I need to construct an interpolating function from a 2D array of data. The reason I need something that returns an actual function is, that I need to be able to evaluate the function as part of an expression that I need to numerically integrate.
For that reason, "interp2" doesn't cut it: it does not return a function.
I could use "TriScatteredInterp", but that's heavy-weight: my grid is equally spaced (and big); so I don't need the delaunay triangularisation.
Are there any alternatives?
(Apologies for the 'late' answer, but I have some suggestions that might help others if the existing answer doesn't help them)
It's not clear from your question how accurate the resulting function needs to be (or how big, 'big' is), but one approach that you could adopt is to regress the data points that you have using a least-squares or Kalman filter-based method. You'd need to do this with a number of candidate function forms and then choose the one that is 'best', for example by using an measure such as MAE or MSE.
Of course this requires some idea of what the form underlying function could be, but your question isn't clear as to whether you have this kind of information.
Another approach that could work (and requires no knowledge of what the underlying function might be) is the use of the fuzzy transform (F-transform) to generate line segments that provide local approximations to the surface.
The method for this would be:
Define a 2D universe that includes the x and y domains of your input data
Create a 2D fuzzy partition of this universe - chosing partition sizes that give the accuracy you require
Apply the discrete F-transform using your input data to generate fuzzy data points in a 3D fuzzy space
Pass the inverse F-transform as a function handle (along with the fuzzy data points) to your integration function
If you're not familiar with the F-transform then I posted a blog a while ago about how the F-transform can be used as a universal approximator in a 1D case: http://iainism-blogism.blogspot.co.uk/2012/01/fuzzy-wuzzy-was.html
To see the mathematics behind the method and extend it to a multidimensional case then the University of Ostravia has published a PhD thesis that explains its application to various engineering problems and also provides an example of how it is constructed for the case of a 2D universe: http://irafm.osu.cz/f/PhD_theses/Stepnicka.pdf
If you want a function handle, why not define f=#(xi,yi)interp2(X,Y,Z,xi,yi) ?
It might be a little slow, but I think it should work.
If I understand you correctly, you want to perform a surface/line integral of 2-D data. There are ways to do it but maybe not the way you want it. I had the exact same problem and it's annoying! The only way I solved it was using the Surface Fitting Tool (sftool) to create a surface then integrating it.
After you create your fit using the tool (it has a GUI as well), it will generate an sftool object which you can then integrate in (2-D) using quad2d
I also tried your method of using interp2 and got the results (which were similar to the sfobject) but I had no idea how to do a numerical integration (line/surface) with the data. Creating thesfobject and then integrating it was much faster.
It was the first time I do something like this so I confirmed it using a numerically evaluated line integral. According to Stoke's theorem, the surface integral and the line integral should be the same and it did turn out to be the same.
I asked this question in the mathematics stackexchange, wanted to do a line integral of 2-d data, ended up doing a surface integral and then confirming the answer using a line integral!
Related
I have been looking for a Matlab function that can do a nonlinear total least square fit, basically fit a custom function to data which has errors in all dimensions. The easiest case being x and y data-points with different given standard deviations in x and y, for every single point. This is a very common scenario in all natural sciences and just because most people only know how to do a least square fit with errors in y does not mean it wouldn't be extremely useful. I know the problem is far more complicated than a simple y-error, this is probably why most (not even physicists like myself) learned how to properly do this with multidimensional errors.
I would expect that a software like matlab could do it but unless I'm bad at reading the otherwise mostly useful help pages I think even a 'full' Matlab license doesn't provide such fitting functionality. Other tools like Origin, Igor, Scipy use the freely available fortran package "ODRPACK95", for instance. There are few contributions about total least square or deming fits on the file exchange, but they're for linear fits only, which is of little use to me.
I'd be happy for any hint that can help me out
kind regards
First I should point out that I haven't practiced MATLAB much since I graduated last year (also as a Physicist). That being said, I remember using
lsqcurvefit()
in MATLAB to perform non-linear curve fits. Now, this may, or may not work depending on what you mean by custom function? I'm assuming you want to fit some known expression similar to one of these,
y = A*sin(x)+B
y = A*e^(B*x) + C
It is extremely difficult to perform a fit without knowning the form, e.g. as above. Ultimately, all mathematical functions can be approximated by polynomials for small enough intervals. This is something you might want to consider, as MATLAB does have lots of tools for doing polynomial regression.
In the end, I would acutally reccomend you to write your own fit-function. There are tons of examples for this online. The idea is to know the true solution's form as above, and guess on the parameters, A,B,C.... Create an error- (or cost-) function, which produces an quantitative error (deviation) between your data and the guessed solution. The problem is then reduced to minimizing the error, for which MATLAB has lots of built-in functionality.
I have a curve which looks roughly / qualitative like the curves displayed in those 3 images.
The only thing I know is that the first part of the curve is hardware-specific supposed to be a linear curve and the second part is some sort of logarithmic part (might be a combination of two logarithmic curves), i.e. linlog camera. But I couldn't tell the mathematic structure of the equation, e.g. wether it looks like a*log(b)+c or a*(log(c+b))^2 etc. Is there a way to best fit/find out a good regression for this type of curve and is there a certain way to do this specifically in MATLAB? :-) I've got the student version, i.e. all toolboxes etc.
fminsearch is a very general way to find best-fit parameters once you have decided on a parametric equation. And the optimization toolbox has a range of more-sophisticated ways.
