how do i find the extremes of an always decreasing curve?

Hello you all,

so i have this project im working on where i have several angles measured with a robot's head.

I did the polyfit function to find a curve and see how this measurements behave with distance.

Now i want to compare this last curve with the expected curve i am suppose to get:

Of course the practical curve doesn't exactly fit the theoretical one because the measurement process has too much error but especially in the beginning of the practical curve it s possible to see some oscillations which i think are the equivalent of the steps in the theoretical one.

So i want to find some way of knowing where exactly in the practical curve those small oscillations peak.

I tried checking the first derivative but i couldn't get the points because the first derivative is almost always negative.

Another approach i tried to make is to see the variation of the first derivative, but also couldn't get anywhere.

Anyone has any idea how to solve this?

[Star Strider]

It’s obvious that you need more data, or a definitive mathematical model of the robot head movement that you can fit to the data you have.

If you were to take only the values of a parabola described by ‘f(x)=x^2’ on the ‘x’ interval (-4,-3), you would assume it goes to -Inf even though you know it has a minimum at ‘x=0’. With more data, or a mathematical description of the function to which you can fit your data, you could determine the minimum (assuming one exists).

[Pedro Vilas-Boas]

The mathematical model of the robot head is pretty simple. The robot head are 8 photodiodes aligned in 8 different angles, from around 18º to 90º. The robot's head will receive light in max 2 photodiodes and will calculate the angle with which the light is received with a weighted average: sum(angle_of_photodiode*power)/sum(power) Thats why the theoretical curve is a step curve.

[Star Strider]

That really isn’t the sort of model I was hoping for.

You need a model that incorporates the motion of the robot head on any trajectory it could possibly take, and then use that information. Even if had three degrees-of-freedom on the interval (0,2*pi) in each (and I would guess that it does not), it would have some sort of limits that you would incorporate in your model.

[Pedro Vilas-Boas]

You can assume that the head of the robot is always perfectly aligned with the receiving light. The raw data is the result of several static measurements. So i don't understand why it is needed to incorporate the motion in the model.

[Star Strider]

I really can’t assume anything, because I don’t know anything about your experiment.

You imply that the light is then on the interval (0,2*pi) in all three dimensions, and that the robot head isn’t really attached to anything. That doesn’t seem realistic to me.

However, if you managed to perfect levitation of the robot head, then forget about the robot and use levitation for your thesis!

[Pedro Vilas-Boas]

This is the robot's head:

Right now im only using the head so i move it and rotate it myself. And for initial measuremnts basically i point the photodidodes to the light source and leave the head on the ground. Then it calculates the angle with which receives the light. After doing for several positions on the floor i got the raw data represented by the red crosses above. My goal is to show using the red practical data the levels which are expected in the theoretical green data.

[Star Strider]

It seems to me then that you need to regress the theoretical and experimental data with a linear model. Then show that the two are not significantly different. The F statistic is likely best for this. See the regress and related functions (Statistics and Machine Learning Toolbox) for details.

[Pedro Vilas-Boas]

I think i can't use a linear model, otherwise i will never see the levels i am looking for.

Answer 1

[Image Analyst]

Try this:

% Create sample data.
A = 90 : -1 : 40;
% Plot it
plot(A, 'b*-');
grid on;
hold on;
% Define the values that the data SHOULD take on.
levels = [90, 85, 79, 73, 68, 64, 58, 53, 48, 42]
% Find which element of levels is closest to A and assign that to quant_A
for k = 1 : length(A)
  [~, index] = min(abs(A(k)-levels));
  quant_A(k) = levels(index);
end
% Plot the quantized data.
plot(quant_A, '*-', 'Color', [0,0.5,0]);

The blue is your original, noisy data. The green is how the data got rounded so that it got forced to the nearest desired/perfect level that you expect it should be.

[Pedro Vilas-Boas]

It isn't exactly what i was looking for but this way i can compare the 2 datas, thank you very much!

Answer 2

[Walter Roberson]

I would not recommend polyfit() for that kind of data.

As you have physical movement constraints that possibly impose some smoothness to the movement, it might be reasonable to use a spline. You can find the derivative of a spline using http://www.mathworks.com/matlabcentral/fileexchange/24996-spline-derivative

[Pedro Vilas-Boas]

thank you, i will try

Answer 3

[Image Analyst]

I don't think you did a polyfit() on the entire set of data unless you used an order that was ridiculously high, which would be insane. I think you must have done a piecewise/moving polynomial fit, which is what a Savitzky-Golay filter is, and that is done by the function sgolayfilt() in the Signal Processing Toolbox.

Anyway, it seems your smooth blue curve is supposed to actually look like that stair step green function according to what you tell us. But I don't see any substantial oscillations or peaks in the blue curve. There are in the red raw data but I thought you didn't want to use that and that's why you smoothed it to obtain the improved blue curve. So are you now trying to figure out how to change the measured and processed blue curve into the desired green stair steps?

Finally, what is the last graph that shows a single blue curve for?

[Pedro Vilas-Boas]

Yes it s ridiculously high (20). I don't need to work with the blue curve, the main goal here to to process the raw data and prove that it behaves similar to the theoretical green curve.

The blue curve in the end is the result of the polyfit alone just so you could see better.

[Image Analyst]

Just quantize your signal with imquantize():

quant_A = imquantize(A,levels)

You pass in levels = [90, 85, 79, 74, ......] and you'll get out the values of the closest step level.

[Pedro Vilas-Boas]

Ok but i want to get to those levels, just processing the raw data.

[Pedro Vilas-Boas]

The thing is that in real world some of this levels are wrong by few degrees. So i want to find the levels of the practical raw data and compare to the theoretical.

Answer 4

[John D'Errico]

This will be a fairly difficult problem to solve in a stable way. You are trying to fit a curve to pretty noisy data. There are some points with sufficiently high residuals that it will drag your curve around mercilessly. Do NOT however, use a high order polynomial to fit this curve. Down that path lies problems.

I'd like to actually test out some ideas on your data, but you have not posted the data itself. For example, there are some variations of smoothing spline that one might try. As well, there are robust modeling schemes to try to reduce the impact of that noise, because it appears there are some outliers. So it is a bit difficult to know what I would recommend.

No matter what you do though, finding small deviations from a noisy curve will be a difficult task to do at all well.