Geostatistical analysis of compositional data.

*(English)*Zbl 1105.86004A composition is a random vector whose components add up to one. Compositional data are a sample drawn from a composition. This book presents traditional geostatistics for compositional data, but it is intended as a “state-of-the-art” book rather than as a textbook. It gives the general framework for regionalized compositions as well as for cokriging.

Due to the constant sum constraint, there appear spurious spatial correlations among the components of a composition, and, moreover, the covariance matrix becomes singular. The way out of this dilemma is shown by J. Aitchison in his 1986 monograph “The statistical analysis of compositional data” [London: Chapman-Hall, Zbl 0688.62004] : from any nonnegative random vector \(W = (W_1,\dots,W_D)'\) a composition can always be derived by \(Z_i =W_i/(W_1 + \cdots + W_D)\). Then always \(W_i/W_j = Z_i/Z_j\), i.e. compositional data only contain information about ratios. Hence, no correlation among compositional data is understood as no correlation among the ratios. Since stochastics for ratios is difficult to handle, Aitchison’s suggestion is to evaluate logarithmic ratios. So, the present book combines geostatistics in the sense of Matheron’s theory of regionalized variables with Aitchison’s theory of statistical analysis of compositions.

Contents: 1. Introduction; 2. Regionalized compositions; 3. Spatial covariance structure; 4. Concepts of null correlation; 5. Cokriging; 6. Practical aspects of compositional data analysis; 7. Application to real data.

Due to the constant sum constraint, there appear spurious spatial correlations among the components of a composition, and, moreover, the covariance matrix becomes singular. The way out of this dilemma is shown by J. Aitchison in his 1986 monograph “The statistical analysis of compositional data” [London: Chapman-Hall, Zbl 0688.62004] : from any nonnegative random vector \(W = (W_1,\dots,W_D)'\) a composition can always be derived by \(Z_i =W_i/(W_1 + \cdots + W_D)\). Then always \(W_i/W_j = Z_i/Z_j\), i.e. compositional data only contain information about ratios. Hence, no correlation among compositional data is understood as no correlation among the ratios. Since stochastics for ratios is difficult to handle, Aitchison’s suggestion is to evaluate logarithmic ratios. So, the present book combines geostatistics in the sense of Matheron’s theory of regionalized variables with Aitchison’s theory of statistical analysis of compositions.

Contents: 1. Introduction; 2. Regionalized compositions; 3. Spatial covariance structure; 4. Concepts of null correlation; 5. Cokriging; 6. Practical aspects of compositional data analysis; 7. Application to real data.

Reviewer: Wolfgang NĂ¤ther (Freiberg)