A Regression Model for Compositional Data Based on the Shifted-Dirichlet Distribution

Abstract

Using an approach based on the Aitchison geometry of the simplex, a Shifted-Dirichlet covariate model is obtained. Allowing the parameters to change linearly with a set of covariates, their effects on the relative contributions of different components in a composition are assessed. An application of this model to sedimentary petrography is given.

1 Introduction

Compositional data are vectors of parts of some whole which carry relative information. They are frequently represented as proportions or percentages, which are subject to a constant sum, $\kappa$, i.e., $\kappa =1$ or $\kappa =100$. Their sample space is then represented by the simplex, denoted by

$$ {\mathscr {S}}^D=\{\mathbf {x}=(x_1,\ldots ,x_D),\; x_i>0,\;\sum _{i=1}^Dx_i=\kappa \}. $$

Compositional data occur in many applied fields: from geology and biology to election forecast, from medicine and psychology to economic studies.

We recall briefly the essential elements of simplicial algebra, as it will be used later. For any vector of D strictly positive real components,

$$\begin{aligned} \mathbf {z}=\left( z_1,\ldots ,z_D\right) \in \mathbb {R}^D_+ \quad z_i>0, \; \text {for all} \; i=1,\ldots ,D, \end{aligned}$$

the closure operation of $\mathbf {z}$ is defined as

$$\begin{aligned} {\mathscr {C}}(\mathbf {z})=\left( \frac{\kappa \;z_1}{\sum _{i=1}^Dz_i},\ldots ,\frac{ \kappa \; z_D}{\sum _{i=1}^Dz_i}\right) \in {\mathscr {S}}^D\,. \end{aligned}$$

(1)

where $\kappa$ is the sum of the components, i.e., the constraint.

The two basic operations required for a vector space structure of the simplex are perturbation: given two compositions $\mathbf {x}$ and $\mathbf {y}\in {\mathscr {S}}^D$,

$$\begin{aligned} \mathbf {x}\oplus \mathbf {y}={\mathscr {C}}\left( x_1y_1,\ldots ,x_Dy_D\right) , \end{aligned}$$

(2)

and powering: given a composition $\mathbf {x}\in {\mathscr {S}}^D$ and a scalar $\alpha \in \mathbb {R}$,

$$\begin{aligned} \alpha \odot \mathbf {x}={\mathscr {C}}\left( x_1^{\alpha },\ldots ,x_D^{\alpha }\right) \,. \end{aligned}$$

(3)

Furthermore, an inner product $\langle {\cdot },{\cdot }\rangle _a$ is defined as

$$\begin{aligned} \langle {\mathbf {x}},{\mathbf {y}}\rangle _a=\sum _{i=1}^D\ln \frac{x_i}{\mathrm {g}_{\mathrm {m}}(\mathbf {x})}\ln \frac{y_i}{\mathrm {g}_{\mathrm {m}}(\mathbf {y})}, \end{aligned}$$

(4)

where $\mathrm {g}_{\mathrm {m}}(\mathbf {x})$ denotes the geometric mean of the components of $\mathbf {x}$ [4, 22]. As shown in Pawlowsky-Glahn and Egozcue [22] the simplex $({\mathscr {S}}^D,\oplus ,\odot ,\langle {\cdot },{\cdot }\rangle _a)$ has a $(D-1)$-dimensional real Euclidean vector space structure called simplicial or Aitchison geometry.

