Identification of the associations between genes and quantitative traits using entropy-based kernel density estimation

Numerous facets of scientific research implicitly or explicitly call for the estimation of probability densities. Histograms and kernel density estimates (KDEs) are two commonly used techniques for estimating such information, with the KDE generally providing a higher fidelity representation of the probability density function (PDF). Both methods require specification of either a bin width or a kernel bandwidth. While techniques exist for choosing the kernel bandwidth optimally and objectively, they are computationally intensive, since they require repeated calculation of the KDE. A solution for objectively and optimally choosing both the kernel shape and width has recently been developed by Bernacchia and Pigolotti (2011). While this solution theoretically applies to multidimensional KDEs, it has not been clear how to practically do so.A method for practically extending the Bernacchia–Pigolotti KDE to multidimensions is introduced. This multidimensional extension is combined with a recently-developed computational improvement to their method that makes it computationally efficient: a 2D KDE on 105 samples only takes 1 s on a modern workstation. This fast and objective KDE method, called the fastKDE method, retains the excellent statistical convergence properties that have been demonstrated for univariate samples. The fastKDE method exhibits statistical accuracy that is comparable to state-of-the-science KDE methods publicly available in R, and it produces kernel density estimates several orders of magnitude faster. The fastKDE method does an excellent job of encoding covariance information for bivariate samples. This property allows for direct calculation of conditional PDFs with fastKDE. It is demonstrated how this capability might be leveraged for detecting non-trivial relationships between quantities in physical systems, such as transitional behavior.

This paper presents a new control strategy for stochastic distribution shape tracking regarding non-Gaussian stochastic non-linear systems. The objective can be summarised as adjusting the probability density function (PDF) of the system output to any given desired distribution. In order to achieve this objective, the system output PDF has first been formulated analytically, which is time-variant. Then, the PDF vectorisation has been implemented to simplify the model description. Using the vector-based representation, the system identification and control design have been performed to achieve the PDF tracking. In practice, the PDF evolution is difficult to implement in real-time, thus a data-driven extension has also been discussed in this paper, where the vector-based model can be obtained using kernel density estimation (KDE) with the real-time data. Furthermore, the stability of the presented control design has been analysed, which is validated by a numerical example. As an extension, the multi-output stochastic systems have also been discussed for joint PDF tracking using the proposed algorithm, and the perspectives of advanced controller have been discussed. The main contribution of this paper is to propose: (1) a new sampling-based PDF transformation to reduce the modelling complexity, (2) a data-driven approach for online implementation without model pre-training, and (3) a feasible framework to integrate the existing control methods.

Genetic Variations of Ionotropic Glutamate Receptor Pathways on Interferon-α-induced Depression in Patients with Hepatitis C Viral Infection

The Probability Density Function (PDF) is a key concept in statistics. Constructing the most adequate PDF from the observed data is still an important and interesting scientific problem, especially for large datasets. PDFs are often estimated using nonparametric data-driven methods. One of the most popular nonparametric method is the Kernel Density Estimator (KDE). However, a very serious drawback of using KDEs is the large number of calculations required to compute them, especially to find the optimal bandwidth parameter. In this paper we investigate the possibility of utilizing Graphics Processing Units (GPUs) to accelerate the finding of the bandwidth. The contribution of this paper is threefold: (a) we propose algorithmic optimization to one of bandwidth finding algorithms, (b) we propose efficient GPU versions of three bandwidth finding algorithms and (c) we experimentally compare three of our GPU implementations with the ones which utilize only CPUs. Our experiments show orders of magnitude improvements over CPU implementations of classical algorithms.

This paper investigates the stochastic distribution tracking problem while the probability density function (PDF) of the stochastic non-linear system output can be controlled to desired distribution. To achieve the control objective, a data- driven approach is proposed in which no information of the system model is required. The output PDF can be estimated by kernel density estimation (KDE) based on the collected system output data. Using the estimated PDF, the probability states can be obtained by sampling operation which can be used to re-characterise the PDF of the system output. Thus, the tracking performance can be achieved by PID control. The parametric selection of the controller has been analysed following the identified PDF dynamic model to assure the convergence of the system output. The effectiveness of the presented algorithm is illustrated by a numerical example.

