Modelling the Extremely Low Frequencies Magnetic Fields Times Series Exposure by Segmentation

Introduction ELFSTAT project, founded by the French ANSES (2015-2019, Grant agreement n. 2015/1/202), aims at characterizing children's exposure to Extremely Low Frequency Magnetic Fields (ELF-MF) in real exposure scenarios using stochastic approaches. The present paper gives details about the first step of the project: this step aims at developing stochastic models to model personal exposure from a dataset of recorded ELF-MF signals. These recordings are coming from the ARIMMORA project [1]: 331 children has worn an EMDEX and their exposition were measured during about 3 days. The stochastic models will then be used to construct realistic simulations of ELF-MF time series. Figure 1 gives an example of ELF-MF 24 hours-signal. The majority of ELF-MF time series are characterized by abrupt changes in structure, such as sudden jumps in mean level or in dispersion around a mean level. The developed model consists in detecting these changes and in modeling the signal between two consecutive changes by a stationary process. These stationary processes are described by parametric models. We chose Auto-Regressive model because of the possibility of characterizing mean effects, variance effects and time-correlation effects with a weak number of parameters. The full-model is obtained by modeling the distribution of the parameters from the whole dataset of 331 recordings. Figure 2 gives a schematic of the full approach. Data sources The 331 individuals of the database are distributed according the geographical location (Switzerland or Italy) and the time season (Winter or Summer), as shown in the following Table: For each individual, we consider a full day of measurements (0h-24h). As the interval between two successive observations is 30 seconds, 2880 values per recording are considered. Change-point detection and segment modeling We consider changepoints (or equivalently breakpoints) to be the time points that divide a data set into distinct homogeneous segments (or equivalently stationary segments); the boundaries of segments can be interpreted as changes in the physical system. The problem of changepoints estimation has attracted significant attention and is required in a number of applications including financial data [2], climate data [3], biomedical data [4] and signal processing [5]. Different authors propose various approaches to the problem of changepoints detection or segmentation of time series. This issue is thoroughly surveyed in [6], where different methods are proposed and an exhaustive list of references is given. Some authors study the estimation of a single changepoint problem [7], while others extend it to multiple changepoints problem. In this last case, the number of changepoints is unknown and it is a challenge to jointly estimate the number of changepoints, their location, and also provide an estimation of the model representing each segment. In this work we consider the [2] procedure because of its ability to jointly estimate the number of changepoints, their location and the AR model parameter of each segment. To resume, the segmentation aims to: (1) find the periods of stability and homogeneity in the behaviour of the time series; (2) identify the locations of change, called changepoints; (3) represent the regularities and features of each segment (estimate the model of each segment by a parameter set like changepoints location, segment amplitude, segment duration, segment regularity). Figure 1 gives the segmentation results for the given example. Statistical characterization of model parameters In order to produce a completely model of the time series, the marginal and joint probability distributions of all model parameters are required. The set of parameters related to piecewise autoregressive model are, at time t: duration ?t and coefficient of the AR model (amplitude or mean ?t, noise variance ?t2 and the autoregressive coefficients ?t). These 4 parameters can be modeled separately (whether by a parametric probability distribution or not) as well as jointly (rather in a nonparametric way because of the difficulty to find a multidimensional probability distribution that efficiently models the parameters set). Figure 3 shows that parameters are correlated and thus cannot be modeled separately. We used the Multivariate Kernel Density Estimation [9] that is a nonparametric and classical method. The advantage is that once we have an estimate of this density, we can simulate realistic realizations of these parameters. The figure 3 also gives the Multivariate KDE contours « Congrès National de Radioprotection » SFRP - LILLE - 7, 8 & 9 juin 2017 and 5000 simulations of these parameters via the Multivariate KDE. Concerning the fitted distributions, first result shows no significant difference between the summer database and the winter database. Finally, in the figure 4, we present an example simulated of ELF-MF exposure time series based in piecewise AR model.