Detecting Very Weak Signals: A Mixed Strategy to Deal with Biologically Relevant Information

Abstract

In many biological investigations, the relevant information does not coincide with the most powerful signals (most elevated eigenvalues, dominant frequencies, most populated clusters...), but very often hides in minor features that are difficult to discriminate from random noise. Here we propose an algorithm that, by the combined use of a non-linear cluster analysis procedure and a strategy to discriminate minor signal components from noise, allows singling out biologically relevant hidden information. We tested the algorithm on a sparse data set corresponding to single-cell RNA-Seq measures, being able to identify a very small population of cells in charge of the immune response toward cancer tissue.

1. Introduction

The multi-scale character of biological regulation typically produces one (or a few) main modes explaining a major part of variance (size component), squashing potentially interesting (shape) components into the noise floor. These minor components could be erroneously discarded as pure noise by the conventional selection methods, based on identifying the main order parameters that organize the system [1].

This phenomenon is very common in applications of principal component analysis (PCA) to biological data [2]. The component explaining most of the data set’s variance typically often reflects well-known regularities. A classic example is the first principal component (PC) in microarray data sets, which frequently captures the gene expression profile characteristic of a specific tissue [3]. In contrast, biologically relevant information—such as the effects of pathology or cell-fate transitions—may be hidden in minor components that account for only a small fraction of the total variance.

Moreover, systematic variations introduced by laboratory conditions, reagent batches, or personnel differences—commonly referred to as batch effects [4]—can generate artificial major components that obscure biologically meaningful signals. It is important to emphasize, however, that this is not always the case: the presence of one or a few components explaining nearly all system variance is not necessarily an artifact, but may instead reflect the existence of a dominant order parameter that organizes the system’s variability [5]. Similar considerations hold true for biomedical applications like imaging studies, where the main order parameter corresponds to anatomical organization, while diagnostic relevant information is embedded in subtle image details [6].

In a purely ‘supervised learning’ frame—such as in medical diagnosis—this issue can be easily addressed by identifying biologically relevant minor components through their correlation with external variables of interest. These variables correspond to labels assigned to statistical units based on their biological or clinical nature, as determined by independent criteria (e.g., healthy/unhealthy, active/inactive). In such cases, the relevant components are those enabling a classification consistent with the a priori labels (considered as the golden standard), regardless of the amount of variance they explain [7].

In the case of explorative, hypothesis-generating studies, this straightforward approach is not applicable. Alternative strategies are required to assign biological meaning to minor components and/or sparsely populated clusters.

In this study, we propose a novel strategy applied to a data set derived from single-cell RNA sequencing (scRNA-Seq). The innovation lies in the integration of a non-linear clustering procedure (Convex Hull-based) with a structural-semantic interpretation of minor components, and in the introduction of a new statistical index—termed ‘Lacunarity’—to determine the optimal partitioning of the data set.

2. Data Analysis Strategy

The higher the proportion of variance explained by principal components, the higher their clustering efficiency—an empirical observation familiar to any data analyst. This stems from the coherence between K-means and PCA procedures. In fact, the continuous solutions of the discrete K-means clustering membership indicators correspond to the principal eigenvectors of the covariance matrix [8]. Similar considerations hold true for other clustering procedures, both hierarchical and non-hierarchical.

This relationship, however, is purely structural and not related to the ‘biological meaning’ of clusters and components. It reflects intrinsic properties of the data set: higher variance implies a wider space for separating groups. This, combined with the mutual linear independence of components (which guarantees a proper Euclidean metric), explains why clustering procedures in unsupervised learning are commonly applied to the major components of a data set.

Non-hierarchical clustering methods such as K-means require the a priori choice of the number of clusters K. To determine the optimal K, the procedure is repeated for different values of K, and each solution is evaluated using R² or pseudo-F statistics. The optimal K is the one that maximizes the ratio of between-cluster to within-cluster variance [9].

The convex-hull optimization strategy adopted in K-volume clustering paradigm [10] replaces variance with ‘lacunarity’. The underlying idea is that, for a structurally meaningful partition of the data set, the sum of the areas corresponding to the different clusters in the reference space should be smaller than the area occupied by the entire data set before clustering.

In the original version [10], the clustering algorithm was applied in a hierarchical mode. Here, we propose a non-hierarchical version of the algorithm, given that a natural hierarchical structure is not necessarily present in every data set.

