difficult because of the complex structure of EEG signals. To classify the EEG signals with good accuracy, most of the recently published studies have used high-dimensional feature spaces. Our objective is to create a low-dimensional feature space that enables binary classification of EEG records.Methods:The proposed approach is based on our theory of the ϵ-complexity of continuous functions, which is extended here (see Appendix) to the case of vector functions. This extension permits us to handle multichannel-EEG records. The method consists of two steps. Firstly, we estimate the ϵ-complexity coefficients of the original signal and its finite differences. Secondly, we utilize the random forest (RF) or support vector machine (SVM) classifier.Results:We demonstrated the performance of our method on simulated data. We also applied it to the problem of classification of multichannel-EEG records related to a group of healthy adolescents (39 subjects) and a group of adolescents with schizophrenia (45 subjects). We found that the random forest classifier provides a superior result. In particular, out-of-bag accuracy in the case of RF was 85.3%. Using 10-fold cross-validation (CV), RF gave an average accuracy of 84.5% on a test set, whereas SVM gave an accuracy of 81.07%. We note that the highest accuracy on CV was 89.3%. To compare our method with the classical approach, we performed classification using the spectral features. In this case, the best performance was achieved using seven-dimensional feature space, with an average accuracy of 83.6%.Conclusions: We developed a model-free method for binary classification of EEG records. The feature space was reduced to four dimensions. The results obtained indicate the effectiveness of the proposed method.

Selected statistical methods of data analysis for multivariate functional data

Data in the form of a continuous vector function on a given interval are referred to as multivariate functional data. These data are treated as realizations of multivariate random processes. The paper is devoted to three statistical dimension reduction techniques for multivariate data. For the first one, principal components analysis, the authors present a review of a recent paper (Jacques and Preda in, Comput Stat Data Anal, 71:92–106, 2014). For two others one, canonical variables and discriminant coordinates, the authors extend existing works for univariate functional data to multivariate. These methods for multivariate functional data are presented, illustrated and discussed in the context of analyzing real data sets. Each of these techniques is applied on real data set.

A constructive approach to boundary value problems with state-dependent impulses

We investigate the non-linear system of ordinary differential equationsu′(t)=f(t,u(t)),a.e.t∈[a,b],subject to the state-dependent impulse conditionu(t+)−u(t−)=γ(u(t−))fort∈(a,b)suchthatg(t,u(t−))=0and the linear two-point boundary conditionAu(a)+Cu(b)=d.Here, −∞<a<b<∞,f and γ are given continuous vector-functions, g is a continuous scalar function, A, C are constant matrices, and d is a constant vector. The instants of time t where the jump occurs are determined by the equation g(t,u(t−))=0 and, thus, are unknown a priori and essentially depend on the solution u. We discuss a reduction technique allowing one to combine the analysis of existence of solutions with an efficient construction of approximate solutions. At present, according to the authors’ knowledge, no numerical results for boundary value problems with state-dependent impulses are available in the literature.

Classification problems based on regression models for multi-dimensional functional data

Data in the form of a continuous vector function on a given interval are referred to as multivariate functional data. These data are treated as realizations of multivariate random processes. We use multivariate functional regression techniques for the classification of multivariate functional data. The approaches discussed are illustrated with an application to two real data sets.

Classification Problems Based on Regression Models for Multi-Dimensional Functional Data

Abstract Data in the form of a continuous vector function on a given interval are referred to as multivariate functional data. These data are treated as realizations of multivariate random processes. We use multivariate functional regression techniques for the classification of multivariate functional data. The approaches discussed are illustrated with an application to two real data sets.

New Approach to Fractal Approximation of Vector-Functions

This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.

Dimension of graphoids of rational vector-functions

Let F⊂R(x,y) be a countable family of rational functions of two variables with real coefficients. Each rational function f∈F can be thought as a continuous function f:dom(f)→R¯ taking values in the projective line R¯=R∪{∞} and defined on a cofinite subset dom(f) of the torus R¯2. Then the family F determines a continuous vector-function F:dom(F)→R¯F defined on the dense Gδ-set dom(F)=⋂f∈Fdom(F) of R¯2. The closure Γ¯(F) of its graph Γ(F)={(x,f(x)):x∈dom(F)} in R¯2×R¯F is called the graphoid of the family F. We prove that the graphoid Γ¯(F) has topological dimension dim(Γ¯(F))=2. If the family F contains all linear fractional transformations f(x,y)=x−ay−b for (a,b)∈Q2, then the graphoid Γ¯(F) has cohomological dimension dimG(Γ¯(F))=1 for any non-trivial 2-divisible abelian group G. Hence the space Γ¯(F) is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.