Clarification needed in documentation about error estimates (cpme, aexerr...) #46
Comments
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errors are relative errors with respect to background variance (error
variance of analysis divided by error variance of background). We will
clarify that in the documentation.
to map diag(s.P) you can apply the function statevector_unpack to map
into the real domain, but as diag(s.P) it is VERY slow we implemented
the approximations cmpe and aexerr. But if you want to compare you can
use statevector_unpack to do so.
Hope it helps
Le 2019-11-06 à 10:50 a.m., croachutas a écrit :
…
Okay, I've applied DIVAnd to create a time-varying oxygen atlas
(1956-2014, surface to 6000m depth at 0.5deg resolution). But I've run
into some problems understanding and interpreting the error fields...
In DIVAnd_cpme and DIVAnd_aexerr the error estimates are described as
"the clever poor mans error" and "the almost exact error"
respectively, while in DIVAnd_cpme_go the error output is described as
"erri: relative error field using the clever poor man's error approach".
So, I want some simple clarification: *Do DIVAnd_cpme and
DIVAnd_aexerr give the error in units of whichever variable we're
gridding or do all three functions return a relative error?*
If they are all relative errors there also needs to be some clarity
what quantity they're relative to. The now obsolete matlab version
specified the error as: "error variance of the analysis field relative
to the error variance of the background" (per
http://modb.oce.ulg.ac.be/mediawiki/index.php/Divand). However looking
at the various SeaData climatologies they appear to treat the relative
error as relative to the interpolated value at each grid-point. *So,
can you clarify if relative errors are relative to the background
variance (as per the matlab version) or the interpolated signal?*
Thanks in advance (and sorry for bothering you again).
-Chris Roach
(There's also a question-mark over if these error estimates represent
something like the standard deviation or something more like the the
standard error or 95% confidence intervals)
(Why aren't I just pulling variances out of s.P? because diag() is
horribly slow and in a real-world global case with a non-square grid
and bathymetry it's unclear how to map diag(s.P) back to a geographic
grid...)
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Okay, thanks for clarifying that. |
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Sorry for the confusion, but |
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statevector_unpack is just a routine to reshape the result from a calculation array into the topology of your grid. If you apply it to a variance it will provide a variance. Here s.P is the error variance relative to the background variance. So at the end you will still have the relative error variance but on the real grid. So no units here. |
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But in the notebook |
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I see where the misunderstanding stems from. In the notebook when speaking about the scaling, we refer to the boundary effect included in the background variance. The background error variance of diva actually increases when approaching coast and you can decide whether or not you want to take this into account when plotting the relative error. Or said differently Bref is normally one far away from the boundary but increases towards coasts and if you want to take this into account when calculating the relative error you need to devide by this Bref. To have real variances, you have to know the actual average variance of the background. More details here. Approximate and Efficient Methods to Assess Error Fields in Spatial Gridding with Data Interpolating Variational Analysis (DIVA) Beckers, Jean-Marie; Barth, Alexander; Troupin, Charles, Alvera-Azcarate, A. Journal of Atmospheric & Oceanic Technology (2014), 31(2), 515-530 |
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I see. So if there is no boundary then the errors given by |
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yep |
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Great. Thanks for clarifying! |
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Just for reference, I've done a bit more reading and it turns out there are algorithms out there to evaluate terms of the inverse of a sparse matrix without computing the inverse for the entire matrix. Taking one such function (https://github.com/DrTimothyAldenDavis/SuiteSparse/blob/master/MATLAB_Tools/sparseinv/sparseinv.m) I've crunched out the time-mean error fields for oxygen conc at 500m in the Atlantic... And the version using the selective inversion (bottom right) agrees closely with the version using a full inversion to produce the co-variance fields (bottom left). And that's a matter of seconds vs about 5 minutes run-time (for 0.5deg spatial resolution). Of cause, there is some extra overhead in the whole export from Julia, import to Matlab, do the inversion, write out and re-read into Julia...
