Robust Multivariate Methods Research Articles

Evaluation of Genetic Diversity in Sorghum Genotypes for Drought Tolerance Using Mahalanobis’ D2 Analysis

Genetic diversity is vital in plant breeding and crop improvement, enabling the development of varieties resilient to biotic and abiotic stresses. Mahalanobis' D² analysis is a robust multivariate statistical method for assessing genetic diversity, offering precise measures of genetic divergence. This study aimed to evaluate the genetic diversity of twenty sorghum (Sorghum bicolor L. Moench) genotypes under rainfed and irrigated conditions using Mahalanobis’ D² statistics. The experiment was conducted at the Main Agricultural Research Station (MARS), Dharwad, involving twenty genotypes sourced from the AICRP sorghum, Dharwad, India. The genotypes were sown in medium black soil under two moisture levels: rainfed and irrigated. Morphological, physiological, phenological, and yield-related traits were recorded to identify drought-tolerant genotypes. Genetic divergence was estimated using D² statistics, and genotypes were grouped into clusters using Tocher’s method. Under rainfed conditions, the genotypes were classified into six clusters, while under irrigated conditions, they were grouped into three clusters. The intra-cluster distances ranged from 7.79 to 12.14 under rainfed conditions and from 8.14 to 15.74 under irrigated conditions. The inter-cluster distances revealed significant genetic variation, with the highest divergence observed between Clusters III and V under rainfed conditions and Clusters II and III under irrigated conditions. These findings highlight the presence of substantial genetic diversity among sorghum genotypes, facilitating the selection of diverse parental lines for breeding programs aimed at enhancing drought tolerance. The insights gained from this study contribute to the development of resilient sorghum varieties capable of withstanding water scarcity, ensuring sustainable agriculture, and promoting food security in drought-prone regions.

Leveraging multivariate approaches to advance the science of early-life adversity

Since the landmark Adverse Childhood Experiences (ACEs) study, adversity research has expanded to more precisely account for the multifaceted nature of adverse experiences. The complex data structures and interrelated nature of adversity data require robust multivariate statistical methods, and recent methodological and statistical innovations have facilitated advancements in research on childhood adversity. Here, we provide an overview of a subset of multivariate methods that we believe hold particular promise for advancing the field's understanding of early-life adversity, and discuss how these approaches can be practically applied to explore different research questions. This review covers data-driven or unsupervised approaches (including dimensionality reduction and person-centered clustering/subtype identification) as well as supervised/prediction-based approaches (including linear and tree-based models and neural networks). For each, we highlight studies that have effectively applied the method to provide novel insight into early-life adversity. Taken together, we hope this review serves as a resource to adversity researchers looking to expand upon the cumulative approach described in the original ACEs study, thereby advancing the field's understanding of the complexity of adversity and related developmental consequences.

Morphologically hypervariable species hinder our knowledge of biodiversity: Daustinia montana (Convolvulaceae) as a case study

Abstract Cryptic species continue to intrigue taxonomists over time and hamper biodiversity knowledge. An example of what would be considered a cryptic species is Daustinia montana (Convolvulaceae). Its wide leaf morphology plasticity has led to multiple interpretations and contrasting classifications: from a monotypic to a six-taxa hypothesis. For this work, we tested six taxonomic hypotheses, including an explicit test of a monotypic approach, under a robust statistical analysis, using univariate and multivariate methods. Besides that, we performed a niche analysis to verify the niche occupation of the populations recognized here as possible species. Forty-eight micro and macromorphological characters (qualitative and quantitative) from individuals of 16 populations of D. montana were evaluated. The taxonomic hypothesis that recognizes eight distinct species has the highest support as they also have non-overlapping niches. We conclude that the number of species in Daustinia may be greater than its current circumscription. We also highlight the importance of an integrative systematic approach in the study of biodiversity. This research represents a first step in the specific delimitations of the genus and can also serve as a model to study taxa with wide morphological variability.

