Clarification on the Use of QDM in MBCn

[sylvainmarchi]

Setup Information

• Xclim version: 0.52.2

Context

Hello,

I would like to seek some clarification regarding an aspect of MBCn. The documentation mentions that for the training phase of MBCn, quantile delta mapping (QDM) is used for bias adjustment. However, it seems that quantile mapping (QM) is actually used instead, which makes sense since only ref and hist are involved at this stage of the algorithm.

In the adjustment phase of MBCn, the adjustment factors (af_q) derived during training are reused as-is, so QDM does not appear to play a role here either.

Given this, do you think the absence of QDM in the repeated rotation steps impacts the rank structure of the bias-adjusted simulation? The original paper by Cannon (sdba-cannon_multivariate_2018) applies the absolute change form of QDM at each iteration step.

Thank you,

Sylvain

Steps To Reproduce

No response

[coxipi]

Hi Sylvain, could you point to a specific point in the documentation where it is implied that QM (and not QDM) is used for MBCn?

At least looking, at the code, I can confirm QDM is being used in rotating part of the algorithm. In the training, we have the function mbcn_train that uses npdf_train, which does an interpolation on the quantiles, as it should for a QDM adjustment

# _adjustment.py 
def _npdft_train(...
...
          af = u._interp_on_quantiles_1D(
                u._rank_bn(hist[iv]), <<=== apply the interpolation on the ranks
                quantiles, <<=== interpolate the sampled adjustment factors af_q on the quantiles
                af_q[ii, iv],
                method=method,
                extrap=extrap,
            )

The same holds for the adjusting part in mbcn_adjust. If it was QM, then quantiles (2nd argument above) would be replaced by the values at the quantiles (hist_q)

Is there some documentation you could point out where this is not clear, or things are simply stated in a wrong way?

Best,
Éric

[sylvainmarchi]

Hello Éric,

Thank you for pointing me to the relevant part of the code. Things are clearer now. Based on my current understanding of MBCn, I believe we might want to revise the header of the xclim.sdba.adjustment.MBCn(*args, _trained=False, **kwargs) class slightly. Specifically, step 3 suggests that QDM is used to compute the adjustment factors during the training phase. As the calculation of these factors relies exclusively on hist and ref, shouldn't we refer to QM instead?

def _npdft_train(ref, hist, rots, quantiles, method, extrap, n_escore, standardize):
    ...
    ref_q, hist_q = nbu._quantile(ref[iv], quantiles), nbu._quantile(hist[iv], quantiles)
    af_q[ii, iv] = ref_q - hist_q

I now agree with your point that QDM is used in the adjustment section (multivariate part) of the algorithm (where the adjustment factors derived from the historical modeled and observed CDFs are used to correct sim).

def _npdft_adjust(sim, af_q, rots, quantiles, method, extrap):
    ...
    sim[iv] = sim[iv] + af

Questions and Clarifications

Regarding your last message, could you please comment on the following points?

Interpolation on Quantiles

I'm not entirely sure I understand this statement:

We have the function mbcn_train that uses npdf_train, which performs an interpolation on the quantiles, as it should for a QDM adjustment.

Isn’t interpolation on the quantiles and the computation of the transfer function/adjustment factors using QM or QDM two separate things? Could you clarify this?
Related to this, I am also unsure how to interpret this comment in your code:

quantiles, <<=== interpolate the sampled adjustment factors af_q on the quantiles

To me, quantiles is a list of quantiles for which the adjustment factors have been evaluated. So, to adjust hist, we would want to interpolate between these quantiles at the values given by u._rank_bn(hist[iv]). Is this the correct interpretation?

Further Clarification on mbcn_adjust

Finally, if I may, I’d appreciate further clarification on the last part of your post, which remains unclear to me:

The same holds for the adjusting part in mbcn_adjust. If it was QM, then quantiles (the 2nd argument above) would be replaced by the values at the quantiles (hist_q).

Once again, thank you for your assistance and the time you've taken to address my questions. Your help is much appreciated!

Best,

Sylvain

[coxipi]

As the calculation of these factors relies exclusively on hist and ref, shouldn't we refer to QM instead?

The training parts of QM/QDM are indeed identical:

• Find the quantile values: {ref_q, hist_q}.
• Compute adjustment factors (simplest case, additive): {af_q = ref_q = hist_q}

Now, in the training part of MBCn, there is in fact an adjustment part. The Npdf routine that rotates on variables perform a QDM training and adjustment. This would proceed differently is we used QM or QDM:

(QM way: The arg of the function is the values)	{hist_q, af_q} -> af(hist)
(QDM way: The arg of the function is the quantiles)	{q, af_q} -> af(q)

We would then obtain the adjustment using the appropriate quantities
adjQDM_hist = hist + af(hist_ranks)
adjQM_hist = hist + af(hist)

I agree, for the training data, it's essentially the same thing as QM. It doesn't matter if you used QM or QDM, you will get very similar values. The discretization is either over quantile values (hist_q) or quantiles (q), so it can lead to small differences. For this reason, I think it is still good to specify we used a QDM adjustment. Also, this ensures we have a consistent terminology between the MBCn training part and MBCn adjust part. In the latter case where we adjust data oustide the training period, the QM/QDM distinction is more crucial, as you noted.

Isn’t interpolation on the quantiles and the computation of the transfer function/adjustment factors using QM or QDM two separate things? Could you clarify this?

Yes, you are right. However, the npdf part of the algorithm that rotates on variables does the training and the adjustment. So, there is in fact some adjustment going in the training part of MBCn, as noted above.

To me, quantiles is a list of quantiles for which the adjustment factors have been evaluated. So, to adjust hist, we would want to interpolate between these quantiles at the values given by u._rank_bn(hist[iv]). Is this the correct interpretation?

Exact!

Finally, if I may, I’d appreciate further clarification on the last part of your post, which remains unclear to me:
The same holds for the adjusting part in mbcn_adjust. If it was QM, then quantiles (the 2nd argument above) would be replaced by the values at the quantiles (hist_q).

It's related to what I wrote above, the question is: do you interpolate on {hist_q,af_q} and apply this to hist, or interp on {q,af_q} and apply this on hist_ranks. The same holds for the sim in the MBCn adjustment part. Sorry, I understand the phrasing above is confusing. See the pseudo-code below, does that clear things up?

# QM way: interp on {array_q, af_q}
## in mbcn_train > _npdft_train
interpolate( hist,  hist_q, af_q)
## in mbcn_adjust > _npdft_adjust
interpolate( sim, sim_q, af_q)

# QDM way: interp on {q, af_q}
## in mbcn_train > _npdft_train
interpolate( hist_ranks, q, af_q)
## in mbcn_adjust > _npdft_adjust
interpolate( sim_ranks, q, af_q)

Hope this helps!
Cheers,
Éric