## 3 Problem Statement and Approach Overview  
### Problem Statement  
PSG provides a data-rich source for understanding and measuring sleep brain activities [48]. For instance, based on the well-known Rechtschaffen and Kales scoring criteria [49], people can leverage these neurophysiological signals for standard sleep staging. The main stages are wakefulness, rapid eye movement (REM) sleep and non-rapid eye movement (NREM) sleep, where NREM is further divided into four stages from the lightest sleep stages 1 and 2 to the deepest sleep stages 3 and 4. Specifically, sleep stage 1 is characterized by theta waves (the amplitude is 50 to 100 micro volts); the characteristics of stage 2 sleep are termed sleep spindles (the amplitude is 50 to 150 micro volts); stage 3 is considered spindle waves and slow waves sleep (the amplitude is 100 to 150 micro volts); stage 4 has the same attributes as stage 3, but more than 50% of the waves are slow waves and delta waves (the amplitude is 100 to 200 micro volts).  
In this paper, we leverage features extracted from psychological sleep data for ischemic stroke prediction. The formulation below represents the time series of sleep data:  
\[SS=\{sc_{1},sc_{2},\ldots,sc_{n}\}, \tag{1}\]  
where \(n\) is the number of sleep cycles. Specifically, each sleep cycle \(sc_{i}\) (\(i\in[1,\,n]\)) can be denoted as:  
\[sc_{i}=(s0_{i},s1_{i},s2_{i},s3_{i},s4_{i},s5_{i}), \tag{2}\]  
where \(s0_{i}\) is the awake stage of the sleep cycle \(sc_{i}\), \(s1_{i}\) and \(s2_{i}\) represent the shallow sleeping stages, \(s3_{i}\) and \(s4_{i}\) denote the stage 3 and 4 of deep sleeping, and REM is represented by s5\(i\).  
The average time span of a sleep cycle lasts about 90 min [50], and most people have four or five sleep cycles overnight with each starting with NREM sleep (i.e., \(s1\), \(s2\), \(s3\), \(s4\)) and then followed by a short REM sleep (i.e., \(s5\)). As the sleeping process goes on, the duration of stage 3 and stage 4 decreases while the duration of REM increases. As a result, stage 3 to 4 takes up a larger proportion in the earlier part of the night, and REM sleep occupies a greater part of the later part of the night [51]. Figure 2 shows the typical structure of a whole night's sleep stages and cycles.  
A number of studies have proven that more than one fourth of stroke occurs during sleep accompanied with a circadian variation [52]. In addition, the frequency pattern of EEG was very sensitive to changes in neuronal function resulting from ischemia [53]. As a result, it is reasonable to assume that the fusion of sleep stage and EEG change related features can help to characterize ischemic stroke from certain aspects, and even predict the occurrence of stroke events. In order to enable early detection of stroke, we propose to extract effective features from the sleep data series (including sleep cycles, sleep stages, and sleep EEG) to distinguish patients with stroke in early phases from healthy individuals. The problem is defined as follows.  
**Problem Formalization-** Given a time series of sleep data stream of either a healthy individual or a patient TSS = \(\{TSC_{1},TSC_{2},\ldots,TSC_{n}\}\), and \(TSC_{i}\) = \(\{TS0_{i},TS1_{i},TS2_{i},TS3_{i},TS4_{i},TSS_{i}\}\), our aim is to extract features from different dimensions and then build a model to predict ischemia.