Table 10 Distance functions to measure dissimilarity between probability density functions of stochastic observation and stochastic predicted outputs

Bhattacharyya

D _B (P,Q)=−L n(B C(P,Q))

D _B (P,Q)=−L n(B C(P,Q))

, \(BC(P,Q)=\int \sqrt {P(x)Q(x)}dx \)

\(,BC(P,Q)=\sum \sqrt {P(x)Q(x)} \)

Hellinger

\( D_{H}=\sqrt { 2\int {(P(x)-Q(x))^{2}}dx} \)

\( D_{H}(P,Q)=\sqrt { 2\sum _{k=1}^{d} {(P(x_{k})-Q(x_{k}))^{2}}} \)

\(= 2\sqrt {1-\int \sqrt {P(x)Q(x)}dx} \)

\( = 2\sqrt {1-\sum _{k=1}^{d} \sqrt {P(x_{k})Q(x_{k})}} \)

Jaccard

-

D _Jac =1−S _Jac

\( S_{Jac} = \frac {\sum _{k=1}^{d}{P(x_{k})\times Q(x_{k})}}{ \sum _{k=1}^{d}{P(x_{k})^{2}} + \sum _{k=1}^{d}{Q(x_{k})^{2}}- \sum _{k=1}^{d}{P(x_{k}).Q(x_{k})}} \)