# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 1 Introduction
[21] measured the continuous walking force generated on a treadmill by three test subjects (TSs) walking at a self-selected pacing rate with different speeds. They observed that the coefficient of variation of the pacing frequency on a step-by-step basis is about 3%. [PERSON] et al. [22] found that the mean coefficient of variation of the step length ranges from 2% and 5% and depends on the walking speed and gender. A detailed description of the intra-subject variability has been proposed by [PERSON] and [PERSON] [5], who statistically characterized the walking gait parameters of a population of ten TSs using a motion capture system that tracks human body movements. The statistical description of the walking gait parameters can be used as a direct input for stochastic modeling. Based on the probabilistic approach, the vibration response can be expressed as a probability that a certain level of vibration will not be exceeded. Over the last decade, stochastic models for vertical human-induced excitations have been developed both in time [11; 16; 23] and frequency domain [24; 25; 26; 27]. Among these, [PERSON] et al. [11] proposed a probability-based approach to predict the vibration response caused by a single pedestrian crossing the footbridge. The pedestrian force is described in the time domain, and the inter-subject variability is considered using the probability distributions of walking frequency, step length and force amplitude, while the intra-subject variability is included through the phase angle. The stochastic model proposed by [PERSON] and [PERSON] [23] was developed starting from the analysis of 824 walking time histories collected from about 80 TSs and is able to replicate temporal and spectral features of real walking loads. Similarly, [PERSON] et al. [16] presented a model of variable individual forces based on the vertical ground reaction forces measured in an experimental campaign involving 50 TSs walking on a treadmill. In the frequency domain, the random nature of human walking is accounted for by presenting the walking forces as auto-spectral densities [23; 28; 29]. The theory of turbulent wind on linear structures can be adopted to describe the correlation among pedestrians at different locations on the structure [24]. This paper proposes a framework for the statistical modeling of the vertical walking forces induced by single pedestrians on a rigid surface based on a step-by-step variation of gait parameters. Basically, the pedestrian walk is modeled as a series of consecutive steps, each one characterized by the typical M shape [1], where parameters of the given step are independent of those of the previous steps. The proposed mathematical model for the pedestrian gait takes inspiration from the well known application of Markov chains called \"random walk\" [30]. Gait parameters of each step are randomly extracted from their probability distributions. Statistical characteristics of the walking gait parameters are estimated from the experimental tests conducted to measure individual walking forces through the force plate system developed by [PERSON] et al. [31]. Experimental tests involved three TSs and allowed measuring about 700 steps for each TS. The analysis of the experimental measurements demonstrated that walking features are related with each other and characterized by multivariate distributions. In this paper, multivariate distributions of the walking features are described through Gaussian Mixture (GM) models [32]. GM is an interesting approach to the problem of fitting multivariate distributions, since it depends on a limited number of parameters, can accurately approximate several practical distributions and can rely on simple algorithms for parameter estimation. The main benefit of the proposed approach lies in the in-depth investigation of the correlation among the walking features and its accurate characterization through multivariate distributions.