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Polygenic risk scores (PRS) enhance population risk stratification and advance personalized medicine, but existing methods face several limitations, encompassing issues related to computational burden, predictive accuracy, and adaptability to a wide range of genetic architectures. To address these issues, we propose Aggregated L0Learn using Summary-level data (ALL-Sum), a fast and scalable ensemble learning method for computing PRS using summary statistics from genome-wide association studies (GWAS). ALL-Sum leverages a L0L2 penalized regression and ensemble learning across tuning parameters to flexibly model traits with diverse genetic architectures. In extensive large-scale simulations across a wide range of polygenicity and GWAS sample sizes, ALL-Sum consistently outperformed popular alternative methods in terms of prediction accuracy, runtime, and memory usage by 10%, 20-fold, and threefold, respectively, and demonstrated robustness to diverse genetic architectures. We validated the performance of ALL-Sum in real data analysis of 11 complex traits using GWAS summary statistics from nine data sources, including the Global Lipids Genetics Consortium, Breast Cancer Association Consortium, and FinnGen Biobank, with validation in the UK Biobank. Our results show that on average, ALL-Sum obtained PRS with 25% higher accuracy on average, with 15 times faster computation and half the memory than the current state-of-the-art methods, and had robust performance across a wide range of traits and diseases. Furthermore, our method demonstrates stable prediction when using linkage disequilibrium computed from different data sources. ALL-Sum is available as a user-friendly R software package with publicly available reference data for streamlined analysis. [ABSTRACT FROM AUTHOR]

In statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order 1/T up to logarithmic factors on the objective function scale, where T denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository LogConcComp (Log-Concave Computation). [ABSTRACT FROM AUTHOR]

It is increasingly common to encounter prediction tasks in the biomedical sciences for which multiple datasets are available for model training. Common approaches such as pooling datasets before model fitting can produce poor out‐of‐study prediction performance when datasets are heterogeneous. Theoretical and applied work has shown multistudy ensembling to be a viable alternative that leverages the variability across datasets in a manner that promotes model generalizability. Multistudy ensembling uses a two‐stage stacking strategy which fits study‐specific models and estimates ensemble weights separately. This approach ignores, however, the ensemble properties at the model‐fitting stage, potentially resulting in performance losses. Motivated by challenges in the estimation of COVID‐attributable mortality, we propose optimal ensemble construction, an approach to multistudy stacking whereby we jointly estimate ensemble weights and parameters associated with study‐specific models. We prove that limiting cases of our approach yield existing methods such as multistudy stacking and pooling datasets before model fitting. We propose an efficient block coordinate descent algorithm to optimize the loss function. We use our method to perform multicountry COVID‐19 baseline mortality prediction. We show that when little data is available for a country before the onset of the pandemic, leveraging data from other countries can substantially improve prediction accuracy. We further compare and characterize the method's performance in data‐driven simulations and other numerical experiments. Our method remains competitive with or outperforms multistudy stacking and other earlier methods in the COVID‐19 data application and in a range of simulation settings. [ABSTRACT FROM AUTHOR]

Existing SNP-heritability estimators that leverage summary statistics from genome-wide association studies (GWAS) are much less efficient (i.e., have larger standard errors) than the restricted maximum likelihood (REML) estimators which require access to individual-level data. We introduce a new method for local heritability estimation—Heritability Estimation with high Efficiency using LD and association Summary Statistics (HEELS)—that significantly improves the statistical efficiency of summary-statistics-based heritability estimator and attains comparable statistical efficiency as REML (with a relative statistical efficiency >92%). Moreover, we propose representing the empirical LD matrix as the sum of a low-rank matrix and a banded matrix. We show that this way of modeling the LD can not only reduce the storage and memory cost, but also improve the computational efficiency of heritability estimation. We demonstrate the statistical efficiency of HEELS and the advantages of our proposed LD approximation strategies both in simulations and through empirical analyses of the UK Biobank data. The authors propose "HEELS", a new method for precise local heritability estimation. It significantly reduces the variances of summary-statistics-based heritability estimators, offering an REML-like estimator without requiring individual-level data. [ABSTRACT FROM AUTHOR]

