arXiv subjects

Aijun Zhang

Publications and source records attributed to Aijun Zhang.

27 records · Page 2Linked to original sources

arXiv2019

Transition Fronts of Fisher-KPP Equations in Locally Spatially Inhomogeneous Patchy Environments I: Existence and Non-existence

This paper is devoted to the study of spatial propagation dynamics of species in locally spatially inhomogeneous patchy environments or media. For a lattice differential equation with monostable nonlinearity in a discrete homogeneous media, it is well-known that there exists a minimal wave speed such that a traveling front exists if and only if the wave speed is not slower than this minimal wave speed. We shall show that strongly localized spatial inhomogeneous patchy environments may prevent the existence of transition fronts (generalized traveling fronts). Transition fronts may exist in weakly localized spatial inhomogeneous patchy environments but only in a finite range of speeds, which implies that it is plausible to obtain a maximal wave speed of existence of transition fronts.

math.DS↗

arXiv2019

Enhancing Explainability of Neural Networks through Architecture Constraints

Prediction accuracy and model explainability are the two most important objectives when developing machine learning algorithms to solve real-world problems. The neural networks are known to possess good prediction performance, but lack of sufficient model interpretability. In this paper, we propose to enhance the explainability of neural networks through the following architecture constraints: a) sparse additive subnetworks; b) projection pursuit with orthogonality constraint; and c) smooth function approximation. It leads to an explainable neural network (xNN) with the superior balance between prediction performance and model interpretability. We derive the necessary and sufficient identifiability conditions for the proposed xNN model. The multiple parameters are simultaneously estimated by a modified mini-batch gradient descent method based on the backpropagation algorithm for calculating the derivatives and the Cayley transform for preserving the projection orthogonality. Through simulation study under six different scenarios, we compare the proposed method to several benchmarks including least absolute shrinkage and selection operator, support vector machine, random forest, extreme learning machine, and multi-layer perceptron. It is shown that the proposed xNN model keeps the flexibility of pursuing high prediction accuracy while attaining improved interpretability. Finally, a real data example is employed as a showcase application.

stat.ML↗

arXiv2019

Persistence and Extinction of Nonlocal Dispersal Evolution Equations in Moving Habitats

This paper is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with moving speed $c$. It is shown that the species becomes extinct if the moving speed $c$ is larger than the so called spreading speed $c^*$, where $c^*$ is determined by the maximum linearized growth rate function. If the moving speed $c$ is smaller than $c^*$, it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some number $L^*$ and in this case, there is a traveling wave solution with speed $c$, and it becomes extinct if the patch size is {smaller} than $L^*$.

math.DS↗

arXiv2018

Mixed-Level Column Augmented Uniform Designs

Follow-up experimental designs are popularly used in industry. In many follow-up designs, some additional factors with two or three levels may be added in the follow-up stage since they are quite important but may be neglected in the first stage. Such follow-up designs are called mixed-level column augmented designs. In this paper, based on the initial designs, mixed-level column augmented uniform designs are proposed by using the uniformity criterion, wrap-around $L_2$-discrepancy (WD). The multi-stage augmented procedure which adds the additional design points stage by stage is also investigated. We present the analytical expressions and the corresponding lower bounds of the WD of the column augmented designs. It is shown that the column augmented uniform designs are also the optimal designs under the non-orthogonality criterion, $E(f_{NOD})$. Furthermore, a construction algorithm for the column augmented uniform design is provided. Some examples show that the lower bounds are tight and the construction algorithm is effective.

math.ST↗

arXiv2018

Interval-valued Data Prediction via Regularized Artificial Neural Network

A regularized artificial neural network (RANN) is proposed for interval-valued data prediction. The ANN model is selected due to its powerful capability in fitting linear and nonlinear functions. To meet mathematical coherence requirement for an interval (i.e., the predicted lower bounds should not cross over their upper bounds), a soft non-crossing regularizer is introduced to the interval-valued ANN model. We conduct extensive experiments based on both simulation datasets and real-life datasets, and compare the proposed RANN method with multiple traditional models, including the linear constrained center and range method (CCRM), the least absolute shrinkage and selection operator-based interval-valued regression method (Lasso-IR), the nonlinear interval kernel regression (IKR), the interval multi-layer perceptron (iMLP) and the multi-output support vector regression (MSVR). Experimental results show that the proposed RANN model is an effective tool for interval-valued prediction tasks with high prediction accuracy.

stat.CO↗

arXiv2014

Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(ξ)$ in the direction of any given unit vector $ξ$. A variational characterization of $c^*(ξ)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $ξ$ with speed $c>c^*(ξ)$.

math.DS↗

arXiv2013

Competing Interactions and Traveling Wave Solutions in Lattice Differential Equations

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj in[1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.

math.DS↗

arXiv2012

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations

This paper deals with front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In the authors' earlier works, it is shown that a general spatially periodic monostable equation with nonlocal dispersal has a unique spatially periodic positive stationary solution and has a spreading speed in every direction. In this paper, we show that a spatially periodic nonlocal monostable equation with certain spatial homogeneity or small nonlocal dispersal distance has a unique stable periodic traveling wave solutions connecting its unique spatially periodic positive stationary solution and the trivial solution in every direction for all speeds greater than the spreading speed in that direction.

math.DS↗

arXiv2006

Majorization framework for balanced lattice designs

This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for assessing designs, which includes a stringent criterion of majorization via pairwise coincidences and flexible surrogates via convex functions. Classical orthogonality, aberration and uniformity criteria are unified by choosing combinatorial and exponential kernels. A construction method is also sketched out.

math.ST↗