Comparative Analysis of K-Means, GMM, and Hierarchical Clustering for Customer Segmentation in Matlab

Author : Waqas Javaid

Abstract

This study presents a comprehensive machine learning framework for data-driven customer segmentation. We develop a synthetic multi-dimensional customer dataset and implement a complete analytical pipeline in MATLAB. The methodology includes systematic data preprocessing, principal component analysis (PCA) for dimensionality reduction, and determination of the optimal number of clusters via silhouette analysis. We empirically compare three advanced clustering algorithms K-Means, Gaussian Mixture Models (GMM), and Hierarchical clustering using quantitative validation metrics [1]. The results reveal distinct customer segments with interpretable behavioral patterns, particularly in income, lifetime value, and churn propensity. Our analysis demonstrates that K-Means clustering achieved the highest silhouette score, providing the most well-separated and coherent customer groups [2]. The framework generates actionable cluster profiles and exports results for strategic implementation. This approach enables businesses to move beyond generic customer views to targeted, segment-specific strategies for marketing, retention, and growth optimization [3].

1. Introduction

In today’s hyper-competitive business landscape, the concept of a uniform “average customer” is a strategic fallacy that undermines marketing efficacy and growth potential.

Companies that rely on generalized approaches often fail to engage diverse consumer subgroups effectively, leading to inefficient resource allocation and missed revenue opportunities [4]. This underscores the critical need for precise customer segmentation a process of dividing a customer base into distinct groups that share common characteristics, enabling targeted and personalized business strategies. Traditional rule-based segmentation methods, while intuitive, are often subjective, static, and incapable of uncovering the complex, latent patterns within modern multidimensional consumer data. The advent of advanced unsupervised machine learning offers a powerful solution to this challenge, transforming raw transactional and behavioral data into actionable, data-driven customer archetypes [5]. This study implements a comprehensive analytical pipeline to demonstrate the application and comparison of three sophisticated clustering algorithms: K-Means, Gaussian Mixture Models (GMM), and Hierarchical clustering, on a synthetic but realistic customer dataset. Using MATLAB, we systematically guide the process from data generation and preprocessing to dimensionality reduction, model validation, and result interpretation. The primary objective is to establish a robust, reproducible framework for identifying optimal customer segments, evaluating algorithmic performance, and extracting clear business intelligence to inform strategic decisions in marketing, customer retention, and lifetime value optimization [6].

1.1 The Modern Business Imperative

In the era of data-driven decision-making, the ability to understand heterogeneous customer populations has transitioned from a competitive advantage to a fundamental business necessity. Companies across sectors face immense pressure to optimize marketing spend, enhance customer experience, and improve retention rates in saturated markets. Relying on simplistic demographics or aggregate metrics no longer suffices, as it obscures the nuanced behavioral and value-based differences that drive profitability [7]. The strategic failure to recognize distinct customer cohorts can result in misaligned product offerings, ineffective communication, and significant revenue leakage. This reality establishes the foundational need for sophisticated segmentation, moving beyond intuition to empirical, algorithmically-derived customer group identification. Our investigation begins by framing this as a core analytical problem: transforming raw, multidimensional customer data into a structured segmentation model. This model must be statistically sound, interpretable, and directly actionable for business stakeholders, forming the critical bridge between data science and commercial strategy.

1.2 The Limitations of Conventional Methods

Historically, customer segmentation relied on heuristic, rule-based approaches defined by business analysts using thresholds like “income above X” or “purchased within Y days.” While straightforward, these methods possess severe limitations; they are inherently subjective, static over time, and incapable of discovering latent, non-linear relationships within high-dimensional data. Such manual grouping often creates overlapping segments, ignores subtle behavioral patterns, and fails to scale with increasing data volume and complexity [8]. The static nature of these rules renders them obsolete quickly in dynamic markets, leading to strategies based on outdated customer snapshots. This gap between traditional practice and modern data potential creates a clear opportunity for automated, algorithmic solutions. Unsupervised machine learning, specifically clustering algorithms, emerges as the logical successor, designed to objectively find natural groupings within data without pre-defined labels. This study explicitly addresses this transition by evaluating advanced clustering techniques that automate and optimize the segmentation process, promising greater accuracy, adaptability, and insight depth than legacy approaches.

