Interpreting Gumbel Distribution Text Results

Raw Text Output

gumbel distribution parameters fitted to annual maxima: {distribution: gumbel, params: (loc=27.9077, scale=5.5744)}
gumbel distribution parameters fitted to annual minima: {distribution: gumbel, params: (loc=0.0001, scale=0.0001)}
Number of days exceeding 3.0: 2266
Number of days below 33.0: 7830
Number of days between 3.0 and 33.0: 2252
Number of days exceeding the 95.0th percentile: 393
Number of days below the 70.0th percentile: 5491
Probability of exceeding 10.0: 1.0000
Return period for exceeding 10.0: 1.0000
Probability of exceeding 20.0: 0.9839
Return period for exceeding 20.0: 1.0163
Probability of exceeding 30.0: 0.4969
Return period for exceeding 30.0: 2.0123
Probability of exceeding 40.0: 0.1080
Return period for exceeding 40.0: 9.2614

Annual maxima summary (from fitted data):
  - Sample size: 30 years
  - Mean: 31.28
  - Standard deviation: 7.55
  - Minimum: 21.43
  - Maximum: 51.64
  - Range: 30.22
  - Skewness: 1.04
  - Kurtosis: 0.67
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  - 5th percentile value: 22.84
  - 50th percentile (median): 30.54
  - 95th percentile value: 47.71

Goodness-of-Fit Summary (Gumbel Distribution):
  - Kolmogorov-Smirnov test: D = 0.0872, p = 0.9614
  - Anderson-Darling statistic: 1.9515
  - Shapiro-Wilk test: W = 0.8980, p = 0.0075

📊 Interpretation:
- KS test p-value (0.9614):
  Very high p-value ⇒ the null hypothesis (data follows the gumbel distribution) is strongly supported.
  There's no evidence to reject the fit. Your data fits the gumbel distribution very well.

- Shapiro-Wilk test p-value (0.0075):
  Very low p-value ⇒ significant deviation from the gumbel distribution's assumptions.

- Distribution Notes:
  Gumbel: Skewed distribution for modeling maxima or minima in extreme value theory.

Understanding Output from Climate Extremes Analyzer

Distribution Fitting: The tool fits the Gumbel distribution to both annual maxima and minima. The loc and scale values indicate the center and spread of the fitted distribution.

Threshold-Based Statistics: This section quantifies the frequency of values falling above, below, or between user-defined thresholds and statistical percentiles. These metrics help to determine the behavior and rarity of climate events across the full dataset.

• 2266 days exceeded the value 3.0 - suggesting this value is quite common across the observation period.
• 7830 days were below the value 33.0 - indicating that values less than 33.0 dominate the dataset.
• 2252 days fell between 3.0 and 33.0 - this range likely represents the core of the distribution.
• 393 days exceeded the 95th percentile - marking them as statistically rare or extreme high events.
• 5491 days fell below the 70th percentile - capturing the lower-to-middle range of values.

Probability and Return Periods: These values estimate the likelihood of exceeding specific thresholds and how frequently such events might occur in time. This is useful for risk assessment and long-term planning.

• Threshold = 10.0: Probability = 1.0000 => Expected every year (Return Period = 1 year)
• Threshold = 20.0: Probability = 0.9839 => Exceedance is almost certain, roughly every 1.02 years
• Threshold = 30.0: Probability = 0.4969 => Nearly 50/50 chance of being exceeded annually (Return Period ~ 2 years)
• Threshold = 40.0: Probability = 0.1080 => Less likely to be exceeded, about once every 9.26 years

Annual Maxima Summary: This section characterizes the distribution of the annual maxima - one extreme value per year - and provides statistical moments and percentiles. It helps describe the central tendency, variability, and shape of the extreme data distribution.

• Sample Size: 30 years of annual maximum values
• Mean: 31.28 - central tendency of extremes
• Standard Deviation: 7.55 - spread or variability among yearly peaks
• Minimum: 21.43 | Maximum: 51.64 - the full observed range of annual maxima
• Range: 30.22 - max minus min; total variability across years
• Skewness: 1.04 - indicates a right-skewed distribution; extreme high values are more probable than low ones
• Kurtosis: 0.67 - slight departure from normality; moderately heavy tails
• 5th Percentile: 22.84 - only 5% of annual maxima fall below this value
• 50th Percentile (Median): 30.54 - half the values are below and half above
• 95th Percentile: 47.71 - only 5% of annual maxima exceed this threshold

Goodness-of-Fit Test Results: These tests evaluate how well the fitted Gumbel distribution models the observed data. Each test offers different insights:

• Kolmogorov-Smirnov Test: D = 0.0872, p = 0.9614 => The high p-value strongly supports the null hypothesis that the data fits the Gumbel distribution. This suggests excellent agreement.
• Anderson-Darling Statistic: 1.9515 => A moderate statistic value; when referenced against critical thresholds, it supports the fit but may flag tail deviation (context-dependent).
• Shapiro-Wilk Test: W = 0.8980, p = 0.0075 => Low p-value indicates non-normality. This is expected because the Gumbel distribution is inherently non-normal - this result does not contradict a good Gumbel fit.

Conclusion: This statistical summary provides a multi-angle view of the dataset's extremes - showing frequency of threshold exceedance, likelihood of rare events, annual trends, and how well a theoretical distribution (like Gumbel) models the data. These insights are crucial for informed decision-making in hydrology, engineering, agriculture, and climate risk management.