Let $\left( \mathbf {e}_1, \mathbf {e}_2, \ldots , \mathbf {e}_{D-1}\right)$ be an orthonormal basis of the simplex and consider the $(D-1)\times D$ matrix $\varvec{\varPsi }$ which rows are $\varPsi _i = \mathrm {clr}(\mathbf {e}_i)$, $(i = 1,\ldots ,D-1)$. Note that $\mathrm {clr}$ is the centered log-ratio transformation, a function from ${\mathscr {S}}^D$ to $\mathbb {R}^D$ defined as

$$ \mathrm {clr}(\mathbf {x})=\left( \log \frac{x_1}{\mathrm {g}_{\mathrm {m}}(\mathbf {x})}, \ldots , \log \frac{x_D}{\mathrm {g}_{\mathrm {m}}(\mathbf {x})}\right) , $$

where $\mathrm {g}_{\mathrm {m}}(\mathbf {x})$ is the geometric mean of the D components of $\mathbf {x}$. The $\varvec{\varPsi }$ matrix is called contrast matrix associated with the orthonormal basis $\left( \mathbf {e}_1, \ldots , \mathbf {e}_{D-1}\right)$. Each row is called a (log)contrast.

The isometric log-ratio transformation, $\mathrm {ilr}$ for short, of $\mathbf {x}$ is the function $\mathrm {ilr}: {\mathscr {S}}^D\rightarrow \mathbb {R}^{D-1}$, which assigns coordinates $\mathbf {x}^{\star }$, with respect to the given basis, to the composition $\mathbf {x}$. The vector $\mathbf {x}^{\star }$ contains the $D-1$ $\mathrm {ilr}$-coordinates of $\mathbf {x}$ in a Cartesian coordinate system. The inverse of the $\mathrm {ilr}$-transformation is denoted by $\mathrm {ilr}^{-1}$. The function $\mathrm {ilr}$ is an isometry of vector spaces. The $\mathrm {ilr}$ transformation is computed as a simple matrix product:

$$\begin{aligned} \mathbf {x}^{\star }=\mathrm {ilr}(\mathbf {x})=\ln (\mathbf {x}) \varvec{\varPsi }^{\prime }\,. \end{aligned}$$

Inversion of $\mathrm {ilr}$, i.e., recovering the composition from its coordinates, is given by

$$\begin{aligned} \mathbf {x}=\mathrm {ilr}^{-1}(\mathbf {x}^{\star })=\mathscr {C}\left( \exp (\varvec{\varPsi }\; \mathbf {x}^{\star })\right) \,. \end{aligned}$$

Given an orthonormal basis of the simplex, any composition $\mathbf {x} \in {\mathscr {S}}^D$ can be expressed as a linear combination,

$$\begin{aligned} \begin{aligned} \mathbf {x}&= (x^{\star }_1 \odot \mathbf {e}_1)\oplus (x^{\star }_2 \odot \mathbf {e}_2)\oplus \cdots \oplus (x^{\star }_{D-1} \odot \mathbf {e}_{D-1})\\&= \bigoplus _{i=1}^{D-1} (x^{\star }_i \odot \mathbf {e}_i), \end{aligned} \end{aligned}$$

where the symbol $\bigoplus$ represents repeated perturbation. The coefficients of the linear combination, for a fixed basis, are uniquely determined, given that in a Euclidean space any point can always be represented in a unique way by its coordinates with respect to an orthonormal basis. Once an orthonormal basis has been chosen, all standard statistical methods can be applied to coordinates and transferred to the simplex preserving their properties [15].

A natural measure on ${\mathscr {S}}^D$, called Aitchison measure, can be defined using orthonormal coordinates [21, 23], that is, the Aitchison measure of a subset on the simplex is the Lebesgue measure of the subset in the space of orthonormal coordinates. This measure is compatible with the Aitchison geometry and is absolutely continuous with respect to the Lebesgue measure on the D-dimensional real space. The relationship between them is $\sqrt{D}x_1x_2\ldots x_D$. The change of the reference measure has some important implications, for example to compute the expected value (see [16] for an in-depth discussion).

