# RV¶

class hyppo.independence.RV

Rank Value (RV) test statistic and p-value.

RV is the multivariate generalization of the squared Pearson correlation coefficient [1]. The RV coefficient can be thought to be closely related to principal component analysis (PCA), canonical correlation analysis (CCA), multivariate regression, and statistical classification [1]. The statistic can be derived as follows [1] [2]:

Let $$x$$ and $$y$$ be $$(n, p)$$ samples of random variables $$X$$ and $$Y$$. We can center $$x$$ and $$y$$ and then calculate the sample covariance matrix $$\hat{\Sigma}_{xy} = x^T y$$ and the variance matrices for $$x$$ and $$y$$ are defined similarly. Then, the RV test statistic is found by calculating

$\mathrm{RV}_n (x, y) = \frac{\mathrm{tr} \left( \hat{\Sigma}_{xy} \hat{\Sigma}_{yx} \right)} {\mathrm{tr} \left( \hat{\Sigma}_{xx}^2 \right) \mathrm{tr} \left( \hat{\Sigma}_{yy}^2 \right)}$

where $$\mathrm{tr} (\cdot)$$ is the trace operator.

The p-value returned is calculated using a permutation test using hyppo.tools.perm_test.

Methods Summary

 RV.statistic(x, y) Helper function that calculates the RV test statistic. RV.test(x, y[, reps, workers]) Calculates the RV test statistic and p-value.

RV.statistic(x, y)

Helper function that calculates the RV test statistic.

Parameters

x,y (ndarray) -- Input data matrices. x and y must have the same number of samples and dimensions. That is, the shapes must be (n, p) where n is the number of samples and p is the number of dimensions.

Returns

stat (float) -- The computed RV statistic.

RV.test(x, y, reps=1000, workers=1)

Calculates the RV test statistic and p-value.

Parameters
• x,y (ndarray) -- Input data matrices. x and y must have the same number of samples and dimensions. That is, the shapes must be (n, p) where n is the number of samples and p is the number of dimensions.

• reps (int, default: 1000) -- The number of replications used to estimate the null distribution when using the permutation test used to calculate the p-value.

• workers (int, default: 1) -- The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process.

Returns

• stat (float) -- The computed RV statistic.

• pvalue (float) -- The computed RV p-value.

Examples

>>> import numpy as np
>>> from hyppo.independence import RV
>>> x = np.arange(7)
>>> y = x
>>> stat, pvalue = RV().test(x, y)
>>> '%.1f, %.2f' % (stat, pvalue)
'1.0, 0.00'