Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework D.Vidhya1,* , S.Jafari2 and G. Nordo3 1 Department of Science and Humanities, Karpagam Institute of Technology, Coimbatore-641105, Tamilnadu, India; [email protected] 2 Professor of Mathematics, College of Vestsjaelland South Herrestarede 11, Slagelse, Denmark; [email protected] 3 MIFT Department of Mathematical and Computer Science, Physical Sciences and Earth Sciences - University of Messina, 98166 Sant’ Agata, Messina, Italy; [email protected] *Correspondence: [email protected] Abstract: Principal Component Analysis (PCA) is a widely used dimensionality reduction technique that transforms correlated variables into a smaller set of uncorrelated principal components. However, classical PCA assumes precise and crisp data, which may not hold true in real-world scenarios characterized by uncertainty and indeterminacy. To address this limitation, this study integrates Neutrosophic Logic into PCA, forming a robust framework capable of handling truth (T), indeterminacy (I), and falsity (F) values. The proposed methodology first converts neutrosophic data into crisp representations using an aggregation function, then applies PCA to extract principal components. A comparative analysis between normal PCA and Neutrosophic PCA is conducted using Python, highlighting how uncertainty impacts variance capture and eigenvector orientation. Visualization tools such as eigenvector plots, projection lines, and scree plots are employed to illustrate the findings. Results demonstrate that Neutrosophic PCA provides a more reliable representation of uncertain datasets without significant loss of variance information. This framework can be applied in fields such as pattern recognition, machine learning, and data-driven decision-making where uncertainty is inherent. Keywords: Principal Component Analysis (PCA), Neutrosophic Logic, Dimensionality Reduction, Eigenvectors and Eigenvalues, Python Implementation, Uncertainty Modeling, Data Analytics 1. Introduction Reducing the dimensionality of datasets is a key task in data analytics, as it helps retain essential information while minimizing redundancy [27]. A widely used method for this purpose is PCA, which transforms the data into a new set of orthogonal components, with each component capturing the maximum possible variance [26]. Due to this capability, PCA has been successfully employed in numerous domains, including pattern recognition, image analysis, machine learning, and scientific research [17, 28]. Despite its popularity, conventional PCA assumes that input data is exact, consistent, and complete. In practice, however, information gathered from devices, surveys, or expert assessments often suffers from ambiguity, incompleteness, and uncertainty [16, 19]. To address this shortcoming, Neutrosophic Logic—introduced by Smarandache [19, 20]—extends traditional and fuzzy set theories [23] by representing three independent dimensions: truth (T), indeterminacy (I), and falsity (F). Neutrosophic Sets and Systems, Vol. 97, 2026 329 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework Neutrosophic sets and their variants have been successfully employed to manage incomplete or vague data in a wide range of fields, including medical decision-making [1, 2, 11, 22], machine learning applications [5], image segmentation [28], and uncertain data modeling [3, 7, 15]. Each element in such a system can carry degrees of T, I, and F, which provides a flexible way to model uncertainty [6, 10, 14]. The integration of Neutrosophic Logic into