Visualizing 1D Distributions

Understanding the distribution of a single variable is a cornerstone of data analysis and forecast evaluation. Before applying complex visualizations, it’s often crucial to inspect the fundamental characteristics of key metrics, such as prediction errors or the width of uncertainty intervals. The histogram and its smoothed counterpart, the Kernel Density Estimate (KDE) [1], are primary tools for this task. In practice, these visualizations are computed with array and scientific routines [2][3] and rendered with common plotting libraries [4][5].

The kdiagram.utils.hist module provides straightforward functions for creating these essential distribution plots.

Summary of Histogram Functions

Distribution Visualization Functions

Function

Description

plot_hist_kde()

Plots a histogram combined with a smooth Kernel Density Estimate (KDE) curve to visualize a 1D distribution.

Detailed Explanations

Let’s explore the plot_hist_kde function in detail.

Histogram and KDE (plot_hist_kde())

Purpose: This function provides a comprehensive visualization of a one-dimensional data distribution. It combines a histogram, which groups data into bins, with a Kernel Density Estimate (KDE), which provides a smooth, continuous estimate of the probability density function. This dual view is highly effective for understanding the shape, central tendency, and spread of a variable.

Mathematical Concept:

  1. Histogram: The data range is divided into a series of intervals, or bins. The plot displays bars where the height of each bar corresponds to the number of data points that fall into that bin. When density=True, the bar heights are normalized so that the total area of the histogram equals 1.

  2. Kernel Density Estimate (KDE): The KDE is a non-parametric way to estimate the probability density function of a random variable. It creates a smooth curve by placing a kernel function (typically a Gaussian) on each data point, and then summing all these kernels. The resulting curve, \(\hat{f}_h(x)\), is a smooth estimate of the data’s distribution [1].

    (1)\[\hat{f}_h(x) = \frac{1}{nh} \sum_{i=1}^{n} K\left(\frac{x - x_i}{h}\right)\]

    Here, \(K\) is the kernel function, \(h\) is the bandwidth (a smoothing parameter), and \(n\) is the number of data points. Typical implementations rely on numerical routines and array ops from SciPy/NumPy [3][2], while the visualization itself is commonly produced with Matplotlib/Seaborn [4][5].

Interpretation:

  • Shape: The overall shape of the histogram and KDE curve reveals the nature of the distribution. Is it symmetric (like a normal distribution), skewed to one side, or does it have multiple peaks (bimodal or multimodal)?

  • Central Tendency: The location of the highest peak(s) indicates the mode(s) of the data—the most frequently occurring values.

  • Spread: The width of the distribution indicates the variability or dispersion of the data. A narrow plot signifies low variance, while a wide plot signifies high variance.

  • Outliers: Data points that fall far from the central mass of the distribution can be identified in the tails of the plot.

Use Cases:

  • Forecast Error Analysis: This is a primary use case. Plotting the distribution of prediction errors (\(y_{true} - \hat{y}_{pred}\)) is crucial. A good model often has errors that are normally distributed and centered at zero.

  • Uncertainty Assessment: Visualize the distribution of prediction interval widths (\(Q_{up} - Q_{low}\)). A narrow, unimodal distribution is often desirable, as it suggests the model produces consistent uncertainty estimates.

  • Feature Inspection: Before building a model, inspect the distribution of input features to identify skewness or other characteristics that might require transformation.

Example: (See the Histogram with KDE Overlay in the Gallery for code and a plot example)

References