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Blonder et al. (2014, Global Ecology and Biogeography, 23, 595–609) introduced a new multivariate kernel density estimation (KDE) method to infer Hutchinsonian hypervolumes in the modelling of ecological niches. The authors argued that their KDE method matches or outperforms several methods for estimating hypervolume geometries and for conducting species distribution modelling. Further clarification, however, is appropriate with respect to the assumptions and limitations of KDE as a method for species distribution modelling. Using virtual species and controlled environmental scenarios, we show that KDE both under- and overestimates niche volumes depending on the dimensionality of the dataset and the number of occurrence records considered. We suggest that KDE may be a viable approach when dealing with large sample sizes, limited sampling bias and only a few environmental dimensions.

The problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient approach uses the fast Fourier transform. Unfortunately, the existing FFT-based solution suffers from a serious limitation, as it can accurately operate only with the constrained (i.e., diagonal) multivariate bandwidth matrices. In this article, we describe the problem and give a satisfactory solution. The proposed solution may be successfully used also in other research problems, for example, for the fast computation of the optimal bandwidth for KDE. Supplementary materials for this article are available online.

In order to contribute to the lack of statistical anomaly detection and sensitivity visualization of the response parameters from international seismic codes, this study presents a comparative analysis between multiple Chilean codes for the seismic design of reinforced concrete buildings. The seismic response and its similar implications to the US codes are studied from a model of a reinforced concrete building in the city of Viña del Mar, Chile. The first study case is the NCh433 Of.96 Mod.2009, the second case is the NCh433 Of.96 Mod.2009 + D.S. 117 and 118, and the third case is the NCh433 Of.96 Mod.2009 + D.S. 60 and 61. The results show that the seismic response does not present great differences, but they show noticeable dependencies between multiple parameters within the three study cases. The third case presents the most oversized values of the response parameters, which require a design against a higher base shear. An analysis of the bio-seismic profile is included, which is a procedure commonly used for buildings in Chile, and its indexes are analyzed with statistical models that provide an easy overview of the data structure. Additionally, a pier wall with an opening, and a pier with a T-shaped wall are selected, which present an increase in the amount of reinforcing steel from the first code case to the third, and then to the second code case.

The traditional optimization scheduling of distribution networks has often only considered the volatility and randomness of wind and solar output. When estimating the prediction errors of wind and solar output, wind turbines and photovoltaics are typically considered separately, overlooking the correlation between them. Accurate modeling of wind and solar output prediction errors is crucial for enhancing the reliability and economy of distribution network scheduling. To address this, this paper proposes a new modeling method. First, based on the volatility and randomness of wind and solar output, it considers the characteristic that wind and solar outputs in the same region at the same time are correlated. A multivariate nonparametric kernel density estimation is introduced to fit the joint prediction error distribution of wind and solar output using historical data. Next, the impact of joint prediction errors on system scheduling costs is considered by introducing a penalty cost in the economic objective function for the errors caused by wind and solar predictions. Additionally, energy storage devices are integrated into the system to smooth power fluctuations, thereby constructing an economically optimized scheduling model for wind–solar–storage distribution networks based on stochastic correlations. Finally, testing is conducted using an improved IEEE-33 node system. The results indicate that the model considering the correlation between wind and solar output significantly improves the fitting accuracy of prediction errors compared to traditional models that only consider randomness. It also enhances the utilization rate of wind and solar energy and improves the economic performance of the distribution network.

Multivariate kernel density estimation is an important technique in exploratory data analysis. Its utility relies on its ease of interpretation, especially by graphical means. The crucial factor which determines the performance of kernel density estimation is the bandwidth matrix selection. Research in finding optimal bandwidth matrices began with restricted parameterizations of the bandwidth matrix which mimic univariate selectors. Progressively these restrictions were relaxed to develop more flexible selectors. In this paper, we propose the first plug-in bandwidth selector with the unconstrained parameterizations of both the final and pilot selectors. Up till now, the development of unconstrained pilot selectors was hindered by the traditional vectorization of higher-order derivatives which lead to increasingly intractable matrix algebraic expressions. We resolve this by introducing an alternative vectorization which gives elegant and tractable expressions. This allows us to quantify the asymptotic and finite sample properties of unconstrained pilot selectors. For target densities with intricate structure (such as multimodality), our unconstrained selectors show the most improvement over the existing plug-in selectors.

