Computation (original) (raw)

The labels associated with DataArray andDataset objects enables some powerful shortcuts for computation, notably including aggregation and broadcasting by dimension names.

Basic array math#

Arithmetic operations with a single DataArray automatically vectorize (like numpy) over all array values:

In [1]: arr = xr.DataArray( ...: np.random.default_rng(0).random((2, 3)), ...: [("x", ["a", "b"]), ("y", [10, 20, 30])], ...: ) ...:

In [2]: arr - 3 Out[2]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[-2.36303831, -2.73021329, -2.95902648], [-2.98347236, -2.18672976, -2.08724442]]) Coordinates:

In [3]: abs(arr) Out[3]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[0.63696169, 0.26978671, 0.04097352], [0.01652764, 0.81327024, 0.91275558]]) Coordinates:

You can also use any of numpy’s or scipy’s many ufunc functions directly on a DataArray:

In [4]: np.sin(arr) Out[4]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[0.59475567, 0.26652587, 0.04096206], [0.01652688, 0.72653812, 0.79119196]]) Coordinates:

Use where() to conditionally switch between values:

In [5]: xr.where(arr > 0, "positive", "negative") Out[5]: <xarray.DataArray (x: 2, y: 3)> Size: 192B array([['positive', 'positive', 'positive'], ['positive', 'positive', 'positive']], dtype='<U8') Coordinates:

Use @ to compute the dot() product:

In [6]: arr @ arr Out[6]: <xarray.DataArray ()> Size: 8B array(1.97498828)

Data arrays also implement many numpy.ndarray methods:

In [7]: arr.round(2) Out[7]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[0.64, 0.27, 0.04], [0.02, 0.81, 0.91]]) Coordinates:

In [8]: arr.T Out[8]: <xarray.DataArray (y: 3, x: 2)> Size: 48B array([[0.63696169, 0.01652764], [0.26978671, 0.81327024], [0.04097352, 0.91275558]]) Coordinates:

In [9]: intarr = xr.DataArray([0, 1, 2, 3, 4, 5])

In [10]: intarr << 2 # only supported for int types Out[10]: <xarray.DataArray (dim_0: 6)> Size: 48B array([ 0, 4, 8, 12, 16, 20]) Dimensions without coordinates: dim_0

In [11]: intarr >> 1 Out[11]: <xarray.DataArray (dim_0: 6)> Size: 48B array([0, 0, 1, 1, 2, 2]) Dimensions without coordinates: dim_0

Missing values#

Xarray represents missing values using the “NaN” (Not a Number) value from NumPy, which is a special floating-point value that indicates a value that is undefined or unrepresentable. There are several methods for handling missing values in xarray:

Xarray objects borrow the isnull(),notnull(), count(),dropna(), fillna(),ffill(), and bfill()methods for working with missing data from pandas:

isnull() is a method in xarray that can be used to check for missing or null values in an xarray object. It returns a new xarray object with the same dimensions as the original object, but with boolean values indicating where missing values are present.

In [12]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [13]: x.isnull() Out[13]: <xarray.DataArray (x: 5)> Size: 5B array([False, False, True, True, False]) Dimensions without coordinates: x

In this example, the third and fourth elements of ‘x’ are NaN, so the resulting DataArrayobject has ‘True’ values in the third and fourth positions and ‘False’ values in the other positions.

notnull() is a method in xarray that can be used to check for non-missing or non-null values in an xarray object. It returns a new xarray object with the same dimensions as the original object, but with boolean values indicating where non-missing values are present.

In [14]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [15]: x.notnull() Out[15]: <xarray.DataArray (x: 5)> Size: 5B array([ True, True, False, False, True]) Dimensions without coordinates: x

In this example, the first two and the last elements of x are not NaN, so the resultingDataArray object has ‘True’ values in these positions, and ‘False’ values in the third and fourth positions where NaN is located.

count() is a method in xarray that can be used to count the number of non-missing values along one or more dimensions of an xarray object. It returns a new xarray object with the same dimensions as the original object, but with each element replaced by the count of non-missing values along the specified dimensions.

In [16]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [17]: x.count() Out[17]: <xarray.DataArray ()> Size: 8B array(3)

In this example, ‘x’ has five elements, but two of them are NaN, so the resultingDataArray object having a single element containing the value ‘3’, which represents the number of non-null elements in x.

dropna() is a method in xarray that can be used to remove missing or null values from an xarray object. It returns a new xarray object with the same dimensions as the original object, but with missing values removed.

