Exponential Decay of Similarity Scores

Animation Specification: Decaying Similarity Score Visualization

Animation Description and Purpose

• Purpose: Illustrate the decaying similarity score over document rank using a continuous 2D line graph over the interval $1 \leq x \leq 10$.
• Key Message: Visualize how similarity scores decrease as document rank increases, emphasizing exponential decay.

Mathematical Elements

• Function: $f(x) = e^{-0.5x}$
• Domain: $x$ ranges from 1 to 10 (inclusive).
• Axes:
  • X-axis: Document rank ($x$), labeled from 1 to 10.
  • Y-axis: Similarity score ($f(x)$), labeled from 0 to 1 (since $e^{-0.5 \cdot 1} \approx 0.6065$ and $e^{-0.5 \cdot 10} \approx 0.0067$).

Visual Elements

• Graph:
  • A smooth, continuous line graph of $f(x) = e^{-0.5x}$ plotted from $x = 1$ to $x = 10$.
  • Line color: Bright blue (#3498db) with a thickness of 3 pixels.
  • The line should have a slight glow effect to emphasize its decay.
• Axes:
  • X-axis: Horizontal line with ticks at integer values (1, 2, ..., 10). Label: "Document Rank (x)".
  • Y-axis: Vertical line with ticks at 0.1 intervals (0.0, 0.1, ..., 1.0). Label: "Similarity Score".
  • Axis color: Light gray (#95a5a6) with a thickness of 2 pixels.
  • Tick labels: Black (#000000) with a font size of 20.
• Background:
  • Plain white background for the graph area.
  • Optional: Light grid lines (very faint, #ecf0f1) for better readability.
• Title:
  • Text: "Decaying Similarity Score Over Document Rank" at the top of the graph.
  • Font: Bold, size 24, color: Dark gray (#2c3e50).
  • Background: Opaque white rectangle behind the text for readability.

Animation Timing and Transitions

• Total Duration: 15 seconds.
• Sequence:
  • Axes fade in (0 - 2 seconds).
  • Curve draws left-to-right (2 - 8 seconds).
  • Brief highlight of "High similarity" vs "Low similarity" (8 - 12 seconds).
  • Hold final frame (12 - 15 seconds).

Camera Angles and Perspectives

• Initial View: Centered on the graph with equal scaling for x and y axes to avoid distortion.
• Zoom: Slight zoom-in during the graph plotting phase (2 - 8 seconds) to focus on the curve, followed by a zoom-out during the conclusion.
• Perspective: 2D orthogonal view; no 3D effects or rotations.

Additional Details

• Text Readability: All text (title, axis labels, annotations) must have an opaque background to ensure readability against the graph.
• Smoothness: All animations (line drawing, transitions) should be smooth and fluid, with no abrupt changes.
• Focus: The primary focus is the decaying curve; avoid cluttering the visualization with unnecessary elements.

Default Assumptions (if no clarification is provided)

• The function is $f(x) = e^{-0.5x}$.
• The graph is a simple 2D line plot with no additional decorations unless specified.
• The animation duration is kept short (15 seconds) to maintain engagement while clearly illustrating the concept.