Well-formed decompositions of generalized additive independence models

Abstract

Generalized additive independence (GAI) models permit to represent interacting variables in decision making. A fundamental problem is that the expression of a GAI model is not unique as it has several equivalent different decompositions involving multivariate terms. Considering for simplicity 2-additive GAI models (i.e., with multivariate terms of at most 2 variables), the paper examines the different questions (definition, monotonicity, interpretation, etc.) around the decomposition of a 2-additive GAI model and proposes as a basis the notion of well-formed decomposition. We show that the presence of a bi-variate term in a well-formed decomposition implies that the variables are dependent in a preferential sense. Restricting to the case of discrete variables, and based on a previous result showing the existence of a monotone decomposition, we give a practical procedure to obtain a monotone and well-formed decomposition and give an explicit expression of it in a particular case.

Appendix: Monotone decompositions of \(v_{\downarrow \!x}\), for x of the form \(1_{ij}\), \(2_i1_j\), and \(2_{ij}\)

• Case 1 Suppose $m(1_{ij})<0$. We make the substitution

  $$\begin{aligned} m(1_{ij})\mu _{1_{ij}} = -m(1_{ij})(\mu _{1_{ij}}^{-i-j}-\mu _{1_i}-\mu _{1_j}) \end{aligned}$$

  (33)

  which yields

  $$\begin{aligned}{}[v_{\downarrow \! 1_{ij}}] = |m(1_{ij})|\mu _{1_{ij}}^{-i-j} + (m(1_{ij}) + m(1_i))\mu _{1_i} + (m(1_{ij}) + m(1_j))\mu _{1_j}. \end{aligned}$$

  (34)

• Case 2 Suppose $m(2_i1_j)<0$ and $m(2_j1_i)\ge 0$.

  • Case 2a $m(1_{ij})<0$. We make the substitution

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} = -m(2_i1_j)\big (\mu _{2_i1_j}^{-2i-j}-\mu _{2_i}-\mu _{1_j}\big ) \end{aligned}$$

    which yields, combined with (33):

    $$\begin{aligned}{}[v_{\downarrow \! 2_i1_j}]&= |m(2_i1_j)|\mu _{2_i1_j}^{-2i-j} + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} +(m(2_i1_j)+m(2_i))\mu _{2_i} \nonumber \\&\quad + (m(1_{ij}) + m(1_i))\mu _{1_i} + (m(2_i1_j) + m(1_{ij}) + m(1_j))\mu _{1_j}. \end{aligned}$$

    (35)

  • Case 2b $m(1_{ij})+m(2_i1_j)\ge 0$. We make the substitution²

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} = -m(2_i1_j)\big (\mu _{2_i1_j}^{-i-j} - \mu _{1_{ij}} - \mu _{2_i}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned}{}[v_{\downarrow \! 2_i1_j}]&=|m(2_i1_j)|\mu _{2_i1_j}^{-i-j} + (m(2_i1_j) + m(1_{ij})) \mu _{1_{ij}}\nonumber \\&\quad + (m(2_i1_j) + m(2_i)) \mu _{2_i} + m(1_i)\mu _{1_i} + m(1_j)\mu _{1_j}. \end{aligned}$$

    (36)

  • Case 2c $m(1_{ij})\ge 0$ and $m(1_{ij})+m(2_i1_j)< 0$. We make the substitution

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j}&= m(1_{ij})\big (\mu _{2_i1_j}^{-i-j} - \mu _{1_{ij}} - \mu _{2_i}\big ) \\&\quad + (-m(2_i1_j)-m(1_{ij}))\big (\mu _{2_i1_j}^{-2i-j}-\mu _{2_i} - \mu _{1_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned}{}[v_{\downarrow \! 2_i1_j}]&= m(1_{ij})\mu _{2_i1_j}^{-i-j} + (-m(2_i1_j)-m(1_{ij}))\mu _{2_i1_j}^{-2i-j}\nonumber \\&\quad + (m(2_i1_j)+m(2_i))\mu _{2_i} + (m(2_i1_j)+m(1_{ij}) \nonumber \\&\quad + m(1_j)) \mu _{1_j} + m(1_i)\mu _{1_i}. \end{aligned}$$

    (37)

• Case 3 Suppose $m(2_i1_j)<0$ and $m(2_j1_i)<0$. Then the intersection $\downarrow \! 2_i1_j\cap \downarrow \! 2_j1_i$ contains $1_{ij}$, so that we need a monotone decomposition for $v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}$.

