Principal component analysis (pca) is a multivariate technique that analyzes a data table in which observations are described by several inter-correlated quantitative dependent variables. Its goal is to extract the important information from the table, to represent it as a set of new orthogonal variables called principal components, and to display the pattern of similarity of the observations and of the variables as points in maps. The quality of the pca model can be evaluated using cross-validation techniques such as the bootstrap and the jackknife. Pca can be generalized as correspondence analysis (ca) in order to handle qualitative variables and as multiple factor analysis (mfa) in order to handle heterogenous sets of variables. Mathematically, pca depends upon the eigen-decomposition of positive semi-definite matrices and upon the singular value decomposition (svd) of rectangular matrices.

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Multiple factor analysis (MFA, also called multiple factorial analysis) is an extension of principal component analysis (PCA) tailored to handle multiple data tables that measure sets of variables collected on the same observations, or, alternatively, (in dual‐MFA) multiple data tables where the same variables are measured on different sets of observations. MFA proceeds in two steps: First it computes a PCA of each data table and ‘normalizes’ each data table by dividing all its elements by the first singular value obtained from its PCA. Second, all the normalized data tables are aggregated into a grand data table that is analyzed via a (non‐normalized) PCA that gives a set of factor scores for the observations and loadings for the variables. In addition, MFA provides for each data table a set of partial factor scores for the observations that reflects the specific ‘view‐point’ of this data table. Interestingly, the common factor scores could be obtained by replacing the original normalized data tables by the normalized factor scores obtained from the PCA of each of these tables. In this article, we present MFA, review recent extensions, and illustrate it with a detailed example. WIREs Comput Stat 2013, 5:149–179. doi: 10.1002/wics.1246This article is categorized under: Data: Types and Structure > Categorical Data Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

Partial least square (PLS) methods (also sometimes called projection to latent structures) relate the information present in two data tables that collect measurements on the same set of observations. PLS methods proceed by deriving latent variables which are (optimal) linear combinations of the variables of a data table. When the goal is to find the shared information between two tables, the approach is equivalent to a correlation problem and the technique is then called partial least square correlation (PLSC) (also sometimes called PLS-SVD). In this case there are two sets of latent variables (one set per table), and these latent variables are required to have maximal covariance. When the goal is to predict one data table the other one, the technique is then called partial least square regression. In this case there is one set of latent variables (derived from the predictor table) and these latent variables are required to give the best possible prediction. In this paper we present and illustrate PLSC and PLSR and show how these descriptive multivariate analysis techniques can be extended to deal with inferential questions by using cross-validation techniques such as the bootstrap and permutation tests.

International audienc

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Sensitive periods in human development have often been proposed to explain age-related differences in the attainment of a number of skills, such as a second language (L2) and musical expertise. It is difficult to reconcile the negative consequence this traditional view entails for learning after a sensitive period with our current understanding of the brain’s ability for experience-dependent plasticity across the lifespan. What is needed is a better understanding of the mechanisms underlying auditory learning and plasticity at different points in development. Drawing on research in language development and music training, this review examines not only what we learn and when we learn it, but also how learning occurs at different ages. First, we discuss differences in the mechanism of learning and plasticity during and after a sensitive period by examining how language exposure versus training forms language-specific phonetic representations in infants and adult L2 learners, respectively. Second, we examine the impact of musical training that begins at different ages on behavioral and neural indices of auditory and motor processing as well as sensorimotor integration. Third, we examine the extent to which childhood training in one auditory domain can enhance processing in another domain via the transfer of learning between shared neuro-cognitive systems. Specifically, we review evidence for a potential bi-directional transfer of skills between music and language by examining how speaking a tonal language may enhance music processing and, conversely, how early music training can enhance language processing. We conclude with a discussion of the role of attention in auditory learning for learning during and after sensitive periods and outline avenues of future research.