Quantifying uncertainties in data and model: a prediction model for extreme rainfall events with application to Beijing subway.

We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) and Kernel Density (KD) methods. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles is a fixed fraction of the sample size (empirically determined to be ), and that this value is relatively insensitive to distributional form or sample size. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is low even for samples of as few as data points; in contrast, for KD the small sample bias can be as large as and for BC as large as . We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.

The optimal group is an important problem of histogram algorithm, and how to confirm group number has not a quantitative rule. So the concept of the close degree is imported to make the close degree between the upper contour line of histogram and the PDF(probability density function) of parameter as the judging criteria of optimal group. With the unknown of the PDF of parameter, the improved kernel density estimation algorithm can pre-select and estimate the PDF. This improved kernel density estimation algorithm combine the selection of fixed window and variable window's width to achieve the window's width automatic adjustment value between the different estimation points based on the sample distribution. In the parameter analysis of radar emitter signal, the algorithm based on improved kernel density estimation and close degree is used to determine optimal group, and the result indicate that this method is effective and can search the optimal group automatically.

The Fokker Planck (FP) equation for the probability density function (PDF) of crystal size in phase transformations ruled by progressive nucleation and growth, has been derived. Crystals are grouped in sub-sets, we refer to as τ-crystals, where τ is the birth time of the set. It is shown that the size PDF is the superposition of the PDF of the crystal sub-sets (τ-PDFs), with weight given by the nucleation rate. The growth and diffusion coefficients entering the FP equations are estimated as a function of both τ-PDFs and nucleation rate. The functional form of these coefficients is studied for solutions of the FP equation for τ-crystals given by the lognormal, Gaussian and gamma distributions. For the first two distributions, the effect of fluctuations, nucleation rate and growth rate, on the shape of the distribution has been investigated. It is shown that for an exponential decay of the fluctuation term, the shape of the PDF is mainly governed by both the time constant for nucleation and the strength of the fluctuation. It is found that τ-PDFs given by the one-parameter gamma distributions are suitable to deal with KJMA (Kolmogorov Johnson Mehl Avrami) compliant phase transformations, where the fluctuation term is proportional to crystal size. The connection between the FP equation for the size PDF and the evolution equation for the density of crystal populations is also discussed.

In many data stream mining applications, traditional density estimation methods such as kernel density estimation, reduced set density estimation can not be applied to the density estimation of data streams because of their high computational burden, processing time and intensive memory allocation requirement. In order to reduce the time and space complexity, a novel density estimation method Dm-KDE over data streams based on the proposed algorithm m-KDE which can be used to design a KDE estimator with the fixed number of kernel components for a dataset is proposed. In this method, Dm-KDE sequence entries are created by algorithm m-KDE instead of all kernels obtained from other density estimation methods. In order to further reduce the storage space, Dm-KDE sequence entries can be merged by calculating their KL divergences. Finally, the probability density functions over arbitrary time or entire time can be estimated through the obtained estimation model. In contrast to the state-of-the-art algorithm SOMKE, the distinctive advantage of the proposed algorithm Dm-KDE exists in that it can achieve the same accuracy with much less fixed number of kernel components such that it is suitable for the scenarios where higher on-line computation about the kernel density estimation over data streams is required.We compare Dm-KDE with SOMKE and M-kernel in terms of density estimation accuracy and running time for various stationary datasets. We also apply Dm-KDE to evolving data streams. Experimental results illustrate the effectiveness of the proposed method.

