The conflict is divided into three levels, small (trifling) conflict, medium conflict and high conflict. Small conflict is considered when the mass function of the first biomarker over the
condition(
i) and the mass function of the second biomarker over the complementary condition
\(\overline{condition(i)} \) are both less than 0.65 (
i.e.
\(m_1(condition(i)), m_2(\overline{condition(i)}) < 0.65\)). Medium conflict is considered when those masses are between 0.65 to 0.8, and high conflict when both are higher than 0.8. Note that these values can be chosen differently depending on the application and the expert’s point of view. To estimate the threshold
\(\varepsilon \) for the previous considered values, we proceed as follows:
-
Threshold
\(\varepsilon _{Small}\)
for small conflict:
(1)
Threshold
\(\varepsilon _1\)
for the parameter
\(m_{ \oplus } ( \phi )\)
of the conflict pair:
We consider the critical situation between small and medium conflict:
$$\begin{aligned} m_{1}(condition1)&= 0.65 \quad m_{1}(condition2) = a \\ m_{2}(condition1)&= b \quad m_{2}(condition2) = 0.65 \end{aligned}$$
With
\(condition1 = \overline{condition2}\) and a, b
\(\in \) [0, 0.35]. It is obvious that the smallest value of
\(m_{ \oplus } ( \phi )\) is 0.42, whatever the values of a and b are. Hence the threshold
\(\varepsilon _1\) for the first parameter
\(m_{ \oplus } ( \phi )\) of the conflict pair is
\(\varepsilon _1\) = 0.42.
(2)
Threshold
\(\varepsilon _2\)
for the second parameter
\(DifBet^{m1}_{m2}\)
:
Using the mass functions of the critical case defined in (1) we will have:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \end{aligned}$$
(12)
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \end{aligned}$$
(13)
Note that:
$$\begin{aligned} b + {m_2 (\varOmega )} = 1 - m_2 (condition2) = 0.35 \end{aligned}$$
(14)
Hence:
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \le 0.35 . \end{aligned}$$
(15)
We have also:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \ge 0.65 \end{aligned}$$
(16)
From (
16) and (
15), we conclude then:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = |BetP_{m1} (condition1) - BetP_{m2} (condition1)| \ge 0.3 \end{aligned}$$
(17)
Since we have only two singletons (
i.e. condition1 and condition2), we do not have to look for the maximum of
\(DifBet^{m1}_{m2}\) since it will be the same for condition2. Hence, we obtain:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = DifBet^{m1}_{m2} (condition2) \ge 0.3 \end{aligned}$$
(18)
From (
18) we can see that the threshold
\(\varepsilon _2\) for the second parameter is 0.3.
(3)
Common threshold
\(\varepsilon \)
for both parameters:
For the sake of ease and precaution, we choose a common threshold
\(\varepsilon \) for both parameters of the conflict pair by taking the smallest value between
\(\varepsilon _1\) and
\(\varepsilon _2\) as follows:
$$\begin{aligned} \varepsilon _{Small} = argmin \lbrace \varepsilon _1, \varepsilon _2 \rbrace = 0.3 \end{aligned}$$
(19)
-
Threshold
\(\varepsilon _{Medium}\)
for Medium conflict:
Following the same reasoning, we obtain the threshold
\(\varepsilon _{Medium} = 0.6\).
Threshold
\(\varepsilon _{Small}\)
for small conflict:
(1)
Threshold
\(\varepsilon _1\)
for the parameter
\(m_{ \oplus } ( \phi )\)
of the conflict pair:
We consider the critical situation between small and medium conflict:
$$\begin{aligned} m_{1}(condition1)&= 0.65 \quad m_{1}(condition2) = a \\ m_{2}(condition1)&= b \quad m_{2}(condition2) = 0.65 \end{aligned}$$
With
\(condition1 = \overline{condition2}\) and a, b
\(\in \) [0, 0.35]. It is obvious that the smallest value of
\(m_{ \oplus } ( \phi )\) is 0.42, whatever the values of a and b are. Hence the threshold
\(\varepsilon _1\) for the first parameter
\(m_{ \oplus } ( \phi )\) of the conflict pair is
\(\varepsilon _1\) = 0.42.
