Semi-continuous data comes from a distribution that is a mixture of the point mass at zero and a continuous distribution with support on the positive real line. A clear example is the daily rainfall data. In this paper, we present a novel algorithm to estimate the density function for semi-continuous data using the principle of maximum entropy. Unlike existing methods in the literature, our algorithm needs only the sample values of the constraint functions in the entropy maximization problem and does not need the entire sample. Using simulations, we show that the estimate of the entropy produced by our algorithm has significantly less bias compared to existing methods. An application to the daily rainfall data is provided.

Predictions obtained by, e.g., artificial neural networks have a high accuracy but humans often perceive the models as black boxes. Insights about the decision making are mostly opaque for humans. Particularly understanding the decision making in highly sensitive areas such as healthcare or fifinance, is of paramount importance. The decision-making behind the black boxes requires it to be more transparent, accountable, and understandable for humans. This survey paper provides essential definitions, an overview of the different principles and methodologies of explainable Supervised Machine Learning (SML). We conduct a state-of-the-art survey that reviews past and recent explainable SML approaches and classifies them according to the introduced definitions. Finally, we illustrate principles by means of an explanatory case study and discuss important future directions.

Contemporary machine learning applications often involve classification tasks with many classes. Despite their extensive use, a precise understanding of the statistical properties and behavior of classification algorithms is still missing, especially in modern regimes where the number of classes is rather large. In this paper, we take a step in this direction by providing the first asymptotically precise analysis of linear multiclass classification. Our theoretical analysis allows us to precisely characterize how the test error varies over different training algorithms, data distributions, problem dimensions as well as number of classes, inter/intra class correlations and class priors. Specifically, our analysis reveals that the classification accuracy is highly distribution-dependent with different algorithms achieving optimal performance for different data distributions and/or training/features sizes. Unlike linear regression/binary classification, the test error in multiclass classification relies on intricate functions of the trained model (e.g., correlation between some of the trained weights) whose asymptotic behavior is difficult to characterize. This challenge is already present in simple classifiers, such as those minimizing a square loss. Our novel theoretical techniques allow us to overcome some of these challenges. The insights gained may pave the way for a precise understanding of other classification algorithms beyond those studied in this paper.

Generally, the criteria involved in a decision making problem are interactive or inter-dependent, and therefore aggregating them by the use of traditional operators which are based on additive measures is not logical. This verifies that we have to implement fuzzy measures for modelling the interaction phenomena among the criteria.On the other hand, based on the recent extension of hesitant fuzzy set, called higher order hesitant fuzzy set (HOHFS) which allows the membership of a given element to be defined in forms of several possible generalized types of fuzzy set, we encourage to propose the higher order hesitant fuzzy (HOHF) Choquet integral operator. This concept not only considers the importance of the higher order hesitant fuzzy arguments, but also it can reflect the correlations among those arguments. Then,a detailed discussion on the aggregation properties of the HOHF Choquet integral operator will be presented.To enhance the application of HOHF Choquet integral operator in decision making, we first assess the appropriate energy policy for the socio-economic development. Then, the efficiency of the proposed HOHF Choquet integral operator-based technique over a number of exiting techniques is further verified by employing another decision making problem associated with the technique of TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making).

This contribution reviews critically the existing entropy measures for probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy measures fail to effectively distinguish a variety of different PHFSs in some cases. In the sequel, we develop a new axiomatic framework of entropy measures for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of uncertainty associated with PHFEs which are known as fuzziness and non-specificity. Respect to each kind of uncertainty, a number of formulae are derived to permit flexible selection of PHFE entropy measures. Moreover, based on the proposed PHFE entropy measures, we introduce some entropy-based distance measures which are used in the portion of comparative analysis. Eventually, the proposed PHFE entropy measures and PHFE entropy-based distance measures are applied to decision making in the strategy initiatives where their reliability and effectiveness are verified. Keywords: Probabilistic hesitant fuzzy set, Entropy measure, Multiple criteria decision making.

