# Gradient-Based Multi-Objective Feature Selection for Gait Mode Recognition of Transfemoral Amputees

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Collection and Experimental Protocol

#### 2.2. Data Windowing

#### 2.3. Feature Extraction

#### 2.4. Feature Selection

#### 2.5. Classification

#### 2.6. Filter

## 3. Feature Selection Algorithm Development

#### 3.1. Biogeography-Based Multi-Objective Optimization

#### 3.2. Gradient-Based Multi-Objective Feature Selection

Algorithm 1: The outline of gradient-based multi-objective feature selection (GMOFS), where ${x}_{i}$ is the i-th feature in the training set X, and Y is the corresponding set of output classes. |

Initialization:$\lambda ={\lambda}_{l}\le {\lambda}_{u}$, Population = ∅, $k=1$While$\lambda \le {\lambda}_{u}$Step 1:Use the training data $\{X,Y\}$ to train the constrained MLP network in Equation (4) by solving Equation (5) Step 2:Sort the input weights $\left\{{\beta}_{i}\right\}$ in descending order Use Equation (7) to select subset ${S}_{k}\subset X$ where size(${S}_{k}$) ≤ size(X) Step 3:Population ← Population $+\phantom{\rule{3.33333pt}{0ex}}{S}_{k}$ $k\leftarrow k+1$ Next $\lambda \leftarrow \lambda +\u25b3\lambda $Step 4:For each subset ${S}_{k}$ in PopulationUse cross-validation to train and test a classifier with dataset $\{{S}_{k},Y\}$ Calculate objective functions ${f}_{1}^{k}$ and ${f}_{2}^{k}$ using Equation (3) Next subset ${S}_{k}$Step 5:Find the Pareto set using Equation (8) |

#### 3.3. Evaluation of Multi-Objective Optimization Pareto Fronts

## 4. Results and Discussion

#### 4.1. Effect of Frame Length on Classification Performance

#### 4.2. Multi-Objective Feature Selection

#### 4.3. Comparison Results of Classification Algorithms

#### 4.4. Performance Assessment of Selected Subset

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

List of acronyms in order of appearance | |||

Acronym | Definition | Acronym | Definition |

UIR | User intent recognition | ZC | Zero crossing |

MOO | Multi-objective optimization | WL | Waveform length |

GMOFS | Gradient-based multi-objective feature selection | VAR | Variance |

MOBBO | Multi-objective biogeography-based optimization | MAV | Mean absolute value |

SVM | Support vector machine | RMS | Root mean square |

RBF | Radial basis function | WAMP | Willison amplitude |

MVF | Majority voting filter | SK | Skewness |

sEMG | Surface electromyography | KU | Kurtosis |

LDA | Linear discriminant analysis | COR | Correlation |

QDA | Quadratic discriminant analysis | ANG | Angle |

GMM | Gaussian mixture model | PSD | Periodogram spectrum density |

ANN | Artificial neural network | MNF | Mean frequency |

BBO | Biogeography-based optimization | MDF | Median frequency |

VEBBO | Vector evaluated BBO | MAXF | Maximum frequency |

NSBBO | Non-dominated sorting BBO | AR | Auto-regressive model |

NPBBO | Niched Pareto BBO | CV | Cross validation |

SPBBO | Strength Pareto BBO | AB01 | Able-bodied subject 01 |

EA | Evolutionary algorithm | AM01 | Amputee subject 01 |

MLP | Multilayer perceptron | PS | Preferred speed |

TD | Time domain | ST | Standing |

FD | Frequency domain | NW | Normal walking |

FLDA | Fisher’s linear discriminant analysis | SW | Slow walking |

PCA | Principal component analysis | FW | Fast walking |

DT | Decision tree | ||

SSC | Slope sign change |

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**Figure 1.**Architecture of user intent recognition system. The double-lined box indicates that an evolutionary algorithm is used for optimization.

