A. Preprocessing of raw datasets.

See -A of panel-‘RgCop framework for gene selection’ of Fig 1. Raw scRNA-seq datasets are obtained from public data sources. The counts matrix , where c is number of cells and g represents the number of genes, is normalized using a transformation method (Linnorm) [16]. We choose that cells which have more than a thousand genes expression values (non zero values) and choose that genes which have the minimum read count greater than 5 in at least 10% among all the cells. log₂ normalization is employed on the transformed matrix by adding one as a pseudo count.

B. RgCop framework for feature selection.

See -B of panel-‘RgCop framework for gene selection’ of Fig 1. The preprocessed data is used in the proposed copula-correlation (Ccor) based feature/gene selection models. First, a feature ranking is performed based on the Ccor scores between all features and class labels. We assume the feature having a larger Ccor value is the most relevant one and we include it in the selected list. Next, Ccor is computed between the selected relevant features and the remaining features. The feature with a minimum score is called the most essential (and not redundant) feature and included in the selected list. The process continued in an iterative way by including the most relevant and minimum redundant features in each step every time in the list. Feature selection in this way ensures the list of genes will be optimal (see proof of correctness). An l₁ regularization term is added with the objective function to penalize the large coefficient of relevancy term. The resulting matrix with selected features is utilized for further downstream analysis.

C. Validation through clustering.

See panel-‘validation-A’ of the Fig 1. We adopt the conventional clustering steps of scanpy [17] package to cluster the resulting matrix obtained from the previous step. We employed two clustering techniques (SC3 [4], and Leiden clustering [18]) for clustering the neighborhood graph of cells. To validate the clusters we utilize the Adjusted Rand Index (ARI) metric which is usually used as a measure of agreement between two partitions. We compare the ARI score of RgCop with different state-of-the-art unsupervised feature selection method.

D. Validation through classification.

See panel-‘validation-A’ of the Fig 1. We validate the selected features by employing several classifiers to train the resulting matrix obtained from step-B of the RgCop workflow. The features are selected by four widely used supervised feature selection algorithm and the classification accuracy are compared with RgCop (see the subsection Comparison with State-of-the-art).

E. Annotating unknown cells.

See panel-‘validation-B’ of the Fig 1. The selected genes obtained from RgCop is able to cluster/classify cells of unknown type. The filtered and preprocessed data is divided into train-test ratio 8:2 and the train set is utilized to obtain the selected features using RgCop. Several classifier models are trained on the train set with the selected feature set and applied to the test set. The test data with selected features are also used for clustering. This provides the validation of our approach to work in practice.

F. Marker identification.

We detect highly differentially expressed (DE) genes within each cluster obtained from step-C in the workflow. Here we utilize Wilcoxon Ranksum test to identify DE genes in each cluster. The top five DE genes are chosen from each cluster according to their p-values.

Clustering performance on real dataset using unsupervised method.

Here single cell Consensus clustering (SC3) method [4] is employed for clustering expression matrix with selected features. Fig 2, panel-A illustrates the boxplots of ARI Values of the clustering results on Yan, Muraro, and Pollen datasets. We vary the number of selected features in the range from 500 to 1000 and compute the ARI scores for each method. It can be seen from the figure that RgCop achieves high ARI values in almost all the three datasets. For the Yan dataset, while the performance of other methods is relatively low, RgCop achieves a good ARI value, demonstrating the capability of RgCop to perform in small sample data. We also create a visualization of the clustering performance of RgCop in Muraro, yan, and Pollen datasets. Fig 2B, shows two dimensional t-SNE plot of predicted clusters and their original labels. Fig 2C shows heatmaps of cell × cell consensus matrix representing how often a pair of cells is assigned to the same cluster considering the average of clustering results from all combinations of clustering parameter [4]. Zero score (blue) means two cells are always assigned to different clusters, while score ‘1’ (red) represents two cells are always within the same cluster. The clustering will be perfect if all diagonal blocks are completely red and off-diagonals are blue. A perfect match between the predicted clusters and the original labels can be seen from panel-B and panel-C of Fig 2.

