Ordination methods (skbio.stats.ordination)

This module contains several ordination methods, including Principal Coordinate Analysis, Correspondence Analysis, Redundancy Analysis and Canonical Correspondence Analysis.

Ordination Functions

ca(X[, scaling])

Compute correspondence analysis, a multivariate statistical

pcoa(distance_matrix[, method, …])

Perform Principal Coordinate Analysis.

pcoa_biplot(ordination, y)

Compute the projection of descriptors into a PCoA matrix

cca(y, x[, scaling])

Compute canonical (also known as constrained) correspondence

rda(y, x[, scale_Y, scaling])

Compute redundancy analysis, a type of canonical analysis.

Classes

OrdinationResults(short_method_name, …[, …])

Store ordination results, providing serialization and plotting support.

Utility Functions

mean_and_std(a[, axis, weights, with_mean, …])

Compute the weighted average and standard deviation along the

corr(x[, y])

Computes correlation between columns of x, or x and y.

scale(a[, weights, with_mean, with_std, …])

Scale array by columns to have weighted average 0 and standard

svd_rank(M_shape, S[, tol])

Matrix rank of M given its singular values S.

e_matrix(distance_matrix)

Compute E matrix from a distance matrix.

f_matrix(E_matrix)

Compute F matrix from E matrix.

Examples

This is an artificial dataset (table 11.3 in 1) that represents fish abundance in different sites (Y, the response variables) and environmental variables (X, the explanatory variables).

>>> import numpy as np
>>> import pandas as pd

First we need to construct our explanatory variable dataset X.

>>> X = np.array([[1.0, 0.0, 1.0, 0.0],
...               [2.0, 0.0, 1.0, 0.0],
...               [3.0, 0.0, 1.0, 0.0],
...               [4.0, 0.0, 0.0, 1.0],
...               [5.0, 1.0, 0.0, 0.0],
...               [6.0, 0.0, 0.0, 1.0],
...               [7.0, 1.0, 0.0, 0.0],
...               [8.0, 0.0, 0.0, 1.0],
...               [9.0, 1.0, 0.0, 0.0],
...               [10.0, 0.0, 0.0, 1.0]])
>>> transects = ['depth', 'substrate_coral', 'substrate_sand',
...              'substrate_other']
>>> sites = ['site1', 'site2', 'site3', 'site4', 'site5', 'site6', 'site7',
...          'site8', 'site9', 'site10']
>>> X = pd.DataFrame(X, sites, transects)

Then we need to create a dataframe with the information about the species observed at different sites.

>>> species = ['specie1', 'specie2', 'specie3', 'specie4', 'specie5',
...            'specie6', 'specie7', 'specie8', 'specie9']
>>> Y = np.array([[1, 0, 0, 0, 0, 0, 2, 4, 4],
...               [0, 0, 0, 0, 0, 0, 5, 6, 1],
...               [0, 1, 0, 0, 0, 0, 0, 2, 3],
...               [11, 4, 0, 0, 8, 1, 6, 2, 0],
...               [11, 5, 17, 7, 0, 0, 6, 6, 2],
...               [9, 6, 0, 0, 6, 2, 10, 1, 4],
...               [9, 7, 13, 10, 0, 0, 4, 5, 4],
...               [7, 8, 0, 0, 4, 3, 6, 6, 4],
...               [7, 9, 10, 13, 0, 0, 6, 2, 0],
...               [5, 10, 0, 0, 2, 4, 0, 1, 3]])
>>> Y = pd.DataFrame(Y, sites, species)

We can now perform canonical correspondence analysis. Matrix X contains a continuous variable (depth) and a categorical one (substrate type) encoded using a one-hot encoding.

>>> from skbio.stats.ordination import cca

We explicitly need to avoid perfect collinearity, so we’ll drop one of the substrate types (the last column of X).

>>> del X['substrate_other']
>>> ordination_result = cca(Y, X, scaling=2)

Exploring the results we see that the first three axes explain about 80% of all the variance.

>>> ordination_result.proportion_explained
CCA1    0.466911
CCA2    0.238327
CCA3    0.100548
CCA4    0.104937
CCA5    0.044805
CCA6    0.029747
CCA7    0.012631
CCA8    0.001562
CCA9    0.000532
dtype: float64

References

1

Legendre P. and Legendre L. 1998. Numerical Ecology. Elsevier, Amsterdam.