Linear regression is a fundamental modeling tool in statistics and related fields. In this paper, we study an important variant of linear regression in which the predictor-response pairs are partially mismatched. We use an optimization formulation to simultaneously learn the underlying regression coefficients and the permutation corresponding to the mismatches. The combinatorial structure of the problem leads to computational challenges. We propose and study a simple greedy local search algorithm for this optimization problem that enjoys strong theoretical guarantees and appealing computational performance. We prove that under a suitable scaling of the number of mismatched pairs compared to the number of samples and features, and certain assumptions on problem data; our local search algorithm converges to a nearly-optimal solution at a linear rate. In particular, in the noiseless case, our algorithm converges to the global optimal solution with a linear convergence rate. Based on this result, we prove an upper bound for the estimation error of the parameter. We also propose an approximate local search step that allows us to scale our approach to much larger instances. We conduct numerical experiments to gather further insights into our theoretical results, and show promising performance gains compared to existing approaches. [ABSTRACT FROM AUTHOR]

Learning Compact High-Dimensional Models in Noisy Environments Building compact, interpretable statistical models where the output depends upon a small number of input features is a well-known problem in modern analytics applications. A fundamental tool used in this context is the prominent best subset selection (BSS) procedure, which seeks to obtain the best linear fit to data subject to a constraint on the number of nonzero features. Whereas the BSS procedure works exceptionally well in some regimes, it performs pretty poorly in out-of-sample predictive performance when the underlying data are noisy, which is quite common in practice. In this paper, we explore this relatively less-understood overfitting behavior of BSS in low-signal noisy environments and propose alternatives that appear to mitigate such shortcomings. We study the theoretical statistical properties of our proposed regularized BSS procedure and show promising computational results on various data sets, using tools from integer programming and first-order methods. We study a seemingly unexpected and relatively less understood overfitting aspect of a fundamental tool in sparse linear modeling—best subset selection—which minimizes the residual sum of squares subject to a constraint on the number of nonzero coefficients. Whereas the best subset selection procedure is often perceived as the "gold standard" in sparse learning when the signal-to-noise ratio (SNR) is high, its predictive performance deteriorates when the SNR is low. In particular, it is outperformed by continuous shrinkage methods, such as ridge regression and the Lasso. We investigate the behavior of best subset selection in the high-noise regimes and propose an alternative approach based on a regularized version of the least-squares criterion. Our proposed estimators (a) mitigate, to a large extent, the poor predictive performance of best subset selection in the high-noise regimes; and (b) perform favorably, while generally delivering substantially sparser models, relative to the best predictive models available via ridge regression and the Lasso. We conduct an extensive theoretical analysis of the predictive properties of the proposed approach and provide justification for its superior predictive performance relative to best subset selection when the noise level is high. Our estimators can be expressed as solutions to mixed-integer second-order conic optimization problems and, hence, are amenable to modern computational tools from mathematical optimization. Funding: R. Mazumder acknowledges research funding from the Office of Naval Research [Grants N000141512342 and N000141812298] and the National Science Foundation [Grant NSF-IIS-1718258]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2276. [ABSTRACT FROM AUTHOR]

We introduce a statistical procedure that integrates datasets from multiple biomedical studies to predict patients' survival, based on individual clinical and genomic profiles. The proposed procedure accounts for potential differences in the relation between predictors and outcomes across studies, due to distinct patient populations, treatments and technologies to measure outcomes and biomarkers. These differences are modeled explicitly with study‐specific parameters. We use hierarchical regularization to shrink the study‐specific parameters towards each other and to borrow information across studies. The estimation of the study‐specific parameters utilizes a similarity matrix, which summarizes differences and similarities of the relations between covariates and outcomes across studies. We illustrate the method in a simulation study and using a collection of gene expression datasets in ovarian cancer. We show that the proposed model increases the accuracy of survival predictions compared to alternative meta‐analytic methods. [ABSTRACT FROM AUTHOR]

