Kernel density estimation

M. C. Jones, J.S. Marron and S.J. Sheather, Progress in datarbased bandwidth selection for kernel density estimation, Comput. Statist. 11 (1996), 337-381. 

H. van der Voet and D.A. Doornbos, The improvement of SIMCA classification by using kernel density estimation. Part 1. Anal. Chim. Acta, 161 (1984), 115-123 Part 2. Anal. Chim. Acta, 161 (1984) 125-134. 

In the absence of an assumed underlying normal distribution, simple bivariate plotting does not lead to an estimate of the true extent of the parent isotope field. This is particularly a problem if only relatively few samples are available, as is usually the case. Kernel density estimation (KDE Baxter et al., 1997) offers the prospect of building up an estimate of the true shape and size of an isotope field whilst making few extra assumptions about the data. Scaife et al. (1999) showed that lead isotope data can be fully described using KDE without resort to confidence ellipses which assume normality, and which are much less susceptible to the influence of outliers. The results of this approach are discussed in Section 9.6, after the conventional approach to interpreting lead isotope data in the eastern Mediterranean has been discussed. 

Figure 9.12 Kernel density estimate of the lead isotope data for part of the Troodos ore field, Cyprus (data from Gale et al., 1997). The superimposed Oxford ellipse has been used to represent the Cyprus ore field in several publications. (From Scaife et al., 1999 Figure 6, with permission from the first author.)...

Baxter, M.J., Beardah, C.C. and Wright, R.V.S. (1997). Some archaeological applications of kernel density estimates. Journal of Archaeological Science 24 347-354. 

Kernel density estimate of the lead isotope data for part... 

The method CLASSY attempts to bring together the appealing ideas of SIMCA and the Kernel density estimation of ALLOC (CLassification by ALLOC and SIMCA SYnergy) It has been applied to the classification of French wines (Bourgogne and Bordeaux) by classical chemica( and physical variables and by peak height of head-space chromatography. 

Particle size distribution and classifier selectivity have been determined, using kernel density estimations, to data from (two) classifier flow streams. The procedure has been applied to hydrocyclones using platey particles whose sizes were determined with a Sedigraph 5100 and spheroidal particles whose size distributions were determine using the Malvern Mastersizer and the Coulter Counter [8] Svarovsky s equation was used [9,10]. 

The smoothed bootstrap has been proposed to deal with the discreteness of the empirical distribution function (F) when there are small sample sizes (A < 15). For this approach one must smooth the empirical distribution function and then bootstrap samples are drawn from the smoothed empirical distribution function, for example, from a kernel density estimate. However, it is evident that the proper selection of the smoothing parameter (h) is important so that oversmoothing or undersmoothing does not occur. It is difficult to know the most appropriate value for h and once the value for h is assigned it influences the variability and thus makes characterizing the variability terms of the model impossible. There are few studies where the smoothed bootstrap has been applied (21,27,28). In one such study the improvement in the correlation coefficient when compared to the standard non-parametric bootstrap was modest (21). Therefore, the value and behavior of the smoothed bootstrap are not clear. 

If one does not wish to bias the boundaries of the NO region of a system, kernel density estimation (KDE) can be used to find the contours underneath the joint probability density of the PC pair, starting from the one that captures most of the information. Below, a brief review of KDE is presented first that will be used as part of the robust monitoring technique discussed in Section 7.7. Then, the use of kernel-based methods for formulating nonlinear Fisher s discriminant analysis (FDA) is discussed. 

M Rudemo. Empirical choice of histograms and kernel density estimators. Scand. J. Statistics, 9 65-78, 1982. 

The standard way of creating probability distributions from histograms is only optimal for dense data. For this reason, an alternative method is employed that is based on kernel density estimation [54]. These probability distributions estimated from this approach are subsequently used in the formula for the calculation of the mutual information, I(x,y) [53]. 

F -) in place of the true c.d.f. Ff). We will now consider the population to be the observed data having c.d.f. which places mass 1 jn on each of the observed data values X,. Thus, we select M random samples of size n (sampling with replacement) from this new population and compute 0i, i,.. .,6m- We now have M realizations of d from which we can estimate the p.d.f. (using a kernel density estimator), the quantile function, or specific parameters like its mean. 

A plot of the quantile function, kernel density estimator of the p.d.f., a box plot, and a normal reference distribution plot for the sampling distribution of the sample quantile are given in Figures 2.13 and 2.14 for 200 and 20,000 bootstrap samples. We note that there are considerable differences in the plots. The plots for 20,000 bootstrap samples reveal the discreteness of the possible values for the median when the sample size (n = 11 in our case) is very small. Also, we note that n = 11 is too small for the sampling distribution for the median to achieve its asymptotic result (n large), an approximate normal distribution. 

Gray A, Moore A. Very fast multivariate kernel density estimation using via computational geometry. In Proceedings of Joint Statistics Meeting 2003. Alexandria The American Statistical Association 2003. 

To define the NO region (NOR) of the plant, kernel density estimation (KDE) is used. The joint probability density of the first and second, and... 

Figure 2. Spatial distribution of El attacks in the period 1980-2011 (n = 8549). The upper panel (a) shows attacks by decade, and the lower panel (b) displays hotspots areas based on point Kernel density estimation.

Kernel density estimator for a given set of d-dimensional points is ... 

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