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

From: A framework for evaluating epidemic forecasts

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))
  , \(BC(P,Q)=\int \sqrt {P(x)Q(x)}dx \) \(,BC(P,Q)=\sum \sqrt {P(x)Q(x)} \)
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
   \( S_{Jac} = \frac {\sum _{k=1}^{d}{P(x_{k})\times Q(x_{k})}}{ \sum _{k=1}^{d}{P(x_{k})^{2}} + \sum _{k=1}^{d}{Q(x_{k})^{2}}- \sum _{k=1}^{d}{P(x_{k}).Q(x_{k})}} \)