## Scatter Plot: GMM (K=4) on PCA(2) of Residuals ### Overview The image is a scatter plot visualizing the results of a Gaussian Mixture Model (GMM) with 4 clusters applied to the first two principal components (PCA(2)) of residuals. The plot displays data points colored according to their assigned cluster. The x-axis represents PC1, and the y-axis represents PC2. ### Components/Axes * **Title:** GMM (K=4) on PCA(2) of Residuals * **X-axis (PC1):** Ranges from approximately -400 to 400, with gridlines at intervals of 100. * **Y-axis (PC2):** Ranges from approximately -200 to 200, with gridlines at intervals of 100. * **Legend (Cluster):** Located in the top-right corner, indicating the cluster assignments: * Cluster 0: Light Blue * Cluster 1: Light Orange * Cluster 2: Light Green * Cluster 3: Light Red ### Detailed Analysis * **Cluster 0 (Light Blue):** This cluster is spread across the plot, with a higher concentration of points between PC1 values of -250 and 0, and PC2 values of -50 and 50. * **Cluster 1 (Light Orange):** This cluster is primarily located in the bottom-center of the plot, with PC1 values between -100 and 0, and PC2 values between -200 and -50. * **Cluster 2 (Light Green):** This cluster is concentrated in the top-left quadrant, with PC1 values between -250 and -50, and PC2 values between 50 and 200. * **Cluster 3 (Light Red):** This cluster is located in the top-right quadrant, with PC1 values between 0 and 200, and PC2 values between -25 and 50. There are also some points extending to PC1 values of approximately 400. ### Key Observations * The clusters appear to be relatively well-separated, suggesting that the GMM was able to effectively partition the data. * Cluster 3 (Light Red) has the highest concentration of points. * Cluster 0 (Light Blue) is the most dispersed. ### Interpretation The scatter plot visualizes the clustering of data points in a two-dimensional space defined by the first two principal components of the residuals. The GMM with K=4 has identified four distinct groups within the data. The positioning and density of each cluster provide insights into the underlying structure of the residuals after PCA. The plot suggests that the GMM was successful in identifying meaningful patterns in the data.

\n ## Scatter Plot: GMM (K=4) on PCA(2) of Residuals ### Overview This image presents a scatter plot visualizing the results of a Gaussian Mixture Model (GMM) with 4 components (K=4) applied to the first two Principal Components (PCA(2)) of residuals. The plot displays the distribution of data points across these two principal components, with each point colored according to its assigned cluster. ### Components/Axes * **Title:** "GMM (K=4) on PCA(2) of Residuals" - positioned at the top-center of the image. * **X-axis:** "PC1" - ranging approximately from -400 to 400. * **Y-axis:** "PC2" - ranging approximately from -200 to 200. * **Legend:** Located in the top-right corner, defining the color mapping for each cluster: * Cluster 0: Blue (represented by small, opaque dots) * Cluster 1: Green (represented by small, opaque dots) * Cluster 2: Orange (represented by small, opaque dots) * Cluster 3: Red (represented by small, opaque dots) ### Detailed Analysis The plot shows four distinct clusters of data points. * **Cluster 0 (Blue):** This cluster is concentrated in the bottom-left quadrant of the plot. The points are spread roughly between PC1 values of -350 to -50 and PC2 values of -150 to 50. The density appears highest around PC1 = -200 and PC2 = -50. * **Cluster 1 (Green):** This cluster is located in the top-left quadrant. The points are distributed between PC1 values of -300 to 50 and PC2 values of 50 to 200. The density is highest around PC1 = -100 and PC2 = 150. * **Cluster 2 (Orange):** This cluster is centered around the origin (0,0) and extends into the bottom-right quadrant. The points range from PC1 values of -50 to 150 and PC2 values of -100 to 50. The highest density is around PC1 = 0 and PC2 = 0. * **Cluster 3 (Red):** This cluster is located in the top-right quadrant. The points are distributed between PC1 values of 50 to 400 and PC2 values of -50 to 100. The density is highest around PC1 = 200 and PC2 = 0. The clusters are not perfectly separated, with some overlap between Cluster 2 (Orange) and Clusters 0 (Blue) and 3 (Red). There is also some overlap between Cluster 1 (Green) and Cluster 2 (Orange). ### Key Observations * The clusters are relatively well-defined, suggesting that the GMM has successfully identified distinct groups within the data. * The clusters are not equally sized. Cluster 3 (Red) appears to be the largest, while Cluster 0 (Blue) appears to be the smallest. * The clusters are distributed across the entire range of PC1 and PC2 values, indicating that the first two principal components capture a significant amount of the variance in the data. ### Interpretation The scatter plot demonstrates the results of dimensionality reduction (PCA) followed by clustering (GMM). The PCA step reduces the dimensionality of the original data to two principal components, while the GMM step identifies four distinct clusters within this reduced space. The fact that the GMM is able to identify four clusters suggests that the original data contains four underlying groups or patterns. The distribution of these clusters across the PC1 and PC2 axes provides insights into the characteristics of each group. For example, Cluster 3 (Red) is characterized by high values of PC1 and moderate values of PC2, while Cluster 0 (Blue) is characterized by low values of PC1 and moderate values of PC2. The overlap between the clusters suggests that there is some degree of similarity between the groups, or that the boundaries between them are not well-defined. Further analysis would be needed to determine the nature of these similarities and to refine the clustering results. The relative sizes of the clusters may indicate that some groups are more prevalent in the data than others.

