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This chapter examines the related objectives of defining spatial clusters and delineating spatial boundaries in discontinuous data. The former often proceeds by grouping together adjacent locations when they have the most similar characteristics; the latter proceeds by estimating boundaries between locations that are most different. For this, there are several methods available that suggest ’boundary elements’ as possible components of a final division or complete boundary, depending on the kind of data (e.g. binary versus qualitative versus continuous quantitative) and the arrangement of the measured locations (e.g. regular lattice versus irregular spatial network). Once boundaries have been established, statistics are available to evaluate them, including boundary overlap measures. Clusters and boundaries represent two aspects of the same phenomenon, with the same challenge of formalizing similarity and difference in continuous spatial data.
The theory of kernels offers a rich mathematical framework for the archetypical tasks of classification and regression. Its core insight consists of the representer theorem that asserts that an unknown target function underlying a dataset can be represented by a finite sum of evaluations of a singular function, the so-called kernel function. Together with the infamous kernel trick that provides a practical way of incorporating such a kernel function into a machine learning method, a plethora of algorithms can be made more versatile. This chapter first introduces the mathematical foundations required for understanding the distinguished role of the kernel function and its consequence in terms of the representer theorem. Afterwards, we show how selected popular algorithms, including Gaussian processes, can be promoted to their kernel variant. In addition, several ideas on how to construct suitable kernel functions are provided, before demonstrating the power of kernel methods in the context of quantum (chemistry) problems.
The development of more sophisticated and, especially, approximate sampling algorithms aimed at improving scalability in one or more of the senses already discussed in this book raises important considerations about how a suitable algorithm should be selected for a given task, how its tuning parameters should be determined, and how its convergence should be as- sessed. This chapter presents recent solutions to the above problems, whose starting point is to derive explicit upper bounds on an appropriate distance between the posterior and the approximation produced by MCMC. Further, we explain how these same tools can be adapted to provide powerful post-processing methods that can be used retrospectively to improve approximations produced using scalable MCMC.
This chapter gives a more comprehensive treatment of nonparametric methods for estimating density functions and dynamic regression models. We also consider the emerging material on the case where there are many explanatory variables and how selection methods can be used to apply estimation and inference techniques to this case.
Maize is one of the three staple foods in the world. The white variety represents 60% of the maize importation with a world consumption of 1125 million tons in 2019/2020. Currently, new technologies could contribute to the analysis of this seed, supporting quality control and improvement. This study aims to carry out the morphological and proteomic comparison between the hybrid MR2008 and its parental lines LUG03 and CML491 through mass spectrometry and bioinformatics analysis. Herein, we identified that 34.8% of the hybrid proteome differs from the parental proteome. Also, ontological and morphological analyses determined that the hybrid exhibits more characteristics related to CML491 than LUG03, for example, metabolic pathways and enzymes, such as anthocyanidin 3-O-glucosyltransferase (UniProt P16166). This analysis allowed the identification of dominant characters, metabolic pathways and confirms the utility of this methodology in agricultural practices, mainly in processes of selection and quality control of a crop.
The appendix contains various Pohozaev identities, some preliminary properties of Sobolev spaces, certain fundamental estimates on elliptic equations, the Kelvin transformation, the kernel of some linear operators and the estimate for the Green’s function.
We consider linear maps between normed spaces. We define what it means for a linear map to be bounded and show that this is equivalent to continuity. We define the norm of a linear operator and show that the space of all linear maps from X to Y is a vector space, which is complete if Y is complete. We give a number of examples and then discuss inverses and invertibility in some detail.
This chapter extends the presentation ofGaussian measures, cylinder measures, and fields fromSobolev spaces to Banach spaceswith quadratic energynorm. The relationship between weak distributions and cylinder measures is elucidated as is the relationship between Gaussian cylinder measures and Gaussian fields.
Hierarchical optimalrecovery games are defined using a hierarchy of measurement functions. The sequence of optimal mixed minmax solutions is shown to be a martingale. Sparse rank-revealing representations of Gaussian fields are established.
