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Über dieses Buch

This volume presents state of the art theories, new developments, and important applications of Partial Least Square (PLS) methods. The text begins with the invited communications of current leaders in the field who cover the history of PLS, an overview of methodological issues, and recent advances in regression and multi-block approaches. The rest of the volume comprises selected, reviewed contributions from the 8th International Conference on Partial Least Squares and Related Methods held in Paris, France, on 26-28 May, 2014. They are organized in four coherent sections: 1) new developments in genomics and brain imaging, 2) new and alternative methods for multi-table and path analysis, 3) advances in partial least square regression (PLSR), and 4) partial least square path modeling (PLS-PM) breakthroughs and applications. PLS methods are very versatile methods that are now used in areas as diverse as engineering, life science, sociology, psychology, brain imaging, genomics, and business among both academics and practitioners. The selected chapters here highlight this diversity with applied examples as well as the most recent advances.





Chapter 1. Partial Least Squares for Heterogeneous Data

Large-scale data, where the sample size and the dimension are high, often exhibits heterogeneity. This can arise for example in the form of unknown subgroups or clusters, batch effects or contaminated samples. Ignoring these issues would often lead to poor prediction and estimation. We advocate the maximin effects framework (Meinshausen and Bühlmann, Maximin effects in inhomogeneous large-scale data. Preprint arXiv:1406.0596, 2014) to address the problem of heterogeneous data. In combination with partial least squares (PLS) regression, we obtain a new PLS procedure which is robust and tailored for large-scale heterogeneous data. A small empirical study complements our exposition of new PLS methodology.

Peter Bühlmann

Chapter 2. On the PLS Algorithm for Multiple Regression (PLS1)

Partial least squares (PLS) was first introduced by Wold in the mid 1960s as a heuristic algorithm to solve linear least squares (LS) problems. No optimality property of the algorithm was known then. Since then, however, a number of interesting properties have been established about the PLS algorithm for regression analysis (called PLS1). This paper shows that the PLS estimator for a specific dimensionality S is a kind of constrained LS estimator confined to a Krylov subspace of dimensionality S. Links to the Lanczos bidiagonalization and conjugate gradient methods are also discussed from a somewhat different perspective from previous authors.

Yoshio Takane, Sébastien Loisel

Chapter 3. Extending the Finite Iterative Method for Computing the Covariance Matrix Implied by a Recursive Path Model

Given q + p variables (q endogenous variables and p exogenous variables) and the covariance matrix among exogenous variables, how to compute the covariance matrix implied by a given recursive path model connecting these q + p variables? The finite iterative method (FIM) was recently introduced by El Hadri and Hanafi (Electron J Appl Stat Anal 8:84–99, 2015) to perform this task but only when all the variables are standardized (and so the covariance matrix is actually a correlation matrix). In this paper, the extension of FIM to the general case of a covariance matrix case is introduced. Moreover, the computational efficiency of FIM and the well-known Jöreskog’s method is discussed and illustrated.

Zouhair El Hadri, Mohamed Hanafi

Chapter 4. Which Resampling-Based Error Estimator for Benchmark Studies? A Power Analysis with Application to PLS-LDA

Resampling-based methods such as k-fold cross-validation or repeated splitting into training and test sets are routinely used in the context of supervised statistical learning to assess the prediction performances of prediction methods using real data sets. In this paper, we consider methodological issues related to comparison studies of prediction methods which involve several real data sets and use resampling-based error estimators as the evaluation criteria. In the literature papers often claim that, say, “Method 1 performs better than Method 2 on real data” without applying any proper statistical inference approach to support their claims and without clearly explaining what they mean by “perform better.” We recently proposed a new statistical testing framework which provides a statistically correct formulation of such paired tests—which are often performed in the machine learning community—to compare the performances of two methods on several real data sets. However, the behavior of the different available resampling-based error estimation procedures in this statistical framework is unknown. In this paper we empirically assess this behavior through an exemplary benchmark study based on 50 microarray data sets and formulate tentative recommendations regarding the choice of resampling-based error estimation procedures in light of the results.

