The labels associated with DataArray andDataset objects enables some powerful shortcuts forcomputation, notably including aggregation and broadcasting by dimensionnames.
Basic array math#
Arithmetic operations with a single DataArray automatically vectorize (likenumpy) over all array values:
In [1]: arr = xr.DataArray( ...: np.random.RandomState(0).randn(2, 3), [("x", ["a", "b"]), ("y", [10, 20, 30])] ...: ) ...: In [2]: arr - 3Out[2]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[-1.23594765, -2.59984279, -2.02126202], [-0.7591068 , -1.13244201, -3.97727788]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30In [3]: abs(arr)Out[3]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[1.76405235, 0.40015721, 0.97873798], [2.2408932 , 1.86755799, 0.97727788]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30
You can also use any of numpy’s or scipy’s many ufunc functions directly ona DataArray:
In [4]: np.sin(arr)Out[4]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[ 0.9813841 , 0.38956314, 0.82979374], [ 0.78376151, 0.95628847, -0.82897801]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30
Use where() to conditionally switch between values:
In [5]: xr.where(arr > 0, "positive", "negative")Out[5]: <xarray.DataArray (x: 2, y: 3)> Size: 192Barray([['positive', 'positive', 'positive'], ['positive', 'positive', 'negative']], dtype='<U8')Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30
Use @ to compute the dot() product:
In [6]: arr @ arrOut[6]: <xarray.DataArray ()> Size: 8Barray(13.69438174)
Data arrays also implement many numpy.ndarray methods:
In [7]: arr.round(2)Out[7]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[ 1.76, 0.4 , 0.98], [ 2.24, 1.87, -0.98]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30In [8]: arr.TOut[8]: <xarray.DataArray (y: 3, x: 2)> Size: 48Barray([[ 1.76405235, 2.2408932 ], [ 0.40015721, 1.86755799], [ 0.97873798, -0.97727788]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30In [9]: intarr = xr.DataArray([0, 1, 2, 3, 4, 5])In [10]: intarr << 2 # only supported for int typesOut[10]: <xarray.DataArray (dim_0: 6)> Size: 48Barray([ 0, 4, 8, 12, 16, 20])Dimensions without coordinates: dim_0In [11]: intarr >> 1Out[11]: <xarray.DataArray (dim_0: 6)> Size: 48Barray([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 aspecial 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 valuesindicating 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: 5Barray([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 xarrayobject. It returns a new xarray object with the same dimensions as the original object, but with booleanvalues 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: 5Barray([ 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 thethird and fourth positions where NaN is located.
count() is a method in xarray that can be used to count the number ofnon-missing values along one or more dimensions of an xarray object. It returns a new xarray object withthe same dimensions as the original object, but with each element replaced by the count of non-missingvalues 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: 8Barray(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 representsthe 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 valuesremoved.
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: 24Barray([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 theoriginal order.
fillna() is a method in xarray that can be used to fill missing or null values in an xarray object with aspecified 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: 40Barray([ 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 andreturns 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 anxarray object along one or more dimensions. It returns a new xarray object with the same dimensions as theoriginal 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: 40Barray([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 lastnon-null value in the same dimension, which are 0 and 1, respectively. The resulting DataArray object hasfive 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 anxarray object along one or more dimensions. It returns a new xarray object with the same dimensions as the original object, butwith 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: 40Barray([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 nextnon-null value in the same dimension, which are 2 and 2, respectively. The resulting DataArray object hasfive 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 representmissing values.
Xarray objects also have an interpolate_na() methodfor filling missing values via 1D interpolation. It returns a new xarray object with the same dimensionsas 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: 40Barray([0. , 1. , 1.05, 1.45, 2. ])Coordinates: xx (x) float64 40B 0.0 1.0 1.1 1.9 3.0Dimensions 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 linearinterpolation 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 valuesare calculated based on the values of the ‘xx’ coordinate, which has non-integer values, resulting innon-integer interpolated values.
Note that xarray slightly diverges from the pandas interpolate syntax byproviding the use_coordinate keyword which facilitates a clear specificationof which values to use as the index in the interpolation.Xarray also provides the max_gap keyword argument to limit the interpolation todata 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 areapplied along particular dimension(s):
In [28]: arr.sum(dim="x")Out[28]: <xarray.DataArray (y: 3)> Size: 24Barray([4.00494555e+00, 2.26771520e+00, 1.46010423e-03])Coordinates: * y (y) int64 24B 10 20 30In [29]: arr.std(["x", "y"])Out[29]: <xarray.DataArray ()> Size: 8Barray(1.09038344)In [30]: arr.min()Out[30]: <xarray.DataArray ()> Size: 8Barray(-0.97727788)
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: 8Barray(2.)
