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This chapter examines methodologies for dimensional analysis and linear decomposition of multivariate data sets, and discusses their implicit hypotheses and interpretations for muscle coordination of movement. It presents tutorials to compare how two common methods—principal components analysis (PCA) and non-negative matrix factorization (NMF)—decompose electromyographic signals into underlying components. To facilitate the integration of such mathematical techniques with physiological hypothesis testing, the chapter focuses on developing an intuitive understanding to the two techniques. It provides a simple two-dimensional tutorial, focusing on how orthogonality constraints in PCA and non-negativity constraints in NMF impact the resulting data decomposition and physiological relevance. Examples are presented using real data sets from human balance control and locomotion. The chapter examines the structure of the resulting components, their robustness across tasks, and their implications for various muscle synergy hypotheses. The chapter addresses practical issues and caveats in organizing datasets, the selection of the appropriate number of components, and considerations and pitfalls of experimental design and analysis, as well as offering suggestions and cautions for interpreting results. Based on these comparisons and on the work in the visual system over the last decade, evidence is presented for the increased neurophysiological relevance of the factors derived from NMF compared to PCA.

Languages are traditionally subdivided into geographically distinct dialects, although any such division is just a coarse approximation of a more fine-grained variation. This underlying variation is usually visualised in the form of maps, where the distribution of various features is shown as isoglosses. Component models such as factor analysis can be used to analyse spatial distributions of a large number of different features — such as the isogloss data in a dialect atlas or the distributions of ethnological or archaeological phenomena — with the goal of finding dialects or similar cultural aggregates. However, there are several such methods, and it is not obvious how their differences affect their usability for computational dialectology. This chapter addresses this question by comparing five such methods (factor analysis, non-negative matrix factorisation, aspect Bernoulli, independent component analysis, and principal components analysis) with two data sets describing Finnish dialectal variation. There are some fundamental differences between these methods, and some of these have implications that affect the dialectological interpretation of the results.