Variable Kernel Estimates of Multivariate Densities

By Leo Breiman, W. Meisel, Edward Purcell. 1977

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Citations in the corpus

One-Class Support Measure Machines for Group Anomaly Detection (2013, background): “This type of KDE is known in the statistics as variable kernel density estimators (VKDEs) (Abramson 1982, Breiman et al. 1977, Terrell and Scott 1992).”
Robust kernel density estimation (2008, methodology): “There is another implementation of the VKDE where σi is based on the distance to its k-th nearest neighbor (Breiman et al., 1977).”
Vicinal Risk Minimization (2000, background): “It is known in density estimation theory that the quality of t he density estimate can be improved using variable kernel widths ( Breiman, Meisel and Purcell, 1977 ).”
MolProbity’s Ultimate Rotamer-Library Distributions for Model Validation (2016, methodology): “To achieve this we calculated smooth distributions in the multi-dimensional v space for each residue type, using an adaptive local-density-dependent kernel density estimation (KDE).”
Uniform in bandwidth consistency of kernel-type function estimators (2005, unknown): “We cite, for instance, [[ 2 , 17, 21, 22, 23, 27]].”
Motion-based background subtraction using adaptive kernel density estimation (2004, methodology): “This type of estimator was introduced by [4] who suggest using H(x i ) = h(x i )I where h(x i ) is the distance from x i to the k-th nearest data point.”
Weighted samples, kernel density estimators and convergence (2003, background): “While the implementation of adaptive kernel estimators involves a great deal of practical questions it is su‰cient for our purposes to note (i) that the final estimate seems insensitive to the fine detail of the pilot estimate ( Breiman, Meisel and Purcell (1977) , Abramson (1982)), so h can be main”
Variable bandwidth and one-step local M-estimator (2000, background): “Thequantity hn= (Xj) is called variable bandwidth by Breiman et al. (1977).”
Wavelet Shrinkage: Asymptopia? (1995, methodology): “Interesting spatially adaptive methods include CART (Breiman, Friedman, Olshen, and Stone, 1982), TURBO (Friedman and Silverman, 1989), MARS (Friedman, 1991), and Variable Bandwidth Kernel methods (Breiman et al., 1977; M uller and Stadtmuller, 1987; Terrell and Scott, 1990; Brockmann et al. 1991).”
Maximum likelihood estimation of a multi‐dimensional log‐concave density (2008, background): “Extensions that adapt to local smoothness began with Breiman et al. (1977) and Abramson (1982).”
Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow (2007, background): “It is well known, however, that better density estimation is possible using anisotropic kernels, as it is shown, e.g., in [58].”
A tutorial on kernel density estimation and recent advances (2017, methodology): “(Adaptive smoothing) The amount of smoothing can depend on the location x (Loftsgaarden and Quesenberry, 1965) or the data point Xi (Breiman et al., 1977). In the former case, we use h = h(x) so KDE becomes p̂n(x) = 1 nhd(x) ∑n i=1K ( x−Xi h(x) ) , which is referred to as the balloon estimator (Terr”
Multivariate locally adaptive density estimation (2002, methodology): “This type of estimator was introduced by Breiman et al. (1977), who suggested using H(xi)= hk(xi)I. Asymptotically, this is equivalent to choosing bandwidths proportional to f(xi)−1=d where d is the dimension of the data.”
Outlier Detection with Kernel Density Functions (2007, methodology): “The usage of the kth nearest neighbor in kernel density estimation was first proposed in [38] (see also [37]).”