Should we close this issue and continue the discussion in #6293?

For anyone who wants to track the progress on this topic: https://github.com/pydata/xarray/projects/1

well, both "contain the origin dims" or just "generate another one" have its benefit.

Agreed, and both are supported by xarray actually. In case we want to keep the original dimensions like ("x", "y") in the example above, it's better to use masking.

This discussion is broader than the topic covered in this issue so I'd suggest you start a new discussion if you want to further discuss this with the xarray community. Thanks.

So I think maintain the origin dims may do less broken on current code.

For such case you could already do ds.stack(z=("t", "x")).set_index(z="C2").sel(z=["a", "e", "h"]).

After the explicit index refactor, we could imagine a custom index that supports multi-dimension coordinates such that you would only need to do something like

```python

S_res = S4.sel(C2=("z", ["a", "e", "h"])) S_res <xarray.Dataset> Dimensions: (z: 3) Coordinates: * C2 (z) <U1 'a' 'e' 'h' Data variables: A1 (z) float64 4 3 3 ```

or without explicitly providing the name of the packed dimension:

```python

S_res = S4.sel(C2=["a", "e", "h"]) S_res <xarray.Dataset> Dimensions: (C2: 3) Coordinates: * C2 (C2) <U1 'a' 'e' 'h' Data variables: A1 (C2) float64 4 3 3 ```

well, both "contain the origin dims" or just "generate another one" have its benefit. if we contain origin dims, we can ensure that: - less difference between 1d coordinate and multi dims ones, both can run like S1.sel(C1=["a", "e", "h"]) S4.sel(C2=["a", "e", "h"]) and return a new data set with origin dims ( that's why I highly not recommended the implicit one ) - return a new data set have original dims which means if you change C1 to C2, and the rest code have S_res.sel(x=[1,2,3]) still work.

For such case you could already do ds.stack(z=("t", "x")).set_index(z="C2").sel(z=["a", "e", "h"]).

After the explicit index refactor, we could imagine a custom index that supports multi-dimension coordinates such that you would only need to do something like

```python

S_res = S4.sel(C2=("z", ["a", "e", "h"])) S_res <xarray.Dataset> Dimensions: (z: 3) Coordinates: * C2 (z) <U1 'a' 'e' 'h' Data variables: A1 (z) float64 4 3 3 ```

or without explicitly providing the name of the packed dimension:

```python

S_res = S4.sel(C2=["a", "e", "h"]) S_res <xarray.Dataset> Dimensions: (C2: 3) Coordinates: * C2 (C2) <U1 'a' 'e' 'h' Data variables: A1 (C2) float64 4 3 3 ```

well, here are my ideas on how to define coordinates with multi dims.(because of github's bug, the characters of 1st image are white, I can not fix it)

Sorry but this is confusing. To me It still looks like you want implicit broadcasting of the A3 variable along the x dimension. In your last comment you depict A3 inconsistently with a 2-d shape but with only the t dimension. I'm also not sure how your suggestion relates to the issue here.

Thanks for the detailed description @weipeng1999. For the first 4 slides I don't see how this is different from how does S_res = S1.sel(C1=['a', 'b'] and S_res = S2.sel(C1=['a', 'b']) currently? And for the last 2 slides, I don't think that we always want such implicit broadcasting for dimensions that are not involved in the indexed coordinates.

thank you for figuring out the wrong things what I done. Well, it' is hard to explain the idea because it is a bit complicated, the last two picture is wrong and make misunderstanding, here are two images explain what I actuarily mean:

Thanks for the detailed description @weipeng1999. For the first 4 slides I don't see how this is different from how does S_res = S1.sel(C1=['a', 'b'] and S_res = S2.sel(C1=['a', 'b']) currently? And for the last 2 slides, I don't think that we always want such implicit broadcasting for dimensions that are not involved in the indexed coordinates.

Hi @weipeng1999,

I'm not sure to fully understand your suggestion, would you mind sharing some illustrative examples?

It is useful to have two distinct coordinate variable vs data variable concepts. Although both are data arrays, the former is used to locate data in the dimensional space(s) defined by all dimensions in the dataset while the latter is used to store field data.

It also helps to have a clear separation between the coordinate variable and index concepts. An index is a specific data structure or object that allows efficient data extraction or alignment based one or more coordinate labels. Sometimes an index object may be handled like a data array (like pandas indexes) but this is not always the case (e.g., a KD-Tree).

Currently in Xarray the index concept is hidden behind "dimension" coordinate variables. The goal of the explicit index refactor is to bring it to the light and make it available to any coordinate (and also open it to custom index structures, not only pandas indexes).

It looks like what you suggest is some kind of implicit (co-)indexes hidden behind any dataset variable(s)? We actually took the opposite direction, trying to make everything explicit.

Try to explain my idea, I make a PPT.

Hi @weipeng1999,

I'm not sure to fully understand your suggestion, would you mind sharing some illustrative examples?

It is useful to have two distinct coordinate variable vs data variable concepts. Although both are data arrays, the former is used to locate data in the dimensional space(s) defined by all dimensions in the dataset while the latter is used to store field data.

It also helps to have a clear separation between the coordinate variable and index concepts. An index is a specific data structure or object that allows efficient data extraction or alignment based one or more coordinate labels. Sometimes an index object may be handled like a data array (like pandas indexes) but this is not always the case (e.g., a KD-Tree).

Currently in Xarray the index concept is hidden behind "dimension" coordinate variables. The goal of the explicit index refactor is to bring it to the light and make it available to any coordinate (and also open it to custom index structures, not only pandas indexes).

It looks like what you suggest is some kind of implicit (co-)indexes hidden behind any dataset variable(s)? We actually took the opposite direction, trying to make everything explicit.

Well, maybe we can consider the coordinates in a more generic way.

Let us define coordinate an array in data set cause co-indexed when we index its data set. It means that:

• If A1,A2,A3 are in a same data set S, we index S[ {'A1':I} ] will return a new data set which not only have indexed A1, but they also been Indexed that the A2 A3 which have dims shared with A1. This behavior I call it co-index.

Use dims to determined the way how other array of the data set will be co-indexed.

• If all dims of A1(as coordinate) are also in A2(as regular array co-indexed), obviously the behavior can simply follow the old behavior, just change at the same dim and contain others.
• If A1 has a dim which not in A2, we should broadcast A2 at the dim, because the older behavior is to consider None dim as broadcast-able dim during other operation so co-index should follow it.

Some compatibility issues:

• maybe need a New Type like DataArray but only have dims instead of both dims and coordinate
• just define how Dataset to deal with index, maybe DataArray is simlar.

Many array types do have implicit indices. For example, sparse arrays do have their coordinates / CSR representation as primary index (.sel()) while dense array's primary index is the position (.isel()). Every labeled dimension is therefore just a separate mapping of a string to the index position in the array.

Going one step further, one could have continuous dimensions where positional indexing (.isel()) does not really make sense. Looking at TileDB's dimensions provides an example for this.

