Arousal Scaling

Arousal scaling was performed using wavelet transform (WT).15,16 WT is particularly superior for analyzing nonstationary signals such as EEG where other traditional techniques (based on Fourier transform) are not as effective. WT is defined by its unique wavelet and scaling functions.17 Several WTs have been proposed in the literature. In this study, we used Daubechies wavelets18 of order 4. Daubechies wavelets are known for their orthogonality and efficient implementation and their order 4 has been found to be the most effective for the analysis of EEG.19 WT was performed in MATLAB (Math-Works, Natick, MA, USA). Because our signal of interest (EEG) is discrete, we used discrete wavelet transform (DWT), which is obtained by taking the wavelet and scaling functions at discrete values. The DWT of a signal can be efficiently calculated by passing the signal through a series of cascade filters. Figure 2 shows a two-level wavelet decomposition of a signal using DWT.

As shown in Figure 2, in each level of decomposition, the signal (or the approximation coefficients) is passed through two special filters: a high-pass (or wavelet) filter (h(n)) and a low-pass (or scaling) filter (g(n)). Figure 3 shows the wavelet and scaling filters for Daubechies wavelet order 4. The high-pass and low-pass filters are related to each other and they are quadrature mirror filters. The frequency ranges corresponding to different levels of decomposition depend on the number of levels and sampling frequency. Table 1 shows the frequency range of detail and approximation coefficients for five levels of decomposition and sampling frequency of 128 Hz.

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We calculated all the wavelet coefficients shown in Table 1 for two EEG signals (C3/A2 and C4/A1). The five-level decomposition coefficients (D1-D5 and A5) were calculated for the period between arousal onset and end, and for an equal period preceding arousal onset (Figure 4). From these coefficients we calculated average power (Pavg, six features total), mean of absolute value (MABS, in total six features), ratio of MABS for all combinations of coefficients (e.g.:

…etc; 15 features total) and total variation20 of coefficients in each level (TV, six features total). TV is the average of all point-to-point absolute amplitude differences in the relevant coefficient over the period of analysis. Thus, 33 features were calculated from every arousal. All features were divided by their prearousal values, resulting in 33 normalized features per arousal.

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The 33 features were obtained for each of the 271 training arousals. One-way analysis of variance (ANOVA) was used to determine which of these features discriminated between the different visual scales. Fourteen features were highly signifi-cant. These were:

Next, we built several classifiers to classify (scale) a new arousal based on the training set. We built three k-nearest neighbor classifiers21 (classifier 1: k = 3, classifier 2: k = 4, classifier 3: k = 5), three discriminant classifiers21 (classifier 4: linear discriminant, classifier 5: quadratic discriminant, classifier 6: Mahalanobis discriminant), and one tree classifier21 with pruning at level 6. In total, we built seven classifiers using our training set. This was done to remove the effect of overfitting to the training dataset and to achieve higher predictability over the new testing data. Each classifier generated a scale for an arousal, resulting in seven scales per arousal. The average of all seven scales was calculated and rounded to obtain a single integer scale between 0 and 9. As mentioned, we used two EEG channels for calculating the intensity of arousals. The final scale for a given arousal was the higher of the two EEG channels.

As a control, for each file we randomly selected 10-14 9-sec intervals from periods with stable sleep (i.e., no arousals scored by the sleep technologist). These intervals are referred to as sham arousals.

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