  • Case 3a $m(1_{ij})<0$. We make the substitution

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} + m(2_j1_i)\mu _{2_j1_i}&= -m(2_i1_j)\big (\mu _{2_i1_j}^{-2i-j}-\mu _{2_i}-\mu _{1_j}\big ) \\&\quad +(-m(2_j1_i))\big (\mu _{2_j1_i}^{-2j-i}-\mu _{2_j}-\mu _{1_i}\big ) \end{aligned}$$

    which yields, together with (33)

    $$\begin{aligned}{}[v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}]&= |m(2_i1_j)|\mu _{2_i1_j}^{-2i-j} + |m(2_j1_i)|\mu _{2_j1_i}^{-2j-i} + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} \nonumber \\&\quad + (m(2_i1_j) + m(2_i))\mu _{2_i} + (m(2_j1_i) + m(2_j))\mu _{2_j} \nonumber \\&\quad + (m(2_j1_i)+m(1_{ij}) + m(1_i))\mu _{1_i} + (m(2_i1_j)+m(1_{ij}) \nonumber \\&\quad + m(1_j))\mu _{1_j}. \end{aligned}$$

    (38)

  • Case 3b $m(2_i1_j)+m(1_{ij})\ge 0$ and $m(2_j1_i)+m(1_{ij})\ge 0$. Then we write, assuming $m(2_i1_j)\le m(2_j1_i)$

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} + m(2_j1_i)\mu _{2_j1_i}&= -m(2_j1_i)\big (\mu _{ij}^{\wedge \wedge } - \mu _{2_i} - \mu _{2_j} - \mu _{1_{ij}}\big ) \\&\quad + (m(2_j1_i) - m(2_i1_j))\big (\mu _{2_i1_j}^{-i-j}-\mu _{1_{ij}} - \mu _{2_i}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned}{}[v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}]&= |m(2_j1_i)|\mu _{ij}^{\wedge \wedge } + (m(2_j1_i) - m(2_i1_j))\mu _{2_i1_j}^{-i-j} \nonumber \\&\quad +(m(1_{ij}) + m(2_i1_j))\mu _{1_{ij}} \nonumber \\&\quad + (m(2_i1_j) + m(2_i))\mu _{2_i} + (m(2_j1_i) + m(2_j))\mu _{2_j} \nonumber \\&\quad + m(1_i)\mu _{1_i} + m(1_j)\mu _{1_j}. \end{aligned}$$

    (39)

  • Case 3c $m(1_{ij})\ge 0$, $m(2_i1_j)+m(1_{ij})<0$ and $m(2_j1_i) + m(1_{ij})\ge 0$. Then we write, assuming $m(2_i1_j)\le m(2_j1_i)$

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} + m(2_j1_i)\mu _{2_j1_i}&= -m(2_j1_i)\big (\mu _{ij}^{\wedge \wedge } - \mu _{2_i} - \mu _{2_j} - \mu _{1_{ij}}\big ) \\&\quad + (m(2_j1_i) + m(1_{ij}))\big (\mu _{2_i1_j}^{-i-j} - \mu _{2_i} - \mu _{1_{ij}}\big ) \\&\quad + (-m(2_i1_j) - m(1_{ij}))\big (\mu _{2_i1_j}^{-2i-j} - \mu _{2_i} - \mu _{1_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned}{}[v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}]&= |m(2_j1_i)|\mu _{ij}^{\wedge \wedge } + (-m(2_i1_j) - m(1_{ij}))\mu _{2_i1_j}^{-2i-j} + (m(2_j1_i) \nonumber \\&\quad + m(1_{ij}))\mu _{2_i1_j}^{-i-j} + (m(2_i1_j)+m(2_i))\mu _{2_i} + (m(2_j1_i)+m(2_j))\mu _{2_j} \nonumber \\&\quad +(m(2_i1_j)+m(1_{ij})+m(1_j))\mu _{1_j}+m(1_i)\mu _{1_i}. \end{aligned}$$

