atmos_flux_inversion.correlations module

Generating correlation operators

class atmos_flux_inversion.correlations.HomogeneousIsotropicCorrelation(*args, **kwargs)

Bases: atmos_flux_inversion.linalg.SelfAdjointLinearOperator

Homogeneous isotropic correlations using FFTs.

This embeds the physical domain passed in a larger computational domain, which allows for treatment of domains that should not be considered periodic.

Note

Do not invoke this directly. Use HomogeneousIsotropicCorrelation.from_function() or HomogeneousIsotropicCorrelaiton.from_array() instead.

See also

scipy.linalg.solve_circulant()

I stole the idea from here.

classmethod from_array(corr_array, is_cyclic=True)

Create an instance with the given correlations.

Parameters

• corr_array (array_like) – The correlation of the first element of the domain with each other element.

• is_cyclic (bool) –

Returns

Return type

HomogeneousIsotropicCorrelation

classmethod from_function(corr_func, shape, is_cyclic=True)

Create an instance to apply the correlation function.

Parameters

• corr_func (callable(dist) -> float) – The correlation of the first element of the domain with each other element.

• shape (tuple of int) – The state is formally a vector, but the correlations are assumed to depend on the layout in some other shape, usually related to the physical layout. This is the other shape.

• is_cyclic (bool) – Whether to assume the domain is periodic in all directions.

Returns

Return type

HomogeneousIsotropicCorrelation

inv()

Construct the matrix inverse of this operator.

Returns

Return type

LinearOperator

Raises

ValueError – if the instance is acyclic.

kron(other)

Construct the Kronecker product of this operator and other.

Parameters

other (HomogeneousIsotropicCorrelation) – The other operator for the Kronecker product. This implementation will accept other objects, passing them along to linalg.SchmidtKroneckerProduct.

Returns

Return type

scipy.sparse.linalg.LinearOperator

solve(vec)

Solve A @ x = vec.

Parameters

vec (array_like[N]) –

Returns

Solution of self @ x = vec

Return type

array_like[N]

Raises

• NotImplementedError – if the instance is acyclic.

• ValueError – If vec is the wrong shape

sqrt()

Compute an S such that S.T @ S == self.

Returns

S

Return type

HomogeneousIsotropicCorrelation

atmos_flux_inversion.correlations.make_matrix(corr_func, shape)

Make a correlation matrix for a domain with shape shape.

Parameters

• corr_func (callable(float) -> float[-1, 1]) – Function giving correlation between two indices a distance d from each other.

• shape (tuple of int) – The underlying shape of the domain. It is viewed as a vector here, but may be more naturally seen as an N-D array. This is the shape of that array. N = prod(shape)

See also

statsmodels.stats.correlation_tools.corr_clipped()

Does something similar, and refers to other functions that may give more accurate results.

Returns

corr – Positive definite dense array, entirely in memory

Return type

np.ndarray[N, N]

Correlation shapes

Interface

class atmos_flux_inversion.correlations.DistanceCorrelationFunction(length)

Bases: object

A correlation function that depends only on distance.

correlation_from_index(*indices)

Find the correlation between the indices.

Should be independent of the length of the underlying shape, but indices must still be even.

Parameters

indices (tuple of int) –

Returns

Find the correlation of the point contained in the first half of the indices with that contained in the second.

Return type

float

Provided shapes

class atmos_flux_inversion.correlations.ExponentialCorrelation(length)

Bases: atmos_flux_inversion.correlations.DistanceCorrelationFunction

A exponential correlation structure.

Notes

Correlation given by exp(-dist/length) where dist is the distance between the points.

class atmos_flux_inversion.correlations.BalgovindCorrelation(length)

Bases: atmos_flux_inversion.correlations.DistanceCorrelationFunction

A Balgovind 3D correlation structure.

Follows Balgovind et al. 1983 recommendations for a 3D field, modified so the correlation length better matches that used by other correlation functions.

Notes

Correlation given by \((1 + 2*dist/length) exp(-2*dist/length)\)

This implementation has problems for length == 10. I have no idea why. 3 and 30 are fine.

References

Balgovind, R and Dalcher, A. and Ghil, M. and Kalnay, E. (1983). A Stochastic-Dynamic Model for the Spatial Structure of Forecast Error Statistics Monthly Weather Review 111(4) 701–722. doi:10.1175/1520-0493(1983)111<0701:asdmft>2.0.co;2

class atmos_flux_inversion.correlations.MaternCorrelation(length, kappa=1)

Bases: atmos_flux_inversion.correlations.DistanceCorrelationFunction

A Matern correlation structure.

Follows Matern (1986) Spatial Variation. The specific definition seems to be similar to the parameterization in Stern’s Interpolation of Spatial Data

Notes

Correlation given by \([2^{\kappa-1}\Gamma(\kappa)]^{-1} (d/L)^{\kappa} K_{\kappa}(d/L)\) where \(\kappa\) is a smoothness parameter and \(K_{\kappa}\) is a modified Bessel function of the third kind.

References

Stein, Michael L. Interpolation of Spatial Data: Some Theory for Kridging Springer-Verlag New York. Part of Springer Series in Statistics (issn:0172-7397) isbn:978-1-4612-7166-6. doi:10.1007/978-1-4612-1494-6

class atmos_flux_inversion.correlations.GaussianCorrelation(length)

Bases: atmos_flux_inversion.correlations.DistanceCorrelationFunction

A gaussian correlation structure.

Notes

Correlation given by exp(-dist**2 / (length**2)) where dist is the distance between the points.