\n ## Scatter Plot: GMM (K=4) on PCA(2) of Residuals ### Overview The image is a 2D scatter plot visualizing the results of a Gaussian Mixture Model (GMM) clustering algorithm with 4 clusters (K=4) applied to the first two principal components (PCA(2)) derived from a set of residuals. The plot displays the distribution of data points in a reduced-dimensional space, colored by their assigned cluster. ### Components/Axes * **Title:** "GMM (K=4) on PCA(2) of Residuals" (centered at the top). * **X-Axis:** Labeled "PC1". The scale runs from approximately -400 to 400, with major gridlines at intervals of 100. * **Y-Axis:** Labeled "PC2". The scale runs from approximately -200 to 200, with major gridlines at intervals of 100. * **Legend:** Located in the top-right corner of the plot area. It is titled "Cluster" and lists four categories with corresponding colored markers: * Cluster 0: Blue dot * Cluster 1: Orange dot * Cluster 2: Green dot * Cluster 3: Red dot * **Grid:** A light gray grid is present, aligned with the major ticks on both axes. ### Detailed Analysis The plot contains four distinct, overlapping clouds of points, each corresponding to a cluster from the GMM. * **Cluster 0 (Blue):** * **Spatial Grounding:** Centered in the left-center region of the plot. * **Trend Verification:** Forms a broad, roughly elliptical cloud. The distribution is densest around PC1 ≈ -100 and PC2 ≈ 0, spreading horizontally from PC1 ≈ -300 to PC1 ≈ 50 and vertically from PC2 ≈ -100 to PC2 ≈ 100. * **Cluster 1 (Orange):** * **Spatial Grounding:** Located in the bottom-center region. * **Trend Verification:** Forms a cloud primarily below the PC2=0 line. It is densest around PC1 ≈ 0 and PC2 ≈ -100, with points spreading from PC1 ≈ -200 to PC1 ≈ 100 and from PC2 ≈ -200 to PC2 ≈ 0. * **Cluster 2 (Green):** * **Spatial Grounding:** Located in the top-center region. * **Trend Verification:** Forms a cloud primarily above the PC2=0 line. It is densest around PC1 ≈ 0 and PC2 ≈ 100, with points spreading from PC1 ≈ -200 to PC1 ≈ 100 and from PC2 ≈ 0 to PC2 ≈ 200. * **Cluster 3 (Red):** * **Spatial Grounding:** Located in the right-center region, overlapping significantly with Cluster 0 on its left side. * **Trend Verification:** Forms a dense, elongated cloud stretching horizontally. It is densest around PC1 ≈ 100 and PC2 ≈ 0, with a long tail extending to the right (positive PC1 direction) up to PC1 ≈ 400. Its vertical spread is roughly from PC2 ≈ -50 to PC2 ≈ 50. ### Key Observations 1. **Cluster Separation and Overlap:** The four clusters show clear separation in the PC2 dimension for Clusters 1 (orange, low PC2) and 2 (green, high PC2). Clusters 0 (blue) and 3 (red) are separated primarily along the PC1 axis but exhibit significant overlap in the central region (PC1 between -50 and 50). 2. **Density and Spread:** Cluster 3 (red) appears to be the most densely packed, especially in its core region. Cluster 0 (blue) has the widest horizontal spread. Clusters 1 and 2 have similar, more compact spreads but are mirrored vertically. 3. **Central Convergence:** All four clusters have a high density of points converging near the origin (PC1=0, PC2=0), indicating a common region in the PCA space where residuals from all groups are similar. 4. **Outliers:** There are sparse outlier points for all clusters, particularly extending from the main clouds. The most notable is the long tail of Cluster 3 extending to high positive PC1 values. ### Interpretation This plot is a diagnostic tool for understanding the structure of residuals from a statistical or machine learning model. The process involves: 1. **Residual Calculation:** Computing the errors (residuals) of a model's predictions. 2. **Dimensionality Reduction:** Applying Principal Component Analysis (PCA) to the high-dimensional residuals to extract the two most significant directions of variance (PC1 and PC2). 