If desired, you can disable this behavior by invoking the aggregation methodwith skipna=False.
Rolling window operations#
DataArray objects include a rolling() method. Thismethod supports rolling window aggregation:
In [33]: arr = xr.DataArray(np.arange(0, 7.5, 0.5).reshape(3, 5), dims=("x", "y"))In [34]: arrOut[34]: <xarray.DataArray (x: 3, y: 5)> Size: 120Barray([[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 thename 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: 120Barray([[ 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, yIn [38]: r.mean()Out[38]: <xarray.DataArray (x: 3, y: 5)> Size: 120Barray([[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 bydefault, 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: 120Barray([[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 thearray produce nans. Setting min_periods in the call to rollingchanges the minimum number of observations within the window required to havea 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: 120Barray([[ 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, yIn [43]: r = arr.rolling(y=3, center=True, min_periods=2)In [44]: r.mean()Out[44]: <xarray.DataArray (x: 3, y: 5)> Size: 120Barray([[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: 120Barray([[ 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 supportsan 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 aview of the original DataArray with the windowed dimension inthe 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 strideIn [47]: rolling_da = r.construct(x="x_win", y="y_win", stride=2)In [48]: rolling_daOut[48]: <xarray.DataArray (x: 2, y: 3, x_win: 2, y_win: 3)> Size: 288Barray([[[[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. ]]]])Dimensions without coordinates: x, y, x_win, y_winIn [49]: rolling_da.mean(["x_win", "y_win"], skipna=False)Out[49]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[ nan, nan, nan], [ nan, 4.25, 5.25]])Dimensions without coordinates: x, y
Because the DataArray given by r.construct('window_dim') is a viewof 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: 120Barray([[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 currentlysupport 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_precOut[56]: DataArrayWeighted with weights along dimensions: month
Calculate the weighted sum:
In [57]: weighted_prec.sum()Out[57]: <xarray.DataArray ()> Size: 8Barray(90.)
Calculate the weighted mean:
In [58]: weighted_prec.mean(dim="month")Out[58]: <xarray.DataArray ()> Size: 8Barray(1.)
Calculate the weighted quantile:
In [59]: weighted_prec.quantile(q=0.5, dim="month")Out[59]: <xarray.DataArray ()> Size: 8Barray(1.)Coordinates: quantile float64 8B 0.5
The weighted sum corresponds to:
In [60]: weighted_sum = (prec * weights).sum()In [61]: weighted_sumOut[61]: <xarray.DataArray ()> Size: 8Barray(90.)
the weighted mean to:
In [62]: weighted_mean = weighted_sum / weights.sum()In [63]: weighted_meanOut[63]: <xarray.DataArray ()> Size: 8Barray(1.)
the weighted variance to:
In [64]: weighted_var = weighted_prec.sum_of_squares() / weights.sum()In [65]: weighted_varOut[65]: <xarray.DataArray ()> Size: 8Barray(0.00688889)
and the weighted standard deviation to:
In [66]: weighted_std = np.sqrt(weighted_var)In [67]: weighted_stdOut[67]: <xarray.DataArray ()> Size: 8Barray(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: 8Barray(3.)
Using (data * weights).sum() / weights.sum() would (incorrectly) resultin 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: 8Barray(0.)
and mean, std and var return NaN:
In [74]: data.weighted(weights).mean()Out[74]: <xarray.DataArray ()> Size: 8Barray(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]: daOut[78]: <xarray.DataArray (time: 364, x: 300)> Size: 874kBarray([[ 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]])Coordinates: * time (time) datetime64[ns] 3kB 1999-12-15 1999-12-16 ... 2000-12-12 * x (x) float64 2kB 0.0 0.03344 0.06689 0.1003 ... 9.9 9.933 9.967 10.0
In order to take a block mean for every 7 days along time dimension andevery 2 points along x dimension,
In [79]: da.coarsen(time=7, x=2).mean()Out[79]: <xarray.DataArray (time: 52, x: 150)> Size: 62kBarray([[ 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 ]])Coordinates: * time (time) datetime64[ns] 416B 1999-12-18 1999-12-25 ... 2000-12-09 * x (x) float64 1kB 0.01672 0.08361 0.1505 0.2174 ... 9.849 9.916 9.983
coarsen() raises an ValueError if the datalength is not a multiple of the corresponding window size.You can choose boundary='trim' or boundary='pad' options for trimmingthe 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: 14kBarray([[ 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]])Coordinates: * time (time) datetime64[ns] 96B 1999-12-29T12:00:00 ... 2000-11-23T12:... * x (x) float64 1kB 0.01672 0.08361 0.1505 0.2174 ... 9.849 9.916 9.983
If you want to apply a specific function to coordinate, you can pass thefunction 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: 62kBarray([[ 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 ]])Coordinates: * time (time) datetime64[ns] 416B 1999-12-15 1999-12-22 ... 2000-12-06 * x (x) float64 1kB 0.01672 0.08361 0.1505 0.2174 ... 9.849 9.916 9.983
You can also use coarsen to reshape without applying a computation.
