linear combination of unknown waves

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Hello!
I am working on electrophysiological data and measure a current probably coming from two different channels (let's call their single currents G(V) and H(V) ). I have data showing the current after 8 different timepoints ( I(V,t) ) and would like to separate the two components with a ratio (A vs (1-A)) changing over time:
I(V,t) = A(t)*G(V) + (1-A(t)*H(V))
As I don't know G(V) and H(V), simpleICA doesn't seem to work, if I understood it correctly. (http://www.igorexchange.com/node/3021)

A reasonable assumption would be, that after 4 seconds, the first component is completely gone, meaning A=0, leading to I(V,4s)=H(V).
This would lead to G(V) = (I(V,t)-I(V,4s))/A(t) + I(V,4s)
But this is an assumption that I would prefer not to use, because I don't wanna run into bs-in, bs-out trap. ;)

I do have waves containing I(V,t) and I(V,4s), but would still have two functions that are unknown.
So, my question is, how could I determine these functions, G(V) and A(t)? Is it possible from a mathematical point of view, at all?

Yours, Dominik
That's a tricky one. My approach would be an iteration loop. Make an initial guess for either G(V) or H(V). In your case H(V)=I(V, 4s) could be a good initial guess. Use this initial value for H(V) to calculate G(V) for the 7 other points in time. Average those 7 G(V) channels to create a best guess for G(V). Use this best guess to calculate H(V) for all 8 points in time. Average those 8 channels to get a better estimate for H(V). Go through this loop over and over until H(V) and G(V) converges... (fingers crossed).

The main problem I see is that for any solution to G(V) you can always add X times H(V) and get another solution with another A(t). You somehow need to limit the solutions. Maybe by requiring all values of G(V) and H(V) to be positive.

Another approach would be to assume H(V) and G(V) can be described by a mathematical function which you can then fit to the data.

October 13, 2017 at 07:53 am - Permalink

Let's separate the two aspects: "I am convinced that I can solve this problem" and "I am convinced that Igor Pro will do the job".

I doubt that anyone can solve the problem as it is stated. It gives three unknowns (A, G, and H) but only ONE EQUATION. OK, make an assumption about ONE of the components A. We now have three unknowns and two equations.

We will have to define an equation for G, an equation for H, or a relationship between G to H. We will also have to define the functional form of A before the time where A = 0. Otherwise, even with one of the additional requirements defined, we will still have three unknowns and two equations until A = 0, and only AFTER A = 0 will we be able to solve anything.

Once we have additional equations for G, H, or their relationship over all time, the rest is programming.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

October 13, 2017 at 11:29 am - Permalink

It's true that we have three unknowns: A(t), G(V) and H(V), but we have the solution for eight separate points in time. Simply looking on the problem as one equation with three unknowns is too simple. Nevertheless, we still need more information to solve it.

I did a somewhat similar analysis recently to separate measured Auger-yield NEXAFS spectra into photoemission and NEXAFS contributions, see https://doi.org/10.1016/j.elspec.2017.05.012. I think your problem is similar. I had 2D data with photon energy and kinetic energy axes. You have 2D data with voltage and time axes.

October 14, 2017 at 07:34 am - Permalink

olelytken wrote:
It's true that we have three unknowns: A(t), G(V) and H(V), but we have the solution for eight separate points in time. Simply looking on the problem as one equation with three unknowns is too simple. Nevertheless, we still need more information to solve it.


We have I at eight different points in time. At any one of those eight points, we still do not know A, G, or H. So, even at points where we measured I, we have three unknowns (A, G, and H) and only one equation I = A*G + (1 - A)*H.

An alternative approach is to say that we have four total parameters to characterize the system ... I_theory, A, G, and H. We have two relationships, I_theory = ... (above) ... and I_theory = I_measured. We still need two more equations.

One difference in the photoemission analysis referenced is, photoemission and Auger emission rates have first-principle relationships (through their respective excitation cross-sections, all else being the same). That is equivalent to providing either one additional equation that relates G and H or possible two additional equations that define both G and H separately (I'd have to think about this in reference to the approach taken in the paper ... I suspect the former approach was applied if the goal was to extract A through iterative analysis).

This actually defines the problem in this way:

* When a relationship between G and H is defined and only one valid solution for A is acceptable (e.g. 0 <= A <= 1 for all times), the analysis can be done through an iterative optimization or a find-roots approach.
* When both G and H are defined from first principles, then A is also defined from first principles.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

October 14, 2017 at 02:25 pm - Permalink