=> Having explicit and implicit indices on arrays would be awesome, even if they don't support all xarray features!

Explicitly propagating indexes requires going through most of xarray's source code and auditing each time we create a Dataset or DataArray object with low-level operations. We have some pretty decent testing functions for this in the form of xarray.testing._assert_internal_invariants, so this is now a pretty mechanical process -- you know it's working if you're now setting indexes explicitly and xarray's test suite passes.

Here's our current progress: - [x] most of dataset.py - [x] alignment.py - [x] merge.py (#3234) - [ ] concat.py - [x] dataarray.py (#3519, #3481) - [ ] computation.py - [ ] groupby.py - [ ] resample.py - [ ] rolling.py - [ ] everything else!

I'll make an example of this when I find some free time, along with a contrasting one in Pandas. :)

👍

Thanks @NowanIlfideme for your feedback.

Could you perhaps share a gist of code related to your use case?

The first example in this comment is similar to my use case: https://github.com/pydata/xarray/issues/3213#issuecomment-520741706 . There are several "core" dimensions, but some part of the coordinates may be hierarchical or cross-defined (e.g. country > province > city > building, but also country > province > voting district > building). We might have a full or nearly-full panel in the MultiIndex representation, but have a huge cross product (even if we keep strictly hierarchical dimensions out).

Meanwhile using a true COO sparse representation (as I understand it) will likely end up with slower operations overall, since nearly all machine learning models (think: linear regression) require a dense array input anyways.

I'll make an example of this when I find some free time, along with a contrasting one in Pandas. :)

My first attempt was to just assume each dimension was orthogonal, which resulted in out-of-memory errors

We have experimental support for https://sparse.pydata.org/en/latest/index.html that may help but no documentation unfortunately. There are some details here: https://github.com/pydata/xarray/issues/3213 and https://github.com/pydata/xarray/issues/3484

Thanks @NowanIlfideme for your feedback.

Could you perhaps share a gist of code related to your use case?

I've noticed that basically all my current troubles with xarray lead to this issue (lack of MultiIndex support). I use xarray for machine learning/data science/econometrics. My current problem requires a semi-hierarchical indexing on one of the dimensions, and slicing/aggregation along some levels of those dimensions.

My first attempt was to just assume each dimension was orthogonal, which resulted in out-of-memory errors. I ended up using a MultiIndex for the hierarchy dimension to have a "dense" representation of a sparse subspace. Unfortunately, currently .sel() and such will cut out MultiIndex dimensions, and I've had to do boolean masking to keep all the dimensions I need.

Multidimensional groupby, especially within the MultiIndex, is a headache as it currently stands. I had to resort to making auxilliary dimensions with one-hot encoded levels (dummy variables) and doing multiply-aggregate operations by hand.

xarray is really beautiful and should be used more by data scientists, but it's really difficult to recommend it to colleagues when not all the familiar pandas-style operations are supported.

I'm not working on any of these right now. You might start with a few of the dataarray.py methods (no need to do them all at once) to get a sense of what piping these arguments around looks like. I suspect you could get quite a few of these working just by handling indexes in _to_temp_dataset/_from_temp_dataset.

@shoyer I was thinking of starting on one of the listed files. Do you have any tips? Are you working on any of those at present? What might be the easiest one to begin?

After spending a few hours on the issue tracker yesterday, it became clear to me that the issue--more flexible indexes--is a major blocker on many high-priority features going forward.

In #2639, @shoyer started to address this. In that now merged-PR, he outlined the following steps, each of which needs its own PR:

• [x] In #2639 Indexes are recreated from coordinates every time a new DataArray or Dataset is created.
• [ ] Refactor indexes to be propagated explicitly in xarray operations. This will facilitate future API changes, when indexes will no longer only be associated with dimensions. I will probably add some testing decorator that can be used to mark part of a test as including no creation of default indexes.
• [ ] Add explicit entries into indexes for MultiIndex levels that are checked instead of MultiIndex variables. Still no public API changes (aside from adding more entries to .indexes).
• [ ] Support arbitrary coordinates in indexes.

So the best way to make progress on all manner of higher-level xarray feature requests is to start working through the next three items in this list.

There's now a good few dozen issues that reference this PR.

Wondering if there's any particular help needed (in the form of coding, discussion, or any other fashion), so as to try and speed it up and unblock those issues?

(I'm personally interested in resolving problems like #934 myself - allowing selection on non-dim coords, which seems to be a major hassle for a lot of use cases.)

I'm starting to make these changes incrementally -- the first step is in https://github.com/pydata/xarray/pull/2639.

I guess the error is probably the best idea.

Agreed. It seems very strict indeed, but it will be easier to relax this later than the other way. There is also a (very rare?) case where the two indexed coordinates have the same labels but are named differently in the two datasets (e.g., station_name and sname). In that case an error is probably better too. It would be a sort of indication that the most useful thing to do for future operations is to rename one of those coordinates first.

Sorry for maybe asking this again but I'm a bit confused now: is there any good reason of supporting "multiple single indexes" along the same dimension?

After all, perhaps better defaults would be to set indexes (pandas.Index) only for 1-d coordinates matching dimension names, like it is the case now.

If you want a different behavior, then you need to use .set_index(), which would raise if it results in multiple single indexes along a dimension. We could also add a new indexes argument to the Dataset / DataArray constructors to save some typing (and avoid the creation of in-memory pandas.Index for very long coordinates if an out-of-core alternative is later supported).

I discussed this is a little bit above in https://github.com/pydata/xarray/issues/1603#issuecomment-442661526, under "MultiIndex as part of the data schema".

I agree that the default behavior should still be to create automatic indexes only for 1d coordinates matching dimension names. But we still will have (rare?) cases where "multiple single indexes" could arise from combining arguments with different indexes.

For example, suppose the station dimension has an index for station_name in one dataset and city in another. Should the result be: - A MultiIndex with levels station_name and city? This would be most useful for future operations. - Two individual indexes for station_name and city? This would be the cheapest result to construct. - An error? This is arguably too strict, because there are no conflicts in either of the indexes.

I guess the error is probably the best idea.

Where does come from array([0, 1])? I wouldn't have been surprised if a KeyError was raised instead. Perhaps this specific case was initially for backward compatibility when the "dimensions without indexes" feature has been introduced, but it was a long time ago and I'm not sure this is still necessary.

This is indeed the historical genesis, but I agree that this is confusing and we should deprecate/remove it.

I think that one big source of confusion has been so far mixing coordinates/variables and indexes. These are really two separate concepts, and the indexes refactoring should address that IMHO.

For example, I think that da[some_name] should never return indexes but only coordinates (and/or data variables for Dataset). That would be much simpler.

Can't claim to be following every detail here, but this sounds very sensible to me FWIW.

It occurs to me that for the case of "multiple single indexes" along the same dimension there is no good way to use them simultaneously for indexing/reindexing at the same time.

Sorry for maybe asking this again but I'm a bit confused now: is there any good reason of supporting "multiple single indexes" along the same dimension?

