Interpretation of multivariate outliers for compositional data

Abstract

Compositional data—and most data in geochemistry are of this type—carry relative rather than absolute information. For multivariate outlier detection methods this implies that not the given data but appropriately transformed data need to be used. We use the isometric logratio (ilr) transformation, which seems to be generally the most proper one for theoretical and practical reasons. In this space it is difficult to interpret the outliers, because the reason for outlyingness can be complex. Therefore we introduce tools that support the interpretation of outliers by representing multivariate information in biplots, maps, and univariate scatterplots.

Introduction

In many practical applications from geosciences one has to deal with compositional data, i.e., with multivariate observations describing quantitatively the parts of some whole. Thus, their components carry exclusively relative information about the parts (Aitchison, 1986). Typically these observations are expressed as data with a constant sum constraint such as proportions, percentages, or mg/kg. Standard statistical methods usually fail when they are applied directly to compositional data (Filzmoser and Hron, 2008, Filzmoser et al., 2009, Hron et al., 2010). Many authors appear to be under the impression that the main reason lies in a nonnormal distribution of the samples (for example, of chemical elements in a rock) and thus recommend applying a logarithmic transformation in order to achieve normality (Reimann et al., 2008) of the data set. Depending on the transformation chosen, normality can often be reached, so that the data pass a statistical test. However, the reality is more difficult. The original data follow, in fact, another geometry (usually called the Aitchison geometry; see, e.g., Egozcue and Pawlowsky-Glahn, 2006, for details) on the sample space of compositions, the simplex, defined for a D-part composition $\mathbf{x}=(x_1,\dots ,x_D)\prime$ as

$${\mathcal{S}}^D=\left\{\mathbf{x}=(x_1,\dots ,x_D)\prime ,x_i>0,i=1,\dots ,D,\sum _{i=1}^D{x}_i=\kappa \right\}\text{.}$$

The positive constant $\kappa$ stands for 1 in the case of proportions, 100 for percentages, or 10⁶ for mg/kg.

Due to the fact that geochemical data follow the Aitchison geometry, standard statistical methods that rely mostly on the Euclidean geometry cannot be used for raw compositional data. Whether or not the data follow a normal distribution is of no importance at all. To transform the data to the Euclidean space, the family of logratio transformations from the simplex S^D to the Euclidean real space was proposed. Only by following such a transformation is the use of the standard statistical methods possible. The three main types in this family of transformations are the additive logratio (alr), the centered logratio (clr), and the isometric logratio (ilr) transformation. The alr (Aitchison, 1986) is simple and could be used in the context of outlier detection. However, it is not recommended because it does not result in an orthogonal basis system, which is necessary for diagnostic tools following outlier detection. The clr (Aitchison, 1986) results in data singularity, which is in conflict with the usual tools for outlier detection. The ilr (Egozcue et al., 2003) is recommended because it forms a one-to-one relation between the Aitchison geometry on the simplex and the standard Euclidean geometry, with excellent geometrical properties.

The D−1 ilr variables are coordinates of an orthonormal basis on the simplex (with respect to the Aitchison geometry); thus a proper choice of the basis seems to be crucial for their interpretation. Here the big step ahead is the sequential binary partition procedure (Egozcue and Pawlowsky-Glahn, 2005), which enables interpretation of the orthonormal coordinates in the sense of balances between groups of compositional parts. Additionally, each ilr variable explains all the logratios, i.e., terms of type $\ln (x_i/x_j),i,j=1,\dots ,D$, between parts of the corresponding groups (Fišerová and Hron, in press); conversely, each logratio in the composition is exclusively explained by one balance. This point of view seems to be meaningful, because the definition of compositions implies that the only relevant information is contained in (log)ratios of compositional parts. Although the sequential binary partition can also be made to measure for the concrete geochemical problems (Buccianti et al., 2008), in practice the following D choices of the orthonormal bases seem to be the most useful (Egozcue et al., 2003, Hron et al., 2010). Explicitly, we obtain (D−1)-dimensional real vectors ${\mathbf{z}}^{(l)}=(z_1^{(l)},\dots ,z_{D-1}^{(l)})\prime ,l=1,\dots ,D$,

$$z_i^{(l)}=\sqrt{\frac{D-i}{D-i+1}}\ln \frac{x_i^{(l)}}{\sqrt[D-i]{{\prod }_{j=i+1}^D{x}_j^{(l)}}},i=1,\dots ,D-1\text{,}$$

