"""
Income mobility measures.
"""
__author__ = "Wei Kang <weikang9009@gmail.com>, Sergio J. Rey <sjsrey@gmail.com>"
__all__ = ["markov_mobility"]
import numpy as np
import numpy.linalg as la
[docs]
def markov_mobility(p, measure="P", ini=None):
r"""
Markov-based mobility index.
Parameters
----------
p : array
(k, k), Markov transition probability matrix.
measure : string
If measure= "P",
:math:`M_{P} = \\frac{m-\sum_{i=1}^m P_{ii}}{m-1}`;
if measure = "D",
:math:`M_{D} = 1 - |\det(P)|`,
where :math:`\det(P)` is the determinant of :math:`P`;
if measure = "L2",
:math:`M_{L2} = 1 - |\lambda_2|`,
where :math:`\lambda_2` is the second largest eigenvalue of
:math:`P`;
if measure = "B1",
:math:`M_{B1} = \\frac{m-m \sum_{i=1}^m \pi_i P_{ii}}{m-1}`,
where :math:`\pi` is the initial income distribution;
if measure == "B2",
:math:`M_{B2} = \\frac{1}{m-1} \sum_{i=1}^m \sum_{
j=1}^m \pi_i P_{ij} |i-j|`,
where :math:`\pi` is the initial income distribution.
ini : array
(k,), initial distribution. Need to be specified if
measure = "B1" or "B2". If not,
the initial distribution would be treated as a uniform
distribution.
Returns
-------
mobi : float
Mobility value.
Notes
-----
The mobility indices are based on :cite:`Formby:2004fk`.
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> import mapclassify as mc
>>> from giddy.markov import Markov
>>> from giddy.mobility import markov_mobility
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
>>> q5 = np.array([mc.Quantiles(y).yb for y in pci]).transpose()
>>> m = Markov(q5)
The Markov Chain is irreducible and is composed by:
1 Recurrent class (indices):
[0 1 2 3 4]
0 Transient classes.
The Markov Chain has 0 absorbing states.
>>> m.p
array([[0.91011236, 0.0886392 , 0.00124844, 0. , 0. ],
[0.09972299, 0.78531856, 0.11080332, 0.00415512, 0. ],
[0. , 0.10125 , 0.78875 , 0.1075 , 0.0025 ],
[0. , 0.00417827, 0.11977716, 0.79805014, 0.07799443],
[0. , 0. , 0.00125156, 0.07133917, 0.92740926]])
(1) Estimate Shorrock1 mobility index:
>>> mobi_1 = markov_mobility(m.p, measure="P")
>>> print("{:.5f}".format(mobi_1))
0.19759
(2) Estimate Shorrock2 mobility index:
>>> mobi_2 = markov_mobility(m.p, measure="D")
>>> print("{:.5f}".format(mobi_2))
0.60685
(3) Estimate Sommers and Conlisk mobility index:
>>> mobi_3 = markov_mobility(m.p, measure="L2")
>>> print("{:.5f}".format(mobi_3))
0.03978
(4) Estimate Bartholomew1 mobility index (note that the initial
distribution should be given):
>>> ini = np.array([0.1,0.2,0.2,0.4,0.1])
>>> mobi_4 = markov_mobility(m.p, measure = "B1", ini=ini)
>>> print("{:.5f}".format(mobi_4))
0.22777
(5) Estimate Bartholomew2 mobility index (note that the initial
distribution should be given):
>>> ini = np.array([0.1,0.2,0.2,0.4,0.1])
>>> mobi_5 = markov_mobility(m.p, measure = "B2", ini=ini)
>>> print("{:.5f}".format(mobi_5))
0.04637
"""
p = np.array(p)
k = p.shape[1]
if measure == "P":
t = np.trace(p)
mobi = (k - t) / (k - 1)
elif measure == "D":
mobi = 1 - abs(la.det(p))
elif measure == "L2":
w, v = la.eig(p)
eigen_value_abs = abs(w)
mobi = 1 - np.sort(eigen_value_abs)[-2]
elif measure == "B1":
if ini is None:
ini = 1.0 / k * np.ones(k)
mobi = (k - k * np.sum(ini * np.diag(p))) / (k - 1)
elif measure == "B2":
mobi = 0
if ini is None:
ini = 1.0 / k * np.ones(k)
for i in range(k):
for j in range(k):
mobi = mobi + ini[i] * p[i, j] * abs(i - j)
mobi = mobi / (k - 1)
return mobi