In [18]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [19]: x.dropna(dim="x") Out[19]: <xarray.DataArray (x: 3)> Size: 24B array([0., 1., 2.]) Dimensions without coordinates: x

In this example, on calling x.dropna(dim=”x”) removes any missing values and returns a newDataArray object with only the non-null elements [0, 1, 2] of ‘x’, in the original order.

fillna() is a method in xarray that can be used to fill missing or null values in an xarray object with a specified value or method. It returns a new xarray object with the same dimensions as the original object, but with missing values filled.

In [20]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [21]: x.fillna(-1) Out[21]: <xarray.DataArray (x: 5)> Size: 40B array([ 0., 1., -1., -1., 2.]) Dimensions without coordinates: x

In this example, there are two NaN values in ‘x’, so calling x.fillna(-1) replaces these values with -1 and returns a new DataArray object with five elements, containing the values [0, 1, -1, -1, 2] in the original order.

ffill() is a method in xarray that can be used to forward fill (or fill forward) missing values in an xarray object along one or more dimensions. It returns a new xarray object with the same dimensions as the original object, but with missing values replaced by the last non-missing value along the specified dimensions.

In [22]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [23]: x.ffill("x") Out[23]: <xarray.DataArray (x: 5)> Size: 40B array([0., 1., 1., 1., 2.]) Dimensions without coordinates: x

In this example, there are two NaN values in ‘x’, so calling x.ffill(“x”) fills these values with the last non-null value in the same dimension, which are 0 and 1, respectively. The resulting DataArray object has five elements, containing the values [0, 1, 1, 1, 2] in the original order.

bfill() is a method in xarray that can be used to backward fill (or fill backward) missing values in an xarray object along one or more dimensions. It returns a new xarray object with the same dimensions as the original object, but with missing values replaced by the next non-missing value along the specified dimensions.

In [24]: x = xr.DataArray([0, 1, np.nan, np.nan, 2], dims=["x"])

In [25]: x.bfill("x") Out[25]: <xarray.DataArray (x: 5)> Size: 40B array([0., 1., 2., 2., 2.]) Dimensions without coordinates: x

In this example, there are two NaN values in ‘x’, so calling x.bfill(“x”) fills these values with the next non-null value in the same dimension, which are 2 and 2, respectively. The resulting DataArray object has five elements, containing the values [0, 1, 2, 2, 2] in the original order.

Like pandas, xarray uses the float value np.nan (not-a-number) to represent missing values.

Xarray objects also have an interpolate_na() method for filling missing values via 1D interpolation. It returns a new xarray object with the same dimensions as the original object, but with missing values interpolated.

In [26]: x = xr.DataArray( ....: [0, 1, np.nan, np.nan, 2], ....: dims=["x"], ....: coords={"xx": xr.Variable("x", [0, 1, 1.1, 1.9, 3])}, ....: ) ....:

In [27]: x.interpolate_na(dim="x", method="linear", use_coordinate="xx") Out[27]: <xarray.DataArray (x: 5)> Size: 40B array([0. , 1. , 1.05, 1.45, 2. ]) Coordinates: xx (x) float64 40B 0.0 1.0 1.1 1.9 3.0 Dimensions without coordinates: x

In this example, there are two NaN values in ‘x’, so calling x.interpolate_na(dim=”x”, method=”linear”, use_coordinate=”xx”) fills these values with interpolated values along the “x” dimension using linear interpolation based on the values of the xx coordinate. The resulting DataArray object has five elements, containing the values [0., 1., 1.05, 1.45, 2.] in the original order. Note that the interpolated values are calculated based on the values of the ‘xx’ coordinate, which has non-integer values, resulting in non-integer interpolated values.

Note that xarray slightly diverges from the pandas interpolate syntax by providing the use_coordinate keyword which facilitates a clear specification of which values to use as the index in the interpolation. Xarray also provides the max_gap keyword argument to limit the interpolation to data gaps of length max_gap or smaller. See interpolate_na()for more.

Aggregation#

Aggregation methods have been updated to take a dim argument instead ofaxis. This allows for very intuitive syntax for aggregation methods that are applied along particular dimension(s):

In [28]: arr.sum(dim="x") Out[28]: <xarray.DataArray (y: 3)> Size: 24B array([0.65348932, 1.08305695, 0.9537291 ]) Coordinates:

In [29]: arr.std(["x", "y"]) Out[29]: <xarray.DataArray ()> Size: 8B array(0.35793963)

In [30]: arr.min() Out[30]: <xarray.DataArray ()> Size: 8B array(0.01652764)

If you need to figure out the axis number for a dimension yourself (say, for wrapping code designed to work with numpy arrays), you can use theget_axis_num() method:

In [31]: arr.get_axis_num("y") Out[31]: 1

These operations automatically skip missing values, like in pandas:

In [32]: xr.DataArray([1, 2, np.nan, 3]).mean() Out[32]: <xarray.DataArray ()> Size: 8B array(2.)

