# Bispectral Analysis of Heart Rate Variability to Characterize and Help Diagnose Pediatric Sleep Apnea

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## Abstract

**:**

## 1. Introduction

_{2}) or airflow, among others [8]. Then, medical experts evaluate these signals following well-established rules to extract indices of respiratory disturbance [10]. Among those indices, the common choice to illustrate and report OSA severity is the Apnea–Hypopnea Index (AHI), which reflects the number of total apneic and hypopneic events per hour (e/h) of sleep [8,10]. Despite the usefulness of PSG to diagnose pediatric OSA, the procedure needs a specialized laboratory and is resource-intensive and time-consuming. Moreover, the high number of sensors connected to the child along with having to sleep outside of home makes PSG also especially uncomfortable and inconvenient for patients and caretakers [11,12]. These drawbacks have motivated the search for alternatives to diagnose pediatric OSA and to study its consequences while reducing the number of signals required for the diagnosis [11,12,13]. A recently published systematic review [14] analyzed and conducted a meta-analysis on machine learning techniques employed to automatically diagnose pediatric OSA. Among the shortcomings identified in the literature, the authors highlighted that most of the studies were exclusively based on the analysis of SpO

_{2}signals. Thus, there exists a lack of studies performing machine learning techniques using other physiological signals as alternative approaches to the gold standard PSG [14].

## 2. Subjects and Signals under Study

## 3. Methods

#### 3.1. Bispectrum Estimation

_{1}and f

_{2}are the frequency indices, and f

_{N}is the Nyquist frequency. The resultant matrix reflects the phase coupling degree between frequency components for each pair f

_{1},f

_{2}[29].

_{1}, f

_{2}and f

_{3}frequencies and phase angles φ

_{1}, φ

_{2}and φ

_{3}is described as f

_{3}= f

_{1}+ f

_{2}and φ

_{3}= φ

_{1}+ φ

_{2}[29]. Thus, if phase coupling exists, it means that there are nonlinear interactions between harmonic components [29].

^{10}samples with 50% overlapping and an FFT of 2

^{11}samples. After bispectrum matrix computation, a normalization was applied by dividing each element of the matrix by the sum of all matrix elements as [29,46]

#### 3.2. Determination of Bispectral Regions

_{1}≤ f

_{2}≤ f

_{1}+ f

_{2}≤ f

_{N}[29,34]. As it was mentioned in the Introduction section, previous studies analyzing HRV bispectra focused their analysis along the whole ROI [30,34,35]. The analysis of HRV in the frequency domain, however, is commonly performed along the classic HRV spectral bands, i.e., the very low frequency (VLF, 0–0.04 Hz), low frequency (LF, 0.04–0.15 Hz) and high frequency (HF, 0.15–0.4 Hz) bands [18]. Past studies defined sub-band regions inside the bispectral ROI, bound by those frequencies [36,37,38], which we have termed classic bispectral regions. Furthermore, in a previous study applying a common spectral analysis, three pediatric OSA-related spectral ranges for HRV analysis were identified, which outperformed the classic spectral bands for pediatric OSA characterization and diagnosis [27]: BW1 (0.001–0.005 Hz), BW2: (0.028–0.074 Hz) and BWRes (0.04 Hz around HF peak). A detailed explanation of the process that led us to obtain those frequency ranges can be found in Appendix A. Following a similar reasoning to those studies that analyzed classic bispectral regions, three OSA-specific bispectral regions can be defined as bound by those OSA-related frequency ranges.

#### 3.3. Feature Extraction Stage

_{1}≠ f

_{2}); therefore, these features were not computed over the BWRes region in this study.

#### 3.3.1. Bispectral Region Amplitude Features

- Maximum amplitude (B
_{max}), measured as the maximum magnitude value inside each of the regions considered [46]:$${B}_{max}=\mathrm{max}\left(\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|\right),{f}_{1},{f}_{2}\in \Omega ,$$ - Minimum amplitude (B
_{min}), measured as the minimum magnitude value inside each of the regions considered [46]:$${B}_{min}=\mathrm{min}\left(\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|\right),{f}_{1},{f}_{2}\in \Omega $$ - Total bispectral power (B
_{total}), measured as the sum of all magnitudes inside each of the regions considered [46]:$${B}_{total}={\displaystyle \sum}_{{f}_{1},{f}_{2}\in \Omega}\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|.$$

#### 3.3.2. Bispectral Entropy Features

- Normalized bispectral entropy (BE
_{1}), normalized squared bispectral entropy (BE_{2}) and normalized cubed bispectral entropy (BE_{3}). These parameters, based on Shannon’s entropy, quantify the irregularity of the bispectral distribution in each region and are computed as [29,34]$$B{E}_{i}=-{\displaystyle \sum}_{j\in \Omega}{p}_{j}\xb7\mathrm{log}\left({p}_{j}\right),i=1,2,3$$$${p}_{j}=\frac{{\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|}^{i}}{{{\displaystyle \sum}}_{{f}_{1},{f}_{2}\in \Omega}{\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|}^{i}},i=1,2,3$$

