Tutorial for mrCOSTS

[roxyboy]

Hi PyDMD developers,

I am quite curious about the recent implementation of COSTS and mrCOSTS but are there any plans on having a tutorial on them?
Regarding specific questions, I'm unsure about how to decide on the window_lengths and step_sizes input arguments.

[klapo]

Hi @roxyboy, Tutorial 20 contains two tutorials on COSTS/mrCOSTS: one for synthetic data and one demonstrating its use on real data.

Let me know if you have any other questions!

[roxyboy]

Apologies for one more question: Is concatenating m.periods() over enumerations of mrc.costs_array supposed become equivalent to:

da = mrc._da_omega
x = da.values
x = x.reshape(
    len(da.window_length), len(da.window_time_means) * len(da.svd_rank)
)
x_trans = np.abs(x.imag) / (2 * np.pi)

[klapo]

Hi @roxyboy, unfortunately, we aren't quite ready for you yet. But, we should hopefully have an arxiv paper with a nice diagram and a formal explanation of the algorithm in the next few weeks.

I'll address your questions in order:

1. The low-frequency component of each decomposition level explicitly includes dynamics which are longer than the window and thus cannot be well-resolved. Additionally, I do not recommend iterating across COSTS results manually except in some specific cases like assessing the fit. The reason is that dynamics tend to have a "leaked" component between decomposition levels. The global scale separation of mrCOSTS was designed to rectify this situation (as in the real data tutorial).

2. The cluster label of 0 should correspond to the slow component of each decomposition level.

3. I don't quite know what you mean here, unfortunately. What is your goal with the operation?

[roxyboy]

@klapo Thank you very much for your detailed reply! :)

1. I've been trying to use DMD and now COSTS to decompose sea-surface fields of the ocean, which generally consists of high-frequency and low-frequency dynamics. Rather than a global reconstruction, my interest has been in extracting the low-frequency modes and hence, I went down the route of manually iterating across COSTS. I went back to the tutorial after your comments and my understanding as it stands now is that I can extract the low-frequency coherent modes via indexing mrc.global_scale_reconstruction() corresponding to the cluster with the period of interest? E.g.

ds_global_separation.frequency_bands.isel(decomp_level=2
                    ).where(ds_global_separation.frequency_bands.isel(
                                    decomp_level=2)!=0.,
                            drop=True).isel(cluster=slice(None,2)).sum('cluster')

2. Understood, thank you.

3. I've been trying to plot the frequency spectra of the COSTS amplitudes and for this, I wanted the period/frequency corresponding to each m.amplitudes_array. For other DMD operations, I would usually get the frequencies as np.abs( np.log(lamb).imag / (2 * np.pi * delta_t) ) where lamb = dmd.eigs is the exponent which has the dimensions of inverse time. My idea was to try the same thing by `np.abs(x.imag) / (2 * np.pi * delta_t)

[roxyboy]

Regarding my third bullet point, essentially I'm trying to recreate the figure attached below (taken from the tutorial) with the x axis being replaced with frequency. The easiest thing would be to simply replace the x axis with mode_periods ** -1 but I wanted to check if my understanding of m.periods() was consistent with what I was doing for other DMD operations.

[klapo]

Regarding my third bullet point, essentially I'm trying to recreate the figure attached below (taken from the tutorial) with the x axis being replaced with frequency. The easiest thing would be to simply replace the x axis with mode_periods ** -1 but I wanted to check if my understanding of m.periods() was consistent with what I was doing for other DMD operations.

mrCOSTS doesn't share a lot of the framework with the rest of the DMD package due to its sliding multi-resolution approach. In this case your best bet is to dig into the code and make sure I did the operation that you are looking for. Also, just a heads up that this plot is not meant to be comparable to a spectra. Instead, I wanted to emphasize the frequency leaking and that the leaked component is generally small amplitude. In fact, the mrCOSTS framework largely loses the connection between DMD spectra and other time-frequency analysis by decomposing the input signal into a small number of oscillators instead of a continuum of oscillators.

[roxyboy]

Hi @klapo and @mtezzele, I was wondering if you'd be interested in co-authoring an academic-journal manuscript I've started typing up in applying mrCoSTS in an oceanographic context. Your input would be greatly appreciated as experts in DMD. If yes, I will personally send you a link to the draft in LaTeX.

[klapo]

Hi @roxyboy, go ahead and send me a link. I already have a paper using oceanographic data in progress as part of the paper describing mrCOSTS, so let's see if we can find a way to navigate that together.

[roxyboy]

@klapo That would be great! I sent you an email with the link. The focus of my draft is not so much on describing mrCoSTS but its application. Hopefully the two manuscripts can be complimentary to each other.