Capacity Random Forest for Correlative Multiple Criteria Decision Pattern Learning

The Choquet capacity and integral is an eminent scheme to represent the interaction knowledge among multiple decision criteria and deal with the independent multiple sources preference information. In this paper, we enhance this scheme’s decision pattern learning ability by combining it with another powerful machine learning tool, the random forest of decision trees. We first use the capacity fitting method to train the Choquet capacity and integral-based decision trees and then compose them into the capacity random forest (CRF) to better learn and explain the given decision pattern. The CRF algorithms of solving the correlative multiple criteria based ranking and sorting decision problems are both constructed and discussed. Two illustrative examples are given to show the feasibilities of the proposed algorithms. It is shown that on the one hand, CRF method can provide more detailed explanation information and a more reliable collective prediction result than the main existing capacity fitting methods; on the other hand, CRF extends the applicability of the traditional random forest method into solving the multiple criteria ranking and sorting problems with a relatively small pool of decision learning data.

Generalized Hausdorff capacities and their applications

Let {\mathfrak{P}({\mathbb{R}^{n}})} be the power set of {\mathbb{R}^{n}} and let {\varphi:\mathfrak{P}({\mathbb{R}^{n}})\rightarrow[0,\infty]} be a set function. In this paper, the authors introduce a class of generalized Hausdorff capacities {H_{\varphi}} with respect to φ. Some basic properties of {H_{\varphi}} including the strong subadditivity are obtained. An equivalent variant of {H_{\varphi}} defined via dyadic cubes is also introduced and proved to be Choquet capacity. The authors then prove the boundedness of some maximal operators, such as the Hardy–Littlewood maximal operator, on Lebesgue spaces with respect to {H_{\varphi}}. As an application, the predual spaces of weighted Morrey spaces are described via these capacities.

Sobolev capacity on the spaceW1, p(⋅)(ℝn)

We define Sobolev capacity on the generalized Sobolev spaceW1, p(⋅)(ℝn). It is a Choquet capacity provided that the variable exponentp:ℝn→[1,∞)is bounded away from 1 and∞. We discuss the relation between the Hausdorff dimension and the Sobolev capacity. As another application we study quasicontinuous representatives in the spaceW1, p(⋅)(ℝn).