Ordered weighted averaging

In applied mathematics, specifically in fuzzy logic, the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager. ^{[1] [2]} Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions.

Definition

An OWA operator of dimension ${\displaystyle \ n}$ is a mapping ${\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ that has an associated collection of weights ${\displaystyle \ W=[w_{1},\ldots ,w_{n}]}$ lying in the unit interval and summing to one and with

${\displaystyle F(a_{1},\ldots ,a_{n})=\sum _{j=1}^{n}w_{j}b_{j}}$

where ${\displaystyle b_{j}}$ is the j^th largest of the ${\displaystyle a_{i}}$.

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.

Notable OWA operators

${\displaystyle \ F(a_{1},\ldots ,a_{n})=\max(a_{1},\ldots ,a_{n})}$ if ${\displaystyle \ w_{1}=1}$ and ${\displaystyle \ w_{j}=0}$ for ${\displaystyle j\neq 1}$

${\displaystyle \ F(a_{1},\ldots ,a_{n})=\min(a_{1},\ldots ,a_{n})}$ if ${\displaystyle \ w_{n}=1}$ and ${\displaystyle \ w_{j}=0}$ for ${\displaystyle j\neq n}$

${\displaystyle \ F(a_{1},\ldots ,a_{n})=\mathrm {average} (a_{1},\ldots ,a_{n})}$ if ${\displaystyle \ w_{j}={\frac {1}{n}}}$ for all ${\displaystyle j\in [1,n]}$

Properties

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

Bounded	${\displaystyle \min(a_{1},\ldots ,a_{n})\leq F(a_{1},\ldots ,a_{n})\leq \max(a_{1},\ldots ,a_{n})}$
Monotonic	${\displaystyle F(a_{1},\ldots ,a_{n})\geq F(g_{1},\ldots ,g_{n})}$ if ${\displaystyle a_{i}\geq g_{i}}$ for ${\displaystyle \ i=1,2,\ldots ,n}$
Symmetric	${\displaystyle F(a_{1},\ldots ,a_{n})=F(a_{\boldsymbol {\pi (1)}},\ldots ,a_{\boldsymbol {\pi (n)}})}$ if ${\displaystyle {\boldsymbol {\pi }}}$ is a permutation map
Idempotent	${\displaystyle \ F(a_{1},\ldots ,a_{n})=a}$ if all ${\displaystyle \ a_{i}=a}$

Characterizing features

Two features have been used to characterize the OWA operators. The first is the attitudinal character, also called orness. ^[1] This is defined as

${\displaystyle A-C(W)={\frac {1}{n-1}}\sum _{j=1}^{n}(n-j)w_{j}.}$

It is known that ${\displaystyle A-C(W)\in [0,1]}$.

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

${\displaystyle H(W)=-\sum _{j=1}^{n}w_{j}\ln(w_{j}).}$

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

Type-1 OWA aggregation operators

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. ^{[3] [4]} 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 ${\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}}$ in the form of fuzzy sets defined on the domain of discourse ${\displaystyle U=[0,\;\;1]}$, then for each ${\displaystyle \alpha \in [0,\;1]}$, an ${\displaystyle \alpha }$-level type-1 OWA operator with ${\displaystyle \alpha }$-level sets ${\displaystyle \left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}}$ to aggregate the ${\displaystyle \alpha }$-cuts of fuzzy sets ${\displaystyle \left\{{A^{i}}\right\}_{i=1}^{n}}$ is given as

${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}}$

where ${\displaystyle W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}}$, and ${\displaystyle \sigma$ :\{\;1,\ldots ,n\;\}\to \{\;1,\ldots ,n\;\}} is a permutation function such that ${\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\ldots ,n-1}$, i.e., ${\displaystyle a_{\sigma (i)}}$ is the ${\displaystyle i}$th largest element in the set ${\displaystyle \left\{{a_{1},\ldots ,a_{n}}\right\}}$.

The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals ${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)}$: ${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)_{-}}$ and ${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)_{+},}$ where ${\displaystyle A_{\alpha }^{i}=[A_{\alpha -}^{i},A_{\alpha +}^{i}],W_{\alpha }^{i}=[W_{\alpha -}^{i},W_{\alpha +}^{i}]}$. Then membership function of resulting aggregation fuzzy set is:

${\displaystyle \mu _{G}(x)=\mathop {\vee } _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha }$

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

${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\min \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}}$

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

${\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\max \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}}$

This paper ^[5] has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.

OWA for committee voting

Amanatidis, Barrot, Lang, Markakis and Ries ^[6] present voting rules for multi-issue voting, based on OWA and the Hamming distance. Barrot, Lang and Yokoo ^[7] study the manipulability of these rules.

References

1. 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.
2. • Yager, R. R. and Kacprzyk, J., The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer: Norwell, MA, 1997.
3. 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
4. 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.
5. S.-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.
6. Amanatidis, Georgios; Barrot, Nathanaël; Lang, Jérôme; Markakis, Evangelos; Ries, Bernard (2015-05-04). "Multiple Referenda and Multiwinner Elections Using Hamming Distances: Complexity and Manipulability". Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. AAMAS '15. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 715–723. ISBN   978-1-4503-3413-6.
7. Barrot, Nathanaël; Lang, Jérôme; Yokoo, Makoto (2017-05-08). "Manipulation of Hamming-based Approval Voting for Multiple Referenda and Committee Elections". Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems. AAMAS '17. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 597–605.

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