The use of tensors is progressively widespread in the realms of data analytics and machine learning. As an extension of vectors and matrices, a tensor is a multi-dimensional array of numbers that provides a means to represent data across multiple dimensions. As an illustration, Figure 1 shows an image stream that can be seen as a three-dimensional tensor, where the first two dimensions denote the pixels within each image, while the third dimension represents the distinct images in the sequence. One of the advantages of representing data as a tensor, as opposed to reshaping it into a vector or matrix, lies in its ability to capture intricate relationships within the data, especially when interactions occur across multiple dimensions. For instance, the image stream depicted in Figure 1 exhibits a spatiotemporal correlation structure. Specifically, pixels within each image have spatial correlation, and pixels at the same location across multiple images are temporally correlated. Transforming the image stream into a vector or matrix would disrupt the spatiotemporal correlation structure, whereas representing it as a three-dimensional tensor preserves this correlation. In addition to capturing intricate relationships, other benefits of using tensors include compatibility with multi-modal data (i.e., accommodating diverse types of data in a unified structure) and facilitating parallel processing (i.e., enabling the parallelization of operations), etc. As a result, the volume of research in tensor-based data analytics has been rapidly increasing in recent years (Shen et al., 2022; Gahrooei et al., 2021; Yan et al., 2019; Hu et al., 2023; Zhen et al., 2023; Zhang et al., 2023).

This paper proposes a supervised dimension reduction methodology for tensor data which has two advantages over most image-based prognostic models. First, the model does not require tensor data to be complete which expands its application to incomplete data. Second, it utilizes time-to-failure (TTF) to supervise the extraction of low-dimensional features which makes the extracted features more effective for the subsequent prognostic. Besides, an optimization algorithm is proposed for parameter estimation and closed-form solutions are derived under certain distributions.