Historically, there are essentially two different approaches to regression models which relate a compositional response variable with a system of covariates: Simplicial regression and Dirichlet regression. The former is based on the Aitchison’s theoretical result that if a compositional vector follows an additive logistic normal distribution, the log-ratio transformed vector will follow a normal distribution [2, 3, 8]; the latter follows the stay-in-the-simplex approach. It assumes that the response variable follows a Dirichlet distribution whose parameters are a log-linear function of a set of covariates [6, 12, 14, 17]. Other solutions, present in the literature but less used, involve models based on the generalized Liouville distribution [25].

The Scaled-Dirichlet distribution is an extension of the Dirichlet one. Given that we work here with the Aitchison geometry of the simplex, and that within this framework it is a perturbation of a standard Dirichlet [18], we will refer hereafter to it as the Shifted-Dirichlet distribution. The reason to change this terminology is twofold. On the one hand, working within the Aitchison geometry implies a change of the reference measure; on the other hand, scaling in this geometry is achieved using a power transformation, which allows another extension already studied in Monti et al. [19]. In summary, the name of the distribution indicates the sample space of the corresponding random vector and its structure. For the Scaled-Dirichlet distribution this is the simplex as a subset of real space with the induced Euclidean geometry, while for the Shifted-Dirichlet distribution it is the simplex as a Euclidean space endowed with the Aitchison geometry. Although in the first case the Lebesgue reference measure is used, and in the second the Aitchison measure, the probability assigned to any measurable subset of the simplex is the same.

The Shifted-Dirichlet covariate model is an extension of the Dirichlet one, based on the algebraic geometric structure of the simplex. The assumption is that $\mathbf {x} = (x_1,\ldots , x_D)$ is a compositional response vector, with D components having a Shifted-Dirichlet distribution, in which the parameters $\varvec{\alpha }$ are allowed to change with a set of covariates.

The paper is structured as follows. Section 2 defines the two existing approaches: Simplicial regression and Dirichlet regression. Section 3 gives a brief overview of the Shifted-Dirichlet distribution and describes the Shifted-Dirichlet covariate model, dealing with the issue of parameter estimation. Section 4 presents an example of application of the proposed regression model to sedimentary petrography, in particular bulk petrography and heavy-mineral data of Pleistocene sands (Regione Lombardia cores; Po Plain); this dataset is described in Garzanti et al. [11].

2 Regression Models for Compositional Response Variable

2.1 Simplicial Regression

Linear regression with compositional response variable can be stated as follows. A compositional sample of n independent observations, denoted by $\mathbf {x}_1,\ldots , \mathbf {x}_n$, is available. Each data point, $\mathbf {x}_j,\; (j=1,\ldots ,n)$ is associated with one or more external variables or covariates grouped in the vector $\mathbf {s}_j= (s_{j0}, s_{j1}, \ldots , s_{jm},\ldots ,s_{jp})$, where $s_{j0}=1$ by convention.

The basic idea of Simplicial regression [8] relies on the principle of working on coordinates: once a basis is chosen, the associated $\mathrm {ilr}$ coordinates are computed and the classical regression of the $\mathrm {ilr}$ coordinates on the covariates is performed. Through the inverse $\mathrm {ilr}$-transformation, the back-transformed coefficient vectors, as well as predictions and confidence intervals are obtained.

The general model can be expressed as

$$\begin{aligned} \hat{\mathbf {x}} (\mathbf {s})=\left( s_0 \odot \varvec{\delta }_0 \right) \oplus \left( s_1 \odot \varvec{\delta }_1 \right) \oplus \cdots \oplus \left( s_p \odot \varvec{\delta }_p \right) =\bigoplus _{m=0}^p s_m \odot \varvec{\delta }_m\,. \end{aligned}$$

(5)

Note that there are $p+1$ coefficient vectors $\delta _m$, as many as covariates, and that they are vectors with $(D-1)$ components, as many as coordinates. The goal of estimating the coefficients $\varvec{\delta }$ of a curve or surface in ${\mathscr {S}}^D$ is solved by translating it into a $(D-1)$ least square problem, i.e., for each coordinate

$$\begin{aligned} \hat{\mathbf {x}}_i^{\star } (\mathbf {s})= \delta ^{\star }_{0i}s_{0}+ \delta ^{\star }_{1i}s_1+\cdots + \delta ^{\star }_{pi}s_p ,\quad i=1,\ldots ,D-1, \end{aligned}$$