Anne-Laure Boulesteix

Chapter 5. Path Directions Incoherence in PLS Path Modeling: A Prediction-Oriented Solution

PLS-PM presents some inconsistencies in terms of coherence with the direction of the relationships specified in the path diagram (i.e., the path directions). The PLS-PM iterative algorithm analyzes interdependence among blocks and misses to distinguish explicitly between dependent and explanatory blocks in the structural model. This inconsistency of PLS-PM is illustrated using the simple two-blocks model. For the case of more than two blocks of variables, it is necessary to have a close look at the different criteria optimized by PLS-PM to show this issue. In general, the role of latent variables in the structural model depends on the way the outer weights are calculated. A recently proposed method, called Non-Symmetrical Component-based Path Modeling, which is based on the optimization of a redundancy-related criterion in a multi-block framework, respects the direction of the relationships specified in the structural model. In order to assess the quality of the model, we provide a new goodness-of-fit index based on redundancy criterion and prediction capability. Furthermore, we provide a procedure to address the problem of multicollinearity within blocks of variables.

Pasquale Dolce, Vincenzo Esposito Vinzi, Carlo Lauro

New Developments in Genomics and Brain Imaging


Chapter 6. Imaging Genetics with Partial Least Squares for Mixed-Data Types (MiMoPLS)

“Imaging genetics” studies the genetic contributions to brain structure and function by finding correspondence between genetic data—such as single nucleotide polymorphisms (SNPs)—and neuroimaging data—such as diffusion tensor imaging (DTI). However, genetic and neuroimaging data are heterogenous data types, where neuroimaging data are quantitative and genetic data are (usually) categorical. So far, methods used in imaging genetics treat all data as quantitative, and this sometimes requires unrealistic assumptions about the nature of genetic data. In this article we present a new formulation of Partial Least Squares Correlation (PLSC)—called Mixed-modality Partial Least Squares (MiMoPLS)—specifically tailored for heterogeneous (mixed-) data types. MiMoPLS integrates features of PLSC and Correspondence Analysis (CA) by using special properties of quantitative data and Multiple Correspondence Analysis (MCA). We illustrate MiMoPLS with an example data set from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) with DTI and SNPs.

Derek Beaton, Michael Kriegsman, Joseph Dunlop, Francesca M. Filbey, Hervé Abdi

Chapter 7. PLS and Functional Neuroimaging: Bias and Detection Power Across Different Resampling Schemes

Correlation Partial-Least Squares (PLSC) provides a robust model for analyzing functional neuroimaging data, which is used to identify functional brain networks that show the largest covariance with task stimuli. However, neuroimaging data tend to be high-dimensional (i.e., there are far more variables P than samples N), with significant noise confounds and variability in brain response. It is therefore challenging to identify the significant, stable components of PLSC analysis. Empirical significance estimators are widely used, as they make minimal assumptions about data structure. The most common estimator in neuroimaging PLS is Bootstrapped Variance (BV), which tests whether bootstrap-stabilized mean component eigenvalues (i.e., covariance) are significantly different from a permuted null distribution; however, recent studies have highlighted issues with this model. Two alternatives were proposed that instead focus on reliability of the PLSC saliences (i.e., singular vectors): a Split-half Stability (SS) model that measures the consistency of reconstructed components for split-half data, and Split-half Reproducibility (SR) which measures the reliability across independent split-half analyses. We compare BV, SS, and SR estimators on functional Magnetic Resonance Imaging (f MRI) data, for both simulated and experimental datasets. The SS and SR methods have comparable sensitivity in detecting “brain” components for most simulated and experimental conditions. However, SR shows consistently greater sensitivity for “task” components. We demonstrate that this is due to relative bias in the SS model: both “brain” and “task” components have biased null distributions, but for the low-dimensional “task” vectors, this bias becomes sufficiently high that it is often impossible to distinguish a significant effect from the null distribution.

Nathan Churchill, Babak Afshin-Pour, Stephen Strother

Chapter 8. Estimating and Correcting Optimism Bias in Multivariate PLS Regression: Application to the Study of the Association Between Single Nucleotide Polymorphisms and Multivariate Traits in Attention Deficit Hyperactivity Disorder

In studies involving genetic data, the correlations between X and Y scores obtained from PLS regression models can be used as measures of association between genome-level measurements, X, and phenotype-level measurements, Y. These correlations may be overestimated due to potential overfitting (i.e., they may be vulnerable to optimism bias). We evaluate the optimism bias through simulations and examine the effect of increasing sample size and strength of correlation. We assess the effectiveness of bootstrap-based and permutation-based bias correction methods. We also investigate the selection of the appropriate number of components for PLS regression. We include an analysis of genetic data consisting of genotypes and phenotypes related to Attention Deficit Hyperactivity Disorder (ADHD).

