Marchenko-Pastur Distribution

mp_distr.py

# Marchenko-Pastur distribution
# see https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution
def marchenko_pastur(x, c, var=1.0):
    '''
    Marchenko-Pastur PDF.
    
    Arguments:
    
        x:   independent variable,
        c:   phase transition parameter,
        var: variance of the data matrix.
    '''
    
    N = 2.0 * np.pi * c * x * var       # denominator
    a = var * np.square(1 - np.sqrt(c)) # inf of the support
    b = var * np.square(1 + np.sqrt(c)) # sup of the support
    
    # piecewise definition
    if a <= x <= b:
        return np.sqrt((b - x) * (x - a)) / N
    else:
        return 0.0

# convenience redefinition to use array-like objects as independent variable
# see https://numpy.org/doc/stable/reference/generated/numpy.vectorize.html
marchenko_pastur = np.vectorize(marchenko_pastur)