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Fig 2. Figure shows the comparisons of clustering performance.

Panel-A shows the boxplot of ARI values computed from the clustering results of each competing method. Each box represents ten ARI scores of clustering results for selected 6 sets of features ranging from 500 to 1000. Panel-B shows the 2 dimensional UMAP visualization of clustering results of three datasets for RgCop. Panel-C shows the consensus clustering plots of obtained clusters from RgCop.

https://doi.org/10.1371/journal.pcbi.1009464.g002

Classification performance on real dataset using supervised method.

We compare RgCop with four well known supervised feature selection methods and compute the classification accuracy. Three widely used classifiers are considered in our work, Neural Network (NNET), Support Vector Machine (SVM), and Gradient Boosting Machine (GBM) for learning the expression matrix with selected features. Table 3 shows the average test accuracy and the corresponding standard errors over 50 runs for each of the competing methods. Results demonstrate that RgCop outperforms the other existing supervised feature selection methods.

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Table 3. Classification results on real datasets using supervised methods.

https://doi.org/10.1371/journal.pcbi.1009464.t003

Single cell RNA sequence datasets.

The study used the following single-cell RNA sequence datasets: Yan [31], Pollen [32], Muraro [33] and PBMC68k [1] (see Table 7). A detailed description of the data is provided in the following text.

• Yan: The dataset consists of a transcriptome of 124 individual cells from a human preimplantation embryo and embryonic stem cell. The 7 unique cell types accommodates labelled 4-cell, 8-cell, zygote, Late blastocyst, and 16-cell.[GEO under accession no. GSE36552; [31]]. We downloaded the processed data from https://hemberg-lab.github.io/scRNA.seq.datasets/human/edev/ which contains 20214 features and 90 samples.
• Pollen: Single cell RNA seq pair-end 100 reads from single cell cDNA libraries were quality trimmed using Trim Galore with the flags. It contains 11 cell types. [GEO accession no GEO1832359; [32]]
• Muraro: Single-cell transcriptomics was carried out on live cells from a mixture using an automated version of CEL-seq2 on live, FACS sorted cells. It contains 2126 number of cells. It is a human pancreas cell tissue with 10 cell types. The dataset was downloaded from GEO under accession no GSE85241 [33].
• PBMC68k: The dataset [1], is downloaded from 10x genomics website https://support.10xgenomics.com/single-cell-gene expression/datasets. The data is sequenced on Illumina NextSeq 500 high output with 20,000 reads per cell.

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Table 7. A brief summary of the real scRNA sequence dataset.

https://doi.org/10.1371/journal.pcbi.1009464.t007

Short description on Copula.

The ‘Copula’ term [34] is originated from a Latin word copulare, which joins multivariate distributions to its one dimensional distribution function. The copula is considerably employed in high dimensional datasets to obtain joint distributions using uniform marginal distributions and vice versa. See S1 Text sec-2 for a detailed descriptions of copula and its related measures.

Copula correlation measure.

Let, Y = {y₁, y₂} and Z = {z₁, z₂} are two bivariate random variables and their joint and marginal distributions are H_YZ, F_Y(y) and, F_Z(z) respectively. Now H_YZ can be expressed as: H_YZ(y, z) = C(F_Y(y), F_Z(z)), where C is a copula function.

Kendall tau(τ), the measure of association, [35] can be expressed in terms of concordance and discordance between random variables. Kendall tau is the difference between probability of concordance and discordance of (y₁, y₂) and (z₁, z₂). It can be described as

(1)

According to Nelson [36] Kendall tau can be expressed using copula function:

(2)

Where, u ∈ F_Y(y) and v ∈ F_Z(z). τ(C_Y,Z) is termed as copula-correlation (Ccor) in our study.

A note on regularization.

