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"""
Markov based methods for spatial dynamics.
"""
__author__ = "Sergio J. Rey <sjsrey@gmail.com>, Wei Kang <weikang9009@gmail.com>"
__all__ = ["Markov", "LISA_Markov", "Spatial_Markov", "kullback",
"prais", "homogeneity", "FullRank_Markov", "sojourn_time",
"GeoRank_Markov"]
import numpy as np
from .ergodic import fmpt
from .ergodic import steady_state as STEADY_STATE
from .components import Graph
from scipy import stats
from scipy.stats import rankdata
from operator import gt
from libpysal import weights
from esda.moran import Moran_Local
import mapclassify as mc
import itertools
# TT predefine LISA transitions
# TT[i,j] is the transition type from i to j
# i = quadrant in period 0
# j = quadrant in period 1
# uses one offset so first row and col of TT are ignored
TT = np.zeros((5, 5), int)
c = 1
for i in range(1, 5):
for j in range(1, 5):
TT[i, j] = c
c += 1
# MOVE_TYPES is a dictionary that returns the move type of a LISA transition
# filtered on the significance of the LISA end points
# True indicates significant LISA in a particular period
# e.g. a key of (1, 3, True, False) indicates a significant LISA located in
# quadrant 1 in period 0 moved to quadrant 3 in period 1 but was not
# significant in quadrant 3.
MOVE_TYPES = {}
c = 1
cases = (True, False)
sig_keys = [(i, j) for i in cases for j in cases]
for i, sig_key in enumerate(sig_keys):
c = 1 + i * 16
for i in range(1, 5):
for j in range(1, 5):
key = (i, j, sig_key[0], sig_key[1])
MOVE_TYPES[key] = c
c += 1
class Markov(object):
"""
Classic Markov transition matrices.
Parameters
----------
class_ids : array
(n, t), one row per observation, one column recording the
state of each observation, with as many columns as time
periods.
classes : array
(k, 1), all different classes (bins) of the matrix.
Attributes
----------
p : array
(k, k), transition probability matrix.
steady_state : array
(k, ), ergodic distribution.
transitions : array
(k, k), count of transitions between each state i and j.
Examples
--------
>>> import numpy as np
>>> from giddy.markov import Markov
>>> c = [['b','a','c'],['c','c','a'],['c','b','c']]
>>> c.extend([['a','a','b'], ['a','b','c']])
>>> c = np.array(c)
>>> m = Markov(c)
>>> m.classes.tolist()
['a', 'b', 'c']
>>> m.p
array([[0.25 , 0.5 , 0.25 ],
[0.33333333, 0. , 0.66666667],
[0.33333333, 0.33333333, 0.33333333]])
>>> m.steady_state
array([0.30769231, 0.28846154, 0.40384615])
US nominal per capita income 48 states 81 years 1929-2009
>>> import libpysal
>>> import mapclassify as mc
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
set classes to quintiles for each year
>>> q5 = np.array([mc.Quantiles(y).yb for y in pci]).transpose()
>>> m = Markov(q5)
>>> m.transitions
array([[729., 71., 1., 0., 0.],
[ 72., 567., 80., 3., 0.],
[ 0., 81., 631., 86., 2.],
[ 0., 3., 86., 573., 56.],
[ 0., 0., 1., 57., 741.]])
>>> 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]])
>>> m.steady_state
array([0.20774716, 0.18725774, 0.20740537, 0.18821787, 0.20937187])
Relative incomes
>>> pci = pci.transpose()
>>> rpci = pci/(pci.mean(axis=0))
>>> rq = mc.Quantiles(rpci.flatten()).yb.reshape(pci.shape)
>>> mq = Markov(rq)
>>> mq.transitions
array([[707., 58., 7., 1., 0.],
[ 50., 629., 80., 1., 1.],
[ 4., 79., 610., 73., 2.],
[ 0., 7., 72., 650., 37.],
[ 0., 0., 0., 48., 724.]])
>>> mq.steady_state
array([0.17957376, 0.21631443, 0.21499942, 0.21134662, 0.17776576])
"""
def __init__(self, class_ids, classes=None):
if classes is not None:
self.classes = classes
else:
self.classes = np.unique(class_ids)
n, t = class_ids.shape
k = len(self.classes)
js = list(range(t - 1))
classIds = self.classes.tolist()
transitions = np.zeros((k, k))
for state_0 in js:
state_1 = state_0 + 1
state_0 = class_ids[:, state_0]
state_1 = class_ids[:, state_1]
initial = np.unique(state_0)
for i in initial:
ending = state_1[state_0 == i]
uending = np.unique(ending)
row = classIds.index(i)
for j in uending:
col = classIds.index(j)
transitions[row, col] += sum(ending == j)
self.transitions = transitions
row_sum = transitions.sum(axis=1)
self.p = np.dot(np.diag(1 / (row_sum + (row_sum == 0))), transitions)
@property
def steady_state(self):
if not hasattr(self, '_steady_state'):
self._steady_state = STEADY_STATE(self.p)
return self._steady_state
class Spatial_Markov(object):
"""
Markov transitions conditioned on the value of the spatial lag.
Parameters
----------
y : array
(n, t), one row per observation, one column per state of
each observation, with as many columns as time periods.
w : W
spatial weights object.
k : integer, optional
number of classes (quantiles) for input time series y.
Default is 4. If discrete=True, k is determined
endogenously.
m : integer, optional
number of classes (quantiles) for the spatial lags of
regional time series. Default is 4. If discrete=True,
m is determined endogenously.
permutations : int, optional
number of permutations for use in randomization based
inference (the default is 0).
fixed : bool, optional
If true, discretization are taken over the entire n*t
pooled series and cutoffs can be user-defined. If
cutoffs and lag_cutoffs are not given, quantiles are
used. If false, quantiles are taken each time period
over n. Default is True.
discrete : bool, optional
If true, categorical spatial lags which are most common
categories of neighboring observations serve as the
conditioning and fixed is ignored; if false, weighted
averages of neighboring observations are used. Default is
false.
cutoffs : array, optional
users can specify the discretization cutoffs for
continuous time series. Default is None, meaning that
quantiles will be used for the discretization.
lag_cutoffs : array, optional
users can specify the discretization cutoffs for the
spatial lags of continuous time series. Default is
None, meaning that quantiles will be used for the
discretization.
variable_name : string
name of variable.
Attributes
----------
class_ids : array
(n, t), discretized series if y is continuous. Otherwise
it is identical to y.
classes : array
(k, 1), all different classes (bins).
lclass_ids : array
(n, t), spatial lag series.
lclasses : array
(k, 1), all different classes (bins) for
spatial lags.
p : array
(k, k), transition probability matrix for a-spatial
Markov.
s : array
(k, 1), ergodic distribution for a-spatial Markov.
transitions : array
(k, k), counts of transitions between each state i and j
for a-spatial Markov.
T : array
(k, k, k), counts of transitions for each conditional
Markov. T[0] is the matrix of transitions for
observations with lags in the 0th quantile; T[k-1] is the
transitions for the observations with lags in the k-1th.
