## 4 Discussion  
In the present paper, an innovative approach that combined universal kriging with polynomial chaos expansions technique (called polynomial chaos kriging PCK method) was used in the electromagnetic dosimetry framework in order to assess for the first time the exposure levels of a single user in a future indoor 5G scenario that deploys an antenna array with beamforming capability. This method represents a further option to the approaches previously used in stochastic dosimetry [19; 20], which typically focused on the use of surrogate models to describe the variability of exposure in uncertain conditions. Although there are not yet at present commercial solutions to 5G indoor access points with 3D adaptive beamforming, the roll-out of small cells will be a key point for success of 5G networks as reported in Reference [16]. The massive deployment of low-power AP will be primarily in the frequency band of 3.5-3.7 GHz, and in the future, in higher frequency bands above the 24 GHz in order to bring higher data capacities and higher data rate transmission. In this work, we modeled a 64 patch element UPA antenna at 3.7 GHz with 3D beamforming capability for a total input power of 100 mW, in line with the maximum allowed input power for indoor hotspot reported in the 3 rd Generation Partnership Project [23], i.e., 250 mW. The distance between the UPA antenna and the model head was simulated at 50 cm to evaluate the highest \(SAR\) levels. Greater distances between the AP and the human body will lead to lower \(SAR\) levels. The aim of the work was to validate the stochastic dosimetry methodology to obtain a complete description of the exposure levels for any beamforming patterns of the UPA antenna in terms of \(SAR\) distributions with an affordable computational effort respect to deterministic simulation methods; this aim was reached by identifying the effectiveness of the PCK method. Even if the distance between the antenna and the human subject was closer compared to a possible real implementation, the estimated exposure levels still led to low levels of \(SAR\) that respect the EMF exposure limitations. In fact, all the highest observed values for the \(SAR\) averaged on the whole body, the whole head, and the whole brain, and for the \(SAR_{10g}\) on tissues were significantly below the reference levels that ICNIRP guidelines introduce as basic restrictions for general public exposure, i.e., 0.08 W/kg for the whole body average \(SAR\) and 2 W/kg for the local head and torso \(SAR_{10g}\)[21]. These results are also in line with those of previous studies regarding 5G networks deployments, which estimated the exposure levels in indoor environments and highlighted that the ICNIRP guidelines will be well respected [13; 14]. Furthermore, the analysis conducted on the QDC values showed QDC values always higher than 67% for all the six examined \(SAR\) distributions. This means that, although low, the exposure levels change consistently as a function of the azimuth and elevation angles of the antenna beam that describes the space of variability of the 3D beamforming of the UPA antenna. This was an expected result, as 3D beamforming techniques inherently focus the radiation in narrow beams to increase the gain in the target direction and not everywhere [5; 6; 7] thus introducing a high variability on the induced \(SAR\) in the child model tissues. This consideration is reinforced by the analysis of the \(SAR\) values higher than the 70% of their maximum values, that showed how the highest values of the \(SAR\) concentrate in small ranges of the elevation and azimuth angles. The analysis highlighted also how only the beamforming patterns directed primarily towards the model area led to high \(SAR\) levels, whereas main antenna beams directed in other directions cause a rapidly decrease of \(SAR\) values.