Interpolation with Curve Fitting Toolbox

Interpolation is a process for estimating values that lie between known data points.

Interpolation involves creating of a function f  that matches given data values y_i at given data sites x_i where f(x_i) = y_i, for all i.

Most interpolation methods create the interpolant f as the unique function of the formula

$$f(x)={\displaystyle \sum _j{f}_j(x)a_j},$$

where the form of the functions f_j depends on the interpolation method.

For spline interpolation, the f_j are the n consecutive B-splines B_j(x) = B(x|t_j,...,t_j+k), j = 1:n, of order k for a knot sequence t₁ ≤ t₂ ≤ ... ≤ t_n + k.

About Interpolation Methods

Curve Fitting Toolbox™ supports the interpolation methods described in the following table.

Method	Description
Nearest neighbor	Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point.
Linear	Linear interpolation. This method fits a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces.
Natural neighbor	Natural neighbor interpolation. This method sets the value of an interpolated point to a weighted average of the nearest data points. The interpolating surface is C1 continuous, except at the sample points.
Shape-preserving (PCHIP)	Piecewise cubic Hermite interpolation (PCHIP). This method preserves monotonicity and the shape of the data (for curves only).
Cubic spline	Cubic spline interpolation. This method fits a different cubic polynomial between each pair of data points for curves, or between sets of three points for surfaces.
Biharmonic (v4)	MATLAB® 4 griddata method. This method fits surfaces that also extrapolate well (for surfaces only).
Thin-plate spline	Thin-plate spline interpolation. This method fits smooth surfaces that also extrapolate well (for surfaces only).

Interpolant surface fits use the MATLAB function scatteredInterpolant for the linear, nearest neighbor, and natural neighbor methods, and the MATLAB function griddata for the cubic spline and biharmonic methods. The thin-plate spline method uses the tpaps function.

The interpolant method you use depends on several factors, including the characteristics of the data being fit, the required smoothness of the curve, speed considerations, and post-fit analysis requirements. The linear and nearest neighbor methods fit models efficiently, and the resulting curves are not very smooth. The natural neighbor, cubic spline, shape-preserving, and biharmonic methods take longer to fit models, and the resulting curves are very smooth.

For example, the following plot shows a nearest neighbor interpolant fit and a shape-preserving (PCHIP) interpolant fit for the nuclear reaction data from the carbon12alpha.mat sample data set. The nearest neighbor interpolant is not as smooth as the shape-preserving interpolant.

Note

Goodness-of-fit statistics, prediction bounds, and weights are not defined for interpolants. Additionally, the fit residuals are always 0 (within computer precision) because interpolants pass through the data points.

Biharmonic interpolant fits consist of radial basis function interpolants. All other interpolants supported by Curve Fitting Toolbox are piecewise polynomials and consist of multiple polynomials defined between data points. For cubic spline and PCHIP interpolation, four coefficients describe each piece. Curve Fitting Toolbox uses a cubic (third-degree) polynomial to calculate the four coefficients. Refer to the following for more information:

• spline for cubic spline interpolation

• pchip for shape-preserving (PCHIP) interpolation, and for a comparison of PCHIP and cubic spline interpolation

• scatteredInterpolant, griddata, and tpaps for surface interpolation

  It is possible to fit a single polynomial interpolant to data, with a degree one less than the number of data points. However, the behavior of such fits is unpredictable between data points. Piecewise polynomials with lower-order segments do not diverge significantly from the fitting data domain, so they are useful for analyzing a wider range of data sets.

Selecting an Interpolant Fit

• Select Interpolant Fit Interactively

• Fit Linear Interpolant Model Using the fit Function

Select Interpolant Fit Interactively

Open the Curve Fitter app by entering curveFitter at the MATLAB command line. Alternatively, on the Apps tab, in the Math, Statistics and Optimization group, click Curve Fitter.

On the Curve Fitter tab, in the Fit Type section, select an Interpolant fit. The app fits an interpolating curve or surface that passes through every data point.

In the Fit Options pane, you can specify the Interpolation method value.

For curve data, you can set Interpolation method to Linear, Nearest neighbor, Cubic spline, or Shape-preserving (PCHIP). For surface data, you can set Interpolation method to Linear, Nearest neighbor, Natural neighbor, Cubic spline, Biharmonic (v4), or Thin-plate spline.

For surfaces, the Interpolant fit uses the scatteredInterpolant function for the Linear, Nearest neighbor, and Natural neighbor methods, the griddata function for the Cubic Spline and Biharmonic (v4) methods, and the tpaps function for the Thin-plate spline method. Try the Thin-plate spline method when you require both smooth surface interpolation and good extrapolation properties.

Tip

If your data variables have very different scales, clear the Center and scale check box to see the difference in the fit. Normalizing the inputs might influence the results of the piecewise Linear and Cubic Spline interpolation methods, and the Nearest neighbor and Natural neighbor surface interpolation methods.

Fit Linear Interpolant Model Using the fit Function

Load the census sample data set.

load census

The variables pop and cdate contain data for the population size and the year the census was taken, respectively.

You can use the fit function to fit any of the interpolant models described in Interpolant Model Names. In this case, fit a linear interpolant model using the 'linearinterp' option, and then plot the result.

f = fit(cdate,pop,'linearinterp');
plot(f,cdate,pop);

Compare Linear Interpolant Models

Load the carbon12alpha sample data set. Create both nearest neighbor and PCHIP interpolant fits using the 'nearestinterp' and 'pchip' options.

load carbon12alpha
f1 = fit(angle,counts,'nearestinterp');
f2 = fit(angle,counts,'pchip');

Compare the fitted curves f1 and f2 by plotting them in the same figure.

p1 = plot(f1,angle,counts);
xlim([min(angle),max(angle)])
hold on

p2 = plot(f2,'b');
hold off
legend([p1;p2],'Counts per Angle','Nearest Neighbor','PCHIP',...
    'Location','northwest')