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Excerpt from The Theory of Matrices, Vol. 2Determination of the index of an arbitrary rational frac tion by the coefficients of numerator and denominator.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

As long ago as 1897 Karl Pearson, in a now classic paper on spurious correlation, first pointed out dangers that may befall the analyst who attempts to interpret correlations between ratios whose numerators and denominators contain common parts. He thus implied that the analysis of compositional data, with its concentration on relation­ ships between proportions of some whole, is likely to be fraught with difficulty. History has proved him correct: over the succeeding years and indeed right up to the present day, there has been no other form of data analysis where more confusion has reigned and where more improper and inadequate statistical methods have been applied. The special and intrinsic feature of compositional data is that the proportions of a composition are naturally subject to a unit-sum constraint. For other forms of constrained data, in particular for directional data where there is a unit-length constraint on each direction vector, scientist and statistician alike have readily appre­ ciated that new statistical methods, appropriate to the special nature of the data, are required; and there now exists an extensive literature on the successful statistical analysis of directional data. It is paradox­ ical that for compositional data, subject to an apparently simpler constraint, such an appreciation and development have been slower to emerge. In applications the unit-sum constraint has been widely ignored or wished away and inappropriate 'standard' statistical methods, devised for and successfully applied to unconstrained data, have been used with disastrous consequences.

As long ago as 1897 Karl Pearson, in a now classic paper on spurious correlation, first pointed out dangers that may befall the analyst who attempts to interpret correlations between ratios whose numerators and denominators contain common parts. He thus implied that the analysis of compositional data, with its concentration on relation­ ships between proportions of some whole, is likely to be fraught with difficulty. History has proved him correct: over the succeeding years and indeed right up to the present day, there has been no other form of data analysis where more confusion has reigned and where more improper and inadequate statistical methods have been applied. The special and intrinsic feature of compositional data is that the proportions of a composition are naturally subject to a unit-sum constraint. For other forms of constrained data, in particular for directional data where there is a unit-length constraint on each direction vector, scientist and statistician alike have readily appre­ ciated that new statistical methods, appropriate to the special nature of the data, are required; and there now exists an extensive literature on the successful statistical analysis of directional data. It is paradox­ ical that for compositional data, subject to an apparently simpler constraint, such an appreciation and development have been slower to emerge. In applications the unit-sum constraint has been widely ignored or wished away and inappropriate 'standard' statistical methods, devised for and successfully applied to unconstrained data, have been used with disastrous consequences.