We consider the least squares regression problem, penalized with a combination of the ℓ 0 and squared ℓ 2 penalty functions (a.k.a. ℓ 0 ℓ 2 regularization). Recent work shows that the resulting estimators enjoy appealing statistical properties in many high-dimensional settings. However, exact computation of these estimators remains a major challenge. Indeed, modern exact methods, based on mixed integer programming (MIP), face difficulties for problems where the number of features p ∼ 10 4 . In this work, we present a new exact MIP framework for ℓ 0 ℓ 2 -regularized regression that can scale to p ∼ 10 7 , achieving speedups of at least 5000x, compared to state-of-the-art exact methods. Unlike recent work, which relies on commercial MIP solvers, we design a specialized nonlinear branch-and-bound (BnB) framework, by critically exploiting the problem structure. A key distinguishing component in our framework lies in efficiently solving the node relaxations using a specialized first-order method, based on coordinate descent (CD). Our CD-based method effectively leverages information across the BnB nodes through using warm starts, active sets, and gradient screening. In addition, we design a novel method for obtaining dual bounds from primal CD solutions, which certifiably works in high dimensions. Experiments on synthetic and real high-dimensional datasets demonstrate that our framework is not only significantly faster than the state of the art, but can also deliver certifiably optimal solutions to statistically challenging instances that cannot be solved with existing methods. We open source the implementation through our toolkit L0BnB. [ABSTRACT FROM AUTHOR]

The Frank--Wolfe method is a popular algorithm for solving large-scale convex optimization problems appearing in structured statistical learning. However, the traditional Frank--Wolfe method can only be applied when the feasible region is bounded, which limits its applicability in practice. Motivated by two applications in statistical learning, the l1 trend filtering problem and matrix optimization problems with generalized nuclear norm constraints, we study a family of convex optimization problems where the unbounded feasible region is the direct sum of an unbounded linear subspace and a bounded constraint set. We propose two new Frank--Wolfe methods, the unbounded Frank--Wolfe method and the unbounded away-step Frank--Wolfe method, for solving a family of convex optimization problems with this class of unbounded feasible regions. We show that under proper regularity conditions, the unbounded Frank--Wolfe method has an O(1/k) sublinear convergence rate, and the unbounded away-step Frank--Wolfe method has a linear convergence rate, matching the best-known results for the Frank--Wolfe method when the feasible region is bounded. Furthermore, computational experiments indicate that our proposed methods appear to outperform alternative solvers. [ABSTRACT FROM AUTHOR]

Dual Bounds of Sparse Principal Component Analysis Sparse principal component analysis (PCA) is a widely used dimensionality reduction tool in machine learning and statistics. Compared with PCA, sparse PCA enhances the interpretability by incorporating a sparsity constraint. However, unlike PCA, conventional heuristics for sparse PCA cannot guarantee the qualities of obtained primal feasible solutions via associated dual bounds in a tractable fashion without underlying statistical assumptions. In "Using L1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA," Santanu S. Dey, Rahul Mazumder, and Guanyi Wang present a convex integer programming (IP) framework of sparse PCA to derive dual bounds. They show the worst-case results on the quality of the dual bounds provided by the convex IP. Moreover, the authors empirically illustrate that the proposed convex IP framework outperforms existing sparse PCA methods of finding dual bounds. Principal component analysis (PCA) is one of the most widely used dimensionality reduction tools in scientific data analysis. The PCA direction, given by the leading eigenvector of a covariance matrix, is a linear combination of all features with nonzero loadings; this impedes interpretability. Sparse principal component analysis (SPCA) is a framework that enhances interpretability by incorporating an additional sparsity requirement in the feature weights (factor loadings) while finding a direction that explains the maximal variation in the data. However, unlike PCA, the optimization problem associated with the SPCA problem is NP-hard. Most conventional methods for solving SPCA are heuristics with no guarantees, such as certificates of optimality on the solution quality via associated dual bounds. Dual bounds are available via standard semidefinite programming (SDP)–based relaxations, which may not be tight, and the SDPs are difficult to scale using off-the-shelf solvers. In this paper, we present a convex integer programming (IP) framework to derive dual bounds. At the heart of our approach is the so-called ℓ 1 -relaxation of SPCA. Although the ℓ 1 -relaxation leads to convex optimization problems for ℓ 0 -sparse linear regressions and relatives, it results in a nonconvex optimization problem for the PCA problem. We first show that the ℓ 1 -relaxation gives a tight multiplicative bound on SPCA. Then, we show how to use standard integer programming techniques to further relax the ℓ 1 -relaxation into a convex IP for which there are good commercial solvers. We present worst-case results on the quality of the dual bound provided by the convex IP. We empirically observe that the dual bounds are significantly better than the worst-case performance and are superior to the SDP bounds on some real-life instances. Moreover, solving the convex IP model using commercial IP solvers appears to scale much better that solving the SDP-relaxation using commercial solvers. To the best of our knowledge, we obtain the best dual bounds for real and artificial instances for SPCA problems involving covariance matrices of size up to 2,000 × 2,000. [ABSTRACT FROM AUTHOR]