Regularization is a type of regression that penalizes the coefficient of redundant feature towards zero [37] (see S1 Text sec-3 for detailed description). The simplest regularization is l₁ norm or Lasso Regression, which adds “absolute value of magnitude” of coefficient as penalty term to the loss function. For any vector , the l₁ norm is , where γ is a tuning parameter, controls penalization. For γ = 0 regularization effect is none. When γ value increases, it starts to penalizes the larger coefficients to zero. However, after a certain value of γ, the model starts losing important properties, increasing bias in the model and thus causes under-fitting. We tuned γ using simulated single cell datasets generated by the well known algorithm Splatter [19].

Max relevancy.

A gene g_i is more relevant to class labels C_D than another gene g_j, if g_i has higher Ccor score with C_D than g_j. This is called Relevancy test for the genes [38] and is used to select most relevant gene from a gene set. Formally it can be described as: g_i ≺ g_j if the following is true, (3) Where τ[C(x, y)] represents copula correlation between two random variable X and Y. For estimating the copula density we have used empirical copula. The maximum-relevancy method choose the gene (feature) subset among gene set G as

(4)

G_{max−relevancy} may contains genes that are mutually dependent. This is because we only consider the Ccor between a gene and class labels, overlooking the mutual dependency among the selected and non-selected genes. This results spurious genes in the selected list.

Min redundancy.

Redundancy is a measure that computes the mutual dependence among set random variables. Here we used the same definition of Ccor to compute multivariate dependency between selected gene (g_i) and non selected gene sets (g_s). Formally it can be expressed as

(5)

Objective function.

RgCop utilizes a forward selection wrapper approach to select gene iteratively from a gene set. It uses multivariate copula-based dependency instead of the classical information measure. The objective function integrates the relevancy and the redundancy terms defined using the Ccor. Mathematically, it can be expressed as follows.

Let us assume genes (g₁, …, g_i) are in the selected list G_s. The next gene g_i+1 ∈ (G − G_s) in at (i+1) iteration is using the objective function

(6)

Where, is Kendall tau dependency score of Empirical copula between two genes g_i and g_j. Here, γ represents the regularization coefficient. An overview of the RgCop algorithm is given in algorithm-1

Algorithm 1 l₁ Regularized Copula Based Feature Selection (RgCop)

Input: Preprocessed Data Matrix D, Cell Type C_D, Number of Selected Features, d.

Output: Optimal Feature subset,(G_s).

 Initialisation:

 G_s = ∅, {T will hold sub-set feature indices.}

 g_1s ← argmax_gi τ (C_gi,CD), {Maximum Relevancy}

 for all i = 0 to (d − 1) do

  E = ∅

  R ← τ(C_gi;CD), {Relevancy Criterion}

  S ← τ (C_gi;g1s;…_;gds), {Redundancy Criterion}

  E ← (R − S)

  G_s ← {G_s ⋃ arg max (lim_gi {E})

  G ← G − {g_i}

 end for

 return G_s

Proof of correctness.

Suppose G_s = {g₁, g₂, …, g_i, g_i+1, …g_d} denotes a subset of genes obtained from a gene set G using RgCop. Here g_i represents selected gene at iteration i. We claim that the set G_s is optimal.

Proof.

Let us prove this by the method of contradiction. If we assume the claim is not true, then there should exist some another optimal gene set . Without loss of generality, let us assume has a maximum number of initial genes (i number genes) common with G_s.

Now can be written as . So, contains {g₁, g₂, …, g_i} from G_s, but not g_i+1. Following our assumption g_i+1 cannot be included in any of the optimal gene lists ( has maximum i number of initial genes overlapped with G_s).

Now we claim that k > (i+1). This is because k cannot be equal to i+ 1, otherwise would have (i + 1) genes overlapped with G_s. Similarly, k ≰ i, because otherwise will contains redundant genes.

Now by the definition of our objective function (f) we can write: f(g_k) < f(g_i+1). So we can substitute g_k with g_i+1 in the list, and the list will be still optimal. This contradicts our assumption that g_i+1 cannot be included in any optimal list. This proves our claim.