Invert a series of data as if inverting a function

Question

Is it possible to "invert" a series of data like when a function is inverted?

Let me explain. If I have a function that I can plot in MATLAB, I can always (almost) find the inverse of such function with the finverse and the plot would provide this:

But I do not have the a function for my data, and I would still like to find a sort of inverse of such series of data, like for example y=f(x) where

x=[0.0050879   0.0056528   0.0062176   0.0067822   0.0073467   0.0079111   0.0084752   0.0090392   0.0096030   0.0101666   0.0107299   0.0112930   0.0118558   0.0124184   0.0129807   0.0135428   0.0141045   0.0146659   0.0152270   0.0157877   0.0163481];

and

y=[0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   4.4901e-05   4.4901e-05   8.9801e-05   1.3470e-04   1.7960e-04   2.2450e-04   2.6940e-04   3.5920e-04   4.0411e-04   4.9391e-04   5.8371e-04];

Is this kind of operation on a series of data possible in MATLAB? Is there a technique/method/function that handles this?

NOTE: I previously tried to "fit" the series of data with a polynomial via the polyfit function and then invert it via the finverse function; but it results in an f^-1 function that is dependent on both x and y at the same time. And because x is not even spaced homogeneously, I do not even know how to plot it...

Answer 1

Since the inverse of a function is simply the reflection of the function about axis y=x, you can achieve this by transformation.

clear;close all

x=[0.0050879   0.0056528   0.0062176   0.0067822   0.0073467   0.0079111   0.0084752   0.0090392   0.0096030   0.0101666   0.0107299   0.0112930   0.0118558   0.0124184   0.0129807   0.0135428   0.0141045   0.0146659   0.0152270   0.0157877   0.0163481];
y=[0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   0.0000e+00   4.4901e-05   4.4901e-05   8.9801e-05   1.3470e-04   1.7960e-04   2.2450e-04   2.6940e-04   3.5920e-04   4.0411e-04   4.9391e-04   5.8371e-04];
x_ori = x; y_ori = y;

figure; hold on;axis equal;grid on;axis tight;
plot(x,y);ls={'ori'};
plot(x,x,'-.');ls{end+1}='y=x';

%rotate x and y about z by -45 degree
rotz1=[cosd(45) sind(45) 0; -sind(45) cosd(45) 0; 0 0 1];
data = [x(:) y(:) ones(size(x(:)))]*rotz1;
x = data(:,1); y = data(:,2);
%plot(x,y,'--');ls{end+1}='rotated';

%flip x
x = -x;
%plot(x,y,'--');ls{end+1}='flipped';

%rotate x and y back
rotz2=[cosd(-45) sind(-45) 0; -sind(-45) cosd(-45) 0; 0 0 1];
data = [x(:) y(:) ones(size(x(:)))]*rotz2;
x = data(:,1); y = data(:,2);
plot(x,y);ls{end+1}='result';

%% Or a transformation matrix.
flipx = eye(3); flipx(1) = -1;
TM = rotz1*flipx*rotz2;

data = [x_ori(:) y_ori(:) ones(size(x(:)))]*TM;
x = data(:,1); y = data(:,2);
plot(x,y,'--');ls{end+1}='result by TM';
legend(ls);

---

Update

I was so into transformation...Then I found the transformation matrix is simply

    | 0 1 0 |
T = | 1 0 0 |
    | 0 0 1 |

which is basically swapping x and y....and yes, that is the fundamental thing about the inverse of a function.

So simply:

TM = [0,1,0;1,0,0;0,0,1];
data = [x_ori(:) y_ori(:) ones(size(x(:)))]*TM;
x = data(:,1); y = data(:,2);

and yes, a blow to my pride, @m7913d's answer is much simpler.

Answer 2

Inverting a function represented by a series of (x, y) data points is done by changing the position of x and y: (y, x). This can be visualised in Matlab as follows:

figure
hold on
plot(x, y, 'red')
plot(y, x, 'blue')
from = min(min(x), min(y));
to = max(max(x), max(y));
plot([from to], [from to], 'k--')
legend('Original', 'Inverted')