Forecasting time trend of road traffic crashes in Iran using the macro-scale traffic flow characteristics

BackgroundThe serial correlation in the time series datasets should be considered to prevent biased estimates for coefficients. Nonetheless, the current models almost cannot explicitly handle autocorrelation and seasonality, and they focus mainly on the discrete nature of data. Nonetheless, the crash time series follows a normal distribution at the macro-scale. Moreover, the influential exogenous variables have been overlooked in Iran, employing univariate models. There are also contradictory results in the literature regarding the effect of average speed on crash frequency. ObjectiveThis study is aimed to evaluate the distinct impacts of mean speed on total and fatal accident time series at the national level. Besides, the SARIMAX modeling framework is introduced as a robust multivariate method for short-term crash frequency prediction. MethodTo this end, monthly total and fatal crash counts were aggregated for all rural highways in Iran. Besides, the time trends of traffic exposure, and average speed recorded by loop detectors, were aggregated at the same level as covariates. The Box-Jenkins methodology was employed for time series analysis. ResultsThe results illustrated that the seasonal autoregressive integrated moving average with explanatory variable (SARIMAX) model outperformed the univariate ARIMA and SARIMA models. Also, SARIMA was more appropriate than the simple ARIMA when seasonality existed in the time series. Besides, the average speed had a negative linear association with the total crashes. In contrast, it revealed an increasing effect on fatal crashes. ConclusionAverage speed has a dissimilar effect on the different traffic crash severities. Besides, the seasonal nature of data and the dynamic effects of the influential underlying factors should be considered to prevent underfitting issues and to predict future time trends accurately. ApplicationsThe developed instruments could be employed by policymakers to evaluate the intervention's effectiveness and to forecast the future time trends of accidents in Iran.

Nonmetric multidimensional scaling and probabilistic ecological risk assessment of trace metals in surface sediments of Daya Bay (China) using diffusive gradients in thin films

This research is one main objective to assess combined toxicity of trace metal mixtures in aquatic biota in coastal sediments. Coastal sediments around the world are a major reservoir of trace metals from industrial wastewater discharge. Our case study site, Daya Bay in southern China, was selected because it has been under severe man-made impacts. Diffusive gradients in thin films (DGT) technique has proven to be a good method for measuring the bioavailability of trace metals. The bioavailability and distribution of trace metals in surface sediments were investigated along with their possible biological risks. The average bioavailable (DGT-labile) concentrations (μg/L) were 0.44 (V), 0.51 (Cr), 52.49 (Mn), 0.10 (Co), 1.36 (Ni), 0.74 (Cu), 14.53 (Zn), 0.97 (As), 0.14 (Se), 6.73 (Mo), 0.17 (Cd), 0.27 (Sb), 0.10 (W), and 1.32 (Pb). Nonmetric multidimensional scaling (NMS) is a robust multivariate ordination method that makes no assumptions about the distribution of the underlying data. NMS was used to explore that DGT-labile concentrations of trace metals were influenced by sediment properties. NMS results indicated that most DGT- labile trace metals influenced by sediment properties. Risk assessment of single trace metal toxicity revealed that risk quotient (RQ) values for Mn, Cu, Zn and Pb significantly exceeded 1, demonstrating that the toxic effects of these trace metals should be not ignored. The probabilistic ecological risk assessment for integral toxicity of one mixture of 14 trace metals revealed that Daya Bay surface sediments had a low probability (9.04 %) of adverse effects on aquatic biota.

Optimization of SFCC as Mineral Filler in Asphalt–Concrete Mixture Using Combined Methods of Taguchi and PCA

Spent fluid catalytic cracking (SFCC) is an industrial waste from oil refineries that is increasing nonstop in Vietnam and many places globally, and needs to be recycled. However, SFCC has various compositions depending on the fluid catalytic cracking (FCC) types and additives in the petroleum-refining industry. Thus, an experimental optimization procedure is needed to find the optimum addition of SFCC as a filler powder for asphalt–concrete mixtures. For this requirement, the study objective was an optimization procedure based on Taguchi parameter design employing an MMDC parameter. MMDC, developed using a robust multivariate statistical method called principal component analysis (PCA), is a representative parameter of required performance quality proposed by the Marshall mixture design. The Marshall mixture design was used to verify the asphalt–concrete mixture in which SFCC filler partly replaced traditional limestone (LS) filler. The effect of LS filler rate on mixture characteristics was used to determine the required range of values of MMDC. Also, Marshall characteristics based on various SFCC filler rates verified the optimum methodology result. The confirmation experiment results showed that the approach could find an optimum case with advantages in reducing asphalt binder content in the mixture and the number of specimens in the laboratory. Furthermore, the experimental results indicated that SFCC filler could enhance the Marshall characteristics of an asphalt–concrete mixture.

