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Normalized recall and precision

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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