**Figure 2.**Experimental setup for data collection. The

**left**figure shows an able-bodied subject and the

**right**figure shows an amputee subject with an Ottobock prosthesis on the right leg.

**Figure 3.**Sample walking trial with four different gait modes for able-bodied subject AB01. Although data is available for both legs, we require only one side for gait mode recognition. The data from the two legs look similar because of gait symmetry.

**Figure 4.**Data windowing. ${S}_{i}$ represents the i-th data segment, ${L}_{f}$ is the frame length, $\tau $ is the required processing time, I is the increment length for overlapped windowing, and ${O}_{i}$ is the detected gait mode corresponding to frame ${S}_{i}$.

**Figure 5.**Mean LDA performance for the able-bodied subjects for different data windowing methods and frame lengths. On the horizontal axis, a single value indicates the frame length of disjoint windowing, and a pair of values indicates the frame length and increment length of overlapped windowing. For instance, 200–50 indicates ${L}_{f}=200$ ms and $I=50$ ms for overlapped windowing.

**Figure 6.**Two-dimensional scatter plot for visualization using principal component analysis (PCA) (

**left column**) and Fisher linear discriminant analysis (FLDA) (

**right column**) for able-bodied subject AB01.

**Figure 7.**Mean classification accuracy of three able-bodied subjects, and processing ratio of 17 feature types trained by LDA using 10-fold cross validation.

**Figure 8.**(

**a**) Pareto fronts obtained from MOO methods with an SVM classifier with linear kernels using AB01 training data; (

**b**) combined Pareto front obtained from non-dominated Pareto points in (

**a**).

**Figure 9.**Selection frequency of 44 features by VEBBO and GMOFS. The plots show how many times each feature appears in the Pareto points of the given method. For instance, feature 6 is present in all 10 GMOFS Pareto points.

**Figure 10.**Classification performance of QDA, SVM-Linear, and SVM-RBF with feature subset ${p}_{9}$ for able-bodied subjects (AB01, AB02, and AB03) and amputee subjects (AM01, AM02, and AM03).

**Table 1.**Physical characteristics of the six human test subjects. AB and AM represent able-bodied and amputee subject, respectively.

Gender | Age | Weight | Height | Walking Speed (m/s) | |||
---|---|---|---|---|---|---|---|

(years) | (kg) | (cm) | SW | PS | FW | ||

AB01 | Male | 37 | 79.5 | 188 | 0.98 | 1.30 | 1.63 |

AB02 | Male | 20 | 73.9 | 172 | 0.86 | 1.15 | 1.44 |

AB03 | Male | 28 | 80.9 | 179 | 0.75 | 1.00 | 1.25 |

AM01 | Male | 32 | 79.1 | 174 | 0.60 | 1.00 | - |

AM02 | Male | 64 | 99.2 | 177 | 0.56 | 0.94 | - |

AM03 | Male | 35 | 81.7 | 176 | 0.60 | 0.90 | - |

**Table 2.**Comparison of mean classification performance for different frame lengths (row values versus column values) using Wilcoxon signed-rank tests at a 10% significance level. Mean classification performance is considered as the average of all linear discriminant analysis (LDA) classifiers trained individually with every single time-domain (TD) and frequency-domain (FD) feature type. ≈ indicates that the two compared frame lengths tie (T) with similar performance and are not statistically significantly different. + indicates that the two frame lengths are statistically significantly different, and B or W indicates that the row frame length performs better or worse than the column frame length, respectively. * indicates that the lower triangular half of the table is equal to its upper triangular half.

Frame Length (ms) | 150 | 200 | 200–50 | 200–150 | 250 | 250–50 | 300–200 | |
---|---|---|---|---|---|---|---|---|

