Regression for compositions based on a generalization of the Dirichlet distribution

Abstract

The simplex is the geometrical locus of D-dimensional positive data with constant sum, called compositions. A possible distribution for compositions is the Dirichlet. In Dirichlet models, there are no scale parameters and the D shapes are assumed dependent on auxiliary variables. This peculiar feature makes Dirichlet models difficult to apply and to interpret. Here, we propose a generalization of the Dirichlet, called the simplicial generalized Beta (SGB) distribution. It includes an overall shape parameter, a scale composition and the D Dirichlet shapes. The SGB is flexible enough to accommodate many practical situations. SGB regression models are applied to data from the United Kingdom Time Use Survey. The R-package SGB makes the methods accessible to users.

Appendices

Proofs

Proof of Theorem 1

• 1. $\{a_k,\, k=1,\ldots ,D\}$ not constant implies dependence on $\theta$.

  Making the change of variables defined by $t = \sum _{j=1}^D y_j$ and $u_k = y_k/t$, $k=1,\ldots ,D-1$, and setting $u_D=1-\sum _{j=1}^{D-1} u_j$, we obtain

  $$\begin{aligned}&f({\mathbf {u}},t | \theta ) = \prod _{k=1}^D \left[ \frac{a_k}{\Gamma (p_k) \theta ^{1/a_k}b_k}\left( \frac{t u_k}{\theta ^{1/a_k}b_k}\right) ^{a_k p_k-1} \exp \left( -\left( \frac{t u_k}{\theta ^{1/a_k}b_k}\right) ^{a_k}\right) \right] t^{D-1} \nonumber \\&= \left[ \prod _{k=1}^D \frac{a_k}{\Gamma (p_k) b_k }\left( \frac{u_k}{b_k}\right) ^{a_k p_k-1}\right] \exp \left[ -\sum _{k=1}^D \left( \frac{t}{\theta ^{1/a_k}}\frac{u_k}{b_k}\right) ^{a_k}\right] \prod _{k=1}^D \left( \frac{t}{\theta ^{1/a_k}}\right) ^{a_kp_k } \frac{1}{t} \nonumber \\&=f({\mathbf {u}}|\theta )f(t|{\mathbf {u}},\theta ). \end{aligned}$$

  We want to find the constant of integration C, such that

  $$\begin{aligned} C\int _0^{\infty }f(t|{\mathbf {u}},\theta ) dt&= \int _0^{\infty } \exp \left[ -\sum _{k=1}^D \left( \frac{t}{\theta ^{1/a_k}}\frac{u_k}{b_k}\right) ^{a_k}\right] \prod _{k=1}^D \left( \frac{t}{\theta ^{1/a_k}}\right) ^{a_kp_k } \frac{1}{t} dt \\&= \int _0^{\infty } \exp \left[ -\theta ^{-1}\sum _{k=1}^D \left( \frac{t\,u_k}{b_k}\right) ^{a_k}\right] \theta ^{-P} \prod _{k=1}^D t^{a_kp_k } \frac{1}{t} dt. \end{aligned}$$

  It is clear that, if the parameters $a_k$ are not constant, the result still depends on $\theta$. This implies that in this case the distribution of the composition depends on the mixing scheme.

• 2. $\{a_k,\, k=1,\ldots ,D\}$ constant implies independence on $\theta$.

  If $a_k=a$ for all $k=1,\ldots ,D$, $f(t|{\mathbf {u}},\theta )$ is easily integrated. Setting

  $$\begin{aligned}c_k= (u_k/b_k)^{a}, \quad v=\left( \sum _{k=1}^D c_k\right) \frac{t^{a}}{\theta } \text { and } dv=a\left( \sum _{k=1}^D c_k\right) \frac{t^{a}}{\theta }\frac{1}{t}dt, \end{aligned}$$

  we have

  $$\begin{aligned} C\int _0^{\infty }f(t|{\mathbf {u}},\theta ) dt&= \frac{\Gamma (P)}{a\left( \sum _{k=1}^D (u_k/b_k)^{a}\right) ^P}. \end{aligned}$$

  (16)

  Thus the constant C in Eq. (16) does not depend on $\theta$. Thus this distribution does not depend on $\theta$.