- Phase entropy (PE), which quantifies the phase regularity of the region [29]. PE, as with the bispectral entropies, is higher as the randomness of a process increases, meaning it would be zero for a harmonic, periodic and predictable process [34]. PE computation is performed applying Shannon’s entropy to the normalized distribution of the region phase angles [29,46]:$$PE=-{\displaystyle \sum}_{n\in \Omega}p\left({\Psi}_{n}\right)\xb7\mathrm{log}\left(p\left({\Psi}_{n}\right)\right)$$$$p\left({\Psi}_{n}\right)=\frac{1}{L}{\displaystyle \sum}_{{f}_{1},{f}_{2}\in \Omega}Ind\left(\phi \left[{B}_{N}\left({f}_{1},{f}_{2}\right)\right]\in {\Psi}_{n}\right),$$$${\Psi}_{n}=\left\{\phi |-\pi +\frac{2\pi n}{N}\le \phi -\pi +\frac{2\pi \left(n+1\right)}{N}\right\},n=0,1,\dots ,N-1$$

#### 3.3.3. Bispectral Region Moment Features

- The sum of the logarithmic magnitude values of the region (H
_{1}), sum of the logarithmic magnitude values of the diagonal of the region (H_{2}) and first- and second-order spectral moments of the magnitude values of the diagonal elements of the region (H_{3}and H_{4}, respectively). These features were included as they allow characterizing the nonlinearity of the regions and are computed as follows [46]:$${H}_{1}={\displaystyle \sum}_{{f}_{1},{f}_{2}\in \Omega}\mathrm{log}\left(\left|{B}_{N}\left({f}_{1},{f}_{2}\right)\right|\right).$$$${H}_{2}={\displaystyle \sum}_{{f}_{k},\in {\mathrm{\Gamma}}_{diag}}\mathrm{log}\left(\left|{B}_{N}\left({f}_{k},{f}_{k}\right)\right|\right).$$$${H}_{3}={\displaystyle \sum}_{{f}_{k},\in {\mathrm{\Gamma}}_{diag}}k\xb7\mathrm{log}\left(\left|{B}_{N}\left({f}_{k},{f}_{k}\right)\right|\right).$$$${H}_{4}={\displaystyle \sum}_{{f}_{k},\in {\mathrm{\Gamma}}_{diag}}{\left(k-H3\right)}^{2}\xb7\mathrm{log}\left(\left|{B}_{N}\left({f}_{k},{f}_{k}\right)\right|\right)$$

#### 3.3.4. Bispectral WCOB Features

- WCOB allows reflecting the interaction of different frequency components through the assignment of a weight to each bispectral point of the region [46]. The weighted center of each region is composed of two vectors, f1m and f2m, which indicate the coupling focus of the region as a summary of the frequency interaction [46]. Those components of WCOB are computed as [46]$$f1m=\frac{{{\displaystyle \sum}}_{{f}_{1},{f}_{2}\in \Omega}{f}_{1}\xb7{B}_{N}\left({f}_{1},{f}_{2}\right)}{{{\displaystyle \sum}}_{{f}_{1},{f}_{2}\in \Omega}{B}_{N}\left({f}_{1},{f}_{2}\right)},$$$$f2m=\frac{{{\displaystyle \sum}}_{{f}_{1},{f}_{2}\in \Omega}{f}_{2}\xb7{B}_{N}\left({f}_{1},{f}_{2}\right)}{{{\displaystyle \sum}}_{{f}_{1},{f}_{2}\in \Omega}{B}_{N}\left({f}_{1},{f}_{2}\right)}$$

#### 3.3.5. Relative Power of the Diagonal, a Novel Bispectral Feature

- The relative power of the diagonal (RP
_{Diag}), computed as the sum of the bispectral amplitudes of the diagonal elements of the region, after a normalization applied over the whole diagonal. This novel parameter evaluates the relative bispectral magnitude value inside the diagonal of the region with respect to the complete bispectral diagonal magnitude:$$R{P}_{Diag}={\displaystyle \sum}_{{f}_{k}\in {\mathrm{\Gamma}}_{diag}}\left|Dia{g}_{N}\left({f}_{k}\right)\right|,$$$$Dia{g}_{N}\left({f}_{k}\right)=\frac{Diag\left({f}_{k}\right)}{DP},{f}_{k}=0,\cdots ,{f}_{N}$$

_{1}= f

_{2}; therefore, this parameter, as well as H2, is intended to measure the phase coupling between the harmonic components of HRV signals, such that f

_{3}= 2f

_{1}and φ

_{3}= 2φ

_{1}[30,46].

_{Diag}and H2 present two important differences. First, a normalization over the whole diagonal is applied in this novel feature. As a result of this normalization, the sum of all bispectral amplitudes of the diagonal elements is equal to 1, meaning RP

_{Diag}evaluates the proportion of the total diagonal bispectral power contained in the region. Then, as the normalization is scaling the values of the diagonal elements, we use a linear scale to compute the sum of the relative power instead of the logarithmic scale applied in H2. The rationale of this parameter lies in the normalization considering all of the frequency range. When OSA occurs, there is an alteration in the synchronization of the heart rhythm [30], leading to a redistribution of HRV activity to frequency components associated with the occurrence of apneic events. Thus, the normalization applied here considers not only the bispectral power in the diagonal of the region evaluated but also that in other diagonal elements. This influence of the redistribution to other frequency ranges is lost when applying a logarithmic scale.

