Problem description

.sel is an xarray method I use a lot and I would have expected it to fairly efficient. However, even on tiny DataArrays, it takes seconds.

Code Sample, a copy-pastable example if possible

```python import timeit

setup = """ import itertools import numpy as np import xarray as xr import string

a = list(string.printable) b = list(string.ascii_lowercase) d = xr.DataArray(np.random.rand(len(a), len(b)), coords={'a': a, 'b': b}, dims=['a', 'b']) d.load() """

run = """ for _a, _b in itertools.product(a, b): d.sel(a=_a, b=_b) """ running_times = timeit.repeat(run, setup, repeat=3, number=10) print("xarray", running_times) # e.g. [14.792144000064582, 15.19372400001157, 15.345327000017278] ```

Expected Output

I would have expected the above code to run in milliseconds. However, it takes over 10 seconds! Adding an additional d = d.stack(aa=['a'], bb=['b']) makes it even slower, about twice as slow.

For reference, a naive dict-indexing implementation in Python takes 0.01 seconds: ```python setup = """ import itertools import numpy as np import string

a = list(string.printable) b = list(string.ascii_lowercase)

d = np.random.rand(len(a), len(b)) indexers = {'a': {coord: index for (index, coord) in enumerate(a)}, 'b': {coord: index for (index, coord) in enumerate(b)}} """

run = """ for _a, _b in itertools.product(a, b): index_a, index_b = indexers['a'][_a], indexers['b'][_b] item = d[index_a][index_b] """ running_times = timeit.repeat(run, setup, repeat=3, number=10) print("dicts", running_times) # e.g. [0.015355999930761755, 0.01466800004709512, 0.014295000000856817] ```

Output of xr.show_versions()

this is a follow-up from #2438

Grouping by multiple dimensions is not yet supported (#324):

python d = DataAssembly([[1, 2, 3], [4, 5, 6]], coords={'a': ('multi_dim', ['a', 'b']), 'c': ('multi_dim', ['c', 'c']), 'b': ['x', 'y', 'z']}, dims=['multi_dim', 'b']) d.groupby(['a', 'b']) # TypeError: `group` must be an xarray.DataArray or the name of an xarray variable or dimension

An inefficient solution is to run the for loops manually:

```python a, b = np.unique(d['a'].values), np.unique(d['b'].values) result = xr.DataArray(np.zeros([len(a), len(b)]), coords={'a': a, 'b': b}, dims=['a', 'b']) for a, b in itertools.product(a, b): cells = d.sel(a=a, b=b) merge = cells.mean() result.loc[{'a': a, 'b': b}] = merge

result = DataArray (a: 2, b: 2)> array([[2., 3.], [5., 6.]])

Coordinates:

* a (a) <U1 'x' 'y'

* b (b) int64 0 1

```

This is however horribly slow for larger arrays. Is there a more efficient / straight-forward work-around?

Output of xr.show_versions()

Related: #324, https://stackoverflow.com/questions/52453426/grouping-by-multiple-dimensions

I have a list of DataArrays with three dimensions. For each item in the list, two of the dimensions are a single value but the combination of all items would yield the full combinatorial values.

Code Sample

```python import itertools import numpy as np import xarray as xr

ds = []
for vals_dim1, vals_dim2 in itertools.product(list(range(2)), list(range(3))):
    d = xr.DataArray(np.random.rand(1, 1, 4),
                     coords={'dim1': [vals_dim1], 'dim2': [vals_dim2], 'dim3': range(4)},
                     dims=['dim1', 'dim2', 'dim3'])
    ds.append(d)

```

Expected Output

I then want to combine these complimentary DataArrays but none of what I tried so far seems to work. The result should be a DataArray with shape |2x3x4| and dimensions dim1: |2|, dim2: |3|, dim3: |4|.

The following do not work: ```python # does not automatically infer dimensions and fails with # "ValueError: conflicting sizes for dimension 'concat_dim': length 2 on 'concat_dim' and length 6 on <this-array>" ds = xr.concat(ds, dim=['dim1', 'dim2'])

# will still try to insert a new `concat_dim` and fails with
# "ValueError: conflicting MultiIndex level name(s): 'dim1' (concat_dim), (dim1) 'dim2' (concat_dim), (dim2)"
import pandas as pd
dims = [[0] * 3 + [1] * 3, list(range(3)) * 2]
dims = pd.MultiIndex.from_arrays(dims, names=['dim1', 'dim2'])
ds = xr.concat(ds, dim=dims)

# fails with
# AttributeError: 'DataArray' object has no attribute 'data_vars'
ds = xr.auto_combine(ds)

```

Output of xr.show_versions()