# Learning Using Concave and Convex Kernels: Applications in Predicting Quality of Sleep and Level of Fatigue in Fibromyalgia

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## Abstract

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## 1. Introduction

## 2. Dataset

## 3. Signal Processing: Preprocessing, Filtering, and Feature Extraction

#### 3.1. Preprocessing

#### 3.2. Feature Extraction

- Denoised (filtered) BVP signal, i.e., the output of the Epsilon Tube algorithm, with sampling frequency of 64 Hz.
- Low-band, mid-band, and high-band pass filters applied to the denoised BVP signal.
- Interpolated accelerometer signal, from 32 HZ to 64 Hz.
- Tube sizes from the Epsilon Tube filtering method, another output of the Epsilon Tube algorithm that has the time-varying tube size signal.
- Temperature signal, with sampling frequency of 4 Hz.
- EDA signal, with sampling frequency of 4 Hz.
- The calculated breaths per minute (BPM) signal based on the denoised BVP signal.
- The calculated HRV signal based on the denoised BVP signal.

## 4. Machine Learning: Learning Using Concave and Convex Kernels

#### 4.1. Notation

#### 4.2. Classification Using a Similarity Function

- $Q\left(\mathbf{x}\right)>0$ for all $\mathbf{x}\in {\mathbb{R}}^{n}$;
- $Q\left(\mathbf{x}\right)=Q(-\mathbf{x})$ for all $\mathbf{x}\in {\mathbb{R}}^{n}$;
- $Q\left(\lambda \mathbf{x}\right)>Q\left(\mathbf{x}\right)$ if $\mathbf{x}\in {\mathbb{R}}^{n}$ is non-zero and $\left|\lambda \right|<1$.

#### 4.3. Choosing the Similarity Function

- One or more of the features is prone to large errors —The value of $Q(\mathbf{x}-\mathbf{y})$ is close to 0 even if $\mathbf{x}$ and $\mathbf{y}$ only differ significantly in a few of the features. This choice of $Q\left(\mathbf{x}\right)$ is therefore very sensitive to small subsets of bad features.
- The curse of dimensionality—For the training data to properly represent the probability distribution function underlying the data, the number of training vectors should be exponential in n, the number of features. In practice, it usually is much smaller. Thus, if $\mathbf{x}$ is a test vector in class ${C}_{k}$, there may not be a training vector $\mathbf{y}$ in ${C}_{k}$ for which $Q(\mathbf{x}-\mathbf{y})$ is not small.

#### 4.4. Choosing the Parameters

#### 4.5. Reweighting the Classes

## 5. Experiments

#### 5.1. UCI Machine Learning Repository

#### 5.2. Fibromyalgia Dataset

#### 5.2.1. Results with Conventional Machine Learning Methods

#### 5.2.2. Results with Our Machine Learning Method: Machine Learning Using Concave and Convex Kernels

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LUCCK | Learning Using Concave and Convex Kernels |

BVP | Blood Volume Pulse |

EDA | Electrodermal Activity |

ANS | Autonomic Nervous System |

HRV | Heart Rate Variability |

FFT | Fast Fourier transform |

BPM | Breaths Per Minute |

AUROC | Area Under Receiver Operating Characteristic Curve |

## Appendix A Classification as Maximum a Posteriori Estimation

## Appendix B. Examples

**Example**

**A1.**

**Figure A1.**$Q\left(x\right)={(1+{\lambda}_{1}{x}^{2})}^{-1/{\lambda}_{1}}$ with for ${\lambda}_{1}=0.4,0.8,\cdots ,4$ (blue curves) and ${\lambda}_{1}=0$ (red curve).

**Example**

**A2.**

**Figure A2.**$Q\left(\mathbf{x}\right)={(1+{x}_{1}^{2})}^{-1}{(1+{x}_{2}^{2})}^{-1}=\alpha $ with $0<\alpha <1$.

**Example**

**A3.**

**Figure A3.**$Q\left(\mathbf{x}\right)={(1+2{x}_{1}^{2})}^{-\frac{1}{2}}{(1+{x}_{2}^{2})}^{-1}=\alpha $ with $0<\alpha <1$.

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**Figure 1.**Schematic Diagram of the Proposed Processing System for BVP, accelerometer, EDA and temperature signals.

Detail Coefficients Level | Threshold |
---|---|

8 | 94.38 |

7 | 147.8 |

6 | 303.1 |

5 | 329.9 |

4 | 90.16 |

3 | 30.67 |

2 | 0 |

1 | 0 |

Signals | Features |
---|---|

Denoised BVP | Mean, Standard deviation, Variance, Power, Median, Frequency with the highest peak, |

Amplitude of the frequency with highest peak, FFT power, Mean of FFT amplitudes, | |

Mean of the FFT frequencies, Median of FFT amplitudes (11 features) | |

Low-band denoised | Mean, Standard deviation, Variance, Power, Median, Frequency with the highest peak, |

BVP | Amplitude of the frequency with highest peak, FFT power, Mean of FFT amplitudes, |

Mean of the FFT frequencies, Median of FFT amplitudes (11 features) | |

Mid-band denoised | Mean, Standard deviation, Variance, Power, Median, Frequency with the highest peak, |

BVP | Amplitude of the frequency with highest peak, FFT power, Mean of FFT amplitudes, |

Mean of the FFT frequencies, Median of FFT amplitudes (11 features) | |

High-band denoised | Mean, Standard deviation, Variance, Power, Median, Frequency with the highest peak, |