(6)

where $\varvec{\delta }^{\star }_{m}=\left( \delta ^{\star }_{m1},\ldots ,\delta ^{\star }_{m,D-1}\right)$ is the coordinate vector associated with $\varvec{\delta }_{m}$. In the case of simple regression $m = 1$ and $\mathbf {s}=s$, which is a straight-line in the simplex.

2.2 Dirichlet Regression

The Dirichlet distribution is one of the well known probability models suitable for random compositions. A random vector $\mathbf {X}= (X_1, \ldots , X_D)\in {\mathscr {S}}^D$ has a Dirichlet distribution, indicated by $\mathbf {X} \sim \mathscr {D}^D(\varvec{\alpha })$, with $\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _D) \in \mathbb {R}^D_+$, when its density function (with respect to the Aitchison measure) is

$$\begin{aligned} f(\mathbf {x};\varvec{\alpha })=\frac{\sqrt{D}\varGamma (\alpha _+)}{\prod _{i=1}^D\varGamma (\alpha _i)} \prod _{i=1}^Dx_i^{\alpha _i}, \end{aligned}$$

(7)

where $\alpha _+=\sum _{i=1}^D\alpha _i$, and $\varGamma$ denotes the gamma function [18]. The Dirichlet distribution has D parameters $\alpha _i$, which are assumed to be positive. Note that the density (7) is obtained by changing the measure to a Dirichlet density with respect to the Lebesgue measure.

In the Dirichlet regression model the $\alpha _i$ parameters are reparameterized in terms of explanatory variables and coefficients through an exponential function as described in Eq. (12). The log-likelihood of the reparameterized Dirichlet distribution can be optimized via an iterative method such as the Newton–Raphson algorithm.

When the variable of interest is continuous and restricted to the unit interval (0, 1), i.e., when $D=2$, the Dirichlet regression is called Beta regression [10].

3 Shifted-Dirichlet Covariate Model

3.1 Shifted-Dirichlet Distribution

One of the generalizations of the Dirichlet distribution is the Shifted-Dirichlet distribution. A random vector $\mathbf {X}\in {\mathscr {S}}^D$ has a Shifted-Dirichlet distribution with parameters $\varvec{\alpha }$ and $\varvec{\beta }=(\beta _1,\ldots ,\beta _D) \in {\mathscr {S}}^D$ if its density function is

$$\begin{aligned} f(\mathbf {x},;\varvec{\alpha },\varvec{\beta }) = \frac{\varGamma (\alpha _+)\sqrt{D}}{\prod _{i=1}^D\varGamma (\alpha _i)} \frac{\prod _{i=1}^D \left( \frac{x_i}{\beta _i}\right) ^{\alpha _i}}{\left( \sum _{i=1}^D \frac{x_i}{\beta _i} \right) ^{\alpha _+}}, \end{aligned}$$

(8)

The density (8) is expressed with respect to the Aitchison probability measure [21]. See Monti et al. [18] for a detailed discussion about the reasons and implications to use the Aitchison measure. This distribution will be denoted by $\mathbf {X}\sim \mathscr {SD}^D(\varvec{\alpha },\varvec{\beta })$.

