Compute the Pearson correlation coefficients.
JAX implementation of numpy.corrcoef().
This is a normalized version of the sample covariance computed by jax.numpy.cov(). For a sample covariance \(C_{ij}\), the correlation coefficients are
\[R_{ij} = \frac{C_{ij}}{\sqrt{C_{ii}C_{jj}}}\]
they are constructed such that the values satisfy \(-1 \le R_{ij} \le 1\).
x (ArrayLike) – array of shape (M, N) (if rowvar is True), or (N, M) (if rowvar is False) representing N observations of M variables. x may also be one-dimensional, representing N observations of a single variable.
y (ArrayLike | None) – optional set of additional observations, with the same form as m. If specified, then y is combined with m, i.e. for the default rowvar = True case, m becomes jnp.vstack([m, y]).
rowvar (bool) – if True (default) then each row of m represents a variable. If False, then each column represents a variable.
A covariance matrix of shape (M, M).
Examples
Consider these observations of two variables that correlate perfectly. The correlation matrix in this case is a 2x2 matrix of ones:
>>> x = jnp.array([[0, 1, 2],
... [0, 1, 2]])
>>> jnp.corrcoef(x)
Array([[1., 1.],
[1., 1.]], dtype=float32)
Now consider these observations of two variables that are perfectly anti-correlated. The correlation matrix in this case has -1 in the off-diagonal:
>>> x = jnp.array([[-1, 0, 1],
... [ 1, 0, -1]])
>>> jnp.corrcoef(x)
Array([[ 1., -1.],
[-1., 1.]], dtype=float32)
Equivalently, these sequences can be specified as separate arguments, in which case they are stacked before continuing the computation.
>>> x = jnp.array([-1, 0, 1])
>>> y = jnp.array([1, 0, -1])
>>> jnp.corrcoef(x, y)
Array([[ 1., -1.],
[-1., 1.]], dtype=float32)
The entries of the correlation matrix are normalized such that they lie within the range -1 to +1, where +1 indicates perfect correlation and -1 indicates perfect anti-correlation. For example, here is the correlation of 100 points drawn from a 3-dimensional standard normal distribution:
>>> key = jax.random.key(0) >>> x = jax.random.normal(key, shape=(3, 100)) >>> with jnp.printoptions(precision=2): ... print(jnp.corrcoef(x)) [[1. 0.03 0.12] [0.03 1. 0.01] [0.12 0.01 1. ]]
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