Choquet Integral - Non-additive aggregation

Overview

Apply F.steps in order.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Rank aggregation, social choice, preference fusion

How it works

1. 1

   Specify fuzzy densities g_i = g_λ({x_i}) for each singleton criterion.

   $$g_i=g_{\lambda }(\{x_i\}),i=1,\dots ,n$$

   Sugeno 1974, p.15

2. 2

   Solve 1 + λ = Π (1 + λ g_i) for the Sugeno λ-parameter.

   $$1+\lambda =\prod _{i=1}^n(1+\lambda g_i),-1\le \lambda <\infty$$

   Sugeno 1974, p.16 Eq.(2)

3. 3

   Build λ-fuzzy measure g_λ on every subset.

   $$g_{\lambda }(H_i)=\frac{1}{\lambda }[\prod _{x\in H_i}(1+\lambda g_x)-1]$$

   Sugeno 1974, p.16 Eq.(3)

4. 4

   Order h(x_1)≥…≥h(x_n) and compute Choquet integral.

   $$\int hdg=\sum _{i=1}^n[h(x_i)-h(x_{i-1})]\cdot g_{\lambda }(H_i),h(x_0)=0$$

   Murofushi-Sugeno 1989, p.5 Eq.(1)

Look elsewhere when

Assumptions to verify

• Singleton fuzzy densities g_i are elicited from domain experts or estimated from data
• Criteria exhibit interaction (superadditive if Σg_i<1, subadditive if Σg_i>1, additive if Σg_i=1)
• All decision matrix entries are real-valued, finite, non-missing

Edge cases and pitfalls

• •ties g_i = g_λ({x_i}) for each singleton criterion.

• Applying CHOQUET-INTEGRAL without verifying this assumption.

  Requirement: Singleton fuzzy densities g_i are elicited from domain experts or estimated from data

• Applying CHOQUET-INTEGRAL without verifying this assumption.

  Requirement: Criteria exhibit interaction (superadditive if Σg_i<1, subadditive if Σg_i>1, additive if Σg_i=1)

• Applying CHOQUET-INTEGRAL without verifying this assumption.

  Requirement: All decision matrix entries are real-valued, finite, non-missing

• Using CHOQUET-INTEGRAL when: Criteria are independent → use SAW/WPM (Choquet collapses to additive weighted sum anyway).

  An alternative method is recommended in this situation.

• Using CHOQUET-INTEGRAL when: Fuzzy densities g_i cannot be estimated.

  An alternative method is recommended in this situation.

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