Table 10.

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

Distance function	Formula (continuous form)	Formula (discrete form)
Bhattacharyya	D B(P,Q)=−L n(B C(P,Q))	D B(P,Q)=−L n(B C(P,Q))
	, $\mathit{\text{BC}}(P,Q)=\int \sqrt{P\left(x\right)Q\left(x\right)}\mathit{\text{dx}}$	$,\mathit{\text{BC}}(P,Q)=\sum \sqrt{P\left(x\right)Q\left(x\right)}$
Hellinger	$D_H=\sqrt{2\int {\left(P\right(x)-Q(x\left)\right)}^2\mathit{\text{dx}}}$	$D_H(P,Q)=\sqrt{2\sum _{k=1}^d{\left(P\right(x_k)-Q(x_k\left)\right)}^2}$
	$=2\sqrt{1-\int \sqrt{P\left(x\right)Q\left(x\right)}\mathit{\text{dx}}}$	$=2\sqrt{1-\sum _{k=1}^d\sqrt{P\left(x_k\right)Q\left(x_k\right)}}$
Jaccard	-	D Jac=1−S Jac
		$S_{\mathit{\text{Jac}}}=\frac{\sum _{k=1}^{d}P\left(x_k\right)\times Q\left(x_k\right)}{\sum _{k=1}^{d}P{\left(x_k\right)}^2+\sum _{k=1}^{d}Q{\left(x_k\right)}^2-\sum _{k=1}^{d}P\left(x_k\right)\mathrm{.Q}\left(x_k\right)}$