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C. Valsecchi and R. Todeschini
To measure the degree of total ordering provided by PWR or, in other words the
degree of not-distinguishable objects, we used the standardized Shannon entropy H:
H =
−
q
k=1
p k · log 2 (p k )
log 2 (n)
0 ≤ H ≤ 1
( 4 )
where q is the number of equivalent ranking levels (i.e., the number of the different
PWR values, rounded to two decimal digits), and p k is the fraction of objects in the
k-th level. If all the objects have different PWR values, a total order without draws
is obtained and the following relationships hold:
q = n
p k = 1/n
∀k, k = 1, 2, . . . , q
H = 1
On the other side, if all the objects have the same PWR value, no ranking is
obtained, and the following relationships hold:
q = 1
p 1 = n/n = 1
H = 0
In general, the standardized Shannon entropy is expected to decrease with the
increase of the threshold value, since increasing the smoothing of the tournament
table leads to have more objects placed at the same ranking level.
2.1.2 Deep Ranking Analysis
The set of families of tournament tables obtained considering different thresholds
t*(w) is automatically pruned by eliminating those matrices whose ranking has a
correlation higher than 0.995 (settled as default value) with those of the previous
ones.
By exploiting the PWR rankings obtained with the different thresholds of the
tournament table, a deep ranking analysis is performed by Principal Component
Analysis (PCA) on the matrix having n rows (the objects) and K columns (the
different PWR rankings). The first component is enough to explain almost the whole
variability of the data. The scores of the first component represent the consensus
ranking (CPWR) of the objects, while the loadings explain the role played by each
threshold in determining the global ranking scores. Thus, the result is a deep ranking
analysis able to summarize all the different PWR rankings.
2.1.3 Retro-regression Analysis
Given the CPWR or a PWR ranking of the objects, a regression can be carried
out to evaluate how the original ordering criteria encoded in the X data matrix are
C. Valsecchi and R. Todeschini
To measure the degree of total ordering provided by PWR or, in other words the
degree of not-distinguishable objects, we used the standardized Shannon entropy H:
H =
−
q
k=1
p k · log 2 (p k )
log 2 (n)
0 ≤ H ≤ 1
( 4 )
where q is the number of equivalent ranking levels (i.e., the number of the different
PWR values, rounded to two decimal digits), and p k is the fraction of objects in the
k-th level. If all the objects have different PWR values, a total order without draws
is obtained and the following relationships hold:
q = n
p k = 1/n
∀k, k = 1, 2, . . . , q
H = 1
On the other side, if all the objects have the same PWR value, no ranking is
obtained, and the following relationships hold:
q = 1
p 1 = n/n = 1
H = 0
In general, the standardized Shannon entropy is expected to decrease with the
increase of the threshold value, since increasing the smoothing of the tournament
table leads to have more objects placed at the same ranking level.
2.1.2 Deep Ranking Analysis
The set of families of tournament tables obtained considering different thresholds
t*(w) is automatically pruned by eliminating those matrices whose ranking has a
correlation higher than 0.995 (settled as default value) with those of the previous
ones.
By exploiting the PWR rankings obtained with the different thresholds of the
tournament table, a deep ranking analysis is performed by Principal Component
Analysis (PCA) on the matrix having n rows (the objects) and K columns (the
different PWR rankings). The first component is enough to explain almost the whole
variability of the data. The scores of the first component represent the consensus
ranking (CPWR) of the objects, while the loadings explain the role played by each
threshold in determining the global ranking scores. Thus, the result is a deep ranking
analysis able to summarize all the different PWR rankings.
2.1.3 Retro-regression Analysis
Given the CPWR or a PWR ranking of the objects, a regression can be carried
out to evaluate how the original ordering criteria encoded in the X data matrix are