Visualizing Relationships

Understanding the relationship between observed (true) values and model predictions is fundamental to evaluation. While standard scatter plots are common, visualizing this relationship in a polar context can sometimes reveal different patterns or allow for comparing multiple prediction series against the true values in a compact format.

k-diagram provides the plot_relationship function to explore these connections using a flexible polar scatter plot where the angle is derived from the true values and the radius from the predicted values.

Summary of Relationship Functions

This section focuses on the primary function for visualizing these true-vs-predicted relationships in polar coordinates:

Relationship Visualization Functions

Function

Description

plot_relationship()

Creates a polar scatter plot mapping true values to angle and (normalized) predicted values to radius.

Detailed Explanations

Let’s dive into the kdiagram.plot.relationship function.

True vs. Predicted Polar Relationship (plot_relationship())

Purpose: This function generates a polar scatter plot designed to visualize the relationship between a single set of true (observed) values and one or more sets of corresponding predicted values. It maps the true values to the angular position and the predicted values (normalized) to the radial position, allowing comparison of how different predictions behave across the range of true values.

Mathematical Concept:

  1. Angular Mapping ( \(\theta\) ): Let’s consider \(\upsilon\) as the angular_angle. The angle \(\theta_i\) for each data point \(i\) is determined by its corresponding true value \(y_{\text{true}_i}\) based on the theta_scale parameter:

    • 'proportional' (Default): Linearly maps the range of

      y_true values to the specified angular coverage (acov).

      \[\theta_i = \theta_{offset} + \upsilon \cdot \frac{y_{\text{true}_i} - \min(y_{\text{true}})} {\max(y_{\text{true}}) - \min(y_{\text{true}})}\]
    • 'uniform': Distributes points evenly across the angular

      range based on their index \(i\), ignoring the actual y_true value for positioning (useful if y_true isn’t strictly ordered or continuous).

      \[\theta_i = \theta_{offset} + \upsilon \cdot \frac{i}{N-1}\]

    Where \(\upsilon\) is determined by acov (e.g., \(2\pi\) for ‘default’, \(\pi\) for ‘half_circle’) and \(\theta_{offset}\) is an optional rotation.

  2. Radial Mapping \(r\): For each prediction series y_pred, its values are independently normalized to the range [0, 1] using min-max scaling. This normalized value determines the radius \(r_i\) for that prediction series at angle \(\theta_i\).

    \[r_i = \frac{y_{\text{pred}_i} - \min(y_{\text{pred}})} {\max(y_{\text{pred}}) - \min(y_{\text{pred}})}\]

  3. Custom Angle Labels \(z_{values}\): If \(z_{values}\) are provided, the angular tick labels are replaced with these values (scaled to match the angular range), providing a way to label the angular axis with a variable other than the raw y_true values used for positioning.

Interpretation:

  • Angle: Represents the position within the range of y_true values

    (if theta_scale=’proportional’) or simply the sample index (if theta_scale=’uniform’). If z_values are used, the tick labels refer to that variable.

  • Radius: Represents the normalized predicted value for a specific

    model/series. A radius near 1 means the prediction was close to the maximum prediction made by that specific model. A radius near 0 means it was close to the minimum prediction made by that model.

  • Comparing Models: Look at points with similar angles (i.e., similar

    y_true values). Compare the radial positions of points from different models (different colors). Does one model consistently predict higher normalized values than another at certain y_true ranges (angles)?

  • Relationship Pattern: Observe the overall pattern. Does the radius

    (normalized prediction) tend to increase as the angle (y_true) increases? Is the relationship linear, cyclical, or scattered? How does the pattern differ between models?

Use Cases:

  • Comparing the relative response patterns of multiple models across

    the observed range of true values, especially when absolute scales differ.

  • Visualizing potential non-linear relationships between true values

    (angle) and normalized predictions (radius).

  • Exploring data using alternative angular representations by providing

    custom labels via z_values.

  • Displaying cyclical relationships if y_true represents a cyclical

    variable (e.g., day of year, hour of day) and acov=’default’.

Advantages (Polar Context):

  • Can effectively highlight cyclical patterns when y_true is mapped

    proportionally to a full circle (acov=’default’).

  • Allows overlaying multiple normalized prediction series against a

    common angular axis derived from the true values.

  • Flexible angular labeling using z_values provides context beyond the

    raw y_true mapping.

  • Normalization focuses the comparison on response patterns rather than

    absolute prediction magnitudes.

Example: (See the Gallery section below for a runnable code example and plot)