P : array
(k, k, k), transition probability matrix for spatial
Markov first dimension is the conditioned on the lag.
S : array
(k, k), steady state distributions for spatial Markov.
Each row is a conditional steady_state.
F : array
(k, k, k),first mean passage times.
First dimension is conditioned on the lag.
shtest : list
(k elements), each element of the list is a tuple for a
multinomial difference test between the steady state
distribution from a conditional distribution versus the
overall steady state distribution: first element of the
tuple is the chi2 value, second its p-value and the third
the degrees of freedom.
chi2 : list
(k elements), each element of the list is a tuple for a
chi-squared test of the difference between the
conditional transition matrix against the overall
transition matrix: first element of the tuple is the chi2
value, second its p-value and the third the degrees of
freedom.
x2 : float
sum of the chi2 values for each of the conditional tests.
Has an asymptotic chi2 distribution with k(k-1)(k-1)
degrees of freedom. Under the null that transition
probabilities are spatially homogeneous.
(see chi2 above)
x2_dof : int
degrees of freedom for homogeneity test.
x2_pvalue : float
pvalue for homogeneity test based on analytic.
distribution
x2_rpvalue : float
(if permutations>0)
pseudo p-value for x2 based on random spatial
permutations of the rows of the original transitions.
x2_realizations : array
(permutations,1), the values of x2 for the random
permutations.
Q : float
Chi-square test of homogeneity across lag classes based
on :cite:`Bickenbach2003`.
Q_p_value : float
p-value for Q.
LR : float
Likelihood ratio statistic for homogeneity across lag
classes based on :cite:`Bickenbach2003`.
LR_p_value : float
p-value for LR.
dof_hom : int
degrees of freedom for LR and Q, corrected for 0 cells.
Notes
-----
Based on :cite:`Rey2001`.
The shtest and chi2 tests should be used with caution as they are based on
classic theory assuming random transitions. The x2 based test is
preferable since it simulates the randomness under the null. It is an
experimental test requiring further analysis.
Examples
--------
>>> import libpysal
>>> from giddy.markov import Spatial_Markov
>>> import numpy as np
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
>>> pci = pci.transpose()
>>> rpci = pci/(pci.mean(axis=0))
>>> w = libpysal.io.open(libpysal.examples.get_path("states48.gal")).read()
>>> w.transform = 'r'
Now we create a `Spatial_Markov` instance for the continuous relative per
capita income time series for 48 US lower states 1929-2009. The current
implementation allows users to classify the continuous incomes in a more
flexible way.
(1) Global quintiles to discretize the income data (k=5), and global
quintiles to discretize the spatial lags of incomes (m=5).
>>> sm = Spatial_Markov(rpci, w, fixed=True, k=5, m=5, variable_name='rpci')
We can examine the cutoffs for the incomes and cutoffs for the spatial lags
>>> sm.cutoffs
array([0.83999133, 0.94707545, 1.03242697, 1.14911154])
>>> sm.lag_cutoffs
array([0.88973386, 0.95891917, 1.01469758, 1.1183566 ])
Obviously, they are slightly different.
We now look at the estimated spatially lag conditioned transition
probability matrices.
>>> for p in sm.P:
... print(p)
[[0.96341463 0.0304878 0.00609756 0. 0. ]
[0.06040268 0.83221477 0.10738255 0. 0. ]
[0. 0.14 0.74 0.12 0. ]
[0. 0.03571429 0.32142857 0.57142857 0.07142857]
[0. 0. 0. 0.16666667 0.83333333]]
[[0.79831933 0.16806723 0.03361345 0. 0. ]
[0.0754717 0.88207547 0.04245283 0. 0. ]
[0.00537634 0.06989247 0.8655914 0.05913978 0. ]
[0. 0. 0.06372549 0.90196078 0.03431373]
[0. 0. 0. 0.19444444 0.80555556]]
[[0.84693878 0.15306122 0. 0. 0. ]
[0.08133971 0.78947368 0.1291866 0. 0. ]
[0.00518135 0.0984456 0.79274611 0.0984456 0.00518135]
[0. 0. 0.09411765 0.87058824 0.03529412]
[0. 0. 0. 0.10204082 0.89795918]]
[[0.8852459 0.09836066 0. 0.01639344 0. ]
[0.03875969 0.81395349 0.13953488 0. 0.00775194]
[0.0049505 0.09405941 0.77722772 0.11881188 0.0049505 ]
[0. 0.02339181 0.12865497 0.75438596 0.09356725]
[0. 0. 0. 0.09661836 0.90338164]]
[[0.33333333 0.66666667 0. 0. 0. ]
[0.0483871 0.77419355 0.16129032 0.01612903 0. ]
[0.01149425 0.16091954 0.74712644 0.08045977 0. ]
[0. 0.01036269 0.06217617 0.89637306 0.03108808]
[0. 0. 0. 0.02352941 0.97647059]]
The probability of a poor state remaining poor is 0.963 if their
neighbors are in the 1st quintile and 0.798 if their neighbors are
in the 2nd quintile. The probability of a rich economy remaining
rich is 0.976 if their neighbors are in the 5th quintile, but if their
neighbors are in the 4th quintile this drops to 0.903.
The global transition probability matrix is estimated:
>>> print(sm.p)
[[0.91461837 0.07503234 0.00905563 0.00129366 0. ]
[0.06570302 0.82654402 0.10512484 0.00131406 0.00131406]
[0.00520833 0.10286458 0.79427083 0.09505208 0.00260417]
[0. 0.00913838 0.09399478 0.84856397 0.04830287]
[0. 0. 0. 0.06217617 0.93782383]]
The Q and likelihood ratio statistics are both significant indicating
the dynamics are not homogeneous across the lag classes:
>>> "%.3f"%sm.LR
'170.659'
>>> "%.3f"%sm.Q
'200.624'
>>> "%.3f"%sm.LR_p_value
'0.000'
>>> "%.3f"%sm.Q_p_value
'0.000'
>>> sm.dof_hom
60
The long run distribution for states with poor (rich) neighbors has
0.435 (0.018) of the values in the first quintile, 0.263 (0.200) in
the second quintile, 0.204 (0.190) in the third, 0.0684 (0.255) in the
fourth and 0.029 (0.337) in the fifth quintile.
>>> sm.S
array([[0.43509425, 0.2635327 , 0.20363044, 0.06841983, 0.02932278],
[0.13391287, 0.33993305, 0.25153036, 0.23343016, 0.04119356],
[0.12124869, 0.21137444, 0.2635101 , 0.29013417, 0.1137326 ],
[0.0776413 , 0.19748806, 0.25352636, 0.22480415, 0.24654013],
[0.01776781, 0.19964349, 0.19009833, 0.25524697, 0.3372434 ]])
States with incomes in the first quintile with neighbors in the
first quintile return to the first quartile after 2.298 years, after
leaving the first quintile. They enter the fourth quintile after
80.810 years after leaving the first quintile, on average.
