## 1 Introduction  
The warships scheduled maintenance (WSM) refers to the preventive repairment that is carried out by comprehensively considering the combat readiness and training tasks of the ship equipment, as well as the protective performance of the ship hull, the main equipment technical status and other related factors. Since the WSM costs constitute a large proportion of the total life cycle costs of warships, the cost prediction of WSM naturally becomes an important research topic in guaranteeing naval equipment maintenance [1]. Due to the complexity of cost-driving factors and the difficulty of obtaining the cost variation knowledge with small data size, existing studies mainly use methods such as grey prediction [2, 3, 4, 5, 6, 7], case-based reasoning [8, 9, 10, 11, 12, 13], support vector regression (SVR) [14, 15, 16, 17] and so on, to model and predict cost in different fields. However, the above methods such as grey prediction and SVR are often based on a single output structure when predicting the costs, and output results are single-dimensional, which does not take advantage of the correlation information between different WSM subentry costs. The total costs of WSM can be further subdivided into subentry costs such as material costs, labour costs, manufacturing costs, etc. There is a strong correlation between different subentry costs. However, the grey prediction or SVR can only be used as a univariate modelling technique due to its inherent single-output structure. Consequently, the cost prediction model based on grey theory or SVR constructs a different predicted engine for each subentry cost separately, which ignores the inherent information of the mutual relationships among different subentry costs, which may lead to a poor prediction performance of the total costs. With the burgeoning development of the neural network, different cost prediction models based on the neural network have been proposed [18; 19; 20; 21; 22]. Different types of artificial neural networks (ANN), such as back propagation neural networks (BPNN) [23], radial basis function neural networks (RBFNN) [20], and deep convolutional neural networks (DCNN) [19] are effective in cost prediction for different projects. Although the neural network has a multi-dimensional output structure, it is prone to underfitting or overfitting in the case of a small size of samples, which also leads to poor performance in the cost prediction of small samples. Moreover, the training of neural networks is vulnerable to anomalous samples and noise, which restrict their application in the cost prediction of WSM problems. Multi-target regression (MTR) is the prediction of multiple continuous variables by a set of common input variables [24]. MTR not only exploits the correlation between input and output variables but also explores the correlation information between different output targets so that it can obtain better prediction performance than single-output regression when predicting multiple correlated variables. In this paper, the MTR algorithm based on multi-layer sparse structure (MTR-MLS) is proposed to model and predict the subentry costs of WSM simultaneously. The multi-output structure of the proposed algorithm is utilized to achieve accurate subentry costs prediction. Furthermore, the total cost of WSM is obtained by summing up the predicted subentry costs, which can efficiently improve the prediction accuracy and stability of WSM cost. The contributions of this paper are summarized as follows:  
1. Based on the traditional MTR framework, the latent variable space is introduced to form a multi-layer learning structure, and the sparsity constraint is imposed so that the same latent variables can be shared among the associated targets, thus improving the performance of the algorithm with multiple outputs. 2.