## 1 Introduction  
Colorectal cancer originates in the tissues of the colon or rectum. Like all cancers, it is a consequence of the uncontrolled growth of cells, in this case the epithelial cells of the mucosa lining the intestine. It is the third most common cancer worldwide. In 2022, more than 1.9 million cases were diagnosed [1]. The first step towards a reliable diagnosis is to collect information about the patient's medical history, followed by a complete examination and several blood and urine chemistry tests. However, additional instrumental examinations are necessary to complete the diagnosis, such as colonoscopy, which is based on the collection of a piece of tissue (biopsy) and subsequent histological analysis, after a proper staining procedure. Histologic analysis is a diagnostic procedure that involves microscope observation of tissue samples and specifically the cells that make up the tissue. This examination is carried out by a pathologist (a physician who specializesin the interpretation of cellular and tissue changes), who provides detailed information on the sample structure, composition and any abnormalities in order to obtain an accurate diagnosis. Training and skills may vary among clinicians, meaning that differences in expertise among pathologists can affect the accuracy of cancer diagnosis [2]. Therefore, the interest of researchers has recently focused on cancer diagnosis by means of analytical chemical techniques; in fact, biochemical alterations characterize the evolution of a cell from a normal to a cancerous state [3; 4; 5]. Besides analytical techniques based on the detection of biochemical markers [6; 7], it has been shown that Raman spectroscopy can be considered as a possible diagnostic technique providing information about differences in the relative content of biochemical components of healthy and cancerous cellular samples [8; 9; 10]. However, the adoption of Raman spectroscopy in the clinical diagnostics has not been fully realized yet, because of technical difficulties (relatively low signal-to-noise ratio, excitation power cannot be very high because of possible thermal destruction of the sample, difficulty of measurements for paraffin-embedded samples because of disturbance by paraffin, etc.) and also because Raman spectra of healthy and cancerous cells are often only slightly different and almost overlapping. In this latter case, spectral differences between cell samples of different types can be highlighted by means of suitable multivariate statistical methods [11; 12; 13]. These are mathematical methods that address and solve the problem of classifying unknown spectra in two successive stages. During the first stage, spectra belonging to known classes are used to develop a classification method based on the spectral differences. In the second stage, such a method that has been built and optimized is used to classify unknown spectra. We have previously used logistic regression, support vector machines, k nearest neighbours and neural networks to successfully classify Raman [14] and FTIR [15] spectra of healthy and cancer colon cells. However, the achievement of reliable results by means of such methods depends on optimization of the values of some parameters: consequently, high skills of the analyst are required. Moreover, the classification by means of the above methods relies on the intensity values of several spectral peaks, which should be properly selected. Therefore, it is preferable to use simple algorithms in order to minimize the analyst skills and choices. In recent years, linear discriminant analysis of principal components (PCA-LDA) and least squares discriminant analysis (PLS-DA) have been used to classify unknown spectra with good accuracy [16; 17]. Such techniques operate so that the selection of spectral features used for classification is made by the model itself, during the optimization procedure. In this work, we investigated the classification of Raman spectra of healthy and cancerous colon cells using PCA-LDA and PLS-DA methods.