    (40)

  • Case 3d $m(1_{ij})\ge 0$, $m(2_i1_j)+m(1_{ij})<0$ and $m(2_j1_i) + m(1_{ij})< 0$. Then we write, assuming $m(2_i1_j)\le m(2_j1_i)$

    $$\begin{aligned} m(2_i1_j)\mu _{2_i1_j} + m(2_j1_i)\mu _{2_j1_i}&+ m(1_{ij})\mu _{1_{ij}} = -m(2_j1_i)\big (\mu _{ij}^{\wedge \wedge }-\mu _{2_i}-\mu _{2_j}\big )\\&\quad + (-m(2_j1_i) -m(1_{ij}))\big (\mu _{1_{ij}}^{-i-j} -\mu _{1_i} - \mu _{1_j} \big )\\&\quad +(m(2_j1_i)-m(2_i1_j))\big (\mu _{2_i1_j}^{-2i-j} - \mu _{2_i} - \mu _{1_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned}{}[v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}]&= |m(2_j1_i)|\mu _{ij}^{\wedge \wedge } + (-m(2_j1_i) - m(1_{ij}))\mu _{1_{ij}}^{-i-j} \nonumber \\&\quad + (m(2_j1_i)-m(2_i1_j))\mu _{2_i1_j}^{-2i-j} \nonumber \\&\quad +(m(2_i1_j) + m(2_i))\mu _{2_i} + (m(2_j1_i) + m(2_j))\mu _{2_j} \nonumber \\&\quad + (m(2_j1_i) + m(1_{ij}) + m(1_i))\mu _{1_i} + (m(2_i1_j) \nonumber \\&\quad +m(1_{ij}) + m(1_j))\mu _{1_j}. \end{aligned}$$

    (41)

• Case 4 Suppose $m(2_{ij})<0$.

  • Case 4a $m(2_i1_j)<0$ and $m(2_j1_i)<0$. We make the substitution

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}} = -m(2_{ij})\big (\mu _{2_{ij}}^{-2i-2j} - \mu _{2_i} - \mu _{2_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}= & {} m(2_{ij})\mu _{2_{ij}} + v_{\downarrow \!2_i1_j \cup \downarrow \!2_j1_i} = -m(2_{ij})\big (\mu _{2_{ij}}^{-2i-2j} - \mu _{2_i} - \mu _{2_j}\big ) \nonumber \\&\quad +v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}. \end{aligned}$$

    (42)

    It remains to substitute in the above the expression of $v_{\downarrow \!2_i1_j\cup \downarrow \!2_j1_i}$ by (38) to (41) successively. Observe that only the coefficients of $\mu _{2_i},\mu _{2_j}$ are affected, and they are changed $m(2_{ij})+m(2_i1_j)+m(2_i)$ and $m(2_{ij})+m(2_j1_i)+m(2_j)$ respectively, which are nonnegative.

  • Case 4b $m(2_i1_j)<0$ and $m(2_{ij}) + m(2_j1_i)\ge 0$. We make the substitution

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}} = -m(2_{ij})\big (\mu _{2_{ij}}^{-2j-i} - \mu _{2_i} - \mu _{2_j1_i}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}&= -m(2_{ij})\big (\mu _{2_{ij}}^{-2j-i} - \mu _{2_i} - \mu _{2_j1_i}\big ) + v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_j1_i} + m(2_j)\mu _{2_j}\nonumber \\&= v_{\downarrow 2_i1_j} +|m(2_{ij})|\mu _{2_{ij}}^{-2j-i} + (m(2_j1_i) + m(2_{ij}))\mu _{2_j1_i} \nonumber \\&\quad + m(2_j)\mu _{2_j} + m(2_{ij})\mu _{2_i}. \end{aligned}$$

    (43)

    It remains to substitute in the above the expression of $v_{\downarrow 2_i1_j}$ by (35) to (37) successively. Observe that only the coefficient of $\mu _{2_i}$ is affected.