Poor states within neighbors in the fourth quintile return to the
first quintile, on average, after 12.88 years, and would enter the
fourth quintile after 28.473 years.
>>> for f in sm.F:
... print(f)
...
[[ 2.29835259 28.95614035 46.14285714 80.80952381 279.42857143]
[ 33.86549708 3.79459555 22.57142857 57.23809524 255.85714286]
[ 43.60233918 9.73684211 4.91085714 34.66666667 233.28571429]
[ 46.62865497 12.76315789 6.25714286 14.61564626 198.61904762]
[ 52.62865497 18.76315789 12.25714286 6. 34.1031746 ]]
[[ 7.46754205 9.70574606 25.76785714 74.53116883 194.23446197]
[ 27.76691978 2.94175577 24.97142857 73.73474026 193.4380334 ]
[ 53.57477715 28.48447637 3.97566318 48.76331169 168.46660482]
[ 72.03631562 46.94601483 18.46153846 4.28393653 119.70329314]
[ 77.17917276 52.08887197 23.6043956 5.14285714 24.27564033]]
[[ 8.24751154 6.53333333 18.38765432 40.70864198 112.76732026]
[ 47.35040872 4.73094099 11.85432099 34.17530864 106.23398693]
[ 69.42288828 24.76666667 3.794921 22.32098765 94.37966594]
[ 83.72288828 39.06666667 14.3 3.44668119 76.36702977]
[ 93.52288828 48.86666667 24.1 9.8 8.79255406]]
[[ 12.87974382 13.34847151 19.83446328 28.47257282 55.82395142]
[ 99.46114206 5.06359731 10.54545198 23.05133495 49.68944423]
[117.76777159 23.03735526 3.94436301 15.0843986 43.57927247]
[127.89752089 32.4393006 14.56853107 4.44831643 31.63099455]
[138.24752089 42.7893006 24.91853107 10.35 4.05613474]]
[[ 56.2815534 1.5 10.57236842 27.02173913 110.54347826]
[ 82.9223301 5.00892857 9.07236842 25.52173913 109.04347826]
[ 97.17718447 19.53125 5.26043557 21.42391304 104.94565217]
[127.1407767 48.74107143 33.29605263 3.91777427 83.52173913]
[169.6407767 91.24107143 75.79605263 42.5 2.96521739]]
(2) Global quintiles to discretize the income data (k=5), and global
quartiles to discretize the spatial lags of incomes (m=4).
>>> sm = Spatial_Markov(rpci, w, fixed=True, k=5, m=4, variable_name='rpci')
We can also examine the cutoffs for the incomes and cutoffs for the spatial
lags:
>>> sm.cutoffs
array([0.83999133, 0.94707545, 1.03242697, 1.14911154])
>>> sm.lag_cutoffs
array([0.91440247, 0.98583079, 1.08698351])
We now look at the estimated spatially lag conditioned transition
probability matrices.
>>> for p in sm.P:
... print(p)
[[0.95708955 0.03544776 0.00746269 0. 0. ]
[0.05825243 0.83980583 0.10194175 0. 0. ]
[0. 0.1294964 0.76258993 0.10791367 0. ]
[0. 0.01538462 0.18461538 0.72307692 0.07692308]
[0. 0. 0. 0.14285714 0.85714286]]
[[0.7421875 0.234375 0.0234375 0. 0. ]
[0.08550186 0.85130112 0.06319703 0. 0. ]
[0.00865801 0.06926407 0.86147186 0.05627706 0.004329 ]
[0. 0. 0.05363985 0.92337165 0.02298851]
[0. 0. 0. 0.13432836 0.86567164]]
[[0.95145631 0.04854369 0. 0. 0. ]
[0.06 0.79 0.145 0. 0.005 ]
[0.00358423 0.10394265 0.7921147 0.09677419 0.00358423]
[0. 0.01630435 0.13586957 0.75543478 0.0923913 ]
[0. 0. 0. 0.10204082 0.89795918]]
[[0.16666667 0.66666667 0. 0.16666667 0. ]
[0.03488372 0.80232558 0.15116279 0.01162791 0. ]
[0.00840336 0.13445378 0.70588235 0.1512605 0. ]
[0. 0.01171875 0.08203125 0.87109375 0.03515625]
[0. 0. 0. 0.03434343 0.96565657]]
We now obtain 4 5*5 spatial lag conditioned transition probability
matrices instead of 5 as in case (1).
The Q and likelihood ratio statistics are still both significant.
>>> "%.3f"%sm.LR
'172.105'
>>> "%.3f"%sm.Q
'321.128'
>>> "%.3f"%sm.LR_p_value
'0.000'
>>> "%.3f"%sm.Q_p_value
'0.000'
>>> sm.dof_hom
45
(3) We can also set the cutoffs for relative incomes and their
spatial lags manually.
For example, we want the defining cutoffs to be [0.8, 0.9, 1, 1.2],
meaning that relative incomes:
2.1 smaller than 0.8 : class 0
2.2 between 0.8 and 0.9: class 1
2.3 between 0.9 and 1.0 : class 2
2.4 between 1.0 and 1.2: class 3
2.5 larger than 1.2: class 4
>>> cc = np.array([0.8, 0.9, 1, 1.2])
>>> sm = Spatial_Markov(rpci, w, cutoffs=cc, lag_cutoffs=cc, variable_name='rpci')
>>> sm.cutoffs
array([0.8, 0.9, 1. , 1.2])
>>> sm.k
5
>>> sm.lag_cutoffs
array([0.8, 0.9, 1. , 1.2])
>>> sm.m
5
>>> for p in sm.P:
... print(p)
[[0.96703297 0.03296703 0. 0. 0. ]
[0.10638298 0.68085106 0.21276596 0. 0. ]
[0. 0.14285714 0.7755102 0.08163265 0. ]
[0. 0. 0.5 0.5 0. ]
[0. 0. 0. 0. 0. ]]
[[0.88636364 0.10606061 0.00757576 0. 0. ]
[0.04402516 0.89308176 0.06289308 0. 0. ]
[0. 0.05882353 0.8627451 0.07843137 0. ]
[0. 0. 0.13846154 0.86153846 0. ]
[0. 0. 0. 0. 1. ]]
[[0.78082192 0.17808219 0.02739726 0.01369863 0. ]
[0.03488372 0.90406977 0.05813953 0.00290698 0. ]
[0. 0.05919003 0.84735202 0.09034268 0.00311526]
[0. 0. 0.05811623 0.92985972 0.01202405]
[0. 0. 0. 0.14285714 0.85714286]]
[[0.82692308 0.15384615 0. 0.01923077 0. ]
[0.0703125 0.7890625 0.125 0.015625 0. ]
[0.00295858 0.06213018 0.82248521 0.10946746 0.00295858]
[0. 0.00185529 0.07606679 0.88497217 0.03710575]
[0. 0. 0. 0.07803468 0.92196532]]
[[0. 0. 0. 0. 0. ]
[0. 0. 0. 0. 0. ]
[0. 0.06666667 0.9 0.03333333 0. ]
[0. 0. 0.05660377 0.90566038 0.03773585]
[0. 0. 0. 0.03932584 0.96067416]]
(4) Spatial_Markov also accept discrete time series and calculate
categorical spatial lags on which several transition probability matrices
are conditioned.
