\n ## Scatter Plot Grid: Fraction of Variance Explained by Principal Components (PCs) ### Overview The image displays a 2x3 grid of six scatter plots. The collective title is "Fraction of variance in centered and averaged activations explained by PCs". Each subplot shows the explained variance (y-axis) for the first 10 principal components (x-axis) for different combinations of linguistic conditions. The plots share a common visual style: blue circular data points on a white background with gray grid lines. ### Components/Axes * **Main Title:** "Fraction of variance in centered and averaged activations explained by PCs" * **Common Y-axis Label (Left side of grid):** "Explained variance" * **Common X-axis Label (Bottom of grid):** "PC index" * **Subplot Titles (Top of each plot):** 1. Top-left: "affirmative" 2. Top-middle: "affirmative, negated" 3. Top-right: "affirmative, negated, conjunctions" 4. Bottom-left: "affirmative, affirmative German" 5. Bottom-middle: "affirmative, affirmative German, negated, negated German" 6. Bottom-right: "affirmative, negated, conjunctions, disjunctions" * **Axes Scales:** * **X-axis (PC index):** Linear scale from 1 to 10, with major ticks at 2, 4, 6, 8, 10. * **Y-axis (Explained variance):** Linear scale. The range varies by subplot: * "affirmative": 0.0 to 0.6 * All other subplots: 0.0 to 0.3 or 0.0 to 0.4 (see detailed analysis). ### Detailed Analysis **Trend Verification:** In all six subplots, the data series follows the same fundamental trend: a steep, monotonic decrease in explained variance from PC1 to PC2, followed by a more gradual, asymptotic decline towards zero by PC10. This is the classic "scree plot" pattern expected from PCA. **Subplot 1: "affirmative" (Top-left)** * **Y-axis Range:** 0.0 to 0.6. * **Approximate Data Points:** * PC1: ~0.60 * PC2: ~0.15 * PC3: ~0.11 * PC4: ~0.07 * PC5: ~0.05 * PC6: ~0.03 * PC7: ~0.02 * PC8: ~0.01 * PC9: ~0.01 * PC10: ~0.01 **Subplot 2: "affirmative, negated" (Top-middle)** * **Y-axis Range:** 0.0 to 0.35 (approx). * **Approximate Data Points:** * PC1: ~0.34 * PC2: ~0.30 * PC3: ~0.09 * PC4: ~0.07 * PC5: ~0.06 * PC6: ~0.04 * PC7: ~0.04 * PC8: ~0.03 * PC9: ~0.02 * PC10: ~0.02 **Subplot 3: "affirmative, negated, conjunctions" (Top-right)** * **Y-axis Range:** 0.0 to 0.35 (approx). * **Approximate Data Points:** * PC1: ~0.34 * PC2: ~0.25 * PC3: ~0.08 * PC4: ~0.07 * PC5: ~0.06 * PC6: ~0.05 * PC7: ~0.04 * PC8: ~0.04 * PC9: ~0.03 * PC10: ~0.03 **Subplot 4: "affirmative, affirmative German" (Bottom-left)** * **Y-axis Range:** 0.0 to 0.5 (approx). * **Approximate Data Points:** * PC1: ~0.50 * PC2: ~0.13 * PC3: ~0.10 * PC4: ~0.07 * PC5: ~0.05 * PC6: ~0.03 * PC7: ~0.03 * PC8: ~0.02 * PC9: ~0.02 * PC10: ~0.02 **Subplot 5: "affirmative, affirmative German, negated, negated German" (Bottom-middle)** * **Y-axis Range:** 0.0 to 0.3. * **Approximate Data Points:** * PC1: ~0.29 * PC2: ~0.28 * PC3: ~0.09 * PC4: ~0.06 * PC5: ~0.05 * PC6: ~0.04 * PC7: ~0.03 * PC8: ~0.03 * PC9: ~0.02 * PC10: ~0.02 **Subplot 6: "affirmative, negated, conjunctions, disjunctions" (Bottom-right)** * **Y-axis Range:** 0.0 to 0.35 (approx). * **Approximate Data Points:** * PC1: ~0.33 * PC2: ~0.24 * PC3: ~0.08 * PC4: ~0.07 * PC5: ~0.05 * PC6: ~0.05 * PC7: ~0.04 * PC8: ~0.04 * PC9: ~0.03 * PC10: ~0.03 ### Key Observations 1. **Dominance of PC1:** The first principal component (PC1) consistently explains the largest fraction of variance in every condition, ranging from ~0.29 to ~0.60. 2. **Impact of Condition Complexity:** Adding more linguistic conditions (negation, conjunctions, disjunctions, translations) generally reduces the variance explained by PC1. The "affirmative" only condition has the highest PC1 value (~0.60), while the most complex condition (bottom-middle) has the lowest (~0.29). 3. **Two-Component Structure:** In several plots ("affirmative, negated"; "affirmative, affirmative German, negated, negated German"), PC1 and PC2 explain nearly equal, substantial portions of variance, suggesting a strong two-dimensional structure in the underlying data for those conditions. 4. **Rapid Drop-off:** After the first 2-3 components, the explained variance per component becomes very small (<0.10) and decays slowly, indicating that most meaningful variance is captured by the top few PCs. ### Interpretation This figure presents a **scree analysis** of principal components applied to neural activation patterns under different linguistic manipulations. The data suggests: * **Core Semantic Dimension:** The high variance explained by PC1, especially in the simple "affirmative" condition, likely corresponds to a primary, dominant axis of meaning or representation in the model's activations (e.g., a general "semantic strength" or "activation magnitude" dimension). * **Effect of Linguistic Operations:** Introducing negation ("affirmative, negated") dramatically splits the variance between PC1 and PC2. This implies that negation creates a second major, orthogonal axis of variation in the activation space, possibly representing a "truth value" or "polarity" dimension. * **Cross-Linguistic Stability:** The pattern for "affirmative, affirmative German" closely mirrors the English-only "affirmative" plot, suggesting the core representational structure is stable across these two languages for affirmative statements. * **Increased Dimensionality with Complexity:** As more logical operations (conjunctions, disjunctions) and language variants are combined, the variance becomes slightly more distributed across the first few components, but the overall scree shape remains. This indicates that while the representational space becomes more nuanced, it is still dominated by a small number of principal directions. In essence, the visualization demonstrates how the intrinsic dimensionality of a model's semantic representation, as captured by PCA, expands and reorganizes when processing increasingly complex linguistic constructs.