One-dimensional Interpolation - Untitled
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### 28.1 One-dimensional Interpolation

Octave supports several methods for one-dimensional interpolation, most of which are described in this section. Polynomial Interpolation and Interpolation on Scattered Data describes further methods.

— Function File: yi = interp1 (x, y, xi)
— Function File: yi = interp1 (..., method)
— Function File: yi = interp1 (..., extrap)
— Function File: pp = interp1 (..., 'pp')

> One-dimensional interpolation. Interpolate y, defined at the points x, at the points xi. The sample points x must be strictly monotonic. If y is an array, treat the columns of y separately.

> Method is one of:

> 'nearest'
>   Return the nearest neighbour.
> 'linear'
>   Linear interpolation from nearest neighbours
> 'pchip'
>   Piece-wise cubic hermite interpolating polynomial
> 'cubic'
>   Cubic interpolation from four nearest neighbours
> 'spline'
>   Cubic spline interpolation–smooth first and second derivatives throughout the curve

> Appending '*' to the start of the above method forces `interp1` to assume that x is uniformly spaced, and only x`(1)` and x`(2)` are referenced. This is usually faster, and is never slower. The default method is 'linear'.

> If extrap is the string 'extrap', then extrapolate values beyond the endpoints. If extrap is a number, replace values beyond the endpoints with that number. If extrap is missing, assume NA.

> If the string argument 'pp' is specified, then xi should not be supplied and `interp1` returns the piece-wise polynomial that can later be used with `ppval` to evaluate the interpolation. There is an equivalence, such that `ppval (interp1 (`x`,` y`,` method`, 'pp'),` xi`) == interp1 (`x`,` y`,` xi`,` method`, 'extrap')`.

> An example of the use of `interp1` is

>                  xf=[0:0.05:10]; yf = sin(2*pi*xf/5);
>                  xp=[0:10];      yp = sin(2*pi*xp/5);
>                  lin=interp1(xp,yp,xf);
>                  spl=interp1(xp,yp,xf,'spline');
>                  cub=interp1(xp,yp,xf,'cubic');
>                  near=interp1(xp,yp,xf,'nearest');
>                  plot(xf,yf,"r",xf,lin,"g",xf,spl,"b", ...
>                       xf,cub,"c",xf,near,"m",xp,yp,"r*");
>                  legend ("original","linear","spline","cubic","nearest")

> See also: interpft.

There are some important differences between the various interpolation methods. The 'spline' method enforces that both the first and second derivatives of the interpolated values have a continuous derivative, whereas the other methods do not. This means that the results of the 'spline' method are generally smoother. If the function to be interpolated is in fact smooth, then 'spline' will give excellent results. However, if the function to be evaluated is in some manner discontinuous, then 'pchip' interpolation might give better results.

This can be demonstrated by the code

         t = -2:2;
         dt = 1;
         ti =-2:0.025:2;
         dti = 0.025;
         y = sign(t);
         ys = interp1(t,y,ti,'spline');
         yp = interp1(t,y,ti,'pchip');
         ddys = diff(diff(ys)./dti)./dti;
         ddyp = diff(diff(yp)./dti)./dti;
         figure(1);
         plot (ti, ys,'r-', ti, yp,'g-');
         legend('spline','pchip',4);
         figure(2);
         plot (ti, ddys,'r+', ti, ddyp,'g*');
         legend('spline','pchip');

Fourier interpolation, is a resampling technique where a signal is converted to the frequency domain, padded with zeros and then reconverted to the time domain.

— Function File: interpft (x, n)
— Function File: interpft (x, n, dim)

> Fourier interpolation. If x is a vector, then x is resampled with n points. The data in x is assumed to be equispaced. If x is an array, then operate along each column of the array separately. If dim is specified, then interpolate along the dimension dim.

> `interpft` assumes that the interpolated function is periodic, and so assumptions are made about the end points of the interpolation.

> See also: interp1.

There are two significant limitations on Fourier interpolation. Firstly, the function signal is assumed to be periodic, and so non periodic signals will be poorly represented at the edges. Secondly, both the signal and its interpolation are required to be sampled at equispaced points. An example of the use of `interpft` is

         t = 0 : 0.3 : pi; dt = t(2)-t(1);
         n = length (t); k = 100;
         ti = t(1) + [0 : k-1]*dt*n/k;
         y = sin (4*t + 0.3) .* cos (3*t - 0.1);
         yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1);
         plot (ti, yp, 'g', ti, interp1(t, y, ti, 'spline'), 'b', ...
               ti, interpft (y, k), 'c', t, y, 'r+');
         legend ('sin(4t+0.3)cos(3t-0.1','spline','interpft','data');

which demonstrates the poor behavior of Fourier interpolation for non periodic functions.

In additional the support function `spline` and `lookup` that underlie the `interp1` function can be called directly.

— Function File: pp = spline (x, y)
— Function File: yi = spline (x, y, xi)

> Returns the cubic spline interpolation of y at the point x. Called with two arguments the piece-wise polynomial pp that may later be used with `ppval` to evaluate the polynomial at specific points.

> The variable x must be a vector of length n, and y can be either a vector or array. In the case where y is a vector, it can have a length of either n or n`+ 2`. If the length of y is n, then the 'not-a-knot' end condition is used. If the length of y is n`+ 2`, then the first and last values of the vector y are the first derivative of the cubic spline at the end-points.

> If y is an array, then the size of y must have the form `[`s1`,` s2`, ...,` sk`,` n`]` or `[`s1`,` s2`, ...,` sk`,` n`+ 2]`. The array is then reshaped internally to a matrix where to leading dimension is given by s1`*` s2`* ... *` sk and each row this matrix is then treated separately. Note that this is exactly the opposite treatment than `interp1` and is done for compatibility.

> Called with a third input argument, `spline` evaluates the piece-wise spline at the points xi. There is an equivalence between `ppval (spline (`x`,` y`),` xi`)` and `spline (`x`,` y`,` xi`)`.

> See also: ppval, mkpp, unmkpp.

The `lookup` is used by other interpolation function to identify the points of the original data that are closest to the current point of interest.

— Function File: idx = lookup (table, y)

> Lookup values in a sorted table. Usually used as a prelude to interpolation.

> If table is strictly increasing and `idx = lookup (table, y)`, then `table(idx(i)) <= y(i) < table(idx(i+1))` for all `y(i)` within the table. If `y(i)` is before the table, then `idx(i)` is 0. If `y(i)` is after the table then `idx(i)` is `table(n)`.

> If the table is strictly decreasing, then the tests are reversed. There are no guarantees for tables which are non-monotonic or are not strictly monotonic.

> To get an index value which lies within an interval of the table, use:

>               idx = lookup (table(2:length(table)-1), y) + 1

> This expression puts values before the table into the first interval, and values after the table into the last interval.