## 1 Introduction  
Atrial fibrillation (AF) is the most common cardiac arrhythmia diagnosed in clinical practice, with an estimated prevalence of about 1-2% of the general population and above 10% in the elderly [1]. Moreover, its prevalence is likely to double in the next 50 years as the population ages [2]. At least 15% of the budget in cardiac disease healthcare is earmarked to AF [3;4]. This disease is associated with an increased risk of stroke and congestive heart failure, so that AF patients have twice the risk of death as compared with healthy persons [5]. These facts make AF a major public health challenge, and its medical and economic aspects could worsen in the future [2]. From an electrophysiological viewpoint, AF is characterized by rapid and chaotic contractions of the atria, originating in disorganized atrial electrical activation [6]. As with many other arrhythmias, AF may require therapeutic intervention, even in patients who suffer no subjective discomfort [7]. Since [PERSON] et al. reported the paramount relevance of the pulmonary veins (PV) in the initiation and maintenance of AF [8], the procedure of catheter ablation (CA) targeting PV foci, namely pulmonary vein isolation, has become such an effective therapy for AF [9], it is considered the first-line therapeutical alternative to pharmacological treatments. Despite its high prevalence, the physiological mechanisms underlying AF are still pending to be completely understood, and the present therapeutic approaches to AF have major limitations [10]. During recent years, many efforts have addressed personalizing CA treatments by mapping the atrial electrophysiological substrate [11] and have been introduced with the aim to identify the arrhyththmogenic atrial sites responsible for AF generation [12]. In this respect, atrial substrate characterization has been one of the most recent approaches aimed at reducing the limited clinical efficacy of the current therapeutic intervention, due to the major knowledge gaps in the mechanisms for sustaining AF [13]. In clinical practice, one of the goals for the characterization of the atrial substrate aims at discerning patients with paroxysmal AF (ParAF) versus persistent AF (PerAF) [14], as statistics have shown that the success rate of CA is dependent on the area of ablation and the AF type [15]. In fact, a high success rate is reached in ParAF just ablating the pulmonary veins (PVs), while for PerAF, the use of further ablations is required to achieve similar results [12]. This defines a challenge for the precise characterization of the atrial substrate aimed at optimally guiding the CA, where methodologies to distinguish between the complex fractionated atrial electrograms (CFAEs) of ParAF versus PerAF would be very interesting and useful for fast and efficient atrial substrate mapping methods [16]. CFAEs can be identified by the presence of multiple electrogram deflections without interruption, a baseline perturbation with continuous deflection [17], or a cycle length \(\geq\)120 ms that includes isoelectric intervals between deflections [18]. In the attempt to personalize AF treatment, nonlinear indices have been applied to CFAEs, which is aimed at quantifying atrial remodeling and the atrial electrophysiological substrate, supporting clinical management decisions, and suggesting the most appropriate approach for ablation procedures [19]. In this regard, different works have been published proposing several classification strategies based on nonlinear metrics as assessed via statistical tests, aimed at classifying ParAF versus PerAF via a CFAE analysis. In this respect, [PERSON] et al. measured the CFAE repetitiveness [16] and quantified the degree of morphological heterogeneity in CFAE deflections [20]; [PERSON] et al. adopted recurrence plots, the Recurrence Quantification Analysis, and entropy measures, proving that the underlying signal generation process of CFAEs in AF is somehow repetitive, even for sequences as short as one second [21]; [PERSON] et al. and [PERSON] et al.