If desired, you can disable this behavior by invoking the aggregation method with skipna=False.

Rolling window operations#

DataArray objects include a rolling() method. This method supports rolling window aggregation:

In [33]: arr = xr.DataArray(np.arange(0, 7.5, 0.5).reshape(3, 5), dims=("x", "y"))

In [34]: arr Out[34]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[0. , 0.5, 1. , 1.5, 2. ], [2.5, 3. , 3.5, 4. , 4.5], [5. , 5.5, 6. , 6.5, 7. ]]) Dimensions without coordinates: x, y

rolling() is applied along one dimension using the name of the dimension as a key (e.g. y) and the window size as the value (e.g. 3). We get back a Rolling object:

In [35]: arr.rolling(y=3) Out[35]: DataArrayRolling [y->3]

Aggregation and summary methods can be applied directly to the Rollingobject:

In [36]: r = arr.rolling(y=3)

In [37]: r.reduce(np.std) Out[37]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[ nan, nan, 0.40824829, 0.40824829, 0.40824829], [ nan, nan, 0.40824829, 0.40824829, 0.40824829], [ nan, nan, 0.40824829, 0.40824829, 0.40824829]]) Dimensions without coordinates: x, y

In [38]: r.mean() Out[38]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[nan, nan, 0.5, 1. , 1.5], [nan, nan, 3. , 3.5, 4. ], [nan, nan, 5.5, 6. , 6.5]]) Dimensions without coordinates: x, y

Aggregation results are assigned the coordinate at the end of each window by default, but can be centered by passing center=True when constructing theRolling object:

In [39]: r = arr.rolling(y=3, center=True)

In [40]: r.mean() Out[40]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[nan, 0.5, 1. , 1.5, nan], [nan, 3. , 3.5, 4. , nan], [nan, 5.5, 6. , 6.5, nan]]) Dimensions without coordinates: x, y

As can be seen above, aggregations of windows which overlap the border of the array produce nans. Setting min_periods in the call to rollingchanges the minimum number of observations within the window required to have a value when aggregating:

In [41]: r = arr.rolling(y=3, min_periods=2)

In [42]: r.mean() Out[42]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[ nan, 0.25, 0.5 , 1. , 1.5 ], [ nan, 2.75, 3. , 3.5 , 4. ], [ nan, 5.25, 5.5 , 6. , 6.5 ]]) Dimensions without coordinates: x, y

In [43]: r = arr.rolling(y=3, center=True, min_periods=2)

In [44]: r.mean() Out[44]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[0.25, 0.5 , 1. , 1.5 , 1.75], [2.75, 3. , 3.5 , 4. , 4.25], [5.25, 5.5 , 6. , 6.5 , 6.75]]) Dimensions without coordinates: x, y

From version 0.17, xarray supports multidimensional rolling,

In [45]: r = arr.rolling(x=2, y=3, min_periods=2)

In [46]: r.mean() Out[46]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[ nan, 0.25, 0.5 , 1. , 1.5 ], [1.25, 1.5 , 1.75, 2.25, 2.75], [3.75, 4. , 4.25, 4.75, 5.25]]) Dimensions without coordinates: x, y

Tip

Note that rolling window aggregations are faster and use less memory when bottleneck is installed. This only applies to numpy-backed xarray objects with 1d-rolling.

We can also manually iterate through Rolling objects:

for label, arr_window in r: # arr_window is a view of x ...

While rolling provides a simple moving average, DataArray also supports an exponential moving average with rolling_exp(). This is similar to pandas’ ewm method. numbagg is required.

arr.rolling_exp(y=3).mean()

The rolling_exp method takes a window_type kwarg, which can be 'alpha','com' (for center-of-mass), 'span', and 'halflife'. The default isspan.

Finally, the rolling object has a construct method which returns a view of the original DataArray with the windowed dimension in the last position. You can use this for more advanced rolling operations such as strided rolling, windowed rolling, convolution, short-time FFT etc.

rolling with 2-point stride

In [47]: rolling_da = r.construct(x="x_win", y="y_win", stride=2)

In [48]: rolling_da Out[48]: <xarray.DataArray (x: 2, y: 3, x_win: 2, y_win: 3)> Size: 288B array([[[[nan, nan, nan], [nan, nan, 0. ]],

    [[nan, nan, nan],
     [0. , 0.5, 1. ]],

    [[nan, nan, nan],
     [1. , 1.5, 2. ]]],


   [[[nan, nan, 2.5],
     [nan, nan, 5. ]],

    [[2.5, 3. , 3.5],
     [5. , 5.5, 6. ]],

    [[3.5, 4. , 4.5],
     [6. , 6.5, 7. ]]]], shape=(2, 3, 2, 3))

