A few aggregation operators

Aggregation Operators:

Definition of a few aggregation operators:

(Direct links to the operators: AM, WM, OWA, WOWA)

• Arithmetic mean:

  AM(a₁,...,a_N) =  Σ_i=1^N a_i / N

• Weighted mean:

  WM(a₁,...,a_N) =  Σ_i=1^N p_i a_i

  In this definition, p_i corresponds to the weight or relevance of the i-th information source. Weights should be positive and add to one.

• OWA Operator:

  OWA(a₁,...,a_N) =  Σ_i=1^N w_i a_σ(i)

  where σ corresponds to a permutation of the a_i so that they are ordered from the largest one to the lowest one. So, a₁ will be the largest of the a_i and a_N will be the lowest of the a_i.In this definition, p_i correspond to the weight of the ith data after ordering them. In this way, weights permits to express us whether we want to give importance to low, high or central data. Weights should be positive and add to one.
  References: The OWA operator was introduced by R. R. Yager in 1988. The full reference is:
  

  Yager, R.R., "On Ordered Weighted Averaging Aggregation Operators in Multi-Criteria Decision Making," IEEE Transactions on Systems, Man and Cybernetics 18, 183-190, 1988.

• WOWA Operator:

  WOWA(a₁,...,a_N) =  Σ_i=1^N ω_i a_σ(i)

  where σ corresponds to a permutation of the a_i so that they are ordered from the largest one to the lowest one,and where &omega is defined by:

  ω_i = w^*(Σ_j≤i p_σ(j)) - w^*(Σ_j<i p_σ(j))

  where, as in the case of the OWA operator, w_i correspond to the weight of the ith data after ordering them (w is a weighting vector as in the weighted mean), and where w^* is a function that interpolates the points (i/n, Σ_j≤ip_j) and the point (0,0), and that should be a straight line if the points can be interpolated in this way.
  Properties: The WOWA operator generalizes the weighted mean and the OWA operator. By means of considering the two weighting vectors p and w, the WOWA operator permits us the fusion of a given set of numerical values considering both the importance of the sources that have supplied the values (as in the weighted mean, i.e., the weight p in our definition) and the degree of compensation or tradeoff between large and small values (as in the OWA operator, the weight w in our definition).
  From a formal point of view, the WOWA operator is a particular case of Choquet integral (using a particular type of measure: a distorted probability).
  References: The WOWA (Weighted Ordered Weighted Aggregation) operator was introduced by V. Torra. See:

  V. Torra, The Weighted OWA operator, Int. J. of Intel. Systems, 12 (1997) 153-166. Paper @ interscience.wiley

  The interpolation method used to build w^* is described in:

  V. Torra, The WOWA operator and the interpolation function W*: Chen and Otto's interpolation method revisited, Fuzzy Sets and Systems, 113:3 (2000) 389-396. Paper @ Sciencedirect

  We present a discussion of different alternative interpolation methods in:

  V. Torra, Z. Lv, On the WOWA operator and its interpolation function, Int. J. of Intelligent Systems, 24:10 (2009) 1039-1056. Paper @ Wiley online

  Implementation: