Splines

Overview

Spline interpolation is a powerful mathematical technique for constructing smooth curves through a set of data points. Unlike polynomial interpolation, which uses a single high-degree polynomial that can exhibit wild oscillations (Runge’s phenomenon), spline methods piece together multiple low-degree polynomials to create a function that is both smooth and well-behaved. The term “spline” originated from the flexible wooden or metal strips that draftsmen used to draw smooth curves through control points.

Splines are fundamental in computer graphics (defining smooth curves and surfaces), data science (smoothing noisy measurements), engineering (modeling aerodynamic surfaces), and statistics (nonparametric regression). The power of splines lies in their ability to balance two competing goals: fidelity to the data and smoothness of the resulting curve.

B-Splines and Basis Functions

B-splines (basis splines) provide the mathematical foundation for modern spline methods. Rather than defining a spline directly through the data points, B-splines represent curves as weighted sums of simpler basis functions. Each basis function is a piecewise polynomial with compact support (nonzero only over a limited range), making B-splines computationally efficient and numerically stable.

The key parameters of a B-spline are: - Degree (k): The degree of the polynomial pieces (e.g., k=3 for cubic splines) - Knot vector: A sequence of parameter values that define where the polynomial pieces join - Control points: The weights that determine the shape of the curve

B-splines offer excellent flexibility for both interpolation (passing exactly through data points) and approximation (creating a smooth curve that balances fit and smoothness). The MAKE_INTERP_SPLINE function constructs interpolating B-splines using this framework.

Least-Squares Spline Fitting

When data contains noise or when you need a smoother representation than exact interpolation provides, least-squares spline fitting finds the best-fit spline by minimizing the sum of squared residuals. This approach is particularly valuable when: - You have more data points than desired spline knots - Measurement error makes exact interpolation inappropriate - You want to reduce data while preserving essential features

The MAKE_LSQ_SPLINE function implements this technique, allowing you to specify the knot locations independently of the data points. This provides explicit control over the spline’s complexity and flexibility.

Smoothing Splines

Smoothing splines automatically balance goodness of fit and smoothness by solving an optimization problem. For a smoothing parameter \lambda, the method minimizes:

\sum_{i=1}^{n} (y_i - f(x_i))^2 + \lambda \int (f''(x))^2 \, dx

The first term ensures the curve stays close to the data points, while the second term penalizes excessive curvature (measured by the second derivative). When \lambda = 0, you get exact interpolation; as \lambda \to \infty, the result approaches a straight line (linear regression).

This framework is extremely powerful because it adapts to the data automatically. The SMOOTH_SPLINE and UNIVARIATE_SPLINE functions provide smoothing spline capabilities with different interfaces and control options. Smoothing splines are closely related to cubic splines and form the basis for many modern statistical techniques like generalized additive models (GAMs).

Interpolating Splines

Traditional interpolating splines pass exactly through every data point while maintaining smoothness constraints between them. The most common is the cubic spline, which constructs piecewise cubic polynomials such that: - The curve passes through all data points - The first and second derivatives are continuous everywhere - Boundary conditions (e.g., “natural” splines with zero second derivative at the endpoints) are satisfied

The INTERP_UV_SPLINE function provides this classical interpolation approach, making it ideal for applications requiring exact data reproduction, such as digitizing curves or creating lookup tables.

Native Excel capabilities

Excel offers limited native support for spline-based curve fitting: - Trendlines: Excel charts support polynomial trendlines (up to degree 6) but do not offer true spline fitting. Polynomial trendlines are global fits that can suffer from oscillation and poor extrapolation behavior. - Chart Smoothing: Excel can smooth line charts visually, but this is a display feature, not a mathematical interpolation method. - Manual Implementation: Users can construct cubic splines manually using worksheet formulas, but this requires substantial mathematical knowledge and becomes impractical for datasets with more than a few points.

Python spline functions provide access to production-quality implementations from SciPy, with robust numerical algorithms, automatic parameter selection, and extensive validation.

Third-party Excel add-ins

• XLSTAT: A comprehensive statistical package that includes spline smoothing and interpolation as part of its curve fitting toolkit.
• DataFit: Specialized curve-fitting software that integrates with Excel and offers spline regression alongside parametric and nonparametric models.
• NumXL: Time series add-in that provides cubic spline interpolation for filling missing data and resampling irregular time series.

Tools

Tool	Description
INTERP_UV_SPLINE	1-D interpolating spline for data.
MAKE_INTERP_SPLINE	Compute interpolating B-spline and evaluate at new points.
MAKE_LSQ_SPLINE	Compute LSQ-based fitting B-spline.
SMOOTH_SPLINE	Smoothing cubic spline.
UNIVARIATE_SPLINE	1-D smoothing spline fit to data.