giddy.mobility.markov_mobility

giddy.mobility.markov_mobility(p, measure='P', ini=None)

Markov-based mobility index.

Parameters:

parray

(k, k), Markov transition probability matrix.

measurestring

If measure= “P”, \(M_{P} = \\frac{m-\sum_{i=1}^m P_{ii}}{m-1}\); if measure = “D”, \(M_{D} = 1 - |\det(P)|\), where \(\det(P)\) is the determinant of \(P\); if measure = “L2”, \(M_{L2} = 1 - |\lambda_2|\), where \(\lambda_2\) is the second largest eigenvalue of \(P\); if measure = “B1”, \(M_{B1} = \\frac{m-m \sum_{i=1}^m \pi_i P_{ii}}{m-1}\), where \(\pi\) is the initial income distribution; if measure == “B2”, \(M_{B2} = \\frac{1}{m-1} \sum_{i=1}^m \sum_{ j=1}^m \pi_i P_{ij} |i-j|\), where \(\pi\) is the initial income distribution.

iniarray

(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:

mobifloat

Mobility value.

Notes

The mobility indices are based on [FSZ04].

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