#### 3.4. Feature Selection Stage

#### 3.5. Classification Stage

_{H}) is a parameter to be optimized. To deal with overfitting, we also introduced a regularization parameter (λ) in the tuning of the network weights, which were randomly initialized [53].

_{H}and λ) was performed, again, by means of 1000 bootstrap replicates from the training dataset, but different from the replicates employed in the feature selection stage. We computed Cohen’s kappa (k) for each N

_{H/}λ combination and selected those values where k was maximum [42,46,51].

#### 3.6. Statistical Analysis

_{S}), controlling the possible effect of age. The polysomnographic indices, related to OSA, as well as sleep structure and quality, were the same as those in [27]: AHI, Obstructive AHI (OAHI), obstructive apnea index (OAI), oxygen desaturation index (ODI), wake after sleep onset (WASO), number of awakenings during total sleep time (#Awakenings), percentage of total sleep spent in sleep stages N1, N2, N3 and rapid eye movement (%N1, %N2, %N3 and %REM, respectively) and total arousal index per hour of sleep (TAI). Correlation analysis was performed on the test set.

## 4. Results

#### 4.1. Feature Selection in the Training Set

_{Classic}) was formed by three features, one of each region: VLF_f2m, LF_BE

_{2}and HF_PE. Regarding the OSA-specific region set (Figure 2b), the optimum subset (BISP

_{Specific}) was composed of four features selected more than 500 times: BW2_RP

_{Diag}, BW2_BE

_{1}, BWRes_B

_{min}and BWRes_BE

_{3}. None of the BW1 region features considered were selected over 500 times.

#### 4.2. Descriptive Analysis of the Features Selected

_{lassic}and BISP

_{Specific}subsets, respectively. The p-value resulting from the Kruskal–Wallis test is also depicted in these figures. It can be appreciated that, in the BISP

_{Classic}subset, VLF_f2m experienced an increase with OSA severity, while a decrease in LF_BE

_{2}and HF_PE occurred as the OSA severity increases. For the features included in BISP

_{Specific}, there was a clear rise in the BW2_RP

_{Diag}values, along with a slight increase in BWRes_BE

_{3}with OSA severity. In contrast, the BW2_BE

_{1}values experienced a decrease with the disease. BWRes_B

_{min}was the only parameter showing an unclear tendency among the severity groups, which led it to be the only one that did not show statistically significant differences. The remaining six parameters showed statistically significant differences among the four OSA severity groups (p-value < 0.01 after Bonferroni correction).

#### 4.3. MLP Network Optimization and Training

_{lassic}features as input (MLP1

_{Classic}, MLP5

_{Classic}and MLP10

_{Classic}, with AHI = 1, 5 and 10 e/h as thresholds for binary classification, respectively), and three models with BISP

_{Specific}as input features (MLP1

_{Specific}, MLP5

_{Specific}and MLP10

_{Specific}, with AHI = 1, 5 and 10 e/h as a threshold for binary classification, respectively). For each model, N

_{H}varied from 2 to 20 in steps of 1, and from 22 to 50 in steps of 2. Similarly, λ varied from 0.5 to 10 in steps of 0.5. Each N

_{H/}λ pair resulted in an averaged k through 1000 bootstrap replicates of the training set; therefore, we selected the N

_{H}/λ combination with the higher averaged k. N

_{H}= 2 and λ = 5 were the optimized design parameters selected in four out of six MLP models: MLP1

_{Classic}, MLP5

_{Classic}, MLP5

_{Specific}and MLP10

_{Specific}. For the MLP10

_{Classic}model, the optimized design parameters were N

_{H}= 34 and λ = 5. Finally, the optimized parameters in MLP1

_{Specific}were N

_{H}= 38 and λ = 5.

#### 4.4. Correlation Analysis in the Test Set

_{S}| obtained were generally low, some of the correlations evaluated were statistically significant (p-value < 0.01 after Bonferroni correction). VLF_f2m and BW2_RP

_{Diag}showed similar behaviors, with a statistically significant positive ρ

_{S}with the four respiratory indices (AHI, OAHI, OAI and ODI) and TAI, as well as a negative ρ

_{S}with %REM. In the opposite way, BW2_BE

_{1}obtained a negative ρ

_{S}with the four respiratory indices and TAI. LF_BE

_{2}also reached a statistically significant negative ρ

_{S}with AHI, OAHI and TAI. BW2_RP

_{Diag}reached the highest absolute correlation values among almost all of these statistically significant correlations, only being equaled by the ρ

_{S}reached between VLF_f2m and OAHI. None of the selected BWRes features nor HF_PE obtained statistically significant correlations with any of the polysomnographic indices considered in this study.

#### 4.5. Diagnostic Ability Assessments

_{Diag}obtained the highest results in terms of Acc and AUC. For the 5 e/h threshold, again, BW2_RP

_{Diag}showed a higher Acc than the other features selected, being only slightly surpassed by VLF_f2m in terms of AUC. When considering 10 e/h as severity cutoff for binary classification, LF_BE

_{2}was the feature showing the higher Acc, but with a lower AUC and a more unbalanced Se/Sp pair than BW2_RP

_{Diag}.

_{Classic}model was the only one that achieved a higher diagnostic performance than the OSA-specific models in terms of Acc and AUC at the cost of a strongly unbalanced Se/Sp pair and a very low Se value (43.5%).

