## Chart Type: Multiple Scatter Plots ### Overview The image contains six scatter plots arranged in a 2x3 grid. Each plot shows the fraction of variance explained by principal components (PCs) for different sentence types. The x-axis represents the PC index (from 1 to 10), and the y-axis represents the explained variance. All plots exhibit a similar trend: a rapid decrease in explained variance for the first few PCs, followed by a gradual leveling off. ### Components/Axes * **Title:** Fraction of variance in centered and averaged activations explained by PCs * **X-axis:** PC index (values from 1 to 10) * **Y-axis:** Explained variance (values from 0.0 to approximately 0.5, depending on the plot) * **Plot Titles (Top Row):** * affirmative * affirmative, negated * affirmative, negated, conjunctions * **Plot Titles (Bottom Row):** * affirmative, affirmative German * affirmative, affirmative German, negated, negated German * affirmative, negated, conjunctions, disjunctions ### Detailed Analysis **Plot 1: affirmative** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.48 * PC 2: ~0.16 * PC 3: ~0.13 * PC 4: ~0.10 * PC 6: ~0.03 * PC 8: ~0.00 * PC 10: ~0.00 **Plot 2: affirmative, negated** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.28 * PC 2: ~0.09 * PC 3: ~0.07 * PC 4: ~0.06 * PC 6: ~0.05 * PC 8: ~0.04 * PC 10: ~0.03 **Plot 3: affirmative, negated, conjunctions** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.24 * PC 2: ~0.07 * PC 3: ~0.05 * PC 4: ~0.04 * PC 6: ~0.03 * PC 8: ~0.03 * PC 10: ~0.02 **Plot 4: affirmative, affirmative German** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.46 * PC 2: ~0.13 * PC 3: ~0.10 * PC 4: ~0.09 * PC 6: ~0.04 * PC 8: ~0.03 * PC 10: ~0.02 **Plot 5: affirmative, affirmative German, negated, negated German** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.28 * PC 2: ~0.08 * PC 3: ~0.06 * PC 4: ~0.05 * PC 6: ~0.03 * PC 8: ~0.03 * PC 10: ~0.02 **Plot 6: affirmative, negated, conjunctions, disjunctions** * Trend: Explained variance decreases rapidly from PC 1 to PC 4, then levels off. * Data Points: * PC 1: ~0.24 * PC 2: ~0.07 * PC 3: ~0.06 * PC 4: ~0.04 * PC 6: ~0.03 * PC 8: ~0.03 * PC 10: ~0.02 ### Key Observations * The first principal component (PC 1) consistently explains the largest fraction of variance across all sentence types. * The explained variance decreases sharply for the first few PCs, indicating that these components capture the most significant patterns in the data. * The plots for "affirmative" and "affirmative, affirmative German" start with a higher explained variance compared to the other plots. * The plots with more complex sentence structures (e.g., "affirmative, negated, conjunctions, disjunctions") tend to have lower explained variance for the first PC compared to simpler sentence structures. ### Interpretation The plots suggest that the initial principal components are effective at capturing the primary variance in the activations for all sentence types. The rapid decrease in explained variance indicates that the first few PCs are the most important for representing the data. The differences in initial explained variance between sentence types may reflect the complexity of the underlying linguistic structures and the corresponding neural activations. The "affirmative" and "affirmative, affirmative German" sentence types, having higher initial explained variance, might be considered less complex in terms of the neural representations they elicit. The addition of negation, conjunctions, and disjunctions appears to reduce the fraction of variance explained by the first PC, suggesting that these linguistic elements introduce more complex patterns in the neural activations.

