eofs.xarray

EOF analysis for data in xarray.DataArray arrays.

class eofs.xarray.Eof(array, weights=None, center=True, ddof=1)

EOF analysis (meta-data enabled xarray interface)

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)

neofs = None

The number of EOFs in the solution.

pcs(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 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)

eofs(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 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)

eofsAsCorrelation(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 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)

eofsAsCovariance(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 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)

eigenvalues(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 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)

varianceFraction(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 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)

totalAnomalyVariance()

Total variance associated with the field of anomalies (the sum of the eigenvalues).

Returns:

total_variance

A scalar value (not a DataArray).

Example:

Get the total variance:

total_variance = solver.totalAnomalyVariance()

northTest(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 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)

reconstructedField(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 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 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])

projectField(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 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)

getWeights()

Weights used for the analysis.

Returns:

weights

An array contaning the analysis weights (not a DataArray).

Example:

The weights used for the analysis:

weights = solver.getWeights()