On weighted multivariate sign functions

Multivariate sign functions are often used for robust estimation and inference. We propose using data dependent weights in association with such functions. The proposed weighted sign functions retain desirable robustness properties, while significantly improving efficiency in estimation and inference compared to unweighted multivariate sign-based methods. Using weighted signs, we demonstrate methods of robust location estimation and robust principal component analysis. We extend the scope of using robust multivariate methods to include robust sufficient dimension reduction and functional outlier detection. Several numerical studies and real data applications demonstrate the efficacy of the proposed methodology.

The usefulness of robust multivariate methods: A case study with the menu items of a fast food restaurant chain

Multivariate statistical methods have been playing an important role in statistics and data analysis for a very long time. Nowadays, with the increase in the amounts of data collected every day in many disciplines, and with the raise of data science, machine learning and applied statistics, that role is even more important. Two of the most widely used multivariate statistical methods are cluster analysis and principal component analysis. These, similarly to many other models and algorithms, are adequate when the data satisfies certain assumptions. However, when the distribution of the data is not normal and/or it shows heavy tails and outlying observations, the classic models and algorithms might produce erroneous conclusions. Robust statistical methods such as algorithms for robust cluster analysis and for robust principal component analysis are of great usefulness when analyzing contaminated data with outlying observations. In this paper we consider a data set containing the products available in a fast food restaurant chain together with their respective nutritional information, and discuss the usefulness of robust statistical methods for classification, clustering and data visualization.

Lessons for machine learning from the analysis of porosity-permeability transforms for carbonate reservoirs

Prediction of permeability is one of the most difficult aspects of reservoir characterization because permeability cannot be directly measured by current well logging technology. This is particularly challenging for carbonate rocks. Machine learning (ML) and robust multivariate methods have been developed that have been used in many fields of study to make accurate estimators for variables of interest from both large and small datasets. ML has been criticized for utilizing approaches that are typically not interpretable. That is, it is not clear how the answers are arrived at and what aspects of input data may be resulting in inaccurate results. The current study uses a number of the mathematical algorithms that operate inside ML modules. It applies them to developing porosity-permeability transforms, with or without rock types, to two well-characterized data sets for carbonate reservoirs. One data set is from Jerry Lucia's 1995 study of carbonate rock types, and the other is from a study of the Seminole, West Texas, San Andres Unit. This study of statistical analysis of porosity-permeability transforms includes: transforming the data to normal distributions; performing cross-validation blind testing; and detecting heteroscedasticity by creating plots of residuals. Heteroscedastic data (populations with variable variance) may have an adverse impact on ML algorithms such as Random Forests (RF). We find that including lithofacies information does not greatly improve porosity-permeability transforms. We also propose a number of strategies to make ML analyses of reservoir (and other geosciences) data sets more robust and accurate.

Evaluation of robust outlier detection methods for zero-inflated complex data

ABSTRACTOutlier detection can be seen as a pre-processing step for locating data points in a data sample, which do not conform to the majority of observations. Various techniques and methods for outlier detection can be found in the literature dealing with different types of data. However, many data sets are inflated by true zeros and, in addition, some components/variables might be of compositional nature. Important examples of such data sets are the Structural Earnings Survey, the Structural Business Statistics, the European Statistics on Income and Living Conditions, tax data or – as in this contribution – household expenditure data which are used, for example, to estimate the Purchase Power Parity of a country.In this work, robust univariate and multivariate outlier detection methods are compared by a complex simulation study that considers various challenges included in data sets, namely structural (true) zeros, missing values, and compositional variables. These circumstances make it difficult or impossible to flag true outliers and influential observations by well-known outlier detection methods.Our aim is to assess the performance of outlier detection methods in terms of their effectiveness to identify outliers when applied to challenging data sets such as the household expenditures data surveyed all over the world. Moreover, different methods are evaluated through a close-to-reality simulation study. Differences in performance of univariate and multivariate robust techniques for outlier detection and their shortcomings are reported. We found that robust multivariate methods outperform robust univariate methods. The best performing methods in finding the outliers and in providing a low false discovery rate were found to be the generalized S estimators (GSE), the BACON-EEM algorithm and a compositional method (CoDa-Cov). In addition, these methods performed also best when the outliers are imputed based on the corresponding outlier detection method and indicators are estimated from the data sets.

