Scott, D. W. (1992). Multivariate Density Estimation. New York: John Wiley.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be estimated from the data. To satisfy the low computational cost imposed by real time processing discrete densities, i.e. m bin histograms should be used. Thus, we have target model : qq target candidate : ppy pp u 1: The histogram is not the best nonparametric density estimate [68], but it suffices for our purposes. Other discrete density estimates can be also employed. We will denote by ##y## ppy; qq#1 a similarity function between pp and qq. The function ##y plays the role of a likelihood and its local maxima in the image indicate the presence of objects in ....

D.W. Scott, Multivariate Density Estimation. Wiley, 1992.

....approach is obtained by computing from the data an estimate of and setting f = f( Such an approach is statistically and computationally very ecient but can lead to poor results if none of the family members f( is close to f . In nonparametric density estimation (e.g. [2]) no parametric assumptions about f are made and one assumes instead that f , for example, has some smoothness properties (e.g. two continuous derivatives) or that it is square integrable. The shape of the density estimate is determined by the data and, in principle, given enough data, arbitrary ....

D. Scott, Multivariate Density Estimation, John Wiley & Sons, 1992.

....fuzzy controllers [6] whose e#ectiveness rely on the same approximation principles [44] Closely related to the fuzzy approach some research [11,79,80] proposed to use the RBFN for mapping and refining propositional knowledge. With a very di#erent approach in the mainstream of applied statistics [75], the problem of regression and classification, and more generally density estimation, were faced by means of kernel estimators that were strongly related to RBFN. Finally, RBFN can also be placed in the framework of instance based learning [57] As a consequence RBFNs can be viewed from several ....

....could be inserted and refined in the network. Activated rules can be prompted as an explanation. 24 10 A Statistical Approach to RBFNs The architecture of the RBFNs presents a strong similarity with regression techniques, based on non parametric estimation of an unknown density function [75] and with the Probabilistic Neural Networks [76,77] 10.1 Kernel Regression Estimators This method is known as kernel regression. The basic idea is that an unknown random function f(x) y can be constructed by estimating the joint probability density function g(x, y) f(x) E(Y X = x) ....

D. W. Scott. Multivariate Density Estimation. Wiley, 1992.

.... is of importance, such as in exploratory scientific data analysis, nonparametric methods are used because they make minimal or no assumptions about the distribution of the data, while achieving high accuracy making serious density estimation almost synonymous with nonparametric methods [14, 12]. For example, kernel density estimation (KDE) the most widely used and studied nonparametric density estimation method and thus our focus here, can be shown to converge to the true underlying density with probability 1 as more data are observed, with no distribution assumptions at all, requiring ....

.... x q is p(x q ) N T t=1 VDh kx Gamma x t k (1) where D is the dimensionality of the data and VDh is the volume encompassed by K( Delta) For good estimates, the exact form of K( Delta) turns out to be relatively unimportant, while the correct choice of h is critical [14, 12]. Common choices are the spherical, Gaussian, or Epanechnikov kernels; our method works efficiently with any such standard kernel. The spherical kernel (K(x q ; x t ) 1 if kx q Gamma x t k h, otherwise 0) is simplest. The Epanechnikov kernel (K(x q ; x t ) D 2 2V 0 Dh (1 Gamma kx q ....

D. W. Scott. Multivariate Density Estimation. Wiley, 1992.

....cost imposed by real time processing discrete densities, i.e. bin histograms should be used. Thus we have target model: D E E F E B E G8 target candidate: H #I J 0KE #L E F E B 0KE M8ON The histogram is not the best nonparametric density estimate [68], but it suffices for our purposes. Other discrete density estimates can be also employed. We will denote by #OQ PSR H #L B (1) a similarity function between . The function # plays the role of a likelihood and its local maxima in the image indicate the presence of objects in ....

D. W. Scott, Multivariate Density Estimation. Wiley, 1992.

....perfect agreement between groups and clusters. This is reassuring after all, the data were generated from a Gaussian mixture, and we would hope that model based clustering would do well. The data in this example were obtained by estimating each group density by a kernel density estimate [17] and then sampling from this estimate, again generating 20 times the number of observations in the group. We used a Gaussian kernel with the same covariance matrix as the corresponding group. As in Example 1 this resulted in a dataset of 22,000 observations, but the data are no longer sampled from ....

D. Scott. Multivariate Density Estimation. Wiley, 1992.

....form in Table 3.1. Each of the aforementioned window functions has been well studied and applied to numerous applications. The quality of a density estimate is now widely recog nized to be primarily dependent on the choice of the expansion factor h as opposed to the kernel window function[36]. For this reason, we will limit the investigation to 46 a Gaussian kernel and focus more on the determination of an appropriate expansion factor. Efforts have been made to determine means of switching between kernels without having to reconsider the problem of calibration. Scott [34] provides ....