Network topology identification (TI) is an essential function for distributed energy resources management systems (DERMS) to organize and operate widespread distributed energy resources (DERs). In this paper, discriminant analysis (DA) is deployed to develop a network TI function that relies only on the measurements available to DERMS. The propounded method is able to identify the network switching configuration, as well as the status of protective devices. Following, to improve the TI resiliency against the interruption of communication channels, a quadratic programming optimization approach is proposed to recover the missing signals. By deploying the propounded data recovery approach and Bayes' theorem together, a benchmark is developed afterward to identify anomalous measurements. This benchmark can make the TI function resilient against cyber-attacks. Having a low computational burden, this approach is fast-track and can be applied in real-time applications. Sensitivity analysis is performed to assess the contribution of different measurements and the impact of the system load type and loading level on the performance of the proposed approach.

This work addresses the inverse identification of apparent elastic properties of random heterogeneous materials using machine learning based on artificial neural networks. The proposed neural network-based identification method requires the construction of a database from which an artificial neural network can be trained to learn the nonlinear relationship between the hyperparameters of a prior stochastic model of the random compliance field and some relevant quantities of interest of an ad hoc multiscale computational model. An initial database made up with input and target data is first generated from the computational model, from which a processed database is deduced by conditioning the input data with respect to the target data using the nonparametric statistics. Two-and three-layer feedforward artificial neural networks are then trained from each of the initial and processed databases to construct an algebraic representation of the nonlinear mapping between the hyperparameters (network outputs) and the quantities of interest (network inputs). The performances of the trained artificial neural networks are analyzed in terms of mean squared error, linear regression fit and probability distribution between network outputs and targets for both databases. An ad hoc probabilistic model of the input random vector is finally proposed in order to take into account uncertainties on the network input and to perform a robustness analysis of the network output with respect to the input uncertainties level. The capability of the proposed neural network-based identification method to efficiently solve the underlying statistical inverse problem is illustrated through two numerical examples developed within the framework of 2D plane stress linear elasticity, namely a first validation example on synthetic data obtained through computational simulations and a second application example on real experimental data obtained through a physical experiment monitored by digital image correlation on a real heterogeneous biological material (beef cortical bone).

Deep neural networks have proven extremely efficient at solving a wide range of inverse problems, but most often the uncertainty on the solution they provide is hard to quantify. In this work, we propose a generic Bayesian framework for solving inverse problems, in which we limit the use of deep neural networks to learning a prior distribution on the signals to recover. We adopt recent denoising score matching techniques to learn this prior from data, and subsequently use it as part of an annealed Hamiltonian Monte-Carlo scheme to sample the full posterior of image inverse problems. We apply this framework to Magnetic Resonance Image (MRI) reconstruction and illustrate how this approach not only yields high quality reconstructions but can also be used to assess the uncertainty on particular features of a reconstructed image.

Quality of life assessment represents a key process of deciding treatment success and viability. As such, patients' perceptions of their functional status and well-being are important inputs for impairment assessment. Given that patient completed questionnaires are often used to assess patient status and determine future treatment options, it is important to know the level of agreement of the words used by patients and different groups of medical professionals. In this paper, we propose a measure called the Agreement Ratio which provides a ratio of overall agreement when modelling words through Fuzzy Sets (FSs). The measure has been specifically designed for assessing this agreement in fuzzy sets which are generated from data such as patient responses. The measure relies on using the Jaccard Similarity Measure for comparing the different levels of agreement in the FSs generated.

Noisy labeled data is more a norm than a rarity for crowd sourced contents. It is effective to distill noise and infer correct labels through aggregation results from crowd workers. To ensure the time relevance and overcome slow responses of workers, online label aggregation is increasingly requested, calling for solutions that can incrementally infer true label distribution via subsets of data items. In this paper, we propose a novel online label aggregation framework, BiLA, which employs variational Bayesian inference method and designs a novel stochastic optimization scheme for incremental training. BiLA is flexible to accommodate any generating distribution of labels by the exact computation of its posterior distribution. We also derive the convergence bound of the proposed optimizer. We compare BiLA with the state of the art based on minimax entropy, neural networks and expectation maximization algorithms, on synthetic and real-world data sets. Our evaluation results on various online scenarios show that BiLA can effectively infer the true labels, with an error rate reduction of at least 10 to 1.5 percent points for synthetic and real-world datasets, respectively.