Let's still use the US state income time series to demonstrate. We first
discretize them into categories and then pass them to Spatial_Markov.
>>> import mapclassify as mc
>>> y = mc.Quantiles(rpci.flatten(), k=5).yb.reshape(rpci.shape)
>>> np.random.seed(5)
>>> sm = Spatial_Markov(y, w, discrete=True, variable_name='discretized rpci')
>>> sm.k
5
>>> sm.m
5
>>> for p in sm.P:
... print(p)
[[0.94787645 0.04440154 0.00772201 0. 0. ]
[0.08333333 0.81060606 0.10606061 0. 0. ]
[0. 0.12765957 0.79787234 0.07446809 0. ]
[0. 0.02777778 0.22222222 0.66666667 0.08333333]
[0. 0. 0. 0.33333333 0.66666667]]
[[0.888 0.096 0.016 0. 0. ]
[0.06049822 0.84341637 0.09608541 0. 0. ]
[0.00666667 0.10666667 0.81333333 0.07333333 0. ]
[0. 0. 0.08527132 0.86821705 0.04651163]
[0. 0. 0. 0.10204082 0.89795918]]
[[0.65217391 0.32608696 0.02173913 0. 0. ]
[0.07446809 0.80851064 0.11170213 0. 0.00531915]
[0.01071429 0.1 0.76428571 0.11785714 0.00714286]
[0. 0.00552486 0.09392265 0.86187845 0.03867403]
[0. 0. 0. 0.13157895 0.86842105]]
[[0.91935484 0.06451613 0. 0.01612903 0. ]
[0.06796117 0.90291262 0.02912621 0. 0. ]
[0. 0.05755396 0.87769784 0.0647482 0. ]
[0. 0.02150538 0.10752688 0.80107527 0.06989247]
[0. 0. 0. 0.08064516 0.91935484]]
[[0.81818182 0.18181818 0. 0. 0. ]
[0.01754386 0.70175439 0.26315789 0.01754386 0. ]
[0. 0.14285714 0.73333333 0.12380952 0. ]
[0. 0.0042735 0.06837607 0.89316239 0.03418803]
[0. 0. 0. 0.03891051 0.96108949]]
"""
def __init__(self, y, w, k=4, m=4, permutations=0, fixed=True,
discrete=False, cutoffs=None, lag_cutoffs=None,
variable_name=None):
y = np.asarray(y)
self.fixed = fixed
self.discrete = discrete
self.cutoffs = cutoffs
self.m = m
self.lag_cutoffs = lag_cutoffs
self.variable_name = variable_name
if discrete:
merged = list(itertools.chain.from_iterable(y))
classes = np.unique(merged)
self.classes = classes
self.k = len(classes)
self.m = self.k
label_dict = dict(zip(classes, range(self.k)))
y_int = []
for yi in y:
y_int.append(list(map(label_dict.get, yi)))
self.class_ids = np.array(y_int)
self.lclass_ids = self.class_ids
else:
self.class_ids, self.cutoffs, self.k = self._maybe_classify(
y, k=k, cutoffs=self.cutoffs)
self.classes = np.arange(self.k)
classic = Markov(self.class_ids)
self.p = classic.p
self.transitions = classic.transitions
self.T, self.P = self._calc(y, w)
if permutations:
nrp = np.random.permutation
counter = 0
x2_realizations = np.zeros((permutations, 1))
for perm in range(permutations):
T, P = self._calc(nrp(y), w)
x2 = [chi2(T[i], self.transitions)[0] for i in range(self.k)]
x2s = sum(x2)
x2_realizations[perm] = x2s
if x2s >= self.x2:
counter += 1
self.x2_rpvalue = (counter + 1.0) / (permutations + 1.)
self.x2_realizations = x2_realizations
@property
def s(self):
if not hasattr(self, '_s'):
self._s = STEADY_STATE(self.p)
return self._s
@property
def S(self):
if not hasattr(self, '_S'):
S = np.zeros_like(self.p)
for i, p in enumerate(self.P):
S[i] = STEADY_STATE(p)
self._S = np.asarray(S)
return self._S
@property
def F(self):
if not hasattr(self, '_F'):
F = np.zeros_like(self.P)
for i, p in enumerate(self.P):
F[i] = fmpt(np.asmatrix(p))
self._F = np.asarray(F)
return self._F
# bickenbach and bode tests
@property
def ht(self):
if not hasattr(self, '_ht'):
self._ht = homogeneity(self.T)
return self._ht
@property
def Q(self):
if not hasattr(self, '_Q'):
self._Q = self.ht.Q
return self._Q
@property
def Q_p_value(self):
self._Q_p_value = self.ht.Q_p_value
return self._Q_p_value
@property
def LR(self):
self._LR = self.ht.LR
return self._LR
@property
def LR_p_value(self):
self._LR_p_value = self.ht.LR_p_value
return self._LR_p_value
@property
def dof_hom(self):
self._dof_hom = self.ht.dof
return self._dof_hom
# shtests
@property
def shtest(self):
if not hasattr(self, '_shtest'):
self._shtest = self._mn_test()
return self._shtest
@property
def chi2(self):
if not hasattr(self, '_chi2'):
self._chi2 = self._chi2_test()
return self._chi2
@property
def x2(self):
if not hasattr(self, '_x2'):
self._x2 = sum([c[0] for c in self.chi2])
return self._x2
@property
def x2_pvalue(self):
if not hasattr(self, '_x2_pvalue'):
self._x2_pvalue = 1 - stats.chi2.cdf(self.x2, self.x2_dof)
return self._x2_pvalue
@property
def x2_dof(self):
if not hasattr(self, '_x2_dof'):
k = self.k
self._x2_dof = k * (k - 1) * (k - 1)
return self._x2_dof
def _calc(self, y, w):
'''Helper to estimate spatial lag conditioned Markov transition
probability matrices based on maximum likelihood techniques.
'''
if self.discrete:
self.lclass_ids = weights.lag_categorical(w, self.class_ids,
ties="tryself")
else:
ly = weights.lag_spatial(w, y)
self.lclass_ids, self.lag_cutoffs, self.m = self._maybe_classify(
ly, self.m, self.lag_cutoffs)
self.lclasses = np.arange(self.m)
T = np.zeros((self.m, self.k, self.k))
n, t = y.shape
for t1 in range(t - 1):
t2 = t1 + 1
for i in range(n):
T[self.lclass_ids[i, t1], self.class_ids[i, t1],
self.class_ids[i, t2]] += 1
P = np.zeros_like(T)
for i, mat in enumerate(T):
row_sum = mat.sum(axis=1)
row_sum = row_sum + (row_sum == 0)
p_i = np.matrix(np.diag(1. / row_sum) * np.matrix(mat))
P[i] = p_i
return T, P
def _mn_test(self):
"""
helper to calculate tests of differences between steady state
distributions from the conditional and overall distributions.
