"""EOF analysis for data in `xarray.DataArray` arrays."""
# (c) Copyright 2016 Andrew Dawson. All Rights Reserved.
#
# This file is part of eofs.
#
# eofs is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# eofs is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
# FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License
# along with eofs. If not, see <http://www.gnu.org/licenses/>.
from __future__ import (absolute_import, division, print_function) # noqa
import collections
try:
import xarray as xr
except ImportError:
import xray as xr
from . import standard
from .tools.xarray import (find_time_coordinates, categorise_ndcoords,
weights_array)
[docs]class Eof(object):
"""EOF analysis (meta-data enabled `xarray` interface)"""
def __init__(self, array, weights=None, center=True, ddof=1):
"""Create an Eof object.
The EOF solution is computed at initialization time. Method
calls are used to retrieve computed quantities.
**Arguments:**
*dataset*
An `xarray.DataArray` with two or more dimensions containing
the data to be analysed. The first dimension is assumed to
represent time. Missing values are allowed provided that
they are constant with time (e.g., values of an
oceanographic field over land).
**Optional arguments:**
*weights*
An array of weights whose shape is compatible with those of
the input array *dataset*. The weights can have the same
shape as *dataset* or a shape compatible with an array
broadcast (i.e., the shape of the weights can can match the
rightmost parts of the shape of the input array *dataset*).
If the input array *dataset* does not require weighting then
the value *None* may be used. Defaults to *None* (no
weighting).
*center*
If *True*, the mean along the first axis of *dataset* (the
time-mean) will be removed prior to analysis. If *False*,
the mean along the first axis will not be removed. Defaults
to *True* (mean is removed).
The covariance interpretation relies on the input data being
anomaly data with a time-mean of 0. Therefore this option
should usually be set to *True*. Setting this option to
*True* has the useful side effect of propagating missing
values along the time dimension, ensuring that a solution
can be found even if missing values occur in different
locations at different times.
*ddof*
'Delta degrees of freedom'. The divisor used to normalize
the covariance matrix is *N - ddof* where *N* is the
number of samples. Defaults to *1*.
**Returns:**
*solver*
An `Eof` instance.
**Examples:**
EOF analysis with no weighting::
from eofs.xarray import Eof
solver = Eof(data_array)
"""
if not isinstance(array, xr.DataArray):
raise TypeError('the input must be an xarray DataArray')
# Find a time-like dimension, and check if it is the first.
time_coords = find_time_coordinates(array)
if not time_coords:
raise ValueError('cannot find a time coordinate (must be called '
'"time", have a numpy.datetime64 dtype, or have '
'an attribute named "axis" with value "T")')
if len(time_coords) > 1:
raise ValueError('multiple time dimensions are not allowed')
if array.dims[0] != time_coords[0].name:
raise ValueError('time must be the first dimension, '
'consider using the transpose() method')
self._time = time_coords[0]
# Collect the other dimension coordinates.
self._coords = [array.coords[dim] for dim in array.dims[1:]]
# Collect other non-dimension coordinates and store them categorised
# them according to the dimensions they span.
(self._time_ndcoords,
self._space_ndcoords,
self._time_space_ndcoords) = categorise_ndcoords(array,
self._time.name)
# Determine the required weights.
if weights is None:
wtarray = None
else:
try:
wtarray = weights_array(array, scheme=weights.lower())
except AttributeError:
# Catches exception from applying .lower() to a non-string.
wtarray = weights
try:
wtarray = wtarray.astype(array.dtype)
except AttributeError:
pass
# Construct the EOF solver.
self._solver = standard.Eof(array.data,
weights=wtarray,
center=center,
ddof=ddof)
# Name of the input DataArray.
self._name = array.name
#: The number of EOFs in the solution.
self.neofs = self._solver.neofs
[docs] def pcs(self, pcscaling=0, npcs=None):
"""Principal component time series (PCs).
**Optional arguments:**
*pcscaling*
Set the scaling of the retrieved PCs. The following
values are accepted:
* *0* : Un-scaled principal components (default).
* *1* : Principal components are scaled to unit variance
(divided by the square-root of their eigenvalue).
* *2* : Principal components are multiplied by the
square-root of their eigenvalue.
*npcs*
Number of PCs to retrieve. Defaults to all the PCs. If the
number of requested PCs is more than the number that are
available, then all available PCs will be returned.
**Returns:**
*pcs*
A `~xarray.DataArray` containing the ordered PCs. The PCs
are numbered from 0 to *npcs* - 1.
