# A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion
## 1 Introduction  
Data envelopment analysis (DEA) is a prominent technique for the non-parametric relative efficiency analysis of a set of decision-making units (DMUs) drawn from a similar production process [1]. DEA models are used in both operation research and data mining literature [2]. Some of the traditional properties of production functions, such as the monotonicity and convexity of the inputs and outputs, that are fundamental in DEA models are often found to be attractive in some data mining models where datasets are noisy and model resistance to learning noise is necessary [3]. An important aspect of DEA models is the reliability of DMU efficiency scores. It is generally accepted that the DEA efficiency estimates are reliable when the sample size is large [4]. Since the reliability of the DEA scores is dependent on the sample size, [PERSON] et al. [5] have suggested the following rule for the minimum number (\(n\)) of DMUs for reliable DEA analysis (each DMU has \(m\) inputs and \(s\) outputs):  
\[n\geq max[3(m+s),\ m\times s] \tag{1a}\]  
For small-size datasets, where violations of the minimum number of DMUs specified by Equation (1a) frequently occur, dimensionality reduction (also known as variable reduction or variable selection) approaches are frequently used to select a subset of variables to satisfy Equation (1a). A variety of variable selection approaches are available in the literature. Among these variable selection approaches are statistical [6], regression [7], efficiency contribution measure [8], bootstrapping [9], hypothesis testing [10], variable aggregation [11] and statistical experiment designs [12]. Variable selection approaches are criticized extensively for applying parametric procedures and linear relationship assumptions for selecting variables to determine an unknown non-linear and non-parametric efficiency frontier. [PERSON] and [PERSON] [13] provide a good description of some of these procedures and their pros and cons. [PERSON] [14] proposed a competing approach to the dimensionality reduction/variable selection problem called the ensemble DEA. In his approach, traditional DEA analysis is conducted for all possible input and output combinations, and the efficiency scores of each DEA model for each DMU are averaged as an ensemble efficiency score for a DMU. Drawing from machine learning literature, [PERSON] [14] showed that the ensemble efficiency score is a reliable estimate of the \"true\" efficiency of a DMU. Even for small datasets, certain combinations of inputs will satisfy the criterion set by Equation (1a), while others will violate it, but the average ensemble score will be closer to the true efficiency of the DMU and will be reliable. [PERSON] [14] also proposed an exhaustive search procedure to generate all possible input and output combinations, and proposed a formula to compute the number of unique DEA models that need to be run to compute an average ensemble score. This number \(N\) of unique DEA models may be computed using the following formula:  
\[N=\left(\sum\
olimits_{i=1}^{m}\left( \begin{array}{c}m\\ i\end{array} \right)\right)\times\left(\sum\
olimits_{i=1}^{s}\left( \begin{array} []{c}s\\ i\end{array} \right)\right)=(2^{m}-1)\times(2^{s}-1). \tag{1b}\]  
Using [PERSON] et al. 's [15] variable-returns-to-scale (VRS) DEA BCC model, and data and models obtained from a few studies in the literature, [PERSON] [14] showed that the ensemble DEA model provides a better ranking of DMUs than the models proposed in a few studies from the literature. This research investigates the additional properties and statistical distribution of the ensemble DEA model scores. It is shown that there are added benefits of ensemble efficiency scores.