Day-ahead prediction of wind speed is a basic and key problem of large-scale wind power penetration. Many current techniques fail to satisfy practical engineering requirements because of wind speed's strong nonlinear features, influenced by many complex factors, and the general model's inability to automatically learn features. It is well recognized that wind speed varies in different patterns. In this paper, we propose a deep feature learning (DFL) approach to wind speed forecasting because of its advantages at both multi-layer feature extraction and unsupervised learning. A deep belief network (DBN) model for regression with an architecture of 144 input and 144 output nodes was constructed using a restricted Boltzmann machine (RBM). Day-ahead prediction experiments were then carried out. By comparing the experimental results, it was found that the prediction errors with respect to both size and stability of a DBN model with only three hidden layers were less than those of the other three typical approaches including support vector regression (SVR), single hidden layer neural networks (SHL-NN), and neural networks with three hidden layers (THL-NN). In addition, the DBN model can learn and obtain complex features of wind speed through its strong nonlinear mapping ability, which effectively improves its prediction precision. In addition, prediction errors are minimized when the number of DBN model's hidden layers reaches a threshold value. Above this number, it is not possible to improve the prediction accuracy by further increasing the number of hidden layers. Thus, the DBN method has a high practical value for wind speed prediction.

The theory of fuzzy rough sets is claimed to be a powerful mathematical tool for dealing with uncertainty in data analysis. Unluckily, the classical model of fuzzy rough sets is sensitive to noisy information. This disadvantage limits the applicabili

Monotonic classification is a kind of special task in machine learning and pattern recognition. Monotonicity constraints between features and decision should be taken into account in these tasks. However, most existing techniques are not able to discover and represent the ordinal structures in monotonic datasets. Thus, they are inapplicable to monotonic classification. Feature selection has been proven effective in improving classification performance and avoiding overfitting. To the best of our knowledge, no technique has been specially designed to select features in monotonic classification until now. In this paper, we introduce a function, which is called rank mutual information, to evaluate monotonic consistency between features and decision in monotonic tasks. This function combines the advantages of dominance rough sets in reflecting ordinal structures and mutual information in terms of robustness. Then, rank mutual information is integrated with the search strategy of min-redundancy and max-relevance to compute optimal subsets of features. A collection of numerical experiments are given to show the effectiveness of the proposed technique.

Rough sets, especially fuzzy rough sets, are supposedly a powerful mathematical tool to deal with uncertainty in data analysis. This theory has been applied to feature selection, dimensionality reduction, and rule learning. However, it is pointed out that the classical model of fuzzy rough sets is sensitive to noisy information, which is considered as a main source of uncertainty in applications. This disadvantage limits the applicability of fuzzy rough sets. In this paper, we reveal why the classical fuzzy rough set model is sensitive to noise and how noisy samples impose influence on fuzzy rough computation. Based on this discussion, we study the properties of some current fuzzy rough models in dealing with noisy data and introduce several new robust models. The properties of the proposed models are also discussed. Finally, a robust classification algorithm is designed based on fuzzy lower approximations. Some numerical experiments are given to illustrate the effectiveness of the models. The classifiers that are developed with the proposed models achieve good generalization performance.

In many decision making tasks, values of features and decision are ordinal. Moreover, there is a monotonic constraint that the objects with better feature values should not be assigned to a worse decision class. Such problems are called ordinal classification with monotonicity constraint. Some learning algorithms have been developed to handle this kind of tasks in recent years. However, experiments show that these algorithms are sensitive to noisy samples and do not work well in real-world applications. In this work, we introduce a new measure of feature quality, called rank mutual information (RMI), which combines the advantage of robustness of Shannon's entropy with the ability of dominance rough sets in extracting ordinal structures from monotonic data sets. Then, we design a decision tree algorithm (REMT) based on rank mutual information. The theoretic and experimental analysis shows that the proposed algorithm can get monotonically consistent decision trees, if training samples are monotonically consistent. Its performance is still good when data are contaminated with noise.

