How are the Parkfield interearthquake times distributed?

Case Studies in Statistical Thinking

Justin Bois

Lecturer, Caltech

The Parkfield Prediction

¹ Adapted from Barkun and Lindh, Science, 229, 619-624, 1985

Hypothesis test on the Nankai megathrust earthquakes

Hypothesis: The time between Nankai Trough earthquakes is Normally distributed with a mean and standard deviation as calculated from the data

Test statistic: ??

At least as extreme as: ??

The Kolmogorov-Smirnov statistic

Kolmogorov-Smirnov test

Hypothesis: The time between Nankai Trough earthquakes is Normally distributed with a mean and standard deviation as calculated from the data

Test statistic: Kolmogorov-Smirnov statistic

At least as extreme as: $\ge$ observed K-S statistic

Simulating the null hypothesis

Draw and store lots of (say, 10,000) samples out of the theoretical distribution

Draw n samples out of the theoretical distribution

Compute the K-S statistic from the samples

# Generate samples from theoretical distribution
x_f = np.random.normal(mean_time_gap, std_time_gap, size=10000)

# Initialize K-S replicates
reps = np.empty(1000)

# Draw replicates
for i in range(1000):
    # Draw samples for comparison
    x_samp = np.random.normal(
          mean_time_gap, std_time_gap, size=len(time_gap)
    )

    # Compute K-S statistic
    reps[i] = ks_stat(x_samp, x_f)

# Compute p-value
p_val = np.sum(reps >= ks_stat(time_gap, x_f)) / 1000

Let's practice!

Case Studies in Statistical Thinking