Ordination_sections_1.3+1.4_PCoA_Eng.pdf - 1.3 Principal coordinate analysis Pierre Legendre Département de sciences biologiques Université de

Ordination_sections_1.3+1.4_PCoA_Eng.pdf - 1.3 Principal...

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Pierre Legendre Département de sciences biologiques Université de Montréal 1.3. Principal coordinate analysis © Pierre Legendre 2018
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Principal coordinate analysis ( PCoA ) An ordination method preserving any dissimilarity measure D among the objects, except nonmetric indices 1 . Also called classical multidimensional scaling (cMDScale) or metric multidimensional scaling, by opposition to nonmetric multidimensional scaling (nMDS). Mathematical properties of data Data can be of any mathematical type : quantitative, semi- quantitative, qualitative, or mixed. Definition of principal coordinate analysis 1 Nonmetric D indices, not used in ecology, can produce –Inf values due to division by 0.
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Principal coordinate analysis The objective of PCoA is • to represent the points in a full-dimensional Euclidean space • that allows one to fully reconstruct the original dissimilarities among the objects. Applications of PCoA • Produce ordinations of the objects in reduced 2-D (or sometimes 3-D) space. Act as a data transformation after computation of an appropriately chosen dissimilarity measure. The coordinates of the objects in full-dimensional PCoA space represent the transformed data. They can be used as input to RDA or other methods of data analysis.
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Principal coordinate analysis Definitions Euclidean space : a space in which the distance among points is measured by the Euclidean distance (Pythagora’s formula). Synonym: Cartesian space. Euclidean property : a dissimilarity coefficient is Euclidean if any resulting dissimilarity matrix can be fully represented in a Euclidean space without distortion (Gower & Legendre 1986). Criterion: Principal coordinate analysis (PCoA) of such a dissimilarity matrix does not produce negative eigenvalues. Euclidean space, Euclidean property
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PCoA: Computation steps Example – The same data as for PCA: Y = 2 1 3 4 5 0 7 6 9 2 Compute a Euclidean distance matrix among the objects (rows of Y ) D = 0.000 3.162 3.162 7.071 7.071 3.162 0.000 4.472 4.472 6.325 3.162 4.472 0.000 6.325 4.472 7.071 4.472 6.325 0.000 4.472 7.071 6.325 4.472 4.472 0.000 PCoA can analyse any other type of dissimilarity matrix.
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Principal coordinate analysis Apply the transformation and centring proposed by Gower: 1. Transform each value D ij into 2. Centring: make the sums of rows and columns equal to 0 0.5 D ij 2 G = 12.8 4.8 4.8 11.2 11.2 4.8 6.8 3.2 0.8 9.2 4.8 3.2 6.8 9.2 0.8 11.2 0.8 9.2 14.8 4.8 11.2 9.2 0.8 4.8 14.8
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Eigen-decomposition of matrix G : Eigenvectors Eigenvalues: λ 1 = 36, λ 2 = 20 U = 0.596 0.000 0.224 5.000 0.224 5.000 0.522 5.000 0.522 5.000 Norm each eigenvector to the square root of its eigenvalue: multiply the values in each eigenvector by sqrt(eigenvalue) Matrix of principal coordinates: Pr.coo = 3.578 0.000 1.342 2.236 1.342 2.236 3.130 2.236 3.130 2.236
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-4 -2 0 2 4 -3 -2 -1 0 1 2 3 Axis.2 1 2 3 4 5
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