Kernel machines and rough sets are two classes of commonly exploited learning techniques. Kernel machines enhance traditional learning algorithms by bringing opportunities to deal with nonlinear classification problems, rough sets introduce a human-focused way to deal with uncertainty in learning problems. Granulation and approximation play a pivotal role in rough sets-based learning and reasoning. However, a way how to effectively generate fuzzy granules from data has not been fully studied so far. In this study, we integrate kernel functions with fuzzy rough set models and propose two types of kernelized fuzzy rough sets. Kernel functions are employed to compute the fuzzy T-equivalence relations between samples, thus generating fuzzy information granules in the approximation space. Subsequently fuzzy granules are used to approximate the classification based on the concepts of fuzzy lower and upper approximations. Based on the models of kernelized fuzzy rough sets, we extend the measures existing in classical rough sets to evaluate the approximation quality and approximation abilities of the attributes. We discuss the relationship between these measures and feature evaluation function ReliefF, and augment the ReliefF algorithm to enhance the robustness of these proposed measures. Finally, we apply these measures to evaluate and select features for classification problems. The experimental results help quantify the performance of the KFRS.

Feature selection is an important preprocessing step in pattern analysis and machine learning. The key issue in feature selection is to evaluate quality of candidate features. In this work, we introduce a weighted distance learning algorithm for feature selection via maximizing fuzzy dependency. We maximize fuzzy dependency between features and decision by distance learning and then evaluate the quality of features with the learned weight vector. The features deriving great weights are considered to be useful for classification learning. We test the proposed technique with some classical methods and the experimental results show the proposed algorithm is effective.

Feature selection plays an important role in pattern recognition and machine learning. Feature evaluation and classification complexity estimation arise as key issues in the construction of selection algorithms. To estimate classification complexity in different feature subspaces, a novel feature evaluation measure, called the neighborhood decision error rate (NDER), is proposed, which is applicable to both categorical and numerical features. We first introduce a neighborhood rough-set model to divide the sample set into decision positive regions and decision boundary regions. Then, the samples that fall within decision boundary regions are further grouped into recognizable and misclassified subsets based on class probabilities that occur in neighborhoods. The percentage of misclassified samples is viewed as the estimate of classification complexity of the corresponding feature subspaces. We present a forward greedy strategy for searching the feature subset, which minimizes the NDER and, correspondingly, minimizes the classification complexity of the selected feature subset. Both theoretical and experimental comparison with other feature selection algorithms shows that the proposed algorithm is effective for discrete and continuous features, as well as their mixture.

Some errors in our original paper in defining relative reduct with information measures are pointed out in this paper. It is shown that in our original work, Theorems 10 and 19 hold just under the condition that decision tables are consistent. We also show that an attribute reduction algorithm based on the entropies can be used to select features in practical applications although the two theorems do not hold in inconsistent cases.

Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work.

Data usually exists with hybrid formats in real-world applications, and a unified data reduction for hybrid data is desirable. In this paper a unified information measure is proposed to computing discernibility power of a crisp equivalence relation and a fuzzy one, which is the key concept in classical rough set model and fuzzy rough set model. Based on the information measure, a general definition of significance of nominal, numeric and fuzzy attributes is presented. We redefine the independence of hybrid attribute subset, reduct, and relative reduct. Then two greedy reduction algorithms for unsupervised and supervised data dimensionality reduction based on the proposed information measure are constructed. Experiments show the reducts found by the proposed algorithms get a better performance compared with traditional rough set approaches.

Yager's entropy was proposed to compute the information of fuzzy indiscernibility relation. In this paper we present a novel interpretation of Yager's entropy in discernibility power of a relation point of view. Then some basic definitions in Shannon's information theory are generalized based on Yager's entropy. We introduce joint entropy, conditional entropy, mutual information and relative entropy to compute the information changes for fuzzy indiscerniblity relation operations. Conditional entropy and relative conditional entropy are proposed to measure the information increment, which is interpreted as the significance of an attribute in fuzzy rough set model. As an application, we redefine independency of an attribute set, reduct, relative reduct in fuzzy rough set model based on Yager's entropy. Some experimental results show the proposed approach is suitable for fuzzy and numeric data reduction.