Bimodal Kernel Research Articles

Nonparametric derivative estimation with bimodal kernels under correlated errors

Nonparametric derivative estimation with bimodal kernels under correlated errors

Semiparametric method and theory for continuously indexed spatio-temporal processes

Spatio-temporal processes with a continuous index in space and time are useful for modeling spatio-temporal data in many scientific disciplines such as environmental and health sciences. However, approaches that enable simultaneous estimation of the mean and covariance functions for such spatio-temporal processes are limited. Here, we propose a flexible spatio-temporal model with partially linear regression in the mean function and local stationarity in the covariance function. We develop a profile likelihood method for estimation and an effective bandwidth selection in the presence of spatio-temporally correlated errors. Specifically, we employ a family of bimodal kernels to alleviate bias, which may be of independent interest for semiparametric spatial statistics. The theoretical properties of our profile likelihood estimation, including consistency and asymptotic normality, are established. A simulation study is conducted and suggests sound empirical properties, while a health hazard data example further illustrates the methodology.

Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection

In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction–diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that “standard” kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing–Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations.

Using bimodal kernel for nonparametric regression specification test

For a nonparametric regression model with a fixed design, we consider the model specification test based on a kernel. Interestingly, we find that a bimodal kernel is useful for the model specification test with a correlated error, whereas a conventional unimodal kernel is useful only for an iid error. Another interesting finding is that the model specification test suffers from a convergence rate change whether the errors are correlated or not. These results are verified by deriving an asymptotic null distribution and asymptotic (local) power and by performing a simulation. Based on our findings, we recommend the use of a bimodal kernel for a nonparametric regression model specification test. The validity of the bimodal kernel for testing is demonstrated with the “drum roller” data (see Laslett(1994) and Altman (1994)).

Visible-infrared fusion schemes for road obstacle classification

In this article we propose different fusion schemes using information provided by VISible (VIS) and InfraRed (IR) images for road obstacle SVM (Support Vector Machine)-based classification. Three probabilistic approaches for the fusion of VIS and IR images are presented. The early fusion at the feature level yields a bimodal feature vector integrating both VIS and IR data, used to feed an SVM-based classifier. An intermediate fusion at the kernel level combines two different monomodal kernels in order to obtain a particularly flexible Bimodal Kernel (BK), we believe more appropriate for heterogeneous VIS and IR data classification with SVM. The late fusion combines matching scores provided by VIS and IR obstacle recognition modules in order to improve the system performance. An important advantage of these fusion schemes is their capability to adapt to the environmental illumination changes and specific weather conditions due to a modality weighting parameter which allows to adjust the decision of the system according to the relative importance of the VIS and IR modalities. Experiments performed on the TetraVision image database showed that all our fusion-based obstacle classifiers outperform both monomodal VIS and IR obstacle recognizers. The matching score fusion with a dynamic weighting scheme provides the best results compared with both early and intermediate fusion schemes using static modality weights. The BK scheme we propose for VIS–IR fusion would need a greater and better balanced database for learning improvement, since the BK has much more hyper-parameters to be simultaneously optimized than the matching-score fusion.

Derivative estimation with local polynomial fitting

We present a fully automated framework to estimate derivatives nonparametrically without estimating the regression function. Derivative estimation plays an important role in the exploration of structures in curves (jump detection and discontinuities), comparison of regression curves, analysis of human growth data, etc. Hence, the study of estimating derivatives is equally important as regression estimation itself. Via empirical derivatives we approximate the qth order derivative and create a new data set which can be smoothed by any nonparametric regression estimator. We derive L1 and L2 rates and establish consistency of the estimator. The new data sets created by this technique are no longer independent and identically distributed (i.i.d.) random variables anymore. As a consequence, automated model selection criteria (data-driven procedures) break down. Therefore, we propose a simple factor method, based on bimodal kernels, to effectively deal with correlated data in the local polynomial regression framework.

Kernel Regression in the Presence of Correlated Errors

It is a well-known problem that obtaining a correct bandwidth and/or smoothing parameter in nonparametric regression is difficult in the presence of correlated errors. There exist a wide variety of methods coping with this problem, but they all critically depend on a tuning procedure which requires accurate information about the correlation structure. We propose a bandwidth selection procedure based on bimodal kernels which successfully removes the correlation without requiring any prior knowledge about its structure and its parameters. Further, we show that the form of the kernel is very important when errors are correlated which is in contrast to the independent and identically distributed (i.i.d.) case. Finally, some extensions are proposed to use the proposed criterion in support vector machines and least squares support vector machines for regression.