Computation using Coordinates#
Xarray objects have some handy methods for the computation with theircoordinates. differentiate() computes derivatives bycentral 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]: aOut[83]: <xarray.DataArray (x: 4)> Size: 32Barray([0, 1, 2, 3])Coordinates: * x (x) float64 32B 0.1 0.11 0.2 0.3In [84]: a.differentiate("x")Out[84]: <xarray.DataArray (x: 4)> Size: 32Barray([100. , 91.11111111, 10.58479532, 10. ])Coordinates: * x (x) float64 32B 0.1 0.11 0.2 0.3
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: 64Barray([[200. , 200. ], [182.22222222, 182.22222222], [ 21.16959064, 21.16959064], [ 20. , 20. ]])Coordinates: * x (x) float64 32B 0.1 0.11 0.2 0.3Dimensions without coordinates: y
integrate() computes integration based ontrapezoidal rule using their coordinates,
In [87]: a.integrate("x")Out[87]: <xarray.DataArray (y: 2)> Size: 16Barray([0.78, 0.98])Dimensions without coordinates: y
Note
These methods are limited to simple cartesian geometry. Differentiationand integration along multidimensional coordinate are not supported.
Fitting polynomials#
Xarray objects provide an interface for performing linear or polynomial regressionsusing the least-squares method. polyfit() computes thebest 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]: outOut[91]: <xarray.Dataset> Size: 64BDimensions: (degree: 2)Coordinates: * degree (degree) int64 16B 1 0Data variables: x_matrix_rank int64 8B 2 x_singular_values (degree) float64 16B 1.358 0.3963 polyfit_coefficients (degree) float64 16B 4.0 3.0 polyfit_residuals float64 8B 4.522e-28
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: 80Barray([ 3., 7., 11., 15., 19., 23., 27., 31., 35., 39.])Dimensions without coordinates: x
Note
These methods replicate the behaviour of numpy.polyfit() and numpy.polyval().
Fitting arbitrary functions#
Xarray objects also provide an interface for fitting more complex functions usingscipy.optimize.curve_fit(). curvefit() acceptsuser-defined functions and can fit along multiple coordinates.
For example, we can fit a relationship between two DataArray objects, maintaininga 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: 10kBDimensions: (x: 100, param: 2, cov_i: 2, cov_j: 2)Coordinates: * x (x) float64 800B -5.0 -4.9 -4.8 ... 4.7 4.8 4.9 * param (param) <U2 16B 'a' 'xc' * cov_i (cov_i) <U2 16B 'a' 'xc' * cov_j (cov_j) <U2 16B 'a' 'xc'Data variables: var2_curvefit_coefficients (x, param) float64 2kB 3.0 -5.0 3.0 ... 3.0 4.9 var2_curvefit_covariance (x, cov_i, cov_j) float64 3kB 9.286e-14 ... 1... var3_curvefit_coefficients (x, param) float64 2kB 0.9999 5.0 ... 1.0 -4.9 var3_curvefit_covariance (x, cov_i, cov_j) float64 3kB 5.825e-11 ... 1...
We can also fit multi-dimensional functions, and even use a wrapper function tosimultaneously fit a summation of several functions, such as this field containingtwo 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 = 2In [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: 2kBDimensions: (param: 10, cov_i: 10, cov_j: 10)Coordinates: * param (param) <U7 280B 'a0' 'xc0' ... 'xalpha1' 'yalpha1' * cov_i (cov_i) <U7 280B 'a0' 'xc0' ... 'xalpha1' 'yalpha1' * cov_j (cov_j) <U7 280B 'a0' 'xc0' ... 'xalpha1' 'yalpha1'Data variables: curvefit_coefficients (param) float64 80B 3.0 1.004 1.003 ... 1.007 1.008 curvefit_covariance (cov_i, cov_j) float64 800B 3.362e-05 ... 2.125e-05
Note
This method replicates the behavior of scipy.optimize.curve_fit().