Abstract 4697: Expression variation analysis for tumor heterogeneity in single-cell RNA-sequencing data

Abstract Introduction: Tumor heterogeneity provides a complex challenge to cancer treatment and is a critical component of therapeutic response, disease recurrence, and patient survival. Single-cell RNA-sequencing (scRNA-seq) technologies reveal the prevalence of intra- and inter-tumor heterogeneity. Computational techniques are essential to quantify the differences in variation of these profiles between distinct cell types, tumor subtypes, and patients to fully characterize intra- and inter-tumor molecular heterogeneity. To address this, we devised a new algorithm, Expression Variation Analysis in Single Cells (EVAsc), to perform multivariate statistical analyses of differential variation of expression in gene sets for scRNA-seq. Methods: EVAsc provides a robust statistical test to compare the heterogeneity of transcriptional profiles of genes in a pathway between groups of cells from two phenotypes. Using simulated data, we demonstrated that this method is robust for imputed scRNA-seq data with high sensitivity and specificity to detect pathways with true differential heterogeneity. We then applied EVAsc to public domain scRNA-seq tumor datasets in breast cancer and head and neck squamous cell carcinoma (HNSCC) to quantify the landscape of tumor heterogeneity in several key applications in cancer genomics, i.e. immunogenicity, cancer subtypes, and metastasis. Results: We demonstrated that immune pathway heterogeneity in hematopoietic cell populations in breast tumors corresponded to the amount diversity present in the T-cell repertoire of each individual. In HNSCC patients, we found dramatic differences in pathway dysregulation across basal primary tumors, indicative of inter-tumor heterogeneity within a single subtype. Within the basal primary tumors we also identified increased immune dysregulation in individuals with a high proportion of fibroblasts present in the tumor microenvironment. Moreover, cells in HNSCC primary tumors had significantly more heterogeneity across pathways than their matched metastatic cells, consistent with a model of clonal outgrowth. Conclusions: The results of these analyses demonstrate the broad utility of EVAsc to quantify inter- and intra-tumor heterogeneity from scRNA-seq data without reliance on low dimensional visualization. EVAsc is a robust multivariate statistical method to quantify differential variation of pathway gene expression and provides the ability to explore transcriptional variation in numerous disease and normal contexts at a single cell resolution. Accurate characterization of inter-sample variation from scRNA-seq data of tumors is critical to quantify the cellular heterogeneity that drives tumor progression through dysregulation of key cancer pathways. Thus, identifying dysregulated pathways in individual tumors may be an important biomarker for clinical response to immunotherapy. Citation Format: Emily F. Davis-Marcisak, Pranay Orugunta, Genevieve Stein-O'Brien, Sidharth V. Puram, Evanthia Roussos Torres, Alexander Hopkins, Elizabeth M. Jaffee, Alexander V. Favorov, Bahman Afsari, Loyal A. Goff, Elana J. Fertig. Expression variation analysis for tumor heterogeneity in single-cell RNA-sequencing data [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 4697.

Using permutational and multivariate statistics to understand inorganic well water chemistry and the occurrence of methane in groundwater, southeastern New Brunswick, Canada