Erica Cunningham, Antonio Ciampi, Ridha Joober, Aurélie Labbe

Chapter 9. Discriminant Analysis for Multiway Data

A multiway Fisher Discriminant Analysis (MFDA) formulation is presented in this paper. The core of MFDA relies on the structural constraint imposed to the discriminant vectors in order to account for the multiway structure of the data. This results in a more parsimonious model than that of Fisher Discriminant Analysis (FDA) performed on the unfolded data table. Moreover, computational and overfitting issues that occur with high dimensional data are better controlled. MFDA is applied to predict the long term recovery of patients after traumatic brain injury from multi-modal brain Magnetic Resonance Imaging. As compared to FDA, MFDA clearly tracks down the discrimination areas within the white matter region of the brain and provides a ranking of the contribution of the neuroimaging modalities. Based on cross validation, the accuracy of MFDA is equal to 77 % against 75 % for FDA.

Gisela Lechuga, Laurent Le Brusquet, Vincent Perlbarg, Louis Puybasset, Damien Galanaud, Arthur Tenenhaus

New and Alternative Methods for Multitable and Path Analysis


Chapter 10. Structured Variable Selection for Regularized Generalized Canonical Correlation Analysis

Regularized Generalized Canonical Correlation Analysis (RGCCA) extends regularized canonical correlation analysis to more than two sets of variables. Sparse GCCA (SGCCA) was recently proposed to address the issue of variable selection. However, the variable selection scheme offered by SGCCA is limited to the covariance (τ = 1) link between blocks. In this paper we go beyond the covariance link by proposing an extension of SGCCA for the full RGCCA model (τ ∈ [0, 1]). In addition, we also propose an extension of SGCCA that exploits pre-given structural relationships between variables within blocks. Specifically, we propose an algorithm that allows structured and sparsity-inducing penalties to be included in the RGCCA optimization problem.

Tommy Löfstedt, Fouad Hadj-Selem, Vincent Guillemot, Cathy Philippe, Edouard Duchesnay, Vincent Frouin, Arthur Tenenhaus

Chapter 11. Supervised Component Generalized Linear Regression with Multiple Explanatory Blocks: THEME-SCGLR

We address component-based regularization of a multivariate Generalized Linear Model (GLM). A set of random responses Y is assumed to depend, through a GLM, on a set X of explanatory variables, as well as on a set T of additional covariates. X is partitioned into R conceptually homogeneous blocks X1, …, X R , viewed as explanatory themes. Variables in each X r are assumed many and redundant. Thus, generalized linear regression demands regularization with respect to each X r . By contrast, variables in T are assumed selected so as to demand no regularization. Regularization is performed searching each X r for an appropriate number of orthogonal components that both contribute to model Y and capture relevant structural information in X r . We propose a very general criterion to measure structural relevance (SR) of a component in a block, and show how to take SR into account within a Fisher-scoring-type algorithm in order to estimate the model. We show how to deal with mixed-type explanatory variables. The method, named THEME-SCGLR, is tested on simulated data, and then applied to rainforest data in order to model the abundance of tree-species.

Xavier Bry, Catherine Trottier, Fréderic Mortier, Guillaume Cornu, Thomas Verron

Chapter 12. Partial Possibilistic Regression Path Modeling

This paper introduces structural equation modeling for imprecise data, which enables evaluations with different types of uncertainty. Coming under the framework of variance-based analysis, the proposed method called Partial Possibilistic Regression Path Modeling (PPRPM) combines the principles of PLS path modeling to model the network of relations among the latent concepts, and the principles of possibilistic regression to model the vagueness of the human perception. Possibilistic regression defines the relation between variables through possibilistic linear functions and considers the error due to the vagueness of human perception as reflected in the model via interval-valued parameters. PPRPM transforms the modeling process into minimizing components of uncertainty, namely randomness and vagueness. A case study on the motivational and emotional aspects of teaching is used to illustrate the method.