$\square$

Proof of Theorem 2

Taking the density $f_{{\mathbf {U}}}({\mathbf {u}}_{-D})$ as expressed in Eq. (4), we see that the kernel is, up to a constant factor,

$$\begin{aligned} K({\mathbf {u}}_{-D}) \propto \frac{\left[ (1-\sum _{j=1}^{D-1}u_j)/b_D\right] ^{ap_D-1}}{\left\{ \sum _{j=1}^{D-1}(u_j/b_j)^{a} + \left( 1-\sum _{j=1}^{D-1}u_j)/b_D\right) ^{a}\right\} ^{P}}, \end{aligned}$$

and cannot be put into the form $K({\mathbf {u}}_{-D})=h\left( \sum _{j=1}^{D-1} (u_j/b_j)^{\beta _j}\right)$, except if $a=b_j=1$, in which case it reduces to $K({\mathbf {u}}_{-D};\,a=1,b_j=1,j=1,\ldots ,D) \propto \left[ 1-\sum _{j=1}^{D-1}u_j\right] ^{p_D-1}.$

Thus $h(x;\,a=1,b_j=1,j=1,\ldots ,D)=(1-x)^{p_d-1}.$ $\square$

Proof of Theorem 3

Without loss of generality, suppose that $J=1,\ldots ,r$, where $2\le r<D-1$. Consider the following change of variables: $x = \sum _{j=1}^r u_j$; $v_k = u_k/x$ if $1 \le k \le r-1$; $w_k = u_k/(1-x)$ if $r+1 \le k \le D-1$. The Jacobian is $x^{r-1}(1-x)^{D-r-1}$.

Let ${\mathbf {b}}_1=(b_1,\ldots ,b_r)$ and ${\mathbf {b}}_2=(b_{r+1},\ldots ,b_D)$. We have $\left( \| {\mathbf {u}}/{\mathbf {b}}\| _a\right) ^a=x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a.$ Making the above change of variables in Eq. (5) and setting $P_1=\sum _{j=1}^r p_j$ and $P_2=\sum _{j=r+1}^D p_j,$ we obtain, rearranging terms

$$\begin{aligned}&f_{X,{\mathbf {V}},{\mathbf {W}}}(x,{\mathbf {v}},{\mathbf {w}}) = f_{{\mathbf {U}}}({\mathbf {u}}_{-D}(x,{\mathbf {v}},{\mathbf {w}})x^{r-1}(1-x)^{D-r-1} \nonumber \\& =\frac{\Gamma(P)a^{D-1}}{\prod_{j=1}^D \Gamma(p_j)} \prod_{k=1}^{r} \left\{\frac{xv_k/b_k}{\left[x^a\left(\|{\mathbf {w}}/{\mathbf {b}}_1\|_a\right)^a+(1-x)^a \left(\|{\mathbf {w}}/{\mathbf {b}}_2\|_a\right)^a\right]^{1/a}}\right\}^{ap_k} \times \nonumber \\&\prod _{k=r+1}^{D} \left\{ \frac{(1-x)w_k/b_k}{\left[ x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a\right] ^{1/a}}\right\} ^{ap_k} \times \nonumber \\&\frac{x^{r-1}(1-x)^{D-r-1}}{x^r\prod _{k=1}^{r-1}v_k \left( 1-\sum _{j=1}^{r-1}v_j\right) (1-x)^{D-r}\prod _{k=r+1}^{D-1}w_k \left( 1-\sum _{j=r+1}^{D-1}w_j\right) } \nonumber \\ \nonumber \\&= \frac{\Gamma (P_1)a^{r-1}}{\prod _{j=1}^r \Gamma (p_j)} \prod _{k=1}^{r}\left\{ \frac{v_k/b_k}{\| {\mathbf {v}}/{\mathbf {b}}_1\| _a}\right\} ^{ap_k} \frac{1}{\prod _{k=1}^{r-1}v_k\left( 1-\sum _{j=1}^{r-1}v_j\right) } \times \end{aligned}$$

(17)

$$\begin{aligned}&\frac{\Gamma (P_2)a^{D-r-1}}{\prod _{j=r+1}^D \Gamma (p_j)}\prod _{k=r+1}^{D}\left\{ \frac{w_k/b_k}{\| {\mathbf {w}}/{\mathbf {b}}_2\| _a}\right\} ^{ap_k} \frac{1}{\prod _{k=r+1}^{D-1}w_k\left( 1-\sum _{j=r+1}^{D-1}w_j\right) } \times \nonumber \\&\frac{\Gamma (P)a}{ \Gamma (P_1)\Gamma (P_2)}\left\{ \frac{x^a(\| {\mathbf {v}}/{\mathbf {b}}_1\| _a)^a}{x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a}\right\} ^{P_1} \times \nonumber \\&\left\{ \frac{(1-x)^a(\| {\mathbf {w}}/{\mathbf {b}}_2\| _a)^a}{x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a}\right\} ^{P_2} \times \frac{1}{x(1-x)} \nonumber \\&\nonumber \\&= f_{{\mathbf {V}}}({\mathbf {v}}) f_{{\mathbf {W}}}({\mathbf {w}}) f_{X | {\mathbf {V}}, {\mathbf {W}}}(x;{\mathbf {v}},{\mathbf {w}}). \end{aligned}$$

