"""Meta-data preserving EOF analysis for `iris`."""
# (c) Copyright 2013-2016 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
from copy import copy
from iris.cube import Cube
from iris.coords import DimCoord
from . import standard
from .tools.iris import get_time_coord, weights_array, classified_aux_coords
[docs]class Eof(object):
"""EOF analysis (meta-data enabled `iris` interface)"""
def __init__(self, cube, 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.
**Argument:**
*dataset*
A `~iris.cube.Cube` instance containing the data to be
analysed. Time must be the first dimension. Missing values
are allowed provided that they are constant with time (e.g.,
values of an oceanographic field over land).
**Optional arguments:**
*weights*
Sets the weighting method. The following pre-defined
weighting methods are available:
* *'area'* : Square-root of grid cell area normalized by
total grid area. Requires a latitude-longitude grid to be
present in the `~iris.cube.Cube` *dataset*. This is a
fairly standard weighting strategy. If you are unsure
which method to use and you have gridded data then this
should be your first choice.
* *'coslat'* : Square-root of cosine of latitude. Requires a
latitude dimension to be present in the `~iris.cube.Cube`
*dataset*.
* *None* : Equal weights for all grid points (*'none'* is
also accepted).
Alternatively an array of weights whose shape is compatible
with the `~iris.cube.Cube` *dataset* may be supplied instead
of specifying a weighting method.
*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
anomalies 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 grid-cell-area weighting for the input field::
from eofs.iris import Eof
solver = Eof(cube, weights='area')
"""
# Check that the input is an Iris cube.
if not isinstance(cube, Cube):
raise TypeError('the input must be an iris cube')
# Check for a time coordinate, raise an error if there isn't one.
# The get_time_coord function will raise a ValuerError with a
# useful message so no need to handle it explicitly here.
_time, self._time_dim = get_time_coord(cube)
self._time = copy(_time)
if self._time_dim != 0:
raise ValueError('time must be the first dimension, '
'consider using the transpose() method')
# Get the cube coordinates and remove time, leaving just the other
# dimensions.
self._coords = [copy(coord) for coord in cube.dim_coords]
self._coords.remove(self._time)
if not self._coords:
raise ValueError('one or more non-time dimensions are required')
# Store the auxiliary coordinates from the cube, categorising them into
# coordinates spanning time only, coordinates spanning space only, and
# coordinates spanning both time and space. This is helpful due to the
# natural separation of space and time in EOF analysis. The time and
# space spanning coordinates are only useful for reconstruction, as all
# other methods return either a temporal field or a spatial field.
(self._time_aux_coords,
self._space_aux_coords,
self._time_space_aux_coords) = classified_aux_coords(cube)
# Define the weights array for the cube.
if weights is None:
wtarray = None
else:
try:
scheme = weights.lower()
wtarray = weights_array(cube, scheme=scheme)
except AttributeError:
wtarray = weights
try:
# Ensure weights are the same type as the cube data.
wtarray = wtarray.astype(cube.data.dtype)
except AttributeError:
pass
# Initialize a solver.
self._solver = standard.Eof(cube.data,
weights=wtarray,
center=center,
ddof=ddof)
#: Number of EOFs in the solution.
self.neofs = self._solver.neofs
# Get the name of the cube to refer to later.
self._cube_name = cube.name(default='dataset').replace(' ', '_')
self._cube_var_name = cube.var_name
[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 `~iris.cube.Cube` 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 = DimCoord(list(range(pcs.shape[1])),
var_name='pc',
long_name='pc_number')
coords = [copy(self._time), pcdim]
pcs = Cube(
pcs,
dim_coords_and_dims=list(zip(coords, list(range(pcs.ndim)))),
var_name='pcs',
long_name='principal_components')
# Add any auxiliary coords spanning time back to the returned cube.
for coord, dims in self._time_aux_coords:
pcs.add_aux_coord(copy(coord), dims)
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 `~iris.cube.Cube` 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 = DimCoord(list(range(eofs.shape[0])),
var_name='eof',
long_name='eof_number')
coords = [eofdim] + [copy(coord) for coord in self._coords]
eofs = Cube(
eofs,
dim_coords_and_dims=list(zip(coords, list(range(eofs.ndim)))),
var_name='eofs',
long_name='empirical_orthogonal_functions')
# Add any auxiliary coordinates spanning space to the returned cube.
for coord, dims in self._space_aux_coords:
eofs.add_aux_coord(copy(coord), dims)
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 `~iris.cube.Cube` 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 = DimCoord(list(range(eofs.shape[0])),
var_name='eof',
long_name='eof_number')
coords = [eofdim] + [copy(coord) for coord in self._coords]
eofs = Cube(
eofs,
dim_coords_and_dims=list(zip(coords, list(range(eofs.ndim)))),
var_name='eofs',
long_name='correlation_between_pcs_and_{:s}'.format(
self._cube_name))
