Support basic arithmetic between PredictionEnsemble objects.

[bradyrx]

It might be useful to support basic arithmetic between PredictionEnsemble objects (+, -, *, /).

Code Sample, a copy-pastable example if possible

import climpred

hind = climpred.tutorial.load_dataset('CESM-DP-SST') 
hindcast1 = climpred.HindcastEnsemble(hind)
hindcast2 = climpred.HindcastEnsemble(hind)

hindcast1 + hindcast2

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-29-c0a6615fb41c> in <module>
      5 hindcast2 = climpred.HindcastEnsemble(hind)
      6 
----> 7 hindcast1 + hindcast2

TypeError: unsupported operand type(s) for +: 'HindcastEnsemble' and 'HindcastEnsemble'

Expected Output

This should perform the arithmetic across variables and products in common. Maybe this doesn't make much sense in most cases, but the best example I can think of is if you're aligning observations, etc. with the drift-correction method used for CESM-DPLE.

You set this up by doing

obs = obs - obs.sel(time=slice(1964, 2014)).mean('time')

But you cannot currently do

hindcast = hindcast - hindcast.sel(time=slice(1964, 2014)).mean('time')

which in theory could do this over every dataset that contains the time dimension.

I anticipate the solution would follow something similar to #243

Output of climpred.show_versions()

INSTALLED VERSIONS ------------------ commit: None python: 3.6.7 | packaged by conda-forge | (default, Jul 2 2019, 02:07:37) [GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)] python-bits: 64 OS: Darwin OS-release: 18.7.0 machine: x86_64 processor: i386 byteorder: little LC_ALL: None LANG: en_US.UTF-8 LOCALE: en_US.UTF-8

climpred: 1.1.0.post55
xarray: 0.14.0
pandas: 0.25.2
numpy: 1.17.2
scipy: 1.3.1
cftime: 1.0.4
dask: 2.6.0
distributed: 2.6.0
setuptools: 41.4.0
pip: 19.3.1
conda: None
pytest: 5.2.1
IPython: 7.8.0
sphinx: 2.2.0

[aaronspring]

Applying these operators to HimdcatsEnsemble then means to apply it to all datasets therein, right? But does this make sense? For skill computation not really

[bradyrx]

I guess I'm thinking just of the specific case I mentioned in this PR. But for now I guess it's reasonable to expect the user to do that before hand (e.g. ds - ds.isel(slice(a, b))).

Another use case would be if there is some variable derived from two other variables. But that's actually a different issue I think. Right now I don't have it set up to be able to do:

pm['C'] = pm['A'] + pm['B'] which would be nice at some point.

[ahuang11]

I think this has something to do with https://stackoverflow.com/questions/20507745/overloading-addition-subtraction-and-multiplication-operators

[bradyrx]

Ah nice, of course there's a magic method for that. Cool. I'll keep this in mind for a future PR.

[bradyrx]

@aaronspring here is the issue if you want to post the use case for this. I think your thought was for integrating, e.g. through PredictionEnsemble.sum(['lat', 'lon']). FYI xarray now has a weighted operators method you could leverage through .apply() for our objects. I haven't tried it yet.

[aaronspring]

my use case for +-*/:

he=(he*area).sum([‘lon’,’lat'])