Contents

Piecewise Cubic Hermite InterpolationEdit

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) is a widely used interpolation method in numerical analysis, data visualization, and engineering applications. It constructs a smooth, continuous function by stitching together cubic polynomials on each subinterval between data points, while enforcing that the function takes the given values at the nodes and uses carefully chosen derivatives at those nodes. This yields a shape-preserving interpolant that tends to avoid artificial oscillations and overshoot which can plague some other cubic methods.

From a practical standpoint, PCHIP is valued for its robustness and local control. It provides a good balance between fidelity to sampled data and smoothness of the resulting curve, making it a common choice in software libraries and scientific workflows where interpretability and stability matter. The method sits within the broader family of Hermite interpolation, and it is often contrasted with Spline interpolation approaches that emphasize global curvature properties.

History and context

The idea of constructing interpolants that pass through data points with specified derivatives goes back to the classic theory of Hermite interpolation and its many refinements. The particular monotone, shape-preserving variant that underpins PCHIP owes its development to researchers who sought to prevent new extrema in the interpolant and to maintain fidelity to the data’s local trends. In many software implementations, this method is described as a monotone cubic interpolation built in a piecewise Hermite framework. Notable contributors in this lineage include researchers such as Fritsch and Carlson, whose work on monotone cubic interpolation helped popularize practical, calculation-friendly schemes for derivative selection. Modern descriptions and implementations often refer to this lineage when explaining why PCHIP preserves monotonicity on data sequences. For software examples and historical notes, you may encounter discussions that co-link to SciPy and MATLAB implementations of the same idea.

Mathematical construction

Given a set of data points (x_i, y_i) with strictly increasing x_i, PCHIP forms a cubic polynomial on each interval [x_i, x_{i+1}] that matches the endpoint values y_i and y_{i+1} and also matches specified derivatives d_i and d_{i+1} at the endpoints. The resulting piecewise function is continuous and has a continuous first derivative, i.e., it is C^1 continuous.

  • The value constraints are y_i = S_i(x_i) and y_{i+1} = S_i(x_{i+1}) for each interval i.
  • The derivative constraints require the derivatives d_i at each node to be chosen to prevent the interpolant from introducing spurious extrema and to respect the local slope of the data.
  • A central design idea is to obtain derivatives d_i using a monotonicity-preserving scheme. When neighboring secant slopes on adjacent intervals have opposite signs or one is zero, the method typically sets the interior derivative to zero to avoid reversing the data’s trend. When the neighboring slopes share the same sign, the derivatives are chosen to align with that trend while limiting over- or undershoot. This approach yields a cubic polynomial on each subinterval that stays faithful to the local data shape without creating new peaks or valleys.
  • The result is an interpolant that is locally determined by neighboring data and does not force global smoothness at the expense of fidelity to the observed values.

In practice, the derivatives at the endpoints are determined by a boundary rule consistent with the interior scheme, and the cubic on each interval is constructed from the endpoint values and endpoint derivatives via the Hermite form.

The practical upshot is a function that on each interval behaves like a cubic Hermite polynomial, with the whole piecewise function matching the data and preserving the data’s monotonicity when present.

Properties and comparisons

  • Shape-preserving: PCHIP is designed to avoid introducing new extrema between data points, making it especially suitable for data that are known to be monotone or to follow a gentle curvature.
  • Local control: Changing a single data point affects only nearby intervals, not the entire interpolant, which helps in interactive data analysis and real-time visualization.
  • Smoothness: The interpolant is C^1 continuous, providing a smooth transition between adjacent cubic pieces without discontinuities in the first derivative.
  • Comparison with splines: Unlike natural cubic splines, which minimize a global curvature criterion and can overshoot or oscillate for certain data, PCHIP emphasizes local fidelity and monotonicity. For users who need more aggressive smoothing or curvature minimization, spline-based methods (e.g., Spline interpolation or Natural cubic spline) may be preferable.
  • Appropriate use cases: Data that represent measurements with inherent order and trends—such as sensor readings, geospatial samples, or time-series data—often benefit from PCHIP’s conservative interpolation behavior.

Implementation notes and practical considerations

  • Data requirements: The x-values must be strictly increasing. If the data are irregularly spaced, PCHIP remains applicable, but the derivative selection adapts to the local spacings.
  • Endpoints and derivatives: The boundary derivatives are determined by a scheme that respects the interior monotonicity constraints. This makes the method robust to a variety of data shapes without requiring user-supplied derivatives.
  • Computational cost: Building the interpolant is typically linear in the number of data points (O(n) for n points), and evaluating the interpolant on a new x in a given interval is O(1). This efficiency makes PCHIP suitable for large datasets and real-time visualization.
  • Software availability: PCHIP is implemented in several numerical libraries. For example, in Python, the SciPy project provides a PchipInterpolator as part of its interpolation suite, and in MATLAB, the pchip function implements a closely related monotone cubic interpolant. See practical software references for SciPy and MATLAB for concrete usage patterns.

Extensions and related methods

  • Related to Cubic Hermite interpolation, since the core idea is to construct cubic polynomials that match function values and derivatives at the nodes.
  • Closely related to Monotone cubic interpolation, which shares the goal of preserving the data’s monotonicity and local shape.
  • Within the broader landscape of interpolation, PCHIP contrasts with more global approaches like Spline interpolation and its variants, as well as with more flexible fits that emphasize minimizing a global error or smoothing criterion.

See also