@r-beer I checked back on this and realized I didn't reply to your question: yes re completing #2652, if you're up for giving this a push

@r-beer would be great to finish this off! I think this would be a popular feature. You could take @hrishikeshac 's code (which is close!) and make the final changes.

OK, that means to make #2652 pass, right?

I downloaded the respective branch from @hrishikeshac, and ran the tests locally.

See respective discussion in #2652.

@r-beer would be great to finish this off! I think this would be a popular feature. You could take @hrishikeshac 's code (which is close!) and make the final changes.

I am also highly interested in this function and in contributing to xarray in general!

If I understand correctly, https://github.com/pydata/xarray/pull/2350 and https://github.com/pydata/xarray/pull/2652 do not solve this PR, do they?

How can I help you finishing these PRs?

Guys sorry for dropping the ball on this one. I made some changes to the PR based on the feedback I got, but I couldn't figure out the tests. Would anyone like to take this over?

Would be great to get this in, if anyone wants to have a go. A small, focused, PR would be a good start.

In the meantime you can use one of the solutions above...

I see that this PR never made it through and there is a somewhat similar PR finished here: https://github.com/pydata/xarray/pull/2350 though it doesn't do exactly what was proposed in this PR. Is there a suggested approach for performing cross-correlation on multiple DataArray?

PR done! Changed np.sum() to dataarray.sum()

@hrishikeshac that looks great! Well done for getting an MVP running.

Do you want to do a PR from this? Should be v close from here.

Others can comment from there. I'd suggest we get something close to this in and iterate from there. How abstract do we want the dimensions to be (i.e. currently we can only pass one dimension in, which is fine, but potentially we could enable multiple).

One nit - no need to use np.sum - that may cause issues with dask arrays - .sum will work fine

Okay. Here's what I have come up with. I have tested it against two 1-d dataarrays, 2 N-D dataarrays, and one 1-D, and another N-D dataarrays, all cases having misaligned and having missing values.

Before going forward, 1. What do you think of it? Any improvements? 2. Steps 1 and 2 (broadcasting and ignoring common missing values) are identical in both cov() and corr(). Is there a better way to reduce the duplication while still retaining both functions as standalone?

``` def cov(self, other, dim = None): """Compute covariance between two DataArray objects along a shared dimension.

Parameters
----------
other: DataArray
    The other array with which the covariance will be computed
dim: The dimension along which the covariance will be computed

Returns
-------
covariance: DataArray
"""
# 1. Broadcast the two arrays
self, other     = xr.broadcast(self, other)

# 2. Ignore the nans
valid_values    = self.notnull() & other.notnull()
self            = self.where(valid_values, drop=True)
other           = other.where(valid_values, drop=True)
valid_count     = valid_values.sum(dim)

#3. Compute mean and standard deviation along the given dim
demeaned_self   = self - self.mean(dim = dim)
demeaned_other  = other - other.mean(dim = dim)

#4. Compute  covariance along the given dim
if dim:
    axis = self.get_axis_num(dim = dim)
else:
    axis = None
cov             =  np.sum(demeaned_self*demeaned_other, axis=axis)/(valid_count)

return cov

def corr(self, other, dim = None): """Compute correlation between two DataArray objects along a shared dimension.

Parameters
----------
other: DataArray
    The other array with which the correlation will be computed
dim: The dimension along which the correlation will be computed

Returns
-------
correlation: DataArray
"""
# 1. Broadcast the two arrays
self, other     = xr.broadcast(self, other)

# 2. Ignore the nans
valid_values    = self.notnull() & other.notnull()
self            = self.where(valid_values, drop=True)
other           = other.where(valid_values, drop=True)

# 3. Compute correlation based on standard deviations and cov()
self_std        = self.std(dim=dim)
other_std       = other.std(dim=dim)

return cov(self, other, dim = dim)/(self_std*other_std)

