Improve performance of xarray.corr() on big datasets

Is your feature request related to a problem? Please describe.

I calculated correlation coefficients based on datasets with sizes between 90-180 GB using xarray and Dask distributed and experienced very low performance for the xarray.corr() function. By observing the Dask dashboard it seemed that during the calculation the whole datasets are loaded from disk several times which, given the size of my datasets, became for some of the calculations a major "performance bottleneck".

Describe the solution you'd like

The problem became so annoying that I implemented my own function to calculate the correlation coefficient (thanks @willirath!), which is considerably more performant (especially for the big datasets!), because it only touches the full data once. I have uploaded a Jupyter notebook that shows the equivalence of the xarray.corr() function and my implementation (using an "unaligned data with nan values"-example, which is what xarray.corr() covers) and an example based on Dask arrays, which demonstrates the performance problems that I have stated above, and also that the xarray.corr() function is not fully lazy. (Which I assume is actually not very desirable?)

At the moment, I think, in terms of improving big data performance, a considerable improvement could be achieved by removing the if not valid_values.all() clause here, because that seems to determine that a call of xarray.corr() is not fully lazy and causes the first (of several?) full touches of the datasets? I haven't checked what's going on afterwards, but maybe that is already a useful starting point? 🤔

Start with xarray/core/computation.py around line 1313 and compare that path with the uploaded Jupyter notebook, especially its Dask-array example. Benchmark xarray.corr() on the notebook's unaligned and NaN-containing data; done means the correlation remains equivalent while avoiding repeated full dataset reads and unnecessary eager work.