"""
n0, n1, n2 = self.T.shape
rn = list(range(n0))
mat = [self._ssmnp_test(
self.s, self.S[i], self.T[i].sum()) for i in rn]
return mat
def _ssmnp_test(self, p1, p2, nt):
"""
Steady state multinomial probability difference test.
Arguments
---------
p1 : array
(k, ), first steady state probability distribution.
p1 : array
(k, ), second steady state probability distribution.
nt : int
number of transitions to base the test on.
Returns
-------
tuple
(3 elements)
(chi2 value, pvalue, degrees of freedom)
"""
o = nt * p2
e = nt * p1
d = np.multiply((o - e), (o - e))
d = d / e
chi2 = d.sum()
pvalue = 1 - stats.chi2.cdf(chi2, self.k - 1)
return (chi2, pvalue, self.k - 1)
def _chi2_test(self):
"""
helper to calculate tests of differences between the conditional
transition matrices and the overall transitions matrix.
"""
n0, n1, n2 = self.T.shape
rn = list(range(n0))
mat = [chi2(self.T[i], self.transitions) for i in rn]
return mat
def summary(self, file_name=None):
"""
A summary method to call the Markov homogeneity test to test for
temporally lagged spatial dependence.
To learn more about the properties of the tests, refer to
:cite:`Rey2016a` and :cite:`Kang2018`.
"""
class_names = ["C%d" % i for i in range(self.k)]
regime_names = ["LAG%d" % i for i in range(self.k)]
ht = homogeneity(self.T, class_names=class_names,
regime_names=regime_names)
title = "Spatial Markov Test"
if self.variable_name:
title = title + ": " + self.variable_name
if file_name:
ht.summary(file_name=file_name, title=title)
else:
ht.summary(title=title)
def _maybe_classify(self, y, k, cutoffs):
'''Helper method for classifying continuous data.
'''
rows, cols = y.shape
if cutoffs is None:
if self.fixed:
mcyb = mc.Quantiles(y.flatten(), k=k)
yb = mcyb.yb.reshape(y.shape)
cutoffs = mcyb.bins
k = len(cutoffs)
return yb, cutoffs[:-1], k
else:
yb = np.array([mc.Quantiles(y[:, i], k=k).yb for i in
np.arange(cols)]).transpose()
return yb, None, k
else:
cutoffs = list(cutoffs) + [np.inf]
cutoffs = np.array(cutoffs)
yb = mc.User_Defined(y.flatten(), np.array(cutoffs)).yb.reshape(
y.shape)
k = len(cutoffs)
return yb, cutoffs[:-1], k
def chi2(T1, T2):
"""
chi-squared test of difference between two transition matrices.
Parameters
----------
T1 : array
(k, k), matrix of transitions (counts).
T2 : array
(k, k), matrix of transitions (counts) to use to form the
probabilities under the null.
Returns
-------
: tuple
(3 elements).
(chi2 value, pvalue, degrees of freedom).
Examples
--------
>>> import libpysal
>>> from giddy.markov import Spatial_Markov, chi2
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> years = list(range(1929, 2010))
>>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
>>> rpci = pci/(pci.mean(axis=0))
>>> w = libpysal.io.open(libpysal.examples.get_path("states48.gal")).read()
>>> w.transform='r'
>>> sm = Spatial_Markov(rpci, w, fixed=True)
>>> T1 = sm.T[0]
>>> T1
array([[562., 22., 1., 0.],
[ 12., 201., 22., 0.],
[ 0., 17., 97., 4.],
[ 0., 0., 3., 19.]])
>>> T2 = sm.transitions
>>> T2
array([[884., 77., 4., 0.],
[ 68., 794., 87., 3.],
[ 1., 92., 815., 51.],
[ 1., 0., 60., 903.]])
>>> chi2(T1,T2)
(23.39728441473295, 0.005363116704861337, 9)
Notes
-----
Second matrix is used to form the probabilities under the null.
Marginal sums from first matrix are distributed across these probabilities
under the null. In other words the observed transitions are taken from T1
while the expected transitions are formed as follows
.. math::
E_{i,j} = \sum_j T1_{i,j} * T2_{i,j}/\sum_j T2_{i,j}
Degrees of freedom corrected for any rows in either T1 or T2 that have
zero total transitions.
"""
rs2 = T2.sum(axis=1)
rs1 = T1.sum(axis=1)
rs2nz = rs2 > 0
rs1nz = rs1 > 0
dof1 = sum(rs1nz)
dof2 = sum(rs2nz)
rs2 = rs2 + (rs2 == 0)
dof = (dof1 - 1) * (dof2 - 1)
p = np.diag(1 / rs2) * np.matrix(T2)
E = np.diag(rs1) * np.matrix(p)
num = T1 - E
num = np.multiply(num, num)
E = E + (E == 0)
chi2 = num / E
chi2 = chi2.sum()
pvalue = 1 - stats.chi2.cdf(chi2, dof)
return chi2, pvalue, dof
class LISA_Markov(Markov):
"""
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).
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).
.. table:: Move Types
== == =========
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).
.. Table:: Significant Moves1
=============== ===================
(s1,s2) move_type
=============== ===================
(1,1) [1, 16]
(1,0) [17, 32]
(0,1) [33, 48]
(0,0) [49, 64]
=============== ===================
.. Table:: Significant Moves2
== == == == =========
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
(n, 1) binary array, locations that were not part of a
cluster in period 1 but joined a prexisting cluster in
period 2.
Examples
--------
>>> import libpysal
>>> import numpy as np
>>> from giddy.markov import LISA_Markov
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> years = list(range(1929, 2010))
>>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
>>> w = libpysal.io.open(libpysal.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])
"""
def __init__(self, y, w, permutations=0,
significance_level=0.05, geoda_quads=False):
y = y.transpose()
pml = Moran_Local
gq = geoda_quads
ml = ([pml(yi, w, permutations=permutations, geoda_quads=gq)
for yi in y])
q = np.array([mli.q for mli in ml]).transpose()
classes = np.arange(1, 5) # no guarantee all 4 quadrants are visited
Markov.__init__(self, q, classes)
self.q = q
self.w = w
n, k = q.shape
k -= 1
self.significance_level = significance_level
move_types = np.zeros((n, k), int)
sm = np.zeros((n, k), int)
self.significance_level = significance_level
if permutations > 0:
p = np.array([mli.p_z_sim for mli in ml]).transpose()
self.p_values = p
pb = p <= significance_level
else:
pb = np.zeros_like(y.T)
for t in range(k):
origin = q[:, t]
dest = q[:, t + 1]
p_origin = pb[:, t]
p_dest = pb[:, t + 1]
for r in range(n):
move_types[r, t] = TT[origin[r], dest[r]]
key = (origin[r], dest[r], p_origin[r], p_dest[r])
sm[r, t] = MOVE_TYPES[key]
if permutations > 0:
self.significant_moves = sm
self.move_types = move_types
# null of own and lag moves being independent
ybar = y.mean(axis=0)
r = y / ybar
ylag = np.array([weights.lag_spatial(w, yt) for yt in y])
rlag = ylag / ybar
rc = r < 1.
rlagc = rlag < 1.
markov_y = Markov(rc)
markov_ylag = Markov(rlagc)
A = np.matrix([[1, 0, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
[0, 1, 0, 0]])
kp = A * np.kron(markov_y.p, markov_ylag.p) * A.T
trans = self.transitions.sum(axis=1)
t1 = np.diag(trans) * kp
t2 = self.transitions
t1 = t1.getA()
self.chi_2 = chi2(t2, t1)
self.expected_t = t1
self.permutations = permutations
def spillover(self, quadrant=1, neighbors_on=False):
"""
Detect spillover locations for diffusion in LISA Markov.