After all, perhaps better defaults would be to set indexes (pandas.Index) only for 1-d coordinates matching dimension names, like it is the case now.

If you want a different behavior, then you need to use .set_index(), which would raise if it results in multiple single indexes along a dimension. We could also add a new indexes argument to the Dataset / DataArray constructors to save some typing (and avoid the creation of in-memory pandas.Index for very long coordinates if an out-of-core alternative is later supported).

da[dim_name] should return all the indexes on that dimension

I think that one big source of confusion has been so far mixing coordinates/variables and indexes. These are really two separate concepts, and the indexes refactoring should address that IMHO.

For example, I think that da[some_name] should never return indexes but only coordinates (and/or data variables for Dataset). That would be much simpler.

Take for example

```python

da = xr.DataArray(np.random.rand(2, 2), ... dims=('one', 'two'), ... coords={'one_labels': ('one', ['a', 'b'])}) da <xarray.DataArray (one: 2, two: 2)> array([[ 0.536028, 0.291895], [ 0.682108, 0.926003]]) Coordinates: one_labels (one) <U1 'a' 'b' Dimensions without coordinates: one, two ```

I find it so weird being able to do this:

```python

da['one'] <xarray.DataArray 'one' (one: 2)> array([0, 1]) Coordinates: one_labels (one) <U1 'a' 'b' Dimensions without coordinates: one ```

Where does come from array([0, 1])? I wouldn't have been surprised if a KeyError was raised instead. Perhaps this specific case was initially for backward compatibility when the "dimensions without indexes" feature has been introduced, but it was a long time ago and I'm not sure this is still necessary.

I might be a good thing explicitly requiring da.set_index('one_labels') to enable indexing/alignment (edit: label indexing/alignment) along dimension one in the example above.

How should dimension names interact with index names - i.e. the "Mapping indexes into pandas" in @shoyer 's comment

I'd suggest that option (3) should be invalid, and that da[dim_name] should return all the indexes on that dimension

A couple of thoughts:

If nothing useful can be done in the case of "multiple single indexes", would it make sense to discourage users explicitly creating multiple single indexes along a dimension? "Multiple single indexes" would be just a default situation when nothing specific as been defined yet or resulting from a failback.

For example, why not requiring that set_index(['x', 'y']) (with a list as argument) should always result in a multi-index regardless of the kind argument, i.e., raise if a single index is given? This is close to the current behavior, I think. This would require calling set_index for each single index that we want to (re)define, but I don't think setting a lot of single indexes at the same time is something that often happens.

Hence, would it be possible to avoid append=None and instead change the default to append=True?

I wonder if we should also change the default value of the append argument in set_index() to append=None, which means something like "append if creating a MultiIndex". For most users, keeping a single MultiIndex is the most usable way to use multiple indexes along a dimension, and our default behavior should reflect that.

It occurs to me that for the case of "multiple single indexes" along the same dimension there is no good way to use them simultaneously for indexing/reindexing at the same time. We should explicitly raise if you try to do this.

I guess we have a few options for automatic alignment with multiple single indexes, too: 1. We could only support "exact" indexing 2. We could require that aligning each index separately gives the same result

(2) seems least restrictive and is probably the right choice.

One advantage of not having MultiIndex objects as variables is that the serialization story gets simpler. The rule becomes "multi-indexes don't get saved".

What should the default behavior of set_index(['x', 'y']) without an explicit kind argument be? - Should this mean individual indexes or a combined MultiIndex? The later might be more surprising but is arguably more useful. It would make sense if the model is that set_index() always creates a single index object. - We could potentially automatically pick an index type using simple heuristics. For example, if the arguments are 1D, you get get a MultiIndex by default. If the arguments have two or more dimensions, you get a KDTree.

Looking at the reported issues related to multi-indexes in xarray, I have the same feeling. Simply reusing pandas.MultiIndex in xarray where slightly different semantics are generally expected has shown to be painful. It seems easier to have our own baked solution and deal with differences during xarray<-> pandas conversion if needed.

I think the pandas.MultiIndex is a pretty solid data structure on a fundamental level, it just has some weird semantics for some indexing edge cases. Whether or not we write xarray.MultiIndex structure, we can achieve most of what we want with a thin layer over pandas.MultiIndex.

If a variable for each multi-coordinate index is "just" for data schema consistency, then why not showing all those indexes in a separate section of the repr?

Yes, I like this! Generally I like @benbovy's entire proposal :).

@fujiisoup can you clarity the use-cases you have for a MultiIndex as a variable?

Am I right in thinking the Multi-indexes is only a helpful note to users, rather than conveying anything about how data is accessed?

From a data perspective, the only thing having an Index and/or MultiIndex should change is that the data is immutable.

But by necessity the nature of the index will determine which indexing operations are possible/efficient. For example, if you want to do nearest-neighbor indexing with multiple coordinates you'll need a KDTree. We should not be afraid to raise errors if an indexing operation can't be done efficiently.

With regards to reindexing: I don't think this needs any special handling versus normal indexing (sel()). The rules basically fall out of those for normal indexing, except we handle missing values differently (by filling with NaN).

Another issue: how do automatic alignment with multiple indexes? Let me suggest a straw-man proposal: We always align indexed coordinates. If a coordinate is used in different types of indexes (e.g., a base Index in one argument and a MultiIndex level in another), we can either: 1. create a MultiIndex with the variable on the fly (this could be slightly expensive), or 2. fall back to only supporting "exact" indexing

ds.sel(multi=list_of_pairs) can probably be replaced by ds.sel(x=..., y=...), but how about reindex along MultiIndex?

Indeed I haven't really thought about reindex and alignment in my suggestion above.

How do you currently reindex along a multi-index dimension?

Contrary to .sel, ds.reindex(multi=list_of_pairs) doesn't seem to work (the list of n-length tuples being interpreted as a ~~n-dim~~ 2-d array). The only way I've found to make it work is to pass another pandas.MultiIndex. Wouldn't be it rather confusing if we choose to go with our own implementation of MultiIndex for xarray instead of pandas.MultiIndex?

Wouldn't be possible to easily support ds.reindex(x=..., y=...) within the new data model proposed here?

Am I right in thinking the Multi-indexes is only a helpful note to users, rather than conveying anything about how data is accessed?

This is a good question.

A related question: apart from ds.sel(multi=list_of_pairs) and ds.reindex(multi=list_of_pairs) use cases discussed so far, is there other reasons of having a variable for a multi-index?

I think we can do much of this before adding the ability to set custom indexes, which would be cool but further from where we are, I think.

I agree, although whether or not we will eventually support custom indexes might influence the design choices that we have to do now, IMO.

And broadening out further:

Default behavior: all 1-dimensional coordinates each have their own, single index (pandas.Index), unless explicitly stated.

This is basically how I think of indexes - as a performant lookup data structure, rather than a feature of the schema. An RDBMS in a good corollary there.

Now, maybe there's enough overlap between the data access and the data schema that we should let them couple - e.g. would you want to be able to run .sel on any coord, even 2D? While it's possible in concept, it could guide users to inefficient operations.

