Absolute degree of grey incidences for matrix sequence

With the rapid development of big data and cloud computing technology, a large number of system behaviors can be recorded from multiple dimensions. Thus, the data structure of the system behaviors is represented as a multidimensional cube and recorded in the form of a matrix sequence. In the field of e-commerce, sales manager usually builds multidimensional data model of sales amount from the perspectives of what, where, who, when, why and how. If all dimensions contain system behavior information, when analyzing the similarity between different system behaviors, all data of matrix sequence must be employed. Currently, the similarity measure method of matrix sequence mainly adopts multilinear principal component analysis (MPCA) method. These methods introduce MPCA to reduce the dimension of original data, and construct some statistics to measure the similarity between different system behaviors based on principle components. However, similarity analysis method of matrix sequence based on MPCA is stringent to sample size and data volume. It is commonly subject to certain limitations in small-scale matrix sequence analysis. In order to overcome the shortcomings of similarity measure method for matrix sequence, a new absolute degree of grey incidences for matrix sequence data is proposed by extending the grey incidence analysis theory into multi-dimension space. Firstly, the paper analyzes the data structure of the matrix sequence, explains its physical meaning, and definite the matrix sequence of a system behavior formally. Moreover, the starting zeroing operator for time series is extended for matrix sequence. Secondly, because the grey incidence models are constructed with the geometric feature, in another word, they extract similarity feature from the geometric description of system behavior, the geometric description method of matrix sequence is studied. However, the geometric description of matrix sequence depends on the three-dimensional data model, some 3-D data models are compared such as tetrahedral model, tri-prism model, CSG model and rule hexahedral model. As the result, hexahedral data model, which is convenient to visualize 3D data fields by calling the marching cube algorithm, commonly used in geology is introduced to shape the spatial geometric features of behavioral matrix sequence. With the method, an arbitrary element in a matrix sequence is considered as a behavior value at the point of three-dimensional space. Plus, different colors or grayscale values are used to represent the behavior values of factors at different locations. Furthermore, the definition of matrix sequence data volume is given, and the similarity measure of two factors is transformed into measure the proximity of two data volumes. Thirdly, the classic absolute degree of grey incidences formula was employed to analyze the relationship of two matrix sequences, but the parameters of it was recomputed under the multi-dimension space. In sequence degree of incidences, the parameters in the model represent the area between zeroing zigzagged line and coordinate axis, or area between two zeroing zigzagged lines. In the paper, the parameters in the model represent total behavioral volume among the image of zeroing starting layer of two factors and their differences in behavioral performance. So, the triple integral is adopted to calculate parameters. In addition, the properties of proposed model are reproofed. The properties indicated the model satisfied the axioms of absolute incidence analysis and translation invariance. It belongs to the degree of proximity incidences model. At the last section, a small bicycle company case is studied by the proposed model. The company wants to divide its market into some different market types through regional multi-dimensional data analysis. Because of short time interval and a few production types, the proposed absolute degree of incidences for matrix sequence is employed to analyze the similarity between different sale region. The result shows that the market can be divided into two types, one is transport type sales area, and the other is sports consumption type sales area.