# Statistical Analysis of Gait Maturation in Children Using Nonparametric Probability Density Function Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Data Description

## 3. Gait Analysis

#### 3.1. Signal Preprocessing

**Figure 1.**Original time series of stride interval of the children (

**a**) aged 45 months (in the group aged 3–5 years); (

**b**) aged 80 months (in the group aged 6–8 years); and (

**c**) aged 129 months (in the group aged 10–14 years), respectively. Outliers detected, along with one stride before or after the outliers, are marked with asterisks; Subfigures (

**d**)–(

**f**) plot the corresponding outlier-free time series of stride interval. The first strides in (

**a**)–(

**f**) start after the start-up 60 s (1 min).

**Figure 2.**Histograms and Parzen-window probability density functions (PDFs) estimated for the stride interval time series of the children (

**a**) aged 45 months (in the group aged 3–5 years); (

**b**) aged 80 months (in the group aged 6–8 years); and (

**c**) aged 129 months (in the group aged 10–14 years), respectively.

#### 3.2. Probability Density Function (PDF) Estimation

#### 3.3. Statistical Parameters

**Figure 3.**Bar graphics of (

**a**) averaged stride interval (ASI) and (

**b**) variation stride interval (VSI) computed from the Parzen-window probability density functions (PDFs) of the children in the groups aged 3–5 years, 6–8 years, and 10–14 years, respectively. Vertical lines on the tops of the bars denote the standard deviation (SD) values. The values (mean ± SD) of the bars for the children of 3–5 years old (ASI: $0.904\pm 0.041$ s, VSI: $0.058\pm 0.015$ s), of 6–8 years old (ASI: $0.96\pm 0.056$ s, VSI: $0.035\pm 0.009$ s), and of 10–14 years old (ASI: $1.059\pm 0.063$ s, VSI: $0.027\pm 0.006$ s).

**Figure 4.**Illustration of the probability density functions (PDFs) with three types of skewness (SK): right-skewed PDF, dash-dot curve, SK $>0$; symmetric PDF, solid curve, SK $=0$ (in particular Gaussian distributions); left-skewed PDF, dashed curve, SK $<0$. Dot line represents a central axis up to the mean of the symmetric PDF. au: arbitrary units.

**Figure 5.**Illustration of four typical probability density functions (PDFs) that possess different kurtosis (KU) values: uniform distribution, dot curve, KU $=1.8$; raised cosine distribution, dashed curve, KU $=2.41$; Gaussian distribution, solid curve, KU $=3$; hyperbolic secant distribution, dash-dot curve, KU $=5$. au: arbitrary units.

**Figure 6.**Bar graphics of the mean values of (

**a**) skewness (SK) and (

**b**) kurtosis (KU) computed from the histograms and the Parzen-window probability density functions (PDFs) of the 3- to 5-year-old, 6- to 8-year-old, and 10- to 14-year-old age groups, respectively. Statistics of the skewness and the kurtosis for the three age groups are listed in Table 1.

**Table 1.**Statistical parameters computed from the histograms and the Parzen-window probability density functions (PDFs) of 50 children subjects (equal number of boys and girls). Values are mean ± standard deviation (SD).

Statistical parameters | 3–5 years old | 6–8 years old | 10–14 years old | |
---|---|---|---|---|

Skewness (SK) | Histogram | $0.32\pm 0.26$ | $0.16\pm 0.28$ | $0.1\pm 0.19$ |

Parzen-window PDF | $0.31\pm 0.25$ | $0.14\pm 0.26$ | $0.08\pm 0.16$ | |

Kurtosis (KU) | Histogram | $3.37\pm 0.72$ | $3.15\pm 0.85$ | $2.94\pm 0.26$ |

Parzen-window PDF | $3.33\pm 0.69$ | $3.03\pm 0.72$ | $2.79\pm 0.16$ |

## 4. Discussion

## 5. Conclusion

## Acknowledgements

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**MDPI and ACS Style**

Xiang, N.; Cai, S.; Yang, S.; Zhong, Z.; Zheng, F.; He, J.; Wu, Y.
Statistical Analysis of Gait Maturation in Children Using Nonparametric Probability Density Function Modeling. *Entropy* **2013**, *15*, 753-766.
https://doi.org/10.3390/e15030753

**AMA Style**

Xiang N, Cai S, Yang S, Zhong Z, Zheng F, He J, Wu Y.
Statistical Analysis of Gait Maturation in Children Using Nonparametric Probability Density Function Modeling. *Entropy*. 2013; 15(3):753-766.
https://doi.org/10.3390/e15030753

**Chicago/Turabian Style**

Xiang, Ning, Suxian Cai, Shanshan Yang, Zhangting Zhong, Fang Zheng, Jia He, and Yunfeng Wu.
2013. "Statistical Analysis of Gait Maturation in Children Using Nonparametric Probability Density Function Modeling" *Entropy* 15, no. 3: 753-766.
https://doi.org/10.3390/e15030753