# On Nonlinear Regression for Trends in Split-Belt Treadmill Training

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Dataset

#### 2.2. Exponential Models

#### 2.2.1. Single Exponential Model

#### Common Language Interpretation

#### 2.2.2. Double Exponential Model

#### Common Language Interpretation

#### 2.3. Model Fitting

#### 2.3.1. Parameter Bounds

#### Bounds for c

#### Bounds for b, ${b}_{s}$, ${b}_{f}$

#### Bounds for a, ${a}_{s}$, ${a}_{f}$

^{−3}] and setting c to 0. Additional constraints, such as ${b}_{f}$ < ${b}_{s}$ and ${a}_{s}+{a}_{f}$ ≤ 2, are also applied. Figure 4 shows the results.

#### 2.3.2. Optimisation Problems

^{−3}with the assumption that it is an adequately small number and it does not reduce the generality of the model in any considerable way. We added these constraints as penalty functions [63].;

#### 2.3.3. Optimisation Algorithm

#### 2.4. Model Evaluation and Selection

#### 2.5. Confidence Intervals for Parameters

#### 2.5.1. Based on Linearisation

#### 2.5.2. Without Linearisation

#### 2.6. MATLAB Implementation of Methods

## 3. Results

#### 3.1. Group-Averaged Symmetry Series

#### 3.1.1. Model Selection

#### 3.1.2. Estimated Parameters and Confidence Intervals

#### 3.2. Participant Symmetry Series

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AIC | Akaike’s information criterion |

PSO | Particle swarm optimisation |

SS | Same speed configuration |

DS | Differential speed configuration |

ctDCS | Cerebellar transcranial direct current stimulation |

AUTEC | Auckland University of Technology Ethics Committee |

CIs | Confidence intervals |

UD | Undefined |

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**Figure 1.**Group-averaged (n = 15) step length symmetry series with 95% confidence intervals (z = 1.96) for baseline, adaptation and de-adaptation phases from two consecutive days of training. A 3-point moving average filter was used to smooth the data for aesthetic considerations.

**Figure 2.**Single exponential function with different parameter values. The first two plots in the first column show growing exponential functions. The remaining plots show decaying exponential functions. c determines the final value of the function. a determines the change in the exponential function from the beginning to the end. b determines the relative rate of change.

**Figure 3.**Double exponential functions with an overshoot that is produced by a positive and a negative exponential trend.

**Figure 4.**Numerical solutions for overshoot value (${y}_{crit}$) for the double exponential model with respect to parameters ${a}_{s}$, ${a}_{f}$.

**Figure 6.**Fitted values from the double exponential models for participant (black) and group-averaged (red) symmetry series.

**Table 1.**Summary of parameter bounds for the single and double exponential models. Direction is defined as positive (+ve) or negative (−ve) depending on the direction from which the symmetry series approaches its final value.

Model (Parameters) | Direction | Overshoot | Bounds |
---|---|---|---|

Single (a; b; c) | +ve | – | [0, 2]; [−ln(2), 0]; [−1, 1] |

−ve | – | [−2, 0]; [$-ln$(2), 0]; [−1, 1] | |

Double (${a}_{s}$; ${b}_{s}$; ${a}_{f}$; ${b}_{f}$; c) | +ve | No | [0, 1]; [−ln(2), 0]; [0, 1]; [−ln(2), 0]; [−1, 1] |

Yes | [−1, 0]; [−ln(2), 0]; [0, 1]; [−ln(2), 0]; [−1, 1] | ||

−ve | No | [−1, 0]; [−ln(2), 0]; [−1, 0]; [−ln(2), 0]; [−1, 1] | |

Yes | [0, 1]; [−ln(2), 0]; [−1, 0]; [−ln(2), 0]; [−1, 1] |

Phase | Session | AIC (Single Exp. Model) | AIC (Double Exp. Model) |
---|---|---|---|

Adaptation | I | −2017.63 | −2327.59 |

II | −2472.15 | −2496.23 | |

De-adaptation | I | −1326.61 | −1371.21 |

II | −1409.85 | −1461.28 |

**Table 3.**Common language parameters along with their 95% confidence intervals (CIs) rounded to three decimal places from the double exponential models. Note: UD stands for undefined, std. stands for standard deviation and * denotes statistical significance.

Phase | Parameter | Session | CI Overlap | |
---|---|---|---|---|

Session I | Session II | |||

Adapt | Asymmetry at beginning | −0.114 [−0.126, −0.106] | −0.033 [−1.018, 0.888] | Yes |

Total change | −0.137 [−0.148, −0.133] | −0.063 [−1.048, 0.857] | Yes | |

Strides to 50% changes | 235 [194, 299]; 21 [18, 24] | 30 [25, 52]; 28 [19, 32] | No *, Yes | |

Asymmetry at end | 0.024 [0.021, 0.027] | 0.030 [0.030, 0.031] | No * | |

Overshoot | UD [UD, UD] | 0.034 [0.030, UD] | Yes | |

Residuals std. | 6.386 × 10^{−3} | 5.600 × 10^{−3} | – | |

De-adapt | Asymmetry at beginning | 0.141 [−0.863, 0.227] | 0.125 [0.090, 0.154] | Yes |

Total change | 0.117 [0.080, 0.200] | 0.098 [0.070, 0.125] | Yes | |

Strides to 50% changes | 100 [76, 13203]; 15 [8, 34] | 58 [38, 252]; 4 [2, 11] | Yes, Yes | |

Asymmetry at end | 0.024 [−0.943, 0.027] | 0.027 [0.019, 0.029] | Yes | |

Overshoot | UD [UD, UD] | UD [UD, UD] | Yes | |

Residuals std. | 7.043 × 10^{−3} | 6.161 × 10^{−3} | – |

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**MDPI and ACS Style**

Rashid, U.; Kumari, N.; Signal, N.; Taylor, D.; Vandal, A.C.
On Nonlinear Regression for Trends in Split-Belt Treadmill Training. *Brain Sci.* **2020**, *10*, 737.
https://doi.org/10.3390/brainsci10100737

**AMA Style**

Rashid U, Kumari N, Signal N, Taylor D, Vandal AC.
On Nonlinear Regression for Trends in Split-Belt Treadmill Training. *Brain Sciences*. 2020; 10(10):737.
https://doi.org/10.3390/brainsci10100737

**Chicago/Turabian Style**

Rashid, Usman, Nitika Kumari, Nada Signal, Denise Taylor, and Alain C. Vandal.
2020. "On Nonlinear Regression for Trends in Split-Belt Treadmill Training" *Brain Sciences* 10, no. 10: 737.
https://doi.org/10.3390/brainsci10100737