On the continuous probability distribution attribute weight of belief rule base model

Abstract

In current researches on belief rule base (BRB), input parameters are tended to be expressed in the form of quantitative values through expert knowledge combined with optimization methods. A singular quantitative value fails to capture the statistical properties, leading to irrational outcomes. Therefore, an attempt on attribute weights is made in this paper, and a new model with probability distribution attribute weights (pdw) called BRB-pdw is proposed. The combination of two attributes is in detail discussed, where attribute weights are described as random variables with specific probability distribution. To characterize the output of probability distribution attribute weight, a new concept of expectation of activation weight is proposed. In addition, the BRB-pdw is extended to multiple attributes to demonstrate its universality. Furthermore, fundamental properties and characteristics of the BRB-pdw are further validated by rigorous mathematical derivation. Finally, practicability of the BRB-pdw is validated with NASA lithium battery open dataset, and experiments show that the BRB-pdw model is more robust while maintaining precision.

Appendices

Appendix 1

For the convenience of proof, the infer of BRB’s activation weight based on Eq. (3) can be expressed as $G(\delta_{1} ,\delta_{2} ) = \left[ {\theta_{k} \prod\nolimits_{m = 1}^{M} {(a_{m}^{k} )^{{\overline{\delta }_{m} }} } } \right]/\left[ {\sum\nolimits_{l = 1}^{L} {\theta_{l} \prod\nolimits_{m = 1}^{M} {(a_{m}^{k} )^{{\overline{\delta }_{m} }} } } } \right]$. Meanwhile, $f(\delta_{1} ,\delta_{2} ) = f(\delta_{1} )f(\delta_{2} )$, $f(\delta_{i} ) = 1/(\delta_{i}^{ + } - \delta_{i}^{ - } ), \, \delta_{i}^{ - } < \delta_{i} < \delta_{i}^{ + }$.

1.1 A. Proof of Theorem 2

Proof:$E(w_{k} )$ can be expressed in the following form:

$$E(w_{k} ) = \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )} } d\delta_{1} d\delta_{2}$$

(52)

According to the property of BRB, $0 \le G(\delta_{1} ,\delta_{2} ) < 1$ can be obtained. Based on the property of the PDF, there is

$$\begin{gathered} \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {f(\delta_{1} ,\delta_{2} )\,} } d\delta_{1} d\delta_{2} = \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} \hfill \\ \, = \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}} \times \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\left( 1 \right)} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} \hfill \\ \, = \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}} \times (\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } ) \hfill \\ \, = 1 \hfill \\ \end{gathered}$$

(53)

The following relationship can be obtained

$$\left\{ {\begin{array}{*{20}l} {\left[ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {f(\delta_{1} ,\delta_{2} )} } \times \left( 0 \right){\text{d}}\delta_{1} {\text{d}}\delta_{2} = 0} \right]} \hfill & { \le \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )\,} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill \\ {\left[ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {f(\delta_{1} ,\delta_{2} )} } \times \left( 1 \right){\text{d}}\delta_{1} {\text{d}}\delta_{2} = 1} \right]} \hfill & { > \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill \\ \end{array} } \right.$$

(54)

Then, we can get

$$0 \le \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )\,} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} < 1$$

(55)

Therefore, it can be proven that $E(w_{k} )$ is bounded and always lies within the interval $[0,1)$. The lower bound of $E(w_{k} )$ is 0, while the upper bound of $E(w_{k} )$ needs to be computed specifically based on numerical analysis.

1.2 B. Proof of Theorem 3

Proof: The cumulative sum of $E(w_{k} )$ for $L$ rules can be profiled by

$$\sum\limits_{k = 1}^{L} {E(w_{k} )} = \sum\limits_{k}^{L} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )\,} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} ;}$$

(56)

as mentioned in Theorem 2 for $\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {f(\delta_{1} ,\delta_{2} )} } d\delta_{1} d\delta_{2} = 1$, Eq. 56 can be rewritten as

$$\begin{gathered} \sum\limits_{k = 1}^{L} {E(w_{k} )} = \sum\limits_{k}^{L} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )f(\delta_{1} ,\delta_{2} )} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill \\ \, = 1 \times \sum\limits_{k}^{L} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {G(\delta_{1} ,\delta_{2} )} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill \\ \, = \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{1} (a_{1}^{1} )^{{\delta_{1} }} (a_{2}^{1} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }}} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} + \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{2} (a_{1}^{2} )^{{\delta_{1} }} (a_{2}^{2} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }}} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} + \cdot \cdot \cdot + \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{L} (a_{1}^{L} )^{{\delta_{1} }} (a_{2}^{L} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }}\,} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} = \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\left[ 1 \right]} } \,{\text{d}}\delta_{1} {\text{d}}\delta_{2} \hfill \\ \, = (\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } ) \hfill \\ \end{gathered}$$

(57)