  • Case 4c $m(2_i1_j)<0$, $m(2_j1_i)\ge 0$ and $m(2_{ij}) + m(2_j1_i)< 0$. We make the substitution

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}}&= m(2_j1_i)\big (\mu _{2_{ij}}^{-2j-i} - \mu _{2_i} - \mu _{2_j1_i}\big ) + (-m(2_{ij}) \nonumber \\&\quad - m(2_j1_i))\big (\mu _{2_{ij}}^{-2i-2j} - \mu _{2_i}- \mu _{2_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}&= m(2_j1_i)\big (\mu _{2_{ij}}^{-2j-i} - \mu _{2_i} - \mu _{2_j1_i}\big ) + (-m(2_{ij}) - m(2_j1_i))\nonumber \\&\quad \big (\mu _{2_{ij}}^{-2i-2j} - \mu _{2_i}- \mu _{2_j}\big ) + v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_j1_i} + m(2_j)\mu _{2_j}\nonumber \\&= v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_{ij}}^{-2j-i} + (-m(2_{ij})-m(2_j1_i))\mu _{2_{ij}}^{-2i-2j}\nonumber \\&\quad + m(2_{ij})\mu _{2_i} + (m(2_{ij})+m(2_j1_i) + m(2_j))\mu _{2_j}. \end{aligned}$$

    (44)

    It remains to substitute in the above the expression of $v_{\downarrow 2_i1_j}$ by (35) to (37) successively. Observe that only the coefficient of $\mu _{2_i}$ is affected.

  • Case 4d $m(2_{ij}) + m(2_i1_j)\ge 0$ and $m(2_{ij}) + m(2_j1_i)\ge 0$. We make the substitution

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}} = -m(2_{ij})\big (\mu _{2_{ij}}^{-i-j} - \mu _{2_i1_j} - \mu _{2_j1_i}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}&= -m(2_{ij})\big (\mu _{2_{ij}}^{-i-j} - \mu _{2_i1_j} - \mu _{2_j1_i}\big ) + v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_j1_i} + m(2_j)\mu _{2_j}\nonumber \\&= v_{\downarrow 2_i1_j} + |m(2_{ij})|\mu _{2_{ij}}^{-i-j} + m(2_{ij})\mu _{2_i1_j} + (m(2_j1_i)+m(2_{ij}))\mu _{2_j1_i} + m_{2j}\mu _{2j}. \end{aligned}$$

    We have two subcases:

    $*$:

    If $m(1_{ij})<0$, then using (34), we have

    $$\begin{aligned} v_{\downarrow \!2_i1_j}&=m(2_i1_j)\mu _{2_i1_j} + v_{\downarrow \!1_{ij}} +m(2_i)\mu _{2_i}\\&= m(2_i1_j)\mu _{2_i1_j} + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} + m(2_i)\mu _{2_i} \\&\quad + (m(1_{ij}) + m(1_i))\mu _{1_i} + (m(1_{ij}) + m(1_j))\mu _{1_j} \end{aligned}$$

    which inserted in the expression of $v_{\downarrow \!2_{ij}}$ yields

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&= |m(2_{ij})|\mu _{2_{ij}}^{-i-j} + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} + (m(2_{ij})+m(2_i1_j))\mu _{2_i1_j} \nonumber \\&\quad + (m(2_j1_i)+m(2_{ij}))\mu _{2_j1_i} + m_{2j}\mu _{2j} + + m(2_i)\mu _{2_i} \nonumber \\&\quad + (m(1_{ij}) + m(1_i))\mu _{1_i} + (m(1_{ij}) + m(1_j))\mu _{1_j}. \end{aligned}$$

