AgriMetSoft


Interpreting the P–P Plot with Gumbel Distribution

P-P plot for Gumbel distribution

What is a P–P Plot?

A P–P (Probability–Probability) plot compares the cumulative distribution function (CDF) values of a fitted theoretical distribution (like Gumbel) against the empirical CDF values derived from actual data. It helps evaluate whether the assumed distribution provides a reasonable fit to the observed dataset.

Axes Explanation:

  • X-axis (Empirical CDF): Shows the observed cumulative probabilities from the ordered sample data.
  • Y-axis (Fitted CDF): Represents the cumulative probabilities predicted by the fitted Gumbel distribution.

Visual Interpretation:

In the plot above:

  • Blue Dots: Each point corresponds to one data value, plotted with its empirical and theoretical probabilities.
  • Black Dashed Line (1:1 line): Represents the ideal case where empirical and theoretical CDFs match perfectly.

What the Plot Tells Us:

The P–P plot in this example shows good alignment of points with the 1:1 line, especially in the mid-range. Here's how to interpret it:

  • Points near the line: Indicate that the Gumbel model is consistent with the empirical data.
  • Slight curvature or deviation: At the lower-left or upper-right ends may suggest minor under- or over-estimation in those tails.
  • No major outliers: Suggests stable and valid use of the Gumbel distribution.

Use Cases in Climate and Hydrology:

In extreme value modeling of variables like rainfall, discharge, or temperature, the P–P plot:

  • Validates your choice of distribution.
  • Assists in diagnosing model quality.
  • Supports regulatory, agricultural, or infrastructure planning.

Conclusion:

This P–P plot visually confirms that the Gumbel distribution adequately models the dataset, with cumulative probabilities from both sources closely aligned. For comprehensive fit assessment, this plot should be considered alongside the Q–Q plot, histogram-PDF overlay, and statistical tests like the Kolmogorov–Smirnov test.

Published by AgriMetSoft