**Examples:**
All un-scaled PCs::
pcs = solver.pcs()
First 3 PCs scaled to unit variance::
pcs = solver.pcs(npcs=3, pcscaling=1)
"""
pcs = self._solver.pcs(pcscaling, npcs)
pcdim = xr.IndexVariable('mode', range(pcs.shape[1]),
attrs={'long_name': 'eof_mode_number'})
coords = [self._time, pcdim]
pcs = xr.DataArray(pcs, coords=coords, name='pcs')
pcs.coords.update({coord.name: ('time', coord)
for coord in self._time_ndcoords})
return pcs
[docs] def eofs(self, eofscaling=0, neofs=None):
"""Emipirical orthogonal functions (EOFs).
**Optional arguments:**
*eofscaling*
Sets the scaling of the EOFs. The following values are
accepted:
* *0* : Un-scaled EOFs (default).
* *1* : EOFs are divided by the square-root of their
eigenvalues.
* *2* : EOFs are multiplied by the square-root of their
eigenvalues.
*neofs*
Number of EOFs to return. Defaults to all EOFs. If the
number of EOFs requested is more than the number that are
available, then all available EOFs will be returned.
**Returns:**
*eofs*
A `~xarray.DataArray` containing the ordered EOFs. The EOFs
are numbered from 0 to *neofs* - 1.
**Examples:**
All EOFs with no scaling::
eofs = solver.eofs()
First 3 EOFs with scaling applied::
eofs = solver.eofs(neofs=3, eofscaling=1)
"""
eofs = self._solver.eofs(eofscaling, neofs)
eofdim = xr.IndexVariable('mode', range(eofs.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim] + self._coords
long_name = 'empirical_orthogonal_functions'
eofs = xr.DataArray(eofs, coords=coords, name='eofs',
attrs={'long_name': long_name})
eofs.coords.update({coord.name: (coord.dims, coord)
for coord in self._space_ndcoords})
return eofs
[docs] def eofsAsCorrelation(self, neofs=None):
"""
Empirical orthogonal functions (EOFs) expressed as the
correlation between the principal component time series (PCs)
and the time series of the `Eof` input *dataset* at each grid
point.
.. note::
These are not related to the EOFs computed from the
correlation matrix.
**Optional argument:**
*neofs*
Number of EOFs to return. Defaults to all EOFs. If the
number of EOFs requested is more than the number that are
available, then all available EOFs will be returned.
**Returns:**
*eofs*
A `~xarray.DataArray` containing the ordered EOFs. The EOFs
are numbered from 0 to *neofs* - 1.
**Examples:**
All EOFs::
eofs = solver.eofsAsCorrelation()
The leading EOF::
eof1 = solver.eofsAsCorrelation(neofs=1)
"""
eofs = self._solver.eofsAsCorrelation(neofs)
eofdim = xr.IndexVariable('mode', range(eofs.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim] + self._coords
long_name = 'correlation_between_pcs_and_{!s}'.format(self._name)
eofs = xr.DataArray(eofs, coords=coords, name='eofs',
attrs={'long_name': long_name})
eofs.coords.update({coord.name: (coord.dims, coord)
for coord in self._space_ndcoords})
return eofs
[docs] def eofsAsCovariance(self, neofs=None, pcscaling=1):
"""
Empirical orthogonal functions (EOFs) expressed as the
covariance between the principal component time series (PCs)
and the time series of the `Eof` input *dataset* at each grid
point.
**Optional arguments:**
*neofs*
Number of EOFs to return. Defaults to all EOFs. If the
number of EOFs requested is more than the number that are
available, then all available EOFs will be returned.
*pcscaling*
Set the scaling of the PCs used to compute covariance. The
following values are accepted:
* *0* : Un-scaled PCs.
* *1* : PCs are scaled to unit variance (divided by the
square-root of their eigenvalue) (default).
* *2* : PCs are multiplied by the square-root of their
eigenvalue.
The default is to divide PCs by the square-root of their
eigenvalue so that the PCs are scaled to unit variance
(option 1).
**Returns:**
*eofs*
A `~xarray.DataArray` containing the ordered EOFs. The EOFs
are numbered from 0 to *neofs* - 1.
**Examples:**
All EOFs::
eofs = solver.eofsAsCovariance()
The leading EOF::
eof1 = solver.eofsAsCovariance(neofs=1)
The leading EOF using un-scaled PCs::
eof1 = solver.eofsAsCovariance(neofs=1, pcscaling=0)
"""
eofs = self._solver.eofsAsCovariance(neofs, pcscaling)
eofdim = xr.IndexVariable('mode', range(eofs.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim] + self._coords
long_name = 'covariance_between_pcs_and_{!s}'.format(self._name)
eofs = xr.DataArray(eofs, coords=coords, name='eofs',
attrs={'long_name': long_name})
eofs.coords.update({coord.name: (coord.dims, coord)
for coord in self._space_ndcoords})
return eofs
[docs] def eigenvalues(self, neigs=None):
"""Eigenvalues (decreasing variances) associated with each EOF.