We defined two different ‘lacunarity’ measures: one as the ratio between the total area and the sum of the cluster areas (see the two-dimensional case presented in [10]) and another based on volume (for our extension to the three-dimensional case). Let us call LC_A the two-dimension lacunarity index, A_TOT the total area within the data points and A_i the area of the i-th cluster for I = 1, …, K. The formula for the two-dimension LC_A is:

$${LC}_A={\displaystyle \frac{\left(A_{TOT}- \sum _{i=1}^K{A}_i\right)}{A_{TOT}}}={\displaystyle \frac{B_A}{A_{TOT}}}$$

(1)

where B_A corresponds to the empty space in two dimensions and the optimal number of clusters (K) will correspond to the maximal lacunarity.

Now let us call LC_V the three-dimensional lacunarity index, V_TOT the total volume within the data points and V_i the volume of the i-th cluster for I = 1, …, K. The formula for the three-dimensional LC_V is:

$${LC}_V={\displaystyle \frac{\left(V_{TOT}- \sum _{i=1}^K{V}_i\right)}{V_{TOT}}}={\displaystyle \frac{B_V}{V_{TOT}}}$$

(2)

where B_V corresponds to the empty space in three dimensions and the optimal number of clusters (K) will correspond to the maximal lacunarity.

It is worth noting the resemblance between lacunarity and the presence of distinct ‘attractors’ drastically reducing the phase space accessible to the system at hand. In the case of scRNA-Seq data, these attractors may correspond to different cell types, each characterized by a unique ‘ideal’ gene expression profile.

In K-volume clustering, the clustering cost corresponds to the minimal convex volume enclosing all points within a cluster. Under this definition, a single data point has zero volume, and any set of collinear points also has null volume as well [10]. This second property is very important in biological investigations: assigning zero volume to collinear points enhances the method’s sensitivity to subsets that exhibit high intra-class correlation (close to one), even when the overall data set shows no correlation—as in principal component spaces, which are spanned by axes that are linearly independent by construction.

Given that each cell type typically exhibits a highly specific gene expression pattern, independent samples of the same cell type tend to show near-unity correlation in terms of expression profile [3]. This makes K-volume clustering particularly effective in identifying small groups of cells that differ from the majority of the cell population. These groups, made by cells of a different kind with respect to the majority of the population, are expected to show high intra-class correlation when embedded into principal component spaces mainly defined by the ‘dominant’ cell kind.

Let us call such cells as ‘Type A’; when we analyze a homogeneous ‘Type A’ population by PCA (or by any other dimensionality reduction method), we will generate a dominant principal component reflecting their unique expression profile [3]. Conversely, when these A-cells are sparsely distributed within a large ‘Type B’ population of a different cell kind, the relatively few ‘Type A’ cells are likely to contribute only to minor components, accounting for a small fraction of the explained variance.

This observation inspired the data analysis strategy we propose: after identifying sparsely populated clusters—particularly those with collinear distributions—in the space defined by the major principal components, we examine the elements of these clusters for an excess of extreme absolute values relative to the minor component(s) corresponding to their main order parameter (ideal profile). The enrichment of extreme values of a minor principal component could qualify such a component as the dominant (and thus high variance) mode of a small group of cells coming from a different tissue kind.

When a statistically significant excess of extreme values of a minor component is observed for a specific cluster, we analyze its loading pattern with respect to RNA species. If this pattern aligns with a known cell type profile, we can assign a putative biological identity (cell type) to the cluster.

3. Results

3.1. Simulated Data

As first, we demonstrate the greater flexibility of K-Volume clustering with respect to K-Means algorithm, allowing to overcome both the classical limitations of handling outliers and individuating non-spherical clusters [11]. All the figures were generated through R (version 4.5.1; R Core Team 2025). We performed the analysis on a simulated data set (Figure 1).

For visual comparison, we report the cluster identified with the two algorithms by a 2D-Convex Hull representation. Through a numerical simulation, we generated 35 statistical units divided into K = 4 a priori defined clusters (Table S1). Specifically, the first three clusters were designed to be approximately spherical—thus theoretically more suitable for K-means—while the fourth cluster was constructed as a collinear distribution (i.e., points aligned along a single direction; see Figure 1a).

As expected, the K-Means algorithm struggled to capture the ‘correct’ shape of the clusters, except for Cluster 3 (Figure 1b). In contrast, the K-Volume algorithm successfully assigned the statistical units to their correct clusters, including the collinear one (Figure 1c).