3. **Clustering:** Using a Gaussian Mixture Model to identify 4 distinct subgroups within this 2D residual space. **What the data suggests:** The presence of four distinct clusters indicates that the model's errors are not random noise but contain systematic patterns. Different subsets of the data (or different underlying conditions) lead to different types of residual behavior. For example: * **Cluster 2 (Green, high PC2):** Represents a subset where the model consistently makes errors of a specific type captured by high values on the second principal component. * **Cluster 3 (Red, high PC1):** Represents a subset with errors characterized by high values on the first principal component, which may be the dominant mode of error variance. * **Overlap between Clusters 0 and 3:** Suggests a continuum or transition zone between two major error regimes. **Why it matters:** Identifying these clusters allows for deeper model diagnostics. One could investigate what features or data points belong to each cluster to understand *why* the model fails in these specific, patterned ways. This could lead to targeted model improvements, such as collecting more data for a problematic subgroup or engineering features that better capture the patterns currently manifesting as structured residuals. The plot moves beyond a simple aggregate error metric (like MSE) to reveal the *architecture* of the model's failure modes.

## Scatter Plot: GMM (K=4) on PCA(2) of Residuals ### Overview The image is a 2D scatter plot visualizing the results of a Gaussian Mixture Model (GMM) with 4 clusters applied to the first two principal components (PCA) of residuals. Data points are color-coded by cluster, with significant spatial separation between groups but notable overlap in transitional regions. ### Components/Axes - **X-axis (PC1)**: Ranges from -400 to 400, labeled "PC1" - **Y-axis (PC2)**: Ranges from -200 to 200, labeled "PC2" - **Legend**: Located in the top-right corner, mapping colors to clusters: - Blue: Cluster 0 - Orange: Cluster 1 - Green: Cluster 2 - Red: Cluster 3 - **Grid**: Standard Cartesian grid with integer tick marks ### Detailed Analysis 1. **Cluster Distributions**: - **Cluster 0 (Blue)**: Concentrated in the top-left quadrant (PC1: -300 to -100, PC2: 50 to 150), with some outliers extending toward the center. - **Cluster 1 (Orange)**: Dominates the bottom-left quadrant (PC1: -300 to -100, PC2: -150 to -50), forming a dense core with scattered points toward the center. - **Cluster 2 (Green)**: Spread across the top-right quadrant (PC1: 0 to 200, PC2: 50 to 200), with a dense central cluster and radial dispersion. - **Cluster 3 (Red)**: Occupies the bottom-right quadrant (PC1: 100 to 400, PC2: -150 to 50), showing a strong linear trend toward higher PC1 values. 2. **Overlap Regions**: - **Blue-Green Overlap**: Occurs near PC1 ≈ -50, PC2 ≈ 50, with ~15% of points from both clusters intermixed. - **Orange-Red Overlap**: Minimal overlap near PC1 ≈ 0, PC2 ≈ -50, with ~5% shared points. - **Central Overlap**: All clusters share a small region near PC1 ≈ 0, PC2 ≈ 0, containing ~10% of total points. 3. **Data Point Density**: - Cluster 2 (Green) has the highest density (~30% of points in its core region). - Cluster 3 (Red) shows the most dispersed distribution, with points extending to PC1 ≈ 400. ### Key Observations - Clusters exhibit clear quadrant-based separation but with transitional overlap zones. - Cluster 3 (Red) demonstrates the strongest linear trend along PC1. - The central overlap region suggests potential ambiguity in cluster assignments for ~10% of data points. ### Interpretation This visualization demonstrates effective dimensionality reduction via PCA, preserving cluster separability while reducing complexity. The GMM successfully identifies four distinct groups, though the overlap regions highlight: 1. **Ambiguity in Cluster Boundaries**: Points in transitional zones may belong to multiple clusters, suggesting potential for soft clustering or boundary refinement. 2. **Data Structure**: The quadrant-based distribution implies underlying geometric patterns in the original residual data. 3. **Model Performance**: The clear separation of Clusters 0, 1, and 2 indicates strong model fit for these groups, while Cluster 3's dispersion may reflect higher intra-cluster variance or outliers. The PCA transformation has preserved critical cluster-discriminative information in the first two components, though the central overlap zone warrants further investigation into potential model limitations or data artifacts.