K-Means-Clustering-With-Missing-Data.pptx

K-Means-Clustering-With-Missing-Data.pptx

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  K-Means Clustering With MissingData Using Mahalanobis Distance for Better Clusters
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  The Problem: RealData is Often Incomplete Missing Values are the Norm In surveys, sensor readings, privacy redaction, or system failures, missing values are a prevalent challenge, not an exception. Traditional K-means Limitations Standard K-means algorithms cannot directly handle NaN (Not a Number) values, leading to data processing hurdles. Suboptimal Solutions Dropping rows with missing data reduces valuable information, while simple imputation techniques can distort the inherent structure of clusters.
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  The Standard Approach:Limitations and Consequences Typical Workflow Impute Missing Values Estimate and fill in absent data points using various statistical methods. Run K-means Clustering Apply the K-means algorithm, typically with Euclidean distance, to the now complete dataset. Interpret Clusters Analyze the resulting clusters to derive insights and patterns. Inherent Issues Distorted Data Geometry Imputation methods can subtly (or significantly) alter the original geometric relationships within the data. Spherical Cluster Assumption Euclidean distance inherently assumes clusters are spherical, which often doesn't hold true for real-world data. Ignoring Feature Correlation The standard approach fails to account for the covariance and correlation between different features, leading to misrepresentative clusters.
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  Why Geometry Matters:Euclidean Distance's Flaw Assumes Spherical Clusters K-means minimizes squared Euclidean distance, performing optimally when clusters are perfectly round and isotropic. Real Clusters Are Elliptical In reality, most data clusters exhibit elliptical shapes due to underlying correlations between features. Correlated Features Distort Strong correlations between features stretch clusters into elongated ellipsoids, challenging Euclidean distance's efficacy. Euclidean "Sees" Distortion Euclidean distance perceives these elliptical clusters as distorted, leading to suboptimal assignments and misrepresentations of actual groupings.
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  Mahalanobis Distance: UnderstandingCluster Shape 1 Measures Distance Using Covariance Unlike Euclidean distance, Mahalanobis distance accounts for the variance and covariance of features. 2 Respects Feature Correlation It inherently considers how features are correlated, making it ideal for non-spherical, elliptical clusters. 3 Perfect for Elliptical Clusters This distance metric is perfectly suited to identify and group data points within elliptically shaped distributions. 4 Whitened Space Equivalence Conceptually, it's equivalent to measuring Euclidean distance after transforming the data into a "whitened" space where features are uncorrelated and have unit variance.
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  Two-Step vs. Unified:A Paradigm Shift The core idea presented in this paper challenges the conventional workflow for clustering incomplete datasets. 1 Instead of: Impute → Cluster The traditional, sequential approach where missing values are filled in first, then clustering is performed. 2 Do This: Cluster + Infer Missing Values (Together) A unified, iterative approach where clustering and the inference of missing values occur simultaneously, enhancing each other. Clustering Improves Imputation Better cluster assignments lead to more accurate estimates for missing data points. Imputed Values Improve Clustering More precise inferred values, in turn, contribute to clearer and more coherent cluster formations.
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  Algorithm Intuition: JointOptimization Loop Initialize Clusters Begin with an initial assignment of data points to clusters or random cluster centers. Assign Points Allocate data points to the nearest cluster centroid using Mahalanobis distance, considering their shape. Estimate Parameters Recalculate cluster centers (means) and covariance matrices based on newly assigned points. Infer Missing Values Update and refine the estimated missing values within each cluster based on current cluster parameters. Repeat Until Stable Continue iterating through assignments, parameter estimation, and inference until cluster assignments and missing values converge. Notes: • This process is fundamentally similar to traditional K-means but incorporates a richer distance metric. • Crucially, missing values are not just ignored or pre-filled but treated as active variables within the optimization.
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  Visual Example: HowIt Looks in 2D To illustrate the effectiveness of this approach, consider a visual example using synthetic data: Synthetic Data with Elliptical Clusters We generate synthetic datasets featuring two distinct, elliptically shaped clusters, mimicking real- world data distributions. Simulated Missingness Approximately 20% of the values are randomly removed from this dataset to simulate realistic incomplete data scenarios. Comparative Analysis The performance of the unified approach using Mahalanobis distance is then compared against traditional methods.
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  Metrics Used for Evaluation AdjustedRand Index (ARI) A measure of the similarity between two data clusterings. It corrects for chance and yields a score between -1 and 1, where 1 indicates perfect agreement. Normalized Mutual Information (NMI) Quantifies the mutual dependence between two clusterings. It measures the amount of information shared between the true labels and the discovered clusters, normalized to be between 0 and 1. These metrics provide robust ways to assess how well the clustering algorithms recover the true underlying structure of the data, especially in the presence of missing values.
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  Experimental Setup Datasets Used 1 SyntheticDatasets Generated with varying complexities, including up to 10 distinct elliptical clusters, allowing for controlled experimentation of algorithm performance. 2 IRIS Dataset A classic benchmark dataset, augmented with artificially introduced missing values to simulate real-world data incompleteness. Methods Compared Impute K-means → (Euclidean) The traditional two-step approach with Euclidean distance. Unified Approach (Euclidean) The iterative clustering and imputation, but using Euclidean distance. Unified Approach (Mahalanobis) The novel iterative method incorporating Mahalanobis distance.