In addition to visual inspection, we assessed the quality of the solutions produced by the K-means and K-Area procedures using the Silhouette and Adjusted Rand Index (ARI) metrics, along with the proposed Lacunarity index. The results of this comparison are presented in

Table 1. K-Area demonstrates superior performance, as expected for Lacunarity, and notably also for the other two indices.

	K-Means	K-Area
ARI	0.67	1.00 *
Silhouette	0.52	0.60 *
LCA	0.59	0.78 *

We use * to denote the best results.

.

3.2. Real Data

Moving on to a real experimental data set, we evaluated the clustering performance of the K-Volume algorithm on a complex biological scRNA-Seq data set, obtained from blood cells (CD45⁺) isolated from dissociated mouse lungs (GEO ACCESSION: GSE261385) [12].

Due to the nature of scRNA-Seq experiments, a common challenge before clustering was the inherent sparsity of the data. In fact, the output of scRNA-Seq is a gene-by-cell matrix, where each entry represents the number of transcripts detected for a gene in a single cell. The sparsity arises from both biological factors (not all genes are expressed in every cell) and technical limitations (the process of capturing and sequencing RNA from single cells is inherently noisy due to the low number of transcripts per cell). As a result, the expression matrix contains many zero counts, which complicates computational efficiency, statistical modeling and dimensionality reduction.

Our data set initially consisted of approximately 32k genes and 6k cells, resulting in a highly sparse count matrix. To address this, we first retained only the genes with non-zero counts across all the cells. Then, using the standard Seurat pipeline [13], we reduced the dimensionality of the scRNA-Seq dataset (100 genes and 321 cells, see Table S2), facilitating subsequent analysis. Figure 2 shows the correlation heatmap of the filtered genes.

It is worth noting in Figure 2 the presence of a highly positive correlation zone in the top-left corner, whereas negative correlations are relatively rare across the matrix. This pattern suggests the existence of a dominant ‘size component’ [2] that accounts for the gene expression profile of the majority of the cell population.

In the following analysis, we will identify a minor component whose most strongly loaded genes approximately correspond to the ‘pale red’ correlation block (and its two associated ‘stripes’), located just after the first block in the bottom-right direction of Figure 2.

A subsequent step involves applying PCA to reduce the dimensionality of the data and assign component scores representing the statistical units in a 3D space. As shown in [9], PCA is a valuable tool for highlighting information relevant to the classification of genes or cell types, enabling the discovery of new biologically meaningful groupings at varying levels of detail. The results of PCA are shown in Figure 3, showing that the first three PCs are the most informative, explaining 56% of the explained variance overall.

PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10
0.271	0.164	0.130	0.101	0.070	0.041	0.026	0.021	0.018	0.017

In order to compare the proposed K-Volume clustering approaches with K-means and Louvain clustering (very popular in scRNA-Seq applications, see [14]) procedures, we evaluate their performance in terms of both LC_V and Silhouette methods. Figure 4 reports the space spanned by the first three principal components.

Figure 5 shows the distributions of LC_V and S values for the three methods, along with a 3D visualization of the resulting clusters. Notably, LC_V exhibits a linear increase with the number of clusters, a pattern not observed for S. This trend supports interpreting the proposed Lacunarity index as a non-linear analogue of the proportion of variance explained by clustering solutions as K increases.

The optimal K is 5 for both K-Volume and K-means, while it is 6 for Louvain. In order to check for the biological meaning of the obtained clusters, we look for the association of these clusters with minor components (not used for clusterization) using the method reported in Algorithm 1 (see below).

Algorithm 1 Cluster-wise 2 × 2 Table Testing Significance (K, C, NC, PC)

1: Set K = n° of clusters from K-volume clustering.

2: Define the clusters partitioning S based on K, initialized as S = C.

3: Initialize a matrix MP of dimension NC × K.

4: Set c = 1.

5: while c ≤ NC do

6:      Set current_PC = PC[c];

7:      Retrieve partition S = {C1, C2,..., Ck} related to current_PC;

8:         for k = 1 to K do

9:           Let Ck the cluster k of the partition S;

10:          Extract values V in Ck for current_PC;

11:         a = n° of current units in Ck such that V ∉ (−2,2);

12:          b = n° of current units in Ck such that V ∈ (−2,2);

13:          c = n° of expected units in Ck such that V ∉ (−2,2);

14:         d = n° of expected units in Ck such that V ∈ (−2,2);

15:          Construct 2 × 2 contingency table MP of the form (3);

16:          Compute p-value p from statistical test on MP;

17:          Update MP[c, k] = p;

18:        end for

19:      c = c + 1;

20: end while

21: return MP

Table 2. Statistical significance of minor components across the obtained clusters: (a) K-Volume; (b) K-means; (c) Louvain. The statistical significance refers to the excess of “extreme values” in the components (see Algorithm 1). The table also reports the percentage of variance explained by each component (Var) and the number of elements in each cluster (N).

