Ordered weighted averaging

An OWA operator of dimension $\backslash \; n$ is a mapping $F:\; \backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ that has an associated collection of weights $\backslash \; W\; =\; [w\_1,\; \backslash ldots,\; w\_n]$ lying in the unit interval and summing to one and with

:$F(a\_1,\; \backslash ldots\; ,\; a\_n)\; =\; \backslash sum\_\{j=1\}^n\; w\_j\; b\_j$

where $b\_j$ is the j^th largest of the $a\_i$.

By choosing different W one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the b_j.

:$\backslash \; F(a\_1,\; \backslash ldots,\; a\_n)\; =\; \backslash max(a\_1,\; \backslash ldots,\; a\_n)$ if $\backslash \; w\_1\; =\; 1$ and $\backslash \; w\_j\; =\; 0$ for $j\; \backslash ne\; 1$

:$\backslash \; F(a\_1,\; \backslash ldots,\; a\_n)\; =\; \backslash min(a\_1,\; \backslash ldots,\; a\_n)$ if $\backslash \; w\_n\; =\; 1$ and $\backslash \; w\_j\; =\; 0$ for $j\; \backslash ne\; n$

:

:$\backslash \; F(a\_1,\; \backslash ldots,\; a\_n)\; =\; \backslash mathrm\{average\}(a\_1,\; \backslash ldots,\; a\_n)$ if $\backslash \; w\_j\; =\; \backslash frac\{1\}\{n\}$ for all $j\; \backslash in\; [1,\; n]$

The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below.

Two features have been used to characterize the OWA operators. The first is the attitudinal character, also called orness. This is defined as

:$A-C(W)=\; \backslash frac\{1\}\{n-1\}\; \backslash sum\_\{j=1\}^n\; (n\; -\; j)\; w\_j.$

It is known that $A-C(W)\; \backslash in\; [0,\; 1]$.

In addition A − C(max) = 1, A − C(ave) = A − C(med) = 0.5 and A − C(min) = 0. Thus the A − C goes from 1 to 0 as we go from Max to Min aggregation. The attitudinal character characterizes the similarity of aggregation to OR operation(OR is defined as the Max).

The second feature is the dispersion. This defined as

:$H(W)\; =\; -\backslash sum\_\{j=1\}^n\; w\_j\; \backslash ln\; (w\_j).$

An alternative definition is $E(W)\; =\; \backslash sum\_\{j=1\}^n\; w\_j^2\; .$ The dispersion characterizes how uniformly the arguments are being used.

The above Yager's OWA operators are used to aggregate the crisp values. Can we aggregate fuzzy sets in the OWA mechanism? The

Type-1 OWA operators have been proposed for this purpose.

S.-M. Zhou, F. Chiclana, R. I. John and J. M. Garibaldi, "Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers," Fuzzy Sets and Systems, Vol.159, No.24, pp. 3281–3296, 2008 [https://dx.doi.org/10.1016/j.fss.2008.06.018]

S.-M. Zhou, R. I. John, F. Chiclana and J. M. Garibaldi, "On aggregating uncertain information by type-2 OWA operators for soft decision making," International Journal of Intelligent Systems, vol. 25, no.6, pp. 540–558, 2010.[https://dx.doi.org/10.1002/int.20420]

So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.

The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows:

Given the n linguistic weights $\backslash left\backslash \{\; \{W^i\}\; \backslash right\backslash \}\_\{i\; =1\}^n$ in the form of fuzzy sets defined on the domain of discourse $U\; =\; [0,\backslash ;\backslash ;1]$, then for each $\backslash alpha\; \backslash in\; [0,\backslash ;1]$, an $\backslash alpha$-level type-1 OWA operator with $\backslash alpha$-level sets $\backslash left\backslash \{\; \{W\_\backslash alpha\; ^i\; \}\; \backslash right\backslash \}\_\{i\; =\; 1\}^n$ to aggregate the $\backslash alpha$-cuts of fuzzy sets $\backslash left\backslash \{\; \{A^i\}\; \backslash right\backslash \}\_\{i\; =1\}^n$ is given as

: $$

\Phi_\alpha \left( {A_\alpha ^1 , \ldots ,A_\alpha ^n } \right) =\left\{ {\frac{\sum\limits_{i = 1}^n {w_i a_{\sigma (i)} } }{\sum\limits_{i = 1}^n {w_i } }\left| {w_i \in W_\alpha ^i ,\;a_i } \right. \in A_\alpha ^i ,\;i = 1, \ldots ,n} \right\}

where $W\_\backslash alpha\; ^i=\; \backslash \{w|\; \backslash mu\_\{W\_i\; \}(w)\; \backslash geq\; \backslash alpha\; \backslash \},\; A\_\backslash alpha\; ^i=\backslash \{\; x|\; \backslash mu\; \_\{A\_i\; \}(x)\backslash geq\; \backslash alpha\; \backslash \}$, and $\backslash sigma\; :\backslash \{\backslash ;1,\; \backslash ldots\; ,n\backslash ;\backslash \}\; \backslash to\; \backslash \{\backslash ;1,\; \backslash ldots\; ,n\backslash ;\backslash \}$ is a permutation function such that $a\_\{\backslash sigma\; (i)\}\; \backslash ge\; a\_\{\backslash sigma\; (i\; +\; 1)\}\; ,\backslash ;\backslash forall\; \backslash ;i\; =\; 1,\; \backslash ldots\; ,n\; -\; 1$, i.e., $a\_\{\backslash sigma\; (i)\}$ is the $i$th largest

element in the set $\backslash left\backslash \{\; \{a\_1\; ,\; \backslash ldots\; ,a\_n\; \}\; \backslash right\backslash \}$.