The Gradient Boosting Machine (GBM) introduced by Friedman [J. H. Friedman, Ann. Statist., 29 (2001), pp. 1189-1232] is a powerful supervised learning algorithm that is very widely used in practice--it routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In spite of the usefulness of GBM in practice, our current theoretical understanding of this method is rather limited. In this work, we propose the Randomized Gradient Boosting Machine (RGBM), which leads to substantial computational gains compared to GBM by using a randomization scheme to reduce search in the space of weak learners. We derive novel computational guarantees for RGBM. We also provide a principled guideline towards better step-size selection in RGBM that does not require a line search. Our proposed framework is inspired by a special variant of coordinate descent that combines the benefits of randomized coordinate descent and greedy coordinate descent, and may be of independent interest as an optimization algorithm. As a special case, our results for RGBM lead to superior computational guarantees for GBM. Our computational guarantees depend upon a curious geometric quantity that we call the Minimal Cosine Angle, which relates to the density of weak learners in the prediction space. On a series of numerical experiments on real datasets, we demonstrate the effectiveness of RGBM over GBM in terms of obtaining a model with good training and/or testing data fidelity with a fraction of the computational cost. [ABSTRACT FROM AUTHOR]

In several scientific and industrial applications, it is desirable to build compact, interpretable learning models where the output depends on a small number of input features. Recent work has shown that such best-subset selection-type problems can be solved with modern mixed integer optimization solvers. Despite their promise, such solvers often come at a steep computational price when compared with open-source, efficient specialized solvers based on convex optimization and greedy heuristics. In "Fast Best-Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms," Hussein Hazimeh and Rahul Mazumder push the frontiers of computation for best-subset-type problems. Their algorithms deliver near-optimal solutions for problems with up to a million features—in times comparable with the fast convex solvers. Their work suggests that principled optimization methods play a key role in devising tools central to interpretable machine learning, which can help in gaining a deeper understanding of their statistical properties. The L0-regularized least squares problem (a.k.a. best subsets) is central to sparse statistical learning and has attracted significant attention across the wider statistics, machine learning, and optimization communities. Recent work has shown that modern mixed integer optimization (MIO) solvers can be used to address small to moderate instances of this problem. In spite of the usefulness of L0-based estimators and generic MIO solvers, there is a steep computational price to pay when compared with popular sparse learning algorithms (e.g., based on L1 regularization). In this paper, we aim to push the frontiers of computation for a family of L0-regularized problems with additional convex penalties. We propose a new hierarchy of necessary optimality conditions for these problems. We develop fast algorithms, based on coordinate descent and local combinatorial optimization, that are guaranteed to converge to solutions satisfying these optimality conditions. From a statistical viewpoint, an interesting story emerges. When the signal strength is high, our combinatorial optimization algorithms have an edge in challenging statistical settings. When the signal is lower, pure L0 benefits from additional convex regularization. We empirically demonstrate that our family of L0-based estimators can outperform the state-of-the-art sparse learning algorithms in terms of a combination of prediction, estimation, and variable selection metrics under various regimes (e.g., different signal strengths, feature correlations, number of samples and features). Our new open-source sparse learning toolkit L0Learn (available on CRAN and GitHub) reaches up to a threefold speedup (with p up to 106) when compared with competing toolkits such as glmnet and ncvreg. [ABSTRACT FROM AUTHOR]