(a)
	N°	275						12						15						18						1
Var		C1						C2						C3						C4						C5
10.1%	PC4	5.4 × 10−61						1.00						1.00						4.2 × 10−9						1.00
7%	PC5	2.2 × 10−5						1.00						1.00						1.00						1.00
4.1%	PC6	3.6 × 10−5						1.00						1.00						1.00						1.00
2.6%	PC7	0.01						1.00						1.00						4.2 × 10−9						1.00
2.1%	PC8	0.09						1.00						1.00						1.00						1.00
1.8%	PC9	0.17						1.00						0.59						1.00						1.00
1.7%	PC10	0.58						0.59						1.00						0.08						1.00
(b)
	N°	15						171						12						21						102
Var		C1						C2						C3						C4						C5
10.1%	PC4	1.00						4.9 × 10−4						1.00						8.1 × 10−11						4.91 × 10−4
7%	PC5	1.00						4.9 × 10−4						1.00						1.00						0.70
4.1%	PC6	1.00						4.9 × 10−4						1.00						1.00						1.00
2.6%	PC7	1.00						0.02						1.00						1.61 × 10−9						0.06
2.1%	PC8	1.00						0.66						1.00						1.00						0.09
1.8%	PC9	0.59						0.79						1.00						1.00						0.02
1.7%	PC10	1.00						0.82						0.59						0.04						0.74
(c)
	N°	103					99					70					21					15					13
Var		C0					C1					C2					C3					C4					C5
10.1%	PC4	4.91 × 10−4					4.91 × 10−4					4.91 × 10−4					8.1 × 10−11					1.00					1.00
7%	PC5	0.73					0.41					4.91 × 10−4					1.00					1.00					1.00
4.1%	PC6	1.00					1.00					4.91 × 10−4					1.00					1.00					1.00
2.6%	PC7	0.06					0.06					0.35					1.6 × 10−9					1.00					1.00
2.1%	PC8	0.08					0.21					0.09					1.00					1.00					1.00
1.8%	PC9	0.02					0.21					0.74					1.00					0.59					1.00
1.7%	PC10	0.75					0.22					0.25					0.04					1.00					0.32

reveals a distinctive role for PC7: cluster C4 (18 cells) from K-Volume, cluster C4 (21 cells) from K-Means, and cluster C3 (21 cells) from Louvain exhibit highly significant values for PC7. Notably, these cell groups are nearly identical across the three clustering methods (Figure 5).

To elucidate the biological relevance of these findings, we annotated all data set cells using the SingleR package [15]. This annotation enabled the characterization of each cluster in terms of cell type(s), based on the ImmGenData reference database, which contains mouse bulk expression profiles from the ‘Immunologic Genome Project’ [16] (Figure 6).

The ‘PC7′ clusters emerging from the three clustering procedures (see Figure 5) are made (with no exception) of endothelial cells. It is worth noting that all the endothelial cells out of the 321 initial population (21/321) pertain to the three (one for each method) PC7-specific clusters. To go more in depth into the ‘endothelial’ character of PC7 we performed a Gene Ontology analysis based on the loadings of the genes on PC7 (Figure 7).

Figure 7 confirms the endothelial profile (the relatively low values of loading are a consequence of the fact that PC7 accounts for a minor part of total variance), so ‘closing the circle’ of the biological meaning assignment to the minority (21 cells) cluster.

A last confirmation comes from the equivalence of UMAP and PCA dimensionality reduction. We performed the same analysis described for PCA by means of the UMAP non-linear dimensionality reduction procedure [17].

Figure 8 reports the projection of the 321 cells in the space spanned by the first three UMAP axes. Panel b of the figure reports the annotation of cell kinds by the same method used in the PCA case.

The ‘endothelial’ cluster emerging from UMAP is perfectly coincident with the cluster coming from PCA so giving cogent proof of the robustness of the proposed strategy.

3.3. Step-by-Step Description of the Procedure

The testing phase of the K-Volume algorithm was performed on the first three PC cell scores (Table S3). We tested different cluster solutions (from K = 3 to K = 10) and for each solution, we calculate the lacunarity index based on Equation (2). Then we reported the results of the clustering (

Table 3. Summary of clustering results for different values of K.

