## 1 Introduction
Thus, in the initial part of the fibre, the Kerr effect is significant, whereas the effect of the ISI created by the chromatic dispersion can be neglected. At the end of the fibre, on the other hand, the accumulated dispersion is high, leading to a significant ISI, but given the high transmission loss, the Kerr effect is reduced. Both scenarios can be modelled relatively easily, the initial part by a memoryless nonlinear phase rotation and the last part of the link by a linear time invariant system. The intermediate part of the link, however, should consider the interaction between the two effects, and therefore, it is complicated to model analytically. Several nonlinear mitigation techniques have been proposed in recent years to overcome this issue. Optical techniques, for instance mid-span nonlinear compensation [15], lack flexibility and require a careful design of the distribution network, which is not possible in PON networks. In these networks, electrical compensation techniques are preferable for their higher flexibility and adaptability. Unfortunately, simple deterministic approaches based on non-uniform phase rotation cannot efficiently compensate signal distortion as they neglect the effect of chromatic dispersion and the subsequent ISI. More complex techniques, such as digital back-propagation (DBP) [16; 17] and inverse Volterra series transfer function (IVSTF) based nonlinear equalization [18; 19], were then studied to invert the dynamic nonlinear time invariant behaviour of the fibre link. These techniques are capable of mitigating the effect of the interplay between dispersion and nonlinear distortion, but suffer from a prohibitively high computational cost, making real-time operation if not unfeasible, at least extremely challenging and power consuming. In this scenario, machine learning emerges as a high potential set of tools to analyse and process complex systems where analytical modelling is unfeasible or the computational cost to solve it is excessively high [20]. Thus, several groups have proposed different machine learning based approaches to overcome the degrading effect of nonlinear distortion in fibre communication systems [21; 22; 23; 24]. In [25; 26; 27], artificial neural networks were employed, whereas in [28] and in [29; 30], support vector machines (SVMs) were proposed. These approaches present a good nonlinear compensation performance, but are all supervised and, consequently, require the transmission of a training sequence that reduces the effective data throughput. Unsupervised machine learning, for instance clustering, on the other hand, does not require any training sequence, but learns from the received dataset. Among the proposed clustering algorithms, k-means is by far the most popular due to its simplicity, its convergence speed, and robustness [31; 32; 33; 34]. In k-means, however, each cluster is represented by a centroid, and the decision regions are limited by straight boundaries, which may not be optimal for constellations strongly affected by SPM. In order to find decision regions adapted to arbitrarily shaped clusters, other clustering algorithms have been proposed: In [35], clustering based on affinity propagation was reported. Density based spatial clustering of applications with noise (DBSCAN) has also been applied successfully to improve the performance of systems affected by the combined effect of noise and nonlinear distortion [36]. In [37], classification based on expectation maximization was successfully employed to combat the effect of nonlinear phase noise. These algorithms, however, suffer from heavy computational cost, and their performance strongly depends on the tuning parameters. In DBSCAN, for example, it is necessary to set the values of the parameters \(\epsilon\) and \(k_{min}\) that correspond to the radius of the area and the minimum number of points in this area, respectively. In this paper, we present a novel clustering algorithm denominated histogram based clustering (HBC) that partially mitigates the effect of nonlinear phase noise caused by SPM.