BVP | Amplitude of the frequency with highest peak, FFT power, Mean of FFT amplitudes, |

Mean of the FFT frequencies, Median of FFT amplitudes (11 features) | |

Tube size | Mean, Standard Deviation, Variance, Power (4 features) |

Interpolated | Mean, Standard Deviation, Variance, Power (4 features) |

accelerometer | |

Temperature signal | Mean, Standard Deviation, Variance, Power (4 features) |

EDA signal | Mean, Standard Deviation, Variance, Power (4 features) |

BPM signal | Maximum, Minimum, Range, Mean, Standard deviation, Power (6 features) |

HRV | The Kubios Standard HRV feature set [32] (25 features) |

**Table 3.**Comparison of our proposed method (LUCCK) with other machine learning methods in terms of accuracy and running time, averaged over 10 folds.

Dataset | Method | Accuracy (%) | Time (s) |
---|---|---|---|

Sonar (208 samples) | LUCCK | 87.42 | 1.5082 |

3-NN | 81.66 | 0.0178 | |

5-NN | 81.05 | 0.0178 | |

Adaboost | 82.19 | 1.0239 | |

SVM | 81.00 | 0.0398 | |

Random Forest (10) | 78.14 | 0.1252 | |

Random Forest (100) | 83.39 | 1.1286 | |

LDA | 74.90 | 0.0343 | |

Glass (214 samples) | LUCCK | 82.56 | 0.3500 |

3-NN | 68.72 | 0.0161 | |

5-NN | 67.04 | 0.0162 | |

Adaboost | 50.82 | 0.5572 | |

SVM | 35.57 | 0.0342 | |

Random Forest (10) | 75.31 | 0.1062 | |

Random Forest (100) | 79.24 | 0.9319 | |

LDA | 63.28 | 0.0155 | |

Iris (150 samples) | LUCCK | 95.93 | 0.1508 |

3-NN | 96.09 | 0.0135 | |

5-NN | 96.54 | 0.0135 | |

Adaboost | 93.82 | 0.4912 | |

SVM | 96.52 | 0.0143 | |

Random Forest (10) | 94.81 | 0.0889 | |

Random Forest (100) | 95.29 | 0.7686 | |

LDA | 98.00 | 0.0122 | |

E. coli (336 samples) | LUCCK | 87.61 | 0.5937 |

3-NN | 85.08 | 0.0190 | |

5-NN | 86.43 | 0.0193 | |

Adaboost | 74.13 | 0.6058 | |

SVM | 87.53 | 0.0448 | |

Random Forest (10) | 84.56 | 0.1075 | |

Random Forest (100) | 87.34 | 0.9265 | |

LDA | 81.46 | 0.0182 |

**Table 4.**Model accuracy with standard deviation and execution time for each model, averaged across the four UCI datasets.

Method | Accuracy (%) | Time (s) |
---|---|---|

LUCCK | 88.38 ± 5.55 | 0.6507 |

3-NN | 82.89 ± 11.27 | 0.0166 |

5-NN | 82.77 ± 12.29 | 0.0167 |

Adaboost | 75.24 ± 18.18 | 0.6695 |

SVM | 75.16 ± 27.15 | 0.0333 |

Random Forest (10) | 83.21 ± 8.65 | 0.1070 |

Random Forest (100) | 86.32 ± 6.84 | 0.9389 |

LDA | 79.41 ± 14.49 | 0.0201 |

Method | Sleep | Fatigue | ||
---|---|---|---|---|

Accuracy (%) | AUROC | Accuracy (%) | AUROC | |

AdaBoost - Decision Stump | 62.07 | 0.63 | 46.64 | 0.55 |

AdaBoost - Random Forest | 59.97 | 0.65 | 51.24 | 0.55 |

K-Nearest Neighbor | 60.55 | 0.55 | 51.88 | 0.53 |

Weighted K-Nearest Neighbor | 65.27 | 0.62 | 68.05 | 0.51 |

Neural Network | 63.47 | 0.64 | 54.80 | 0.59 |

Random Forest | 63.32 | 0.63 | 52.46 | 0.57 |

Support Vector Machine | 64.47 | 0.50 | 55.94 | 0.50 |

LUCCK | 66.95 | 0.66 | 87.59 | 0.68 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sabeti, E.; Gryak, J.; Derksen, H.; Biwer, C.; Ansari, S.; Isenstein, H.; Kratz, A.; Najarian, K. Learning Using Concave and Convex Kernels: Applications in Predicting Quality of Sleep and Level of Fatigue in Fibromyalgia. *Entropy* **2019**, *21*, 442.
https://doi.org/10.3390/e21050442

**AMA Style**

Sabeti E, Gryak J, Derksen H, Biwer C, Ansari S, Isenstein H, Kratz A, Najarian K. Learning Using Concave and Convex Kernels: Applications in Predicting Quality of Sleep and Level of Fatigue in Fibromyalgia. *Entropy*. 2019; 21(5):442.
https://doi.org/10.3390/e21050442

**Chicago/Turabian Style**

Sabeti, Elyas, Jonathan Gryak, Harm Derksen, Craig Biwer, Sardar Ansari, Howard Isenstein, Anna Kratz, and Kayvan Najarian. 2019. "Learning Using Concave and Convex Kernels: Applications in Predicting Quality of Sleep and Level of Fatigue in Fibromyalgia" *Entropy* 21, no. 5: 442.
https://doi.org/10.3390/e21050442