(18)

Thus the amalgamation is, conditionally on the two sub-compositions,

$SGB\left( a,\{(\| {\mathbf {v}}/{\mathbf {b}}_1\| _a^{-1},P_1), (\| {\mathbf {w}}/{\mathbf {b}}_2\| _a^{-1},P_2) \}\right) ,$ with conditional density

$$\begin{aligned}&f_{X | {\mathbf {V}}, {\mathbf {W}}_2}(x;{\mathbf {v}},{\mathbf {w}}) = \frac{\Gamma (P)a}{\Gamma (P_1)\Gamma (P_2)} \times \left\{ \frac{x^a(\| {\mathbf {v}}/{\mathbf {b}}_1\| _a)^a}{x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a}\right\} ^{P_1} \times \nonumber \\&\left\{ \frac{(1-x)^a(\| {\mathbf {w}}/{\mathbf {b}}_2\| _a)^a}{x^a\left( \| {\mathbf {v}}/{\mathbf {b}}_1\| _a\right) ^a+(1-x)^a \left( \| {\mathbf {w}}/{\mathbf {b}}_2\| _a\right) ^a}\right\} ^{P_2} \times \frac{1}{x(1-x)} \end{aligned}$$

(19)

The constant of integration does not involve ${\mathbf {v}}$ and ${\mathbf {w}}$. Thus the two sub-compositions ${\mathbf {V}}$ and ${\mathbf {W}}$ are independent SGB.

• 1. The densities of ${\mathbf {V}}$ and ${\mathbf {W}}$ are at the first two rows of Eq. (17). The independence follows directly from Eq. (18).

• 2. The conditional distribution of $(X|{\mathbf {V}},{\mathbf {W}})$ is given in Eq. (19).

• 3. The conditional expectation of $\log (X/(1-X))$ is a direct application of Eq. (21).

• 4. The expression for ${\mathrm {{E}}}_A(X | {\mathbf {V}}={\mathbf {v}}, {\mathbf {W}}={\mathbf {w}})$ is an application of Eq. (8) to the density in Eq. (19).

$\square$

Moments of ratios and log-ratios of parts

1. It is equivalent to compute moment of ratios and log-ratios of parts from the distribution of the composition ${\mathbf {U}}$ or from the initial vector ${\mathbf {Y}}$, because

$$\begin{aligned} U_k/U_j=Y_k/Y_j\qquad \text {for all } j,k=1,\ldots ,D. \end{aligned}$$

The mixed moments of the random vector ${\mathbf {Y}}$ are given by $M_{{\mathbf {Y}}}(t_1,\ldots ,t_D)= \mathrm {{E}}\left( Y_1^{t_1} \ldots Y_{D}^{t_D}\right) .$

Set $t_+=\sum _{j=1}^{D-1} t_j$. Then the mixed moment ratios of the random composition following an SGB distribution ${{SGB(a, \{b_j,p_j\})}},$ $j=1,\ldots ,D$ are given by the corresponding moment of a product of generalized Gamma random variables, namely,

$$\begin{aligned}&M_{{\mathbf {U}}}(t_1,\ldots ,t_{-{D-1}})= M_{{\mathbf {Y}}}(t_1,\ldots ,t_{D-1},-t_+) \nonumber \\&= \mathrm {{E}}\left[ \left( \frac{U_1}{U_{D}}\right) ^{t_1}\ldots \left( \frac{U_{D-1}}{U_{D}}\right) ^{t_{D-1}}\right] \nonumber \\&= \frac{\prod _{k=1}^{D-1} (b_k)^{t_k}}{(b_D)^{t_+}} \frac{\left\{ \prod _{k=1}^{D-1} \Gamma (p_k+t_k/a)\right\} \Gamma (p_D-t_+/a)}{\prod _{j=1}^D \Gamma (p_j)} \nonumber \\&\qquad -ap_k< t_k, k=1,\ldots ,D-1; \, t_+ < ap_D. \end{aligned}$$

(20)

2. The function $M_{{\mathbf {U}}}$ in Eq. (20) is the moment generating function of the log-ratios of parts. By taking the first and second derivative of $M_{{\mathbf {U}}}(t{\mathbf {e}}_i)$ at $t=0$, Eqs. (21) and (22) are obtained.

$$\begin{aligned} \mathrm {{E}}\log \left( U_i/U_{D}\right)&= \log \left( b_i/b_D\right) +\frac{1}{a}\left( \psi (p_i)-\psi (p_D)\right) \end{aligned}$$

(21)

$$\begin{aligned} \mathrm {{E}}\left[ \log (U_i/U_D)\right] ^2&= \left[ \log \left( b_i/b_D\right) + \frac{1}{a}\left( \psi (p_i)-\psi (p_D)\right) \right] ^2 \nonumber \\&+ \frac{1}{a^2}\left( \psi ^{(1)}(p_i)+\psi ^{(1)}(p_D)\right) . \end{aligned}$$

(22)

Distinct pairs of log-ratios of parts are uncorrelated. The technique can be readily applied to log-ratio transforms of any kind. From Eq. (21), we recover Eq. (8).