**Optional argument:**
*neigs*
Number of eigenvalues to return. Defaults to all
eigenvalues.If the number of eigenvalues requested is more
than the number that are available, then all available
eigenvalues will be returned.
**Returns:**
*eigenvalues*
A `~xarray.DataArray` containing the eigenvalues arranged
largest to smallest. The eigenvalues are numbered from 0 to
*neigs* - 1.
**Examples:**
All eigenvalues::
eigenvalues = solver.eigenvalues()
The first eigenvalue::
eigenvalue1 = solver.eigenvalues(neigs=1)
"""
lambdas = self._solver.eigenvalues(neigs=neigs)
eofdim = xr.IndexVariable('mode', range(lambdas.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim]
long_name = 'eigenvalues'
lambdas = xr.DataArray(lambdas, coords=coords, name='eigenvalues',
attrs={'long_name': long_name})
return lambdas
[docs] def varianceFraction(self, neigs=None):
"""Fractional EOF mode variances.
The fraction of the total variance explained by each EOF mode,
values between 0 and 1 inclusive.
**Optional argument:**
*neigs*
Number of eigenvalues to return the fractional variance for.
Defaults to all eigenvalues. If the number of eigenvalues
requested is more than the number that are available, then
fractional variances for all available eigenvalues will be
returned.
**Returns:**
*variance_fractions*
A `~xarray.DataArray` containing the fractional variances
for each eigenvalue. The eigenvalues are numbered from 0 to
*neigs* - 1.
**Examples:**
The fractional variance represented by each eigenvalue::
variance_fractions = solver.varianceFraction()
The fractional variance represented by the first 3 eigenvalues::
variance_fractions = solver.VarianceFraction(neigs=3)
"""
vf = self._solver.varianceFraction(neigs=neigs)
eofdim = xr.IndexVariable('mode', range(vf.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim]
long_name = 'variance_fractions'
vf = xr.DataArray(vf, coords=coords, name='variance_fractions',
attrs={'long_name': long_name})
return vf
[docs] def totalAnomalyVariance(self):
"""
Total variance associated with the field of anomalies (the sum
of the eigenvalues).
**Returns:**
*total_variance*
A scalar value (not a `~xarray.DataArray`).
**Example:**
Get the total variance::
total_variance = solver.totalAnomalyVariance()
"""
return self._solver.totalAnomalyVariance()
[docs] def northTest(self, neigs=None, vfscaled=False):
"""Typical errors for eigenvalues.
The method of North et al. (1982) is used to compute the typical
error for each eigenvalue. It is assumed that the number of
times in the input data set is the same as the number of
independent realizations. If this assumption is not valid then
the result may be inappropriate.
**Optional arguments:**
*neigs*
The number of eigenvalues to return typical errors for.
Defaults to typical errors for all eigenvalues.
*vfscaled*
If *True* scale the errors by the sum of the eigenvalues.
This yields typical errors with the same scale as the values
returned by `Eof.varianceFraction`. If *False* then no
scaling is done. Defaults to *False* (no scaling).
**Returns:**
*errors*
A `~xarray.DataArray` containing the typical errors for each
eigenvalue. The egienvalues are numbered from 0 to
*neigs* - 1.
**References**
North G.R., T.L. Bell, R.F. Cahalan, and F.J. Moeng (1982)
Sampling errors in the estimation of empirical orthogonal
functions. *Mon. Weather. Rev.*, **110**, pp 669-706.
**Examples:**
Typical errors for all eigenvalues::
errors = solver.northTest()
Typical errors for the first 3 eigenvalues scaled by the sum of
the eigenvalues::
errors = solver.northTest(neigs=3, vfscaled=True)
"""
typerrs = self._solver.northTest(neigs=neigs, vfscaled=vfscaled)
eofdim = xr.IndexVariable('mode', range(typerrs.shape[0]),
attrs={'long_name': 'eof_mode_number'})
coords = [eofdim]
long_name = 'typical_errors'
typerrs = xr.DataArray(typerrs, coords=coords, name='typical_errors',
attrs={'long_name': long_name})
return typerrs
[docs] def reconstructedField(self, neofs):
"""Reconstructed data field based on a subset of EOFs.
If weights were passed to the `Eof` instance the returned
reconstructed field will automatically have this weighting
removed. Otherwise the returned field will have the same
weighting as the `Eof` input *dataset*.
Returns the reconstructed field in a `~xarray.DataArray`.
**Argument:**
*neofs*
Number of EOFs to use for the reconstruction.
