Circular Statistics: The Hidden Superpower

🤯 What Statistics Class Never Taught You

Standard statistics breaks horribly on circular data. But vector embedding makes circular statistics trivially easy—and incredibly powerful!

📊 The Problem with Regular Stats

# Wind directions (degrees)
angles = [350, 10, 5, 355]

# Regular mean = DISASTER
mean = sum(angles) / len(angles)  # = 180°
# 180° is the OPPOSITE direction! 🤦‍♂️

# Regular standard deviation = MEANINGLESS
std = numpy.std(angles)  # = 187°
# Standard deviation larger than the range!

✨ Vector Statistics = Magic

# Convert to vectors
vectors = [angle_to_vector(θ) for θ in angles]

# Circular mean = WORKS!
mean_vector = np.mean(vectors, axis=0)
mean_angle = vector_to_angle(mean_vector)  # ≈ 0°
# Correct direction! 🎯

# Circular standard deviation
R = |mean_vector|  # Mean vector length
circular_std = sqrt(-2 * log(R))  # ≈ 22°
# Sensible measure of spread!

🧠 The Key Insight: Mean Vector Length

💡 The Beautiful Truth

Tight cluster: Mean vector is long (|R| → 1)

Spread out: Mean vector is short (|R| → 0)

Uniform circle: Mean vector ≈ 0

The length of the mean vector automatically captures how 'concentrated' your circular data is!

🔬 Advanced Circular Stats That Actually Work

# Circular correlation between two angular variables
def circular_correlation(angles1, angles2):
    v1 = [angle_to_vector(θ) for θ in angles1]
    v2 = [angle_to_vector(θ) for θ in angles2]
    # Standard correlation on the vector components
    return pearsonr(v1, v2)

# Circular regression (yes, this exists!)
def circular_regression(x, circular_y):
    vectors_y = [angle_to_vector(θ) for θ in circular_y]
    # Regress each component separately
    cos_model = LinearRegression().fit(x, [v[0] for v in vectors_y])
    sin_model = LinearRegression().fit(x, [v[1] for v in vectors_y])
    return cos_model, sin_model

# Watson-Williams test (circular ANOVA)
# Test if multiple groups have the same mean direction
def watson_williams_test(group1_angles, group2_angles):
    # Convert to vectors, compute mean directions, test significance
    # (This is the circular equivalent of a t-test!)

🌟 Real-World Applications

🧬 Biology

Animal migration: Test if birds change direction by season

Circadian rhythms: Correlate activity with time of day

🌊 Earth Sciences

Wind patterns: Seasonal wind direction changes

Ocean currents: Correlation with tidal cycles

🎵 Signal Processing

Phase analysis: Correlation between signal phases

Circular convolution: Periodic signal filtering