100 | vs. | W (+) | W (+) | W (+) | W (+) | W (+) | W (+) | W (+) |

150 | vs. | − | W (+) | W (+) | W (+) | W (+) | W (+) | W (+) |

200 | vs. | * | − | W (+) | T (≈) | W (+) | W (+) | W (+) |

200–50 | vs. | * | * | − | T (≈) | T (≈) | W (+) | W (+) |

200–150 | vs. | * | * | * | − | W (+) | W (+) | W (+) |

250 | vs. | * | * | * | * | − | W (+) | W (+) |

250–50 | vs. | * | * | * | * | * | − | T (≈) |

Symbol | Value | |
---|---|---|

MOBBO | ||

Mutation rate | $\mu $ | 0.04 |

Number of elites | E | 2 |

Population size | N | 100 |

Number of generations | $Gen$ | 1000 |

Problem dimension | d | 44 |

Migration model | ${m}_{\mathrm{flag}}$ | sinusoidal |

GMOFS | ||

Number of hidden nodes | p | 5 |

Elastic net parameter | $\alpha $ | 0 |

Bound for shrinkage parameter | [${\lambda}_{l},{\lambda}_{u}$] | [0, 150] |

Bound for neuron weights | $a,b$ | 5 |

Increment of shrinkage parameter | $\u25b3\lambda $ | 1 if $0\le \lambda \le 30$; and 10 if $30<\lambda \le 150$ |

Trust region reflective | ||

Maximum allowable iterations | MaxIter | 100 |

Termination tolerance on the independent variable | TolX | 0.001 |

Termination tolerance on the cost function | TolFun | 0.001 |

Typical values for the independent variable | TypicalX | 0.1 |

Finite difference method | FinDiffType | central |

**Table 4.**Comparison of Pareto fronts using relative coverage (RC). Only 7.2% and 30% of the VEBBO and GMOFS points, respectively, are dominated by other Pareto points; so VEBBO and GMOFS rank first and second, respectively, in terms of RC.

VEBBO | SPBBO | NSBBO | NPBBO | GMOFS | |
---|---|---|---|---|---|

VEBBO | − | 62.5 | 75.0 | 85.7 | 40.0 |

SPBBO | 0.0 | − | 25.0 | 71.4 | 40.0 |

NSBBO | 14.3 | 50.0 | − | 100.0 | 40.0 |

NPBBO | 0.0 | 0.0 | 0.0 | − | 0.0 |

GMOFS | 14.3 | 50.0 | 50.0 | 100.0 | − |

Mean RC $(\%)$ | 7.2 | 40.4 | 37.5 | 89.3 | 30.0 |

**Table 5.**Comparison of Pareto fronts using normalized hypervolume. ${N}_{p}$ is the number of Pareto points obtained by each MOO method. VEBBO and GMOFS rank first and second, respectively, in terms of normalized hypervolume, and GMOFS ranks first in terms of the number of points.

${\mathit{N}}_{\mathit{p}}$ | Normalized Hypervolume | |
---|---|---|

VEBBO | 7 | 0.5026 |

SPBBO | 8 | 0.5814 |

NSBBO | 8 | 0.5676 |

NPBBO | 7 | 0.8013 |

GMOFS | 10 | 0.5332 |

**Table 6.**Mean classification accuracy (ACC) and standard deviation (STD) for AB01 of classifiers trained with 13 different feature subsets. NF is the number of features in each set.

Pareto Point | NF | LDA | QDA | SVM-Linear | SVM-RBF | MLP | DT | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACC | STD | ACC | STD | ACC | STD | ACC | STD | ACC | STD | ACC | STD | ||

${p}_{1}$ | 6 | 93.56 | 0.740 | 94.33 | 0.852 | 95.37 | 1.218 | 98.33 | 0.421 | 97.34 | 0.698 | 96.15 | 1.16 |

${p}_{2}$ | 7 | 95.31 | 0.829 | 96.06 | 0.711 | 96.99 | 0.775 | 98.88 | 0.216 | 98.29 | 0.539 | 96.35 | 1.00 |

${p}_{3}$ | 8 | 96.69 | 0.835 | 96.82 | 0.410 | 97.47 | 0.694 | 98.86 | 0.378 | 98.20 | 0.411 | 96.67 | 1.18 |

${p}_{4}$ | 9 | 96.86 | 0.684 | 96.95 | 0.484 | 98.07 | 0.576 | 99.31 | 0.234 | 98.34 | 0.459 | 96.73 | 1.25 |

