Interpolation – Estimation of Values Between Known Data Points

What is Interpolation?

Interpolation is a fundamental mathematical process used to estimate unknown values that fall between known data points. When a function or measurement is only available at discrete locations or times, interpolation provides a way to fill in the gaps, constructing a continuous curve or function that passes through the given points. Unlike guesswork, interpolation leverages the structure and trends present in the data, ensuring estimates are consistent with known values.

The simplest interpolation assumes a straight line between points (linear interpolation), but more sophisticated techniques—like polynomial or spline interpolation—allow for smooth curves or surfaces that better model real-world phenomena. Interpolation is crucial in engineering, scientific computing, geostatistics, computer graphics, and aviation, especially where direct measurement everywhere is impractical or impossible.

In aviation and environmental modeling, for example, the International Civil Aviation Organization (ICAO) requires accurate interpolation for weather data, emissions modeling, and regulatory reporting, ensuring that estimates of environmental variables are reliable and consistent.

Fundamental Concepts and Terminology

Data Points

Data points are the known values of a function, typically represented as pairs ((x_i, y_i)) in one dimension or as tuples in higher dimensions. The quality and spacing of these points greatly affect the reliability of interpolation. Closely spaced, accurate points yield better results; widely spaced or unevenly distributed data can cause large errors, especially with high-degree polynomials.

Interpolation vs. Extrapolation

• Interpolation estimates values within the domain of known data.
• Extrapolation estimates values outside the known data range and is generally less reliable, as it assumes data trends continue beyond available measurements.

This distinction is critical in regulatory contexts such as ICAO environmental modeling, where extrapolation is discouraged due to its unreliability.

Underlying Function

Interpolation presumes the data points are samples from a continuous, often smooth function (f(x)). The chosen interpolation method should align with the assumed smoothness and behavior of this function.

Order of Interpolation

The order or degree refers to the degree of polynomial used in interpolation:

• Linear (order 1)
• Quadratic (order 2)
• Cubic (order 3)
• Higher-degree polynomials

High-order interpolation can cause instability and oscillations (Runge’s phenomenon), especially with uneven data spacing.

Piecewise Interpolation

Rather than using a single global function, piecewise interpolation constructs low-degree polynomials between successive data points (e.g., splines), providing stability and local adaptability, which is especially important for irregular datasets.

Why Use Interpolation?

Interpolation is indispensable whenever continuous information must be reconstructed from discrete samples:

• Filling missing values in time series and sensor data
• Resampling and grid refinement in computational modeling
• Computer graphics and image processing for smooth curves and resizing
• Aviation and meteorology to estimate weather or environmental parameters (per ICAO guidance)
• Numerical integration and differentiation when analytic functions are not available
• Geosciences and mapping to create continuous surfaces from scattered measurements

Example:
An airport monitors air pollutant concentrations at several locations. If a sensor fails, interpolation (e.g., spline or IDW) estimates the missing value using nearby data—essential for maintaining a complete emissions inventory as required by ICAO.

Common Interpolation Methods

Linear Interpolation

Linear interpolation assumes a straight-line relationship between two data points:

[ y = y_0 + (x - x_0) \frac{y_1 - y_0}{x_1 - x_0} ]

Strengths: Simple, fast, no oscillations
Limitations: Not smooth at data points, poor for nonlinear behavior

Polynomial Interpolation

Fits a single polynomial of degree (n) through (n+1) points. Lagrange interpolation is the most common approach:

[ P(x) = \sum_{i=0}^{n} y_i \ell_i(x) ] with [ \ell_i(x) = \prod_{\substack{j=0 \ j \neq i}}^{n} \frac{x - x_j}{x_i - x_j} ]

Strengths: Smooth, exact fit
Limitations: Oscillates with high degree or uneven points (Runge’s phenomenon), sensitive to noise

Piecewise Polynomial Interpolation (Splines)

Piecewise Linear

Connects each pair of points with a straight line—simple but not smooth.

Cubic Spline

Fits cubic polynomials between each pair, ensuring continuity and smoothness of the curve and its first and second derivatives.

Advantages: Smooth, avoids oscillations
Applications: Graphics, aerodynamics, environmental modeling

Nearest-Neighbor and Inverse Distance Weighted (IDW) Interpolation

• Nearest-Neighbor: Assigns the value of the closest data point (stepwise, discontinuous)
• IDW: Weighted average based on inverse distance (good for scattered data, used in geostatistics and ICAO environmental mapping)

Higher-Order and Specialized Methods

• Hermite interpolation: Uses both function values and derivatives for more controlled, smooth curves
• Trigonometric (Fourier) interpolation: Ideal for periodic data
• Multidimensional methods: Bilinear, trilinear interpolation for 2D/3D data grids (e.g., images, weather models)

Worked Examples

Linear Example

Given points (2, 4) and (5, 10), estimate at (x = 3):

[ y = 4 + (3-2) \frac{10-4}{5-2} = 6 ]

Lagrange Polynomial Example

Given ((2, 1), (3, 5), (4, 13), (6, 61), (7, 125)), interpolate at (x = 5). Applying the Lagrange formula yields (y \approx 28.6).

Cubic Spline Example

Given ((0, 0), (1, 2), (2, 0)), fit a cubic spline and interpolate at (x = 1.5) using computational tools (e.g., SciPy).

Theoretical Considerations and Error

• Assumptions: The underlying function is smooth and continuous.
• Error: Linear interpolation error is (O(h^2)), spline interpolation is more accurate and stable, high-degree polynomial interpolation can be unstable.
• Runge’s phenomenon: Oscillations in high-degree polynomial interpolation.
• Interpolation vs. regression: Interpolation passes through all points; regression finds a best-fit curve.

Glossary of Key Terms

Conclusion

Interpolation is a cornerstone of numerical analysis, data science, engineering, and aviation modeling. By providing mathematically rigorous estimates between known data points, it enables accurate analysis, modeling, and regulatory reporting in countless applications.

If you require robust, accurate interpolation methods for your projects—whether in engineering, environmental modeling, or aviation—contact us or schedule a demo to discover how our solutions can help.