Determine differentiability of a x and y dataset

Hello everybody,

I have a dataset (x old, y old). Now I do some math on it and get a new dataset (x new, y new). To check whether my calculations are correct, the curve of the new datset must be differentiable.

Do you have any ideas, how I could do this "check"?

Thank you

Cheers

Christian

[Christian]

As you can see, the critical spot of the new dataset looks sometimes like this:

And now I need to find out, if the new curve is smooth or looks like above. It would be perfect if the code would give me a result which looks something like this:

new_dataset_1 = false ("not differentiable")

new_dataset_2 = true ("differentiable")

new_dataset_3 = true

new_dataset_4 = false...

Answer 1

[Guillaume]

The concept of diferrentiable does not mean much for a discretised function. By definition dy/dx always exists when you have a vector of x and y values, unless one of y or x is infinite or NaN.

So really the only thing you could test is

assert(all(isfinite(xnew)) & all(isfinite(ynew)))

[Christian]

Well, I know this is gonna be tricky, that is the reason why I Need your help.

If I use "isfinite" on my new datapoints, I won't get anything else than "1" (true). Because there is not a single datapoint which tends towards infinity.

So maybe I don't really get your point?

[Guillaume]

Well, if all the points are finite then your discretised function is differentiable.

For a discreet set of points, the derivative is equivalent to y(i+1)-y(i), that is the diff function in matlab. So for dicreet points, being differentiable simply means that you can calculate diff(y)./diff(x)

Note: I am assuming there's no duplicate x, since otherwise diff(x) is 0 and in this case, the function is indeed not differentiable.

[Christian]

That is a good idea! I might just check diff(x_new). If this returns me only positiv values, the curve should be rather smooth. If it returns any negative values, I have a critical point somewhere.

The only problem which this method does not cover, if I have a "perfect sharp bend" where my x data do not "go backwards".

[Guillaume]

Well,

all(diff(x) > 0) | all(diff(x) < 0)

will tell you if all the x values are strictly monotonically increasing or decreasing.

I'm not sure what you mean by perfect sharp bend.

[Christian]

As you can see at the second picture I added above, the "curve" makes a "U-turn" and then crosses the "curve". In this case, I will get some negative diff(x) values and I can use this fact as an evaluation whether the curve is "differentiable" or not.

But, if I am really unlucky and this case appears:

I won't get any negative diff(x) values but the curve is still not "differentiable".

Do you know what I mean?

[Jan]

Guillaume is right: For a discretized function, the term "differentiable" has no meaning. A line like x=[1,2,3], y=[1,2,100] might or might not represent a differentiable function, because even a smooth function can contain a huge derivative in one point. There is also no to "proove" if sin(1/x) is differentiable in x=0 if all you have is a finite number of its values.

All you can decide based on a set of data points is if the difference between neighboring points exceeds a certain limit or not. But this does not mean differentiablility.

[Christian]

I know what differentiability means, therefore I asked this question to the community.

But I think I might get along with the diff(x) < 0 thing in 99% of the time.