We probably don't need to answer this question to proceed, but I'd be interested whether others see indexes as a property of the schema / I'm missing something.

I broadly agree with @benbovy 's proposal.

One question that I think is worth being clear on is what additional contracts do multiple indexes on a dimension have over individual indexes?

e.g. re: Coordinates: * level_1 (x) object 'a' 'a' 'b' 'b' * level_2 (x) int64 1 2 1 2 Multi-indexes: pandas.MultiIndex [level_1, level_2]

Am I right in thinking the Multi-indexes is only a helpful note to users, rather than conveying anything about how data is accessed?

@fujiisoup 's poses a good case of this question:

ds.sel(multi=list_of_pairs) can probably be replaced by ds.sel(x=..., y=...), but how about reindex along MultiIndex?

(and separately, I think we can do much of this before adding the ability to set custom indexes, which would be cool but further from where we are, I think)

I am late for the party (but still only have time to write a short comment). I am a big fan of MultiIndex and like @shoyer 's idea.

ds.sel(multi=list_of_pairs) can probably be replaced by ds.sel(x=..., y=...), but how about reindex along MultiIndex? I have encountered its use cases several times. I also think it would be nice to have MultiIndex as a variable.

we will definitely have to make some intentional deviations from the behavior of pandas

Looking at the reported issues related to multi-indexes in xarray, I have the same feeling. Simply reusing pandas.MultiIndex in xarray where slightly different semantics are generally expected has shown to be painful. It seems easier to have our own baked solution and deal with differences during xarray<-> pandas conversion if needed.

If we re-design indexes so that we allow 3rd-party indexes, maybe we could support both and let the user choose the one (xarray or pandas baked) that best suits his needs?

Regarding MultiIndex as part of the data schema vs an implementation detail, if we support extending indexes (and already given the different kinds of multi-coordinate indexes: MultiIndex, KDTree, etc.), then I think that it should be transparent to the user.

However, I don't really see why a multi-coordinate index should have its own variable (with tuples of values). I don't want to speak for others, but IMHO ds.sel(multi=list_of_pairs) is rather a edge case and I'm not sure if we really need to support it. Using ds.sel(x=..., y=...) with DataArray objects is certainly more code to write, but this form of indexing is very powerful and it might not be a bad idea to encourage it.

If a variable for each multi-coordinate index is "just" for data schema consistency, then why not showing all those indexes in a separate section of the repr? For example:

Coordinates: * level_1 (x) object 'a' 'a' 'b' 'b' * level_2 (x) int64 1 2 1 2 Multi-indexes: pandas.MultiIndex [level_1, level_2]

It is equally transparent, not more verbose, and it is clear that multi-indexes are not part of the coordinates (in fact there is no need of "virtual" coordinates either, nor to name the index). I don't think single indexes should be shown here as it would results in duplicated, uninformative lines.

More generally, here is how I would see indexes handled in xarray (I might be missing important aspects, though):

• Default behavior: all 1-dimensional coordinates each have their own, single index (pandas.Index), unless explicitly stated.
• Explicit API is used for setting new, possibly multi-coordinate indexes. Note the absence of keyword argument below to specify the variables: This is actually more consistent with the pandas API but this would be a breaking change and I don't know how a smooth transition could look like.
  • set_index(['x', 'y'], kind='multiindex') # xarray built-in index
  • set_index(['x', 'y'], kind='kdtree') # xarray built-in index
  • set_index('x', kind=ASingleIndexWrapperClass) # 3rd-party index
• If a coordinate is removed from the Dataset or if its index is reset or changed:
  • If the coordinate had a single index, no problem
  • If the coordinate was part of a multi-coordinate index: a new index is built from all remaining coordinates that were also part of the original index, if it is supported. Otherwise, the original index is removed and the default behavior (single pandas.Index) is reset for all those remaining coordinates.

Let me make a tentative proposal: we should model a MultiIndex in xarray as exactly equivalent to a sparse multi-dimensional array, except with missing elements modeled implicitly (by omission) instead of explicitly (with NaN).

💯- that very much resonates! And it leaves the implementation flexible if we want to iterate.

I'll try to think of some dissenting cases to the proposal / helpful responses to the above.

There's no need to support indexing like ds.sel(multi=list_of_pairs). Indexing like ds.sel(x=..., y=...) solves the same use case and looks nicer.

This needs an important caveat: it's only true that you use ds.sel(x=..., y=...) to emulate ds.sel(multi=list_of_pairs) if you do explicit vectorized indexing like in @max-sixty's example above (https://github.com/pydata/xarray/issues/1603#issuecomment-442636798). It would be nice to preserve a way to select a list of particular points that didn't require constructing explicit DataArray objects as the indexers. (But maybe this is a somewhat niche use-case and it isn't worth the trouble.)

Let me make a tentative proposal: we should model a MultiIndex in xarray as exactly equivalent to a sparse multi-dimensional array, except with missing elements modeled implicitly (by omission) instead of explicitly (with NaN). If we do this, I think MultiIndex semantics could be defined to be identical to those of separable Index objects.

One challenge is that we will definitely have to make some intentional deviations from the behavior of pandas, at least when dealing with array indexing of a MultiIndex level. Pandas has some strange behaviors with array indexing of a MultiIndex level, and I'm honestly not sure if they are bugs or features: - It ignores missing labels (https://github.com/pandas-dev/pandas/issues/15452) - It drops duplicate labels (https://github.com/pandas-dev/pandas/issues/19414)

Fortunately, the MultiIndex data model is not that complicated, and it is quite straightforward to remap indexing results from sub-Index levels onto integer codes. I suspect we will find it easier to rewrite some of these routines than to change pandas, both because pandas may not agree with different semantics and because the pandas indexing code is an unholy mess.

For example, we can reproduce the above issues: python import pandas as pd index = pd.MultiIndex.from_arrays([['a', 'b', 'c']]) print(index.get_locs((['a', 'a'],))) # [0] print(index.get_locs((['a', 'd'],))) # [0] We actually want something more like: ```python def get_locs(index, key): return index.get_indexer(pd.MultiIndex.from_product(key))

print(get_locs(index, (['a', 'a'],))) # [0, 0] print(get_locs(index, (['a', 'd'],))) # [0, -1] ```

That said, I still don't know how to use public MultiIndex methods for this. Neither index.get_loc_level([1, 2], level=1) nor index.get_loc((slice(None), [1, 2])) work.

The answer is the index.get_locs() method: index.get_locs([slice(None), 1, 2]]) works.

It's painfully slow for large numbers of points due to a Python loop over each point, but presumably that could be optimized: x = np.arange(10000) index = pd.MultiIndex.from_arrays([x]) %timeit index.get_locs((x,)) # 1.31 s per loop %timeit index.levels[0].get_indexer(x) # 93 µs per loop

I've been thinking about this a little more in the context of starting on the implementation (in #2195).

