11.2. Data Interpolation

MATLAB has two functions that will interpolate between known data samples to estimate unknown sample points. The difference between the two functions relates just to the purpose for calling them. The fillmissing function is used when values that should be in the data are missing – often replaced with the NaN symbol. The interp1 function is used when you wish to add data points not contained in the data. MATLAB has similar functions for some of the data interpolation algorithms that can be invoked by fillmissing and interp1. These include pchip, spline, and makima.

Several different interpolation algorithms are available. The default is 'linear', which is good for slowly changing data or when there are a lot of sample points. The methods 'pchip', 'makima', and 'spline' yield very good results. The ‘spline’ method requires the most computation because it uses a matrix computation for each point to determine polynomial coefficients that not only match that data, but maintain constant first and second order derivatives at each data point added, which makes the curves between data points smooth. However, the ‘spline’ method may overshoot data fluctuation causing and an oscillation. The ‘pchip’ and ‘makima’ algorithms give similar results, they maintain consistent first order derivatives at each point, but not second order derivatives. They avoid overshoots and can accurately connect the flat regions. Another advantage of the ‘pchip’ and ‘makima’ algorithms is that the known data points are either monotonically increasing or decreasing, so will the interpolated data points. Because of oscillations, may not always maintain the monotonicity of the data. In the comparison plots below, the ‘pchip’ interpolation is shown, but the ‘makima’ interpolation is not shown because ‘pchip’ gives similar results with slightly less computation.

f = @(x) 1 + x.^2 - x.^3 + 20*sin(x);
x1 = (-3:1.5:3)';  % Limited data points
y1 = f(x1);
x2 = (-3:0.5:3)';  % More data points
subplot(2,2,1)
plot(x1,y1,'o',x2,interp1(x1,y1,x2,'nearest'),'*-'), title('nearest')
subplot(2,2,2)
plot(x1,y1,'o',x2,interp1(x1,y1,x2,'linear'),'*-'), title('Linear')
subplot(2,2,3)
plot(x1,y1,'o',x2,interp1(x1,y1,x2,'pchip'),'*-'), title('pchip')
subplot(2,2,4)
plot(x1,y1,'o',x2,interp1(x1,y1,x2,'spline'),'*-'), title('spline')

Fig. 11.7 The interp1 and fillmissing functions can use different data interpolation algorithms to suite the application needs.

The following example illustrates the difference between ‘pchip’ and ‘spline’ interpolation results. The ‘pchip’ result may be less smooth, but the monotonicity of the data is preserved.

x = -3:3;
y = [-1 -1 -1 0 1 1 1];
t = -3:.01:3;
hold on
plot(x, y, 'o')
plot(t, interp1(x, y, t, 'pchip'), 'g')
plot(t, interp1(x, y, t, 'spline'), '-.r')
legend('data','pchip','spline', 'Location', 'NorthWest')
hold off

Fig. 11.8 Comparison of ‘pchip’ and ‘spline’ data inerpolation.