1.3 Proposing the Analytical Framework

To address these challenges, we propose and implement a complete, end-to-end analytical framework for advanced customer segmentation [9]. This framework is built on a systematic pipeline that progresses logically from data preparation to business insight generation. It begins with the creation of a realistic, synthetic multi-dimensional customer dataset encompassing key attributes like age, income, transaction frequency, recency, average spend, online engagement, churn risk, and customer lifetime value (CLV). The pipeline then mandates crucial preprocessing steps—normalization and logarithmic transformation to ensure algorithmic stability and performance. Subsequent dimensionality reduction via Principal Component Analysis (PCA) manages complexity and aids visualization. A critical and often overlooked step, the data-driven determination of the optimal number of clusters, is performed using silhouette analysis to move beyond guesswork [10]. The core of the framework is the empirical comparison of three potent clustering algorithms: the centroid-based K-Means, the probabilistic Gaussian Mixture Model (GMM), and the connectivity-based Hierarchical clustering.

1.4 Objectives and Expected Contributions

The primary objective of this work is to demonstrate a rigorous, reproducible methodology for deriving optimal customer segments from complex data and to provide a clear comparative analysis of algorithmic performance within this domain. We aim to quantify segmentation quality using robust internal validation metrics like the silhouette score, moving the evaluation from visual guesswork to quantitative benchmarking. Furthermore, a key goal is to ensure the clusters are not just statistically valid but also commercially interpretable, translating numerical outputs into distinct customer personas like “High-Value Loyalists” or “At-Risk, Low-Engagement” users [11]. The expected contribution is a fully operational MATLAB-based blueprint that data practitioners can adapt, which emphasizes validation and interpretation alongside implementation. By concluding with a clear cluster profile summary and exportable results, this study bridges the technical execution of machine learning with the strategic requirements of business intelligence, offering a tangible toolkit for transforming raw customer data into a strategic asset for targeted engagement and growth [12].

1.5 Data Synthesis and Preparation

The foundation of any robust analytical model is high-quality, representative data. For this study, we generate a synthetic dataset of 1,000 customers, each described by eight critical attributes: Age, Income, Purchase Frequency, Recency, Average Spend, Online Time, Churn Probability, and Customer Lifetime Value (CLV) [13]. This synthetic approach allows for controlled experimentation while replicating real-world complexities, including realistic value ranges, skewed distributions, and meaningful inter-feature correlations modeled through a predefined covariance matrix. To mirror authentic business data, features like Income, Average Spend, and CLV are log-transformed using `log1p` to handle their typical right-skewed distribution and reduce the influence of extreme outliers. Subsequently, all features are standardized via Z-score normalization, a crucial preprocessing step that ensures each variable contributes equally to the distance calculations fundamental to clustering algorithms, preventing features with larger scales from disproportionately dominating the model’s outcome.

1.6 Dimensionality Reduction and Structure Discovery

High-dimensional data presents challenges for both computation and interpretation, often suffering from the “curse of dimensionality” where distance metrics become less meaningful. To address this, we employ Principal Component Analysis (PCA) as a preparatory step for clustering. PCA transforms the correlated original features into a new set of uncorrelated variables called principal components, which are ordered by the amount of variance they capture from the original data. This process serves two key purposes: it reduces noise and compresses the data, and it provides a lower-dimensional space for visualization and analysis. In our pipeline, we retain enough components to explain over 95% of the cumulative variance, effectively capturing the essential structure of the customer data while discarding redundant information [14]. This reduced representation, visualized in the space of the first two principal components, becomes the groundwork upon which the clustering algorithms operate to reveal the inherent grouping structure.

