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The best/worst ``envelope'' surrounding an actual Re/Pre curve is
related to a similar comparison known as **NORMALIZED RECALL** [Rocchio66] . Imagine plotting the
*fraction* of relevant documents retrieved as a function of the
fraction of the total number of documents retrieved. Such a function is
plotted in Figure *(figure)* . Comparing the area between the
actual retrieval and worst-case (colored \yellow) to the total area
between best and worst cases (that above the \blue region) is very much
like the best-/worst-case envelope of Figure *(FOAref)* . We can
derive expressions for this area. Let $r_i$ be the hitlist rank of the
$i$-th relevant document. Then (if we define $r_0=0$): \beq
\mathname{Actual} =
\sum_{i=1}^{\mathname{NRel}}(r_i-r_{i-1}){i\over{\mathname{NRel}}} \eeq
In the best case $r_i=i$: \mathname{Best} &=&
\sum_{i=1}^{\mathname{NRel}} {(i-(i-1))\cdot i\over{\mathname{NRel}}}
\nonumber \\ &=&{1\over\mathname{NRel}}\sum_{i=1}^{\mathname{NRel}}i
\nonumber \\ &=&{\mathname{NRel}+1\over2} when $\mathname{NRel}
\rightarrow$. In the worst case $r_i=NDoc-NRel+i$: \mathname{Worst}&=&
\sum_{i=1}^{\mathname{NRel}} {{((\mathname{NDoc}-\mathname{NRel}+i) -
(\mathname{NDoc}-\mathname{NRel}+i-1))\cdot i}\over \hbox{NRel}}
\nonumber \\ &=& \hbox{NDoc}-\mathname{NRel}

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