## 5. Discussion

#### 5.1. Physiological Interpretation of the Features Selected

_{Diag}was one of the two features selected. This parameter, as with VLF_f2m, showed an increasing tendency with OSA severity (Figure 4a). As explained in the Methods section, RP

_{Diag}is intended to measure phase coupling between the harmonic components (f

_{3}= 2f

_{1}and φ

_{3}= 2φ

_{1}) of the HRV. The increment in BW2_RP

_{Diag}with OSA severity would indicate increasing nonlinear interactions between those harmonics of OSA-affected children. Consequently, there is an increment in less random/more periodic harmonics in HRV signals inside the BW2 region due to apneic events. The increase in VLF_f2m and BW2_RP

_{Diag}with OSA severity is supported by the correlation study, showing statistically significant correlations with all respiratory indices, as well as with TAI. Furthermore, BW2_RP

_{Diag}generally obtained the highest individual diagnostic performance. Taken together, these facts highlight the importance of analyzing the BW2 bispectral region in the field of pediatric OSA, especially characterized through our new proposed parameter RP

_{Diag}.

_{1}inside the BW2 region, BE

_{2}inside the LF region and BE

_{3}inside BWRes. These parameters measure the irregularity of the HRV from the bispectral distribution in each region, with the inclusion of quadratic and cubic components scaling the differences in the bispectral amplitude [29]. Bispectral distributions averaged for each severity group in these three regions are shown in Figure A2 (LF region), A4 (BW2 region) and A5 (BWRes region). In the case of BW2_BE1, a decrease with severity was observed, reflecting a reduction in irregularity with apneic events in the HRV components linked to the frequencies of this region. As a result of apneic events, it can be appreciated that the bispectral amplitude in BW2 is more concentrated at low frequencies in the severe group (Figure A4d) and starts to distribute more randomly to other frequencies as OSA decreases. This may be due to the aforementioned increment in the less random harmonics in this range due to OSA, leading to the reduction with severity experimented in BW2_BE

_{1}(Figure 4b). In the case of LF_BE

_{2}, this parameter also decreases with the severity of the disease, again reflecting a reduction in irregularity with OSA in the HRV associated with this frequency region. Figure A2 points to the fact that the bispectral power distribution of no-OSA subjects is more dispersed over the whole LF region and starts to concentrate at lower frequencies as OSA severity increases. Interestingly, this parameter only showed negative statistically significant correlation values when considering respiratory indices that include hypopneas (AHI and OAHI). It seems that, as apneas are less frequent than hypopneas in the database under study, the only effect of apneas is not enough to decrease the bispectral HRV irregularity associated with the LF frequency range. Similarly, collective apneic effects are better captured with the inclusion of the quadratic amplitude, suggesting that BE

_{2}is more accurate in detecting alterations in the bispectral distribution of LF as a result of all apneic events. Regarding BWRes_BE

_{3}, Figure 4d shows that no-OSA, mild OSA and moderate OSA subjects presented lower values than the severe OSA group. As it can be seen in Figure A6, there is a higher bispectral power concentration around the respiratory peak for the first three severity groups, and a lower concentration for severe OSA, whose distribution spreads over other frequencies. The lower coupling around the respiratory peak in the severe group makes sense as OSA results in a redistribution in the bispectral power to frequency ranges related to apneic events, such as the BW2 region. These are milder differences than those observed in BW2 and LF; therefore, an increase in HRV irregularity due to OSA appears to be better captured through BE

_{3}from BWRes.

_{min}was the remaining feature selected. The normalization applied over each bispectral matrix allowed B

_{min}to estimate the minimum coupling within this region [46]. Despite the unclear tendency and the absence of differences obtained in this parameter (Figure 4c), BWRes_B

_{min}was selected by the algorithm. This implies that BWRes_B

_{min}contains information that is complementary to the other features selected in the OSA-specific region feature subset.

_{Diag}, which seems to be more accurate in the characterization of apneic alterations.

#### 5.2. Diagnostic Performance of the Bispectral Models

_{Specific}obtained a higher Acc (63.4% versus 54.7% from the MLP1

_{Classic}model) and a higher AUC (0.627 versus 0.600). In the 5e/h threshold, the Acc obtained by MLP5

_{Specific}and MLP5

_{Classic}was 81.0% in both cases, but with the specific model, showing a more balanced Se/Sp, a higher AUC was found (0.774 versus 0.791). Finally, in the 10e/h severity cutoff, MLP10

_{Classic}obtained the highest Acc and AUC from the study. However, the Se/Sp pair was strongly unbalanced, with a very low Se value of 43.5%. Nevertheless, the MLP10

_{Specific}model, at the cost of a very slight reduction in terms of Acc and AUC (89.3% versus 91.7%, and 0.847 versus 0.841, respectively), resulted in a more balanced Se/Sp pair (66.7%/91.6%). These results reinforce the conclusion from our previous work [27] about the importance of analyzing OSA-specific HRV frequency ranges whenever pediatric OSA is under study.

