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ECDF vs. Theoretical Distribution ​

The Empirical Cumulative Distribution Function (ECDF) is a statistical tool used to estimate the cumulative probability distribution of an observed dataset. It provides a stepwise function that indicates the proportion of data points less than or equal to a specific value.

Key Characteristics ​

  • The ECDF explicitly represents individual data points, distinguishing it from histograms, which aggregate data into bins.
  • It is a non-decreasing function constrained within the range [0,1].
  • It is particularly useful for examining distributional characteristics in small-to-medium-sized datasets.

Mathematical Definition ​

Given a dataset of n observations, the ECDF at a given point x is defined as:

ECDF(x)=Number of observations≤xn

Each step in the ECDF plot corresponds to a unique data point within the dataset.

Comparison with a Theoretical Distribution ​

To compare the ECDF of an empirical dataset with the cumulative distribution function (CDF) of a fitted theoretical distribution, the method .plot_ecdf_distribution() is utilized. The parameters for this method are detailed below.

Parameters ​

General Parameters ​

  • id_distribution (str): Identifier of the theoretical distribution against which the empirical data is compared. A full list of available distributions can be found in the distributions documentation.
  • plot_title (str, optional): Title of the plot. Default is "ECDF".
  • plot_xaxis_title (str, optional): Label for the x-axis. Default is "Domain".
  • plot_yaxis_title (str, optional): Label for the y-axis. Default is "Cumulative Distribution Function".
  • plot_xaxis_min_offset (float, optional): Offset from the minimum domain value on the x-axis. Default is 0.3.
  • plot_xaxis_max_offset (float, optional): Offset from the maximum domain value on the x-axis. Default is 0.3.
  • plot_legend_title (str | None, optional): Title for the legend. If set to None, no title is displayed.
  • plot_height (int, optional): Height of the plot in pixels. Default is 400.
  • plot_width (int, optional): Width of the plot in pixels. Default is 600.

Empirical Distribution Customization ​

  • plot_empirical_line_color (str, optional): Color of the ECDF line in RGBA format. Default is "rgba(128,128,128,1)".
  • plot_empirical_line_width (int, optional): Line width of the ECDF. Default is 4.
  • plot_empirical_line_name (str, optional): Legend label for the ECDF. Default is "Empirical Distribution".
  • plot_empirical_bar_color (str, optional): Color of the ECDF bars in RGBA format. Default is "rgba(128,128,128,1)".

Theoretical Distribution Customization ​

  • plot_distribution_line_color (str, optional): Color of the theoretical distribution line in RGBA format. Default is "rgba(255,0,0,1)".
  • plot_distribution_line_width (int, optional): Line width of the theoretical distribution. Default is 2.

Rendering Options ​

  • plotly_plot_renderer ("png" | "jpeg" | "svg" | None, optional): Export format when using Plotly. If None, Plotly's default renderer is used.
  • plot_engine ("plotly" | "matplotlib", optional): Specifies the plotting library to use. The default is "plotly".

Default Usage ​

To generate a basic ECDF plot comparing an empirical dataset to a theoretical distribution, use the following implementation:

python
phi.plot_ecdf_distribution(id_distribution="weibull")

Replace "weibull" with the desired distribution identifier.

Complete Usage Example ​

For a fully customized ECDF plot with all available parameters, the following implementation can be used:

python
phi.plot_ecdf_distribution(
    id_distribution="normal",
    plot_title="Empirical vs. Theoretical CDF",
    plot_xaxis_title="Values",
    plot_yaxis_title="Probability",
    plot_xaxis_min_offset=0.3,
    plot_xaxis_max_offset=0.3,
    plot_legend_title="Legend",
    plot_height=500,
    plot_width=700,
    plot_empirical_line_color="rgba(0,128,255,1)",
    plot_empirical_line_width=3,
    plot_empirical_line_name="Empirical Data",
    plot_empirical_bar_color="rgba(0,128,255,0.5)",
    plot_distribution_line_color="rgba(255,0,0,1)",
    plot_distribution_line_width=2,
    plotly_plot_renderer="png",
    plot_engine="matplotlib"
)

Example Visualization ​

ECDF Distribution