In this paper, we study the popularly dubbed matrix completion problem, where the task is to "fill in" the unobserved entries of a matrix from a small subset of observed entries, under the assumption that the underlying matrix is of low rank. Our contributions herein enhance our prior work on nuclear norm regularized problems for matrix completion (Mazumder et al. in J Mach Learn Res 1532(11):2287–2322, 2010) by incorporating a continuum of nonconvex penalty functions between the convex nuclear norm and nonconvex rank functions. Inspired by Soft-Impute (Mazumder et al. 2010; Hastie et al. in J Mach Learn Res, 2016), we propose NC-Impute—an EM-flavored algorithmic framework for computing a family of nonconvex penalized matrix completion problems with warm starts. We present a systematic study of the associated spectral thresholding operators, which play an important role in the overall algorithm. We study convergence properties of the algorithm. Using structured low-rank SVD computations, we demonstrate the computational scalability of our proposal for problems up to the Netflix size (approximately, a 500,000 × 20,000 matrix with 10 8 observed entries). We demonstrate that on a wide range of synthetic and real data instances, our proposed nonconvex regularization framework leads to low-rank solutions with better predictive performance when compared to those obtained from nuclear norm problems. Implementations of algorithms proposed herein, written in the R language, are made available on github. [ABSTRACT FROM AUTHOR]

Factor analysis is a classical multivariate dimensionality reduction technique popularly used in statistics, econometrics and data science. Estimation for factor analysis is often carried out via the maximum likelihood principle, which seeks to maximize the Gaussian likelihood under the assumption that the positive definite covariance matrix can be decomposed as the sum of a low-rank positive semidefinite matrix and a diagonal matrix with nonnegative entries. This leads to a challenging rank constrained nonconvex optimization problem, for which very few reliable computational algorithms are available. We reformulate the low-rank maximum likelihood factor analysis task as a nonlinear nonsmooth semidefinite optimization problem, study various structural properties of this reformulation; and propose fast and scalable algorithms based on difference of convex optimization. Our approach has computational guarantees, gracefully scales to large problems, is applicable to situations where the sample covariance matrix is rank deficient and adapts to variants of the maximum likelihood problem with additional constraints on the model parameters. Our numerical experiments validate the usefulness of our approach over existing state-of-the-art approaches for maximum likelihood factor analysis. [ABSTRACT FROM AUTHOR]

We study the nonparametric least squares estimator (LSE) of a multivariate convex regression function. The LSE, given as the solution to a quadratic program with O(n2) linear constraints (n being the sample size), is difficult to compute for large problems. Exploiting problem specific structure, we propose a scalable algorithmic framework based on the augmented Lagrangian method to compute the LSE. We develop a novel approach to obtain smooth convex approximations to the fitted (piecewise affine) convex LSE and provide formal bounds on the quality of approximation. When the number of samples is not too large compared to the dimension of the predictor, we propose a regularization scheme—Lipschitz convex regression—where we constrain the norm of the subgradients, and study the rates of convergence of the obtained LSE. Our algorithmic framework is simple and flexible and can be easily adapted to handle variants: estimation of a nondecreasing/nonincreasing convex/concave (with or without a Lipschitz bound) function. We perform numerical studies illustrating the scalability of the proposed algorithm—on some instances our proposal leads to more than a 10,000-fold improvement in runtime when compared to off-the-shelf interior point solvers for problems with n = 500. [ABSTRACT FROM AUTHOR]

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10–15 years, we present a mixed integer linear optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed by Bertsimas et al. and the continuous nonconvex quadratic optimization framework of Liu et al.. We propose new discrete first-order methods, which when paired with the state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular \ell 1 -based approaches. We illustrate that our approach can handle problem instances with $p =10,\!000$ features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling. [ABSTRACT FROM AUTHOR]

Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to ind uce near-optimal solutions on lowdimensional faces of the feasible region. This is accomplished by a new approach to generating "in-face" directions at each iteration, as well as through new choice rules for selecting between inface and "regular" Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological a pproach at producing nearly optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speedups in computing very low rank nearly optimal solutions as compared to the Frank-Wolfe method (as well as several of its significant variants). [ABSTRACT FROM AUTHOR]

