Octave で関数の内挿

1.　一次元の内挿 interp1()

usage:
yi = interp1(x, y, xi)
yi = interp1(y, xi)
yi = interp1(..., method)
yi = interp1(..., extrap)
yi = interp1(..., "left")
yi = interp1(..., "right")
pp = interp1(..., "pp")

x, y: データポイント。x が省略されたときには x = 1:length(y) と解釈される

xi, yi: 内挿されたデータポイント

method: "nearest", "previous", "next", "linear"(デフォルト), "pchip", "cubric", "spline"

extrap: "extrap" 終点を超えて外挿する
数値 終点以降に指定された数値で外挿する
NA デフォルト

"right", "left": 不連続点の指定

"pp": 内挿値は返さず，ピースワイズ多項式オブジェクトを返す

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");

x = [0:0.05:10];
y = sin(2 * pi * x/5);
xi = 0:10;
interp1(x, y, xi, "linear", "extrap")

ans =

 Columns 1 through 8:

        0   0.9511   0.5878  -0.5878  -0.9511  -0.0000   0.9511   0.5878

 Columns 9 through 11:

  -0.5878  -0.9511  -0.0000

上と同じことを interp1() で "pp" を指定し，戻り値を使って ppval() により xi のときの y を求めることで行う。

pp = interp1(x, y, "linear", "pp");
ppval(pp, xi)

ans =

 Columns 1 through 8:

        0   0.9511   0.5878  -0.5878  -0.9511  -0.0000   0.9511   0.5878

 Columns 9 through 11:

  -0.5878  -0.9511  -0.0000

2.　二次元の内挿 interp2()

usage:
zi = interp2(x, y, z, xi, yi)
zi = interp2(z, xi, yi)
zi = interp2(z, n)
zi = interp2(z)
zi = interp2(..., method)
zi = interp2(..., method, extrap)

method: "nearest", "linear"(デフォルト), "pchip", "cubic", "spline"

clf;
 colormap("default");
 a = [13, -1, 12; 5, 4, 3; 1, 6, 2];
 x = [0, 1, 4];
 y = [10, 11, 12];
 xi = linspace(min(x), max(x), 17);
 yi = linspace(min(y), max(y), 26)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "linear"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 [x, y, a] = peaks(10);
 x = x(1, :)';
 y = y(:, 1);
 xi = linspace(min(x), max(x), 41);
 yi = linspace(min(y), max(y), 41)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "linear"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 a = [13, -1 , 12; 5, 4, 3; 1, 6, 2];
 x = [0, 1, 4];
 y = [10, 11, 12];
 xi = linspace(min(x), max(x), 17);
 yi = linspace(min(y), max(y), 26)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "nearest"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 [x, y, a] = peaks(10);
 x = x(1, :)';
 y = y(:, 1);
 xi = linspace(min(x), max(x), 41);
 yi = linspace(min(y), max(y), 41)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "nearest"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 a = [13, -1, 12; 5, 4, 3; 1, 6, 2];
 x = [0, 1, 2];  y = [10, 11, 12];
 xi = linspace(min(x), max(x), 17);
 yi = linspace(min(y), max(y), 26)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "cubic"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 [x, y, a] = peaks(10);
 x = x(1, :)';
 y = y(:, 1);
 xi = linspace(min(x), max(x), 41);
 yi = linspace(min(y), max(y), 41)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "cubic"));
 [x,y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

clf;
 colormap("default");
 a = [13, -1, 12; 5, 4, 3; 1, 6,2];
 x = [0, 1, 2];
 y = [10, 11, 12];
 xi = linspace(min(x), max(x), 17);
 yi = linspace(min(y), max(y), 26)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "spline"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

 clf;
 colormap("default");
 [x, y, a] = peaks(10);
 x = x(1, :)';
 y = y(:,1);
 xi = linspace(min(x), max(x), 41);
 yi = linspace(min(y), max(y), 41)';
 mesh(xi, yi, interp2(x, y, a, xi, yi, "spline"));
 [x, y] = meshgrid(x, y);
 hold on; plot3(x, y, a, "b*"); hold off;

3.　三次元の内挿 interp3()

usage:
vi = interp3(x, y, z, v, xi, yi, zi)
vi = interp3(v, xi, yi, zi)
vi = interp3(v, n)
vi = interp3(v)
vi = interp3(..., method)
vi = interp3(..., method, extrapval)

method: "nearest", "linear"(デフォルト), "cubic", "spline"

x = y = z = -1:1;
f = @(x, y, z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f(xx, yy, zz);
xi = yi = zi = -1:0.1:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline");
mesh (zi, yi, squeeze (vi(1, :, :)));

4.　$n$ 次元の内挿 interpn()

usage:
vi = interpn(x1, x2, ..., v, y1, y2, ...)
vi = interpn(v, y1, y2, ...)
vi = interpn(v, m)
vi = interpn(v)
vi = interpn(..., method)
vi = interpn(..., method, extrapval)

x = y = z = -1:1;
f = @(x, y, z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f(xx, yy, zz);
xi = yi = zi = -1:0.1:1;
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline");
mesh (zi, yi, squeeze (vi2(1, :, :)));