Dimensions without coordinates: x, y, x_win, y_win

In [49]: rolling_da.mean(["x_win", "y_win"], skipna=False) Out[49]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[ nan, nan, nan], [ nan, 4.25, 5.25]]) Dimensions without coordinates: x, y

Because the DataArray given by r.construct('window_dim') is a view of the original array, it is memory efficient. You can also use construct to compute a weighted rolling sum:

In [50]: weight = xr.DataArray([0.25, 0.5, 0.25], dims=["window"])

In [51]: arr.rolling(y=3).construct(y="window").dot(weight) Out[51]: <xarray.DataArray (x: 3, y: 5)> Size: 120B array([[nan, nan, 0.5, 1. , 1.5], [nan, nan, 3. , 3.5, 4. ], [nan, nan, 5.5, 6. , 6.5]]) Dimensions without coordinates: x, y

Note

numpy’s Nan-aggregation functions such as nansum copy the original array. In xarray, we internally use these functions in our aggregation methods (such as .sum()) if skipna argument is not specified or set to True. This means rolling_da.mean('window_dim') is memory inefficient. To avoid this, use skipna=False as the above example.

Weighted array reductions#

DataArray and Dataset objects include DataArray.weighted()and Dataset.weighted() array reduction methods. They currently support weighted sum, mean, std, var and quantile.

In [52]: coords = dict(month=("month", [1, 2, 3]))

In [53]: prec = xr.DataArray([1.1, 1.0, 0.9], dims=("month",), coords=coords)

In [54]: weights = xr.DataArray([31, 28, 31], dims=("month",), coords=coords)

Create a weighted object:

In [55]: weighted_prec = prec.weighted(weights)

In [56]: weighted_prec Out[56]: DataArrayWeighted with weights along dimensions: month

Calculate the weighted sum:

In [57]: weighted_prec.sum() Out[57]: <xarray.DataArray ()> Size: 8B array(90.)

Calculate the weighted mean:

In [58]: weighted_prec.mean(dim="month") Out[58]: <xarray.DataArray ()> Size: 8B array(1.)

Calculate the weighted quantile:

In [59]: weighted_prec.quantile(q=0.5, dim="month") Out[59]: <xarray.DataArray ()> Size: 8B array(1.) Coordinates: quantile float64 8B 0.5

The weighted sum corresponds to:

In [60]: weighted_sum = (prec * weights).sum()

In [61]: weighted_sum Out[61]: <xarray.DataArray ()> Size: 8B array(90.)

the weighted mean to:

In [62]: weighted_mean = weighted_sum / weights.sum()

In [63]: weighted_mean Out[63]: <xarray.DataArray ()> Size: 8B array(1.)

the weighted variance to:

In [64]: weighted_var = weighted_prec.sum_of_squares() / weights.sum()

In [65]: weighted_var Out[65]: <xarray.DataArray ()> Size: 8B array(0.00688889)

and the weighted standard deviation to:

In [66]: weighted_std = np.sqrt(weighted_var)

In [67]: weighted_std Out[67]: <xarray.DataArray ()> Size: 8B array(0.08299933)

However, the functions also take missing values in the data into account:

In [68]: data = xr.DataArray([np.nan, 2, 4])

In [69]: weights = xr.DataArray([8, 1, 1])

In [70]: data.weighted(weights).mean() Out[70]: <xarray.DataArray ()> Size: 8B array(3.)

Using (data * weights).sum() / weights.sum() would (incorrectly) result in 0.6.

If the weights add up to to 0, sum returns 0:

In [71]: data = xr.DataArray([1.0, 1.0])

In [72]: weights = xr.DataArray([-1.0, 1.0])

In [73]: data.weighted(weights).sum() Out[73]: <xarray.DataArray ()> Size: 8B array(0.)

and mean, std and var return nan:

In [74]: data.weighted(weights).mean() Out[74]: <xarray.DataArray ()> Size: 8B array(nan)

Note

weights must be a DataArray and cannot contain missing values. Missing values can be replaced manually by weights.fillna(0).

Coarsen large arrays#

DataArray and Dataset objects include acoarsen() and coarsen()methods. This supports block aggregation along multiple dimensions,

In [75]: x = np.linspace(0, 10, 300)

In [76]: t = pd.date_range("1999-12-15", periods=364)

In [77]: da = xr.DataArray( ....: np.sin(x) * np.cos(np.linspace(0, 1, 364)[:, np.newaxis]), ....: dims=["time", "x"], ....: coords={"time": t, "x": x}, ....: ) ....:

In [78]: da Out[78]: <xarray.DataArray (time: 364, x: 300)> Size: 874kB array([[ 0. , 0.03343858, 0.06683976, ..., -0.48672119, -0.51565952, -0.54402111], [ 0. , 0.03343845, 0.06683951, ..., -0.48671934, -0.51565756, -0.54401905], [ 0. , 0.03343807, 0.06683875, ..., -0.4867138 , -0.51565169, -0.54401285], ..., [ 0. , 0.0182217 , 0.03642301, ..., -0.26522911, -0.28099849, -0.29645358], [ 0. , 0.01814439, 0.03626848, ..., -0.26410385, -0.27980632, -0.29519584], [ 0. , 0.01806694, 0.03611368, ..., -0.26297658, -0.27861203, -0.29393586]], shape=(364, 300)) Coordinates:

In order to take a block mean for every 7 days along time dimension and every 2 points along x dimension,

In [79]: da.coarsen(time=7, x=2).mean() Out[79]: <xarray.DataArray (time: 52, x: 150)> Size: 62kB array([[ 0.01671847, 0.08349886, 0.14990579, ..., -0.41198807, -0.47195655, -0.52981418], [ 0.01671269, 0.08347003, 0.14985403, ..., -0.41184582, -0.47179359, -0.52963124], [ 0.01670071, 0.08341016, 0.14974655, ..., -0.41155042, -0.47145519, -0.52925136], ..., [ 0.00968205, 0.04835611, 0.0868139 , ..., -0.23859177, -0.2733209 , -0.30682759], [ 0.00941742, 0.04703446, 0.08444113, ..., -0.23207067, -0.26585059, -0.29844148], [ 0.00914929, 0.04569531, 0.08203696, ..., -0.22546326, -0.25828142, -0.2899444 ]], shape=(52, 150)) Coordinates:

coarsen() raises a ValueError if the data length is not a multiple of the corresponding window size. You can choose boundary='trim' or boundary='pad' options for trimming the excess entries or padding nan to insufficient entries,

In [80]: da.coarsen(time=30, x=2, boundary="trim").mean() Out[80]: <xarray.DataArray (time: 12, x: 150)> Size: 14kB array([[ 0.01670121, 0.08341265, 0.14975103, ..., -0.41156272, -0.47146929, -0.52926718], [ 0.0165891 , 0.08285275, 0.14874584, ..., -0.40880017, -0.46830462, -0.52571455], [ 0.01636376, 0.08172729, 0.14672529, ..., -0.40324704, -0.46194319, -0.51857326], ..., [ 0.01183847, 0.05912615, 0.10614938, ..., -0.29173175, -0.33419587, -0.37516528], [ 0.01082401, 0.05405954, 0.09705329, ..., -0.26673283, -0.30555813, -0.34301681], [ 0.00973567, 0.04862391, 0.08729468, ..., -0.23991312, -0.27483458, -0.30852683]], shape=(12, 150)) Coordinates:

If you want to apply a specific function to coordinate, you can pass the function or method name to coord_func option,

In [81]: da.coarsen(time=7, x=2, coord_func={"time": "min"}).mean() Out[81]: <xarray.DataArray (time: 52, x: 150)> Size: 62kB array([[ 0.01671847, 0.08349886, 0.14990579, ..., -0.41198807, -0.47195655, -0.52981418], [ 0.01671269, 0.08347003, 0.14985403, ..., -0.41184582, -0.47179359, -0.52963124], [ 0.01670071, 0.08341016, 0.14974655, ..., -0.41155042, -0.47145519, -0.52925136], ..., [ 0.00968205, 0.04835611, 0.0868139 , ..., -0.23859177, -0.2733209 , -0.30682759], [ 0.00941742, 0.04703446, 0.08444113, ..., -0.23207067, -0.26585059, -0.29844148], [ 0.00914929, 0.04569531, 0.08203696, ..., -0.22546326, -0.25828142, -0.2899444 ]], shape=(52, 150)) Coordinates:

You can also use coarsen to reshape without applying a computation.

Computation using Coordinates#

Xarray objects have some handy methods for the computation with their coordinates. differentiate() computes derivatives by central finite differences using their coordinates,

In [82]: a = xr.DataArray([0, 1, 2, 3], dims=["x"], coords=[[0.1, 0.11, 0.2, 0.3]])

In [83]: a Out[83]: <xarray.DataArray (x: 4)> Size: 32B array([0, 1, 2, 3]) Coordinates:

In [84]: a.differentiate("x") Out[84]: <xarray.DataArray (x: 4)> Size: 32B array([100. , 91.11111111, 10.58479532, 10. ]) Coordinates:

This method can be used also for multidimensional arrays,

In [85]: a = xr.DataArray( ....: np.arange(8).reshape(4, 2), dims=["x", "y"], coords={"x": [0.1, 0.11, 0.2, 0.3]} ....: ) ....:

In [86]: a.differentiate("x") Out[86]: <xarray.DataArray (x: 4, y: 2)> Size: 64B array([[200. , 200. ], [182.22222222, 182.22222222], [ 21.16959064, 21.16959064], [ 20. , 20. ]]) Coordinates:

integrate() computes integration based on trapezoidal rule using their coordinates,

In [87]: a.integrate("x") Out[87]: <xarray.DataArray (y: 2)> Size: 16B array([0.78, 0.98]) Dimensions without coordinates: y

Note

These methods are limited to simple cartesian geometry. Differentiation and integration along multidimensional coordinate are not supported.