K	Vk	LCV
3	202.1	0.89
4	131.1	0.93
5	54.1	0.97
6	52.2	0.97
7	46.2	0.98
8	28.6	0.98
9	26.7	0.99
10	25.9	0.99

).

The choice of the optimal value for K is shown in Figure 9a. It is worth noting that the lacunarity index reaches a plateau at K = 5 (LC_V = 0.97), which leads to the visualization of clustering solution in 3D-space (Figure 9b). The clusters had different shapes and sizes (275 cells for Cluster 1, 12 cells for Cluster 2, 15 cells for Cluster 3, 18 cells for Cluster 4 and one cell for Cluster 5).

After clustering, we assessed the statistical significance of the minor PCs not used in the clustering phase with respect to the obtained clusters. In fact, weak signals may be embedded inside these minor components—despite their low explained variance—and could be associated with one of the less populated clusters. We propose a potential approach to assess the statistical significance of minor principal components by measuring their deviation from what is expected by a standard normal distribution N(0,1) inside a specific cluster. Since component values are expressed as Z-scores across the entire data set, an excess of ‘extreme values’ (e.g., more than 2 Standard Deviation (SD) apart from zero) suggests that the component may be specific to that cluster.

For small clusters (fewer than 30 observations), we will use Fisher’s exact test, whereas for larger clusters we will use the Chi-Squared Test. The procedure focuses on identifying anomalies, for each cluster, within a specific principal component, by counting how many statistical units (i.e., cells) fall outside the ±2 SD range (using the score values associated with the single components). These observed frequencies are then compared to expected frequencies under the normality assumption (95% within the range, 5% outside). A two-way contingency table is constructed for this comparison.

	Outside ± 2 SD	Inside ± 2 SD
Current	a	b	(3)
Expected	c	d

The first row reports the observed number of statistical units falling inside or outside the range of ±2 SD for a given cluster/PC combination, while the second row shows the expected counts under the normality assumption. Applying the Fisher test to (3) yields a p-value indicating whether the cluster significantly deviates from the expected distribution. A low p-value suggests that the minor component carries meaningful information not captured during clustering based on the major components. The compact representation of the proposed approach is presented in Algorithm 1.

Table 4. Summary of p-values for K = 5, along with the number of units in each cluster and the percentage of variance explained by each minor PC.

	N°	275	12	15	18
Var		C1	C2	C3	C4
7.8%	PC4	5.4 × 10−61	1.00	1.00	4.2 × 10−9
5.5%	PC5	2.2 × 10−5	1.00	1.00	1.00
3.2%	PC6	3.6 × 10−5	1.00	1.00	1.00
2%	PC7	0.01	1.00	1.00	4.2 × 10−9
1.6%	PC8	0.09	1.00	1.00	1.00
1.4%	PC9	0.17	1.00	0.59	1.00
1.4%	PC10	0.58	0.59	1.00	0.08

summarizes the results obtained by applying Algorithm 1. Cluster 5 was excluded from the analysis as it contained only one cell and it was not informative. It is worth noting that Cluster 4, which contains 18 cells, shows a very significant link with the minor component PC7 (explaining 2% of the total variance). PC4 too exhibits a significant association with Cluster 4; however, its loading pattern primarily reflects the generic pertinence of the cluster elements to blood cells, making it less informative about the specific nature of the cells. It could be interesting to observe this cluster in terms of the score values of the statistical units.

From

Table 5. Score values for the statistical units in Cluster 4 with respect to PC7.

Cluster 4	PC7
CTGCCTACATCACCCT-1	6.78
GATGAGGGTACTCAAC-1	6.36
GCGAGAACAATGAATG-1	5.72
TGGTTCCGTCGCGGTT-1	5.59
CCTCAGTAGCGGCTTC-1	5.45
CGCTGGAAGCAACGGT-1	4.92
CGTAGGCTCTAACTCT-1	4.89
CTCGTACGTTCACGGC-1	4.75
CTGCCTACATCACCCT-1	3.94
CTGCGGATCGCCATAA-1	−2.82
GACGCGTAGTGGGCTA-1	−3.59
GATGAGGGTACTCAAC-1	−3.61
GCGAGAACAATGAATG-1	−5.87
GTGCGGTGTGTGCCTG-1	−5.93
GTTCATTGTAGGACAC-1	−5.95
TCACGAAAGCGAGAAA-1	−6.81
TGCGGGTAGGTGCAAC-1	−7.04
TGGTTCCGTCGCGGTT-1	−7.42

it is worth noting that the high significance between Cluster 4 and PC7 was found due to the presence of extreme score values, for each statistical unit in the cluster, assumed for this specific minor component. In particular, the range of variation reached extreme values (far away from the usual “range” of ±2 SD) in both directions (positive/negative).

