"Dependency" definition

Question

Origin Lab has in their fitting parameter's statistics "Dependency". Each parameter has a dependency. It's not like the covariance between 2 parameters. I thought it could be defined from all the covariances of the parameter with the others (i.e. with the covariance matrix) but I couldn't find any documentation.

I don't know if it's a general concept because I have zero knowledge in statistics. The main goal it's to implement it on Python, so if the definition is not trivial (e.g. aritmethic expressions), I would appreciate a function that can compute it.

Answer 1

For completness I will write an answer. Thanks to @whuber. It's defined in the documentation as

$$\text{Dependency}_i = 1 - \frac{1}{[\mathbf{C}]_{ii} \cdot [\mathbf{C}^{-1}]_{ii}},$$

where $\mathbf{C}$ is the covariance matrix.

Answer 2

To add to the contributions @whuber and @MacroCiafa, I will include a Python implementation since that was your end goal. stats.SE is not primarily about coding, so consider this post merely a supplement to the accepted answer.

import numpy as np

def dependence(X):
    '''
    Calculate the dependence of a 
    collection of variables from a data
    matrix.
   

    PARAMETERS
        X (2D array-like): Data matrix

    RETURNS
        (1D array): Dependence score of each variable.
    '''
    C = np.cov(X)
    result = np.diag(C) * np.diag(np.linalg.inv(C))
    return 1 - 1 / result

If you're fine with using the Moore-Penrose pseudoinverse, then this can be adjusted slightly into the following.

import numpy as np

def dependence(X):
    '''
    Calculate the dependence of a 
    collection of variables from a data
    matrix.
   

    PARAMETERS
        X (2D array-like): Data matrix

    RETURNS
        (1D array): Dependence score of each variable.
    '''
    C = np.cov(X)
    result = np.diag(C) * np.diag(np.linalg.pinv(C))
    return 1 - 1 / result