Kernel Regression with Correlated Errors

It is a well-known problem that obtaining a correct bandwidth in nonparametric regression is difficult in the presence of correlated errors. There exist a wide variety of methods coping with this problem, but they all critically depend on a tuning procedure which requires accurate information about the correlation structure. Since the errors cannot be observed, the latter is a hard goal to achieve. In this paper, we show the breakdown of several data-driven parameter selection procedures. We also develop a bandwidth selection procedure based on bimodal kernels which successfully removes the error correlation without requiring any prior knowledge about its structure. Some extensions are made to use such a criterion in least squares support vector machines for regression.

Learning Bimodal Structure in Audio–Visual Data

A novel model is presented to learn bimodally informative structures from audio-visual signals. The signal is represented as a sparse sum of audio-visual kernels. Each kernel is a bimodal function consisting of synchronous snippets of an audio waveform and a spatio-temporal visual basis function. To represent an audio-visual signal, the kernels can be positioned independently and arbitrarily in space and time. The proposed algorithm uses unsupervised learning to form dictionaries of bimodal kernels from audio-visual material. The basis functions that emerge during learning capture salient audio-visual data structures. In addition, it is demonstrated that the learned dictionary can be used to locate sources of sound in the movie frame. Specifically, in sequences containing two speakers, the algorithm can robustly localize a speaker even in the presence of severe acoustic and visual distracters.

Using bimodal kernel for inference in nonparametric regression with correlated errors

For nonparametric regression model with fixed design, it is well known that obtaining a correct bandwidth is difficult when errors are correlated. Various methods of bandwidth selection have been proposed, but their successful implementation critically depends on a tuning procedure which requires accurate information about error correlation. Unfortunately, such information is usually hard to obtain since errors are not observable. In this article a new bandwidth selector based on the use of a bimodal kernel is proposed and investigated. It is shown that the new bandwidth selector is quite useful for the tuning procedures of various other methods. Furthermore, the proposed bandwidth selector itself proves to be quite effective when the errors are severely correlated.

Fronts from complex two-dimensional dispersal kernels: Theory and application to Reid’s paradox

Bimodal dispersal probability distributions with characteristic distances differing by several orders of magnitude have been derived and favorably compared to observations by Nathan et al. [Nature (London) 418, 409 (2002)]. For such bimodal kernels, we show that two-dimensional molecular dynamics computer simulations are unable to yield accurate front speeds. Analytically, the usual continuous-space random walks (CSRWs) are applied to two dimensions. We also introduce discrete-space random walks and use them to check the CSRW results (because of the inefficiency of the numerical simulations). The physical results reported are shown to predict front speeds high enough to possibly explain Reid’s paradox of rapid tree migration. We also show that, for a time-ordered evolution equation, fronts are always slower in two dimensions than in one dimension and that this difference is important both for unimodal and for bimodal kernels.

An iterated cochrane-orcutt procedure for nonparametric regression

Altman (1990) and Hart (1991) have shown that kernel regression can be an effective method for estimating an unknown mean function when the errors are correlated. However, the optimal bandwidth for kernel smoothing depends strongly on the correlation function, as do confidence bands for the regression curve. In this paper, the simultaneous estimation of the regression and correlation functions is explored. An iterative technique analogous to the iterated Cochrane-Orcutt method for linear regression (Cochrane and Orcutt, 1949) is shown to perform well. However, for moderate sample sizes, stopping after the first iteration produces better results. An interesting feature of the simultaneous method is that it performs best when different kernels are used to estimate the regression and correlation functions. For the regression function, unimodal kernels are known to be optimal. However, examination of the mean squared error of the correlation estimator suggests that a bimodal kernel will perform better for est...

On the integral equations of the linear theory of recurrent lateral interaction in vision

In a spatially homogeneous visual system with linear, recurrent, lateral interactions that fall off with distance at large distances, the response field is related to the excitation field through a functional equation involving convolution of the response with an “interaction kernel” which characterizes the system. We discuss here the general theory of solutions of such equations in Bochner's F k-spaces, and we present explicit solutions for some special cases in which the interaction kernel is “bimodal,” i.e., achieves its maximum absolute value at a non-zero distance of separation of sensory points. Bimodal kernels are of interest because they appear to be in qualitative accord with physiological data, and, as is known, a linear recurrent system with such a kernel can yield a transfer function whose maximum value is finite and occurs at a finite, non-zero, spatial frequency. Such a recurrent system with a bimodal kernel and a regular transfer function can, however, also show “oscillations,” i.e., multiple turning points, in the response to a simple step-function excitation field. To help study this last phenomenon, we derive resolvent functions for integral equations in a class which generalizes the classical Lalesco-Picard equation.