Parameters
----------
quadrant : int
which quadrant in the scatterplot should form the core
of a cluster.
neighbors_on : binary
If false, then only the 1st order neighbors of a core
location are included in the cluster.
If true, neighbors of cluster core 1st order neighbors
are included in the cluster.
Returns
-------
results : dictionary
two keys - values pairs:
'components' - array (n, t)
values are integer ids (starting at 1) indicating which
component/cluster observation i in period t belonged to.
'spillover' - array (n, t-1)
binary values indicating if the location was a
spill-over location that became a new member of a
previously existing cluster.
Examples
--------
>>> import libpysal
>>> from giddy.markov import LISA_Markov
>>> f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
>>> years = list(range(1929, 2010))
>>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
>>> w = libpysal.io.open(libpysal.examples.get_path("states48.gal")).read()
>>> np.random.seed(10)
>>> lm_random = LISA_Markov(pci, w, permutations=99)
>>> r = lm_random.spillover()
>>> (r['components'][:, 12] > 0).sum()
17
>>> (r['components'][:, 13]>0).sum()
23
>>> (r['spill_over'][:,12]>0).sum()
6
Including neighbors of core neighbors
>>> rn = lm_random.spillover(neighbors_on=True)
>>> (rn['components'][:, 12] > 0).sum()
26
>>> (rn["components"][:, 13] > 0).sum()
34
>>> (rn["spill_over"][:, 12] > 0).sum()
8
"""
n, k = self.q.shape
if self.permutations:
spill_over = np.zeros((n, k - 1))
components = np.zeros((n, k))
i2id = {} # handle string keys
for key in list(self.w.neighbors.keys()):
idx = self.w.id2i[key]
i2id[idx] = key
sig_lisas = (self.q == quadrant) \
* (self.p_values <= self.significance_level)
sig_ids = [np.nonzero(
sig_lisas[:, i])[0].tolist() for i in range(k)]
neighbors = self.w.neighbors
for t in range(k - 1):
s1 = sig_ids[t]
s2 = sig_ids[t + 1]
g1 = Graph(undirected=True)
for i in s1:
for neighbor in neighbors[i2id[i]]:
g1.add_edge(i2id[i], neighbor, 1.0)
if neighbors_on:
for nn in neighbors[neighbor]:
g1.add_edge(neighbor, nn, 1.0)
components1 = g1.connected_components(op=gt)
components1 = [list(c.nodes) for c in components1]
g2 = Graph(undirected=True)
for i in s2:
for neighbor in neighbors[i2id[i]]:
g2.add_edge(i2id[i], neighbor, 1.0)
if neighbors_on:
for nn in neighbors[neighbor]:
g2.add_edge(neighbor, nn, 1.0)
components2 = g2.connected_components(op=gt)
components2 = [list(c.nodes) for c in components2]
c2 = []
c1 = []
for c in components2:
c2.extend(c)
for c in components1:
c1.extend(c)
new_ids = [j for j in c2 if j not in c1]
spill_ids = []
for j in new_ids:
# find j's component in period 2
cj = [c for c in components2 if j in c][0]
# for members of j's component in period 2, check if they
# belonged to any components in period 1
for i in cj:
if i in c1:
spill_ids.append(j)
break
for spill_id in spill_ids:
id = self.w.id2i[spill_id]
spill_over[id, t] = 1
for c, component in enumerate(components1):
for i in component:
ii = self.w.id2i[i]
components[ii, t] = c + 1
results = {}
results['components'] = components
results['spill_over'] = spill_over
return results
else:
return None
def kullback(F):
"""
Kullback information based test of Markov Homogeneity.
Parameters
----------
F : array
(s, r, r), values are transitions (not probabilities) for
s strata, r initial states, r terminal states.
Returns
-------
Results : dictionary
(key - value)
Conditional homogeneity - (float) test statistic for homogeneity
of transition probabilities across strata.
Conditional homogeneity pvalue - (float) p-value for test
statistic.
Conditional homogeneity dof - (int) degrees of freedom =
r(s-1)(r-1).
Notes
-----
Based on :cite:`Kullback1962`.
Example below is taken from Table 9.2 .
Examples
--------
>>> import numpy as np
>>> from giddy.markov import kullback
>>> s1 = np.array([
... [ 22, 11, 24, 2, 2, 7],
... [ 5, 23, 15, 3, 42, 6],
... [ 4, 21, 190, 25, 20, 34],
... [0, 2, 14, 56, 14, 28],
... [32, 15, 20, 10, 56, 14],
... [5, 22, 31, 18, 13, 134]
... ])
>>> s2 = np.array([
... [3, 6, 9, 3, 0, 8],
... [1, 9, 3, 12, 27, 5],
... [2, 9, 208, 32, 5, 18],
... [0, 14, 32, 108, 40, 40],
... [22, 14, 9, 26, 224, 14],
... [1, 5, 13, 53, 13, 116]
... ])
>>>
>>> F = np.array([s1, s2])
>>> res = kullback(F)
>>> "%8.3f"%res['Conditional homogeneity']
' 160.961'
>>> "%d"%res['Conditional homogeneity dof']
'30'
>>> "%3.1f"%res['Conditional homogeneity pvalue']
'0.0'
"""
F1 = F == 0
F1 = F + F1
FLF = F * np.log(F1)
T1 = 2 * FLF.sum()
FdJK = F.sum(axis=0)
FdJK1 = FdJK + (FdJK == 0)
FdJKLFdJK = FdJK * np.log(FdJK1)
T2 = 2 * FdJKLFdJK.sum()
FdJd = F.sum(axis=0).sum(axis=1)
FdJd1 = FdJd + (FdJd == 0)
T3 = 2 * (FdJd * np.log(FdJd1)).sum()
FIJd = F[:, :].sum(axis=1)
FIJd1 = FIJd + (FIJd == 0)
T4 = 2 * (FIJd * np.log(FIJd1)).sum()
T6 = F.sum()
T6 = 2 * T6 * np.log(T6)
s, r, r1 = F.shape
chom = T1 - T4 - T2 + T3
cdof = r * (s - 1) * (r - 1)
results = {}
results['Conditional homogeneity'] = chom
results['Conditional homogeneity dof'] = cdof
results['Conditional homogeneity pvalue'] = 1 - stats.chi2.cdf(chom, cdof)
return results
def prais(pmat):
"""
Prais conditional mobility measure.
