MATLAB: How to find a unique point of a set of data

inflectioninterpolation

I am given a set of data, P and Phase. I have plotted this data and now I want to find a point on the line which can be used to characterize the line – so for example the inflection point, which should be unique. Probably I will have to interpolate the data first, maybe using:

P_intrp    = linspace(min(P),max(P),100)
data_intrp = interp1(P,Phase,P_intrp,'spline');

But what now? I think I can find the inflection point by setting the second derivative to 0. There is no point in my line with first derivative equal to 0.

idx = find(P,diff(diff(data_intrp)) = 0);
P_infl = P(idx);
Phase_infl = data_intrp(idx);

I don't know, if this would work. It's important to me to know which value of P belongs to the inflection point. I also want to know the corresponding Phase.

further information: I want to find such a unique point, in order to find out how the following lines can be obtained from each other just by displacement:

Best Answer

If you want to inflection points in the data you plotted over the range that you plotted, you might not be successful. I don’t see any inflection points.

I would use the gradient function (not diff) to calculate the derivatives, since the output is the same length as the input vector, and this simplifies your code.

Related Solutions

MATLAB: I want to find inflections point for point data input

The inflections points are when the sign of the difference of your y changes. So, whichever way you obtain your y points (as is or using linear or spline interpolation), the indices of the inflection points are:

inflection_idx = find(diff(sign(diff(y)))) + 1;  %+1 to compensate for the index shift caused by the double diff

Note that linear interpolation won't change the location of the inflection points, it'll still be at y=9 and y=6.5, and even with spline interpolation it's not going to change much.

MATLAB: Finding inflection points of a noisy signal

Differentiation is a noise amplification process. Why does that matter here? And what do I mean by that claim, anyway?

First, the claim. Computing a derivative goes back to a finite difference, thus deltay/deltax, taken as a limit as deltax goes to zero. If your data is noisy, then the noise in y is divided by a tiny number, thus amplifying the noise.

Another way to look at it if you prefer, you can view estimation of the derivative from data as trying to solve a first kind integral equation, something known to be ill-posed. ill-posed means here that the solution amplifies any noise in the data heavily.

Worse, higher order derivatives are larger amplifiers of noise in the data. So computing the second derivative (in the presence of noise) is a more difficult problem than computing the first derivative.

Why is any of this relevant to the question at hand? Because an inflection point is a point where the second derivative changes sign! So, you want to infer where that second derivative changes sign, therefore you need to estimate that second derivative function in the presence of significant noise.

Don't expect magic to happen here. It won't. That noise will cause all sorts of nasty stuff to happen to you.

x = linspace(0,2*pi,100);
y = sin(x) + randn(size(x))/5;
plot(x,y,'o')
grid on

Yes. We all know the inflection point happens at x == pi. But this noise is pretty nasty. Any kind of local smoothing you do will be overwhelmed by that noise. Even visually, I can conclude the inflection point is somewhere between 2.5 and 3.5, but I'm not sure I would have much more confidence than that.

The simplest trick may be to use some sort of smoothing spline. I'll use my SLM toolbox here.

slm = slmengine(x,y,'knots',10,'reg','cross','plot','on');

A credibly decent curve fit there, given the amount of noise. Less data or more noise might have been problematic though.

Look at the second derivative function. (This is available as a menu choice form the plot from my SLM toolbox .)

So it appears the inflection point happens just a bit beyond x==3. Although we should also find zero crossings near 0 and 2*pi.

We can find those second derivative crossings using slmsolve, another tool from that toolbox.

slmsolve(slm,0,2)
ans =
     0.099671       3.0826       6.2232

As expected, there were three of them.

While it might seem pretty decent as an estimate. I'll bet if you tried to compute confidence intervals on that estimate based on tools such as a jackknife or bootstrap, they might be moderately wide. Again, fewer data points or more noise would result in a great deal more uncertainty.

Note that I added some capability to slmsolve just to enable this computation. I've posted a new version on the FEX. You should find it there now.

Best Answer

Related Solutions

MATLAB: I want to find inflections point for point data input

MATLAB: Finding inflection points of a noisy signal

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