The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals $\backslash Phi\; \_\backslash alpha\; \backslash left(\; \{A\_\backslash alpha\; ^1\; ,\; \backslash ldots\; ,A\_\backslash alpha\; ^n\; \}\; \backslash right)$:

$\backslash Phi\; \_\backslash alpha\; \backslash left(\; \{A\_\backslash alpha\; ^1\; ,\; \backslash ldots\; ,A\_\backslash alpha\; ^n\; \}\; \backslash right)\_\{-\}$ and $$

\Phi _\alpha \left( {A_\alpha ^1 , \ldots ,A_\alpha ^n } \right)_ {+},

where $A\_\backslash alpha\; ^i=[A\_\{\backslash alpha-\}^i,\; A\_\{\backslash alpha+\}^i],\; W\_\backslash alpha\; ^i=[W\_\{\backslash alpha-\}^i,\; W\_\{\backslash alpha+\}^i]$. Then membership function of resulting aggregation fuzzy set is:

:$\backslash mu\; \_\{G\}\; (x)\; =\; \backslash mathop\; \backslash vee\; \_\{\backslash alpha\; :x\; \backslash in\; \backslash Phi\; \_\backslash alpha\; \backslash left(\; \{A\_\backslash alpha\; ^1\; ,\; \backslash cdots$

,A_\alpha ^n } \right)_\alpha } \alpha

For the left end-points, we need to solve the following programming problem:

:$\backslash Phi\; \_\backslash alpha\; \backslash left(\; \{A\_\backslash alpha\; ^1\; ,\; \backslash cdots\; ,A\_\backslash alpha\; ^n\; \}\; \backslash right)\_\{-\}\; =\; \backslash min\backslash limits\_\{\backslash begin\{array\}\{l\}\; W\_\{\backslash alpha\; -\; \}^i\; \backslash le\; w\_i\; \backslash le\; W\_\{\backslash alpha\; +\; \}^i\; A\_\{\backslash alpha\; -\; \}^i\; \backslash le\; a\_i\; \backslash le\; A\_\{\backslash alpha\; +\; \}^i\; \backslash end\{array\}\}\; \backslash sum\backslash limits\_\{i\; =\; 1\}^n\; \{w\_i\; a\_\{\backslash sigma\; (i)\}\; /\; \backslash sum\backslash limits\_\{i\; =\; 1\}^n\; \{w\_i\; \}\; \}$

while for the right end-points, we need to solve the following programming problem:

:$\backslash Phi\; \_\backslash alpha\; \backslash left(\; \{A\_\backslash alpha\; ^1\; ,\; \backslash cdots\; ,\; A\_\backslash alpha\; ^n\; \}\; \backslash right)\_\{+\}\; =\; \backslash max\backslash limits\_\{\backslash begin\{array\}\{l\}\; W\_\{\backslash alpha\; -\; \}^i\; \backslash le\; w\_i\; \backslash le\; W\_\{\backslash alpha\; +\; \}^i\; A\_\{\backslash alpha\; -\; \}^i\; \backslash le\; a\_i\; \backslash le\; A\_\{\backslash alpha\; +\; \}^i\; \backslash end\{array\}\}\; \backslash sum\backslash limits\_\{i\; =\; 1\}^n\; \{w\_i\; a\_\{\backslash sigma\; (i)\}\; /\; \backslash sum\backslash limits\_\{i\; =$

1}^n {w_i } }

This paperS.-M. Zhou, F. Chiclana, R. I. John and J. M. Garibaldi, "Alpha-level aggregation: a practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments," IEEE Transactions on Knowledge and Data Engineering, vol. 23, no.10, 2011, pp. 1455–1468.[https://dx.doi.org/10.1109/TKDE.2010.191] has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.

Amanatidis, Barrot, Lang, Markakis and Ries{{Cite journal |last=Amanatidis |first=Georgios |last2=Barrot |first2=Nathanaël |last3=Lang |first3=Jérôme |last4=Markakis |first4=Evangelos |last5=Ries |first5=Bernard |date=2015-05-04 |title=Multiple Referenda and Multiwinner Elections Using Hamming Distances: Complexity and Manipulability |url=https://dl.acm.org/doi/abs/10.5555/2772879.2773245 |journal=Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems |series=AAMAS '15 |location=Richland, SC |publisher=International Foundation for Autonomous Agents and Multiagent Systems |pages=715–723 |isbn=978-1-4503-3413-6}} present voting rules for multi-issue voting, based on OWA and the Hamming distance. Barrot, Lang and Yokoo{{Cite journal |last=Barrot |first=Nathanaël |last2=Lang |first2=Jérôme |last3=Yokoo |first3=Makoto |date=2017-05-08 |title=Manipulation of Hamming-based Approval Voting for Multiple Referenda and Committee Elections |url=https://dl.acm.org/doi/abs/10.5555/3091125.3091212 |journal=Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems |series=AAMAS '17 |location=Richland, SC |publisher=International Foundation for Autonomous Agents and Multiagent Systems |pages=597–605}} study the manipulability of these rules.

{{reflist}}

• Liu, X., "The solution equivalence of minimax disparity and minimum variance problems for OWA operators," International Journal of Approximate Reasoning 45, 68–81, 2007.
• Torra, V. and Narukawa, Y., Modeling Decisions: Information Fusion and Aggregation Operators, Springer: Berlin, 2007.
• Majlender, P., "OWA operators with maximal Rényi entropy," Fuzzy Sets and Systems 155, 340–360, 2005.
• Szekely, G. J. and Buczolich, Z., " When is a weighted average of ordered sample elements a maximum likelihood estimator of the location parameter?" Advances in Applied Mathematics 10, 1989, 439–456.

Category:Logic in computer science

Category:Fuzzy logic

Category:Information retrieval techniques