Parameters
----------
pmat : matrix
(k, k), Markov probability transition matrix.
Returns
-------
pr : matrix
(1, k), conditional mobility measures for each of the k classes.
Notes
-----
Prais' conditional mobility measure for a class is defined as:
.. math::
pr_i = 1 - p_{i,i}
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> from giddy.markov import Markov,prais
>>> 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)
>>> m.transitions
array([[729., 71., 1., 0., 0.],
[ 72., 567., 80., 3., 0.],
[ 0., 81., 631., 86., 2.],
[ 0., 3., 86., 573., 56.],
[ 0., 0., 1., 57., 741.]])
>>> 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]])
>>> prais(m.p)
array([0.08988764, 0.21468144, 0.21125 , 0.20194986, 0.07259074])
"""
pmat = np.array(pmat)
pr = 1 - np.diag(pmat)
return pr
def homogeneity(transition_matrices, regime_names=[], class_names=[],
title="Markov Homogeneity Test"):
"""
Test for homogeneity of Markov transition probabilities across regimes.
Parameters
----------
transition_matrices : list
of transition matrices for regimes, all matrices must
have same size (r, c). r is the number of rows in the
transition matrix and c is the number of columns in
the transition matrix.
regime_names : sequence
Labels for the regimes.
class_names : sequence
Labels for the classes/states of the Markov chain.
title : string
name of test.
Returns
-------
: implicit
an instance of Homogeneity_Results.
"""
return Homogeneity_Results(transition_matrices, regime_names=regime_names,
class_names=class_names, title=title)
class Homogeneity_Results:
"""
Wrapper class to present homogeneity results.
Parameters
----------
transition_matrices : list
of transition matrices for regimes, all matrices must
have same size (r, c). r is the number of rows in
the transition matrix and c is the number of columns
in the transition matrix.
regime_names : sequence
Labels for the regimes.
class_names : sequence
Labels for the classes/states of the Markov chain.
title : string
Title of the table.
Attributes
-----------
Notes
-----
Degrees of freedom adjustment follow the approach in :cite:`Bickenbach2003`.
Examples
--------
See Spatial_Markov above.
"""
def __init__(self, transition_matrices, regime_names=[], class_names=[],
title="Markov Homogeneity Test"):
self._homogeneity(transition_matrices)
self.regime_names = regime_names
self.class_names = class_names
self.title = title
def _homogeneity(self, transition_matrices):
# form null transition probability matrix
M = np.array(transition_matrices)
m, r, k = M.shape
self.k = k
B = np.zeros((r, m))
T = M.sum(axis=0)
self.t_total = T.sum()
n_i = T.sum(axis=1)
A_i = (T > 0).sum(axis=1)
A_im = np.zeros((r, m))
p_ij = np.dot(np.diag(1. / (n_i + (n_i == 0) * 1.)), T)
den = p_ij + 1. * (p_ij == 0)
b_i = np.zeros_like(A_i)
p_ijm = np.zeros_like(M)
# get dimensions
m, n_rows, n_cols = M.shape
m = 0
Q = 0.0
LR = 0.0
lr_table = np.zeros_like(M)
q_table = np.zeros_like(M)
for nijm in M:
nim = nijm.sum(axis=1)
B[:, m] = 1. * (nim > 0)
b_i = b_i + 1. * (nim > 0)
p_ijm[m] = np.dot(np.diag(1. / (nim + (nim == 0) * 1.)), nijm)
num = (p_ijm[m] - p_ij)**2
ratio = num / den
qijm = np.dot(np.diag(nim), ratio)
q_table[m] = qijm
Q = Q + qijm.sum()
# only use nonzero pijm in lr test
mask = (nijm > 0) * (p_ij > 0)
A_im[:, m] = (nijm > 0).sum(axis=1)
unmask = 1.0 * (mask == 0)
ratio = (mask * p_ijm[m] + unmask) / (mask * p_ij + unmask)
lr = nijm * np.log(ratio)
LR = LR + lr.sum()
lr_table[m] = 2 * lr
m += 1
# b_i is the number of regimes that have non-zero observations in row i
# A_i is the number of non-zero elements in row i of the aggregated
# transition matrix
self.dof = int(((b_i - 1) * (A_i - 1)).sum())
self.Q = Q
self.Q_p_value = 1 - stats.chi2.cdf(self.Q, self.dof)
self.LR = LR * 2.
self.LR_p_value = 1 - stats.chi2.cdf(self.LR, self.dof)
self.A = A_i
self.A_im = A_im
self.B = B
self.b_i = b_i
self.LR_table = lr_table
self.Q_table = q_table
self.m = m
self.p_h0 = p_ij
self.p_h1 = p_ijm
def summary(self, file_name=None, title="Markov Homogeneity Test"):
regime_names = ["%d" % i for i in range(self.m)]
if self.regime_names:
regime_names = self.regime_names
cols = ["P(%s)" % str(regime) for regime in regime_names]
if not self.class_names:
self.class_names = list(range(self.k))
max_col = max([len(col) for col in cols])
col_width = max([5, max_col]) # probabilities have 5 chars
n_tabs = self.k
width = n_tabs * 4 + (self.k + 1) * col_width
lead = "-" * width
head = title.center(width)
contents = [lead, head, lead]
l = "Number of regimes: %d" % int(self.m)
k = "Number of classes: %d" % int(self.k)
r = "Regime names: "
r += ", ".join(regime_names)
t = "Number of transitions: %d" % int(self.t_total)
contents.append(k)
contents.append(t)
contents.append(l)
contents.append(r)
contents.append(lead)
h = "%7s %20s %20s" % ('Test', 'LR', 'Chi-2')
contents.append(h)
stat = "%7s %20.3f %20.3f" % ('Stat.', self.LR, self.Q)
contents.append(stat)
stat = "%7s %20d %20d" % ('DOF', self.dof, self.dof)
contents.append(stat)
stat = "%7s %20.3f %20.3f" % ('p-value', self.LR_p_value,
self.Q_p_value)
contents.append(stat)
print(("\n".join(contents)))
print(lead)
cols = ["P(%s)" % str(regime) for regime in self.regime_names]
if not self.class_names:
self.class_names = list(range(self.k))
cols.extend(["%s" % str(cname) for cname in self.class_names])
max_col = max([len(col) for col in cols])
col_width = max([5, max_col]) # probabilities have 5 chars
p0 = []
line0 = ['{s: <{w}}'.format(s="P(H0)", w=col_width)]
line0.extend((['{s: >{w}}'.format(s=cname, w=col_width) for cname in
self.class_names]))
print((" ".join(line0)))
p0.append("&".join(line0))
for i, row in enumerate(self.p_h0):
line = ["%*s" % (col_width, str(self.class_names[i]))]
line.extend(["%*.3f" % (col_width, v) for v in row])
print((" ".join(line)))
p0.append("&".join(line))
pmats = [p0]
print(lead)
for r, p1 in enumerate(self.p_h1):
p0 = []
line0 = ['{s: <{w}}'.format(s="P(%s)" %
regime_names[r], w=col_width)]
line0.extend((['{s: >{w}}'.format(s=cname, w=col_width) for cname
in self.class_names]))
print((" ".join(line0)))
p0.append("&".join(line0))
for i, row in enumerate(p1):
line = ["%*s" % (col_width, str(self.class_names[i]))]
line.extend(["%*.3f" % (col_width, v) for v in row])
print((" ".join(line)))
p0.append("&".join(line))
pmats.append(p0)
print(lead)
if file_name:
k = self.k
ks = str(k + 1)
with open(file_name, 'w') as f:
c = []
fmt = "r" * (k + 1)
s = "\\begin{tabular}{|%s|}\\hline\n" % fmt
s += "\\multicolumn{%s}{|c|}{%s}" % (ks, title)
c.append(s)
s = "Number of classes: %d" % int(self.k)
c.append("\\hline\\multicolumn{%s}{|l|}{%s}" % (ks, s))
s = "Number of transitions: %d" % int(self.t_total)
c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
s = "Number of regimes: %d" % int(self.m)
c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
s = "Regime names: "
s += ", ".join(regime_names)
c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
s = "\\hline\\multicolumn{2}{|l}{%s}" % ("Test")
s += "&\\multicolumn{2}{r}{LR}&\\multicolumn{2}{r|}{Q}"
c.append(s)
s = "Stat."
