# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
Third, an along-across similarity model to generate the \(k\)-nearest matrix neighbors (\(k\)NMN) is presented. The along similarity considers the associations of the time stamps. The across similarity focuses on the relation between the variables. Additionally, with the PAA- and SAX-MTSs, the most popular numerical or symbolic metrics can be combined regardless of whether they are similarities or distances. Given a sliding window \(w\), the PAA- and SAX-MTSs can be transformed into \(m-w+1\) temporal subsequences, called instances. \(m\) is the number of time stamps, and all instances are matrices with the shape \(m\times n\). Moreover, the latest state following each instance is denoted as the decision information, called the label. With the optimal \(k\) labels from \(m-w\), the tri-state can be finally predicted using the traditional voting strategy. Fourth, two post-filling strategies called the individual and related ones, are designed to fill the possibly missing symbols of each variable. The reason for which the tri-state may be uncompleted is that no strong, medium or weak symbols occur after all matrix instances. For brevity, given a tri-state, we assume that the strong symbol of its \(i\)-th variable (\(a_{i}\)) is missing. The individual filling strategy (IFS) directly scans the history data of \(a_{i}\) to obtain the most frequently occurring strong symbol. The related filling strategy (RFS) considers the associations between \(a_{i}\) and the other \(n-1\) variables. One of the other variables, which is the most linear related to \(a_{i}\), is its condition. The main contributions of this paper are presented as follows:  
* _Tri-state._ It provides three kinds of symbols for each variable simultaneously. The proposed deviation degree-based alphabet tri-partition strategy makes the outliers more noticeable for experts. Moreover, the IFS and RFS are designed to obtain a completed tri-state. * _Along-across similarity model_. The similarities between time stamps and variables are considered simultaneously. This model provides a framework for the integration of the popular similarity or distance metrics. * _Combination of the popular numerical or symbolic metrics_. The PAA- and SAX-MTSs are simultaneously used in the above similarity model. The PAA-MTS is available for the numerical metrics, while the SAX-MTS fits the symbolic ones. The experimental results undertaken on four real-world datasets show that (1) in terms of precision, the states are 30% to 50% higher than the three kinds of tri-states, while for the recall, the three kinds of tri-state are 10% higher than the state; (2) the IFS and RFS can slightly improve the recall by approximately 1%; and (3) the along-across similarity model composed of the Triangle and Jaccard metrics are first recommended for new datasets. Note that the IFS and RFS are necessary if the tri-state is incomplete. In other words, when the obtained tri-state is fulfilled, no difference is found among the three kinds of tri-states. The rest of this paper is organized as follows. Section 2 reviews the existing work on time-series prediction. Section 3 presents the fundamental definitions of the tri-state. Section 4 proposes the algorithm for tri-state prediction. Section 5 discusses the performance of the prediction algorithm on four real-world datasets. Section 6 lists the conclusions and future work of this paper.