Since $(\delta_{1}^{ + } - \delta_{1}^{ - } ) > 0$ and $(\delta_{2}^{ + } - \delta_{2}^{ - } ) > 0$ always hold, there is $\sum\nolimits_{k = 1}^{L} {E(w_{k} )} > 0$. Since $\delta_{1} \in [\delta_{1}^{ - } ,\delta_{1}^{ + } ], \, \delta_{2} \in [\delta_{2}^{ - } ,\delta_{2}^{ + } ]$, when $\delta_{1}^{ - } = \delta_{2}^{ - } = 0$, $\delta_{1}^{ + } = \delta_{2}^{ + } = 1$, there is $\sum\nolimits_{k = 1}^{L} {E(w_{k} )} \equiv 1$. When $\delta_{1} \in (0,1), \, \delta_{2} \in (0,1)$, there is $\sum\nolimits_{k = 1}^{L} {E(w_{k} )} < 1$. Therefore, $0 < \sum\nolimits_{k = 1}^{L} {E(w_{k} )} \le 1$ is always satisfied.

1.3 C. Proof of Theorem 4

Proof: Suppose that $m$ and $M$ are lower and upper bounds on $G(\delta_{1} ,\delta_{2} )$, because $(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } ) > 0$, the following inequality can be obtained

$$\frac{m}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}} \le \frac{{G(\delta_{1} ,\delta_{2} )}}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}} \le \frac{M}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}$$

(58)

According to the property of double integration, there is

$$\left\{ {\begin{array}{*{20}l} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{m}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill & { \le \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{G(\delta_{1} ,\delta_{2} )}}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{1} {\text{d}}\delta_{2} } } } \hfill \\ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{G(\delta_{1} ,\delta_{2} )}}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill & { \le \int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{M}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{1} {\text{d}}\delta_{2} } } } \hfill \\ \end{array} } \right.$$

(59)

where

$$\left\{ {\begin{array}{*{20}l} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{G(\delta_{1} ,\delta_{2} )}}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill & { = E(w_{k} )} \hfill \\ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{m}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} } \hfill & { = m} \hfill \\ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{M}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{1} {\text{d}}\delta_{2} } } } \hfill & { = M} \hfill \\ \end{array} } \right.$$

(60)

Therefore, $m \le E(w_{k} ) \le M$ can be proven. Hence, $E(w_{k} )$ is a definite value within the interval $[m,M]$, and $G(\delta_{1} ,\delta_{2} )$ is a continuous function. According to the intermediate value theorem for continuous functions on a closed interval, there must be at least one point $(\delta_{\tau 1} ,\delta_{\tau 2} )$ such that $G(\delta_{\tau 1} ,\delta_{\tau 2} ) = E(w_{k} )$. Theorem 4 is proved.

Appendix 2

For $\partial p_{n,e(L)} /\partial E(w_{k} )$, when each rule is complete, $\sum\nolimits_{j = 1}^{N} {\beta_{j,i} } \equiv 1$ can be obtained. Therefore, $\sum\nolimits_{j = 1}^{N} {\beta_{j,i} } \equiv 1$ has no effect on calculation result of the partial derivative, and the output belief degree $p_{n,e(L)}$ can be abbreviated as

$$p_{n,e(L)} = \frac{{\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{n,q} - 1)E(w_{q} ) + 1} \right] - \prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } }}{{\sum\nolimits_{j = 1}^{N} {\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right] - N\prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } } }}$$

(61)

Let $p_{n,e(L)} = (A - B)/(C - D)$, then the partial derivative of $A,B,C \, and{\text{ D }}$ with respect to $E(w_{k} )$ is

$$\left\{ {\begin{array}{*{20}l} {A{\prime} = \prod\limits_{q \ne k}^{L} {\left[ {(\beta_{n,q} - 1)E(w_{q} ) + 1} \right](\beta_{n,k} - 1)} , \, } \hfill & {B{\prime} = \prod\limits_{q \ne k}^{L} {(E(w_{q} ) - 1)} } \hfill \\ {C{\prime} = \sum\limits_{j = 1}^{N} {\prod\limits_{q \ne k}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right](\beta_{j,k} - 1),} } } \hfill & {D{\prime} = N\prod\limits_{q \ne k}^{L} {(E(w_{q} ) - 1)} } \hfill \\ \end{array} } \right.$$

(62)