Alternatively this argument can be an iterable of mode
numbers (where the first mode is 1) in order to facilitate
reconstruction with arbitrary modes.
**Returns:**
*reconstruction*
A `~xarray.DataArray` with the same dimensions `Eof` input
*dataset* containing the reconstruction using *neofs* EOFs.
**Example:**
Reconstruct the input field using 3 EOFs::
reconstruction = solver.reconstructedField(3)
Reconstruct the input field using EOFs 1, 2 and 5::
reconstruction = solver.reconstuctedField([1, 2, 5])
"""
rfield = self._solver.reconstructedField(neofs)
coords = [self._time] + self._coords
if isinstance(neofs, collections.Iterable):
name_part = 'EOFs_{}'.format('_'.join([str(e) for e in neofs]))
else:
name_part = '{}_EOFs'.format(neofs)
long_name = '{!s}_reconstructed_with_{!s}'.format(self._name,
name_part)
rfield = xr.DataArray(rfield, coords=coords, name=self._name,
attrs={'long_name': long_name})
ndcoords = (self._time_ndcoords + self._space_ndcoords +
self._time_space_ndcoords)
rfield.coords.update({coord.name: (coord.dims, coord)
for coord in ndcoords})
return rfield
[docs] def projectField(self, array, neofs=None, eofscaling=0, weighted=True):
"""Project a field onto the EOFs.
Given a data set, projects it onto the EOFs to generate a
corresponding set of pseudo-PCs.
**Argument:**
*field*
An `xarray.DataArray` containing the field to project onto
the EOFs. It must have the same corresponding spatial
dimensions (including missing values in the same places) as
the `Eof` input *dataset*. It may have a different length
time dimension to the `Eof` input *dataset* or no time
dimension at all. If a time dimension exists it must be the
first dimension.
**Optional arguments:**
*neofs*
Number of EOFs to project onto. Defaults to all EOFs. If the
number of EOFs requested is more than the number that are
available, then the field will be projected onto all
available EOFs.
*eofscaling*
Set the scaling of the EOFs that are projected
onto. The following values are accepted:
* *0* : Un-scaled EOFs (default).
* *1* : EOFs are divided by the square-root of their eigenvalue.
* *2* : EOFs are multiplied by the square-root of their
eigenvalue.
*weighted*
If *True* then the field is weighted using the same weights
used for the EOF analysis prior to projection. If *False*
then no weighting is applied. Defaults to *True* (weighting
is applied). Generally only the default setting should be
used.
**Returns:**
*pseudo_pcs*
A `~xarray.DataArray` containing the pseudo-PCs. The PCs are
numbered from 0 to *neofs* - 1.
**Examples:**
Project a field onto all EOFs::
pseudo_pcs = solver.projectField(field)
Project fields onto the three leading EOFs::
pseudo_pcs = solver.projectField(field, neofs=3)
"""
if not isinstance(array, xr.DataArray):
raise TypeError('the input must be an xarray DataArray')
array_name = array.name
time_coords = find_time_coordinates(array)
if len(time_coords) > 1:
raise ValueError('multiple time dimensions are not allowed')
if time_coords:
has_time = True
time_coord = time_coords[0]
if array.dims[0] != time_coord.name:
raise ValueError('time must be the first dimension, '
'consider using the transpose() method')
time_ndcoords, _, _ = categorise_ndcoords(array, time_coord.name)
else:
has_time = False
pcs = self._solver.projectField(array.values,
neofs=neofs,
eofscaling=eofscaling,
weighted=weighted)
# Create the PCs DataArray.
if pcs.ndim == 2:
pcdim = xr.IndexVariable('mode', range(pcs.shape[1]),
attrs={'long_name': 'eof_mode_number'})
pcs = xr.DataArray(
pcs,
coords=[time_coord, pcdim], name='pseudo_pcs',
attrs={'long_name': '{}_pseudo_pcs'.format(array_name)})
else:
pcdim = xr.IndexVariable('mode', range(pcs.shape[0]),
attrs={'long_name': 'eof_mode_number'})
pcs = xr.DataArray(
pcs,
coords=[pcdim], name='pseudo_pcs',
attrs={'long_name': '{}_pseudo_pcs'.format(array_name)})
if has_time:
# Add non-dimension coordinates.
pcs.coords.update({coord.name: (coord.dims, coord)
for coord in time_ndcoords})
return pcs
[docs] def getWeights(self):
"""Weights used for the analysis.
**Returns:**
*weights*
An array contaning the analysis weights (not a
`~xarray.DataArray`).
**Example:**
The weights used for the analysis::
weights = solver.getWeights()
"""
return self._solver.getWeights()