Fitting polynomials#

Xarray objects provide an interface for performing linear or polynomial regressions using the least-squares method. polyfit() computes the best fitting coefficients along a given dimension and for a given order,

In [88]: x = xr.DataArray(np.arange(10), dims=["x"], name="x")

In [89]: a = xr.DataArray(3 + 4 * x, dims=["x"], coords={"x": x})

In [90]: out = a.polyfit(dim="x", deg=1, full=True)

In [91]: out Out[91]: <xarray.Dataset> Size: 64B Dimensions: (degree: 2) Coordinates:

The method outputs a dataset containing the coefficients (and more if full=True). The inverse operation is done with polyval(),

In [92]: xr.polyval(coord=x, coeffs=out.polyfit_coefficients) Out[92]: <xarray.DataArray (x: 10)> Size: 80B array([ 3., 7., 11., 15., 19., 23., 27., 31., 35., 39.]) Dimensions without coordinates: x

Fitting arbitrary functions#

Xarray objects also provide an interface for fitting more complex functions usingscipy.optimize.curve_fit(). curvefit() accepts user-defined functions and can fit along multiple coordinates.

For example, we can fit a relationship between two DataArray objects, maintaining a unique fit at each spatial coordinate but aggregating over the time dimension:

In [93]: def exponential(x, a, xc): ....: return np.exp((x - xc) / a) ....:

In [94]: x = np.arange(-5, 5, 0.1)

In [95]: t = np.arange(-5, 5, 0.1)

In [96]: X, T = np.meshgrid(x, t)

In [97]: Z1 = np.random.uniform(low=-5, high=5, size=X.shape)

In [98]: Z2 = exponential(Z1, 3, X)

In [99]: Z3 = exponential(Z1, 1, -X)

In [100]: ds = xr.Dataset( .....: data_vars=dict( .....: var1=(["t", "x"], Z1), var2=(["t", "x"], Z2), var3=(["t", "x"], Z3) .....: ), .....: coords={"t": t, "x": x}, .....: ) .....:

In [101]: ds[["var2", "var3"]].curvefit( .....: coords=ds.var1, .....: func=exponential, .....: reduce_dims="t", .....: bounds={"a": (0.5, 5), "xc": (-5, 5)}, .....: ) .....: Out[101]: <xarray.Dataset> Size: 10kB Dimensions: (x: 100, param: 2, cov_i: 2, cov_j: 2) Coordinates:

We can also fit multi-dimensional functions, and even use a wrapper function to simultaneously fit a summation of several functions, such as this field containing two gaussian peaks:

In [102]: def gaussian_2d(coords, a, xc, yc, xalpha, yalpha): .....: x, y = coords .....: z = a * np.exp( .....: -np.square(x - xc) / 2 / np.square(xalpha) .....: - np.square(y - yc) / 2 / np.square(yalpha) .....: ) .....: return z .....:

In [103]: def multi_peak(coords, *args): .....: z = np.zeros(coords[0].shape) .....: for i in range(len(args) // 5): .....: z += gaussian_2d(coords, *args[i * 5 : i * 5 + 5]) .....: return z .....:

In [104]: x = np.arange(-5, 5, 0.1)

In [105]: y = np.arange(-5, 5, 0.1)

In [106]: X, Y = np.meshgrid(x, y)

In [107]: n_peaks = 2

In [108]: names = ["a", "xc", "yc", "xalpha", "yalpha"]

In [109]: names = [f"{name}{i}" for i in range(n_peaks) for name in names]

In [110]: Z = gaussian_2d((X, Y), 3, 1, 1, 2, 1) + gaussian_2d((X, Y), 2, -1, -2, 1, 1)

In [111]: Z += np.random.normal(scale=0.1, size=Z.shape)

In [112]: da = xr.DataArray(Z, dims=["y", "x"], coords={"y": y, "x": x})

In [113]: da.curvefit( .....: coords=["x", "y"], .....: func=multi_peak, .....: param_names=names, .....: kwargs={"maxfev": 10000}, .....: ) .....: Out[113]: <xarray.Dataset> Size: 2kB Dimensions: (param: 10, cov_i: 10, cov_j: 10) Coordinates:

Broadcasting by dimension name#

DataArray objects automatically align themselves (“broadcasting” in the numpy parlance) by dimension name instead of axis order. With xarray, you do not need to transpose arrays or insert dimensions of length 1 to get array operations to work, as commonly done in numpy with numpy.reshape() ornumpy.newaxis.

