A Kernel approach for extending nonparametric multivariate analysis of variance in high-dimensional settings

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

ObjectivesJob exposure matrices (JEMs) can be constructed from expert-rated assessments, direct measurement and self-reports. This paper describes the construction of a general population JEM based on self-reported physical exposures, its...

Relationship of medical status, medications, and salivary flow rates in adults of different ages

Forest fires are influenced by several factors, including forest location, species type, age and density, date of fire occurrence, temperatures, and wind speeds, among others. This study investigates the quantitative effects of these factors on the degree of forest fire disaster using nonparametric statistical methods to provide a theoretical basis and data support for forest fire management. Data on forest fire damage from 1969 to 2013 was analyzed. The results indicate that different forest locations and types, fire occurrence dates, temperatures, and wind speeds were statistically significant. The eastern regions of the study area experienced the highest fire occurrence, accounting for 85.0% of the total number of fires as well as the largest average forested area burned. April, May, and October had more frequent fires than other months, accounting for 78.9%, while September had the most extensive forested area burned (63.08 ha) and burnt area (106.34 ha). Hardwood mixed forest and oak forest had more frequent fires, accounting for 31.9% and 26.0%, respectively. Hardwood-conifer mixed forest had the most forested area burned (50.18 ha) and burnt area (65.09 ha). Temperatures, wind speeds, and their interaction had significant impacts on forested area burned and area burnt.

A microcosm experiment was carried out in order to test the effect of the herbicide metolachlor on river periphytic diatoms and to find potential diatom bioindicators of contamination. Effects were investigated at different biological organization levels (biofilm, diatom community, population and individual levels). The colonization of glass substrates by natural biofilm in artificial streams did not vary quantitatively between control and contaminated conditions (5 and 30 μg.L− 1 ). However, non-parametric multivariate analysis of variance revealed a significant difference between contaminated and control diatom communities with regard to species composition. The difference was due to the greater development of probably tolerant species in the presence of the herbicide (e.g. , Planothidium frequentissimum, Planothidium lanceolatum, Amphora montana, Surirella brebissonii and Nitzschia gracilis ). An increase in the occurrence of abnormal forms was observed in relation to metolachlor concentration. In particular, up to 8% of the frustules of the species Surirella angusta exhibited prominent deformities. Monospecific acute toxicity tests were then performed on two species to estimate toxicity parameters based on growth inhibition. These tests also confirmed the teratogenic effect of the herbicide on S. angusta . This study shows that low concentrations of metolachlor in natural streams may significantly alter diatom community structure and that abnormal diatom forms should be taken into account in water contamination assessment.

A new method for non-parametric multivariate analysis of variance

In spite of their ecological and economic importance, reef fishes from the coast of Oaxaca, Mexico are rarely studied, therefore precluding their management and conservation. In order to identify the set of habitat characteristics/environmental conditions that predict major shifts in fish assemblages in space and time, a stationary census (5′, φ = 5 m) was conducted on a semi-monthly basis from 2006 to 2009 at patch reefs along the coast. Habitat configuration was gathered using 25 m long point-intersect transects (data every 25 cm), recording all underlying coral species and substrate characteristics (rocks, sand, algal mats, rubble or dead corals). Recorded were 65 452 fishes grouped in 11 orders, 36 families, 65 genera and 89 species. Labridae (nine species), Pomacentridae (eight species) and Serranidae (seven species) were the most frequent families. Abundance is severely skewed among species; four species Thalassoma lucasanum, Chromis atrilobata, Apogon pacificus and Stegastes acapulcoensis comprise nearly 59% of the fish abundance, 11 species contribute 30%, whereas most of the species (75) can be considered as rare since they contribute <1% each to the total. Species richness and family-level assemblage composition are similar to those recorded elsewhere in the eastern Pacific. Non-parametric multivariate analysis of variance demonstrated that changes of diversity metrics might be associated with environmental differences on the scale of hundreds of meters to kilometers, as well as coupled with major changes on oceanographic variables throughout time, exerting meaningful changes on reef-related fish assemblages.

