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# numpy.corrcoef

numpy.corrcoef(x, y=None, rowvar=True, bias=<no value>, ddof=<no value>, *, dtype=None)

  Return Pearson product-moment correlation coefficients.

  Please refer to the documentation for `cov` for more detail. The relationship between the correlation coefficient matrix, R, and the covariance matrix, C, is

  \[R_{ij} = \frac{ C_{ij} } { \sqrt{ C_{ii} C_{jj} } }\]

  The values of R are between -1 and 1, inclusive.

  Parameters:
    x array_like

      A 1-D or 2-D array containing multiple variables and observations. Each row of x represents a variable, and each column a single observation of all those variables. Also see rowvar below.

    y array_like, optional

      An additional set of variables and observations. y has the same shape as x.

    rowvar bool, optional

      If rowvar is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations.

    bias_NoValue, optional

      Has no effect, do not use.

      Deprecated since version 1.10.0.

    ddof_NoValue, optional

      Has no effect, do not use.

      Deprecated since version 1.10.0.

    dtype data-type, optional

      Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision.

      Added in version 1.20.

  Returns:
    R ndarray

      The correlation coefficient matrix of the variables.

  See also

  `cov`

    Covariance matrix

  Notes

  Due to floating point rounding the resulting array may not be Hermitian, the diagonal elements may not be 1, and the elements may not satisfy the inequality abs(a) <= 1. The real and imaginary parts are clipped to the interval [-1, 1] in an attempt to improve on that situation but is not much help in the complex case.

  This function accepts but discards arguments bias and ddof. This is for backwards compatibility with previous versions of this function. These arguments had no effect on the return values of the function and can be safely ignored in this and previous versions of numpy.

  Examples

      >>> import numpy as np

  In this example we generate two random arrays, `xarr` and `yarr`, and compute the row-wise and column-wise Pearson correlation coefficients, `R`. Since `rowvar` is true by default, we first find the row-wise Pearson correlation coefficients between the variables of `xarr`.

      >>> import numpy as np
      >>> rng = np.random.default_rng(seed=42)
      >>> xarr = rng.random((3, 3))
      >>> xarr
      array([[0.77395605, 0.43887844, 0.85859792],
             [0.69736803, 0.09417735, 0.97562235],
             [0.7611397 , 0.78606431, 0.12811363]])
      >>> R1 = np.corrcoef(xarr)
      >>> R1
      array([[ 1.        ,  0.99256089, -0.68080986],
             [ 0.99256089,  1.        , -0.76492172],
             [-0.68080986, -0.76492172,  1.        ]])

  If we add another set of variables and observations `yarr`, we can compute the row-wise Pearson correlation coefficients between the variables in `xarr` and `yarr`.

      >>> yarr = rng.random((3, 3))
      >>> yarr
      array([[0.45038594, 0.37079802, 0.92676499],
             [0.64386512, 0.82276161, 0.4434142 ],
             [0.22723872, 0.55458479, 0.06381726]])
      >>> R2 = np.corrcoef(xarr, yarr)
      >>> R2
      array([[ 1.        ,  0.99256089, -0.68080986,  0.75008178, -0.934284  ,
              -0.99004057],
             [ 0.99256089,  1.        , -0.76492172,  0.82502011, -0.97074098,
              -0.99981569],
             [-0.68080986, -0.76492172,  1.        , -0.99507202,  0.89721355,
               0.77714685],
             [ 0.75008178,  0.82502011, -0.99507202,  1.        , -0.93657855,
              -0.83571711],
             [-0.934284  , -0.97074098,  0.89721355, -0.93657855,  1.        ,
               0.97517215],
             [-0.99004057, -0.99981569,  0.77714685, -0.83571711,  0.97517215,
               1.        ]])

  Finally if we use the option `rowvar=False`, the columns are now being treated as the variables and we will find the column-wise Pearson correlation coefficients between variables in `xarr` and `yarr`.

      >>> R3 = np.corrcoef(xarr, yarr, rowvar=False)
      >>> R3
      array([[ 1.        ,  0.77598074, -0.47458546, -0.75078643, -0.9665554 ,
               0.22423734],
             [ 0.77598074,  1.        , -0.92346708, -0.99923895, -0.58826587,
              -0.44069024],
             [-0.47458546, -0.92346708,  1.        ,  0.93773029,  0.23297648,
               0.75137473],
             [-0.75078643, -0.99923895,  0.93773029,  1.        ,  0.55627469,
               0.47536961],
             [-0.9665554 , -0.58826587,  0.23297648,  0.55627469,  1.        ,
              -0.46666491],
             [ 0.22423734, -0.44069024,  0.75137473,  0.47536961, -0.46666491,
               1.        ]])