Broadcasting by dimension name#
DataArray objects automatically align themselves (“broadcasting” inthe numpy parlance) by dimension name instead of axis order. With xarray, youdo not need to transpose arrays or insert dimensions of length 1 to get arrayoperations to work, as commonly done in numpy with numpy.reshape() ornumpy.newaxis.
This is best illustrated by a few examples. Consider two one-dimensionalarrays with different sizes aligned along different dimensions:
In [114]: a = xr.DataArray([1, 2], [("x", ["a", "b"])])In [115]: aOut[115]: <xarray.DataArray (x: 2)> Size: 16Barray([1, 2])Coordinates: * x (x) <U1 8B 'a' 'b'In [116]: b = xr.DataArray([-1, -2, -3], [("y", [10, 20, 30])])In [117]: bOut[117]: <xarray.DataArray (y: 3)> Size: 24Barray([-1, -2, -3])Coordinates: * y (y) int64 24B 10 20 30
With xarray, we can apply binary mathematical operations to these arrays, andtheir dimensions are expanded automatically:
In [118]: a * bOut[118]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[-1, -2, -3], [-2, -4, -6]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30
Moreover, dimensions are always reordered to the order in which they firstappeared:
In [119]: c = xr.DataArray(np.arange(6).reshape(3, 2), [b["y"], a["x"]])In [120]: cOut[120]: <xarray.DataArray (y: 3, x: 2)> Size: 48Barray([[0, 1], [2, 3], [4, 5]])Coordinates: * y (y) int64 24B 10 20 30 * x (x) <U1 8B 'a' 'b'In [121]: a + cOut[121]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[1, 3, 5], [3, 5, 7]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30
This means, for example, that you always subtract an array from its transpose:
In [122]: c - c.TOut[122]: <xarray.DataArray (y: 3, x: 2)> Size: 48Barray([[0, 0], [0, 0], [0, 0]])Coordinates: * y (y) int64 24B 10 20 30 * x (x) <U1 8B 'a' 'b'
You can explicitly broadcast xarray data structures by using thebroadcast() function:
In [123]: a2, b2 = xr.broadcast(a, b)In [124]: a2Out[124]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[1, 1, 1], [2, 2, 2]])Coordinates: * x (x) <U1 8B 'a' 'b' * y (y) int64 24B 10 20 30In [125]: b2Out[125]: <xarray.DataArray (x: 2, y: 3)> Size: 48Barray([[-1, -2, -3], [-1, -2, -3]])Coordinates: * y (y) int64 24B 10 20 30 * x (x) <U1 8B 'a' 'b'
Automatic alignment#
Xarray enforces alignment between index Coordinates (that is,coordinates with the same name as a dimension, marked by *) on objects usedin binary operations.
Similarly to pandas, this alignment is automatic for arithmetic on binaryoperations. 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: 16Barray([0, 2])Coordinates: * x (x) int64 16B 0 1
If coordinate values for a dimension are missing on either argument, allmatching 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: 8Barray([0])Coordinates: * x (x) int64 8B 0
Before loops or performance critical code, it’s a good idea to align arraysexplicitly (e.g., by putting them in the same Dataset or usingalign()) to avoid the overhead of repeated alignment with eachoperation. See Align and reindex for more details.
Note
There is no automatic alignment between arguments when performing in-placearithmetic operations such as +=. You will need to usemanual alignment. This ensures in-placearithmetic never needs to modify data types.