```

For testing: ``` # self: Load demo data and trim it's size ds = xr.tutorial.load_dataset('air_temperature') air = ds.air[:18,...] # other: select missaligned data, and smooth it to dampen the correlation with self. air_smooth = ds.air[2:20,...].rolling(time= 3, center=True).mean(dim='time') #. # A handy function to select an example grid def select_pts(da): return da.sel(lat=45, lon=250)

#Test #1: Misaligned 1-D dataarrays with missing values
ts1 = select_pts(air.copy())
ts2 = select_pts(air_smooth.copy())

def pd_corr(ts1,ts2):
    """Ensure the ts are aligned and missing values ignored"""
    # ts1,ts2 = xr.align(ts1,ts2)
    valid_values = ts1.notnull() & ts2.notnull()

    ts1  = ts1.where(valid_values, drop = True)
    ts2  = ts2.where(valid_values, drop = True)

    return ts1.to_series().corr(ts2.to_series())

expected = pd_corr(ts1, ts2)
actual   = corr(ts1,ts2)
np.allclose(expected, actual)

#Test #2: Misaligned N-D dataarrays with missing values
actual_ND = corr(air,air_smooth, dim = 'time')
actual = select_pts(actual_ND)
np.allclose(expected, actual)

# Test #3: One 1-D dataarray and another N-D dataarray; misaligned and having missing values
actual_ND = corr(air_smooth,ts1, dim = 'time')
actual    = select_pts(actual_ND)
np.allclose(actual, expected)

```

Okay. I am writing the simultaneous correlation and covariance functions on dataxarray.py instead of dataset.py- following the pd.Series.corr(self, other, dim) style.

Yes for useful, but not sure whether they should be on the same method. They're also fairly easy for a user to construct (call correlation on a .shift copy of the array).

And increments are easy to build on! I'm the worst offender, but don't let completeness get in the way of incremental improvement

(OK, I'll go and finish the fill_value branch...)

I think lagged correlations would be a useful feature.

Sorry for the radio silence- I will work on this next week. Thanks @max-sixty for the updates, @rabernat for reaching out, will let you know if I need help.

Should we keep it simple following @max-sixty , or should I also add the functionality to handle lagged correlations?

Hey @hrishikeshac -- any progress on this? Need any help / advice from xarray devs?

And one that handles NaNs:

```python

untested!

def covariance(x, y, dim=None): valid_values = x.notnull() & y.notnull() valid_count = valid_values.sum(dim)

demeaned_x = (x - x.mean(dim)).fillna(0)
demeaned_y = (y - y.mean(dim)).fillna(0)

return xr.dot(demeaned_x, demeaned_y, dims=dim) / valid_count

def correlation(x, y, dim=None): # dim should default to the intersection of x.dims and y.dims return covariance(x, y, dim) / (x.std(dim) * y.std(dim)) ```