In recent years, the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case, while multivariate kernel density estimation has been largely neglected. In part, this may be due to the perception that calibrating multivariate densities is substantially more difficult. In this article, we explicitly derive and compare multivariate versions of the bootstrap method of Taylor, the least-squares cross-validation method developed by Bowman and Rudemo, and a biased cross-validation method similar to that of Scott and Terrell for multivariate kernel estimation using the product kernel estimator. The theoretical behavior of these cross-validation algorithms is shown to improve (surprisingly) as the dimension increases, approaching the best rate of O(n −1/2 ). Simulation studies suggest that the new biased cross-validation method performs quite well and with reasonable variability as compared to the other two methods. Bivariate examples with heart disease and ozone data are given to illustrate the behavior of these algorithms.

The gradient approach allows for an innovative representation of landscape composition and configuration not presupposing spatial discontinuities typical of the conventional methods of analysis. Also the urban-rural dichotomy can be better understood through a continuous landscape gradient whose characterization changes accordingly to natural and anthropic variables taken into account and to the spatio-temporal scale adopted for the study. The research was aimed at the analysis of an urban-rural gradient within a study area located in central Italy, using spatial indicators associated with urbanization, agriculture and natural elements. A multivariate spatial analysis (MSA) of such indicators enabled the identification of urban, agricultural and natural dominated areas, as well as specific landscape transitions where the most relevant relationships between agriculture and other landscape components were detected. Landscapes derived from MSA were studied by a set of key landscape pattern metrics within a framework oriented to the structural characterization of the whole urban-rural gradient. The results showed two distinct sub-gradients: one urban-agricultural and one agricultural-natural, both characterized by different fringe areas. This application highlighted how the proposed methodology can represent a reliable approach supporting modern landscape planning and management.

This chapter contains sections titled: Approximating a Density From a Set of Monte Carlo Samples General Concepts Importance Sampling Summary References

Several disciplines, like the social sciences, epidemiology, sentiment analysis, or market research, are interested in knowing the distribution of the classes in a population rather than the individual labels of the members thereof. Quantification is the supervised machine learning task concerned with obtaining accurate predictors of class prevalence, and to do so particularly in the presence of label shift. The distribution-matching (DM) approaches represent one of the most important families among the quantification methods that have been proposed in the literature so far. Current DM approaches model the involved populations using histograms of posterior probabilities. In this paper, we argue that their application to the multiclass setting is suboptimal since the histograms become class-specific, thus missing the opportunity to model inter-class information that may exist in the data. We propose a new representation mechanism based on multivariate densities that we model via kernel density estimation (KDE). The experiments we have carried out show our method, dubbed KDEy, yields superior quantification performance compared to previous DM approaches and other state-of-the-art quantification systems.

Current tools for multivariate density estimation struggle when the density is concentrated near a non-linear subspace or manifold. Most approaches require the choice of a kernel, with the multivariate Gaussian kernel by far the most commonly used. Although heavy-tailed and skewed extensions have been proposed, such kernels cannot capture curvature in the support of the data. This leads to poor performance unless the sample size is very large relative to the dimension of the data. The paper proposes a novel generalization of the Gaussian distribution, which includes an additional curvature parameter. We refer to the proposed class as Fisher–Gaussian kernels, since they arise by sampling from a von Mises–Fisher density on the sphere and adding Gaussian noise. The Fisher–Gaussian density has an analytic form and is amenable to straightforward implementation within Bayesian mixture models by using Markov chain Monte Carlo sampling. We provide theory on large support and illustrate gains relative to competitors in simulated and real data applications.