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## Scatter Plot: Projection of Activations on t_G and t_P
### Overview
The image displays two side-by-side scatter plots under the main title "Projection of activations on t_G and t_P". The left plot is titled "Affirmative Statements" and the right plot is titled "Negated Statements". Each plot visualizes data points projected onto a 2D space defined by two axes: `a_ij^T t_G` (x-axis) and `a_ij^T t_P` (y-axis). The data points are colored according to a binary classification: red for "False" and blue for "True".
### Components/Axes
* **Main Title:** "Projection of activations on t_G and t_P"
* **Subplot Titles:**
* Left: "Affirmative Statements"
* Right: "Negated Statements"
* **X-Axis Label (Both Plots):** `a_ij^T t_G`
* **Y-Axis Label (Both Plots):** `a_ij^T t_P`
* **Legend (Located in the bottom-left corner of the "Negated Statements" plot):**
* Red circle: "False"
* Blue circle: "True"
* **Axis Scales:**
* **Left Plot ("Affirmative Statements"):**
* X-axis: Linear scale from approximately 0 to 1.0 x 10^6 (1,000,000). Major ticks at 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 (all multiplied by 1e6).
* Y-axis: Linear scale from approximately -2.75 x 10^6 to -1.00 x 10^6. Major ticks at -2.75, -2.50, -2.25, -2.00, -1.75, -1.50, -1.25, -1.00 (all multiplied by 1e6).
* **Right Plot ("Negated Statements"):**
* X-axis: Linear scale from approximately 0 to 800,000. Major ticks at 0, 200000, 400000, 600000, 800000.
* Y-axis: Linear scale from approximately -2.8 x 10^6 to -1.4 x 10^6. Major ticks at -2.8, -2.6, -2.4, -2.2, -2.0, -1.8, -1.6, -1.4 (all multiplied by 1e6).
### Detailed Analysis
**Left Plot: Affirmative Statements**
* **Trend Verification:** The data shows two distinct clusters with a clear separation.
* **Red Points ("False"):** This cluster is located in the lower-left quadrant of the plot. The points form a dense, roughly elliptical cloud. The trend is that these points have lower values on both the `a_ij^T t_G` and `a_ij^T t_P` axes.
* **Blue Points ("True"):** This cluster is located in the upper-right quadrant. It is also a dense cloud, positioned diagonally opposite the red cluster. The trend is that these points have higher values on both axes compared to the red cluster.
* **Spatial Grounding & Data Points:**
* The "False" (red) cluster spans approximately from `a_ij^T t_G = 0` to `400,000` and from `a_ij^T t_P = -2.75e6` to `-1.75e6`. The center of mass appears near `(200,000, -2.25e6)`.
* The "True" (blue) cluster spans approximately from `a_ij^T t_G = 400,000` to `1,000,000` and from `a_ij^T t_P = -1.75e6` to `-1.00e6`. The center of mass appears near `(700,000, -1.50e6)`.
* There is minimal overlap between the two clusters.
**Right Plot: Negated Statements**
* **Trend Verification:** The data also shows two clusters, but their spatial relationship is different from the left plot.
* **Red Points ("False"):** This cluster is located in the upper-left region of the plot. The points are more dispersed than in the left plot. The trend is that these points have lower `a_ij^T t_G` values but higher (less negative) `a_ij^T t_P` values.
* **Blue Points ("True"):** This cluster is located in the lower-right region. The points are also dispersed. The trend is that these points have higher `a_ij^T t_G` values but lower (more negative) `a_ij^T t_P` values.
* **Spatial Grounding & Data Points:**
* The "False" (red) cluster spans approximately from `a_ij^T t_G = 0` to `400,000` and from `a_ij^T t_P = -2.2e6` to `-1.4e6`. The center of mass appears near `(200,000, -1.8e6)`.
* The "True" (blue) cluster spans approximately from `a_ij^T t_G = 400,000` to `800,000` and from `a_ij^T t_P = -2.8e6` to `-1.8e6`. The center of mass appears near `(600,000, -2.2e6)`.
* There is a region of moderate overlap between the clusters, roughly in the center of the plot around `a_ij^T t_G = 400,000` and `a_ij^T t_P = -2.0e6`.
### Key Observations
1. **Cluster Separation:** In both plots, the "True" and "False" data points form distinct clusters, indicating that the projection onto the `t_G` and `t_P` axes captures a meaningful difference between the two classes.
2. **Inversion of Relationship:** The spatial relationship between the clusters inverts between the two conditions. For "Affirmative Statements," "True" is associated with high values on both axes. For "Negated Statements," "True" is associated with high `t_G` but low `t_P` values.
3. **Scale Difference:** The numerical ranges on the axes differ between the two plots. The "Affirmative Statements" plot uses a larger scale for the x-axis (up to 1e6) compared to the "Negated Statements" plot (up to 800,000). The y-axis ranges are similar in magnitude but shifted.
4. **Dispersion:** The data points in the "Negated Statements" plot appear more scattered and less tightly clustered than those in the "Affirmative Statements" plot.
### Interpretation
This visualization likely comes from an analysis of a neural network's internal activations (`a_ij`) when processing statements. The vectors `t_G` and `t_P` are probably projection directions (e.g., from a technique like PCA or a learned linear probe) chosen to separate the model's processing of "True" vs. "False" information.
* **What the data suggests:** The model's internal representation, when projected onto these specific directions, cleanly distinguishes between true and false statements. Crucially, the *nature* of this distinction changes based on whether the statement is affirmative or negated.
* **How elements relate:** The inversion of the cluster positions suggests that the model uses a different or transformed coding scheme for truth value when negation is involved. The `t_P` direction seems particularly sensitive to this flip, as its relationship with the "True" class reverses.
* **Notable patterns/anomalies:** The clear separation in the "Affirmative" plot is striking. The increased dispersion in the "Negated" plot might indicate that the model's internal processing is more variable or less consistent when handling negated statements. The overlap in the "Negated" plot suggests some statements are harder for the model to classify definitively as true or false in this projected space.
**Language:** All text in the image is in English.
