# Ordination methods (skbio.stats.ordination)¶

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

## Functions¶

 ca(X[, scaling]) Compute correspondence analysis, a multivariate statistical pcoa(distance_matrix) Perform Principal Coordinate Analysis. 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. 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.

## Classes¶

 OrdinationResults(short_method_name, ...[, ...]) Store ordination results, providing serialization and plotting support.

## Examples¶

This is an artificial dataset (table 11.3 in [R275]) 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¶

 [R275] Legendre P. and Legendre L. 1998. Numerical Ecology. Elsevier, Amsterdam.