${p}_{5}$ | 10 | 97.04 | 0.657 | 96.99 | 0.427 | 98.08 | 0.430 | 98.90 | 0.406 | 98.47 | 0.645 | 96.56 | 1.22 |

${p}_{6}$ | 11 | 96.84 | 0.536 | 97.15 | 0.654 | 98.14 | 0.692 | 98.94 | 0.293 | 98.62 | 0.578 | 96.32 | 1.28 |

${p}_{7}$ | 12 | 96.93 | 0.656 | 97.36 | 0.372 | 98.20 | 0.497 | 99.05 | 0.356 | 98.87 | 0.406 | 97.21 | 1.00 |

${p}_{8}$ | 13 | 96.61 | 0.384 | 97.62 | 0.554 | 98.25 | 0.534 | 99.14 | 0.305 | 95.76 | 9.180 | 96.86 | 1.36 |

${p}_{9}$ | 14 | 96.76 | 0.485 | 97.79 | 0.311 | 98.49 | 0.500 | 98.88 | 0.290 | 98.90 | 0.355 | 97.15 | 0.78 |

${p}_{10}$ | 16 | 96.95 | 0.525 | 97.84 | 0.578 | 98.59 | 0.467 | 98.38 | 0.449 | 98.66 | 0.371 | 96.99 | 0.72 |

${p}_{11}$ | 21 | 97.13 | 0.501 | 97.93 | 0.716 | 98.62 | 0.564 | 98.40 | 0.392 | 99.00 | 0.432 | 97.30 | 0.54 |

${p}_{12}$ | 27 | 97.41 | 0.568 | 97.77 | 0.861 | 98.70 | 0.422 | 97.58 | 0.663 | 99.07 | 0.373 | 96.91 | 0.64 |

**Table 7.**Comparison of classification performance using Wilcoxon signed-rank tests (W.T.) at a 5% significance level. B or W indicates that the row method performs better or worse than the column method, respectively, while T shows that they tie with similar performance. * indicates that the lower triangular half of the table is equal to its upper triangular half. These results are obtained using all the data from Table 6.

DT | SVM-RBF | SVM-linear | QDA | LDA | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

p-Value | W.T. | p-Value | W.T. | p-Value | W.T. | p-Value | W.T. | p-Value | W.T. | ||

MLP | vs. | 2.44 × 10${}^{-4}$ | B | 7.32 × 10${}^{-1}$ | T | 8.50 × 10${}^{-3}$ | B | 5.02 × 10${}^{-3}$ | B | 2.44 × 10${}^{-4}$ | B |

DT | vs. | − | 1.23 × 10${}^{-4}$ | W | 8.20 × 10${}^{-3}$ | W | 1.33 × 10${}^{-1}$ | T | 1.70 × 10${}^{-1}$ | T | |

SVM-RBF | vs. | * | − | 6.70 × 10${}^{-3}$ | B | 2.44 × 10${}^{-4}$ | B | 1.22 × 10${}^{-4}$ | B | ||

SVM-linear | vs. | * | * | − | 1.15 × 10${}^{-4}$ | B | 1.25 × 10${}^{-4}$ | B | |||

QDA | vs. | * | * | * | − | 2.44 × 10${}^{-4}$ | B |

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## Share and Cite

**MDPI and ACS Style**

Khademi, G.; Mohammadi, H.; Simon, D. Gradient-Based Multi-Objective Feature Selection for Gait Mode Recognition of Transfemoral Amputees. *Sensors* **2019**, *19*, 253.
https://doi.org/10.3390/s19020253

**AMA Style**

Khademi G, Mohammadi H, Simon D. Gradient-Based Multi-Objective Feature Selection for Gait Mode Recognition of Transfemoral Amputees. *Sensors*. 2019; 19(2):253.
https://doi.org/10.3390/s19020253

**Chicago/Turabian Style**

Khademi, Gholamreza, Hanieh Mohammadi, and Dan Simon. 2019. "Gradient-Based Multi-Objective Feature Selection for Gait Mode Recognition of Transfemoral Amputees" *Sensors* 19, no. 2: 253.
https://doi.org/10.3390/s19020253