This is best illustrated by a few examples. Consider two one-dimensional arrays with different sizes aligned along different dimensions:

In [114]: a = xr.DataArray([1, 2], [("x", ["a", "b"])])

In [115]: a Out[115]: <xarray.DataArray (x: 2)> Size: 16B array([1, 2]) Coordinates:

In [116]: b = xr.DataArray([-1, -2, -3], [("y", [10, 20, 30])])

In [117]: b Out[117]: <xarray.DataArray (y: 3)> Size: 24B array([-1, -2, -3]) Coordinates:

With xarray, we can apply binary mathematical operations to these arrays, and their dimensions are expanded automatically:

In [118]: a * b Out[118]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[-1, -2, -3], [-2, -4, -6]]) Coordinates:

Moreover, dimensions are always reordered to the order in which they first appeared:

In [119]: c = xr.DataArray(np.arange(6).reshape(3, 2), [b["y"], a["x"]])

In [120]: c Out[120]: <xarray.DataArray (y: 3, x: 2)> Size: 48B array([[0, 1], [2, 3], [4, 5]]) Coordinates:

In [121]: a + c Out[121]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[1, 3, 5], [3, 5, 7]]) Coordinates:

This means, for example, that you always subtract an array from its transpose:

In [122]: c - c.T Out[122]: <xarray.DataArray (y: 3, x: 2)> Size: 48B array([[0, 0], [0, 0], [0, 0]]) Coordinates:

You can explicitly broadcast xarray data structures by using thebroadcast() function:

In [123]: a2, b2 = xr.broadcast(a, b)

In [124]: a2 Out[124]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[1, 1, 1], [2, 2, 2]]) Coordinates:

In [125]: b2 Out[125]: <xarray.DataArray (x: 2, y: 3)> Size: 48B array([[-1, -2, -3], [-1, -2, -3]]) Coordinates:

Automatic alignment#

Xarray enforces alignment between index Coordinates (that is, coordinates with the same name as a dimension, marked by *) on objects used in binary operations.

Similarly to pandas, this alignment is automatic for arithmetic on binary operations. The default result of a binary operation is by the intersection(not the union) of coordinate labels:

In [126]: arr = xr.DataArray(np.arange(3), [("x", range(3))])

In [127]: arr + arr[:-1] Out[127]: <xarray.DataArray (x: 2)> Size: 16B array([0, 2]) Coordinates:

If coordinate values for a dimension are missing on either argument, all matching dimensions must have the same size:

In [128]: arr + xr.DataArray([1, 2], dims="x") ValueError: arguments without labels along dimension 'x' cannot be aligned because they have different dimension size(s) {2} than the size of the aligned dimension labels: 3

However, one can explicitly change this default automatic alignment type (“inner”) via set_options() in context manager:

In [129]: with xr.set_options(arithmetic_join="outer"): .....: arr + arr[:1] .....:

In [130]: arr + arr[:1] Out[130]: <xarray.DataArray (x: 1)> Size: 8B array([0]) Coordinates:

Before loops or performance critical code, it’s a good idea to align arrays explicitly (e.g., by putting them in the same Dataset or usingalign()) to avoid the overhead of repeated alignment with each operation. See Align and reindex for more details.

Note

There is no automatic alignment between arguments when performing in-place arithmetic operations such as +=. You will need to usemanual alignment. This ensures in-place arithmetic never needs to modify data types.

Coordinates#

Although index coordinates are aligned, other coordinates are not, and if their values conflict, they will be dropped. This is necessary, for example, because indexing turns 1D coordinates into scalar coordinates:

In [131]: arr[0] Out[131]: <xarray.DataArray ()> Size: 8B array(0) Coordinates: x int64 8B 0

In [132]: arr[1] Out[132]: <xarray.DataArray ()> Size: 8B array(1) Coordinates: x int64 8B 1

notice that the scalar coordinate 'x' is silently dropped

In [133]: arr[1] - arr[0] Out[133]: <xarray.DataArray ()> Size: 8B array(1)

Still, xarray will persist other coordinates in arithmetic, as long as there are no conflicting values:

only one argument has the 'x' coordinate

In [134]: arr[0] + 1 Out[134]: <xarray.DataArray ()> Size: 8B array(1) Coordinates: x int64 8B 0

both arguments have the same 'x' coordinate

In [135]: arr[0] - arr[0] Out[135]: <xarray.DataArray ()> Size: 8B array(0) Coordinates: x int64 8B 0

