Multivariate

Overview

Multivariate interpolation extends the concept of 1-D interpolation to functions of multiple variables. When your data has more than one independent variable—such as temperature varying across geographic coordinates (x, y) or material properties depending on pressure and temperature (P, T)—univariate methods are insufficient. Multivariate interpolation constructs a continuous function f(x_1, x_2, \ldots, x_n) from discrete observations at scattered or gridded locations.

Applications are widespread across engineering and science. In geospatial analysis, multivariate interpolation creates digital elevation models from survey points. In computational fluid dynamics, it reconstructs flow fields from discrete sensor readings. In image processing, it performs texture mapping and image resampling. In materials science, it builds response surfaces from experimental design data.

The fundamental challenge is the curse of dimensionality: the number of data points needed to represent a function grows exponentially with the number of dimensions. This makes careful selection of interpolation methods critical for performance and accuracy.

Regular Grids vs. Scattered Data

The structure of your input data fundamentally determines which interpolation method is appropriate.

Regular (structured) grids have data points arranged in a rectangular lattice, where each dimension has a fixed set of coordinates. For example, measurements at all combinations of x \in \{0, 1, 2\} and y \in \{0, 1, 2\}. This structure enables fast lookup and efficient tensor-product methods. The INTERPN and GRID_INTERP functions are optimized for this case.

Scattered (unstructured) data has observations at arbitrary locations with no regular pattern—common in field measurements, sensor networks, and experimental data. Interpolating scattered data is computationally more demanding and typically uses triangulation-based methods or radial basis functions. The GRIDDATA function handles scattered data by first triangulating the points (Delaunay triangulation) and then interpolating within each simplex.

Interpolation Methods

Different mathematical approaches offer varying trade-offs between smoothness, accuracy, computational cost, and extrapolation behavior.

Linear interpolation (LINEAR_ND_INTERP) connects neighboring points with hyperplanes. It is fast, local (changing one data point affects only nearby regions), and preserves monotonicity, but produces C^0 continuity (discontinuous derivatives). In 2D, this corresponds to piecewise linear interpolation over a triangulation.

Nearest-neighbor interpolation (NEAREST_ND_INTERP) assigns the value of the closest data point. It is the simplest method and naturally handles categorical data, but produces piecewise-constant (discontinuous) results. It is useful for label propagation and quick prototyping.

Cubic interpolation (available in GRIDDATA via the cubic method) produces smoother (C^1 continuous) surfaces by fitting local cubic polynomials. It is more accurate for smooth functions but can produce overshooting near sharp features and is more computationally expensive.

Radial Basis Function (RBF) interpolation (RBF_INTERPOLATOR) represents the interpolant as a weighted sum of radially symmetric basis functions centered at each data point. Common basis functions include thin-plate splines, multiquadrics, and Gaussian kernels. RBFs produce globally smooth results and naturally handle scattered data in any dimension, but can be computationally intensive for large datasets (O(N^3) for direct methods). They excel at smooth function reconstruction and are widely used in geographic information systems.

Choosing a Method

The optimal method depends on your data structure, smoothness requirements, and computational constraints:

• For regular grids with moderate dimensions: Use INTERPN or GRID_INTERP for fast tensor-product interpolation.
• For scattered data with simple needs: Use GRIDDATA with linear method for efficiency or cubic for smoothness.
• For scattered data requiring maximum smoothness: Use RBF_INTERPOLATOR, particularly for dimensions > 2.
• For quick prototyping or categorical data: Use NEAREST_ND_INTERP.

Native Excel Capabilities

Excel lacks native multivariate interpolation functions. The built-in capabilities are limited to 1-D scenarios:

• Linear interpolation: Can be manually constructed using FORECAST.LINEAR or slope-intercept formulas, but only for one independent variable.
• 2-D table lookups: The XLOOKUP and INDEX/MATCH functions can retrieve values from tables but do not perform true interpolation between grid points.
• Trend surfaces: The Analysis ToolPak supports multiple linear regression, which can fit a planar surface z = ax + by + c, but this is approximation, not interpolation.

For genuine multivariate interpolation (especially on scattered data or with nonlinear surfaces), Excel users must resort to complex manual implementations or external tools. Python functions provide access to production-grade algorithms from SciPy’s interpolate module.

Third-Party Excel Add-ins

• XLSTAT: Offers surface fitting and kriging (a geostatistical interpolation method related to RBFs) for spatial data analysis.
• DataFit: Provides 2D and 3D surface fitting capabilities with various interpolation and smoothing methods.
• XonGrid: A specialized add-in for 2D/3D gridding and contouring from scattered data, commonly used in geosciences.

These add-ins typically focus on 2D or 3D interpolation; higher-dimensional interpolation generally requires dedicated scientific computing environments.

Tools

Tool	Description
GRID_INTERP	Interpolator on a regular grid in 2D.
GRIDDATA	Interpolate unstructured D-D data.
INTERPN	Multidimensional interpolation on regular grids (2D).
LINEAR_ND_INTERP	Piecewise linear interpolator in N > 1 dimensions.
NEAREST_ND_INTERP	Nearest neighbor interpolation in N > 1 dimensions.
RBF_INTERPOLATOR	Radial basis function interpolation in N dimensions.