For posterity, I made a small adjustment to @shoyer 's draft:

```python

untested!

def covariance(x, y, dim=None): # need to ensure the dim lengths are the same - i.e. no auto-aligning # could use count-1 for sample return xr.dot(x - x.mean(dim), y - y.mean(dim), dims=dim) / x.count(dim)

def correlation(x, y, dim=None): # dim should default to the intersection of x.dims and y.dims return covariance(x, y, dim) / (x.std(dim) * y.std(dim)) ```

Great! Ping me / the issues with any questions at all!

@max-sixty thanks!

Then I will start with testing @shoyer 's suggestion and mvstats for the basic implementation.

@hrishikeshac if you'd like to contribute, we can help you along - xarray is a v welcoming project!

And from mvstats it looks like you're already up to speed

Let us know

Sometime back I wrote a package based on xarray regarding this. I would be happy to be involved in implementing it in xarray as well, but I am new to contributing to such a large-scale project and it looks a bit intimidating!

I tend to view the second case as a generalization of the first case. I would also hesitate to implement the n x m array -> m x m correlation matrix version because xarray doesn't handle repeated dimensions well.

I think the basic implementation of this looks quite similar to what I wrote here for calculating the Pearson correlation as a NumPy gufunc: http://xarray.pydata.org/en/stable/dask.html#automatic-parallelization

The main difference is that we might naturally want to support summing over multiple dimensions at once via the dim argument, e.g., something like: ```python

untested!

def covariance(x, y, dim=None): return xarray.dot(x - x.mean(dim), y - y.mean(dim), dim=dim)

def corrrelation(x, y, dim=None): # dim should default to the intersection of x.dims and y.dims return covariance(x, y, dim) / (x.std(dim) * y.std(dim)) ```

If you want to achieve the equivalent of np.corr on an array with dimensions ('n', 'm') with this, you just write something like correlation(x, x.rename({'m': 'm2'}), dim='n').

I'm up for adding .corr to xarray

What do want this to look like? It's a bit different from most xarray functions, which either return the same shape or reduce one dimension. - The basic case here would take a n x m array and return an m x m correlation matrix. We could easily wrap https://docs.scipy.org/doc/numpy/reference/generated/numpy.corrcoef.html - Another case would be take two similarly sized arrays (with the option of broadcasting) and return an array with one dimension reduced. For example 200 x 10 and 200, return a 10 array. - I need to think about how those extrapolate to multiple dimensions

Should I start with the first case and then we can expand as needed?

@hrishikeshac I was just looking for a function doing a regression between two datasets (x, y, time), so thanks for your function! However, I'm still wondering whether there is a much faster C (or Cython) implementation doing these kind of things?

FYI @shoyer @fmaussion , I had to revisit the problem and ended up writing a function to compute vectorized cross-correlation, covariance, regression calculations (along with p-value and standard error) for xr.DataArrays. Essentially, I tried to mimic scipy.stats.linregress() but for multi-dimensional data, and included the ability to compute lagged relationships. Here's the function and its demonstration; please feel free to incorporate it in xarray if deemed useful: https://hrishichandanpurkar.blogspot.com/2017/09/vectorized-functions-for-correlation.html

I'll chime in here to ask a usage question: what is the recommended way to compute correlation maps with xarray? I.e. I have a dataarray of dims (time, lat, lon) and I'd like to correlate every single grid point with a timeseries of dim (time) to get a correlation map of dim (lat, lon). My current strategy is a wonderfully unpythonic double loop over lons and lats, and I wonder if there's better way?

To be clear, I am not say that this does not belong in xarray.

I'm saying that we lack clear general guidelines for how to determine whether a particular function belongs in xarray. The criterion of a "pretty fundamental operation for working with data" is a good starting point. I would add: - used across a wide range of scientific disciplines - clear, unambiguous / uncontroversial definition - numpy implementation already exists

corr meets all of these criteria. Many others (e.g. interpolation, convolution, curve fitting) do as well. Expanding xarray beyond the numpy ufuncs opens the door to supporting these things. I'm just saying it should be conscious, deliberate decision, given the limits on developer time.

Many of these things will be pretty trivial once .apply() is here. So perhaps it's not a big deal.

That said, correlation coefficients are a pretty fundamental operation for working with data. I could see implementing a basic corr in xarray and referring to a separate signal processing package for more options in the docstring.

I agree with @rabernat in the sense that it could be part of another package (e.g., signal processing). This would also allow the computation of statistical test to assess the significance of the correlation (which is useful since correlation may often be misinterpreted without statistical tests).

I agree this would be very useful. But it is also feature creep. There is an extremely wide range of such functions that could hypothetically be put into the xarray package. (all of scipy.signal for example) At some point the community should decide what is the intended scope of xarray itself vs. packages built on top of xarray.

The first step here is to find a library that implements the desired functionality on pure NumPy arrays, ideally in a vectorized fashion. Then it should be pretty straightforward to wrap in xarray.