## Scatter Plots: Fraction of Variance Explained by PCs ### Overview The image presents six scatter plots, each displaying the fraction of variance explained by Principal Components (PCs) for different linguistic conditions. The x-axis represents the PC index (ranging from 1 to 10), and the y-axis represents the explained variance (ranging from 0 to approximately 0.45). Each plot corresponds to a specific combination of linguistic features: affirmative, affirmative/negated, affirmative/negated/conjunctions, affirmative/affirmative German/negated, affirmative/affirmative German/negated German, and affirmative/negated/conjunctions/disjunctions. ### Components/Axes * **Title:** "Fraction of variance in centered and averaged activations explained by PCs" (positioned at the top-center) * **X-axis Label:** "PC index" (appears on all plots) * **Y-axis Label:** "Explained variance" (appears on all plots) * **Plots (6 total):** 1. "affirmative" 2. "affirmative, negated" 3. "affirmative, negated, conjunctions" 4. "affirmative, affirmative German, negated" 5. "affirmative, affirmative German, negated, negated German" 6. "affirmative, negated, conjunctions, disjunctions" ### Detailed Analysis Each plot contains approximately 10 data points (scattered dots). The trend in each plot is a general decrease in explained variance as the PC index increases. **Plot 1: "affirmative"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.42 * PC Index 2: Explained Variance ≈ 0.18 * PC Index 3: Explained Variance ≈ 0.12 * PC Index 4-10: Explained Variance ≈ 0.05 - 0.08 **Plot 2: "affirmative, negated"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.31 * PC Index 2: Explained Variance ≈ 0.25 * PC Index 3: Explained Variance ≈ 0.08 * PC Index 4-10: Explained Variance ≈ 0.04 - 0.06 **Plot 3: "affirmative, negated, conjunctions"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.30 * PC Index 2: Explained Variance ≈ 0.14 * PC Index 3-10: Explained Variance ≈ 0.06 - 0.08 **Plot 4: "affirmative, affirmative German, negated"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.43 * PC Index 2: Explained Variance ≈ 0.14 * PC Index 3: Explained Variance ≈ 0.08 * PC Index 4-10: Explained Variance ≈ 0.04 - 0.06 **Plot 5: "affirmative, affirmative German, negated, negated German"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.32 * PC Index 2: Explained Variance ≈ 0.22 * PC Index 3: Explained Variance ≈ 0.07 * PC Index 4-10: Explained Variance ≈ 0.04 - 0.06 **Plot 6: "affirmative, negated, conjunctions, disjunctions"** * The data points generally decrease from left to right. * PC Index 1: Explained Variance ≈ 0.31 * PC Index 2: Explained Variance ≈ 0.15 * PC Index 3-10: Explained Variance ≈ 0.06 - 0.08 ### Key Observations * The first PC (PC Index 1) consistently explains the largest fraction of variance across all conditions, ranging from approximately 0.30 to 0.43. * The explained variance rapidly decreases after the first few PCs, suggesting that a relatively small number of PCs capture most of the variance in the data. * The "affirmative, affirmative German, negated" condition shows the highest explained variance for the first PC. * The plots are visually similar, all exhibiting a decreasing trend. ### Interpretation The plots demonstrate how the fraction of variance explained by Principal Components differs across various linguistic conditions. The consistent decrease in explained variance as the PC index increases indicates that the data is effectively represented by a lower-dimensional space. The differences in explained variance between conditions suggest that the linguistic features (affirmative, negation, conjunctions, disjunctions, German) influence the underlying structure of the data. The higher explained variance for the first PC in most conditions suggests that the first PC captures the most prominent source of variation in the data. The fact that the plots are generally similar suggests that the underlying patterns are consistent across conditions, despite the differences in linguistic features. This could indicate that the core cognitive processes involved in processing these linguistic structures are similar, but that the specific features modulate the strength or nature of these processes.

\n ## Scree Plots: Fraction of Variance Explained by Principal Components (PCs) ### Overview The image displays a 2x3 grid of six scatter plots (scree plots). Each plot shows the fraction of variance in "centered and averaged activations" explained by the first 10 principal components (PCs) for a different dataset or condition. The overall title is "Fraction of variance in centered and averaged activations explained by PCs". All plots share the same structure: the x-axis represents the PC index (1 through 10), and the y-axis represents the "Explained variance". Data points are plotted as blue circles. ### Components/Axes * **Overall Title:** "Fraction of variance in centered and averaged activations explained by PCs" * **X-axis (Common to all plots):** Label: "PC index". Ticks: 2, 4, 6, 8, 10. The axis spans from 1 to 10. * **Y-axis (Common label, varying scales):** Label: "Explained variance". The scale differs for each subplot to accommodate the data range. * **Subplot Titles (Defining the dataset/condition for each plot):** 1. Top-Left: "affirmative" 2. Top-Center: "affirmative, negated" 3. Top-Right: "affirmative, negated, conjunctions" 4. Bottom-Left: "affirmative, affirmative German" 5. Bottom-Center: "affirmative, affirmative German, negated, negated German" 6. Bottom-Right: "affirmative, negated, conjunctions, disjunctions" * **Data Series:** Each plot contains a single data series represented by blue circles. There is no legend, as the title of each subplot defines the series. ### Detailed Analysis **Trend Verification:** In all six plots, the explained variance is highest for PC1 and decreases monotonically for subsequent PCs, forming a characteristic "scree" shape. The drop from PC1 to PC2 is the most significant. **Plot-by-Plot Data Point Extraction (Approximate Values):** 1. **Top-Left: "affirmative"** * Y-axis scale: 0.0 to ~0.5. * PC1: ~0.45 * PC2: ~0.16 * PC3: ~0.13 * PC4: ~0.10 * PC5: ~0.05 * PC6: ~0.03 * PCs 7-10: ~0.00 (near zero) 2. **Top-Center: "affirmative, negated"** * Y-axis scale: 0.0 to ~0.32. * PC1: ~0.31 * PC2: ~0.26 * PC3: ~0.095 * PC4: ~0.075 * PC5: ~0.06 * PC6: ~0.05 * PC7: ~0.045 * PC8: ~0.04 * PC9: ~0.03 * PC10: ~0.02 3. **Top-Right: "affirmative, negated, conjunctions"** * Y-axis scale: 0.0 to ~0.25. * PC1: ~0.25 * PC2: ~0.23 * PC3: ~0.09 * PC4: ~0.07 * PC5: ~0.06 * PC6: ~0.055 * PC7: ~0.05 * PC8: ~0.045 * PC9: ~0.04 * PC10: ~0.035 4. **Bottom-Left: "affirmative, affirmative German"** * Y-axis scale: 0.0 to ~0.45. * PC1: ~0.45 * PC2: ~0.14 * PC3: ~0.11 * PC4: ~0.09 * PC5: ~0.05 * PC6: ~0.035 * PC7: ~0.03 * PC8: ~0.025 * PC9: ~0.02 * PC10: ~0.015 5. **Bottom-Center: "affirmative, affirmative German, negated, negated German"** * Y-axis scale: 0.0 to ~0.3. * PC1: ~0.29 * PC2: ~0.23 * PC3: ~0.085 * PC4: ~0.065 * PC5: ~0.05 * PC6: ~0.045 * PC7: ~0.04 * PC8: ~0.035 * PC9: ~0.03 * PC10: ~0.02 6. **Bottom-Right: "affirmative, negated, conjunctions, disjunctions"** * Y-axis scale: 0.0 to ~0.25. * PC1: ~0.25 * PC2: ~0.23 * PC3: ~0.085 * PC4: ~0.065 * PC5: ~0.055 * PC6: ~0.05 * PC7: ~0.045 * PC8: ~0.04 * PC9: ~0.035 * PC10: ~0.03 ### Key Observations 1. **Dominance of PC1:** The first principal component explains the largest fraction of variance in every case, ranging from ~0.25 to ~0.45. 2. **Impact of Dataset Complexity:** As more conditions are added to the dataset (moving from "affirmative" alone to combinations with negations, conjunctions, etc.), the variance explained by PC1 generally decreases, and the variance becomes more distributed across the first few PCs. For example, PC1 drops from ~0.45 ("affirmative") to ~0.25 ("affirmative, negated, conjunctions"). 3. **Similarity in Structure:** The plots for "affirmative, negated, conjunctions" (top-right) and "affirmative, negated, conjunctions, disjunctions" (bottom-right) are nearly identical, suggesting adding "disjunctions" does not significantly alter the variance structure compared to the three-condition set. 4. **Cross-Lingual Consistency:** The plot for "affirmative, affirmative German" (bottom-left) closely resembles the "affirmative" plot (top-left), indicating that the variance structure is preserved across English and German for the same (affirmative) condition. 5. **Steep Initial Drop:** All plots show a sharp decline in explained variance from PC1 to PC2, followed by a more gradual decay. This is a classic pattern indicating that a few primary components capture most of the signal. ### Interpretation This analysis investigates the underlying structure of neural activation patterns (likely from a language model) under different linguistic conditions. Principal Component Analysis (PCA) is used to find the directions (PCs) of maximum variance in the data. * **What the data suggests:** The high variance explained by PC1 in the simple "affirmative" case suggests a strong, dominant pattern in the activations for affirmative statements. Adding negation ("affirmative, negated") significantly reduces PC1's dominance and increases the variance explained by PC2, indicating that negation introduces a major, distinct source of variation in the activation space. * **Relationship between elements:** The plots demonstrate how the complexity of the linguistic input (affirmation, negation, logical connectives like conjunctions/disjunctions, and cross-lingual data) affects the dimensionality of the representation. More complex or varied inputs lead to a more distributed representation across multiple principal components. * **Notable patterns/anomalies:** The near-identity of the last two plots is a key finding. It implies that, within this analysis, the activation patterns for "conjunctions" and "disjunctions" may be similar or that adding disjunctions does not introduce a new major axis of variation beyond what is already captured by the combination of affirmation, negation, and conjunctions. The strong similarity between the English-only and English-German affirmative plots suggests a language-invariant core representation for this simple condition.