The number of parameters of this model is $2D-1$, since $\varvec{\beta } \in {\mathscr {S}}^D$. The Shifted-Dirichlet distribution can be obtained by normalizing a vector of D independent, scaled (in the Euclidean geometry of real space), gamma r.v.s $W_i\sim Ga(\alpha _i,\beta _i)$, $i=1,2,\ldots , D$; i.e., if $\mathbf {X}={\mathscr {C}}(\mathbf {W})$, with $\mathbf {W}=(W_1,\ldots , W_D) \in \mathbb {R}^D_+$, then $\mathbf {X}\sim \mathscr {SD}^D(\varvec{\alpha },\varvec{\beta })$ [18]. For this reason, in the literature, when working with the Lebesgue reference measure, the distribution is called Scaled-Dirichlet. This distribution can also be obtained as a perturbed random composition with a Dirichlet density. Recall that perturbation is, in the Aitchison geometry of the simplex, a shift. Therefore, here it is called Shifted-Dirichlet distribution, understanding that the density is expressed with respect to the Aitchison measure.

Indeed, let $\mathbf {\widetilde{X}}\sim \mathscr {D}^D(\varvec{\alpha })$ be a random composition defined in ${\mathscr {S}}^D$, and let $\varvec{\beta }\in {\mathscr {S}}^D$ be a composition. The random composition $\mathbf {X}=\ominus \varvec{\beta }\oplus \mathbf {\widetilde{X}}$ has a $\mathscr {SD}^D(\varvec{\alpha },\varvec{\beta })$ distribution (note that $\ominus$ is the inverse operation of $\oplus$). Observe that using the Aitchison measure and geometry, $\varvec{\beta }$ can be interpreted as a parameter of location, instead of as a measure of scale. The expected value of $\mathbf {X}\sim \mathscr {SD}^D(\varvec{\alpha },\varvec{\beta })$ with respect to the Aitchison measure is

$$\begin{aligned} \mathrm {E}_a(\mathbf {X})=\ominus \varvec{\beta }\oplus \mathrm {E}_a(\mathbf {\widetilde{X}})\,, \end{aligned}$$

(9)

where $\mathrm {E}_a(\mathbf {\widetilde{X}})$ is the expected value of a Dirichlet composition with respect to the Aitchison measure

$$\begin{aligned} \mathrm {E}_a(\mathbf {\widetilde{X}})={\mathscr {C}}(e^{\psi (\alpha _1)},\ldots ,e^{\psi (\alpha _D)} )\,, \end{aligned}$$

(10)

with $\psi$ the digamma function. The metric variance of $\mathbf {X}$ coincides with the metric variance of a Dirichlet composition, because this measure of dispersion is invariant under perturbation

$$\begin{aligned} \mathrm {Mvar}(\mathbf {X})=\frac{D-1}{D}(\psi ^{\prime }(\alpha _1), \ldots ,\psi ^{\prime }(\alpha _D))\,, \end{aligned}$$

(11)

with $\psi ^{\prime }$ the trigamma function [1].

3.2 Shifted-Dirichlet Regression

Given a sample of n independent compositional observations $(\mathbf {x}_1,\ldots ,\mathbf {x}_n )$, we hypothesize that each observation $\mathbf {x}_j$ follows a conditional Shifted-Dirichlet distribution, given a set of covariates. Polynomial regression on a covariate s is included as a particular case taking $s_{jm}= s_j^m$.

In order to incorporate the covariate effects into the model [6, 14], we reparameterize each parameter $\alpha _i$ of the density written in Eq. (8) in terms of covariates and regression coefficients via the following log-linear model

$$\begin{aligned} \alpha _{ij}=\alpha _{ij}(\mathbf {s}_j)=\exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm} \right\} \,, \end{aligned}$$

(12)

where $\mathbf {s}_j$ is the covariate vector recorded on the j-th observed composition $(j=1,\ldots ,n)$, and $\delta _{im}$ are the coefficients for the m-th covariate. The parameter $\delta _{im}$ theoretically can vary by component, and the covariates may or may not be the same set of explanatory variables for each $\alpha _{ij}$. We augment each vector $\mathbf {s}_j$ with 1 as first position for notation simplicity. Thus, given a sample of independent compositional observations of size n, $\mathbf {x}_1,\ldots , \mathbf {x}_j,\ldots , \mathbf {x}_n$  the log-likelihood function for the reparameterized Shifted-Dirichlet, given the covariates $\mathbf {s}$ and ignoring the constant part that does not involve the parameters, is equal to