Therefore, $\partial p_{n,e(L)} /\partial E(w_{k} )$ can be calculated as follows:

$$\begin{aligned} \frac{{\partial p_{n,e(L)} }}{{\partial E(w_{k} )}} \\ \, & = \frac{{(A - B){\prime} (C - D) - (A - B)(C - D){\prime} }}{{(C - D)^{2} }} \\ \, & = \frac{{\prod\nolimits_{q \ne k}^{L} {\left[ {(\beta_{n,q} - 1)E(w_{q} ) + 1} \right](\beta_{n,k} - 1)} - \prod\nolimits_{q \ne k}^{L} {(E(w_{q} ) - 1)} }}{{\sum\nolimits_{j = 1}^{N} {\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right] - N\prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } } }} \\ \, & \quad - \frac{{\left\{ {\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{n,q} - 1)E(w_{q} ) + 1} \right] - \prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } } \right\}}}{{\left\{ {\sum\nolimits_{j = 1}^{N} {\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right] - N\prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } } } \right\}^{2} }} \\ \, & \quad \times \frac{{\left\{ {\sum\nolimits_{j = 1}^{N} {\prod\nolimits_{q \ne k}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right](\beta_{j,k} - 1)} } - N\prod\nolimits_{q \ne k}^{L} {(E(w_{q} ) - 1)} } \right\}}}{{\left\{ {\sum\nolimits_{j = 1}^{N} {\prod\nolimits_{q = 1}^{L} {\left[ {(\beta_{j,q} - 1)E(w_{q} ) + 1} \right] - N\prod\nolimits_{q = 1}^{L} {(1 - E(w_{q} ))} } } } \right\}^{2} }} \\ \end{aligned}$$

(63)

For $\partial E(w_{k} )/\partial \delta_{i}^{ + }$, first calculate $\partial E(w_{k} )/\partial \delta_{1}^{ + }$, and there is

$$\begin{gathered} \frac{{\partial E(w_{k} )}}{{\partial \delta_{1}^{ + } }} = \frac{{\partial \left[ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } } \right]}}{{\partial \delta_{1}^{ + } }} \hfill \\ \, = \frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1}^{ + } }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1}^{ + } }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )^{2} }} - \frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} \times (a_{1}^{k} )^{{\delta_{1} }} \times (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} \times (a_{1}^{l} )^{{\delta_{1} }} \times (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{2} {\text{d}}\delta_{1} }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )^{2} }} \hfill \\ \end{gathered}$$

(64)

According to the definition of the derivative of an integral, $\partial E(w_{k} )/\partial \delta_{1}^{ - }$ can be calculated by

$$\begin{gathered} \frac{{\partial E(w_{k} )}}{{\partial \delta_{1}^{ - } }} = \frac{{\partial \left[ {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } } \right]}}{{\partial \delta_{1}^{ - } }} \hfill \\ \, = \frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} \times (a_{1}^{k} )^{{\delta_{1} }} \times (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} \times (a_{1}^{l} )^{{\delta_{1} }} \times (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}} } {\text{d}}\delta_{2} {\text{d}}\delta_{1} }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )^{2} }} - \frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1}^{ - } }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1}^{ - } }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } }}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )^{2} }} \hfill \\ \end{gathered}$$

(65)

Similarly, $\partial E(w_{k} )/\partial \delta_{2}^{ + }$ can be calculated by

$$\begin{gathered} \frac{{\partial E(w_{k} )}}{{\partial \delta_{2}^{ + } }} = \frac{{\partial \left[ {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } } \right]}}{{\partial \delta_{2}^{ + } }} \hfill \\ \, = \frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2}^{ + } }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2}^{ + } }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )^{2} }} - \frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} \times (a_{1}^{k} )^{{\delta_{1} }} \times (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} \times (a_{1}^{l} )^{{\delta_{1} }} \times (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )^{2} }} \hfill \\ \end{gathered}$$

(66)

$\partial E(w_{k} )/\partial \delta_{2}^{ - }$ can be calculated by

$$\begin{aligned} \frac{{\partial E(w_{k} )}}{{\partial \delta_{2}^{ - } }} & = \frac{{\partial \left[ {\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )}}{\text{d}}\delta_{2} } } \right]}}{{\partial \delta_{2}^{ - } }} \\ \, & = \frac{{\int_{{\delta_{2}^{ - } }}^{{\delta_{2}^{ + } }} {\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} \times (a_{1}^{k} )^{{\delta_{1} }} \times (a_{2}^{k} )^{{\delta_{2} }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} \times (a_{1}^{l} )^{{\delta_{1} }} \times (a_{2}^{l} )^{{\delta_{2} }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}} } {\text{d}}\delta_{1} {\text{d}}\delta_{2} }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )^{2} }} - \frac{{\int_{{\delta_{1}^{ - } }}^{{\delta_{1}^{ + } }} {\frac{{\theta_{k} (a_{1}^{k} )^{{\delta_{1} }} (a_{2}^{k} )^{{\delta_{2}^{ - } }} }}{{\sum\nolimits_{l = 1}^{L} {\theta_{l} (a_{1}^{l} )^{{\delta_{1} }} (a_{2}^{l} )^{{\delta_{2}^{ - } }} } }} \times \frac{1}{{(\delta_{1}^{ + } - \delta_{1}^{ - } )}}{\text{d}}\delta_{1} } }}{{(\delta_{2}^{ + } - \delta_{2}^{ - } )^{2} }} \\ \end{aligned}$$

(67)

In view of this, $\partial E(w_{k} )/\partial \delta_{i}^{ + }$ can be calculated iteratively.