In particular, I no longer agree with this "Separate indexers without a MultiIndex should be prohibited" from my original proposal. The problem is that the semantics of a MultiIndex are not quite the same as separate indexes, and I don't think all use-cases are well solved by always using a MultiIndex. ~~For example, I don't think it's possible to do point-wise indexing along anything other than the first level of a MultiIndex.~~ (note: this is not true, see https://github.com/pydata/xarray/issues/1603#issuecomment-442662561)

Instead, I think we should make the model transparent by retaining an xarray variable for the MultiIndex, and provide APIs for explicitly converting index types.

e.g., for the repr with a MultiIndex: Coordinates: * x (x) MultiIndex[level_1, level_2] * level_1 (x) object 'a' 'a' 'b' 'b' * level_2 (x) int64 1 2 1 2 and without a MultiIndex: Coordinates: * level_1 (x) object 'a' 'a' 'b' 'b' * level_2 (x) int64 1 2 1 2

The main way in which this could get confusing is if you explicitly mutate the Dataset to remove some but not all of the variables corresponding to the MultiIndex (e.g., x but not level_1 or vise-versa). We have a few potential options here: 1. Don't worry about it: if you mutate objects, you can potentially end up in slightly confusing internal states. If you care about whether level_1 uses a pandas.Index or pandas.MultiIndex, you can find out for sure by checking ds.indexes['level_1']. 2. Prohibit it in our data model: either (a) raise an error if you try to manually delete a single variable or (b) automatically delete all associated variables, too. Encourage using various explicit APIs that return new objects with a new index. 3. Use a different indicator than * for marking "indirect" indexes, so it's more obvious if some coordinates get removed, e.g., Coordinates: * x (x) MultiIndex[level_1, level_2] + level_1 (x) object 'a' 'a' 'b' 'b' + level_2 (x) int64 1 2 1 2

The different indicator might make sense regardless but I am also partial to "Prohibit it in our data model." The main downside is that this adds a little more complexity to the logic for determining indexes resulting from an operation (namely, verifying that all MultiIndex levels still correspond to coordinates).

For example, I don't think it's possible to do point-wise indexing along anything other than the first level of a MultiIndex.

This is clearly not true, since it works in pandas: python import pandas as pd index = pd.MultiIndex.from_product([list('ab'),[1,2]]) series = pd.Series(range(4), index) print(series.loc[:, [1, 2]])

That said, I still don't know how to use public MultiIndex methods for this. Neither index.get_loc_level([1, 2], level=1) nor index.get_loc((slice(None), [1, 2])) work.

@max-sixty I like your schema vs. implementation breakdown. In general, I agree with you that it would be nice to have MultiIndex has an implementation detail rather than part of xarray's schema. But I'm not entirely sure that's feasible.

Let's try to list out the pros/cons. Consider a MultiIndex 'multi' with levels 'x' and 'y': - Advantages of MultiIndex as part of the data schema: - There is an explicit coordinate (of tuples) corresponding to MultiIndex values, which can be returned from ds.coords['multi']. This is inherently not that useful compared to the separable variables, but is a cleaner solution that creating ds.coords['multi'] as a "virtual" variable on the fly (which we would need for backwards compatibility). - We don't need to do full "normalization" when multiple indexes along the same dimension are encountered, e.g., in an operation that combines two different indexes, we would simply put both on the result instead of building a MultiIndex (which would require allocating a whole new array of integer codes). - The nature of the MultiIndex is more transparent as part of the data model. For example, if x and y are numeric, it could make sense to use either a MultiIndex or KDTree for indexing. Explicit APIs (e.g., set_multiindex and set_kdtree) would allow users a high level of control. - For advanced use-cases, it is potentially easier to work around the limitations of a MultiIndex, e.g., the way that some operations require lex-sorted-ness. - Advantages of MultiIndex as an implementation detail: - Simpler data model (for users). There are few good use cases for multiple indexes that aren't a MultiIndex. - Easier to do automatic alignment: we know that indexes will always have the same normalized form (in a MultiIndex). Otherwise, we would have to do this on the fly, or request that users explicitly setup compatible indexes. - More flexibility for xarray: we can potentially swap out indexing without changing the user-facing API. We might have something like a "hybrid" MultiIndex/KDTree that chooses the appropriate index based on the requested operation. - We don't need to create an explicit array of tuples for the MultiIndex variable (but we could still have a variable corresponding to a MultiIndex and only construct the .data array in a "lazy" fashion). - There's no need to name extraneous variables that only exist for the sake of a MultiIndex. - There's no need to support indexing like ds.sel(multi=list_of_pairs). Indexing like ds.sel(x=..., y=...) solves the same use case and looks nicer. That said, this would be a minor backwards compatibility break (this currently works in xarray).

P.S. I haven't made much progress on this yet so there's definitely still time to figure out the right decision -- thanks for your engagement on this!

Potentially this is too much 'stepping back' now we're at the implementation stage - my perception is that @shoyer is leading this without much support, so weighting having some additional viewpoints, some questions:

Is a MultiIndex a feature of the schema or the implementation?

I had thought of an MI being an implementation detail in code, rather than in the data schema. We use it as a container for all the indexes along a dimension, rather than representing any properties about the data it contains.

One exception to that would be if we wanted multiple groups of indexes along the same dimension, for example:

``` Coordinates: * xa (x) MultiIndex[level_a_1, level_a_2] * level_a_1 (x) object 'a' 'a' 'b' 'b' * level_a_2 (x) int64 1 2 1 2

• xb (x) MultiIndex[level_b_1, level_b_2]
• level_b_1 (x) object 'a' 'a' 'b' 'b'
• level_b_2 (x) int64 1 2 1 2 ```

But is that common / required?

MultiIndex as an implementation detail

If it's an implementation detail, is there a benefit to investing in allowing both separate and MIs? While it may not be possible to do pointwise indexing with the current implementation of MI, am I mistaken that it's not an API issue, assuming we pass in index names? e.g.:

```python [ins] In [22]: da = xr.DataArray(np.arange(12).reshape((3, 4)), dims=['x', 'y'], coords=dict(x=list('abc'), y=pd.MultiIndex.from_product([list('ab'),[1,2]])))

[ins] In [23]: da Out[23]: <xarray.DataArray (x: 3, y: 4)> array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) Coordinates: * x (x) <U1 'a' 'b' 'c' * y (y) MultiIndex - y_level_0 (y) object 'a' 'a' 'b' 'b' - y_level_1 (y) int64 1 2 1 2

[ins] In [26]: da.sel(x=xr.DataArray(['a','c'],dims=['z']), y_level_0=xr.DataArray(['a','b'],dims=['z']) y_level_1=xr.DataArray([1,1],dims=['z']))

Out[80]: # hypothetical <xarray.DataArray (z: 3)> array([ 0, 10]) Dimensions without coordinates: z ```

If that's the case, could we instead force all indexes along a dimension to be in a MI, tolerate the short-term constraints of the current MI implementation, and where needed build out additional features?