#### 5.3. Comparison with Previous Work

_{BW1}individually) and 0.592 AUC (LDA band of interest model). With 5 e/h as the severity threshold, the highest results were 76.6% Acc (RP

_{BW2}) and 0.688 AUC (LDA band of interest model). Finally, in the 10 e/h severity cutoff, the highest results reached were 82.8% Acc and 0.796 AUC (both using the LDA band of interest model). It can be observed from Table 4 that, in the present work, the best results achieved with the MLP optimized models surpassed, by far, those achieved previously, and even some of the individual bispectral features eventually outperformed several of these results. It can be argued that the improvement in the diagnostic outcomes may be mediated by the increased complexity of the classification algorithm. To deal with this issue, and also for a fair comparison, we have included in Appendix B the results obtained using the same classification methodology as in [27]. Table A1 shows the classification results obtained including the features selected from each approach using an LDA classifier. The results obtained with both the LDA

_{Classic}and LDA

_{Specific}models also outperform the diagnostic performance obtained in the previous work, being only surpassed in terms of Acc in the 1 e/h threshold. Thus, the diagnostic utility of the features extracted from the bispectral HRV region analysis is clearly demonstrated, with the MLP models reaching the highest diagnostic performance in the literature when using HRV features exclusively to generate an automated classification of pediatric subjects into the presence or absence of OSA and to estimate the severity grouping. Moreover, the higher diagnostic performance reached by the bispectral analysis highlights the usefulness of this analysis in the pediatric OSA context, which seems to be more accurate than traditional frequency analysis to evaluate HRV alterations. The presence of HRV nonlinear dynamics demonstrated in Appendix C and captured through the HRV bispectrum estimation may be behind this improvement over the traditional techniques.

_{Specific}obtained similar Se results. It is worth noting that, among the studies included in the meta-analysis, none of them considered HRV signals. Furthermore, the sample size from seventeen out of nineteen studies included in the systematic review was markedly smaller than the databases analyzed here, only being surpassed by the cohort included in the studies of Hornero et al. [42] and Vaquerizo-Villar et al. [58]. Therefore, these considerations along with the similar diagnostic performance reached in the 5 and 10 e/h severity thresholds reinforce the support for the use of HRV bispectral analysis as a potential alternative to overnight PSG for pediatric OSA diagnosis.

#### 5.4. Limitations and Outlook

## 6. Conclusions

_{Specific}, MLP5

_{Specific}and MLP10

_{Classic}models achieving the highest diagnostic yield from the study for each severity cutoff. These results highlight the usefulness of bispectral HRV analysis in the pediatric OSA context, especially when analyzing bispectral regions bounded by OSA-specific frequency ranges. Thus, we conclude that information extracted from HRV bispectra allows for the characterization and diagnosis of pediatric OSA, leading us to propose this approach as a potential alternative to PSG.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Determination of HRV OSA-Specific Frequency Ranges and Averaged Bispectral Regions in the Training Set

**Figure A1.**Averaged bispectrum magnitude in the very low frequency region (0–0.04 Hz) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure A2.**Averaged bispectrum magnitude in the low-frequency region (0.04–0.15 Hz) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure A3.**Averaged bispectrum magnitude in the high-frequency region (0.15–0.40 Hz) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure A4.**Averaged bispectrum magnitude in the BW1 frequency region (0.001–0.005 Hz) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure A5.**Averaged bispectrum magnitude in the BW2 frequency region (0.028–0.074 Hz) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure A6.**Averaged bispectrum magnitude in the BWRes frequency region (0.04 Hz around the respiratory peak inside HF) for the four severity groups in the training set. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

## Appendix B. Diagnostic Performance of Bispectral Region Models with a Linear Discriminant Analysis Classifier

_{Classic}and BISP

_{Specific}feature subsets was evaluated through two models based on LDA (LDA

_{Classic}and LDA

_{Specific}models, respectively). Both classifiers were trained in the training set for each severity threshold, and the diagnostic performance was evaluated in the test set. Table A1 shows the results achieved by each model, along with the results from our previous work [27]. It can be observed that, as it is commented on in the Discussion section, the new diagnostic performance surpassed our previous classification results in terms of Acc and AUC in almost all the parameters. Only the individual Acc in 1 e/h for RP

_{BW1}outperformed the LDA results presented here, but with a lower AUC.

**Table A1.**Diagnostic performance in the test set achieved following a spectral and a bispectral approach. The individual performances correspond to each relative power extracted in our previous work. Joint performance was assessed by constructing LDA classifiers.

Feature/Model | AHI Threshold = 1 e/h | AHI Threshold = 5 e/h | AHI Threshold = 10 e/h | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Se | Sp | Acc | AUC | Se | Sp | Acc | AUC | Se | Sp | Acc | AUC | ||

RP_{VLF} | 68.9 | 31.6 | 56.3 | 0.518 | 33.0 | 65.0 | 60.2 | 0.456 | 40.6 | 64.2 | 62.1 | 0.495 | |

Previous work (frequency analysis approach) | RP_{LF} | 43.5 | 62.9 | 50.1 | 0.557 | 52.7 | 58.4 | 57.6 | 0.590 | 59.4 | 58.4 | 58.5 | 0.666 |