Rosaria Romano, Francesco Palumbo

Chapter 13. Assessment and Validation in Quantile Composite-Based Path Modeling

The paper aims to introduce assessment and validation measures in Quantile Composite-based Path modeling. A quantile approach in the Partial Least Squares path modeling framework overcomes the classical exploration of average effects and highlights how and if the relationships among observed and unobserved variables change according to the explored quantile of interest. A final evaluation of the quality of the obtained results both from a descriptive (assessment) and inferential (validation) point of view is needed. The functioning of the proposed method is shown through a real data application in the area of the American Customer Satisfaction Index.

Cristina Davino, Vincenzo Esposito Vinzi, Pasquale Dolce

Advances in Partial Least Square Regression


Chapter 14. PLS-Frailty Model for Cancer Survival Analysis Based on Gene Expression Profiles

Partial least squares (PLS) and gene expression profiling are often used in survival analysis for cancer prognosis; but these approaches show only limited improvement over conventional survival analysis. In this context, PLS has mainly been used in dimension reduction to alleviate the overfitting and collinearity issues arising from the large number of genomic variables. To further improve the cancer survival analysis, we developed a new PLS-frailty model that considers frailty as a random effect when modeling the risk of death. We used PLS regression to generate K PLS components from genomic variables and added the frailty of censoring as a random effect variable. The statistically significant PLS components were used in the frailty model for survival analysis. The genomic components representing the frailty followed a Gaussian distribution. Ten-fold cross-validation was used to evaluate the risk discrimination (between high risk and low risk) and survival prediction based on two breast cancer datasets. The PLS-frailty model performed better than the traditional PLS-Cox model in discriminating between the high and low risk clinical groups. The PLS-frailty model also outperformed the conventional Cox model in discriminating between high and low risk breast cancer patients according to their gene expression profiles.

Yi Zhou, Yanan Zhu, Siu-wai Leung

Chapter 15. Functional Linear Regression Analysis Based on Partial Least Squares and Its Application

Functional linear model with functional predictors and scalar response is a simple and popular model in the field of functional data analysis. The slope function is usually expanded on some basis functions, such as spline and functional principal component (FPC) basis, and then the model can be converted into a multivariate linear model. The FPC basis can keep most variance information of the functional data, but the correlation with response is not considered. Motivated by this, we use partial least square basis to expand the slope function. Meanwhile, considering the functional predictors are not all significant and variable selection procedure is implemented. In this process, group variable selection is introduced to identify the significant predictors. Then the proposed method is used to analyse the relationship between number of monthly emergency patients and some environmental factors in functional form, and some meaningful results are obtained.

Huiwen Wang, Lele Huang

Chapter 16. Multiblock and Multigroup PLS: Application to Study Cannabis Consumption in Thirteen European Countries

We address the problem of investigating the relationships between (K + 1) blocks of variables (i.e., K blocks of independent variables and one block of dependent variables), where the observations are a priori divided into several known groups. We propose a simple procedure called multiblock and multigroup PLS regression—which is a straightforward extension of multiblock PLS regression—that takes into account the group structure of the observations. This method of analysis is illustrated with a large, questionnaire based, survey exploring, in 2011, the cannabis consumption of teenagers of thirteen European countries (the European School Survey Project on Alcohol and other Drugs).

Aida Eslami, El Mostafa Qannari, Stéphane Legleye, Stéphanie Bougeard

Chapter 17. A Unified Framework to Study the Properties of the PLS Vector of Regression Coefficients

In this paper we propose a new approach to study the properties of the Partial Least Squares (PLS) vector of regression coefficients. This approach relies on the link between PLS and discrete orthogonal polynomials. In fact many important PLS objects can be expressed in terms of some specific discrete orthogonal polynomials, called the residual polynomials. Based on the explicit analytical expression we have stated for these polynomials in terms of signal and noise, we provide a new framework for the study of PLS. We show that this approach allows to simplify and retrieve independent proofs of many classical results (proved earlier by different authors using various approaches and tools). This general and unifying approach also sheds light on PLS and helps to gain insight on its properties.