There is a proliferation of methods of point estimation other than ML. First, MLEs may not have an explicit formula and may be computationally more demanding than alternatives. Second, MLEs typically require the specification of a distribution. Third, optimization of criteria other than the likelihood may have some justification. The first argument has become less relevant with the advent of fast computers, and the alternative estimators based on it usually entail a loss of optimality properties. The second can be countered to some extent with large-sample invariance arguments or with the nonparametric MLE and empirical likelihood seen earlier. However, the third reason can be more fundamental.This chapter presents a selection of four common methods of point estimation, addressing the reasons outlined earlier, to varying degrees: method of moments, least squares, nonparametric (density and regression), and Bayesian estimation methods. In addition to these reasons for alternative estimators, point estimation itself may not be the most informative way to summarize what the data indicate about the parameters. Therefore, the chapter also introduces interval estimation and its multivariate generalization, a topic that leads quite naturally to the subject matter of Chapter 14.
The notion of coaxers is introduced in a pseudo-complemented distributive lattice. Boolean algebras are characterized in terms of coaxer ideals and congruences. The concept of coaxer lattices is introduced in pseudo-complemented distributive lattices and characterized in terms of coaxer ideals and maximal ideals. Finally, the coaxer lattices are also characterized in topological terms.
This paper presents a practical and simple fully nonparametric multivariate smoothingprocedure that adapts to the underlying smoothness of the true regression function. Ourestimator is easily computed by successive application of existing base smoothers (withoutthe need of selecting an optimal smoothing parameter), such as thin-plate spline or kernelsmoothers. The resulting smoother has better out of sample predictive capabilities thanthe underlying base smoother, or competing structurally constrained models (MARS, GAM) forsmall dimension (3 ≤ d ≤7) and moderate sample size n ≤ 1000. Moreover our estimator is still usefulwhen d > 10and to our knowledge, no other adaptive fully nonparametric regression estimator isavailable without constrained assumption such as additivity for example. On a realexample, the Boston Housing Data, our method reduces the out of sample prediction error by20%. An R package ibr, available at CRAN, implements the proposedmultivariate nonparametric method in R.
In this paper we investigate some subclasses of strongly regular congruences on an $E$-inversive semigroup $S$. We describe the minimum and the maximum strongly orthodox congruences on $S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on $S$ with characteristic trace $\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an $E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence.
A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.
Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.
We obtain a generalized discrete Hilbert and Hardy-Hilbert inequality with non-conjugate parameters by means of an Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our findings with existing results. We improve some earlier results and apply the results to some special homogeneous functions.
Internal cracks caused by high temperature or excessive moisture during maize (Zea mays L.) kernel development were characterized, and their effects on kernel quality were assessed. Pre-harvest stress cracks are often located near the middle of the kernel along the embryo axis, but they were also detected in other positions, irrespective of the shape of the kernel. X-ray analysis enabled visualisation of stress cracks that are invisible to the human eye and, therefore, gave a better estimate of the percentage of cracks. However, low temperature scanning electron microscopy of the surface of milled kernels revealed small cracks not noticed by visual or X-ray inspection. All kernels tested in this way had a crack of some sort in the endosperm tissue. Cracks were also frequent in the scutellum, but rare in the embryo axis. Endosperm cracks followed the boundary of the starch granules, but did not extend into the pericarp tissue. In contrast to external cracks caused by mechanical impact, pre-harvest internal stress cracks generally are not detrimental to germination and vigour. However, if the crack is located inside or perpendicular to the embryo axis, it may affect the quality of the kernel, probably by impeding nutrient translocation to the embryo.
We consider a diffusion process Xt smoothed with (small)sampling parameter ε. As in Berzin, León and Ortega(2001), we consider a kernel estimate$\widehat{\alpha}_{\varepsilon}$ with window h(ε) of afunction α of its variance. In order to exhibit globaltests of hypothesis, we derive here central limit theorems forthe Lp deviations such as\[ \frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p\right).\]
This study examines the types of interactional
trouble that arise from narrative variation in institutional
interviews. Specifically, we examine protective order interviews
in which Latina women tell of domestic violence to paralegal
interviewers charged with the duty of helping them obtain a
protective order. Victims' narratives are shown to take
different shapes, and paralegals respond to them in different
pragmalinguistic ways, depending on how they diverge from
institutional needs. The factors found most heavily to influence
narrative outcomes are contextual ones, related to participant
social roles, the type of communicative activity interlocutors
perceive themselves to be engaged in, and their interactional goals.
An additional finding is that when expectations of what constitutes
appropriate speech behavior differ, the interlocutor holding greater
institutional power will try to constrain the speech of the other,
despite the fact that both appear to share an extralinguistic goal,
in this case obtaining a protective order.