Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices

Chacón, José E.; Duong, Tarn

doi:10.1007/s11749-009-0168-4

Cited by 113 publications

(96 citation statements)

References 26 publications

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“…The reasoning follows similar lines to computing Equations (9) and (10) in the proof of Theorem 2 in Chacón and Duong (2008).…”

Section: A13 Proof Of Theoremmentioning

confidence: 91%

“…where the last line follows by using a similar reasoning to determine Equation (11) in the proof of Theorem 2 in Chacón and Duong (2008). The differential of AIV(H) is…”

Section: A13 Proof Of Theoremmentioning

confidence: 95%

“…However for r > 2, it appears that the matrix analysis using a generalization of the vector half operator becomes intractable. Other authors have used the same vectorization we propose to develop results for higher order derivatives: Holmquist (1996a) computes derivatives of the multivariate normal density function and Chacón and Duong (2008) compute kernel estimators of multivariate integrated density derivative functionals. Suppose now that f : R d → R is a density and we want to estimate D ⊗r f (x) for some r ∈ N. To this end, we use the kernel estimator D ⊗r f (x; H) = D ⊗rf (x; H), where given a random sample X 1 , X 2 , .…”

Section: Kernel Density Derivative Estimationmentioning

confidence: 99%

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Asymptotics for general multivariate kernel density derivative estimators

Chacón¹,

Duong²,

Wand³

2011

STAT Sinica

117

103

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We investigate general kernel density derivative estimators, that is, kernel estimators of multivariate density derivative functions using general (or unconstrained) bandwidth matrix selectors. These density derivative estimators have been relatively less well researched than their density estimator analogues. A major obstacle for progress has been the intractability of the matrix analysis when treating higher order multivariate derivatives. With an alternative vectorization of these higher order derivatives, these mathematical intractabilities are surmounted in an elegant and unified framework. The finite sample and asymptotic analysis of squared errors for density estimators are generalized to density derivative estimators. Moreover, we are able to exhibit a closed form expression for a normal scale bandwidth matrix for density derivative estimators. These normal scale bandwidths are employed in a numerical study to demonstrate the gain in performance of unconstrained selectors over their constrained counterparts.

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“…The reasoning follows similar lines to computing Equations (9) and (10) in the proof of Theorem 2 in Chacón and Duong (2008).…”

Section: A13 Proof Of Theoremmentioning

confidence: 91%

“…where the last line follows by using a similar reasoning to determine Equation (11) in the proof of Theorem 2 in Chacón and Duong (2008). The differential of AIV(H) is…”

Section: A13 Proof Of Theoremmentioning

confidence: 95%

Section: Kernel Density Derivative Estimationmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotics for general multivariate kernel density derivative estimators

Chacón¹,

Duong²,

Wand³

2011

STAT Sinica

117

103

View full text Add to dashboard Cite

show abstract

“…The monographs [21,28] provide overview of the research in the area of multivariate KDE. oosing the best value of H is not a trivial task and this probwas and still is extensively studied in literature [3,4,5]. Currently available selectors can be roughly divided into ee classes [12,28].…”

Section: The Plugin Methods and Data Preprocessingmentioning

confidence: 99%

FPGA-based bandwidth selection for kernel density estimation using high level synthesis approach