Comparing the merits of one parametric equation against another, however, is a deep topic. The main thing to be aware of is that you can always tweak the equation, adding another term or parameter or whatever, and get a better fit in terms of lower sum-squared-error or whatever other goodness-of-fit metric you decide is appropriate. That doesn't mean it's a good thing to keep adding parameters: your solution might be becoming overly complex. In the end the most reliable way to compare how well two different parametric models are doing is to cross-validate: optimize the parameters on a subset of the data, and evaluate only on data that the optimization procedure has not yet seen.
You can try the "function finder" on my curve fitting web site zunzun.com and see what it comes up with - it is free. If you have any trouble please email me directly and I'll do my best to help.
James Phillips
zunzun#zunzun.com
I am looking for numerical integration with matlab. I know that there is a trapz function in matlab but the precision is not good enough. By searching it online, I found there is a quad function there it seems only accept symbolic expression as input. My data is all discrete and one-dimensional. Is that any way to use quad on my data? Thanks.
An answer to your question would be no. The only way to perform numerical integration for data with no expression in Matlab is by using the trapz function. If it's not accurate enough for you, try writing your own quad function as Li-aung said, it's very simple, this may help.
Another method you may try is to use the powerful Curve Fitting Tool cftool to make a fit then use the integrate function which can operate on cfit objects (it has a weird convention, the upper limit is the first argument!). I don't think you will get much accurate answers than trapz, it depends on the fit.
Use the spline function in MATLAB to interpolate your data, then integrate this data. This is the standard method for integrating data in discrete form.
You can use quadl() to integrate your data if you first create a function in which you interpolate them.
function f = int_fun(x,xdata,ydata)
f = interp1(xdata,ydata,x);
And then feed it to the quadl() function:
integral = quadl(#int_fun,A,B,[],[],x,y) % syntax to pass extra arguments
% to the function
Integration of a function of one variable is the computation of the area under the curve of the graph of the function. For this answer I'll leave aside the nasty functions and the corner cases and all the twists and turns that trip up writers of numerical integration routines, most of which are probably not relevant here.
Simpson's rule is an approach to the numerical integration of a function for which you have a code to evaluate the function at points within its domain. That's irrelevant here.
Let's suppose that your data represents a time series of values collected at regular intervals. Then you can plot your data as a histogram with bars of equal width. The integrand you seek is the sum of the areas of the bars in the histogram between the limits you are interested in.
You should be able to apply this approach to data sets where the x-axis (ie the width of the bars in the histogram) does not show time, to the situation where the bars are not of equal width, to the situation where the data crosses the x-axis, and most reasonable data sets, quite easily.
The discretisation of your data establishes a limit to the accuracy of the result you can get. If, for example, your time series is sampled at 1sec intervals you can't integrate over an interval which is not a whole number of seconds by this approach. But then, you don't really have the data on which to compute a figure with any more accuracy by any approach. Sure, you can use Matlab (or anything else) to generate extra digits of precision but they don't carry any meaning.
I want to write an implementation of a (not a binary) tree and and run some algorithms on it. The reason for using the matlab is that the rest of all programs are in matlab and it would be usful for some analysis and plotting. From an initial search in matlab i found that there aren't thing like pointers in matlab. So I'd like to know the best ( in terms on convinience) possible way to do this in matlab ? or any other ways ?
You can do this with MATLAB objects but you must make sure you use handle objects and not value objects because your nodes will contain cross-references to other nodes (i.e. parent, next sibling, first child).
This question is very old but still open. So I would just like to point readers to this implementation in plain MATLAB made by yours truly. Here is a tutorial that walks you through its use.
Matlab is very well suited to handle any kind of graphs (not only trees) represented as adjacency matrix or incidence matrix.
Matrices (representing graphs) can be either dense or sparse, depending on the properties of your graphs.
Last but not least, graph theory and linear algebra are in very fundamental ways related to each other see for example, so Matlab will be able to provide for you a very nice platform to harness such relationships.
Does MATLAB have a built-in function to find general properties like center of mass & moments of inertia for a polygon defined as a list of (non-integer valued) points?
regionprops performs this task for integer valued points, on the assumption that these represent indices of pixels in an image. But the only functions I can find that treat non integral point lists are polyarea and inpolygon.
My kludge for now is to create a bwconncomp structure with all the points multiplied by some large value (like 10,000), then feeding it in to regionprops, but wondered if there is a more elegant solution.
You should check out the submission POLYGEOM by H.J. Sommer on the MathWorks File Exchange. It looks like it has all the property measurements you want, and nice documentation describing the formulae used in the code.
I don't know of a function in MATLAB that would do this for you.
However, poly2mask might be of use for you to create the pixel masks to feed into regionprops. I also suggest that, should you decide to go this route, you carefully test how much the discretization affects the results, so that you don't create crazy large arrays (and waste time) for no real gain in accuracy.
One possibility is to farm out the calculations to the Java Topology Suite. I don't know about "moments of inertia", but it does at least have a centroid method.