Empirical Cumulative Distribution Function

The Empirical Cumulative Distribution Function (ECDF) provides a non-parametric estimation of the cumulative distribution of a given dataset. Unlike parametric approaches, the ECDF does not impose assumptions regarding the underlying distribution, allowing for an unbiased representation of the observed data.

Properties

• The ECDF is a non-decreasing function that is bounded within the interval $[0,1]$.
• It represents the cumulative probability of observations up to a specified value.
• It is particularly useful for visualizing data distributions, especially in smaller datasets where individual observations significantly influence the overall distribution.

Calculation

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

$$\text{ECDF}(x)=\frac{\text{Number of observations }\le x}{n}$$

ECDF Plotting

The ECDF can be visualized using the .plot_ecdf() method. This function offers several customizable parameters.

Parameters

• plot_title (str, optional): The title of the plot. Default: "ECDF".
• plot_xaxis_title (str, optional): The label for the horizontal axis. Default: "Domain".
• plot_yaxis_title (str, optional): The label for the vertical axis. Default: "Cumulative Distribution Function".
• plot_xaxis_range (tuple[float, float], optional): The range of values for the horizontal axis. If not specified, the range adjusts automatically to the data.
• plot_xaxis_margin (float, optional): The margin applied around the data domain. Default: 0.3.
• plot_legend_title (str or None, optional): The title of the legend. If None, no legend title is displayed.
• plot_height (int, optional): The height of the plot in pixels. Default: 400.
• plot_width (int, optional): The width of the plot in pixels. Default: 400.
• plot_line_color (str, optional): The color of the ECDF line, specified in RGBA format.
• plot_line_width (int, optional): The thickness of the ECDF line. Default: 2.
• plot_line_name (str, optional): The label for the ECDF in the plot legend. Default: "Empirical Distribution".
• plotly_plot_renderer ("png" | "jpeg" | "svg" | None, optional): The output format when using Plotly as the rendering engine. Default: None, which uses the default renderer.
• plot_engine ("plotly" | "matplotlib", optional): Specifies the plotting library to be used. Default: "plotly".

Default Usage

For a basic implementation without additional customization, the ECDF plot can be generated as follows:

phi.plot_ecdf()

Complete Usage

For a more detailed customization of the ECDF plot, the following implementation demonstrates the use of multiple parameters:

phi.plot_ecdf(
    plot_title="Empirical CDF of Sample Data",
    plot_xaxis_title="Observed Values",
    plot_yaxis_title="Cumulative Probability",
    plot_xaxis_range=(0, 100),
    plot_xaxis_margin=0.2,
    plot_legend_title="ECDF Curve",
    plot_height=600,
    plot_width=800,
    plot_line_color="rgba(255,0,0,0.8)",
    plot_line_width=3,
    plot_line_name="Empirical CDF",
    plotly_plot_renderer="png",
    plot_engine="plotly"
)

Example Visualization

The following figure provides an example of an ECDF plot generated using the .plot_ecdf() method.

For further details on additional functionalities and extended capabilities, refer to the subsequent sections of this documentation.