PCA—referred to as Neutrosophic PCA—creates a more resilient approach to dimensionality reduction, explicitly accounting for uncertainty. Recent progress in Python-based libraries for neutrosophic operations [4, 6, 18, 21] has further enabled the design and implementation of such hybrid methods. In this paper, we introduce a Python-driven framework for Neutrosophic PCA. The data points, expressed as neutrosophic triplets (T,I,F) were converted into crisp equivalents using an aggregation function [12, 13]: Xcrisp=T+αI−F, where α represents the weight assigned to indeterminacy [16]. The processed dataset is then subjected to PCA, and its outcomes are evaluated against those of traditional PCA. 2. Methodology 2.1 Dataset Preparation: A sample two-dimensional dataset is selected for analysis. 2.2 Neutrosophic Data Representation: Each data point is converted into a neutrosophic triplet (𝑇,𝐼,𝐹) by introducing controlled indeterminacy and falsity. 2.3 Crisp Conversion: The triplet values are aggregated using the formula, Xcrisp=T+0.5I−F , to obtain a single numeric value. 2.4 Principal Component Analysis: PCA is applied to both the normal dataset and the neutrosophic-crisp dataset. Eigenvalues, eigenvectors, and explained variance ratios are computed. 2.5 Visualization: Scatter plots, eigenvector direction plots, projection lines, and scree plots are generated to compare the two approaches. 3. Normal PCA vs Neutrosophic PCA This section presents a comparative analysis between Normal PCA and Neutrosophic PCA. The section highlights how uncertainty in data is handled differently, impacting variance retention and data representation 3.1. Normal PCA: Numerical Explanation Step 1: Standardization We standardize the data: Neutrosophic Sets and Systems, Vol. 97, 2026 330 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework X_scaled = (X - μ)/σ Example for the first value of X1 = 2.5: μ_X1 = 1.81, σ_X1 = 0.83 X_scaled = (2.5 - 1.81) / 0.83 = 0.83. Where X → Original data value, μ → Mean (average) of that feature, σ → Standard deviation of that feature, X_scaled → Standardized value after transformation. Step 2: Covariance Matrix: Cov = [[1.111, 0.916], [0.916, 1.111]] Step 3: Eigenvalues & Eigenvectors: Eigenvalues: λ1 = 2.028, λ2 = 0.194 Eigenvector for λ1 (PC1): [0.707, 0.707] Step 4: Dimensionality Reduction: Project data onto PC1 to capture 91.2% of variance 3.2. Normal PCA: Python Code import numpy as np import matplotlib.pyplot as plt from sklearn.decomposition import PCA from sklearn.preprocessing import StandardScaler X = np.array([[2.5,2.4],[0.5,0.7],[2.2,2.9],[1.9,2.2],[3.1,3.0], [2.3,2.7],[2.0,1.6],[1.0,1.1],[1.5,1.6],[1.1,0.9]]) scaler = StandardScaler() X_scaled = scaler.fit_transform(X) cov_matrix = np.cov(X_scaled.T) eig_vals, eig_vecs = np.linalg.eig(cov_matrix) pca = PCA(n_components=1) X_pca = pca.fit_transform(X_scaled) Neutrosophic Sets and Systems, Vol. 97, 2026 331 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework plt.figure(figsize=(10,5)) plt.subplot(1,2,1) plt.scatter(X_scaled[:,0], X_scaled[:,1], color='blue') plt.title("Original 2D Data") plt.subplot(1,2,2) plt.scatter(X_pca, np.zeros(len(X_pca)), color='red') plt.title("Data After PCA (1D)") plt.show() 3.3. Normal PCA: Output & Visualization Original Data: [[2.5 2.4] [0.5 0.7] [2.2 2.9] [1.9 2.2] [3.1 3.0] [2.3 2.7] [2.0 1.6] [1.0 1.1] [1.5 1.6] [1.1 0.9]] Standardized Data: [[ 0.93 0.61] [-1.76 -1.51] [ 0.52 1.23] [ 0.12 0.36] [ 1.73 1.36] [ 0.66 0.98] [ 0.26 -0.39] [-1.09 -1.01] [-0.42 -0.39] [-0.95 -1.26]] Covariance Matrix: [[1.111 1.029] [1.029 