D. Scott, Multivariate Density Estimation, John Wiley, New York, 1992.

....a decision about how much influence a point does have, and involves an unavoidable trade o# between bias and variance in our estimate. If we make h too large (oversmoothing) we might fail to identify structure in our data. Methods of choosing h fall into two approximate categories (see Scott [Sco92]) 1) Theoretical error bounds based on some assumptions about the density f , such as assuming a bound on the derivatives or using the normal density as a reference. This gives an optimal h that minimises the error bound. Empirical studies suggest that histograms and kernel estimators are ....

....everything we might try. They are all symptoms of a deeper problem called the curse of dimensionality . This phrase is used to describe a range of e#ects that arise from the basic nature of R itself. A full discussion is beyond our scope, and the reader is referred to the discussion in Scott [Sco92, chapters 1, 7], and to a paper by Aggarwal et al. appropriately titled On the surprising behaviour of distance metrics in high dimensional space [AHK01] The latter is especially interesting because it reports empirical results about this e#ect in some data mining data sets. What follows is a summary of some ....

[Article contains additional citation context not shown here]

David W. Scott. Multivariate Density Estimation. John Wiley & Sons, 1992.

....be used to embed every configuration as a single point in configuration space such that the distances between the points match the dissimilarities previously calculated. The interactions in a complex system typically confine it to a tiny fraction of the full high dimensional configuration space [54]. In the case of a molecular system, the nominal dimensionality is three times the number of atoms minus some degrees of freedom for rotation, translation and possible constraints. The e#ective dimensionality, however, is much lower due to the specific physics and chemistry of a molecule; for ....

....which our low dimensional geometric intuition and spatial perception are unaccustomed. Noteworthy in the context of density estimation (step 6) is the vast volume of high dimensional spaces, quickly making a cloud of points highly sparse (the following examples are taken from [57] chapter 4. 5 and [54], chapter 7) the number of sample points required to achieve a constant bias and variance rises dramatically (at least exponentially) with the dimensionality. Furthermore, most of the probability mass quickly becomes concentrated in the tails, even of distributions with very light tails. As an ....

[Article contains additional citation context not shown here]

D. W. Scott. Multivariate Density Estimation. Wiley, New York, 1992.

....to each other but far from all others, their local density is twice that of a single isolated point and their contribution to the above equation is that of a single point at the same position, which is what was desired. In practice, f(x) can be obtained from a kernel density estimate (e.g. [26, 27, 28]) that is, a sum over kernels centered at each point. Likely candidates for these kernels are the covariance function or its square, though we currently ignore which is more appropriate. The criterion corresponding to the above iterative algorithm is (x)# 1 #(x) 8) This algorithm is a ....

....further information, we currently know of no compelling objective reason to prefer one criterion over the other. However, we would like to give our subjective view on the matter. Often six and more descriptors are used to span the chemical space, meaning that the curse of dimensionality (e.g. [27, 26]) makes itself felt. One of its symptoms is that the Voronoi cells corresponding to the design points become extremely spiky . That is, the vertices of the Voronoi polyhedra (called the deep holes in lattice theory) become very remote from the design points while the volume or probability mass ....

[Article contains additional citation context not shown here]

Scott, D. W. Multivariate Density Estimation. Wiley, New York, 1992.

No context found.

Scott, D. W. (1992). Multivariate Density Estimation. New York: John Wiley.

No context found.

D. W. Scott. Multivariate Density Estimation. Wiley, 1992.

No context found.

Scott, D. (1992). Multivariate density estimation. Wiley-NY.

No context found.

Scott, D. (1992). Multivariate density estimation. New York: Wiley.

No context found.

Scott,D.W., Multivariate Density Estimation, New York:Wiley (1992)

No context found.

Scott, D. (1992). Multivariate density estimation. Wiley-NY.

No context found.

D. W. Scott. Multivariate Density Estimation. Wiley, 1992.

No context found.

D.W. Scott. Multivariate Density Estimation. Wiley, 1992.

No context found.

D.W. Scott, Multivariate Density Estimation, Wiley, 1992.

No context found.

D. W. Scott, Multivariate Density Estimation. John Wiley & Sons, New York, 1992.

No context found.

Scott, D.W. (1992). Multivariate density estimation. Theory, practice and visualization. John Wiley & Sons, INC.

No context found.

Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York.

No context found.

D. Scott. Multivariate Density Estimation. John Wiley & Sons, 1992.

No context found.

Scott D. W. Multivariate Density Estimation. Wiley, New York, 1992.

No context found.

D. W. Scott, Multivariate Density Estimation. Wiley, 1992.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC