# Motion Symmetry Evaluation Using Accelerometers and Energy Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Data Acquisition

#### 2.2. Data Processing

- relative energy components at selected wavelet decomposition levels for signals recorded by an accelerometer,
- relative energy components at selected wavelet decomposition levels for signals recorded by a gyrometer,

## 3. Results

- acquiring signals recorded by the accelerometer and gyrometer during selected physical activities using handheld devices,
- transferring signals by wired or wireless communication links to a mathematical environment (of MATLAB 2018b in this case),
- mathematical data analysis including their resampling and digital filtering,
- applying the wavelet transform and evaluating the relative signal energy at selected decomposition levels,
- defining a pattern matrix with Q column vectors for each signal segment and associated vector of target values,
- optimizing and then verifying the classification model.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Principle of raw data acquisition (with the sampling frequency of 100 Hz) from a mobile device located on (

**A**) the right limb and (

**B**) the left limb, using (

**C**) the accelerometer and gyrometer during an exercise lasting 60 s, and (

**D**) orientations of the axes for data acquisition.

**Figure 2.**Shannon wavelet functions presenting (

**a**) their dilation in the time domain, (

**b**) corresponding spectra compression, and (

**c**) distribution of scalogram coefficients.

**Figure 3.**Distribution of features evaluated as the relative energy for the second decomposition $db2$ wavelet level of signals recorded by the accelerometer and gyrometer on the left and the right legs.

**Figure 4.**A two-layer neural network with sigmoidal and softmax transfer functions for classification of a pattern vector ${[p(1,k),p(2,k),\cdots ,p(R,k)]}^{\prime}$ evaluating probabilities of their affiliation into classes ${c}_{1},{c}_{2},\cdots ,{c}_{S2}$.

**Figure 5.**Results of classification for a selected experiment 180 s long for wavelet features evaluated for the left leg (class 1) and the right leg (class 2) evaluated by the (

**a**) Bayesian method, (

**b**) support vector machine, (

**c**) neural networks, and (

**d**–

**f**) associated confusion matrices.

**Figure 6.**The comparison of (

**a**) the mean energy at the second Haar decomposition level for signals recorded by the accelerometer (feature $F1$) and (

**b**) gyrometer (feature $F2$), and (

**c**) classification accuracy for different methods for the left leg (class 1) and the right leg (class 2) with respect to the load during the exercise.

**Figure 7.**Receiver operating characteristic (ROC) curves for Bayes, support vector machine (SVM) and neural network (NN) classification methods for the medium load using (

**a**) two and (

**b**) six features with areas under individual curves.

**Figure 8.**Precision-recall plots for Bayes, SVM and NN classification methods for the medium load using (

**a**) two and (

**b**) six features with areas under individual curves.

**Table 1.**Parameters of cluster centers of the left and the right legs for feature 1 ($F1$: the mean acceleration energy at the second wavelet level) and feature 2 ($F2$: the mean gyrometer energy at the second wavelet level) for different loads.

Wavelet | Load | Mean Energy [%] | Standard Deviation | ||||||
---|---|---|---|---|---|---|---|---|---|

Left Leg | Right Leg | Left Leg | Right Leg | ||||||

$\mathit{F}\mathbf{1}$ | $\mathit{F}\mathbf{2}$ | $\mathit{F}\mathbf{1}$ | $\mathit{F}\mathbf{2}$ | $\mathit{F}\mathbf{1}$ | $\mathit{F}\mathbf{2}$ | $\mathit{F}\mathbf{1}$ | $\mathit{F}\mathbf{2}$ | ||

db2 | 1a | 3.82 | 3.50 | 3.14 | 4.71 | 0.67 | 0.76 | 0.64 | 0.78 |

1b | 3.50 | 3.36 | 3.25 | 4.17 | 0.69 | 0.71 | 0.63 | 0.65 | |

2a | 3.50 | 3.32 | 2.78 | 3.93 | 0.59 | 0.56 | 0.52 | 0.52 | |

2b | 3.68 | 3.42 | 3.05 | 3.95 | 0.74 | 0.58 | 0.67 | 0.55 | |

3a | 3.34 | 3.50 | 3.07 | 3.58 | 0.63 | 0.48 | 0.41 | 0.49 | |

3b | 3.34 | 3.59 | 3.39 | 3.54 | 0.48 | 0.45 | 0.42 | 0.40 | |

Haar | 1a | 3.27 | 8.33 | 2.78 | 11.90 | 0.60 | 1.10 | 0.48 | 1.55 |

1b | 3.12 | 8.13 | 2.90 | 10.67 | 0.54 | 1.28 | 0.55 | 1.36 | |

2a | 2.90 | 8.45 | 2.51 | 10.69 | 0.44 | 1.12 | 0.30 | 1.42 | |

2b | 2.77 | 9.22 | 2.59 | 10.23 | 0.35 | 0.88 | 0.47 | 1.66 | |

3a | 2.69 | 9.23 | 2.46 | 9.85 | 0.40 | 0.89 | 0.29 | 1.57 | |

3b | 2.50 | 9.64 | 2.51 | 8.97 | 0.37 | 1.01 | 0.33 | 1.61 |

**Table 2.**Classification accuracies ($AC$) and cross-validation ($CV$) errors to evaluate experiments under different loads and using different methods: k-nearest neighbors (NN), Bayesian, support vector machine, and neural networks, using features evaluated by different wavelet functions.

