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# Inverse Least Squares with Matlab

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See the attached files.

The data for these calculations are:
knowns.xyz contains the spectra of twenty known samples, concentrations.xyz contains the concentrations of a target chemical T in each of the twenty samples, and unknowns.xyz gives the spectra of fifteen more samples whose concentrations of T you will predict. Concentrations2xyz is just to compare values afterwards.

Also, no particular reason for long rows vs. long columns.

Data to apply to regression is
*I thought I was on the right track here, but my plot is not really comparable to what it should be* Any help is greatly appreciated!

The fundamental relationship between a measured spectral data matrix S and the associ-
ated analyte concentration matrix C is
S = CP + N;
spectra are represented as matrix rows
The ordinary least squares (OLS) estimate of the concentration values is
C = SP^T(PP^T)^-1:
If we want to predict the concentration of just one target chemical T in a new sample, we
could simply use the first column of Pt(PPt)-1, call it a vector b^t, and calculate
c = sb^t (1)
where s is the sample's spectrum.

1. Inverse least squares. The simplest thing you can do is to say, I can just solve
Eq. 1 by least squares to get an estimate of b." This is called Inverse Least Squares (ILS).
(a) Solve Eq. 1 for an algebraic expression for b.
(b) There is an inverse in your answer. State what constraint that places on your ex-perimental design, i.e. , what balance is required between numbers of samples, wave-lengths, pure components, etc. to guarantee that the inverse will exist?
(c) Select the intensities at the discete pixels 100, 200...1000 from the spectra in knowns
and call this your S matrix. (In Matlab, this would mean typing something like
>>S = S(100:100:end).) Use ILS to generate b, and then predict the concentration
of T in unknowns. Scatter-plot your predictions versus the true values, which are
provided in concentrations2.xyz.

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#### Solution Preview

The attached script reg.m shows how to solve the problem and just in case I also attach a jpg plot as I recall you had some problems with your Matlab.

There are more points of confusion, such as the explanation starting with S = CP + N and C = SP^T(PP^T)^-1 goes in the wrong direction, but I have figured what you in ...

#### Solution Summary

Inverse least squares with Matlab are featured and discussed in the solution.

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