Is it possible to do in MATLAB? Of course it is. It of course depends on the skill of the person writing the code. It depends on the signal to noise ratio. High noise problems will be problematic for any code. And any heuristic you devise will fail on SOME set of carefully chosen data. It depends on your requirements for doing this automatically. Should the code be able to know automatically how many lines to fit? Again, if you say yes to that, then I can easily devise a set of data that will cause insolvable problems. Sometimes that will be simple, but not always.
And whether this can be done in MATLAB is a silly question. (Sorry, but it is.) MATLAB is just a programming language. It is the skill of the programmer that matters, NOT the language. How robust is the algorithm depends on the programmer, their understanding of the problem, and their knowledge of statistics, of numerical methods, etc. And of course, it depends on how well the programmer understands the data which they will be seeing. How much noise should they expect?
x=[5705 5690 5671 5667 5604 5585 5555 5542 5502 5501 5495];
y=[12644 12612 12570 12560 12420 12361 12278 12240 12098 12078 12005];
plot(x,y,'o')
When I look at that plot, I might see one line that can be fit. With a little more care, I might decide the bottom three points seem to follow a different slope from the rest. But is there a third segment up high? Perhaps. How noisy is the data? I don't know. Only you know that.
Anyway, what might I do?
Just looking at the plot of the data, we might make the decision from that picture that the first three points belong to one group. Then the next 4 points seem to form a cluster with a common slope. Finally, the last three points might have a slightly lower slope, and they MIGHT fall on a different line.
One simple trick is to simply compute the slope of the line segments between each consecutive pair of points. If we do so, we see this:
[xsort,tags] = sort(x);
ysort = y(tags);
segslopes = diff(ysort)./diff(xsort) % slope between each consecutive pair of points
xmid = conv(xsort,[1/2,1/2],'valid') % midpoints of each segment in x
plot(xmid,segslopes,'o')
However, if I look at this plot, it becomes clear that if we consider the variability of the slopes in blocks 2 and 3, then compare to the variability in slopes in that first "block" I might decide that we actually might have 4 lines, NOT 3 to consider. Which is it?
Any heuristic you will write will suffer from exactly these issues. How many lines are there to be found?
The comomon tool to estimate such a model is called a broken stick regression. But even there, we can find issues. For example, I used a tool of my own creation to fit that data. You can find it on the file exchange, as my SLM toolbox. Here is what it did though:
Even though I told it to break the curve into three segments, then decide where the breaks should go based on your data, do you see it made a decision that is not the same as what I first guessed? The problem is, those first three points just don't fall on a straight line very well.