## Scatter Plot Grid: Fraction of Variance in Centered and Averaged Activations Explained by PCs ### Overview The image displays a 2x3 grid of scatter plots comparing the fraction of variance explained by principal components (PCs) across different linguistic conditions. Each subplot visualizes how explained variance decreases with increasing PC index (1–10) for specific combinations of linguistic features (e.g., affirmative/negated terms, German language variants, conjunctions/disjunctions). All plots use blue data points with no explicit legend. ### Components/Axes - **X-axis**: "PC index" (1–10), labeled uniformly across all subplots. - **Y-axis**: "Explained variance" (0–0.4), labeled uniformly across all subplots. - **Subplot Titles**: 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" ### Detailed Analysis 1. **Top-left ("affirmative")**: - Highest explained variance at PC 1 (~0.45). - Sharp decline to near-zero by PC 10. - Data points: (1, ~0.45), (2, ~0.18), (3, ~0.15), (4, ~0.12), (5–10, ~0.02–0.05). 2. **Top-middle ("affirmative, negated")**: - Explained variance starts at ~0.3 at PC 1. - Gradual decline to ~0.05 by PC 10. - Data points: (1, ~0.3), (2, ~0.2), (3, ~0.15), (4, ~0.1), (5–10, ~0.05–0.08). 3. **Top-right ("affirmative, negated, conjunctions")**: - Starts at ~0.25 at PC 1. - Steady decline to ~0.05 by PC 10. - Data points: (1, ~0.25), (2, ~0.2), (3, ~0.15), (4, ~0.1), (5–10, ~0.05–0.07). 4. **Bottom-left ("affirmative, affirmative German")**: - Explained variance peaks at ~0.4 at PC 1. - Rapid drop to ~0.05 by PC 10. - Data points: (1, ~0.4), (2, ~0.15), (3, ~0.1), (4, ~0.08), (5–10, ~0.03–0.06). 5. **Bottom-middle ("affirmative, affirmative German, negated, negated German")**: - Starts at ~0.3 at PC 1. - Gradual decline to ~0.05 by PC 10. - Data points: (1, ~0.3), (2, ~0.2), (3, ~0.15), (4, ~0.1), (5–10, ~0.05–0.08). 6. **Bottom-right ("affirmative, negated, conjunctions, disjunctions")**: - Explained variance begins at ~0.25 at PC 1. - Slow decline to ~0.05 by PC 10. - Data points: (1, ~0.25), (2, ~0.2), (3, ~0.15), (4, ~0.1), (5–10, ~0.05–0.07). ### Key Observations - **Decreasing Trend**: All subplots show a consistent decline in explained variance as PC index increases, indicating diminishing returns in variance explained by higher-order PCs. - **Highest Variance**: The "affirmative" condition (top-left) explains the most variance (~0.45 at PC 1), followed by "affirmative, affirmative German" (~0.4 at PC 1). - **Lowest Variance**: The "affirmative, negated, conjunctions, disjunctions" subplot (bottom-right) explains the least variance (~0.25 at PC 1). - **German Variants**: Subplots involving German language features (bottom row) generally show lower explained variance compared to their non-German counterparts. ### Interpretation The data suggests that simpler linguistic conditions (e.g., "affirmative") capture the largest fraction of variance in the dataset, likely reflecting dominant patterns in the activations. Adding negated terms, conjunctions, or disjunctions reduces explained variance, indicating these features introduce complexity or noise. The inclusion of German-specific variants further reduces explained variance, possibly due to language-specific idiosyncrasies or smaller sample sizes. The consistent decay across PCs across all subplots aligns with the typical behavior of principal component analysis, where early components capture the most significant structure in the data. This pattern underscores the importance of focusing on early PCs for modeling or interpretation tasks.