Math with datasets#

Datasets support arithmetic operations by automatically looping over all data variables:

In [136]: ds = xr.Dataset( .....: { .....: "x_and_y": (("x", "y"), np.random.randn(3, 5)), .....: "x_only": ("x", np.random.randn(3)), .....: }, .....: coords=arr.coords, .....: ) .....:

In [137]: ds > 0 Out[137]: <xarray.Dataset> Size: 42B Dimensions: (x: 3, y: 5) Coordinates:

Datasets support most of the same methods found on data arrays:

In [138]: ds.mean(dim="x") Out[138]: <xarray.Dataset> Size: 48B Dimensions: (y: 5) Dimensions without coordinates: y Data variables: x_and_y (y) float64 40B -0.1779 0.449 -0.6525 0.2515 0.09179 x_only float64 8B -0.371

In [139]: abs(ds) Out[139]: <xarray.Dataset> Size: 168B Dimensions: (x: 3, y: 5) Coordinates:

Datasets also support NumPy ufuncs (requires NumPy v1.13 or newer), or alternatively you can use map() to map a function to each variable in a dataset:

In [140]: np.sin(ds) Out[140]: <xarray.Dataset> Size: 168B Dimensions: (x: 3, y: 5) Coordinates:

In [141]: ds.map(np.sin) Out[141]: <xarray.Dataset> Size: 168B Dimensions: (x: 3, y: 5) Coordinates:

Datasets also use looping over variables for broadcasting in binary arithmetic. You can do arithmetic between any DataArray and a dataset:

In [142]: ds + arr Out[142]: <xarray.Dataset> Size: 168B Dimensions: (x: 3, y: 5) Coordinates:

Arithmetic between two datasets matches data variables of the same name:

In [143]: ds2 = xr.Dataset({"x_and_y": 0, "x_only": 100})

In [144]: ds - ds2 Out[144]: <xarray.Dataset> Size: 168B Dimensions: (x: 3, y: 5) Coordinates:

Similarly to index based alignment, the result has the intersection of all matching data variables.

Wrapping custom computation#

It doesn’t always make sense to do computation directly with xarray objects:

For these reasons, it is often well-advised to write low-level routines that work with NumPy arrays, and to wrap these routines to work with xarray objects. However, adding support for labels on both Dataset andDataArray can be a bit of a chore.

To make this easier, xarray supplies the apply_ufunc() helper function, designed for wrapping functions that support broadcasting and vectorization on unlabeled arrays in the style of a NumPyuniversal function (“ufunc” for short).apply_ufunc takes care of everything needed for an idiomatic xarray wrapper, including alignment, broadcasting, looping over Dataset variables (if needed), and merging of coordinates. In fact, many internal xarray functions/methods are written using apply_ufunc.

Simple functions that act independently on each value should work without any additional arguments:

In [145]: squared_error = lambda x, y: (x - y) ** 2

In [146]: arr1 = xr.DataArray([0, 1, 2, 3], dims="x")

In [147]: xr.apply_ufunc(squared_error, arr1, 1) Out[147]: <xarray.DataArray (x: 4)> Size: 32B array([1, 0, 1, 4]) Dimensions without coordinates: x

For using more complex operations that consider some array values collectively, it’s important to understand the idea of “core dimensions” from NumPy’sgeneralized ufuncs. Core dimensions are defined as dimensions that should not be broadcast over. Usually, they correspond to the fundamental dimensions over which an operation is defined, e.g., the summed axis innp.sum. A good clue that core dimensions are needed is the presence of anaxis argument on the corresponding NumPy function.

With apply_ufunc, core dimensions are recognized by name, and then moved to the last dimension of any input arguments before applying the given function. This means that for functions that accept an axis argument, you usually need to set axis=-1. As an example, here is how we would wrapnumpy.linalg.norm() to calculate the vector norm:

def vector_norm(x, dim, ord=None): return xr.apply_ufunc( np.linalg.norm, x, input_core_dims=[[dim]], kwargs={"ord": ord, "axis": -1} )

In [148]: vector_norm(arr1, dim="x") Out[148]: <xarray.DataArray ()> Size: 8B array(3.74165739)

Because apply_ufunc follows a standard convention for ufuncs, it plays nicely with tools for building vectorized functions, likenumpy.broadcast_arrays() and numpy.vectorize. For high performance needs, consider using Numba’s vectorize and guvectorize.

In addition to wrapping functions, apply_ufunc can automatically parallelize many functions when using dask by setting dask='parallelized'. SeeParallelize custom functions with apply_ufunc and map_blocks for details.

apply_ufunc() also supports some advanced options for controlling alignment of variables and the form of the result. See the docstring for full details and more examples.