Concerns over possible impacts from the rapid expansion of unconventional oil and natural gas (ONG) resource development prompted a regional domestic well sampling program focusing on the Carboniferous Maritimes Basin bedrock in southeastern New Brunswick, Canada. This work applies recent developments in robust multivariate statistical methods to overcome issues with highly non-Gaussian data and support the development of a conceptual model for the regional groundwater chemistry and the occurrence of methane. Principal component analysis reveals that the redox-sensitive species, DO, NO3, Fe, Mn, methane, As and U are the most important parameters that differentiate the samples. Permutation-based MANOVA and ANOVA testing revealed that geology was more important than geographic location and topography in influencing groundwater composition. The statistical inferences are supported by chemistry trends observed in relation to road de-icing salt and other saline sources. However, source differentiation between Carboniferous brines, entrapped post-glacial marine water and modern seawater cannot be made. Furthermore, Cl:Br ratios lower than those of seawater or regional brines suggest an origin related to the diagenesis of organic-rich sediment and that the groundwater may be influenced by local low permeability units.Combined spatial, statistical and chemical analysis shows that, while trace or low levels of methane, <1 mg/L, are found ubiquitously throughout the Maritimes Basin, elevated concentrations, >1 mg/L, are associated with the Horton Group, consistent with it being the host and inferred source of ONG resources in the province. The highest methane concentrations (14–29 mg/L) were detected in the region with a complex history of cycles of uplift and erosion which, in some locations, resulted in the juxtaposition at the surface of the Horton Group with several other groups of the Maritimes Basin. It is thought that proximity to the Horton Group can lead to naturally high methane concentrations in non-ONG-bearing units.

Geometric morphometrics reveals shifts in flower shape symmetry and size following gene knockdown of CYCLOIDEA and ANTHOCYANIDIN SYNTHASE

BackgroundWhile floral symmetry has traditionally been assessed qualitatively, recent advances in geometric morphometrics have opened up new avenues to specifically quantify flower shape and size using robust multivariate statistical methods. In this study, we examine, for the first time, the ability of geometric morphometrics to detect morphological differences in floral dorsoventral asymmetry following virus-induced gene silencing (VIGS). Using Fedia graciliflora Fisch. & Meyer (Valerianaceae) as a model, corolla shape of untreated flowers was compared using canonical variate analysis to knockdown phenotypes of CYCLOIDEA2A (FgCYC2A), ANTHOCYANIDIN SYNTHASE (FgANS), and empty vector controls.ResultsUntreated flowers and all VIGS treatments were morphologically distinct from each other, suggesting that VIGS may cause subtle shifts in floral shape. Knockdowns of FgCYC2A were the most dramatic, affecting the position of dorsal petals in relation to lateral petals, thereby resulting in more actinomorphic-like flowers. Additionally, FgANS knockdowns developed larger flowers with wider corolla tube openings.ConclusionsThese results provide a method to quantify the role that specific genes play in the developmental pathway affecting the dorsoventral axis of symmetry in zygomorphic flowers. Additionally, they suggest that ANS may have an unintended effect on floral size and shape.

Neural correlates of visuospatial bias in patients with left hemisphere stroke: a causal functional contribution analysis based on game theory

Stroke patients frequently display spatial neglect, an inability to report, or respond to, relevant stimuli in the contralesional space. Although this syndrome is widely considered to result from the dysfunction of a large-scale attention network, the individual contributions of damaged grey and white matter regions to neglect are still being disputed. Moreover, while the neuroanatomy of neglect in right hemispheric lesions is well studied, the contributions of left hemispheric brain regions to visuospatial processing are less well understood. To address this question, 128 left hemisphere acute stroke patients were investigated with respect to left- and rightward spatial biases measured as severity of deviation in the line bisection test and as Center of Cancellation (CoC) in the Bells Test. Causal functional contributions and interactions of nine predefined grey and white matter regions of interest in visuospatial processing were assessed using Multi-perturbation Shapley value Analysis (MSA). MSA, an inference approach based on game theory, constitutes a robust and exact multivariate mathematical method for inferring functional contributions from multi-lesion patterns. According to the analysis of performance in the Bells test, leftward attentional bias (contralesional deficit) was associated with contributions of the left superior temporal gyrus and rightward attentional bias with contributions of the left inferior parietal lobe, whereas the arcuate fascicle was contributed to both contra- and ipsilesional bias. Leftward and rightward deviations in the line bisection test were related to contributions of the superior longitudinal fascicle and the inferior parietal lobe, correspondingly. Thus, Bells test and line bisection tests, as well as ipsi- and contralesional attentional biases in these tests, have distinct neural correlates. Our findings demonstrate the contribution of different grey and white matter structures to contra- and ipsilesional spatial biases as revealed by left hemisphere stroke. The results provide new insights into the role of the left hemisphere in visuospatial processing.