Coordinates#
Although index coordinates are aligned, other coordinates are not, and if theirvalues conflict, they will be dropped. This is necessary, for example, becauseindexing turns 1D coordinates into scalar coordinates:
In [131]: arr[0]Out[131]: <xarray.DataArray ()> Size: 8Barray(0)Coordinates: x int64 8B 0In [132]: arr[1]Out[132]: <xarray.DataArray ()> Size: 8Barray(1)Coordinates: x int64 8B 1# notice that the scalar coordinate 'x' is silently droppedIn [133]: arr[1] - arr[0]Out[133]: <xarray.DataArray ()> Size: 8Barray(1)
Still, xarray will persist other coordinates in arithmetic, as long as thereare no conflicting values:
# only one argument has the 'x' coordinateIn [134]: arr[0] + 1Out[134]: <xarray.DataArray ()> Size: 8Barray(1)Coordinates: x int64 8B 0# both arguments have the same 'x' coordinateIn [135]: arr[0] - arr[0]Out[135]: <xarray.DataArray ()> Size: 8Barray(0)Coordinates: x int64 8B 0
Math with datasets#
Datasets support arithmetic operations by automatically looping over all datavariables:
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 > 0Out[137]: <xarray.Dataset> Size: 42BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) bool 15B True True False True ... True False False False x_only (x) bool 3B False True False
Datasets support most of the same methods found on data arrays:
In [138]: ds.mean(dim="x")Out[138]: <xarray.Dataset> Size: 48BDimensions: (y: 5)Dimensions without coordinates: yData variables: x_and_y (y) float64 40B -0.1779 0.449 -0.6525 0.2515 0.09179 x_only float64 8B -0.371In [139]: abs(ds)Out[139]: <xarray.Dataset> Size: 168BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) float64 120B 0.4281 0.9399 0.884 ... 0.7523 0.1212 0.3989 x_only (x) float64 24B 0.5093 0.2509 0.8548
Datasets also support NumPy ufuncs (requires NumPy v1.13 or newer), oralternatively you can use map() to map a functionto each variable in a dataset:
In [140]: np.sin(ds)Out[140]: <xarray.Dataset> Size: 168BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) float64 120B 0.4152 0.8075 -0.7733 ... -0.1209 -0.3884 x_only (x) float64 24B -0.4875 0.2483 -0.7544In [141]: ds.map(np.sin)Out[141]: <xarray.Dataset> Size: 168BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) float64 120B 0.4152 0.8075 -0.7733 ... -0.1209 -0.3884 x_only (x) float64 24B -0.4875 0.2483 -0.7544
Datasets also use looping over variables for broadcasting in binaryarithmetic. You can do arithmetic between any DataArray and a dataset:
In [142]: ds + arrOut[142]: <xarray.Dataset> Size: 168BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) float64 120B 0.4281 0.9399 -0.884 ... 1.248 1.879 1.601 x_only (x) float64 24B -0.5093 1.251 1.145
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 - ds2Out[144]: <xarray.Dataset> Size: 168BDimensions: (x: 3, y: 5)Coordinates: * x (x) int64 24B 0 1 2Dimensions without coordinates: yData variables: x_and_y (x, y) float64 120B 0.4281 0.9399 -0.884 ... -0.1212 -0.3989 x_only (x) float64 24B -100.5 -99.75 -100.9
Similarly to index based alignment, the result has the intersection of allmatching data variables.
Wrapping custom computation#
It doesn’t always make sense to do computation directly with xarray objects:
In the inner loop of performance limited code, using xarray can addconsiderable overhead compared to using NumPy or native Python types.This is particularly true when working with scalars or small arrays (lessthan ~1e6 elements). Keeping track of labels and ensuring their consistencyadds overhead, and xarray’s core itself is not especially fast, because it’swritten in Python rather than a compiled language like C. Also, xarray’shigh level label-based APIs removes low-level control over how operationsare implemented.
Even if speed doesn’t matter, it can be important to wrap existing code, orto support alternative interfaces that don’t use xarray objects.
For these reasons, it is often well-advised to write low-level routines thatwork 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() helperfunction, designed for wrapping functions that support broadcasting andvectorization 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 (ifneeded), and merging of coordinates. In fact, many internal xarrayfunctions/methods are written using apply_ufunc.
Simple functions that act independently on each value should work withoutany additional arguments:
In [145]: squared_error = lambda x, y: (x - y) ** 2In [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: 32Barray([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 dimensionsthat should not be broadcast over. Usually, they correspond to the fundamentaldimensions 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 tothe last dimension of any input arguments before applying the given function.This means that for functions that accept an axis argument, you usually needto 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: 8Barray(3.74165739)
Because apply_ufunc follows a standard convention for ufuncs, it playsnicely with tools for building vectorized functions, likenumpy.broadcast_arrays() and numpy.vectorize. For high performanceneeds, consider using Numba’s vectorize and guvectorize.
In addition to wrapping functions, apply_ufunc can automatically parallelizemany functions when using dask by setting dask='parallelized'. SeeAutomatic parallelization with apply_ufunc and map_blocks for details.
apply_ufunc() also supports some advanced options forcontrolling alignment of variables and the form of the result. See thedocstring for full details and more examples.