In this paper, a novel CFAR algorithm for detecting layover and shadow areas in Interferometric synthetic aperture radar (InSAR) images is proposed. Firstly, the probability density function (PDF) of the square root amplitude of InSAR image is estimated by the kernel density estimation. Then, a CFAR algorithm combining with the morphological method for detecting both layover and shadow is presented. Finally, the proposed algorithm is evaluated on a real InSAR image obtained by TerraSAR-X system. The experimental results have validated the effectiveness of the proposed method.

This paper studies how to sample load more realistically and efficiently for security constraint unit commitment (SCUC) problems in order to achieve a high degree of robustness of the unit commitment (UC) solution. For example, given the UC solution, 95% of load profiles can be supplied. Principal component analysis (PCA) is introduced to find a clear feature of the historical load in two-dimensional space rather than the original high-dimensional load space, whose feature is hard to capture. Kernel density estimation (KDE) with Gaussian kernel is applied to form the probability density function (PDF), from which hourly load profiles are sampled. The load profiles based on sparse sampling are used to find the UC solution with high robustness while the load profiles based on dense sampling are used to verify the robustness of the obtained UC solution.

Objective: To analyze the adjustments of the weibull, gamma, normal and logistic probability density distributions of the historical series of hospitalizations for respiratory diseases (childhood and adult pneumonia) from 2011 to 2015, in Campo Grande, MS. Methods: The shape and scale parameters of the distributions were determined to verify the quality of the data fit. Results: Four probability density functions (Table 2) were fitted and the R2, MAE, RSME, MAPE tests were used to verify the best density function for hospitalization data. Conclusion: The best fit was the Gamma distribution; the distribution can be used as an alternative distribution that adequately describes the data on hospital admissions for respiratory diseases in Campo Grande.

BackgroundSingle nucleotide polymorphism (SNP) variants are abundant, persistent and widely distributed across the genome and are frequently linked to the development of genetic diseases. Identifying SNPs that underpin complex diseases can aid scientists in the discovery of disease-related genes by allowing for early detection, effective medication and eventually disease prevention.Main bodyVarious SNP or polymorphism-based studies were used to categorize different SNPs potentially related to three quantitative traits: body mass index (BMI), intelligence quotient (IQ) and blood pressure, and then uncovered common SNPs for these three traits. We employed SNPedia, RefSNP Report, GWAS Catalog, Gene Cards (Data Bases), PubMed and Google Scholar search engines to find relevant material on SNPs associated with three quantitative traits. As a result, we detected three common SNPs for all three quantitative traits in global populations: SNP rs6265 of the BDNF gene on chromosome 11p14.1, SNP rs131070325 of the SL39A8 gene on chromosome 4p24 and SNP rs4680 of the COMT gene on chromosome 22q11.21.ConclusionIn our review, we focused on the prevalent SNPs and gene expression activities that influence these three quantitative traits. These SNPs have been used to detect and map complex, common illnesses in communities for homogeneity testing and pharmacogenetic studies. High blood pressure, diabetes and heart disease, as well as BMI, schizophrenia and IQ, can all be predicted using common SNPs. Finally, the results of our work can be used to find common SNPs and genes that regulate these three quantitative features across the genome.

This work focuses on the application of Preisach-Mayergoyz (PM) model applied to the evaluation of elastic properties of hysteretic material. In the first part, essential concepts are explained such as material hysteresis, probability density functions of PM space, optimization algorithm used, and phi-divergence measures applied in the PM identification process. Next, two different characterizations of PM space of hysterons based on kernel density estimators are proposed, either for one-dimensional PM space projection or for fully two-dimensional pyramid kernel. Finally, new index of elasticity is built up by means of the squared Le Cam divergence between probability density corresponding to the super-elastic distribution and the probability density found by our optimized PM identification process, respectively. This elasticity index describes ability of the material to absorb mechanical deformation, or alternatively, it gives an evidence about the certain degree of damage of the material. This proposed index of elasticity (IE) is evaluated for the case of experimental data measured on the steel dampers used for the protection against earthquakes.