Advanced Equipment Statistics Documentation

Overview

This document describes the mathematical calculations and interpretation logic behind the advanced equipment statistics system in the Archery Apprentice application. These statistics provide insights into shooting performance, equipment effectiveness, and form consistency through mathematical analysis of shot coordinate data.

Data Requirements

Minimum Data Thresholds

Basic Statistics: 1+ arrows
Advanced Grouping Analysis: 5+ arrows with coordinates
Fatigue Analysis: 20+ arrows with recent shot data
Consistency Analysis: 5+ completed ends

Coordinate System

Coordinates are normalized to target-face units (-1 to 1 range)
Target center is always (0, 0)
Distance measurements are in target radii (1.0 = full target radius)

Statistical Calculations

1. Basic Shot Grouping

Group Center Calculation

centerX = average(all shot x-coordinates)
centerY = average(all shot y-coordinates)

Average Group Size

distances = sqrt((x - centerX)² + (y - centerY)²) for each shot
averageGroupSize = mean(distances)

Group Tightness (Standard Deviation)

variance = mean((distance - averageGroupSize)² for each distance)
groupTightness = sqrt(variance)

Bias Calculation

horizontalBias = centerX (positive = right bias, negative = left bias)
verticalBias = centerY (positive = up bias, negative = down bias)

2. Advanced Grouping Analysis

Eccentricity Analysis (Covariance Matrix Method)

Covariance Matrix Components:

deltaX = x-coordinates - centerX
deltaY = y-coordinates - centerY
varX = mean(deltaX²)
varY = mean(deltaY²) 
covXY = mean(deltaX * deltaY)

Eigenvalue Calculation:

trace = varX + varY
determinant = varX * varY - covXY²
discriminant = sqrt(trace² - 4 * determinant)
eigenvalue1 = (trace + discriminant) / 2
eigenvalue2 = (trace - discriminant) / 2

Eccentricity:

eccentricity = eigenvalue1 / eigenvalue2 (capped at 10.0)

Interpretation:

1.0 - 1.2: Circular grouping (ideal)
1.2 - 2.0: Slightly elliptical
2.0 - 3.0: Moderately elliptical
3.0+: Highly directional (systematic issue)

Primary Axis Calculation

primaryAxis = atan2(eigenvalue1 - varX, covXY) * 180/π

Angle of the major axis of the elliptical grouping pattern.

Radial Consistency

radialDistances = sqrt(x² + y²) for each shot from actual target center
mean = average(radialDistances)
stdDev = sqrt(mean((distance - mean)² for each distance))
radialConsistency = 1 / (stdDev / mean) (capped at 10.0)

Purpose: Measures how consistent the distance from target center is across shots, independent of grouping pattern.

3. Fatigue Analysis

Shot Segmentation

recentShotCount = shotCount * 0.2 (minimum 5)
recentShots = last 20% of shots in chronological order
earlierShots = remaining 80% of shots

Performance Drop Calculation

recentAverageScore = mean(recent shot scores)
earlierAverageScore = mean(earlier shot scores)
performanceDrop = earlierAverageScore - recentAverageScore

Grouping Deterioration

recentGroupTightness = stdDev(recent shot distances from center)
earlierGroupTightness = stdDev(earlier shot distances from center)
groupingDeterioration = recentGroupTightness - earlierGroupTightness

Fatigue Score Calculation

scoreFactor = (performanceDrop / 2.0).clamp(0.0, 1.0)
groupingFactor = (groupingDeterioration / 0.2).clamp(0.0, 1.0)
fatigueScore = (scoreFactor + groupingFactor) / 2.0

Interpretation:

0.0 - 0.2: No fatigue detected
0.2 - 0.4: Mild fatigue
0.4 - 0.6: Moderate fatigue
0.6 - 0.8: Significant fatigue
0.8 - 1.0: High fatigue

4. Consistency Analysis

Score Variation

endScores = total score for each completed end
mean = average(endScores)
variance = mean((score - mean)² for each end)
scoreVariation = sqrt(variance)

Consistency Percentage

consistencyPercentage = (1 - (scoreVariation / mean)) * 100

Clamped to 0-100% range.

Trend Analysis (Linear Regression)

x = [0, 1, 2, ..., n-1] (end sequence numbers)
y = end average scores
n = number of ends
sumX = sum(x)
sumY = sum(y) 
sumXY = sum(x[i] * y[i])
sumXX = sum(x[i]²)

slope = (n * sumXY - sumX * sumY) / (n * sumXX - sumX²)

Trend Interpretation:

slope > 0.5: Improving significantly
slope > 0.1: Slightly improving
slope < -0.5: Declining significantly
slope < -0.1: Slightly declining
else: Stable

Confidence Assessment

Data Confidence Levels

confidence = switch(totalArrows) {
    0: NONE
    1-19: LOW
    20-49: MEDIUM
    50-99: HIGH
    100+: VERY_HIGH
}

Statistical Reliability

Low Confidence: Statistics may fluctuate significantly with new data
Medium Confidence: Statistics becoming stable, trends emerging
High Confidence: Reliable patterns, meaningful for equipment decisions
Very High Confidence: Highly stable statistics, suitable for detailed analysis

Practical Applications

Equipment Tuning Indicators

High Eccentricity (>2.5)

Check arrow spine compatibility
Verify rest alignment and centershot
Inspect sight mounting stability
Review bow tuning parameters

Consistent Bias Pattern

Horizontal bias: Windage adjustment needed
Vertical bias: Elevation adjustment needed
Combined bias: Anchor point consistency issue

Poor Radial Consistency

Arrow spine mismatch
Inconsistent release technique
Bow balance issues
Environmental factors (wind patterns)

Form Analysis Insights

Circular Grouping + Good Radial Consistency

Excellent shooting form
Proper equipment setup
Consistent execution

Directional Grouping + Poor Radial Consistency

Form inconsistencies (anchor point, release)
Equipment problems (rest, spine)
Systematic shooting errors

High Fatigue Detection

Reduce practice session length
Focus on quality over quantity
Consider physical conditioning

Implementation Notes

Error Handling

All calculations include bounds checking and null safety
Insufficient data returns null rather than invalid statistics
Division by zero protection throughout
Outlier detection could be added in future versions

Performance Considerations

Calculations are O(n) where n = number of shots
Eigenvalue calculation is O(1) for 2x2 matrices
Large datasets (>1000 shots) may need optimization
Consider caching for frequently accessed statistics

Future Enhancements

Outlier detection and removal
Moving window analysis for trend detection
Environmental factor correlation
Machine learning performance prediction
Cross-equipment comparative analysis

Mathematical References

Covariance Matrix Analysis: Standard multivariate statistics
Eigenvalue Decomposition: Linear algebra for pattern detection
Linear Regression: Basic trend analysis
Statistical Moments: Mean, variance, standard deviation calculations

Testing and Validation

Unit Test Cases

Known circular patterns should yield eccentricity ≈ 1.0
Known linear patterns should yield high eccentricity
Performance decline patterns should trigger fatigue detection
Consistent end scores should yield high consistency percentage

Integration Testing

Verify calculations with real shooting data
Compare results with manual calculations
Test edge cases (single shot, identical coordinates)
Validate performance with large datasets

This documentation covers Phase 2 implementation (August 2025). Future phases may expand these calculations with additional statistical methods and machine learning integration.

Target Face Visualization - Visual representation of grouping statistics
Project Overview
Test Coverage

Archery Apprentice Documentation

Explorer

Advanced Equipment Statistics