That would (ideally) leave us uncoupled to MIs - if we built a better in-memory data structure, we could transition. The contract would be around the cases above.

--

...and as mentioned above, these are intended as questions rather than high-confident views.

Sure, this is as good a time as any. But we'll probably need to refinish this refactoring before it makes sense to implement anything.

On Tue, May 29, 2018 at 8:59 AM Alistair Miles notifications@github.com wrote:

Ok, cool. Was wondering if now was right time to revisit that, alongside the work proposed in this PR. Happy to participate in that discussion, still interested in implementing some alternative index classes.

On Tue, 29 May 2018, 15:45 Stephan Hoyer, notifications@github.com wrote:

Yes, the index API still needs to be determined. But I think we want to support something like that. On Tue, May 29, 2018 at 1:20 AM Alistair Miles <notifications@github.com

wrote:

I see this mentions an Index API, is that still to be decided?

On Tue, 29 May 2018, 05:28 Stephan Hoyer, notifications@github.com wrote:

I started thinking about how to do this incrementally, and it occurs to me that a good place to start would be to write some of the utility functions we'll need for this:

1. Normalizing and creating default indexes in the Dataset/DataArray constructor.
2. Combining indexes from all xarray objects that are inputs for an operations into indexes for the outputs.
3. Extracting MultiIndex objects from arguments into Dataset/DataArray and expanding them into multiple variables.

I drafted up docstrings for each of these functions and did a little bit of working starting to think through implementations in #2195 https://github.com/pydata/xarray/pull/2195. So this would be a great place for others to help out. Each of these could be separate PRs.

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Ok, cool. Was wondering if now was right time to revisit that, alongside the work proposed in this PR. Happy to participate in that discussion, still interested in implementing some alternative index classes.

On Tue, 29 May 2018, 15:45 Stephan Hoyer, notifications@github.com wrote:

Yes, the index API still needs to be determined. But I think we want to support something like that. On Tue, May 29, 2018 at 1:20 AM Alistair Miles notifications@github.com wrote:

I see this mentions an Index API, is that still to be decided?

On Tue, 29 May 2018, 05:28 Stephan Hoyer, notifications@github.com wrote:

I started thinking about how to do this incrementally, and it occurs to me that a good place to start would be to write some of the utility functions we'll need for this:

1. Normalizing and creating default indexes in the Dataset/DataArray constructor.
2. Combining indexes from all xarray objects that are inputs for an operations into indexes for the outputs.
3. Extracting MultiIndex objects from arguments into Dataset/DataArray and expanding them into multiple variables.

I drafted up docstrings for each of these functions and did a little bit of working starting to think through implementations in #2195 https://github.com/pydata/xarray/pull/2195. So this would be a great place for others to help out. Each of these could be separate PRs.

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Yes, the index API still needs to be determined. But I think we want to support something like that. On Tue, May 29, 2018 at 1:20 AM Alistair Miles notifications@github.com wrote:

I see this mentions an Index API, is that still to be decided?

On Tue, 29 May 2018, 05:28 Stephan Hoyer, notifications@github.com wrote:

I started thinking about how to do this incrementally, and it occurs to me that a good place to start would be to write some of the utility functions we'll need for this:

1. Normalizing and creating default indexes in the Dataset/DataArray constructor.
2. Combining indexes from all xarray objects that are inputs for an operations into indexes for the outputs.
3. Extracting MultiIndex objects from arguments into Dataset/DataArray and expanding them into multiple variables.

I drafted up docstrings for each of these functions and did a little bit of working starting to think through implementations in #2195 https://github.com/pydata/xarray/pull/2195. So this would be a great place for others to help out. Each of these could be separate PRs.

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I see this mentions an Index API, is that still to be decided?

On Tue, 29 May 2018, 05:28 Stephan Hoyer, notifications@github.com wrote:

I started thinking about how to do this incrementally, and it occurs to me that a good place to start would be to write some of the utility functions we'll need for this:

1. Normalizing and creating default indexes in the Dataset/DataArray constructor.
2. Combining indexes from all xarray objects that are inputs for an operations into indexes for the outputs.
3. Extracting MultiIndex objects from arguments into Dataset/DataArray and expanding them into multiple variables.

I drafted up docstrings for each of these functions and did a little bit of working starting to think through implementations in #2195 https://github.com/pydata/xarray/pull/2195. So this would be a great place for others to help out. Each of these could be separate PRs.

— You are receiving this because you commented. Reply to this email directly, view it on GitHub https://github.com/pydata/xarray/issues/1603#issuecomment-392649605, or mute the thread https://github.com/notifications/unsubscribe-auth/AAq8QvMauEPa6hfgorDoShZ2PwyYWk6Tks5t3M6AgaJpZM4PtACU .

I started thinking about how to do this incrementally, and it occurs to me that a good place to start would be to write some of the utility functions we'll need for this: 1. Normalizing and creating default indexes in the Dataset/DataArray constructor. 2. Combining indexes from all xarray objects that are inputs for an operations into indexes for the outputs. 3. Extracting MultiIndex objects from arguments into Dataset/DataArray and expanding them into multiple variables.

I drafted up docstrings for each of these functions and did a little bit of working starting to think through implementations in https://github.com/pydata/xarray/pull/2195. So this would be a great place for others to help out. Each of these could be separate PRs.

I've been thinking about getting started on this. Here are my current thoughts on the right design approach.

Data model

Dataset.indexes and DataArray.indexes

My current thinking is that indexes should simply be a dictionary mapping from coordinate and/or dimension names to pandas.Index objects. Mapping from label-based to integer-based then becomes simply a matter of looking up the appropriate indexes for each coordinate/dimension (i.e., the keyword argument names in .sel()), and using the corresponding index(es) to transform label-based indexers into integer indexers.

If multiple coordinates are part of the same index, they should point to the same MultiIndex/KDTree object. The MultiIndex would be responsible for resolving the combined indexing operation along the coordinate dimension(s).

By default, indexes is populated with an Index/MultiIndex for each dimension of all indexes along that dimension. Additional indexes may be set manually, e.g., using set_index().

Indexes keyed by a dimension name are used for axis-positional indexing with .loc and for alignment with reindex/align. However, if the index is a MultiIndex with a level name matching a coordinate, then only that level will be used for indexing/alignment. In other words: the coordinate name corresponding to indexing request takes precedence, but if it isn't found, we use all indexes along the dimension.

Separate indexers without a MultiIndex should be prohibited

It should be impossible to express inconsistent and/or confusing states in xarray's data model. This sort of inconsistency (e.g., levels not being stored directly in Dataset.variables) is the major source of our issues with the current MultiIndex data model.

I'm particularly concerned about the clearly showing difference between coordinates that are part of a MultiIndex and coordinates that are separately indexed. I suspect we could make indexing operations nearly equivalent from a user perspective, but there would likely remain small differences that would be a source of confusion and bugs. Preserving indexes in the form in which they are created is not also not really an option, because there are lots of xarray operations that would probably normalize indexes into standard forms, such as groupby, stack/unstack and to/from_pandas.