Partial derivatives of the pseudo-log-likelihood

Let n be the sample size, D the number of parts and p the number of explanatory variables. Set

$$\begin{aligned} K_{bi}= \sum _{j=1}^D z_j({\mathbf {u}}_i) \log \left( \frac{u_{ij}}{b_{ij}}\right) , \end{aligned}$$

where ${\mathbf {u}}_i=(u_{i1},u_{i2},\ldots , u_{iD}), i=1,\ldots ,n$ are the observed compositions, $b_{ij}, j=1,\ldots ,D$ the corresponding scales and $z_j({\mathbf {u}}_i)$ the j-th component of the vector defined in Eq. (6).

The partial derivatives of the pseudo-log-likelihood in Eq. (12) are

$$\begin{aligned} \frac{\partial \ell }{\partial a}&= \frac{n(D-1)}{a} + \sum _{i=1}^n w_i\sum _{k=1}^D p_k \left[ \log \left( \frac{u_{ik}}{b_{ik}}\right) -K_{bi}\right] \\ \frac{\partial \ell }{\partial b_{ik}}&= w_i \sum _{j=1}^D p_j \frac{\partial \log z_j({\mathbf {u}}_i)}{\partial b_{ik}} \nonumber \\&= w_i \frac{a}{b_{ik}}\left( P z_k({\mathbf {u}}_i)-p_k\right) \quad k=1,\ldots ,D, \quad i=1,\ldots ,n\\ \frac{\partial \ell }{\partial p_k}&= n(\psi (P) - \psi (p_k)) + \sum _{i=1}^n w_i \log z_k({\mathbf {u}}_i) \quad k=1,\ldots ,D, \end{aligned}$$

where $\psi$ is the digamma function.

The derivatives with respect to the regression parameters are given by

$$\begin{aligned} \frac{\partial \ell }{\partial \beta _{j m}} = \sum _{i=1}^n \sum _{k=1}^D \frac{\partial \ell }{\partial b_{ik}} \frac{\partial b_{ik}}{\partial \beta _{j m}},\quad j=1,\ldots ,p;\, m=1,\ldots ,D-1. \end{aligned}$$

The partial derivatives $\partial b_{ik}/\partial \beta _{j \ell }$ are computed in two steps.

Setting ${\mathbf {s}}_i={\mathbf {x}}_i^t {\mathbf {B}}{\mathbf {V}}^+ = (s_{ir}),r=1,\ldots ,D$ in Eq. (14), we have

$$\begin{aligned} \frac{\partial b_{ik}}{\partial s_{ir}}&= b_{ik}(\delta _{kr}-b_{ir}) \, \qquad \qquad \text {where } \delta _{kr}=1 \text { if } r=k, 0 \text { otherwise}.\\ \frac{\partial s_{ir}}{\partial \beta _{jm}}&= x_{ij}\,({\mathbf {V}}^+)_{rm}. \end{aligned}$$

Thus, denoting by $({\mathbf {V}}^+)_{m}$ the m-th column of ${\mathbf {V}}^+$, we have

$$\begin{aligned} \frac{\partial b_{ik}}{\partial \beta _{j m}}&= \sum _{r=1}^D \frac{\partial b_{ik}}{\partial s_{ir}} \frac{\partial s_{ir}}{\partial \beta _{jm}} = x_{ij}b_{ik}\left[ ({\mathbf {V}}^+)_{km}-{\mathbf {b}}_i^t ({\mathbf {V}}^+)_{m}\right] , \\ \frac{\partial \ell }{\partial \beta _{j m}}&= \sum _{i=1}^n \sum _{k=1}^D w_i \frac{a}{b_{ik}}\left( P z_k({\mathbf {u}}_i)-p_k\right) x_{ij}b_{ik}\left[ ({\mathbf {V}}^+)_{km}-{\mathbf {b}}_i^t ({\mathbf {V}}^+)_{m}\right] \\&= a\sum _{i=1}^n w_i x_{ij} \sum _{k=1}^D \left( P z_k({\mathbf {u}}_i)-p_k\right) ({\mathbf {V}}^+)_{km}, \end{aligned}$$

because $\left( P z_k({\mathbf {u}}_i)-p_k\right) \mathbf {1}_D = 0$.