We next aim to identify the genes that play a major role in characterizing the cells within Cluster 4. To this end, we analyze the loadings (associated with the genes, see Table S4) for PC7 (Figure 10a). The bar plot highlights the genes with the highest loadings on PC7. Loadings for all 100 original genes were sorted in ascending order. As expected for a minor component, the loading values are relatively low (ranging between −0.4 and 0.4).

After examining the loading distribution for PC7, we proceed with the biological characterization of Cluster 4. Classical bioinformatics analysis was performed using Gene Ontology (GO) [18]. The input for this analysis consists of genes selected according to biological criteria related to specific signatures of different cell types. Based on the distribution of loadings, we select as input all genes/sequences with absolute loading values above the ±0.1 threshold.

The results of the Gene Ontology analysis are shown in Figure 10b and in Table S5. The most significant biological pathways (p-value < 0.05) are reported and categorized into three groups: biological processes, molecular functions, and cellular components. The most highly loaded genes reveal a clear immune-related signature for the cells in Cluster 4. This assignment is supported by the presence of biological functions associated with membrane remodeling (linked to lipid pathways), communication with the extracellular matrix, angiogenesis, and the production of cytokines, chemokines, and interleukins.

To conclude this section, the entire analytical procedure presented here is summarized in Algorithm 2 with the method called K-Volume Biological Signal Characterization (KBSC).

Algorithm 2 K-Volume Biological Signal Characterization (MEXP, KLOW, KMAX)

1: Let MEXP be the original gene-by-cell expression matrix of scRNA-Seq object.

2: Apply standard Seurat pipeline to reduce the sparsity of MEXP.

3: Let MEXPF be the resulting filtered gene-by-cell expression matrix.

4: Apply PCA to MEXPF. 5: Set C = n° of cells of MEXPF, G = n° of genes of MEXPF and T = n° of PCs.

6: Let A = {PC1, PC2, PC3} be the set containing the three PCs with highest explained variance. 7: Let AC = {PC4, PC5,…, PCT} be the set containing the remaining PCs

8: Let MS be the scores matrix (C × T).

9: Set KLOW = minimum n° of clusters and KMAX = maximum n° of clusters.

10:    for each k in [KLOW, KMAX] do

11:      Apply K-Volume clustering algorithm taking as input MS[C,A];

12:      Calculate Lacunarity Index LCV;

13:    end for

14: Select optimal K ∈ [KLOW, KMAX] clustering solution based on LCV values.

15: Define the optimal clustering partition S = {C1, C2, …, CK}.

16: Apply Algorithm 1 with the following input parameters (K, S, AC, T).

17: Set n = n° of {AC, S} combinations where p-value < 0.05.

18:    if n is empty do

19:      stop;

20:    else

21:      for each element in n do

22:         Set PC0 = current PC in AC and C0 = current cluster in S;

23:         Loading analysis for PC0 and selection of relevant genes LC;

24:         Gene Ontology analysis with LC as input;

25:         Biological interpretation of cluster C0;

26:      end for

4. Conclusions

The proposed strategy aims to amplify potentially interesting signals that are difficult to distinguish from randomness. This scenario is common in biomedical sciences but is also relevant in other application domains. The core idea is to focus on sparsely populated clusters identified through a non-linear optimization procedure independent of the clusters’ geometrical shape. The presence of intra-cluster correlation within a space spanned by linearly independent axes is particularly noteworthy, as it suggests a heterogeneous pattern of relationships among the measured variables. This feature is often revealed by components that explain only a minor portion of the total variance, corresponding to the ‘signature’ of a small, heterogeneous group embedded within a larger population. In the specific case analyzed, this is especially relevant, as subpopulations of immune cells—represented by the heterogeneous ‘rare’ Cluster 4—may crucially influence patient outcome in many medical conditions. The same method may find a useful application in oncology by identifying rare cell populations such as cancer stem cells, chemoresistant cells and dormant cells, whose elusiveness often frustrates the success of anticancer therapies [19]. Parallel advancements in biomedical technologies and statistical approaches will likely pave the way for an improved management of diseases involving the perturbation of specific cell populations.