$$\begin{aligned} \begin{aligned} l(\varvec{\beta },\varvec{\delta }|\mathbf {x},\mathbf {s}) =&\sum _{j=1}^n \Bigg \lbrace \log \varGamma \left( \sum _{i=1}^D\exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm}\right\} \right) - \sum _{i=1}^D \log \varGamma \left( \exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm}\right\} \right) \\&- \sum _{i=1}^D \exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm}\right\} \log \beta _i + \sum _{i=1}^D\exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm}\right\} \log x_{ij}\\&- \left( \sum _{i=1}^D\exp \left\{ \sum _{m=0}^p\delta _{im}s_{jm}\right\} \right) \log \left( \sum _{i=1}^D \frac{x_i}{\beta _i } \right) \Bigg \rbrace \,. \end{aligned} \end{aligned}$$

(13)

Equation (13) can be estimated using the maximum likelihood method via some optimization algorithm, e.g., the Newton–Raphson algorithm. The choice of the starting values for the algorithm is of fundamental importance to get fast convergence.

For the Dirichlet regression in Hijazi and Jernigan [14] a method based on resampling from the original data is proposed; for each resample a Dirichlet model with constant parameters is fitted and the mean of the corresponding covariates is computed. After that, D models of the form $\sum _{m=0}^p\delta _{im}s_{jm}$ are fitted by least squares. The fitted coefficients $\hat{\delta }_{im}$ are used as starting values. For the Shifted-Dirichlet covariate model we have followed the same principle; as starting point for the vector $\varvec{\beta }$ we have chosen the closed geometric mean of the components of $\mathbf {x}$ given by

$$\begin{aligned} \mathbf {g}(\mathbf {x})={\mathscr {C}}\left( \left( \prod _{j=1}^nx_{1j}\right) ^{1/n}, \ldots , \left( \prod _{j=1}^nx_{Dj}\right) ^{1/n}\right) \,, \end{aligned}$$

(14)

Model selection is performed by testing

$$\begin{aligned} H_0:\;\delta _{im}=0\;, \end{aligned}$$

(15)

for some pair (i, m), $i=1,\ldots ,D$ and $m=1,\ldots ,p$. For it, the traditional likelihood ratio test is implemented.

4 Example from Sedimentology

4.1 Data Description

In Garzanti et al. [11] the authors studied the paleogeographic and paleodrainage changes during Pleistocene glaciations of Po Plain by compositional signatures of Pleistocene sands. In particular we consider here Cilavegna and Ghedi cores of Regione Lombardia, with 18 and 19 compositional observations, respectively. In this section we compare the above-mentioned approaches to regression with a composition as dependent variable. The goal is to model the effect of the depth covariate on compositional signatures of Pleistocene sands taking the fact into account that the cores may have separate effects on the response. The three compositional parts are: Q ($=$ quartz), F ($=$ feldspar), and L ($=$ lithic grains) represented in the usual ternary diagram in Fig. 1.

Fig. 1

Quaternary Po Plain sediments: ternary plot. Points are distinguished by core

Table 1 Regression output of the Dirichlet covariate model for Quaternary Po Plain sediments

4.2 Estimated Models Comparison

For each of the three components we have fitted a regression model considering the depth as covariate, including in the model one dummy variable representing the core provenance (0 for Cilavegna and 1 for Ghedi), as well as the interaction term. The categorical covariate core has been included to account for variation in proportions that is a function of a group-specific factor. The Dirichlet and Shifted-Dirichlet covariate models are expressed with respect to the Aitchison measure.

Tables 1 and 2 summarize the estimated regression coefficients, together with results from the inference for the Dirichlet and Shifted-Dirichlet covariate model, respectively.

