If we take a look at the plot we see that the out of sample performance is best for the order 1ДЄnd keeps getting worse with increasing degree. However if we consider the out of sample performance one sees that too high orders tend to overfit and therefore perform badly out of sample. The interpolation with order 5 and order 10 polynomials has almost no apprroximation error.

While the approximation of the straight line in this example has larger errors the order 3 polynomial approximates the sinus function in this intervall relatively good. One can see that the approximation error for the sample gets smaller when the degree of the polynomial increases. We now set the grid we want to compute our interpolating function on: zx = 0:0.1:10 ДЄnd calculate the y-values: zy1 = polyval(p1,zx) Hereby is x the x-value and y the y-value of our data points and the third number is the order/degree of the polynomial. Now the coefficients of the approximating polynГіmials are being calculated: p1 = polyfit(x,y,1) This means the underlying function for the data in the interval is sinusoidal. We initialize the data we want to interpolate: x = 0:0.5:10 Hereby are nearst, next and previous interpolation piecewise constant interpolations. Shape-preserving piecewise cubic interpolation, z_y = interp1(x,y,z, 'pchip') ДЄnd spline interpolation z_y = interp1(x,y,z, 'spline') Previous interpolation, z_y = interp1(x,y,z, 'previous') Next interpolation, z_y = interp1(x,y,z, 'next') Interp1 provides other options too like nearest interpolation, z_y = interp1(x,y,z, 'nearest') Hereby one calculates the line between two adjacent points and gets z_y by assuming that the point would be an element of those lines. One way to find the y-values of z is piecewise linear interpolation. Hereby x and y are the coordinates of the data points and z are the points we need information about. We will use the following data: x = 1:5:50 Type help interp1 in order to read more about the different options. The default is linear interpolation, but there are other types available, such as: linear nearest spline cubic etc. Or spline interpolation: Vz = interp2(X,Y,Z,Vx,Vy,'spline') In MATLAB we can use the interp1()function.

Linear interpolation, Vz = interp2(X,Y,Z,Vx,Vy,'linear') Д¬ubic interpolation Vz = interp2(X,Y,Z,Vx,Vy,'cubic') We now can perform nearest interpolation, Vz = interp2(X,Y,Z,Vx,Vy,'nearest') Now we set the points where we want to interpolate: = meshgrid(1:0.25:10)

We initialize the data: = meshgrid(1:2:10) zy = interp1(x,y,'method','extrapolation').

Matlab interp1