pysal.explore.giddy.markov.LISA_Markov

class pysal.explore.giddy.markov.LISA_Markov(y, w, permutations=0, significance_level=0.05, geoda_quads=False)[source]

Markov for Local Indicators of Spatial Association

Parameters:
y : array

(n, t), n cross-sectional units observed over t time periods.

w : W

spatial weights object.

permutations : int, optional

number of permutations used to determine LISA significance (the default is 0).

significance_level : float, optional

significance level (two-sided) for filtering significant LISA endpoints in a transition (the default is 0.05).

geoda_quads : bool

If True use GeoDa scheme: HH=1, LL=2, LH=3, HL=4. If False use PySAL Scheme: HH=1, LH=2, LL=3, HL=4. (the default is False).

Examples

>>> import pysal.lib
>>> import numpy as np
>>> from pysal.explore.giddy.markov import LISA_Markov
>>> f = pysal.lib.io.open(pysal.lib.examples.get_path("usjoin.csv"))
>>> years = list(range(1929, 2010))
>>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
>>> w = pysal.lib.io.open(pysal.lib.examples.get_path("states48.gal")).read()
>>> lm = LISA_Markov(pci,w)
>>> lm.classes
array([1, 2, 3, 4])
>>> lm.steady_state
array([0.28561505, 0.14190226, 0.40493672, 0.16754598])
>>> lm.transitions
array([[1.087e+03, 4.400e+01, 4.000e+00, 3.400e+01],
       [4.100e+01, 4.700e+02, 3.600e+01, 1.000e+00],
       [5.000e+00, 3.400e+01, 1.422e+03, 3.900e+01],
       [3.000e+01, 1.000e+00, 4.000e+01, 5.520e+02]])
>>> lm.p
array([[0.92985458, 0.03763901, 0.00342173, 0.02908469],
       [0.07481752, 0.85766423, 0.06569343, 0.00182482],
       [0.00333333, 0.02266667, 0.948     , 0.026     ],
       [0.04815409, 0.00160514, 0.06420546, 0.88603531]])
>>> lm.move_types[0,:3]
array([11, 11, 11])
>>> lm.move_types[0,-3:]
array([11, 11, 11])

Now consider only moves with one, or both, of the LISA end points being significant

>>> np.random.seed(10)
>>> lm_random = LISA_Markov(pci, w, permutations=99)
>>> lm_random.significant_moves[0, :3]
array([11, 11, 11])
>>> lm_random.significant_moves[0,-3:]
array([59, 43, 27])

Any value less than 49 indicates at least one of the LISA end points was significant. So for example, the first spatial unit experienced a transition of type 11 (LL, LL) during the first three and last tree intervals (according to lm.move_types), however, the last three of these transitions involved insignificant LISAS in both the start and ending year of each transition.

Test whether the moves of y are independent of the moves of wy

>>> "Chi2: %8.3f, p: %5.2f, dof: %d" % lm.chi_2
'Chi2: 1058.208, p:  0.00, dof: 9'

Actual transitions of LISAs

>>> lm.transitions
array([[1.087e+03, 4.400e+01, 4.000e+00, 3.400e+01],
       [4.100e+01, 4.700e+02, 3.600e+01, 1.000e+00],
       [5.000e+00, 3.400e+01, 1.422e+03, 3.900e+01],
       [3.000e+01, 1.000e+00, 4.000e+01, 5.520e+02]])

Expected transitions of LISAs under the null y and wy are moving independently of one another

>>> lm.expected_t
array([[1.12328098e+03, 1.15377356e+01, 3.47522158e-01, 3.38337644e+01],
       [3.50272664e+00, 5.28473882e+02, 1.59178880e+01, 1.05503814e-01],
       [1.53878082e-01, 2.32163556e+01, 1.46690710e+03, 9.72266513e+00],
       [9.60775143e+00, 9.86856346e-02, 6.23537392e+00, 6.07058189e+02]])

If the LISA classes are to be defined according to GeoDa, the geoda_quad option has to be set to true

>>> lm.q[0:5,0]
array([3, 2, 3, 1, 4])
>>> lm = LISA_Markov(pci,w, geoda_quads=True)
>>> lm.q[0:5,0]
array([2, 3, 2, 1, 4])
Attributes:
chi_2 : tuple

(3 elements) (chi square test statistic, p-value, degrees of freedom) for test that dynamics of y are independent of dynamics of wy.

classes : array

(4, 1) 1=HH, 2=LH, 3=LL, 4=HL (own, lag) 1=HH, 2=LL, 3=LH, 4=HL (own, lag) (if geoda_quads=True)

expected_t : array

(4, 4), expected number of transitions under the null that dynamics of y are independent of dynamics of wy.

move_types : matrix

(n, t-1), integer values indicating which type of LISA transition occurred (q1 is quadrant in period 1, q2 is quadrant in period 2).

q1 q2 move_type
1 1 1
1 2 2
1 3 3
1 4 4
2 1 5
2 2 6
2 3 7
2 4 8
3 1 9
3 2 10
3 3 11
3 4 12
4 1 13
4 2 14
4 3 15
4 4 16
p : array

(k, k), transition probability matrix.

p_values : matrix

(n, t), LISA p-values for each end point (if permutations > 0).

significant_moves : matrix

(n, t-1), integer values indicating the type and significance of a LISA transition. st = 1 if significant in period t, else st=0 (if permutations > 0).

(s1,s2) move_type
(1,1) [1, 16]
(1,0) [17, 32]
(0,1) [33, 48]
(0,0) [49, 64]
q1 q2 s1 s2 move_type
1 1 1 1 1
1 2 1 1 2
1 3 1 1 3
1 4 1 1 4
2 1 1 1 5
2 2 1 1 6
2 3 1 1 7
2 4 1 1 8
3 1 1 1 9
3 2 1 1 10
3 3 1 1 11
3 4 1 1 12
4 1 1 1 13
4 2 1 1 14
4 3 1 1 15
4 4 1 1 16
1 1 1 0 17
1 2 1 0 18
. . . . .
. . . . .
4 3 1 0 31
4 4 1 0 32
1 1 0 1 33
1 2 0 1 34
. . . . .
. . . . .
4 3 0 1 47
4 4 0 1 48
1 1 0 0 49
1 2 0 0 50
. . . . .
. . . . .
4 3 0 0 63
4 4 0 0 64
steady_state : array

(k, ), ergodic distribution.

transitions : array

(4, 4), count of transitions between each state i and j.

spillover : array

Detect spillover locations for diffusion in LISA Markov.

Methods

spillover([quadrant, neighbors_on]) Detect spillover locations for diffusion in LISA Markov.
__init__(y, w, permutations=0, significance_level=0.05, geoda_quads=False)[source]

Initialize self. See help(type(self)) for accurate signature.

Methods

__init__(y, w[, permutations, …]) Initialize self.
spillover([quadrant, neighbors_on]) Detect spillover locations for diffusion in LISA Markov.

Attributes

steady_state