MultispatiPCA

class multispaeti.MultispatiPCA(n_components=None, *, connectivity)

MULTISPATI-PCA

In contrast to Principal component analysis (PCA), MULTISPATI-PCA does not optimize the variance explained of each component but rather the product of the variance and Moran’s I. This can lead to negative eigenvalues i.e. in the case of negative auto-correlation.

The problem is solved by diagonalizing the symmetric matrix \(H=1/(2n)*X^t(W+W^t)X\) where X is matrix of n observations \(\times\) d features, and W is a matrix of the connectivity between observations.

Parameters:

n_components (int or tuple[int, int], optional) – Number of components to keep. If None, will keep all components (only supported for non-sparse X). If an int, it will keep the top n_components. If a tuple, it will keep the top and bottom n_components respectively.
connectivity (sparray or spmatrix) – Matrix of row-wise neighbor definitions i.e. c_ij is the connectivity of i \(\to\) j. The matrix does not have to be symmetric. It can be a binary adjacency matrix or a matrix of connectivities in which case c_ij should be larger if i and j are close. A distance matrix should be transformed to connectivities by e.g. calculating \(1-d/d_{max}\) beforehand.

Raises:

ValueError – If connectivity is not a square matrix.
ZeroDivisionError – If one of the observations has no neighbors.

components_

The estimated components: Array of shape (n_components, n_features).

Type:: ndarray

variance_

The estimated variance part of the eigenvalues. Array of shape (n_components,).

Type:: ndarray

moransI_

The estimated Moran’s I part of the eigenvalues. Array of shape (n_components,).

Type:: ndarray

eigenvalues_

The eigenvalues corresponding to each of the selected components. Array of shape (n_components,).

Type:: ndarray

n_components_

The estimated number of components.

Type:: int

n_samples_

Number of samples in the training data.

Type:: int

n_features_in_

Number of features seen during fit.

Type:: int

References

Dray, Stéphane, Sonia Saïd, and Françis Débias. “Spatial ordination of vegetation data using a generalization of Wartenberg’s multivariate spatial correlation.” Journal of vegetation science 19.1 (2008): 45-56.

Methods

`fit`	Fit MULTISPATI-PCA projection.
`fit_transform`	Fit the MULTISPATI-PCA projection and transform the data.
`moransI_bounds`	Calculate the minimum and maximum bound for Moran's I given the connectivity and the expected value given the #observations.
`transform`	Transform the data using fitted MULTISPATI-PCA projection.
`transform_spatial_lag`	Transform the data using fitted MULTISPATI-PCA projection and calculate the spatial lag.