Mélanie Blazère, Fabrice Gamboa, Jean-Michel Loubes

Chapter 18. A New Bootstrap-Based Stopping Criterion in PLS Components Construction

We develop a new universal stopping criterion in components construction, in the sense that it is suitable both for Partial Least Squares Regressions (PLSR) and its extension to Generalized Linear Regressions (PLSGLR). This criterion is based on a bootstrap method and has to be computed algorithmically. It allows to test each successive components on a significant level α. In order to assess its performances and robustness with respect to different noise levels, we perform intensive datasets simulations, with a preset and known number of components to extract, both in the case N > P (N being the number of subjects and P the number of original predictors), and for datasets with N < P. We then use t-tests to compare the predictive performance of our approach to some others classical criteria. Our conclusion is that our criterion presents better performances, both in PLSR and PLS-Logistic Regressions (PLS-LR) frameworks.

Jérémy Magnanensi, Myriam Maumy-Bertrand, Nicolas Meyer, Frédéric Bertrand

PLS Path Modeling: Breaktroughs and Applications


Chapter 19. Extension to the PATHMOX Approach to Detect Which Constructs Differentiate Segments and to Test Factor Invariance: Application to Mental Health Data

In this paper we propose an extension to the PATHMOX segmentation algorithm to detect which endogenous latent variables and predictors are responsible for heterogeneity. We also address the problem of factor invariance in the terminal nodes of PATHMOX. We demonstrate the utility of such methodology on real mental health data by investigating the relationship between dementia, depression and delirium.

Tomas Aluja-Banet, Giuseppe Lamberti, Antonio Ciampi

Chapter 20. Multi-group Invariance Testing: An Illustrative Comparison of PLS Permutation and Covariance-Based SEM Invariance Analysis

This paper provides a didactic example of how to conduct multi-group invariance testing distribution-free multi-group permutation procedure used in conjunction with Partial Least Squares (PLS).To address the likelihood that methods such as covariance-based SEM (CBSEM) with chi-square difference testing can enable group effects that mask noninvariance at lower levels of analysis problem, a variant of CBSEM invariance testing that focuses the evaluation on one parameter at a time (i.e. single parameter invariance testing) is proposed. Using a theoretical model from the field of Information Systems, with three exogenous constructs (routinization, infusion, and faithfulness of appropriation) predicting the endogenous construct of deep usage, the results show both techniques yield similar outcomes for the measurement and structural paths. The results enable greater confidence in the permutation-based procedure with PLS. The pros and cons of both techniques are also discussed.

Wynne W. Chin, Annette M. Mills, Douglas J. Steel, Andrew Schwarz

Chapter 21. Brand Nostalgia and Consumers’ Relationships to Luxury Brands: A Continuous and Categorical Moderated Mediation Approach

This study investigates the role of nostalgia in the consumer-brand relationships in the luxury sector. Results indicate that the nostalgic luxury car brands (vs. futuristic luxury car brands) lead to stronger consumer-brand relationships. Moreover, brand nostalgia has a direct positive effect on brand attachment and separation distress. Brand attachment is also a partial mediator between brand nostalgia and separation distress. In addition, the influence of two moderating variables is examined. We show that past temporal orientation reinforces the relationship between (1) brand nostalgia and brand attachment, and between (2) brand nostalgia and separation distress. Finally, consumers’ need for uniqueness reinforces the relationship between brand attachment and separation distress. On a methodological side, the study shows the ability of the PLS approach to handle higher order latent variables both in the context of continuous and categorical latent moderated mediation variables.

Aurélie Kessous, Fanny Magnoni, Pierre Valette-Florence

Chapter 22. A Partial Least Squares Algorithm Handling Ordinal Variables

The partial least squares (PLS) is a popular path modeling technique commonly used in social sciences. The traditional PLS algorithm deals with variables measured on interval scales while data are often collected on ordinal scales. A reformulation of the algorithm, named Ordinal PLS (OrdPLS), is introduced, which properly deals with ordinal variables. Some simulation results show that the proposed technique seems to perform better than the traditional PLS algorithm applied to ordinal data as they were metric, in particular when the number of categories of the items in the questionnaire is small (4 or 5) which is typical in the most common practical situations.

Gabriele Cantaluppi, Giuseppe Boari


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