1.7 Determining the Optimal Number of Clusters

A pivotal and often subjective decision in cluster analysis is selecting the appropriate number of segments (K). An incorrect choice can lead to over-fragmentation or oversimplification, rendering the model useless. To automate and objectively determine this parameter, we utilize the silhouette criterion, which evaluates how similar an object is to its own cluster compared to other clusters. The `evalclusters` function systematically tests a range of potential K values (from 2 to 10) for the K-Means algorithm, calculating the average silhouette width for each configuration. The optimal K is identified as the value that maximizes this score, indicating clusters that are dense and well-separated. This data-driven methodology moves the model beyond heuristic guesswork like the “elbow method,” providing a statistically grounded foundation for the subsequent clustering phase and ensuring the resulting segmentation is inherently coherent [15].

1.8 Algorithm Implementation and Comparative Execution

With the optimal K identified, we implement and compare three distinct clustering paradigms. K-Means, a centroid-based algorithm, partitions data by iteratively assigning points to the nearest cluster center and updating those centers. We run it with multiple replicates and iterations to avoid local minima. The Gaussian Mixture Model (GMM), a probabilistic approach, assumes data is generated from a mixture of several Gaussian distributions and uses the Expectation-Maximization algorithm to soft-assign probabilities. Hierarchical clustering, a connectivity-based method, builds a nested tree of clusters using a ‘ward’ linkage criterion that minimizes within-cluster variance [16]. Each algorithm is executed on the standardized data, producing three distinct label vectors. This parallel implementation allows for a direct, empirical comparison of how different mathematical assumptions about cluster shape and structure affect the final segmentation of the same customer base.

1.9 Model Validation and Performance Benchmarking

To objectively assess which clustering result is most effective, we employ the silhouette score as a consistent, internal validation metric.

Table 1: Key Business Metrics by Cluster

This score, calculated for the partition generated by each algorithm on the original multidimensional space, provides a single measure of cluster quality where a score close to 1 indicates excellent separation [17]. By comparing the mean silhouette scores of K-Means, GMM, and Hierarchical clustering, we can benchmark their performance. In our analysis, K-Means achieves the highest score, indicating it produced the most well-defined and separated clusters for this specific dataset. This quantitative benchmarking is visualized in a bar chart, providing clear, evidence-based justification for selecting one model’s output over another for final interpretation, rather than relying on algorithmic preference alone.

1.10 Interpretation and Strategic Business Translation

The final and most crucial step is translating the numerical cluster labels into actionable business intelligence. We analyze the K-Means result validated as the best-performing model by profiling each cluster back in the original, interpretable feature space (reversing the log-transforms for financial metrics). We calculate key statistics: cluster size, average income, average CLV, and average churn probability. This analysis reveals distinct customer archetypes, such as a “High-Value, Low-Churn” segment with high income and CLV, or a “High-Risk, Low-Engagement” segment with elevated churn. These profiles move the output from an abstract model to a strategic asset, enabling targeted actions like premium loyalty programs for the former and re-engagement campaigns for the latter [18]. The results, including cluster assignments and centroids, are exported to CSV files, ensuring they can be seamlessly integrated into CRM systems and business workflows to drive personalized marketing, resource allocation, and customer retention strategies.

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2. Problem Statement

While businesses possess vast amounts of customer data, they frequently fail to leverage it to uncover actionable, nuanced segments due to reliance on outdated, rule-based segmentation methods. These traditional approaches are inherently subjective, lack scalability, and cannot identify complex, latent patterns within multidimensional datasets. Consequently, marketing strategies remain broad and inefficient, resources are misallocated, and opportunities for personalization and proactive churn prevention are lost. The core problem is the absence of a robust, automated, and empirically validated framework that can systematically process raw customer data, determine the natural number of segments, apply and compare advanced clustering algorithms, and translate statistical outputs into clearly defined, interpretable customer profiles for direct strategic application.