Table 2 Regression output of the Shifted-Dirichlet covariate model for Quaternary Po Plain sediments

Table 3 Model fit statistics for the nested Shifted-Dirichlet covariate models, where $\varDelta G^2=-2\log L (\text {reduced model}-\text {current model})$

The p-value of the likelihood ratio tests to compare the intercept-only model (e.g., no predictors) with the fitted Shifted Dirichlet covariate model is essentially zero (<0.0001), which provides evidence against the reduced model in favor of the current model, as well as the model with only depth as covariate (see Table 3).

In Table 2 it can be seen that the z-values for the first two components are highly significant, implying that the use of the dummy variable is important; the models appear to have a definite nonzero slope. This consideration is confirmed by the fitted regression lines reported in Fig. 2.

Akaike information criterion (AIC) and Bayesian information criterion (BIC) are usually used to compare adequacy of models of the same family, the preferred model is the one with the minimum AIC value or BIC value. For the Dirichlet regression we obtained that $AIC= 131.2837$ and $BIC=150.6148$, while, for the Shifted-Dirichlet regression the two measures are $AIC= 122.45$ and $BIC=145.053$ (see Full model in Table 3). Therefore we can conclude that the improvement of the model compensates the additional parameters in the Shifted-Dirichlet model.

Fig. 2

Observed and fitted compositions for Quaternary Po Plain sediments using the three models for each level of the core variable (Shifted-Dirichlet covariate model: solid line, Dirichlet covariate model: dashed line; Simplicial regression: dotted line). Red colors refers to Cilavegna core data and blue color refers to Ghedi core data

In order to apply the Simplicial regression, $\mathrm {ilr}$ coordinates of the Quaternary Po Plain sediment dataset are computed (Table 4). The canonical basis in the $\mathrm {clr}$ plane was used as $\mathrm {ilr}$ transform, so that, the two coordinates or balances are expressed as:

$$\begin{aligned} x_1^{\star }=\frac{1}{\sqrt{2}}\log \frac{x_2}{x_1}\;, \quad x_2^{\star }=\frac{1}{\sqrt{6}}\log \frac{x_3^2}{x_1\;x_2}\;. \end{aligned}$$

(16)

Table 4 Regression output for the first and second coordinate of Simplicial regression for the Quaternary Po Plain sediments

Predictions of the three coordinates can be back-transformed with the inverse $\mathrm {ilr}$, to obtain a prediction of the proportions themselves.

In order to assess the adequacy of the different regression approaches, we examine some goodness of fit measures. One suitable determination coefficient for the regression model to evaluate the proportion of explained variation in the compositions by the covariate is connected with the total variability [2, 13], based on the variation matrix of the transformed log-ratio data,

$$\begin{aligned} \mathbf {T}(\mathbf {x})= [t_{ir}]=\left[ \text{ var }\left( \ln \frac{x_i}{x_r}\right) \right] \quad i,r=1,\ldots ,D. \end{aligned}$$

(17)

Each element $t_{ir}$ is the usual variance of the log ratio of parts i and r. Aitchison’s total variability measure $\text{ totvar }(\mathbf {x})$, a measure of global dispersion of a compositional sample, is defined as

$$\begin{aligned} \text{ totvar }(\mathbf {x})=\frac{1}{2D}\sum _{i,r} \text{ var }\left( \ln \frac{x_i}{x_r}\right) =\frac{1}{2D}\sum _{i,r} t_{ir}\,, \end{aligned}$$

(18)

Table 5 Goodness of fit measures for the three different regression models

Fig. 3

In the left column coordinate residual plots associated to the estimated simplicial regression. In the right column Q–Q plots for residuals of the corresponding regression models are displayed

Fig. 4

In the left column coordinate residual plots associated to the estimated Dirichlet regression. In the right column Q–Q plots for residuals of the corresponding regression models are displayed

Fig. 5

In the left column coordinate residual plots associated to the estimated Shifted-Dirichlet regression. In the right column Q–Q plots for residuals of the corresponding regression models are displayed

The determination coefficient $R^2_T$ is defined as

$$\begin{aligned} R^2_T=\frac{\text{ totvar }({\hat{\mathbf{x}}})}{\text{ totvar }(\mathbf {x})}\,; \end{aligned}$$

(19)

it compares the total variability of the observed with the fitted data.