Robust reduced-rank regression

SummaryIn high-dimensional multivariate regression problems, enforcing low rank in the coefficient matrix offers effective dimension reduction, which greatly facilitates parameter estimation and model interpretation. However, commonly used reduced-rank methods are sensitive to data corruption, as the low-rank dependence structure between response variables and predictors is easily distorted by outliers. We propose a robust reduced-rank regression approach for joint modelling and outlier detection. The problem is formulated as a regularized multivariate regression with a sparse mean-shift parameterization, which generalizes and unifies some popular robust multivariate methods. An efficient thresholding-based iterative procedure is developed for optimization. We show that the algorithm is guaranteed to converge and that the coordinatewise minimum point produced is statistically accurate under regularity conditions. Our theoretical investigations focus on non-asymptotic robust analysis, demonstrating that joint rank reduction and outlier detection leads to improved prediction accuracy. In particular, we show that redescending -functions can essentially attain the minimax optimal error rate, and in some less challenging problems convex regularization guarantees the same low error rate. The performance of the proposed method is examined through simulation studies and real-data examples.

Abstract 3811: Defining the relevant genome in solid tumors

Abstract Decades of cancer research including comprehensive molecular profiling combined with the development of a broad array of targeted therapies have created the opportunity to transform cancer care by implementing precision oncology based approaches. An important element of this system is the widespread availability of robust and cost-effective multi-variate profiling methods to characterize relevant cancer associated molecular alterations in the future. Current commercially available multi-variate profiling methods vary dramatically with regard to the number of cancer genes that are interrogated. Given that many large scale and detailed molecular profiling studies have been completed, the landscape of somatic alterations in solid tumors is reasonably well-known. Furthermore, the specific gene variants that are relevant to application of targeted therapies are also a matter of record. Therefore, we set out to define the number of relevant cancer genes for precision oncology research based on the currently available empirical evidence. To define recurrent somatic alterations in solid tumors, we created a compendium of variant calls from &amp;gt; 15,000 exomes, defined focal amplifications and deletions from &amp;gt; 30,000 arrays, and defined recurrent fusions from several thousand RNAseq profiles. Statistical approaches were implemented to define genes containing recurrent missense mutations (i.e., hotspots), enriched in truncating mutations or subject to recurrent copy number gain/loss or translocation. This gene set was then used to comprehensively search approved cancer drug labels, clinical practice guidelines, and clinical trials to identify records containing published evidence that specific recurrent somatic gene variants were used as part of the indication statement of an approved targeted therapy, were recommended for testing as part of clinical practice for therapeutic decisions, or were used as enrollment criteria in clinical trials. The relevant cancer genome thus defined consists of &amp;lt; 100 genes. These results suggest that targeted multi-variate profiling approaches may be sufficient to support precision oncology goals in the future. Citation Format: Habib R. Hamidi, Santoshi Bandla, Nick Khazanov, Paul William, Nikki Bonevich, Chris Zurenko, Sarah Anstead, Chris Taylor, Reuban Richmonds, David Galimberti, Ken Kopp, Dinesh Cyanam, Michael Hogan, Vinay Mittal, Seth Sadis. Defining the relevant genome in solid tumors [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 3811. doi:10.1158/1538-7445.AM2017-3811

Depth‐weighted robust multivariate regression with application to sparse data

AbstractA robust method for multivariate regression is developed based on robust estimators of the joint location and scatter matrix of the explanatory and response variables using the notion of data depth. The multivariate regression estimator possesses desirable affine equivariance properties, achieves the best breakdown point of any affine equivariant estimator, and has an influence function which is bounded in both the response as well as the predictor variable. To increase the efficiency of this estimator, a re‐weighted estimator based on robust Mahalanobis distances of the residual vectors is proposed. In practice, the method is more stable than existing methods that are constructed using subsamples of the data. The resulting multivariate regression technique is computationally feasible, and turns out to perform better than several popular robust multivariate regression methods when applied to various simulated data as well as a real benchmark data set. When the data dimension is quite high compared to the sample size it is still possible to use meaningful notions of data depth along with the corresponding depth values to construct a robust estimator in a sparse setting. The Canadian Journal of Statistics 45: 164–184; 2017 © 2017 Statistical Society of Canada