1.111]] Neutrosophic Sets and Systems, Vol. 97, 2026 332 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework Eigenvalues: [2.14 0.082] Eigenvectors: [[ 0.707 -0.707] [ 0.707 0.707]] Transformed Data (1D PCA scores): [[ 1.086] [-2.309] [ 1.242] [ 0.341] [ 2.184] [ 1.161] [-0.093] [-1.482] [-0.567] [-1.563]] Explained Variance Ratio: [0.963] 3.4. Neutrosophic PCA: Numerical Explanation Step 1: Crisp Conversion: x_crisp = T + 0.5I - F Example for (2.5, 0.1, 0.05): x = 2.5 + 0.5*0.1 - 0.05 = 2.5 Neutrosophic Sets and Systems, Vol. 97, 2026 333 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework Step 2: Covariance Matrix: Cov = [[1.11, 0.91], [0.91, 1.12]] Eigenvalues: λ1 = 2.03, λ2 = 0.20 Step 3: Dimensionality Reduction: Project onto PC1 and capture 91% variance 3.5. Neutrosophic PCA: Python Code data_neutro = [[(2.5, 0.1, 0.05), (2.4, 0.1, 0.05)], [(0.5, 0.2, 0.05), (0.7, 0.2, 0.05)], [(2.2, 0.15, 0.05), (2.9, 0.15, 0.05)], [(1.9, 0.1, 0.05), (2.2, 0.1, 0.05)], [(3.1, 0.1, 0.05), (3.0, 0.1, 0.05)], [(2.3, 0.1, 0.05), (2.7, 0.1, 0.05)], [(2.0, 0.15, 0.05), (1.6, 0.15, 0.05)], [(1.0, 0.2, 0.05), (1.1, 0.2, 0.05)], [(1.5, 0.15, 0.05), (1.6, 0.15, 0.05)], [(1.1, 0.2, 0.05), (0.9, 0.2, 0.05)]] data_crisp = np.array([[T + 0.5*I - F for (T, I, F) in row] for row in data_neutro]) scaler = StandardScaler() Xn_scaled = scaler.fit_transform(data_crisp) cov_matrix_n = np.cov(Xn_scaled.T) eig_vals_n, eig_vecs_n = np.linalg.eig(cov_matrix_n) pca = PCA(n_components=1) Xn_pca = pca.fit_transform(Xn_scaled) plt.figure(figsize=(10,5)) plt.subplot(1,2,1) plt.scatter(Xn_scaled[:,0], Xn_scaled[:,1], color='blue') plt.title("Neutrosophic Crisp 2D Data") plt.subplot(1,2,2) plt.scatter(Xn_pca, np.zeros(len(Xn_pca)), color='red') plt.title("Data After Neutrosophic PCA (1D)") plt.show() 3.6. Neutrosophic PCA: Output & Visualization Crisp Neutrosophic Data: [[2.5 2.4 ] Neutrosophic Sets and Systems, Vol. 97, 2026 334 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework [0.55 0.75] [2.23 2.92] [1.9 2.2 ] [3.1 3.0 ] [2.3 2.7 ] [2.03 1.62] [1.05 1.15] [1.52 1.62] [1.15 0.95]] Covariance Matrix: [[1.111 1.025] [1.025 1.111]] Eigenvalues: [2.136 0.086] Eigenvectors: [[ 0.707 -0.707] [ 0.707 0.707]] PCA Transformed Data (1D): [[ 1.07 ] [-2.312] [ 1.275] [ 0.306] [ 2.194] [ 1.146] [-0.09 ] [-1.466] [-0.576] [-1.548]] Explained Variance Ratio: [0.961] Neutrosophic Sets and Systems, Vol. 97, 2026 335 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework 3.7. Comparison of Normal PCA Vs Neutrosophic PCA 3.7.1. Scree Plot (Explained Variance)-Python Code & Visualization plt.figure(figsize=(6,4)) components = np.arange(1, len(eig_vals)+1) plt.bar(components, eig_vals/sum(eig_vals)*100, color='purple') plt.ylabel("Variance Explained (%)") plt.xlabel("Principal Components") plt.title("Scree Plot") plt.show() Shows that PC1 captures ~91% variance in both normal and neutrosophic cases. Neutrosophic Sets and Systems, Vol. 97, 2026 336 D.Vidhya, S.Jafari and G.Nordo, Integrating Neutrosophic Logic into Principal Component Analysis: A Python-Based Framework 3.7.2. Eigenvector Direction Plot-Python Code & Visualization # Draw eigenvectors on the scatter plot colors = ['red', 'green'] # Colors for PC1 and PC2 for i in range(len(eig_vecs)): vec = eig_vecs[:, i] plt.arrow(origin[0], origin[1], vec[0]*2, vec[1]*2, head_width=0.1, head_length=0.1, color=colors[i], label=f'PC{i+1}') plt.title("Eigenvector Directions") plt.xlabel("Feature 1") plt.ylabel("Feature 2") plt.legend() plt.show() 3.7.3. 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