Statistics

Statistics for Geoscience Research

When to Use This Skill

Invoke when:

• Calculating z-scores or anomaly detection
• Quantifying dating uncertainty
• Analyzing spatial clustering of events
• Computing recurrence intervals
• Performing Monte Carlo simulations
• Assessing statistical significance

Core Methods

1. Z-Score Analysis

Standard approach for anomaly detection in geochemical time series:

z = (x - μ) / σ

where:
  x = observed value
  μ = mean of reference period
  σ = standard deviation of reference period

Interpretation thresholds:

Best practices:

• Use rolling window for mean/std (50-100 data points)
• Exclude the test point from reference calculation
• Report both positive and negative z-scores
• Consider autocorrelation in time series

2. Dating Uncertainty

U-Th dating (typical for speleothems):

• Precision: ±50-100 years for Holocene
• Report as: 1285 ± 85 yr (U-Th: 1237-1322 CE)

Radiocarbon dating:

• Raw dates in 14C years BP
• Calibrate using IntCal20/SHCal20
• Report 2σ range: 830 ± 50 14C yr BP (cal. 720-940 CE)

Monte Carlo uncertainty propagation:

# Pseudocode for age-depth model uncertainty
for i in range(10000):
    ages_sample = sample_from_age_distributions(age_points)
    model = fit_age_depth_model(ages_sample, depths)
    interpolated_ages[i] = model.interpolate(target_depth)

uncertainty = np.percentile(interpolated_ages, [2.5, 97.5])

3. Recurrence Interval Statistics

For earthquake recurrence from event dates:

intervals = [t[i+1] - t[i] for i in range(len(events)-1)]
mean_recurrence = mean(intervals)
std_recurrence = std(intervals)
COV = std / mean  # Coefficient of variation

Interpretation:

• COV < 0.5: Quasi-periodic behavior
• COV ~ 1.0: Random (Poisson) process
• COV > 1.0: Clustered behavior

MCP Tool: calc_recurrence computes these statistics

4. Spatial Clustering Analysis

Orphan earthquake detection: Identify seismicity far from mapped faults using nearest-neighbor distance.

# For each earthquake, calculate distance to nearest fault
distances = [min_distance_to_fault(eq) for eq in earthquakes]

# Earthquakes > 50 km from any fault are "orphans"
orphans = [eq for eq, d in zip(earthquakes, distances) if d > 50]

Kernel density estimation:

• Use for seismicity hotspot mapping
• Bandwidth selection: Scott's rule or cross-validation

5. Significance Testing

When comparing distributions:

• Kolmogorov-Smirnov test: Compare CDFs
• Mann-Whitney U: Non-parametric comparison of medians
• Permutation tests: Distribution-free significance

Correlation analysis:

• Pearson r: Linear relationship (assumes normality)
• Spearman ρ: Monotonic relationship (rank-based)
• Cross-correlation: Time-lagged relationships

P-value interpretation:

Warning: With small samples (n < 20), be cautious about p-values.

6. Time Series Analysis

Detrending:

1. Remove long-term trend (linear, polynomial, or LOESS)
2. Calculate residuals for anomaly detection
3. Avoid over-fitting the trend

Autocorrelation:

• Check lag-1 autocorrelation before independence tests
• Effective sample size: n_eff = n × (1 - ρ) / (1 + ρ)

Change point detection:

• PELT algorithm for multiple change points
• Bayesian change point detection for uncertainty

Available MCP Calculation Tools

Common Pitfalls

1. Multiple comparisons: Testing many anomalies inflates false positive rate. Apply Bonferroni correction or FDR.

2. Small sample size: With n < 10 events, statistical power is low. Report effect sizes, not just p-values.

3. Circular logic: Don't use detected anomalies to define the reference period.

4. Ignoring uncertainty: Always propagate dating errors through calculations.

5. Cherry-picking: Report ALL tests performed, not just significant ones.

Reporting Standards

Always report:

• Sample size (n)
• Central tendency AND spread (mean ± std, or median + IQR)
• Uncertainty ranges (95% CI or 2σ)
• Effect size, not just significance
• Method used for calculation