    (45)

    $*$:

    If $m(1_{ij})\ge 0$, then we get

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&= |m(2_{ij})|\mu _{2_{ij}}^{-i-j} + (m(2_{ij})+m(2_i1_j))\mu _{2_i1_j} + (m(2_j1_i)+m(2_{ij}))\mu _{2_j1_i} \nonumber \\&\quad +m(1_{ij})\mu _{1_{ij}} + m(2_i)\mu _{2_i} +m_{2j}\mu _{2j} + m(1_i)\mu _{1_i} + m(1_j)\mu _{1_j}. \end{aligned}$$

    (46)

  • Case 4e $m(2_{ij}) + m(2_i1_j)\ge 0$, $m(2_j1_i)\ge 0$ and $m(2_{ij}) + m(2_j1_i)< 0$. We make the substitution

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}}&= m(2_j1_i)\big (\mu _{2_{ij}}^{-i-j} - \mu _{2_i1_j} - \mu _{2_j1_i}\big ) + (-m(2_{ij}) - m(2_j1_i))\\&\quad \times \big (\mu _{2_{ij}}^{-2i-j}-\mu _{2_j} - \mu _{2_i1_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}&= m(2_j1_i)\big (\mu _{2_{ij}}^{-i-j} - \mu _{2_i1_j} - \mu _{2_j1_i}\big ) + (-m(2_{ij}) - m(2_j1_i))\big (\mu _{2_{ij}}^{-2i-j} -\mu _{2_j}-\mu _{2_i1_j}\big )\\&\quad + v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_j1_i} + m(2_j)\mu _{2_j}\\&= v_{\downarrow 2_i1_j} + (-m(2_{ij}) - m(2_j1_i))\mu _{2_{ij}}^{2i-j} + m(2_j1_i)\mu _{2_{ij}}^{-i-j} \\&\quad + m(2_{ij})\mu _{2_i1_j} + (m(2_{ij})+m(2_j1_i)+m(2_j))\mu _{2_j} \end{aligned}$$

    As above, we have two cases, which yield:

    $*$:

    If $m(1_{ij})<0$:

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&=(-m(2_{ij}) - m(2_j1_i))\mu _{2_{ij}}^{2i-j} + m(2_j1_i)\mu _{2_{ij}}^{-i-j} + (m(2_{ij})+m(2_i1_j))\mu _{2_i1_j} \nonumber \\&\quad + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} + (m(2_{ij})+m(2_j1_i)+m(2_j))\mu _{2_j} + m(2_i)\mu _{2_i} \nonumber \\&\quad + (m(1_{ij}) +m(1_i))\mu _{1_i} + (m(1_{ij}) + m(1_j))\mu _{1_j}. \end{aligned}$$

    (47)

    $*$:

    If $m(1_{ij})\ge 0$:

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&= (-m(2_{ij}) - m(2_j1_i))\mu _{2_{ij}}^{2i-j} + m(2_j1_i)\mu _{2_{ij}}^{-i-j} + (m(2_{ij})+m(2_i1_j))\mu _{2_i1_j} \nonumber \\&\quad + m(1_{ij})\mu _{1_{ij}} + (m(2_{ij})+m(2_j1_i)+m(2_j))\mu _{2_j} \nonumber \\&\quad + m(2_i)\mu _{2_i} + m(1_i)\mu _{1_i} + m(1_j)\mu _{1_j}. \end{aligned}$$

    (48)

