New approach based on ANN and RBF for analyzing the spatial distribution of electromagnetic field from an exposure standpoint

Abstract

This paper shows the versatility of artificial neural networks (ANN) and radial basis functions (RBF) applied to electromagnetic analyses with Medium Wave samples. A new approach is given from an exposure point of view, analyzing field levels in wide areas. In particular supervised ANN are used to distinguish electromagnetic field areas, if levels are above or below a certain threshold. The novel application Electric-Frontier ANN (EF-ANN) allows us to compare the performances of different neural network topologies (linear and recurrent), and it also gives us the possibility to create maps via geographic information systems (GISs). Furthermore another designed application, Electric-Interpolation RBF (EI-RBF), is used to obtain interpolated maps of electric field levels in the environment under study, with excellent performances, and having the possibility of overlapping the results with the local cartography in GIS. The interest of these developed tools is to identify the areas where the levels are higher from an exposure standpoint, and to know how the distribution of levels is in a real environment, but using exclusively experimental samples which are more reliable than simulations.

Appendices

Appendix 1: Parameters to measure the ANN suitability

When presenting the results, we can use different parameters to measure the suitability of the network. On the one hand, for the training process, we use the following ones:

• MSE: Mean squared normalized error function given by Eq. (6), where r refers to the number of neurons in the output layer, N computes the number of patterns which are being used in the network, d is the desired output for each pattern in that layer, and y is the output of the network obtained for each one. It measures the performances of the network according to the mean of squared errors.

$$e_{mse}^{TRAIN} = \frac{1}{N}\mathop \sum \limits_{k = 1}^{N} \mathop \sum \limits_{p = 1}^{r} \left( {d_{k} (p) - y_{k} (p)} \right)^{2}$$

(6)

On the other hand, if we are interesting in calculating the performance of the network, by comparing how the correspondence between original values and those predicted or simulated by the network for those same locations is, the next parameters are used [49]:

• 1. MSE: with similar expression as indicated in Eq. (6), but in this case handling the targets, t, and the outputs of the network, y, as Eq. (7) shows.

$$e_{mse}^{SIM} = \frac{1}{N}\mathop \sum \limits_{k = 1}^{N} \left( {t_{k} - y_{k} } \right)^{2}$$

(7)

• 2. MAE: mean absolute error function. The error is calculated by subtracting the target (t) from the output of the system (y); see Eq. (8).

$$e_{mae}^{SIM} = \frac{1}{N}\mathop \sum \limits_{k = 1}^{N} \left| {t_{k} - y_{k} } \right|$$

(8)

Additionally, in some cases, we calculate the standard deviation and the mean value.

Appendix 2: Parameters to measure the RBF suitability

If we are interesting in calculating the performances of the system by comparing the correspondence between original and predicted values, besides the mean value (μ) and standard deviation (σ), the following measures will be used [49]:

• 1. e _mse : mean squared error as Eq. (9) shows.

$$e_{mse} = \frac{1}{N}\mathop \sum \limits_{k = 1}^{N} \left( {t_{k} - y_{k} } \right)^{2}$$

(9)

• 2. e _mae : mean absolute error as Eq. (10) shows.

$$e_{mae} = \frac{1}{N}\mathop \sum \limits_{k = 1}^{N} \left| {t_{k} - y_{k} } \right|$$

(10)

The error is calculated by subtracting the target (t), or experimental value, from the system’s output (y), or predicted value.