## 1 Introduction  
Walnut is the fruit of the _Juglans regia_ L. tree. It is an economically interesting arboreal species, appreciated for its wood and edible fruits, which grows in temperate climate areas. Its seed is an important source of phospholipids,tocopherol, proteins, and mono- and polyunsaturated fatty acids. Overall, it is a noticeable source for some microelements, such as iron, copper, selenium, and zinc. In addition, as it has been observed that their consumption is capable to reduce the incidence of coronary diseases, walnuts have been deeply studied in the last few years [1]. In Italy, walnuts are harvested throughout the national territory; nevertheless, the fruits produced in the Sorrento area (South Italy) are commercially valuable for their peculiar organoleptic characteristics, confirmed by genetic criteria established by [PERSON] et al. [2]. From this reason arises the necessity of characterizing Sorrento walnuts, in order to discriminate it from common fruits (having lower market value), preventing possible commercial frauds (e.g., counterfeits). In general, many analytical techniques have been exploited in walnut analysis: chromatographic approaches have been involved in studies on polyphenolic fraction [3, 4], to quantify bioactive compounds [5], or lipids [6]; moreover, thermogravimetric analysis coupled with gas chromatography/mass spectroscopy (GC/MS) has been used to study the thermal behavior of walnut shell [7], while Inductively Coupled Plasma Optical Emission Spectroscopy (ICP-OES) was exploited to study the mineral composition of walnuts and walnut oils [8]. In some works, also Simple Sequence Repeats (SSRs) is used to study different walnuts [1, 9], while [PERSON] et al. used image processing to distinguish different cultivars on the basis of dimensional, gravimetric, and morphological features of the fruits [10]. Eventually, [PERSON] and [PERSON] discriminated walnuts varieties on the basis of morphological and organoleptic characteristics [11]. Near-infrared spectroscopy (NIRS) presents several advantages: if coupled with an integrating sphere, it allows carrying out analysis on raw samples, without any physical pre-treatment, resulting in a fast, green, relatively cheap, noninvasive, and automatable technique. Moreover, it is possible to analyze samples before selling them, in a time range compatible with their shelf-life and avoiding any loss of product. For this reason, NIRS was elected as the tool of choice to address the Sorrento walnut authentication problem. There are many examples in literature about the use of NIRS coupled with chemometric classification tools in food-related authentication issues; for instance, on dried foodstuff (cereal, fruits, and nuts) such as rice [12, 13], tea [14, 15], macadamia [16], hazelnuts [17], almonds [18, 19], and several others [20] can be found. NIR (coupled with linear discriminant analysis) has been used for varietal discrimination on Portuguese walnuts, providing indication of the feasibility of this approach [21]. Under this perspective, NIRS has been paired with two different classification strategies, with the aim of distinguishing Sorrento from non-Sorrento walnuts. In particular, it has been coupled with Partial Least Squares-Discriminant Analysis (PLS-DA), which has been applied on data collected on the shell and on spectra of the kernels. This classifier is one of the most used in different fields [22, 23, 24, 25], and it has been chosen because it had performed well in similar situations [26, 27, 28, 29], and it is therefore a common choice in this context. Eventually, in order to investigate whether a simultaneous analysis of both sets of spectra could improve predictions, two multi-block strategies have been tested. In particular, Sequential and Orthogonalized Partial Least Squares Linear Discriminant Analysis (SO-PLS-LDA) [30] and Sequential and Orthogonalized Covariance Selection Linear Discriminant Analysis (SO-CovSel-LDA) [31] have been exploited, handling together both data blocks.