Page 48 - Contributed Paper Session (CPS) - Volume 3

CPS1943 Nandish C. et al.
There are two major aspects that one needs to consider. Firstly from a pool of
available attributes, one has to choose variables which are logically important,
and pertain to non-overlapping information. Secondly the individual weights
of the attributes are to be estimated in such a way which would maximally
differentiate the players. Therefore, it is necessary that the estimated weights
maximize the ratio of the variation of performances among different players
to the variation of performances of individual players, i.e., the ratio of
between-group variation to the within group variation considering the
performances of a particular player as a group. This can be formally written as
the following optimization problem:

and it has a closed form solution ( , . . . , ) = ̂ , where ̂ is the eigen-
̂
̂

1
1
1
vector corresponding to the greatest among the non-zero eigen-values of
2
2
−1 , = ∑ (̅ − ̅) and = ∑ ∑ ( − ̅ ) . See Johnson

=1
=1
=1
and Wichern (2007, p-610) for more details.
Note that the solution to this unconstrained optimization problem may
lead to some of the weights having a negative value which may not be the
appropriate sign in many cases depending upon the nature of the associated
variable. Though that would maximally discriminate the players, it is not
guarantee that the sign is appropriate for the highest score is associated with
the best player. Since we are interested in obtaining ranks using the scores,
our purpose would be defeated. Hence, it is necessary to add some constraints
that ensure that all the weights are positive, for all those variables that
positively impact the player’s abilities. Similarly, for those variables that
negatively impact the player’s abilities, the corresponding weights should be
negative. Without loss of generality, we consider that all the performance
variables under consideration have positive impact on the player’s abilities for
further discussion. In order to fulfill our purpose, we consider the following
constrained optimization problem:

However, there is no closed form solution for this, and one needs to employ
numerical optimisation methods for finding optimal solution (Nocedal and
Wright, 1999).
For the ease of interpretation, we consider the normalized weights
= , so that the score = ∑ () , for = 1, . . . , , and

∑ =1 =1
= 1, . . . , , lies in the range [0, 1]. The rating is therefore obtained by sorting

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