## 1 Introduction
\(\widehat{X}\) is the optimal choice with some evaluation indexes for classifiers; the feature subset has the lowest dimension for a given accuracy; the conditional probability distribution function for the data and that of the selected features remain the same; the error rate of the classifier would not be reduced by not increasing or decreasing the number of features. By such a selection process, we could get rid of either redundant or irrelevant features without incurring much loss of information. The distinguishing features can be found, and in this way, the dimension of data space declines, the complexity of data reduces and, especially, the performance of classification and prediction can be improved. Because of the direct interpretability of the data, feature selection is widely used in many fields such as genomics, medical image analysis, computer vision, speech recognition, computer vision, information retrieval, time series prediction [11; 12; 13], etc. According to different ways of combining the evaluation criteria and classifiers, feature selection methods can be divided into five types, i.e., filter, wrapper, embedded, ensemble and hybrid methods [14]. Filter methods mainly depend on the attribute of features, and the evaluation criteria depend only on the original data, but not on classifiers [15]. Wrapper methods directly take the performance of the classifiers as the evaluation criterion for the selection of feature subsets; thus, the results of wrapper methods are related to specific classifiers [16]. Methods of embedding filter methods and wrapper methods are called embedded methods. For embedded methods, they are usually composed of two stages. Firstly, filter methods are used to eliminate most of the irrelevant and noise features, so as to reduce the data dimension of the subsequent search process effectively. The second stage adopts wrapper methods to carry out the further feature selection process [17]. Ensemble methods are based on different sampling strategies to extract multiple sample sets, and then, they use a specific feature selection algorithm to obtain multiple sets of feature subsets. These feature subsets are further integrated to obtain a more stable feature subset [18]. Compared with the above three methods, the performance of the ensemble methods no longer depends merely on a single subset selected, but it is still limited since it uses only one specific feature learner. Hybrid methods can be combined with some different feature selection methods. Hybrid approaches combine two or more well-studied feature selection algorithms to form a new strategy and achieve a complementary advantage of different feature selection methods to solve a particular problem [19; 20]. The hybrid approach usually capitalizes on the advantages from the sub-algorithms and therefore is more robust compared with single approaches. The feature selection techniques mentioned above have been applied to many fields of dimensionality reduction analysis [21; 22; 23]. In addition to the above five types of feature selection methods, some traditional statistical methods can also be used to reduce dimensionality, such as hypothesis testing, correlation coefficients, etc. These methods can obtain features with higher distinguishing ability, so as to improve the discriminative capacity of different classes [24; 25]. Motivated by identifying biomarkers of SZ that are associated with cognitive composite ability and specific cognitive domains such as attention, working memory and verbal learning, in this paper, by proposing a hybrid feature selection method combining both machine learning and traditional statistical approaches, we explore the brain abnormalities of SZ. The data have 410 features, including both functional and structural MRI, i.e., functional network connectivity (FNC) and source-based morphometric (SBM) of 40 patients with SZ and 46 healthy controls (HCs).