A cautionary note on robust covariance plug-in methods

Many multivariate statistical methods rely heavily on the sample covariance matrix. It is well known though that the sample covariance matrix is highly non-robust. One popular alternative approach for "robustifying" the multivariate method is to simply replace the role of the covariance matrix with some robust scatter matrix. The aim of this paper is to point out that in some situations certain properties of the covariance matrix are needed for the corresponding robust "plug-in" method to be a valid approach, and that not all scatter matrices necessarily possess these important properties. In particular, the following three multivariate methods are discussed in this paper: independent components analysis, observational regression and graphical modeling. For each case, it is shown that using a symmetrized robust scatter matrix in place of the covariance matrix results in a proper robust multivariate method.

Modeling soot formation in premixed flames using an Extended Conditional Quadrature Method of Moments

The scope of this study is the application of the recently developed univariate moment method Extended Quadrature Method of Moments (EQMOM) (Yuan et al., 2012) to model soot formation in flames. Furthermore, it is combined with another advanced moment approach, called the Conditional Quadrature Method of Moments (CQMOM) (Yuan and Fox, 2011), and this extension leads to a bivariate model.Retaining the efficiency of a moment method, EQMOM enables the reconstruction of the number density function. CQMOM is a numerically robust multivariate moment method which allows a bivariate soot particle description in terms of particle volume and surface to take into account aggregation. The joint Extended Conditional Quadrature Method of Moments (ECQMOM) model combines the advantages of the two methods to arrive at a numerically efficient bivariate moment method which captures both the particle size distribution and the formation of aggregates.Both the EQMOM and the ECQMOM model are validated against experimental results for premixed burner-stabilized ethylene flames. Thereby, the gas phase is modeled using a modified version of a very detailed, well-established kinetic scheme, which is adapted to be consistent with the moment methods introduced. The results demonstrate the suitability of the applied models to describe both soot precursors and soot evolution in flames. Furthermore, the ability of the moment approaches to represent the statistical soot model accurately is evaluated comparing EQMOM and ECQMOM to other numerical approaches, which are based on the Monte Carlo method, the standard Gaussian Quadrature Method of Moments and the Gaussian-Radau Quadrature Method of Moments, respectively.

Linking watershed geomorphic characteristics to sediment yield: Evidence from the Loess Plateau of China

The geomorphic characteristics of a watershed affect the energy fluxes, mass movement, and water and sediment dispersion within the watershed. This paper examines how watershed complexity affects sediment yield in terms of rainfall and geomorphic characteristics. The geomorphic characteristics include primary, secondary and compound topographic attributes; watershed shape characteristics; relief parameters; and stream network characteristics. Because of the high co-dependence among these characteristics, partial least-squares regression (PLSR) was used to identify the relationships between the sediment yield and 29 selected watershed characteristics. The PLSR combines the features of a principal component analysis and multiple linear regression and is a robust multivariate regression method that is appropriate when the predictors exhibit multiple co-linearity. The first-order factors were determined by calculating the variable importance for the projection (VIP). Those variables with high VIP values are the most relevant for explaining the dependent variable. The results showed that the watershed shape and relief parameters have large influences on the sediment yield. The VIP values revealed that the sediment yield is primarily controlled by the plan curvature (VIP=1.87) and the highest order channel length (VIP=1.53), followed by the hypsometric integral (VIP=1.49), rainfall (VIP=1.44), basin relief (VIP=1.19), slope (VIP=1.15), sediment transport capacity index (VIP=1.13), length ratio (VIP=1.06), profile curvature (VIP=1.01) and divide average relief (VIP=1.00). This paper quantified the effects and relative importance of different geomorphic attributes on sediment yield. The insight provided by these results can be used in the selection of appropriate geomorphic variables for watershed erosion and hydrological models. Thus, this study is intended to elucidate the internal dynamics of sediment transport and storage in a watershed and provide a guide for watershed management.