  • Case 4f $m(2_i1_j)\ge 0$, $m(2_j1_i)\ge 0$, $m(2_{ij}) + m(2_i1_j)< 0$ and $m(2_{ij}) + m(2_j1_i)< 0$. Then we write, assuming $m(2_i1_j)\le m(2_j1_i)$,

    $$\begin{aligned} m(2_{ij})\mu _{2_{ij}}&= m(2_i1_j)\big (\mu _{2_{ij}}^{-i-j}- \mu _{2_i1_j}-\mu _{2_j1_i} \big ) + (m(2_j1_i)-m(2_i1_j))\\&\quad \times \big (\mu _{2_{ij}}^{-i-2j}-\mu _{2_j1_i}-\mu _{2_i}\big ) \\&\quad +(-m(2_{ij})-m(2_j1_i))\big (\mu _{2_{ij}}^{-2i-2j}- \mu _{2_i} - \mu _{2_j}\big ) \end{aligned}$$

    which yields

    $$\begin{aligned} v_{\downarrow \!2_{ij}}&= m(2_i1_j)\big (\mu _{2_{ij}}^{-i-j} - \mu _{2_i1_j} - \mu _{2_j1_i}\big ) + (m(2_j1_i)-m(2_i1_j))\big (\mu _{2_{ij}}^{-i-2j} - \mu _{2_j1_i} - \mu _{2_i}\big ) \\&\quad +(-m(2_{ij})-m(2_j1_i))\big (\mu _{2_{ij}}^{-2i-2j} - \mu _{2_i} - \mu _{2_j}\big )\\&\quad + v_{\downarrow 2_i1_j} + m(2_j1_i)\mu _{2_j1_i} + m(2_j)\mu _{2_j}\\&= v_{\downarrow 2_i1_j} +m(2_i1_j)\mu _{2_{ij}}^{-i-j} + (m(2_j1_i)-m(2_i1_j)) \mu _{2_{ij}}^{-i-2j} \\&\quad +(-m(2_{ij})-m(2_j1_i))\mu _{2_{ij}}^{-2i-2j} + (-m(2_i1_j))\mu _{2_i1_j} \\&\quad +(m(2_{ij}) + m(2_i1_j))\mu _{2_i} +(m(2_{ij}) + m(2_j1_i) + m(2_j))\mu _{2_j} . \end{aligned}$$

    Distinguishing the two cases yields, assuming $m(2_i1_j)\le m(2_j1_i)$

    $*$:

    If $m(1_{ij})<0$:

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&=m(2_i1_j)\mu _{2_{ij}}^{-i-j} + (m(2_j1_i)-m(2_i1_j))\mu _{2_{ij}}^{-i-2j} \nonumber \\&\quad +(-m(2_{ij})-m(2_j1_i))\mu _{2_{ij}}^{-2i-2j} + (m(2_{ij}) + m(2_i1_j) + m(2_i))\mu _{2_i}\nonumber \\&\quad + (m(2_{ij}) + m(2_j1_i) + m(2_j))\mu _{2_j} + |m(1_{ij})|\mu _{1_{ij}}^{-i-j} \nonumber \\&\quad +(m(1_{ij})+m(1_i))\mu _{1_i} + (m(1_{ij})+m(1_j))\mu _{1_j}. \end{aligned}$$

    (49)

    $*$:

    If $m(1_{ij})\ge 0$:

    $$\begin{aligned}{}[v_{\downarrow \!2_{ij}}]&=m(2_i1_j)\mu _{2_{ij}}^{-i-j} + (m(2_j1_i)-m(2_i1_j))\mu _{2_{ij}}^{-i-2j} \nonumber \\&\quad +(-m(2_{ij})-m(2_j1_i))\mu _{2_{ij}}^{-2i-2j} + (m(2_{ij}) + m(2_i1_j) + m(2_i))\mu _{2_i}\nonumber \\&\quad + (m(2_{ij}) + m(2_j1_i) + m(2_j))\mu _{2_j} + m(1_{ij})\mu _{1_{ij}} \nonumber \\&\quad +m(1_i)\mu _{1_i} + m(1_j)\mu _{1_j}. \end{aligned}$$

    (50)

In all formulas (34) to (50), it can be checked using (20) to (22) that all coefficients are nonnegative, hence they are indeed monotone decompositions.