AgriMetSoft


Understanding the Gumbel PDF vs Empirical Histogram Chart

Gumbel PDF vs Histogram

Interpreting the Gumbel Distribution Fit

This chart displays the Gumbel probability density function (PDF) fitted to a dataset of annual maximum values, overlaid on an empirical histogram. The goal is to evaluate how well the theoretical distribution matches the observed behavior of extreme values (e.g., rainfall, temperature).

X-axis: Represents annual maximum values.
Y-axis: Represents the normalized frequency (density).

What the Histogram Tells Us

The light blue bars show how often annual maximum values fall into each range. Key features:

  • The distribution peaks between 26–30, indicating that most extreme events are centered here.
  • The right-hand tail (values above 40) includes some low bars-these represent rare extremes.
  • A spike near 50 may suggest an anomaly, such as an unusually high event or an outlier.

What the Green Gumbel Curve Means

The smooth green line is the theoretical Gumbel PDF fitted to the data:

  • The peak (mode) of the curve aligns with the histogram's highest bars - this supports a good fit.
  • The curve is skewed right - a characteristic shape of Gumbel for maxima modeling.
  • Tail behavior shows decreasing density for higher values, consistent with how extreme events behave in nature.

Interpretation Tips

  • Goodness-of-fit: When the green curve closely follows the shape of the histogram, the distribution fit is good.
  • Return period estimation: From the fitted curve, analysts can calculate exceedance probabilities and estimate how often rare values like 45 or 50 might occur.
  • Outlier detection: Large deviation between bars and the curve may signal anomalies or suggest the need for data cleaning or using another distribution.

In summary, this chart enables analysts to visually verify that the Gumbel distribution is appropriate for modeling observed annual maxima and assess the fit's quality. It supports applications like hazard mapping, flood risk estimation, or climate resilience planning.