# Add any auxiliary coordinates spanning space to the returned cube.
for coord, dims in self._space_aux_coords:
eofs.add_aux_coord(copy(coord), dims)
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 `~iris.cube.Cube` 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 = DimCoord(list(range(eofs.shape[0])),
var_name='eof',
long_name='eof_number')
coords = [eofdim] + [copy(coord) for coord in self._coords]
eofs = Cube(
eofs,
dim_coords_and_dims=list(zip(coords, list(range(eofs.ndim)))),
var_name='eofs',
long_name='covariance_between_pcs_and_{:s}'.format(
self._cube_name))
# Add any auxiliary coordinates spanning space to the returned cube.
for coord, dims in self._space_aux_coords:
eofs.add_aux_coord(copy(coord), dims)
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 `~iris.cube.Cube` 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 = DimCoord(list(range(lambdas.shape[0])),
var_name='eigenvalue',
long_name='eigenvalue_number')
coords = [eofdim]
lambdas = Cube(
lambdas,
dim_coords_and_dims=list(zip(coords, list(range(lambdas.ndim)))),
var_name='eigenvalues',
long_name='eigenvalues')
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 `~iris.cube.Cube` 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)
"""
vfrac = self._solver.varianceFraction(neigs=neigs)
eofdim = DimCoord(list(range(vfrac.shape[0])),
var_name='eigenvalue',
long_name='eigenvalue_number')
coords = [eofdim]
vfrac = Cube(
vfrac,
dim_coords_and_dims=list(zip(coords, list(range(vfrac.ndim)))),
var_name='variance_fractions',
long_name='variance_fractions')
return vfrac
[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 `~iris.cube.Cube`).
**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 `~iris.cube.Cube` 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 = DimCoord(list(range(typerrs.shape[0])),
var_name='eigenvalue',
long_name='eigenvalue_number')
coords = [eofdim]
typerrs = Cube(
typerrs,
dim_coords_and_dims=list(zip(coords, list(range(typerrs.ndim)))),
var_name='typical_errors',
long_name='typical_errors')
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 `~iris.cube.Cube`.
**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 `~iris.cube.Cube` 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 = [copy(self._time)] + [copy(coord) for coord in self._coords]
if isinstance(neofs, collections.Iterable):
name_part = 'EOFs_{}'.format('_'.join([str(e) for e in neofs]))
else:
name_part = '{}_EOFs'.format(neofs)
rfield = Cube(
rfield,
dim_coords_and_dims=list(zip(coords, list(range(rfield.ndim)))),
var_name=self._cube_var_name or 'dataset',
long_name='{:s}_reconstructed_with_{:s}'.format(
self._cube_name, name_part))
rfield.attributes.update({'neofs': neofs})
# Add any auxiliary coordinates to the returned cube.
for coord, dims in (self._time_aux_coords +
self._space_aux_coords +
self._time_space_aux_coords):
rfield.add_aux_coord(copy(coord), dims)
return rfield
[docs] def projectField(self, cube, 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 `iris.cube.Cube` 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 `~iris.cube.Cube` 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)
"""
# Check that the input is an Iris cube.
if not isinstance(cube, Cube):
raise TypeError('the input must be an iris cube')
cube_name = cube.name(default='dataset').replace(' ', '_')
has_time = False
try:
# A time dimension must be first.
time, time_dim = get_time_coord(cube)
has_time = True
except ValueError:
# No time dimension is also acceptable.
has_time = False
if has_time:
if time_dim != 0:
raise ValueError('time must be the first dimension, '
'consider using the transpose() method')
_time_aux_coords, _, _ = classified_aux_coords(cube)
pcs = self._solver.projectField(cube.data,
neofs=neofs,
eofscaling=eofscaling,
weighted=weighted)
# Create the PCs cube.
pcs = Cube(pcs,
long_name='{}_pseudo_pcs'.format(cube_name),
var_name='pseudo_pcs')
# Construct the required dimensions.
if pcs.ndim == 2:
# 2D PCs require a time axis and a PC axis.
pcdim = DimCoord(list(range(pcs.shape[1])),
var_name='pc',
long_name='pc_number')
pcs.add_dim_coord(copy(time), 0)
pcs.add_dim_coord(pcdim, 1)
# Add any time-spanning auxiliary coordinates from the input cube
# to the returned PCs.
for coord, dims in _time_aux_coords:
pcs.add_aux_coord(copy(coord), dims)
else:
# 1D PCs require only a PC axis.
pcdim = DimCoord(list(range(pcs.shape[0])),
var_name='pc',
long_name='pc_number')
pcs.add_dim_coord(pcdim, 0)
return pcs
[docs] def getWeights(self):
"""Weights used for the analysis.
**Returns:**
*weights*
An array contaning the analysis weights (not a
`~iris.cube.Cube`).
**Example:**
The weights used for the analysis::
weights = solver.getWeights()
"""
return self._solver.getWeights()
