# Ensembles of Random SHAPs
## 1 Introduction
The second approach consists of the well-known SHAP (SHapley Additive exPlanations) method [10; 11] and its modifications. The method is inspired by game-theoretic Shapley values [12] which can be interpreted as average expected marginal contributions over all possible subsets (coalitions) of features to the black-box model prediction. The SHAP has many advantages, for example, it can be used for local and global explanations in contrast to LIME, but there are also two important shortcomings. The first one is a question as to how to add or remove features in order to implement their subsets as inputs for the black-box model. There are many approaches to removing features, exhaustively described by [13], but the SHAP may be too sensitive to each of them, and there are no strong justifications for their use. Nevertheless, the SHAP can be regarded as the most promising and efficient explanation method. The second shortcoming is that the SHAP is computationally expensive when there is a large number of features due to considering all possible coalitions whose number is \(2^{m}\), where \(m\) is the number of features. Therefore, the computational time grows exponentially. Several simplifications and approximations have been proposed in order to overcome this difficulty. Some of them are presented by [11; 14; 15]. One of the simplifications is based on using the ordered permutations of the feature indices and the probability distributions of the features [14]. Another approximation is the quasi-random and adaptive sampling which includes two improvements [15]. The first one is based on exploiting the Monte Carlo integration. The second improvement is based on the optimal number of the perturbations of every feature in accordance with its variance to minimize the overall approximation error. Ref. [15] also proposed to average the local contributions of the values of each feature across all instances. Another interesting approach to simplify the SHAP is the Kernel SHAP [10] which can be regarded as a computationally efficient approximation to the Shapley values in higher dimensions. In order to relax the assumption of the feature independence accepted in the Kernel SHAP, ref. [16] extended the Kernel SHAP method to handle dependent features. Ref. [17] proposed the polynomial-time approximation of the Shapley values, called the Deep Approximate Shapley Propagation method. In spite of the many approaches to simplify the SHAP, it is difficult to expect a significant simplification from the above modifications of the SHAP. Therefore, a new approach is proposed for simplifying the SHAP method and for reducing the computational expenses for calculating the Shapley values. A key idea behind the proposed approach is to apply a modification of the random subspace method [18] and to consider an ensemble of random SHAPs, called the ensemble of random SHAPs (ER-SHAP). The approach is very similar to the random forests, when an ensemble of randomly built decision trees is used to obtain some average classification or regression measures. Random SHAPs are constructed by a random selection of \(t\) features with indices \(J_{k}=(i_{1},\ldots,i_{t})\) from the instance for an explanation, and the obtained subset of the instance features is analyzed by the SHAP as a separate instance. Repeating this procedure \(N\) times, we obtain a set \(\mathcal{S}=\{S_{1},\ldots,S_{N}\}\) of the Shapley values corresponding to the input subsets of the features, where the \(k\)-th subset is \(S_{k}=\{\phi_{i}:i\in J_{k}\}\). By applying some combination rule for combining subsets \(S_{k}\) from \(\mathcal{S}\), we obtain the final Shapley values. The above general approach considering an ensemble of random SHAPs has several extensions which form the corresponding methods and algorithms. First of all, we can generate points around the analyzed instance and construct \(S_{k}\) for the \(k\)-th generated point. In this case, every point is assigned by a weight, depending on the distance from the analyzed point.