Thomas, C.W. and Aitchison, J.

68. Can we identify possible compositional processes from the limestone data?Aitchison and Thomas (1998) explained how modern compositional data analysiscould be used to investigate identify and analyse possible compositional processes,using two simple examples: Arctic lake sediments to identify the process in relation todepth and simple olivine data to identify possible equilibrium relationships. Thestatistical technique used depended largely on compositional regression analysis andlogcontrast principal component analysis. In a complementary paper Aitchison andBarceló-Vidal (2002) used compositional singular value decompositions to provide astaying-in-the-simplex approach where the compositions of the data set are expressedas power-perturbation combinations. Thus for an N × D compositional data matrixwith nth row the composition x nthe singular value composition provides theexpressionx = ξ$ ⊕ ( u1p1 ⊗ b1 ) ⊕ . .. ⊕ ( u p ⊗ b ) ,n n nR R Rwhere ξ $ is the estimate of the centre of the data set, and pi ( i = 1 , . . . , R)are positive‘singular values’ in descending order of magnitude, the bi ( i = 1 , . . . , R)areorthogonal compositions, R is a readily defined rank of the compositional data set andthe u’s are power components specific to each composition. . In practice R iscommonly D − 1, the full dimension of the simplex. In a way similar to that for dataDsets in R we may consider an approximation of order r < R to the compositionaldata set given bySuch an approximation retains a proportionx ( r ) = $ ⊕ ( u p ⊗ b ) ⊕ . .. ⊕ ( u p ⊗ b ).nξn1 1 1nr r r2 2 2 2( p + .. . + p ) / ( p + . .. + p )1rof the total variability of the N × D compositional data matrix as measured by thetrace of the estimated centered logratio covariance matrix or equivalently in terms ofthe total mutual squared distances as−1 2{ N ( N − 1)} ∆ ( x , x ).D∑m