We propose a method to test the correlation of two random fields when they are both spatially autocorrelated. In this scenario, the assumption of independence for the pair of observations in the standard test does not hold, and as a result we reject in many cases where there is no effect (the precision of the null distribution is overestimated). Our method recovers the null distribution taking into account the autocorrelation. It uses Monte-Carlo methods, and focuses on permuting, and then smoothing and scaling one of the variables to destroy the correlation with the other, while maintaining at the same time the initial autocorrelation. With this simulation model, any test based on the independence of two (or more) random fields can be constructed. This research was motivated by a project in biodiversity and conservation in the Biology Department at Stanford University. [ABSTRACT FROM AUTHOR]

We address the problem of sparse selection in linear models. A number of nonconvex penalties have been proposed in the literature for this purpose, along with a variety of convex-relaxation algorithms for finding good solutions. In this article we pursue a coordinate-descent approach for optimization, and study its convergence properties. We characterize the properties of penalties suitable for this approach, study their corresponding threshold functions, and describe a df-standardizing reparametrization that assists our pathwise algorithm. The MC + penalty is ideally suited to this task, and we use it to demonstrate the performance of our algorithm. Certain technical derivations and experiments related to this article are included in the Supplementary Materials section. [ABSTRACT FROM AUTHOR]

This paper aims at identifying statistically different circulation patterns characterising fluid flow in the trough region between two adjacent asymmetric waveforms, using the velocity data collected by 3D acoustic Doppler velocimeter. Statistical clustering has been performed using ideas originating from information theory and scale space theory in computer vision for splitting the trough region into different spatially connected segments (identifying the circulation bubble in the process) on the basis of circulation patterns. The paper attempts to visualise the fluid fluctuations in the trough region, with emphasis on the circulation region, by simulating the directional fluctuations of fluid particles from the kernel density estimates learned from the experimental data. The image representation of the estimate of the spatial turbulent kinetic energy (TKE) function reveals interesting features corresponding to the regions of high TKE, suggesting the possibilities for further research in this area along the lines of feature extraction and image analysis. [ABSTRACT FROM AUTHOR]

This paper aims at understanding turbulence occurring due to fluid flow in the trough region between two artificial adjacent asymmetric waveforms in an open-channel using velocity data collected by 3-D Acoustic Doppler Velocimeter (ADV), concentrating on interactions among the turbulence bursting phenomena across different spatial locations. The Statistical Learning stage of the analysis begins with the identification and extraction of statistically informative and physically interpretable features in the geometry of the Bursting Rate Processes (BRP). Statistical measures characterising the differences among the concerned processes have been developed and used for splitting the trough region into different regions based on the geometry, structure and randomness in the BRP. using the principles of statistical clustering involving parametric, non-parametric techniques and ideas of information theoretic entropy. Experimental observations support the existence of certain BRP which may be considered to be dominant over the others in an almost global sense. The issue of identifying a single bursting phenomenon that changes its orientation strongly relative to others or its closest neighbour across all spatial locations or stretches of vertical heights for every horizontal location and its importance in the entire physical scenario in the light of the spatial clustering problem has been addressed and settled too. [ABSTRACT FROM AUTHOR]

The objective of this paper is to analyze statistically the turbulence characteristics of flow over an artificial waveform geometry in an open channel, using velocity data collected by 3-D acoustic doppler velocimeter (ADV). The joint distribution of velocity fluctuations have been studied and modeled using multivariate normal distribution, and its effectiveness is analyzed in the light of explaining turbulent bursting events, like sweeping and ejection. The relationship between the randomness in the directional variations of the 3-D velocity fluctuations and the intensity of circulation has been studied. We have exploited this relationship and used it in conjunction with the principles of statistical clustering for splitting the region of flow, into different spatially connected segments based on circulation patterns. Directional data analysis has been adopted to visualize the flow pattern over the entire test section, in particular, the circulation bubbles in the region of separated flow. [ABSTRACT FROM AUTHOR]