(octave.info)One-dimensional Interpolation

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29.1 One-dimensional Interpolation
==================================

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

 -- : YI = interp1 (X, Y, XI)
 -- : YI = interp1 (Y, XI)
 -- : YI = interp1 (..., METHOD)
 -- : YI = interp1 (..., EXTRAP)
 -- : YI = interp1 (..., "left")
 -- : YI = interp1 (..., "right")
 -- : PP = interp1 (..., "pp")

     One-dimensional interpolation.

     Interpolate input data to determine the value of YI at the points
     XI.  If not specified, X is taken to be the indices of Y (‘1:length
     (Y)’).  If Y is a matrix or an N-dimensional array, the
     interpolation is performed on each column of Y.

     The interpolation METHOD is one of:

     "nearest"
          Return the nearest neighbor.

     "previous"
          Return the previous neighbor.

     "next"
          Return the next neighbor.

     "linear" (default)
          Linear interpolation from nearest neighbors.

     "pchip"
          Piecewise cubic Hermite interpolating
          polynomial—shape-preserving interpolation with smooth first
          derivative.

     "cubic"
          Cubic interpolation (same as "pchip").

     "spline"
          Cubic spline interpolation—smooth first and second derivatives
          throughout the curve.

     Adding ’*’ to the start of any method above 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 using the current METHOD.  If EXTRAP is a number,
     then replace values beyond the endpoints with that number.  When
     unspecified, EXTRAP defaults to ‘NA’.

     If the string argument "pp" is specified, then XI should not be
     supplied and ‘interp1’ returns a piecewise polynomial object.  This
     object 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")’.

     Duplicate points in X specify a discontinuous interpolant.  There
     may be at most 2 consecutive points with the same value.  If X is
     increasing, the default discontinuous interpolant is
     right-continuous.  If X is decreasing, the default discontinuous
     interpolant is left-continuous.  The continuity condition of the
     interpolant may be specified by using the options "left" or "right"
     to select a left-continuous or right-continuous interpolant,
     respectively.  Discontinuous interpolation is only allowed for
     "nearest" and "linear" methods; in all other cases, the X-values
     must be unique.

     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);
          near = interp1 (xp, yp, xf, "nearest");
          pch = interp1 (xp, yp, xf, "pchip");
          spl = interp1 (xp, yp, xf, "spline");
          plot (xf,yf,"r", xf,near,"g", xf,lin,"b", xf,pch,"c", xf,spl,"m",
                xp,yp,"r*");
          legend ("original", "nearest", "linear", "pchip", "spline");

     See also: Note: pchip, Note: spline, Note:
     interpft, Note: interp2, *note interp3:
     XREFinterp3, Note: interpn.

   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.

 -- : interpft (X, N)
 -- : 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 a matrix or an N-dimensional
     array, the interpolation is performed on each column of X.

     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 endpoints of the interpolation.

     See also: Note: interp1.

   There are two significant limitations on Fourier interpolation.
First, the function signal is assumed to be periodic, and so
non-periodic signals will be poorly represented at the edges.  Second,
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 addition, the support functions ‘spline’ and ‘lookup’ that
underlie the ‘interp1’ function can be called directly.

 -- : PP = spline (X, Y)
 -- : YI = spline (X, Y, XI)
     Return the cubic spline interpolant of points X and Y.

     When called with two arguments, return the piecewise polynomial PP
     that may be used with ‘ppval’ to evaluate the polynomial at
     specific points.

     When called with a third input argument, ‘spline’ evaluates the
     spline at the points XI.  The third calling form ‘spline (X, Y,
     XI)’ is equivalent to ‘ppval (spline (X, Y), XI)’.

     The variable X must be a vector of length N.

     Y can be either a vector or array.  If Y is a vector it must 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 values of
     the first derivative of the cubic spline at the endpoints.

     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 reshaped
     internally to a matrix where the leading dimension is given by ‘S1
     * S2 * ... * SK’ and each row of this matrix is then treated
     separately.  Note that this is exactly the opposite of ‘interp1’
     but is done for MATLAB compatibility.

     See also: Note: pchip, Note: ppval, Note:
     mkpp, Note: unmkpp.