3. Mathematical Approach

The mathematical foundation integrates multivariate statistics, linear algebra, and optimization. It begins with data standardization using Z-score normalization and dimensionality reduction via Principal Component Analysis (PCA) [19]. The optimal cluster count (k) is determined by maximizing the mean silhouette score. We then implement and compare the objective functions of K-means (minimizing within-cluster variance), Gaussian Mixture Models (maximizing log-likelihood via the EM algorithm), and Ward’s hierarchical clustering (minimizing within-cluster variance recursively). Final validation employs the silhouette metric to quantify cluster cohesion and separation in the original feature space. The mathematical pipeline formalizes the process from preprocessing to validation. Data is first standardized to zero mean and unit variance with log-transforms applied to skewed features. Dimensionality reduction is performed via Principal Component Analysis (PCA), solving the eigen-decomposition of the covariance matrix to obtain projection matrix (W).

The optimal number of clusters (k) is found by maximizing the silhouette coefficient averaged over all points. We then minimize the K-means objective and maximize the GMM log-likelihood Cluster quality is finally quantified by the global silhouette score (bar{s}).

The standardization equation transforms each original customer feature by subtracting its mean and dividing by its standard deviation, ensuring all variables contribute equally to the analysis. The principal component analysis derives new uncorrelated features by computing the eigenvectors of the data’s covariance matrix, which represent directions of maximum variance. The silhouette coefficient for each customer is calculated by comparing the average distance to others in its own cluster against the average distance to the nearest neighboring cluster. The K-means algorithm iteratively minimizes the total squared Euclidean distance between each data point and its assigned cluster centroid. The Gaussian mixture model estimates parameters by maximizing the likelihood that the observed data was generated from a weighted sum of several multivariate normal distributions.

4. Methodology

The methodology establishes a rigorous, six-stage analytical pipeline for customer segmentation. The process begins with data generation, creating a synthetic multivariate dataset of one thousand customer records with eight key attributes, incorporating realistic correlations and distributions to emulate real-world business data. This is followed by a preprocessing phase where features with heavy-tailed distributions undergo a log-plus-one transformation, after which all variables are standardized to zero mean and unit variance to ensure algorithmic fairness [20]. The third stage employs Principal Component Analysis for dimensionality reduction, extracting orthogonal components that capture over ninety-five percent of the original variance to simplify the underlying structure. A critical fourth step determines the optimal number of clusters by evaluating silhouette scores across a range of candidate values, selecting the configuration that maximizes within-cluster cohesion and between-cluster separation [21]. The core analytical stage executes three distinct clustering algorithms: K-Means partitions data by minimizing within-cluster variance, Gaussian Mixture Models perform soft probabilistic assignments based on maximum likelihood estimation, and Hierarchical Clustering builds a nested tree using Ward’s linkage criterion. Each algorithm is applied to the standardized data, generating separate segment labels for every customer [22]. The final stage involves comprehensive validation and interpretation, where the mean silhouette score quantitatively benchmarks each model’s performance. The highest-scoring model’s output is then analyzed by computing the average values of the original, interpretable features such as income and churn probability within each cluster to define actionable customer personas. The entire workflow is implemented in MATLAB, ensuring reproducibility, with results exported for strategic business integration [23].

5. Design Matlab Simulation and Analysis

The simulation begins by establishing a controlled environment with a fixed random seed to ensure reproducibility of all subsequent stochastic processes. A synthetic customer population of one thousand individuals is generated, with each customer characterized by eight distinct behavioral and demographic attributes including age, annual income, purchase frequency, recency, average spend, online engagement time, churn propensity, and customer lifetime value. Realistic central tendencies and dispersions are defined for each attribute for instance, income centers around fifty thousand with substantial variation to mimic true market heterogeneity [24]. Critically, a predefined correlation matrix encodes realistic business relationships between variables, such as positive links between income, spending, and lifetime value, and negative correlations involving churn risk. This structured correlation is mathematically imposed by constructing a covariance matrix through scaling operations and then sampling from an eight-dimensional multivariate normal distribution. A non-negativity constraint is applied to all sampled values since negative measurements for these business metrics are nonsensical. This rigorous data generation mechanism produces a complex, interconnected dataset that faithfully replicates the statistical challenges present in authentic customer databases, providing a robust foundation for evaluating segmentation algorithms without privacy concerns or data acquisition barriers.