The simplest option is to prohibit one of these cases entirely, either: 1. Always group repeated indexes along a dimension into a MultiIndex, or 2. Never use pandas.MultiIndex (keep separate indexes for each coordinate).

From xarray's perspective, it would certainly be cleaner to prohibit MultiIndex. The level order dependent behavior of MultiIndex is not the best fit for xarray's data model, and could be challenging to keep in sync with coordinate order on xarray objects. We would need to ensure that coordinate/level order remains consistent in all operations, or at least ensure that coordinates are always printed in order of their appearence in MultiIndex levels. (We generally preserve coordinate order already, but well behaved programs using xarray currently don't need to rely on this behavior.)

That said, always using MultiIndexes for multiple indexes along the same dimension has it's own clear advantages. First, it's consistent with pandas, which makes it easier to transition data back and forth. Second, simultaneous indexing operations across MultiIndex levels would be difficult to express efficiently with a MultiIndex. This is probably the right choice for xarray.

We could potentially allow for non-consolidated indexes (not part of a MultiIndex) when using the advanced API (e.g., directly setting the indexes parameter). But we'll save this for later.

Functionality

Index variables

Every MultiIndex level must have a corresponding xarray.Variable object in coordinates on each Dataset/DataArray on which they appear. These objects may reference the same pandas.Index/pandas.MultiIndex object used for indexing, but must have immutable data (e.g., flag.writeable = False in NumPy). For now, I expect to reuse the existing IndexVariable class.

Now that levels are xarray.Variable objects, there will no longer be a Variable object in Dataset._variables/DataArray._coords corresponding to a pandas.MultiIndex. However, we will continue to create a "virtual variable" upon indexing consisting of an dtype=object array of MultiIndex values, as a fallback if there is no coordinate matching a dimension name.