Gramacki

Sawerwain

Gramacki

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. Field-programmable gate arrays (FPGA) technology can offer significantly higher performance at much lower power consumption than is available from single and multicore CPUs and GPUs (graphics processing unit) in many computational problems. Unfortunately, the pure programming for FPGA using hardware description languages (HDL), like VHDL or Verilog, is a difficult and not-trivial task and is not intuitive for C/C++/Java programmers. To bring the gap between programming effectiveness and difficulty, the high level synthesis (HLS) approach is promoted by main FPGA vendors. Nowadays, time-intensive calculations are mainly performed on GPU/CPU architectures, but can also be successfully performed using HLS approach. In the paper we implement a bandwidth selection algorithm for kernel density estimation (KDE) using HLS and show techniques which were used to optimize the final FPGA implementation. We are also going to show that FPGA speedups, comparing to highly optimized CPU and GPU implementations, are quite substantial. Moreover, power consumption for FPGA devices is usually much less than typical power consumption of the present CPUs and GPUs.Key words: FPGA, high level synthesis, kernel density estimation, bandwidth selection, plug-in selector.FPGA-based bandwidth selection for kernel density estimation using high level synthesis approach In the paper we are concerned with FPGA approach. In [10] the author considers a problem how to use FPGA for fast computing of PDFs using direct very high speed integrated circuits hardware description language (VHDL) programming approach. However, the problem we are concerning is of different nature, as we concentrate our attention for computing the optimal bandwidth for PDF (see Section 2).To develop the final FPGA design we use the high level synthesis (HLS) approach [8,16], in which no direct hardware description language (HDL) coding is needed (typically, VHDL or Verilog languages a are used). The remainder of the paper is organized as follows. In Section 2, we turn our attention to give the reader some preliminary information on KDE and bandwidth selection. In Section 3 we provide detailed mathematical formulas for calculating optimal bandwidth using the PLUGIN method. In Section 4 we cover all the necessary details on our FPGA-based implementation. We also present practical experiments carried out and discuss the results. In Section 5, we conclude the paper. Kernel density estimation and bandwidth selectionThe univariate kernel density estimator fˆ for a random sample X i (i = 1, 2, …, n), drawn from a common and usually unknown density function f is given by a It is worth to note that OpenCL framework, which is commonly used by GPU programmers, also becomes available for FPGA devices. Nowadays, OpenCL is offered by Altera SDK for OpenCL to easily implement OpenCL applications for FPGA. Recently, Xilinx announced a similar solution, namely SDAccel Development Environment for OpenCL, C, and C++.

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“…Proof of Theorem 1: To establish the asymptotic sampling distribution ofT, we follow the approach of Chacón and Duong (29). Suppose that the conditions hold.…”

Section: Methodsmentioning

confidence: 99%

Closed-form density-based framework for automatic detection of cellular morphology changes

Duong

Goud

Schauer

2012

Proc. Natl. Acad. Sci. U.S.A.

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A primary method for studying cellular function is to examine cell morphology after a given manipulation. Fluorescent markers attached to proteins/intracellular structures of interest in conjunction with 3D fluorescent microscopy are frequently exploited for functional analysis. Despite the central role of morphology comparisons in cell biological approaches, few statistical tools are available that allow biological scientists without a high level of statistical training to quantify the similarity or difference of fluorescent images containing multifactorial information. We transform intracellular structures into kernels and develop a multivariate two-sample test that is nonparametric and asymptotically normal to directly and quantitatively compare cellular morphologies. The asymptotic normality bypasses the computationally intensive calculations used by the usual resampling techniques to compute the P-value. Because all parameters required for the statistical test are estimated directly from the data, it does not require any subjective decisions. Thus, we provide a black-box method for unbiased, automated comparison of cell morphology. We validate the performance of our test statistic for finite synthetic samples and experimental data. Employing our test for the comparison of the morphology of intracellular multivesicular bodies, we detect changes in their distribution after disruption of the cellular microtubule cytoskeleton with high statistical significance in fixed samples and live cell analysis. These results demonstrate that density-based comparison of multivariate image information is a powerful tool for automated detection of cell morphology changes. Moreover, the underlying mathematics of our test statistic is a general technique, which can be applied in situations where two data samples are compared.hypothesis test | integrated density functional | optimal bandwidth selection | quantitative cell comparison F luorescent markers attached to proteins of interest in conjunction with modern fluorescent microscopy technologies are a useful proxy for studying subcellular compartments and their behavior after a given manipulation. Treatment with chemical compounds or specific gene silencing by RNA interference are commonly used at the scale of individual experiments to highthroughput studies. Visual inspection by expert biologists has been performed for several decades, ranging from early studies by microscopists like Ramon y Cajal to contemporary large scale, high-throughput screens (1-4). Although human observation may be very accurate, the three major drawbacks are that (i) it lacks quantitative measures, (ii) it may be biased, and (iii) it is time consuming.The structural features of cells and the topological relationships between the numerous intracellular compartments give rise to multivariate data whose unbiased, automatic comparison is a major challenge. Importantly, alterations in cellular morphology also occur in many diseases, including cancer, requiring quantitative tools for their detection. Give...

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Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices

Abstract: Asymptotic MISE, Multivariate kernel density estimation, Plug-in method, Pre-sphering, Unconstrained bandwidth selectors, 62G07,

Cited by 113 publications

References 26 publications

Asymptotics for general multivariate kernel density derivative estimators

Asymptotics for general multivariate kernel density derivative estimators

FPGA-based bandwidth selection for kernel density estimation using high level synthesis approach

Closed-form density-based framework for automatic detection of cellular morphology changes

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