RP_{HF} | 35.5 | 71.9 | 47.8 | 0.523 | 39.3 | 68.1 | 63.8 | 0.540 | 43.5 | 76.7 | 73.7 | 0.605 | |

LF/HF | 37.7 | 70.3 | 48.7 | 0.540 | 45.5 | 66.8 | 63.7 | 0.567 | 49.3 | 70.8 | 68.8 | 0.643 | |

RP_{BW1} | 66.3 | 45.3 | 59.2 | 0.559 | 65.2 | 54.0 | 55.6 | 0.621 | 69.6 | 52.3 | 53.9 | 0.624 | |

RP_{BW2} | 32.7 | 78.1 | 48.1 | 0.591 | 45.5 | 82.0 | 76.6 | 0.670 | 58.0 | 78.2 | 76.4 | 0.751 | |

RP_{BWRes} | 45.5 | 56.6 | 49.3 | 0.532 | 44.6 | 64.0 | 61.2 | 0.571 | 49.3 | 64.0 | 62.6 | 0.628 | |

LDA Classic Bands | 25.7 | 81.3 | 44.5 | 0.559 | 46.4 | 72.2 | 68.4 | 0.633 | 50.7 | 75.3 | 73.1 | 0.685 | |

LDA Bands of Interest | 42.5 | 72.3 | 52.6 | 0.592 | 50.0 | 80.9 | 76.4 | 0.688 | 63.8 | 84.7 | 82.8 | 0.796 | |

Present work (bispectral analysis approach) | LDA_{Classic} | 30.1 | 81.3 | 47.4 | 0.601 | 53.6 | 85.3 | 80.6 | 0.779 | 66.7 | 89.7 | 87.6 | 0.847 |

LDA_{Specific} | 37.9 | 77.3 | 51.3 | 0.615 | 63.4 | 82.8 | 79.9 | 0.792 | 71.0 | 85.9 | 84.5 | 0.842 |

## Appendix C. Surrogate Data Approach

#### Appendix C.1. Testing for Nonlinearities

#### Appendix C.2. Bispectrum with Surrogate

**Figure A7.**Bispectrum magnitude estimation with and without surrogate method for a no-OSA subject. (

**a**) 2D bispectrum estimation; (

**b**) 2D BWS estimation; (

**c**) 3D bispectrum estimation; (

**d**) 3D BWS estimation.

**Figure A8.**Bispectrum magnitude estimation with and without surrogate method for a mild OSA subject. (

**a**) 2D bispectrum estimation; (

**b**) 2D BWS estimation; (

**c**) 3D bispectrum estimation; (

**d**) 3D BWS estimation.

**Figure A9.**Bispectrum magnitude estimation with and without surrogate method for a moderate OSA subject. (

**a**) 2D bispectrum estimation; (

**b**) 2D BWS estimation; (

**c**) 3D bispectrum estimation; (

**d**) 3D BWS estimation.

**Figure A10.**Bispectrum magnitude estimation with and without surrogate method for a severe OSA subject. (

**a**) 2D bispectrum estimation; (

**b**) 2D BWS estimation; (

**c**) 3D bispectrum estimation; (

**d**) 3D BWS estimation.

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**Figure 1.**Averaged bispectrum magnitude in the range 0–0.4 Hz in the training set for the four severity groups. To improve the visualization of the coupling focus at very low frequencies, an amplification between 0 and 0.05 Hz is depicted in the upper right corner for each figure. (

**a**) No-OSA; (

**b**) mild OSA; (

**c**) moderate OSA; (

**d**) severe OSA.

**Figure 2.**Features selected in each feature set after applying FCBF algorithm over 1000 bootstrap replicates of the training set. (

**a**) Features selected in the classic region set; (

**b**) features selected in the OSA-specific region set.

**Figure 3.**Boxplot distribution of the features selected in the bispectral classic region feature subset for the four OSA severity groups in the training set. The p-value obtained with the Kruskal–Wallis test is shown in each subplot. (

**a**) VLF_f2m boxplots and p-value; (

**b**) LF_BE

_{2}boxplots and p-value; (

**c**) HF_PE boxplots and p-value.

**Figure 4.**Boxplot distribution of the features selected in the bispectral OSA-specific region feature subset for the four OSA severity groups in the training set. The p-value obtained with the Kruskal–Wallis test is shown in each subplot. (

**a**) BW2_RP

_{Diag}boxplots and p-value; (

**b**) BW2_BE

_{1}boxplots and p-value; (

**c**) BWRes_B

_{min}boxplots and p-value; (

**d**) BWRes_BE

_{3}boxplots and p-value.

All | Training Set (UofC) | Test Set (CHAT) | |
---|---|---|---|

Subjects (n) | 1738 | 981 | 757 |

Age (years) | 6.4 [3.3] | 6.0 [6.0] | 7.0 [2.4] |

Males (n) | 962 (55.35%) | 602 (61.37%) | 360 (47.95%) |

BMI (kg/m^{2}) | 17.63 [5.37] | 18.02 [5.86] | 17.28 [4.64] |

AHI (e/h) | 2.23 [5.27] | 3.8 [7.76] | 1.46 [2.07] |

AHI ≥ 1 (e/h) | 1309 (75.31%) | 808 (82.36%) | 501 (66.18%) |

AHI ≥ 5 (e/h) | 519 (29.86%) | 407 (41.49%) | 112 (14.80%) |

AHI ≥ 10 (e/h) | 298 (17.15%) | 229 (23.34%) | 69 (9.11%) |

**Table 2.**Summary of the bispectral features initially computed in each region. Features related to the diagonal of the region were excluded in the BWRes region.