EDIT: One word of warning, sparseinv() is sensitive to the type you pass the values in the sparse matrix as... Burnt a few hours trying to sort out why it was working in some cases but not other only to find I'd exported a couple of matrices as Float64s when testing but coded the main script to instead export them as Float32s... Due to the rounding errors the resulting matrices were not positive definite (so couldn't be factorized using chol()) with resulting failure... |
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That looks very interesting. Maybe the SuiteSparse wrapper of Julia can give access to the functions used, if not now but at some point in the future ? |
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Okay, so Sparseinv works well enough for a global time-mean on a single depth level (1/2deg resolution, runtime of 10s-60s depending on the number of ocean grid points on the particular depth level), but turns out to really slow down when moving to the time-varying anomaly (relative to the above time-mean, not surprising given the matrix for a 10 year chunk is two orders of magnitude larger than the time-mean background case).... So, been chasing up other options, so currently looking at a number of methods described in Siden et al. 2018 (paper at: https://www.tandfonline.com/doi/full/10.1080/10618600.2018.1473782, code at: https://github.com/psiden/CovApprox). Still trying to wrap my head around the more sophisticated approaches but looks like their simple Rao–Blackwellized Monte |
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Thanks for the references. For the moment we are testing some ideas along these lines. If you are ready to test, we can send you a julia code (hopefully in a few days) |
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If you want to try an experimental version: https://github.com/gher-ulg/DIVAnd.jl/blob/JMB/src/DIVAnd_diagapp.jl just use it after you made an analysis retrieving the structure s from the analysis and using s.P and s.sv Only interesting if length scale is small compared to domain size (which seems to be the case in your images) |
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Sorry about the lack of reply... Was away on holiday and had other priorities recently. JM, been unable to get your code working. Didn't proceed with the methods described in Siden et al. as run-time gains were minimal. Looked at SelInv by Lin Lin et al. (code at: https://math.berkeley.edu/~linlin/software.html, paper at: https://web.stanford.edu/~lexing/diagex.pdf), runtime to invert a matrix was moderately better than Sparseinv() in matlab, but considerable effort to convert matricies to the input format; considered the parallelized version of it (included in PEXSI: https://pexsi.readthedocs.io/), but some of the dependencies required admin level access... so can't set them up on the local compute server (thus, haven't taken it any further). Looking further, if anyone else is interested in exact solutions, it seems that Simon Etter from the National University of Singapore has implemented a (supposedly relatively efficient, if one believes his paper...) selective sparse matrix inversion algorithm in Julia (arxiv preprint at https://arxiv.org/abs/2001.06211, and Julia package at https://github.com/ettersi/IncompleteSelectedInversion.jl). Will try it out and get back if it actually works. Edit: Public version hasn't been maintained in the last three years (requirement is for Julia 0.5...). |
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And something I'd noticed but not paid much attention to that was mentioned in the above papers, is a "sampling-based approaches for estimating the selected inverse", based on ideas discussed in Bekas, Kokiopoulou,and Saad (2007 https://www-users.cs.umn.edu/~saad/PDF/umsi-2005-082.pdf) and Hutchinson (1990 https://www.tandfonline.com/doi/abs/10.1080/03610919008812866). Implementations can be found in the Siden et al paper discussed above and MacCarthy et al (2011 JGR, https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2011JB008234). Former used PCG to solve the inv(A).v(j) terms, latter uses Least-squares (without preconditioning, so takes a f***-ton of iterations...). When implemented in series this approach is rather slow but it's trivial to parallelize (in matlab literally just a mater of changing a for statement to parfor), so if you've got a dozen CPUs to throw at things runtimes should be manageable, and doesn't place too many demands on RAM. Of cause, it's only an approximation and accuracy is limited by how many random first guess vectors you throw at the problem... |
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And tested for a chunk of the North Atlantic, North Sea and Med. Runtime on the Hutchison/Sampling approach (described in papers in last post) about 1000s on four threads for 200 random 'sampling' vectors. Maps, from left to right, relative error obtained by inversion of precision matrix; relative error from CPME; relative error from Hutchinson approach, and relative error from Hutchinson approach with low values (<1E-3) eliminated and smoothing (2.5degx2.5degx3year window) applied: CPME produces spatial patterns similar to the full error but almost universally under estimates values by a factor of 2-10. The Hutchinson approach produces error values and spatial patterns consistent the full error, but produces a scatter of individual grid-points with values much less than the full error (hence smoothing in the last panel), this scatter reduces with an increase in the number of 'sampling' vectors, but that means increased runtime (assuming constant number of threads, going to 500 sampling vectors increases runtime to about 2000s in my test case). Haven't tried implementing the Hutchinson approach in Julia yet, so still passing the precision matrix to matlab for inversion... |
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The cpme underestimation for good data coverage has been adressed in the experimental routine DIVAnd_scalecpme! from my branch. |
Okay, I've applied DIVAnd to create a time-varying oxygen atlas (1956-2014, surface to 6000m depth at 0.5deg resolution). But I've run into some problems understanding and interpreting the error fields...
In DIVAnd_cpme and DIVAnd_aexerr the error estimates are described as "the clever poor mans error" and "the almost exact error" respectively, while in DIVAnd_cpme_go the error output is described as "erri: relative error field using the clever poor man's error approach".
So, I want some simple clarification: Do DIVAnd_cpme and DIVAnd_aexerr give the error in units of whichever variable we're gridding or do all three functions return a relative error?
If they are all relative errors there also needs to be some clarity what quantity they're relative to. The now obsolete matlab version specified the error as: "error variance of the analysis field relative to the error variance of the background" (per http://modb.oce.ulg.ac.be/mediawiki/index.php/Divand). However looking at the various SeaData climatologies they appear to treat the relative error as relative to the interpolated value at each grid-point. So, can you clarify if relative errors are relative to the background variance (as per the matlab version, and would explain why the confidence intervals on my oxygen inventory time series look unreasonably small) or the interpolated signal?
Thanks in advance (and sorry for bothering you again).
-Chris Roach
(There's also a question-mark over if these error estimates represent something like the standard deviation or something more like the the standard error or 95% confidence intervals)
(Why aren't I just pulling variances out of s.P? because diag() is horribly slow and in a real-world global case with a non-square grid and bathymetry it's unclear how to map diag(s.P) back to a geographic grid...)
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