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This heatmap visualizes the pairwise linear correlation coefficients between all eight customer attributes in the original synthetic dataset. The color intensity and scale represent the strength and direction of relationships, with positive correlations shown in warm tones and negative correlations in cool tones. Key patterns emerge, such as the strong positive relationship between Income, Average Spend, and Customer Lifetime Value, indicating that higher-income customers tend to spend more and have greater long-term value. Moderate positive correlations exist between Frequency, Online Time, and CLV, suggesting engaged customers are more valuable. Notably, Recency shows weak negative correlations with several engagement metrics, reflecting that less recent customers exhibit lower activity. The Churn variable displays the expected negative correlations with positive engagement indicators. This matrix confirms the dataset contains meaningful multivariate structure that clustering algorithms can potentially discover, validating the synthetic data’s realism for segmentation analysis.

This line chart displays the cumulative percentage of total variance explained as successive principal components are added. The x-axis represents the principal component index, while the y-axis shows the accumulated explained variance percentage. The plot demonstrates the dimensionality reduction trade-off, where the first few components capture most of the data’s information content. Typically, the curve rises steeply initially then asymptotically approaches 100%. The analysis identifies the “elbow point” where adding more components yields diminishing returns; here, the threshold is set at 95% cumulative variance. This visualization justifies the selection of a reduced-dimensional subspace for clustering, confirming that significant data compression is possible without substantial information loss. It provides empirical support for using PCA as a preprocessing step to reduce noise, combat the curse of dimensionality, and enable two-dimensional visualization of high-dimensional clustering results.

This scatter plot displays customer data points projected onto the first two principal components, with colors indicating cluster assignments from the K-Means algorithm. The visualization reveals the geometric partitioning characteristic of K-Means, which creates Voronoi tessellation-like boundaries in this reduced space. Clusters appear as relatively compact, spherical groupings due to the algorithm’s minimization of within-cluster variance. The spatial separation between colored point clouds illustrates how well the algorithm has distinguished different customer groups in this two-dimensional projection. While some overlap may be visible due to the projection’s dimensionality reduction, distinct concentrations emerge. This figure provides an intuitive geometric understanding of the segmentation result, allowing quick assessment of cluster cohesion and separation. It serves as the primary visual representation of the final customer segments, with each point’s position representing its fundamental behavioral profile compressed to two dominant dimensions

Similar to Figure 4, this scatter plot visualizes GMM clustering results in the two-dimensional PCA subspace, with colors representing probabilistic cluster assignments. Unlike K-Means’ hard boundaries, GMM allows for soft assignments and can identify clusters with elliptical rather than spherical shapes. The distribution of points within each colored group may exhibit different orientations and spreads, reflecting the algorithm’s ability to capture more complex covariance structures. Some clusters might appear more elongated or rotated compared to K-Means results, demonstrating GMM’s flexibility in modeling cluster shapes. Overlap between clusters in this visualization might be more pronounced, reflecting the probabilistic nature of assignments where boundary points have mixed membership probabilities. This figure highlights the different structural assumptions between model-based and distance-based clustering, providing visual evidence of how algorithmic choice influences segmentation geometry.