Classic Region Feature Set | OSA-Specific Region Feature Set | |||||
---|---|---|---|---|---|---|

Features | VLF | LF | HF | BW1 | BW2 | BWRes |

RP_{Diag} | VLF_RP_{Diag} | LF_RP_{Diag} | HF_RP_{Diag} | BW1_RP_{Diag} | BW2_RP_{Diag} | - |

B_{max} | VLF_B_{max} | LF_B_{max} | HF_B_{max} | BW1_B_{max} | BW2_B_{max} | BWRes_B_{max} |

B_{min} | VLF_B_{min} | LF_B_{min} | HF_B_{min} | BW1_B_{min} | BW2_B_{min} | BWRes_B_{min} |

B_{Total} | VLF_B_{Total} | LF_B_{Total} | HF_B_{Total} | BW1_B_{Total} | BW2_B_{Total} | BWRes_B_{Total} |

BE_{1} | VLF_BE_{1} | LF_BE_{1} | HF_BE_{1} | BW1_BE_{1} | BW2_BE_{1} | BWRes_BE_{1} |

BE_{2} | VLF_BE_{2} | LF_BE_{2} | HF_BE_{2} | BW1_BE_{2} | BW2_BE_{2} | BWRes_BE_{2} |

BE_{3} | VLF_BE_{3} | LF_BE_{3} | HF_BE_{3} | BW1_BE_{3} | BW2_BE_{3} | BWRes_BE_{3} |

PE | VLF_PE | LF_PE | HF_PE | BW1_PE | BW2_PE | BWRes_PE |

H_{1} | VLF_H_{1} | LF_H_{1} | HF_H_{1} | BW1_H_{1} | BW2_H_{1} | BWRes_H_{1} |

H_{2} | VLF_H_{2} | LF_H_{2} | HF_H_{2} | BW1_H_{2} | BW2_H_{2} | - |

H_{3} | VLF_H_{3} | LF_H_{3} | HF_H_{3} | BW1_H_{3} | BW2_H_{3} | - |

H_{4} | VLF_H_{4} | LF_H_{4} | HF_H_{4} | BW1_H_{4} | BW2_H_{4} | - |

f1m | VLF_f1m | LF_f1m | HF_f1m | BW1_f1m | BW2_f1m | BWRes_f1m |

f2m | VLF_f2m | LF_f2m | HF_f2m | BW1_f2m | BW2_f2m | BWRes_f2m |

**Table 3.**Results of the partial correlation study in the test set between features selected for each subset and the polysomnographic indices considered.