(2)
Threshold
\(\varepsilon _2\)
for the second parameter
\(DifBet^{m1}_{m2}\)
:
Using the mass functions of the critical case defined in (1) we will have:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \end{aligned}$$
(12)
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \end{aligned}$$
(13)
Note that:
$$\begin{aligned} b + {m_2 (\varOmega )} = 1 - m_2 (condition2) = 0.35 \end{aligned}$$
(14)
Hence:
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \le 0.35 . \end{aligned}$$
(15)
We have also:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \ge 0.65 \end{aligned}$$
(16)
From (
16) and (
15), we conclude then:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = |BetP_{m1} (condition1) - BetP_{m2} (condition1)| \ge 0.3 \end{aligned}$$
(17)
Since we have only two singletons (
i.e. condition1 and condition2), we do not have to look for the maximum of
\(DifBet^{m1}_{m2}\) since it will be the same for condition2. Hence, we obtain:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = DifBet^{m1}_{m2} (condition2) \ge 0.3 \end{aligned}$$
(18)
From (
18) we can see that the threshold
\(\varepsilon _2\) for the second parameter is 0.3.
(3)
Common threshold
\(\varepsilon \)
for both parameters:
For the sake of ease and precaution, we choose a common threshold
\(\varepsilon \) for both parameters of the conflict pair by taking the smallest value between
\(\varepsilon _1\) and
\(\varepsilon _2\) as follows:
$$\begin{aligned} \varepsilon _{Small} = argmin \lbrace \varepsilon _1, \varepsilon _2 \rbrace = 0.3 \end{aligned}$$
(19)
Threshold
\(\varepsilon _1\)
for the parameter
\(m_{ \oplus } ( \phi )\)
of the conflict pair:
We consider the critical situation between small and medium conflict:
$$\begin{aligned} m_{1}(condition1)&= 0.65 \quad m_{1}(condition2) = a \\ m_{2}(condition1)&= b \quad m_{2}(condition2) = 0.65 \end{aligned}$$
With
\(condition1 = \overline{condition2}\) and a, b
\(\in \) [0, 0.35]. It is obvious that the smallest value of
\(m_{ \oplus } ( \phi )\) is 0.42, whatever the values of a and b are. Hence the threshold
\(\varepsilon _1\) for the first parameter
\(m_{ \oplus } ( \phi )\) of the conflict pair is
\(\varepsilon _1\) = 0.42.
Threshold
\(\varepsilon _2\)
for the second parameter
\(DifBet^{m1}_{m2}\)
:
Using the mass functions of the critical case defined in (1) we will have:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \end{aligned}$$
(12)
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \end{aligned}$$
(13)
Note that:
$$\begin{aligned} b + {m_2 (\varOmega )} = 1 - m_2 (condition2) = 0.35 \end{aligned}$$
(14)
Hence:
$$\begin{aligned} BetP_{m2}(condition1) = b + ( \frac{m_2 (\varOmega )}{2} ) \le 0.35 . \end{aligned}$$
(15)
We have also:
$$\begin{aligned} BetP_{m1}(condition1) = 0.65 + ( \frac{m_1 (\varOmega )}{2} ) \ge 0.65 \end{aligned}$$
(16)
From (
16) and (
15), we conclude then:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = |BetP_{m1} (condition1) - BetP_{m2} (condition1)| \ge 0.3 \end{aligned}$$
(17)
Since we have only two singletons (
i.e. condition1 and condition2), we do not have to look for the maximum of
\(DifBet^{m1}_{m2}\) since it will be the same for condition2. Hence, we obtain:
$$\begin{aligned} DifBet^{m1}_{m2} (condition1) = DifBet^{m1}_{m2} (condition2) \ge 0.3 \end{aligned}$$
(18)
From (
18) we can see that the threshold
\(\varepsilon _2\) for the second parameter is 0.3.
Common threshold
\(\varepsilon \)
for both parameters:
For the sake of ease and precaution, we choose a common threshold
\(\varepsilon \) for both parameters of the conflict pair by taking the smallest value between
\(\varepsilon _1\) and
\(\varepsilon _2\) as follows:
$$\begin{aligned} \varepsilon _{Small} = argmin \lbrace \varepsilon _1, \varepsilon _2 \rbrace = 0.3 \end{aligned}$$
(19)
Threshold
\(\varepsilon _{Medium}\)
for Medium conflict:
Following the same reasoning, we obtain the threshold
\(\varepsilon _{Medium} = 0.6\).