where $(x_1^{(l)},x_2^{(l)},\dots ,x_l^{(l)},x_{l+1}^{(l)},\dots ,x_D^{(l)})$ stands for a permutation of the parts $(x_1,\dots ,x_D)$ such that the lth compositional part always fills the first position, $(x_l,x_1,\dots ,x_{l-1},x_{l+1},\dots ,x_D)$. In such a configuration, the first ilr variable $z_1^{(l)}$ explains all the relative information (logratios) about the original compositional part x_l; the coordinates $z_2^{(l)},\dots ,z_{D-1}^{(l)}$ then explain the remaining logratios in the composition (Fišerová and Hron, in press). Note that the only important position is that of $x_1^{(l)}$ (because it can be fully explained by $z_1^{(l)}$), the other parts can be chosen arbitrarily, because different ilr transformations are orthogonal rotations of each other (Egozcue et al., 2003). Of course, we cannot say that $z_1^{(l)}$ is the original compositional part x_l, but it explains all the information concerning x_l; thus, it stands for x_l.

An interesting consequence follows for the known clr transformation from ${\mathcal{S}}^D$ to ${\mathbf{R}}^D$, resulting in

$$\mathbf{y}=(y_1,\dots ,y_D)\prime =\left(\ln \frac{x_1}{\sqrt[D]{{\prod }_{i=1}^D{x}_i}},\dots ,\ln \frac{x_D}{\sqrt[D]{{\prod }_{i=1}^D{x}_i}}\right)\prime \text{.}$$

It is easy to see that there exists a linear relationship between y_l and $z_1^{(l)}$, namely

$$y_l=\sqrt{\frac{D}{D-1}}{z}_1^{(l)}\text{.}$$

Thus, up to a constant, the single clr variables have the same interpretation as the corresponding ilr coordinates: they explain all logratios concerning the lth compositional part. However, as a consequence, some of the logratios are explained more than once by the D clr variables (in contrast to the ilr transformation). This is also an intuitive reason for the resulting singularity $y_1+\cdots +y_D=0$ of clr variables, which makes, e.g., the use of robust multivariate statistical methods not possible. On the other hand, the clr transformation is a cornerstone of the compositional biplot (Aitchison and Greenacre, 2002), which will be employed further in the paper.

From the above-mentioned properties of logratio transformations, it is visible that the ordered D-tuple of the ilr coordinates, $z_1^{(l)},l=1,\dots ,D$, can be obtained from clr-transformed data as $\sqrt{(D-1)/D}\mathbf{y}$. Nevertheless, note that it would be not meaningful to interpret the relations between the clr variables or even between the variables $z_1^{(l)}$ using their correlation structure; here the subcompositional incoherence (which means that the results of statistical modeling might be incompatible if only a subset of the parts were used; see, e.g., Aitchison, 1986, Filzmoser et al., 2010, for details) could lead to wrong conclusions. Some kind of “incompatibility” is obtained also for the single $z_1^{(l)},l=1,\dots ,D$, variables; however, here it is as a natural consequence of the fact that the information available in $\mathbf{x}$ was reduced just to a subcomposition.

In contrast to univariate outliers, whose identification as extreme observations is straightforward, for multivariate outliers the covariance structure of the data set needs to be considered as well (Filzmoser et al., 2005). Moreover, when working with compositional data, one has to consider the data structure in view of the Aitchison geometry; see, e.g., Hron et al. (2010) and Filzmoser and Hron (in press). For example, an elliptical point cloud arising from a multivariate normal distribution in the usual Euclidean geometry can look very different in the Aitchison geometry, depending on its position in space (see, for instance, the back-transformed ellipses in Fig. 1B to the Aitchison geometry in Fig. 1A). This is important for multivariate outlier detection methods, which are usually based on distances from an elliptically symmetric distribution. Moreover, each compositional data point can be shifted along the line from the origin through the point without changing the ratios of the compositional parts. Formally, an observed composition $\mathbf{x}=(x_1,\dots ,x_D)\prime$ is defined as a member of the corresponding equivalence class of $\mathbf{x}$,

$$\underline{\mathbf{x}}=\{c\mathbf{x},c\in {\mathbf{R}}^{+}\}\text{.}$$

Thus, two compositions that are elements of the same equivalence class $\underline{\mathbf{x}}$ (we call them also compositionally equivalent; see Egozcue and Pawlowsky-Glahn, 2006) contain the same information and have zero Aitchison (1986) distance. From this point of view, the “extremeness” of the outliers can be even more misleading than in case of standard (Euclidean) multivariate outliers.

The methods for outlier detection of compositional data will be discussed in Section 2, where both theoretical aspects and possibilities for graphical representations will be considered. Section 3 proposes several tools for the interpretation of multivariate outliers that have been implemented in the statistical software environment R (R Development Core Team, 2011). In Section 4 we show how the tool is used for real problems, and how results can be interpreted. Section 5 concludes.