Introd u ction to statistical seismolog y C ASE STU D IE S IN STATISTIC AL TH IN K IN G J u stin Bois Lect u rer , Caltech
California mo v es and shakes 1 Fa u lt data : USGS Q u aternar y Fa u lt and Fold Database of the United States CASE STUDIES IN STATISTICAL THINKING
The Parkfield region 1 Fa u lt data : USGS Q u aternar y Fa u lt and Fold Database of the United States CASE STUDIES IN STATISTICAL THINKING
The Parkfield region 1 Fa u lt data : USGS Q u aternar y Fa u lt Fa u lt and Fold Database of the United 2 States Earthq u ake data : USGS ANSS Comprehensi v e Earthq u ake Catalog CASE STUDIES IN STATISTICAL THINKING
The Parkfield region 1 Image : Linda Tanner , CC - BY -2.0 CASE STUDIES IN STATISTICAL THINKING
Seismic Japan 1 Data so u rce : USGS ANSS Comprehensi v e Earthq u ake Catalog ( ComCat ) CASE STUDIES IN STATISTICAL THINKING
ECDF of magnit u des , Japan , 1990-1999 1 Data so u rce : USGS ANSS Comprehensi v e Earthq u ake Catalog ( ComCat ) CASE STUDIES IN STATISTICAL THINKING
Location parameters ′ m ≡ m − 5 ∼ Exponential ′ m ≡ m − m ∼ Exponential t CASE STUDIES IN STATISTICAL THINKING
The G u tenberg - Richter La w The magnit u des of earthq u akes in a gi v en region o v er a gi v en time period are E x ponentiall y distrib u ted − m , describes earthq u ake m t One parameter , gi v en b y magnit u des for a region CASE STUDIES IN STATISTICAL THINKING
The b -v al u e b = ( − m ) ⋅ ln 10 m t # Completeness threshold mt = 5 # b-value b = (np.mean(magnitudes) - mt) * np.log(10) print(b) 0.9729214742632566 CASE STUDIES IN STATISTICAL THINKING
ECDF of all magnit u des 1 Data so u rce : USGS ANSS Comprehensi v e Earthq u ake Catalog ( ComCat ) CASE STUDIES IN STATISTICAL THINKING
ECDF of all magnit u des 1 Data so u rce : USGS ANSS Comprehensi v e Earthq u ake Catalog ( ComCat ) CASE STUDIES IN STATISTICAL THINKING
Completeness threshold The magnit u de , m , abo v e w hich all earthq u akes in a region t can be detected CASE STUDIES IN STATISTICAL THINKING
Let ' s practice ! C ASE STU D IE S IN STATISTIC AL TH IN K IN G
Timing of major earthq u akes C ASE STU D IE S IN STATISTIC AL TH IN K IN G J u stin Bois Lect u rer , Caltech
Models for earthq u ake timing E x ponential : Earthq u akes happen like a Poisson process Ga u ssian : Earthq u akes happen w ith a w ell - de � ned period CASE STUDIES IN STATISTICAL THINKING
Stable continental region earthq u akes 1 Data so u rce : USGS Earthq u ake Catalog for Stable Continental Regions CASE STUDIES IN STATISTICAL THINKING
The Nankai Tro u gh CASE STUDIES IN STATISTICAL THINKING
Earthq u akes in the Nankai Tro u gh Date Magnit u de 684-11-24 8.4 887-08-22 8.6 1099-02-16 8.0 1361-07-26 8.4 1498-09-11 8.6 1605-02-03 7.9 1707-10-18 8.6 1854-12-23 8.4 CASE STUDIES IN STATISTICAL THINKING
ECDF of time bet w een Nankai q u akes CASE STUDIES IN STATISTICAL THINKING
Formal ECDFs ECDF ( x ) = fraction of data points ≤ x CASE STUDIES IN STATISTICAL THINKING
Formal ECDFs CASE STUDIES IN STATISTICAL THINKING
Formal ECDFs CASE STUDIES IN STATISTICAL THINKING
# time_gap is an array of interearthquake times _ = plt.plot(*dcst.ecdf(time_gap, formal=True)) _ = plt.xlabel('time between quakes (yr)') _ = plt.ylabel('ECDF') CASE STUDIES IN STATISTICAL THINKING
# Compute the mean time gap mean_time_gap = np.mean(time_gap) # Standard deviation of the time gap std_time_gap = np.std(time_gap) # Generate theoretical Exponential distribution of timings time_gap_exp = np.random.exponential(mean_time_gap, size=100000) # Generate theoretical Normal distribution of timings time_gap_norm = np.random.normal( mean_time_gap, std_time_gap, size=100000 ) # Plot theoretical CDFs _ = plt.plot(*dcst.ecdf(time_gap_exp)) _ = plt.plot(*dcst.ecdf(time_gap_norm)) CASE STUDIES IN STATISTICAL THINKING
Model for Nankai Tro u gh CASE STUDIES IN STATISTICAL THINKING
Let ' s practice ! C ASE STU D IE S IN STATISTIC AL TH IN K IN G
Ho w are the Parkfield interearthq u ake times distrib u ted ? C ASE STU D IE S IN STATISTIC AL TH IN K IN G J u stin Bois Lect u rer , Caltech
The Parkfield Prediction 1 Adapted from Bark u n and Lindh , Science , 229, 619-624, 1985 CASE STUDIES IN STATISTICAL THINKING
H y pothesis test on the Nankai megathr u st earthq u akes H y pothesis : The time bet w een Nankai Tro u gh earthq u akes is Normall y distrib u ted w ith a mean and standard de v iation as calc u lated from the data Test statistic : ?? At least as e x treme as : ?? CASE STUDIES IN STATISTICAL THINKING
The Kolmogoro v- Smirno v statistic CASE STUDIES IN STATISTICAL THINKING
The Kolmogoro v- Smirno v statistic CASE STUDIES IN STATISTICAL THINKING
The Kolmogoro v- Smirno v statistic CASE STUDIES IN STATISTICAL THINKING
The Kolmogoro v- Smirno v statistic CASE STUDIES IN STATISTICAL THINKING
The Kolmogoro v- Smirno v statistic CASE STUDIES IN STATISTICAL THINKING
Kolmogoro v- Smirno v test H y pothesis : The time bet w een Nankai Tro u gh earthq u akes is Normall y distrib u ted w ith a mean and standard de v iation as calc u lated from the data Test statistic : Kolmogoro v- Smirno v statistic At least as e x treme as : ≥ obser v ed K - S statistic CASE STUDIES IN STATISTICAL THINKING
Sim u lating the n u ll h y pothesis Dra w and store lots of ( sa y, 10,000) samples o u t of the theoretical distrib u tion Dra w n samples o u t of the theoretical distrib u tion Comp u te the K - S statistic from the samples CASE STUDIES IN STATISTICAL THINKING
# 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 CASE STUDIES IN STATISTICAL THINKING
Let ' s practice ! C ASE STU D IE S IN STATISTIC AL TH IN K IN G
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