What is Interpolation?
Interpolation is a mathematical technique used to estimate values between a set of known data points. It is commonly applied in fields like data visualization, numerical analysis, and physics to create smooth transitions or fill gaps in data.
The most straightforward method of interpolation is linear interpolation, where straight-line segments connect adjacent data points. While effective and easy to compute, linear interpolation lacks smoothness, resulting in a "piecewise" appearance that may not align with the natural trend of the data.
To address this, methods like cubic interpolation use polynomials to produce smooth curves. However, these methods can introduce new challenges, such as violating "monotonicity" in the data.
What is Monotonicity?
Monotonicity refers to a series of values that either consistently increase or decrease in value. For example, a cumulative rainfall total is monotonically increasing.
When interpolating between data points, a curve that preserves monotonicity respects the natural direction of the data, maintaining its increasing or decreasing behavior. While cubic methods of interpolation create smooth, pleasing curves, they often violate monotonicity by producing spurious dips or peaks that are not present in the actual data. In the example below, this problem is particularly noticeable between the 2018 and 2019 data points in the "Cardinal Cubic" and "Natural Cubic" curves.
The Monotone Cubic Solution
To address these challenges, Matthias Steffen’s monotonic cubic interpolation method modifies standard cubic interpolation to preserve monotonicity while maintaining smoothness. Introduced in the paper "A Simple Method for Monotonic Interpolation in One Dimension" (1979), the method ensures:
- Smoothness: Produces curves with continuous first derivatives, avoiding abrupt transitions.
- Monotonicity: Prevents spurious oscillations or extrema between data points while ensuring that any peaks or troughs occur only at the actual data points.
This is achieved by constraining the first derivatives (slopes) at each data point to avoid overshoots or reversals in the curve.
In Mappica, line and area charts can be constructed using Steffen’s interpolation method by selecting the "Monotone Cubic" curve option.