Mapping indexes into pandas

Another concern is how to map all of the new possible indexing states into pandas:

```

case 1 (one indexed variable, same name as dimension):

• time (time)

case 2 (one indexed variable, different name from dimension):

• year (time)

case 3 (multiple indexed variables, one has same name as dimension):

• time (time)
• year (time)

case 4 (multiple indexed variables, all have different names from dimension):

• year (time)
• month (time) ```

For consistency with current behavior, case 1 should correspond to a standard pandas.Index and case 4 should correspond to a pandas.MultiIndex. But what about the intermediate cases 2 and 3, which are currently prohibited by xarray's data model?

I think we should use the rule that all indexed variables are consolidated into a single Index in pandas. If there are multiple indexed variables (case 3 or 4), this would be a MultiIndex; otherwise (cases 2 or 3), this would be a standard Index. This has a virtue of speed and simplicity: we can simply reuse the existing Index or MultiIndex object from indexes.

The other option would be prohibit cases 2 and 3 (like we currently do), because we will not be able to map them into pandas and back faithfully. I think this would be a mistake, because indexes on multiple levels would be useful for xarray, even if one level corresponds to the dimension name.

Indexes for unstack

With the introduction of more flexible and optional index levels, it may not always may sense to unstack() every index coordinate. We should support optionally specifying levels to unstack, possibly with an API mirroring stack(), e.g., perhaps .unstack(dim_name=['level_0', 'level_1']) to unstack coordinates level_0 and level_1 from dimension dim_name.

Overall, I agree with the proposed conclusion. And appreciate the level of thoughtfulness and clarity. I'm happy to help with some of the implementation if we can split this up.

@fujiisoup Yes, we certainly could add a "N-dimensional index", even if it has no function other than a placeholder to mark a variable as an index. This would let us restore index state after selecting/concatenating along a dimension.

However, I'm not sure it would be a satisfactory solution. If we keep these indexes around like coordinates, we could end up with scalar coordinates from different dimensions. Then it's still not clear how they should stack up in the final result -- we would have the same issue we currently have with concatenating coordinates.

The other concern is that existence and behavior of scalar/N-dimensional indexes could be a surprising. What does it mean to index an N-dimensional index? This operations probably cannot be supported in a sensible way, or at least not without significant effort.

@shoyer, thank you for detailing.

I am thinking how can we establish the following selecting-concatenating behavior with MultiIndex(-like) coordinate with our new Indexes machinary, xr.concat([da.isel(x=i) for i in range(len(da['x':))], dim='x') Personally, I think it would be nice if we could recover the original Index structue. We may need to track Indexes object even when the corresponding dimension becomes one dimensional? But scalar index sounds strange...

Or, we may give up to restore the original coordinate structure during the above action, but stil keep them as ordinary coodinates.

@jjpr-mit can you explain your use case a little more? What sort of order dependent queries do you want to do? The one that comes to mind for me are range based queries, e.g, [('bar', 1) : ('foo', 9)].

I think it is still relatively easy to ensure a unique ordering between levels, based on the order of coordinate variables in the xarray dataset.

A bigger challenge is that for efficiency, these sorts of queries depend critically on having an actual MultiIndex. This means that if indexes for each of the levels arise from different arguments that were merged together, we might need to "merge" the separate indexes into a joint MultiIndex. This could potentially be slightly expensive.

Will the new API preserve the order of the levels? One of the features that's necessary for MultiIndex to be truly hierarchical is that there is a defined order to the levels.

Just to say I'm interested in how MultiIndexes are handled also. In our use case, we have two variables conventionally named CHROM (chromosome) and POS (position) which together describe a location in a genome. I want to combine both variables into a multi-index so I can, e.g., select all data from some data variable for chromosome X between positions 100,000-200,000. For all our data variables, this genome location multi-index would be used to index the first dimension.

I am wondering what the advantageous cases which are realized with this Index concept are.

The other advantage is that it solves many of the issues with the current MultiIndex implementation. Making MultiIndex levels their own variables considerably simplifies the data model, and means that many features (including serialization) should "just work".

In principle, this data model would allow for two mostly equivalent indexing schemes: MultiIndex[time, space] vs two indexes Index[time] and Index[space]. I like the latter one, as it is easier to understand even for non-pandas users.

I agree, but there are probably some advantages to using a MultiIndex internally. For example, it allows for looking up on multiple levels at the same time.

What does the actual implementation look like? xr.Dataset.indexes will be an OrderedDict that maps from variable's name to its associated dimension? Actual instance of Index will be one of xr.Dataset.variables?

I think we could get away with making xr.Dataset.indexes simply a dict, with keys given by index names and values given by a pandas.Index instance. We should enforce that Index.name or MultiIndex.names corresponds to coordinate variables.

For KDTree, this means we'll have to write our own wrapper KDTreeIndex that adds a names property, but we would probably need to add special methods like get_indexer anyways.

Thanks for the details. (Sorry for my late responce. It took a long for me to understand what does it look like.)

I am wondering what the advantageous cases which are realized with this Index concept are. As far as my understanding is correct,

1. It will enable more flexible indexing, e.g. more than one Indexes are associated with one dimension and we can select from these coordinate values very flexibly.
2. It will naturally integrate more advanced Indexes such as KDTree

Are they correct?

Probably the most elegant rule would again be to check all indexed variables for exact matches.

That sounds reasonable.

In principle, this data model would allow for two mostly equivalent indexing schemes: MultiIndex[time, space] vs two indexes Index[time] and Index[space].

I like the latter one, as it is easier to understand even for non-pandas users.

What does the actual implementation look like? xr.Dataset.indexes will be an OrderedDict that maps from variable's name to its associated dimension? Actual instance of Index will be one of xr.Dataset.variables?

1. Use cases of the independent Index and dims Would it be general cases where dimension and index are independent? (It is the case only for MultiIndex and KDtree)?

We would still assign default indexes (using a normal pandas.Index) when you assign a 1D coordinate with matching name and dimension. But in general, yes, it seems like you should be able to make an index even for variables that aren't dimensions, including for a 1D variable whose name does not match a dimension. The rule would be that any coordinates can be part of an index.

Another aspect to consider how to handle alignment when you have indexes along non-dimension coordinates. Probably the most elegant rule would again be to check all indexed variables for exact matches.

Directly assigning indexes rather than using this default or set_index() would be an advanced feature, not recommended for everyday use. The main use case is routines which create a new xarray object based on an existing one, and want to re-use old indexes.

For performance reasons, we probably do not want to actually check the values of manually assigned indexes, although we should verify that the shape matches. (We would have a clear disclaimer that if you manually assign an index with mismatched values the behavior is not well defined.)

In principle, this data model would allow for two mostly equivalent indexing schemes: MultiIndex[time, space] vs two indexes Index[time] and Index[space]. We would need to figure out how to propagate and compare indexes like this. (I suppose if the coordinate values match, the result could have the union of all indexes from input arguments.)

1. MultiIndex implementation In MultiIndex case, will a xarray object store a MultiIndex object and also the level variables as Variable objects (there will be some duplicates)?

Yes, this is a little unfortunate. We could potentially make a custom wrapper for use in IndexVariable._data on the level variabless that lazily computes values from the MultiIndex (similar to our LazilyIndexedArray class), but I'm not certain yet that this is necessary.

If indexes[dim] returns multiple Variables, which realizes a MultiIndex-like structure without pd.MultiIndex, indexes would be very different from dim, because a single dimension can have multiple indexes.

Every entry in indexes should be a single pandas.Index or subclass, including MultiIndex (possibly eventually allowing for index-like objects such as something based on a KDTree).

@shoyer, could you add more details of this idea? I think I do not yet fully understand the practical difference between dim and index.

1. Use cases of the independent Index and dims Would it be general cases where dimension and index are independent? (It is the case only for MultiIndex and KDtree)?

2. MultiIndex implementation In MultiIndex case, will a xarray object store a MultiIndex object and also the level variables as Variable objects (there will be some duplicates)? If indexes[dim] returns multiple Variables, which realizes a MultiIndex-like structure without pd.MultiIndex, indexes would be very different from dim, because a single dimension can have multiple indexes.

I think that promoting "Indexes" to a first-class concept is indeed a very good idea, at both internal and public levels, even if at the latter level it would be another concept for users (it should be already familiar for pandas users, though). IMHO the "coordinate" and "index" concepts are different enough to consider them separately.

I like the proposed repr for Dataset.indexes. I wouldn't mind if it is not included in Dataset.__repr__, considering that multi-indexes, kdtree, etc. only represent a few use cases. In too many cases it could result in a long, uninformative list of simple pandas.Index.

I have to think a bit more about the details but I like the idea.

CC @benbovy @fmaussion

Does your proposal means that Dataset will keep an additional attribute indexes, and indexes[dim] gives a pd.Index (or pd.MultiIndex, KDTree)?

Yes, exactly. We actually already have an attribute that works like this, but it's current computed lazily, from either Dataset._variables or DataArray._coords.

I sometimes find it helpful to think about what the right repr() looks right, and then work backwards from there to the right data model.

For example, we might imagine that "Indexes" are no longer coordinates, but instead their own entry in the repr: <xarray.Dataset (exp_time: 5)> Coordinates: * experiment (exp_time) int64 0 0 0 1 1 * time (exp_time) float64 0.0 0.1 0.2 0.0 0.15 Indexes: exp_time: pandas.MultiIndex[experiment, time]

"Indexes" might not even need to be part of the main Dataset.__repr__, but it would certainly be the repr for Dataset.indexes. Other entries could include: time: pandas.Datetime64Index[time] space: scipy.spatial.KDTree[latitude, longitude]

In this model:

1. We would promote "Indexes" to a first-class concept in the xarray data model: (a) The levels of a MultiIndex would have corresponding Variable objects and be found in coords. (b) In contrast, theMultiIndex would not have a corresponding Variable object or be part of coords, though it could still be returned upon __getitem__ access (computed on demand from .indexes). (c) Dataset and DataArray would gain an indexes argument in their constructors, which could be used for passing indexes on to new xarray objects.
2. Coordinates marked with * are part of an index. They can't be modified, unless all corresponding indexes ares removed.
3. Indexes would still be propagated, like coordinates.

I think we currently assume variables[dim] is an Index. Does your proposal means that Dataset will keep an additional attribute indexes, and indexes[dim] gives a pd.Index (or pd.MultiIndex, KDTree)?

It sounds a much cleaner data model.

One API design challenge here is that I think we still want a explicit notation of "indexed" variables. We could possibly allow operations like .sel() on non-indexed variables, but they would be slower, because we would not want to create expensive hash-tables (i.e., pandas.Index) in a non-transparent fashion.

I'm using MultiIndex a lot, but I noticed that it is just a workaround to index along multiple kinds of coordinate.

Consider the following example, ```python In [1]: import numpy as np ...: import xarray as xr ...: da = xr.DataArray(np.arange(5), dims=['x'], ...: coords={'experiment': ('x', [0, 0, 0, 1, 1]), ...: 'time': ('x', [0.0, 0.1, 0.2, 0.0, 0.15])}) ...:

In [2]: da Out[2]: <xarray.DataArray (x: 5)> array([0, 1, 2, 3, 4]) Coordinates: experiment (x) int64 0 0 0 1 1 time (x) float64 0.0 0.1 0.2 0.0 0.15 Dimensions without coordinates: x

```

I want to do something like this python da.sel(experiment=0).sel(time=0.1) but it cannot. MultiIndexing enables this,

python In [2]: da = da.set_index(exp_time=['experiment', 'time']) ...: da ...: Out[2]: <xarray.DataArray (x: 5)> array([0, 1, 2, 3, 4]) Coordinates: * exp_time (exp_time) MultiIndex - experiment (exp_time) int64 0 0 0 1 1 - time (exp_time) float64 0.0 0.1 0.2 0.0 0.15 Dimensions without coordinates: x

If we could make a selection from a non-index coordinate, MultiIndex is not necessary for this case.

I think there should be other important usecases of MultiIndex. I would be happy if anyone could list them in this issue.