Source water inversion with multiple state variables

[ggebbie]

• Start from test "source water inversion: obs at one time, many surface regions, with circulation lag" or the case with an observational timeseries
• construct a random M (water-mass matrix) like before, but construct two "x" variables
• decide how to return the x solution: 1) 3D DimArray with time, surface region, variable type, 2) NamedTuple with 2 fields that each have time, surface region, 3) Two or more DimArrays
• update convolve (i.e., the forward function that does x -> y) to handle the output from the previous step.
• Store convolve output as the synthetic data. Ultimately the data should be corrupted according to Cnn.
• Given knowledge of y and dims(x), produce a first guess x0 that has the same form as step 3 above
• Run x0 through the forward function to make a first prediction of the observation(s) y0
• See if datacost still works to compute half of the cost function
• Update construction of Cxx to handle two variables, assume diagonal in first step, relax this later, probably best to use vec to change x0::Union{DimArray,NamedTuple,etc.} into a vector, Cxx requires the same type of bookkeeping, perhaps a new function to do this automatically would be useful
• Modify controlcost to handle two state variables, must take the Cxx output from the previous step
• If using a timeseries of observations, modify impulseresponse to take M water-mass matrix and return E the matrix used in BLUEs.solve
• Check that E*vec(x0) gives the same thing as convolve(x0,M)
• Pass new problem to BLUEs.solve
• Modify BLUEs.getproperty to handle cases where there are two state variables, if the state variables are in a DimArray then ndims>2 could be checked to see if there are multiple state variables
• Check that output and covariance is appropriately displayed in the REPL in the same form as x and x0, and that x.x contains this information

[b-r-hamilton]

Added testset for two state variables, one observation. I assume that the one observation is some linear combination of the two state variables propagated through the M matrix.
Some choices I made -

• 3D DimArray for x
• convolve method is not quite right - see comment
• any reason not to use copy to generate x_0 template?
• Cxx basically worked in original format...?

[ggebbie]

Awesome work! Everything ran like a dream and seemed to give reasonable output.

[ggebbie]

• 3D DimArrays seem to work well, good choice. I plan to take account (i.e., make an issue) for all the little formatting issues between making a 1-element vs. 1x1-element UnitfulMatrix. Then I can fix them in UnitfulLinearAlgebra so that the code just does what we want. Thanks for handling these things which must have been annoying!

• I saw the convolve comment in src/BLUEs.jl. I had the same issue and resorted to a conditional statement (if ndims ...) which isn't applicable when having multiple state variables. Idea: make coeffs simpler and pass it to convolve to send a signal that there are multiple state variables. coeffs::Vector could simply be the last argument of convolve. Then add a multiplication inside of convolve.

• No problem using copy in this example. I didn't use it because I was pretending that I only knew dims(x) but not parent(x).

• Yes! Cxx works in the original format, yet it calculates a 2x2 block matrix with the proper combination of units in each block. I think this is the way to go. Likewise, the "E" matrix was derived in a pretty snazzy and automated way. This is what I was hoping for.

• Slight bug that I probably introduced: Cxx and Cnn should have squared sigma-x and sigma-n values on the diagonal.

• Consider having a "show" method that gives the name of each state variable and outputs a 2D DimArray for each one.

• Please do try the next case if you get a chance! I think this is going well and we can get this into BLUEs#main. We can collect extra stuff in BLUEs and clean it up later.

[b-r-hamilton]

Did the two state variables + timeseries test! Fixed the following issues, for both new tests

• Updated convolve so that it both convolves and linearly combines for the two state variable case
• made sure Cxx and Cnn are \sigma^2 on the diagonal, not \sigma
• added the show method for the x variable, probably still needs some help

[b-r-hamilton]

Revisiting some of the initial things in Jake's list that I apparently didn't feel like dealing with 2 weeks ago

• controlcost and datacost just work, apparently. No modification needed
• getproperty also works for the two state variable case, returns a vector with the first dimension/state variable info on top, concatenated with the second dimension/state variable. As long as this behavior is okay, I don't think I need to modify this either.