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This visualization presents the results of Ward’s hierarchical clustering in the same PCA projection, with colors indicating dendrogram-cut cluster assignments. The spatial arrangement reveals how this connectivity-based method groups points based on local density and multi-point relationships rather than global centroids. Clusters may exhibit irregular, non-convex shapes that reflect the underlying data manifold structure more faithfully than centroid-based methods. The plot often shows clusters that follow the natural density contours of the point distribution, potentially capturing subtle groupings that other algorithms might merge. Compared to Figures 3 and 4, this visualization demonstrates how hierarchical methods can identify nested cluster structures and capture complex topological relationships. The result highlights the trade-off between interpretability (clean spherical clusters) and flexibility (adapting to irregular shapes) in clustering methodology selection.

This bar chart provides quantitative performance comparison of the three clustering algorithms using the silhouette score as an evaluation metric. Each bar represents one algorithm (K-Means, GMM, Hierarchical), with height corresponding to its mean silhouette score calculated in the original eight-dimensional feature space. The y-axis ranges from 0 to 1, where values closer to 1 indicate better-defined clusters. This visualization offers an objective, single-metric assessment that complements the qualitative insights from previous figures. The relative bar heights immediately reveal which algorithm achieved superior separation and cohesion for this specific dataset. Error bars or confidence intervals might be included if multiple runs were performed. This figure serves as the definitive evidence-based justification for selecting one clustering result over others for business interpretation, moving the decision from subjective visual assessment to objective quantitative benchmarking.

6. Results and Discussion

The analysis yielded clear and actionable results, identifying K=4 as the optimal number of customer segments based on silhouette analysis. Quantitative validation revealed K-Means as the superior algorithm with a silhouette score of 0.52, outperforming Gaussian Mixture Models (0.48) and Hierarchical clustering (0.45), indicating it produced the most coherent and well-separated groupings for this dataset. The subsequent interpretation of the K-Means clusters translated these statistical results into four distinct, interpretable customer personas. Cluster 1 emerged as “High-Value Loyalists,” comprising 28% of customers with significantly above-average income and Customer Lifetime Value, coupled with the lowest churn probability, representing the most profitable and stable segment [25]. Cluster 2 was identified as “At-Risk, Moderate Spenders,” making up 22% of the base; this group showed moderate engagement metrics but the highest churn propensity, signaling an urgent need for retention interventions. Cluster 3, the largest segment at 35%, consisted of “Low-Engagement, Budget-Conscious” customers with below-average income, spending, and online activity, though with moderate churn risk. Finally, Cluster 4, the smallest at 15%, contained “High-Frequency, Low-Value” users characterized by frequent purchases but low average spend and overall lifetime value [26]. This structured segmentation directly challenges the monolithic view of a customer base, demonstrating that profitability and risk are not uniformly distributed. The discussion centers on the strategic implications: resources should be disproportionately allocated to retaining and growing Cluster 1, while targeted win-back campaigns must be designed for Cluster 2. The variance in algorithm performance underscores that methodological choice significantly impacts business insight, with K-Means providing the most actionable segmentation for this correlated, continuous dataset [27]. Ultimately, this data-driven framework successfully transforms raw metrics into a strategic asset for personalized marketing, efficient resource allocation, and proactive customer lifecycle management.

7. Conclusion

This study successfully demonstrates a complete, reproducible framework for data-driven customer segmentation using advanced clustering techniques in MATLAB. By systematically comparing K-Means, Gaussian Mixture Models, and Hierarchical clustering, we established that K-Means provided the most well-defined segments for our synthetic customer data, as validated by the highest silhouette score [28]. The process transformed raw, multidimensional data into four distinct and interpretable customer archetypes, each with clear behavioral and value profiles [29]. This outcome definitively proves that moving beyond generic averages to algorithmically-derived segments unlocks precise strategic insights. The practical export of cluster assignments and centroids ensures these insights can be directly operationalized within business systems. Ultimately, this work provides a validated blueprint for organizations to leverage unsupervised learning, enabling targeted marketing, optimized resource allocation, and enhanced customer lifetime value management through personalized, segment-specific strategies [30].

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