Wavelet | Load | 3-NN | 5-NN | Bayes | SVM | NN | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | ||

db2 | 1a | 83.3 | 0.183 | 86.7 | 0.217 | 83.3 | 0.175 | 82.5 | 0.242 | 85.0 | 0.175 |

1b | 80.8 | 0.325 | 77.5 | 0.325 | 73.3 | 0.292 | 78.3 | 0.283 | 77.5 | 0.242 | |

2a | 85.8 | 0.258 | 84.2 | 0.225 | 82.5 | 0.183 | 85.8 | 0.183 | 86.7 | 0.175 | |

2b | 86.7 | 0.267 | 81.7 | 0.258 | 79.2 | 0.242 | 81.7 | 0.242 | 82.5 | 0.175 | |

3a | 76.7 | 0.475 | 72.5 | 0.433 | 65.0 | 0.383 | 70.0 | 0.392 | 72.5 | 0.308 | |

3b | 75.0 | 0.600 | 60.8 | 0.533 | 52.5 | 0.550 | 66.7 | 0.558 | 69.2 | 0.367 | |

Mean: | 81.4 | 0.351 | 77.2 | 0.332 | 72.6 | 0.304 | 77.5 | 0.317 | 78.9 | 0.240 | |

Haar | 1a | 99.2 | 0.042 | 97.5 | 0.042 | 95.8 | 0.042 | 97.5 | 0.050 | 99.2 | 0.025 |

1b | 90.8 | 0.150 | 88.3 | 0.167 | 82.5 | 0.192 | 88.3 | 0.167 | 96.7 | 0.058 | |

2a | 93.3 | 0.133 | 90.8 | 0.100 | 90.8 | 0.108 | 92.5 | 0.133 | 95.0 | 0.025 | |

2b | 82.5 | 0.367 | 75.8 | 0.300 | 76.7 | 0.242 | 75.0 | 0.283 | 80.8 | 0.200 | |

3a | 75.8 | 0.592 | 68.3 | 0.525 | 62.5 | 0.425 | 65.8 | 0.408 | 72.5 | 0.350 | |

3b | 73.0 | 0.495 | 72.5 | 0.470 | 67.5 | 0.330 | 70.5 | 0.333 | 71.5 | 0.320 | |

Mean: | 85.8 | 0.297 | 82.2 | 0.267 | 79.3 | 0.223 | 82.2 | 0.229 | 86.1 | 0.163 |

**Table 3.**$AC$ and $CV$ errors to evaluate experiments under different loads and using different methods: k-NN, Bayesian, support vector machine, and neural networks, using features evaluated by Haar wavelet functions and different number of features.

Number of Features | Load | 3-NN | 5-NN | Bayes | SVM | NN | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | $\mathit{AC}$ [%] | $\mathit{CV}$ | ||

$R=2$ | 1 | 95.5 | 0.096 | 92.9 | 0.105 | 89.2 | 0.117 | 92.9 | 0.109 | 97.9 | 0.042 |

2 | 87.9 | 0.25 | 83.3 | 0.200 | 83.8 | 0.175 | 83.9 | 0.208 | 87.9 | 0.112 | |

3 | 74.4 | 0.54 | 70.4 | 0.498 | 65.0 | 0.378 | 68.2 | 0.371 | 72.0 | 0.335 | |

Mean: | 85.8 | 0.297 | 82.2 | 0.267 | 79.3 | 0.223 | 82.2 | 0.229 | 86.1 | 0.163 | |

$R=6$ | 1 | 94.6 | 0.100 | 92.1 | 0.104 | 94.6 | 0.071 | 97.5 | 0.117 | 98.4 | 0.002 |

2 | 90.1 | 0.188 | 85.4 | 0.221 | 88.8 | 0.137 | 90.8 | 0.283 | 92.9 | 0.088 | |

3 | 80.0 | 0.404 | 76.3 | 0.383 | 71.7 | 0.304 | 84.2 | 0.317 | 81.7 | 0.183 | |

Mean: | 88.2 | 0.231 | 84.6 | 0.236 | 85.0 | 0.171 | 90.8 | 0.239 | 91.0 | 0.091 |

© 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**

Procházka, A.; Vyšata, O.; Charvátová, H.; Vališ, M.
Motion Symmetry Evaluation Using Accelerometers and Energy Distribution. *Symmetry* **2019**, *11*, 871.
https://doi.org/10.3390/sym11070871

**AMA Style**

Procházka A, Vyšata O, Charvátová H, Vališ M.
Motion Symmetry Evaluation Using Accelerometers and Energy Distribution. *Symmetry*. 2019; 11(7):871.
https://doi.org/10.3390/sym11070871

**Chicago/Turabian Style**

Procházka, Aleš, Oldřich Vyšata, Hana Charvátová, and Martin Vališ.
2019. "Motion Symmetry Evaluation Using Accelerometers and Energy Distribution" *Symmetry* 11, no. 7: 871.
https://doi.org/10.3390/sym11070871