Can you Explain this SVD Application?

[ecastro]

Here is the website:

http://www.columbia.edu/itc/applied/e3101/SVD_applications.pdf

I need help on understanding the second part of the document, page 13 onwards. On page 15, it showed 3 data sets, relative elevation as a function of kilometers across axis, however at page 16, the author constructed a matrix $A$ which is $179 \times 80$. This is where I get lost. How did it come up with such a matrix of such size? Is the 80 here the number of x-axis points on the data set (assuming each point has a frequency of 1 kilometers across axis)? How about the 179? I'm supposed to do something similar, but instead of ocean ridges I need to apply it with spectral reflectances.

I have a basic understanding of what is Singular Value Decomposition (SVD), but I am not completely familiar with it. For example, I do not know how to acquire the eigenvalues acquired from SVD, since I will most likely be using the Matlab built-in function to calculate the SVD of a matrix.

Thank you in advance.

 

[mfb]

What is the spreading rate? Looks like there are 179 profiles. The 179x80 is the input to SVD, and SVD does not care (much) about where the input comes from.

 

[ecastro]

So, you mean there are a total of 179 data sets, and the three data sets shown on page 15 are three of them? If so, I wonder if it's possible to apply the SVD if the y-axis (the spreading rate) is non-numerical; if I were to consider the plot as three dimensional, i.e. the x-axis is kilometers per axis, y-axis is the spreading rate, and the z-axis is the relative elevation, then if I were to apply it with spectral reflectances, the x-axis is the wavelength, y-axis is the sample number (or ID), and the z-axis is the reflectance?

 

[mfb]

So, you mean there are a total of 179 data sets, and the three data sets shown on page 15 are three of them?

That's how I would interpret that, but I'm not a geology expert.

If your samples are all the same, SVD should work very well. If your samples are different in some smooth way, SVD should work as well. If your samples are expected to be completely different, SVD could be ... interesting (but still working if there is some pattern behind them).

 

[ecastro]

Thank you. But I have another question: what if my data sets have different resolutions, i.e. I have a data set which has 100 data points, another one has 50 data points, etc., how will I form my matrix for the SVD?

 

[mfb]

Hmm.. challenging. Does the data span the same range? If yes, some interpolation might work. You'll get wrong results for some eigenvalues related to short-distance fluctuations (because you don't reproduce them properly), but those could be small.

 

[ecastro]

Unfortunately, they don't span the same range. What I did was I reconstructed the data by taking a constant interval and taking the average of the points within the interval, if there is no data within the interval, I consider the value to be zero. What are the consequences for this?

 

[mfb]

I don't think that will work well.
How does your data coverage look like?

 

[ecastro]

I only considered the range of visible (blue to red) and near infrared for my application. Most of my data share the same range, there are some, however, that starts around the red region of the electromagnetic spectrum and some don't have the same resolution as the other data.