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A local–global mixed kernel with reproducing property

Various kernel-based methods have been developed with great success in many fields, but very little research has been published that is concerned with a multiple attribute kernel in reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel elastic kernel called a multiple attribute convolution kernel with reproducing property (MACKRP) and present improved classification results over conventional approaches in the RKHS rather than the more commonly used Hilbert space. The MACKRP consists of two major steps. First, we find the basic solution of a generalized differential operator by the delta function, and then we design a convolution function using this solution. This convolution function is proven to be a specific reproducing kernel called a convolution reproducing kernel (CRK) in H3-space. Second, we prove that the CRK satisfies the condition of Mercer kernel. And the CRK is composed of three attributes (L1-norm, L2-norm and Laplace kernel), and each attribute can capture a different feature, with all attributes generating a novel kernel which we call an MACKRP. The experimental results demonstrate that the MACKRP possesses approximation and regularization performance and that classification results are consistently comparable or superior to a number of other state-of-the-art kernel functions.

Covariance matrices have attracted increasing attention for data representation in many computer vision tasks. The nonsingular covariance matrices are regarded as points on Riemannian manifolds rather than Euclidean space. A common technique for classification on Riemannian manifolds is to embed the covariance matrices into a reproducing kernel Hilbert space (RKHS), and then construct a map from RKHS to Euclidean space, while the explicit map from RKHS to Euclidean space in most kernel-based methods only depends on a linear hypothesis. In this paper, we propose a subspace learning framework to project Riemannian manifolds to Euclidean space, and give the theoretical derivation for it. Specifically, the Euclidean space is isomorphic to the subspace of RKHS. Under the framework, firstly we define an improved Log-Euclidean Gaussian radial basis function kernel for embedding. The first order statistical features of input images are incorporated into the kernel function to increase the discriminative power. After that we seek the optimal projection matrix of the subspace of the RKHS by conducting a graph embedding discriminant analysis. Texture recognition and object categorization experiments with region covariance descriptors demonstrate the considerable effectiveness of the improved Log-Euclidean Gaussian RBK kernel and the proposed method.

In the machine learning based on kernel tricks, people often put one variable of a kernel function on the given samples to produce the basic functions of a solution space of learning problem. If the collection of the given samples deviates from the data distribution, the solution space spanned by these basic functions will also deviate from the real solution space of learning problem. In this paper a multikernel-like learning algorithm based on data probability distribution (MKDPD) is proposed, in which the parameters of a kernel function are locally adjusted according to the data probability distribution, and thus produces different kernel functions. These different kernel functions will generate different Reproducing Kernel Hilbert Spaces (RKHS). The direct sum of the subspaces of these RKHS constitutes the solution space of learning problem. Furthermore, based on the proposed MKDPD algorithm, a new algorithm for labeling new coming data is proposed, in which the basic functions are retrained according to the new coming data, while the coefficients of the retrained basic functions remained unchanged to label the new coming data. The experimental results presented in this paper show the effectiveness of the proposed algorithms.

In this note, we prove that an operator between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets. To be more precise, it is shown that an operator T between reproducing kernel Hilbert spaces is a multiplication operator if and only if (Tf)(z)=0 holds for all f and z satisfying f(z)=0. As possible applications, we deduce a general reflexivity result for multiplier algebras, and furthermore prove fully vector-valued generalizations of mulitplier lifting results of Beatrous and Burbea.

Modeling data relation as a hierarchical structure has proven beneficial for many learning scenarios, and the hyperbolic space, with negative curvature, can encode such data hierarchy without distortion. Several recent studies also show that the representation power of the hyperbolic space can be further improved by endowing the kernel methods. Unfortunately, the known kernel methods, developed in hyperbolic space, are limited by the adaptation capacity or distortion issues. This paper addresses the issues through a novel embedding function. To this end, we propose a curvature-aware isometric embedding, which establishes an isometry from the Poincar\'e model to a special reproducing kernel Hilbert space (RKHS). Then we can further define a series of kernels on this RKHS, including several positive definite kernels and an indefinite kernel. Thorough experiments are conducted to demonstrate the superiority of our proposals over existing-known hyperbolic and Euclidean kernels in various learning tasks, e.g., graph learning and zero-shot learning.

We propose a data analysis method that combines the objectives of nonlinear principal component analysis and nonlinear discriminant analysis with the kernel method in a reproducing kernel Hilbert space. This method addresses nonlinear data analysis problems in high-dimensional spaces, specifically the reproducing kernel Hilbert space, through the use of the kernel trick. Our proposed method can be considered as a semi-supervised data analysis approach. We evaluate our proposed method using various kernel functions and datasets, both visually and quantitatively. The evaluation results demonstrate that our proposal outperforms kernel principal component analysis and generalized discriminant analysis in terms of classification performance. This indicates the advantages and originality of our proposed method. Furthermore, we analyze and discuss our findings based on the evaluation results, and highlight potential areas for further research and future work related to our proposal.