Concepts
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structures are decomposable
...A further simplification of the measurement function may be achieved by requiring a special kind of non interaction of the components which has become known as additive independence
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P where F is simply the addition function
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P It can be shown that starting at the other end given an additively independent representation the properties defined in 1 and 3,and the Archimedean property are necessary
...Here the term [[Phi]]1 [[Phi]]2 is referred to as the interaction term,its absence accounts for the non interaction in the previous condition
...We are now in a position to state the main representation theorem
...Theorem Suppose <R x P,>>is an additive conjoint structure,then there exist functions,[[Phi]]1 from R,and [[Phi]]2 from P into the real numbers such that,for all R,R [[propersubset]]R and P,P [[propersubset]]P:R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P If [[Phi]]i []are two other functions with the same property,then there exist constants [[Theta]]>0,[[gamma]]1,and [[gamma]]2 such that [[Phi]]1 [][[Theta]][[Phi]]1 [[gamma]]1 [[Phi]]2 [][[Theta]][[Phi]]2 [[gamma]]2 The proof of this theorem may be found in Krantz et al
...Let us stop and take stock of this situation
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The model We start by examining the structure which it is reasonable to assume for the measurement of effectiveness
...If R is the set of possible recall values and P is the set of possible precision values then we are interested in the set R x P with a relation on it
...Definition 1
...1 Connectedness:either e 1 >e 2 or e 2 >e 1 2 Transitivity:if e 1 >e 2 and e 2 >e 3 then e 1 >e 3 We insist that if two pairs can be ordered both ways then R 1,P 1 R 2,P 2,i
...We now turn to a second condition which is commonly called independence
...Definition 2
...All we are saying here is,given that at a constant recall precision we find a difference in effectiveness for two values of precision recall then this difference cannot be removed or reversed by changing the constant value
...We now come to a condition which is not quite as obvious as the preceding ones
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conjoint structure
...The analysis is not limited to the two factors precision and recall,it could equally well be carried out for say the pair fallout and recall
...Presentation of experimental results In my discussion of micro,macro evaluation,and expected search length,various ways of averaging the effectiveness measure of the set of queries arose in a natural way
...In this section the discussion will be restricted to single number measures such as a normalised symmetric difference,normalised recall,etc
...The measurements we have therefore are Za Q 1,Za Q 2,... |
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assume that we have the points on l 1 and l 2 a but wish to deduce the relative ordering in between these two lines
...Definition 3 Thomsen condition
...Intuitively this can b e reasoned as follows
...The fourth condition is one concerned with the continuity of each component
...Definition 4 Restricted Solvability
...1 whenever R,R,R [[propersubset]]R and P,P [[propersubset]]P for which R,P >R,P >R,P then there exists R [[propersubset]]R s
...2 a similar condition holds on the second component
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one unit of precision for an increase of one unit of recall,but will not sacrifice another unit of precision for a further unit increase in recall,i
...R 1,P 1 >R,P but R 1,P >R 2,P 1 We conclude that the interval between R 1 and R exceeds the interval between P and P 1 whereas the interval between R 1 and R 2 is smaller
...Finally,we incorporate into our measurement procedure the fact that users may attach different relative importance to precision and recall
...Definition 6
...Can we find a function satisfying all these conditions?If so,can we also interpret it in an intuitively simple way?The answer to both these questions is yes
...The scale functions are therefore,[[Phi]]1 P [[alpha]]1 P,and [[Phi]]2 R 1 [[alpha]]1 R
...We now have the effectiveness measure
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