Moreover, the Aitchison distance of any two compositions $\mathbf {x}$ and $\mathbf {y} \in {\mathscr {S}}^D$ is defined as

$$\begin{aligned} d_a(\mathbf {x},\mathbf {y})=\sqrt{\frac{1}{2D}\sum _{i=1}^D\sum _{r=1}^D \left( \ln \frac{x_i}{x_r}-\ln \frac{y_i}{y_r}\right) ^2}\;. \end{aligned}$$

(20)

Similarly to the standard least squares regression analysis, the compositional total sum of squares (CSST) and the compositional sum of squared residuals (CSSE) are given by $CSST =\sum _{j=1}^nd_a^2(\mathbf {x}_j,\mathrm {g}_{\mathrm {m}}(\mathbf {x}))$ and $CSSE =\sum _{j=1}^nd_a^2(\mathbf {x}_j,{\hat{\mathbf {x}}}_j)$. In this way another $R^2$-measure based on the compositional sum of squares [13] is

$$\begin{aligned} R^2_A=1-\frac{CSSE}{CSST}\,. \end{aligned}$$

(21)

In Table 5, KL-div refers to the Kullback–Leibler divergence calculated as

$$\sum _{j=1}^n\sum _{i=1}^3 x_{ji}\log \frac{x_{ji}}{\hat{x}_{ji}}\,.$$

The measures of goodness of fit reported in Table 5 show a good performance of the Shifted-Dirichlet model with respect to the other two models. The coefficient of determination based on the Aitchison norm shows that 60 % of the total variability is captured by the Shifted-Dirichlet regression model.

In order to check for absence of trends in central tendency and in variability, diagnostic plots are useful. We have expressed all the regression models in orthonormal coordinates (see Eq. (16)) which is making it possible to apply the standard battery of testing hypotheses for linear regression models, such as testing marginal normality of each coordinate residual. Coordinate residual plots and associated normal Q–Q plots for the three different regression models are displayed in Figs. 3, 4 and 5.

Except for the tails of the distribution (see Fig. 5), the assumption of normality seems to be reasonable. For the first coordinate residuals, whose points are displayed in the upper left of Fig. 5, the p-values of the Anderson–Darling test and of the Lilliefors (Kolmogorov–Smirnov) test for normality are 0.19 and 0.4789, respectively, so that the hypothesis of normal distribution cannot be rejected, while for the second coordinate residuals, lower left of Fig. 5, the two p-values of the two mentioned normality tests are 0.0009 and 0.003, respectively, due to the presence of an upper outlier (25th observation). If we omit such outlying point, normality is confirmed, i.e., the p-value of Anderson–Darling test equals 0.431 while the Lilliefors test p-value is 0.115.

5 Conclusions

Regression models with compositional response were proposed in the eighties. In this work, using the Shifted-Dirichlet distribution a new covariate model on the simplex is proposed. The Shifted-Dirichlet distribution is a generalization of the Dirichlet distribution obtained, within the Aitchison geometry, after applying a perturbation to the standard Dirichlet one. As a probability distribution, it is the same as the Scaled-Dirichlet distribution but the density is expressed with respect to the Aitchison measure on the simplex, and not with respect to the Lebesgue measure in the induced Euclidean geometry from the real space. Consequently, the Shifted-Dirichlet regression model is a generalization of the Dirichlet regression model. Even though the number of parameters to estimate increases, we see that it is a feasible and more flexible model. Using a real data set, we obtain results comparable to those obtained using the Simplicial regression.