BISP_{Classic} Features | ||||||||

PSG Index | VLF_f2m | LF_BE_{2} | HF_PE | |||||

ρ_{S} | p-Value | ρ_{S} | p-Value | ρ_{S} | p-Value | |||

AHI | 0.274 | <<0.01 | −0.185 | <<0.01 | −0.112 | 0.002 * | ||

OAHI | 0.261 | <<0.01 | −0.149 | <<0.01 | −0.097 | 0.008 | ||

OAI | 0.167 | <<0.01 | −0.105 | 0.004 * | −0.064 | 0.079 | ||

ODI | 0.215 | <<0.01 | −0.123 | 0.001 * | −0.054 | 0.138 | ||

#Awakenings | −0.075 | 0.039 | −0.027 | 0.461 | −0.020 | 0.586 | ||

WASO | −0.003 | 0.929 | 0.065 | 0.076 | −0.022 | 0.538 | ||

%N1 | 0.089 | 0.014 | −0.071 | 0.052 | −0.030 | 0.404 | ||

%N2 | −0.034 | 0.357 | 0.099 | 0.007 * | 0.013 | 0.715 | ||

%N3 | 0.034 | 0.355 | −0.025 | 0.497 | −0.044 | 0.23 | ||

%REM | −0.125 | 0.001 | −0.052 | 0.154 | 0.059 | 0.108 | ||

TAI | 0.213 | <<0.01 | −0.158 | <<0.01 | −0.115 | 0.002 * | ||

BISP_{Specific} Features | ||||||||

PSG Index | BW2_RP_{Diag} | BW2_BE_{1} | BWRes_B_{min} | BWRes_BE_{3} | ||||

ρ_{S} | p-Value | ρ_{S} | p-Value | ρ_{S} | p-Value | ρ_{S} | p-Value | |

AHI | 0.308 | <<0.01 | −0.180 | <<0.01 | 0.054 | 0.136 | 0.045 | 0.214 |

OAHI | 0.261 | <<0.01 | −0.180 | <<0.01 | 0.098 | 0.007 * | 0.028 | 0.435 |

OAI | 0.177 | <<0.01 | −0.173 | <<0.01 | 0.071 | 0.051 | 0.058 | 0.112 |

ODI | 0.247 | <<0.01 | −0.139 | 0.001 | 0.019 | 0.61 | 0.072 | 0.047 |

#Awakenings | −0.033 | 0.372 | −0.001 | 0.994 | −0.006 | 0.876 | 0.035 | 0.331 |

WASO | 0.071 | 0.05 | 0.078 | 0.031 | −0.018 | 0.622 | 0.056 | 0.126 |

%N1 | 0.107 | 0.003 * | −0.061 | 0.093 | 0.023 | 0.527 | 0.028 | 0.441 |

%N2 | −0.061 | 0.091 | 0.008 | 0.837 | 0.048 | 0.184 | 0.059 | 0.104 |

%N3 | 0.053 | 0.147 | 0.008 | 0.817 | −0.075 | 0.04 | −0.092 | 0.011 |

%REM | −0.139 | 0.001 | 0.048 | 0.192 | −0.007 | 0.855 | 0.013 | 0.722 |

TAI | 0.225 | <<0.01 | −0.144 | <<0.01 | 0.068 | 0.062 | 0.068 | 0.06 |

^{−4}appear as << 0.01. * Non-significant after Bonferroni correction. Statistically significant correlations (p-value < 0.01 after Bonferroni correction) appear in bold.

**Table 4.**Diagnostic performance achieved in the test set by each feature selected and each MLP optimized model for the binary classification in each severity threshold. Results are shown in terms of sensitivity (Se %), specificity (Sp %), accuracy (Acc %) and AUC.

Threshold: AHI = 1 e/h | ||||

Feature/Model | Se | Sp | Acc | AUC |

VLF_f2m | 44.5 | 72.3 | 53.9 | 0.605 |

LF_BE_{2} | 42.1 | 72.7 | 52.4 | 0.581 |

HF_PE | 42.9 | 63.3 | 49.8 | 0.550 |

BW2_RP_{Diag} | 50.9 | 64.8 | 55.6 | 0.629 |

BW2_BE_{1} | 47.1 | 59.4 | 51.3 | 0.559 |

BWRes_B_{min} | 40.5 | 57.4 | 46.2 | 0.482 |

BWRes_BE_{3} | 41.5 | 57.4 | 46.9 | 0.513 |

MLP1_{Classic} | 52.3 | 59.4 | 54.7 | 0.600 |

MLP1_{Specific} | 76.3 | 38.3 | 63.4 | 0.627 |

Threshold: AHI = 5 e/h | ||||

Feature/Model | Se | Sp | Acc | AUC |

VLF_f2m | 62.5 | 72.2 | 70.8 | 0.749 |

LF_BE_{2} | 56.3 | 74.4 | 71.7 | 0.670 |

HF_PE | 45.5 | 72.1 | 68.2 | 0.628 |

BW2_RP_{Diag} | 60.7 | 77.7 | 75.2 | 0.747 |

BW2_BE_{1} | 56.3 | 70.1 | 68.0 | 0.671 |

BWRes_B_{min} | 58.9 | 45.3 | 47.3 | 0.567 |

BWRes_BE_{3} | 47.3 | 58.4 | 56.8 | 0.569 |

MLP5_{Classic} | 50.9 | 86.2 | 81.0 | 0.774 |

MLP5_{Specific} | 62.5 | 84.2 | 81.0 | 0.791 |

Threshold: AHI = 10 e/h | ||||

Feature/Model | Se | Sp | Acc | AUC |

VLF_f2m | 63.8 | 76.7 | 75.6 | 0.784 |

LF_BE_{2} | 58.0 | 81.5 | 79.4 | 0.740 |

HF_PE | 53.6 | 72.1 | 70.4 | 0.663 |

BW2_RP_{Diag} | 68.1 | 76.0 | 75.3 | 0.789 |

BW2_BE_{1} | 47.8 | 76.0 | 73.4 | 0.692 |

BWRes_B_{min} | 56.5 | 50.6 | 51.1 | 0.557 |

BWRes_BE_{3} | 55.1 | 59.4 | 59.0 | 0.614 |

MLP10_{Classic} | 43.5 | 96.5 | 91.7 | 0.847 |

MLP10_{Specific} | 66.7 | 91.6 | 89.3 | 0.841 |

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## Share and Cite

**MDPI and ACS Style**

Martín-Montero, A.; Gutiérrez-Tobal, G.C.; Gozal, D.; Barroso-García, V.; Álvarez, D.; del Campo, F.; Kheirandish-Gozal, L.; Hornero, R.
Bispectral Analysis of Heart Rate Variability to Characterize and Help Diagnose Pediatric Sleep Apnea. *Entropy* **2021**, *23*, 1016.
https://doi.org/10.3390/e23081016

**AMA Style**

Martín-Montero A, Gutiérrez-Tobal GC, Gozal D, Barroso-García V, Álvarez D, del Campo F, Kheirandish-Gozal L, Hornero R.
Bispectral Analysis of Heart Rate Variability to Characterize and Help Diagnose Pediatric Sleep Apnea. *Entropy*. 2021; 23(8):1016.
https://doi.org/10.3390/e23081016

**Chicago/Turabian Style**

Martín-Montero, Adrián, Gonzalo C. Gutiérrez-Tobal, David Gozal, Verónica Barroso-García, Daniel Álvarez, Félix del Campo, Leila Kheirandish-Gozal, and Roberto Hornero.
2021. "Bispectral Analysis of Heart Rate Variability to Characterize and Help Diagnose Pediatric Sleep Apnea" *Entropy* 23, no. 8: 1016.
https://doi.org/10.3390/e23081016