s = "\\multicolumn{2}{|l}{%s}" % (s)
s += "&\\multicolumn{2}{r}{%.3f}" % self.LR
s += "&\\multicolumn{2}{r|}{%.3f}" % self.Q
c.append(s)
s = "\\multicolumn{2}{|l}{%s}" % ("DOF")
s += "&\\multicolumn{2}{r}{%d}" % int(self.dof)
s += "&\\multicolumn{2}{r|}{%d}" % int(self.dof)
c.append(s)
s = "\\multicolumn{2}{|l}{%s}" % ("p-value")
s += "&\\multicolumn{2}{r}{%.3f}" % self.LR_p_value
s += "&\\multicolumn{2}{r|}{%.3f}" % self.Q_p_value
c.append(s)
s1 = "\\\\\n".join(c)
s1 += "\\\\\n"
c = []
for mat in pmats:
c.append("\\hline\n")
for row in mat:
c.append(row + "\\\\\n")
c.append("\\hline\n")
c.append("\\end{tabular}")
s2 = "".join(c)
f.write(s1 + s2)
class FullRank_Markov:
"""
Full Rank Markov in which ranks are considered as Markov states rather
than quantiles or other discretized classes. This is one way to avoid
issues associated with discretization.
Parameters
----------
y : array
(n, t) with t>>n, one row per observation (n total),
one column recording the value of each observation,
with as many columns as time periods.
Attributes
----------
ranks : array
ranks of the original y array (by columns): higher values
rank higher, e.g. the largest value in a column ranks 1.
p : array
(n, n), transition probability matrix for Full
Rank Markov.
steady_state : array
(n, ), ergodic distribution.
transitions : array
(n, n), count of transitions between each rank i and j
fmpt : array
(n, n), first mean passage times.
sojourn_time : array
(n, ), sojourn times.
Notes
-----
Refer to :cite:`Rey2014a` Equation (11) for details. Ties are resolved by
assigning distinct ranks, corresponding to the order that the values occur
in each cross section.
Examples
--------
US nominal per capita income 48 states 81 years 1929-2009
>>> from giddy.markov import FullRank_Markov
>>> import libpysal as ps
>>> import numpy as np
>>> f = ps.io.open(ps.examples.get_path("usjoin.csv"))
>>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)]).transpose()
>>> m = FullRank_Markov(pci)
>>> m.ranks
array([[45, 45, 44, ..., 41, 40, 39],
[24, 25, 25, ..., 36, 38, 41],
[46, 47, 45, ..., 43, 43, 43],
...,
[34, 34, 34, ..., 47, 46, 42],
[17, 17, 22, ..., 25, 26, 25],
[16, 18, 19, ..., 6, 6, 7]])
>>> m.transitions
array([[66., 5., 5., ..., 0., 0., 0.],
[ 8., 51., 9., ..., 0., 0., 0.],
[ 2., 13., 44., ..., 0., 0., 0.],
...,
[ 0., 0., 0., ..., 40., 17., 0.],
[ 0., 0., 0., ..., 15., 54., 2.],
[ 0., 0., 0., ..., 2., 1., 77.]])
>>> m.p[0, :5]
array([0.825 , 0.0625, 0.0625, 0.025 , 0.025 ])
>>> m.fmpt[0, :5]
array([48. , 87.96280048, 68.1089084 , 58.83306575, 41.77250827])
>>> m.sojourn_time[:5]
array([5.71428571, 2.75862069, 2.22222222, 1.77777778, 1.66666667])
"""
def __init__(self, y):
y = np.asarray(y)
# resolve ties: All values are given a distinct rank, corresponding
# to the order that the values occur in each cross section.
r_asc = np.array([rankdata(col, method='ordinal') for col in y.T]).T
# ranks by high (1) to low (n)
self.ranks = r_asc.shape[0] - r_asc + 1
frm = Markov(self.ranks)
self.p = frm.p
self.transitions = frm.transitions
@property
def steady_state(self):
if not hasattr(self, '_steady_state'):
self._steady_state = STEADY_STATE(self.p)
return self._steady_state
@property
def fmpt(self):
if not hasattr(self, '_fmpt'):
self._fmpt = fmpt(self.p)
return self._fmpt
@property
def sojourn_time(self):
if not hasattr(self, '_st'):
self._st = sojourn_time(self.p)
return self._st
def sojourn_time(p):
"""
Calculate sojourn time based on a given transition probability matrix.
Parameters
----------
p : array
(k, k), a Markov transition probability matrix.
Returns
-------
: array
(k, ), sojourn times. Each element is the expected time a Markov
chain spends in each states before leaving that state.
Notes
-----
Refer to :cite:`Ibe2009` for more details on sojourn times for Markov
chains.
Examples
--------
>>> from giddy.markov import sojourn_time
>>> import numpy as np
>>> p = np.array([[.5, .25, .25], [.5, 0, .5], [.25, .25, .5]])
>>> sojourn_time(p)
array([2., 1., 2.])
"""
p = np.asarray(p)
pii = p.diagonal()
if not (1 - pii).all():
print("Sojourn times are infinite for absorbing states!")
return 1 / (1 - pii)
class GeoRank_Markov:
"""
Geographic Rank Markov.
Geographic units are considered as Markov states.
Parameters
----------
y : array
(n, t) with t>>n, one row per observation (n total),
one column recording the value of each observation,
with as many columns as time periods.
Attributes
----------
p : array
(n, n), transition probability matrix for
geographic rank Markov.
steady_state : array
(n, ), ergodic distribution.
transitions : array
(n, n), count of rank transitions between each
geographic unit i and j.
fmpt : array
(n, n), first mean passage times.
sojourn_time : array
(n, ), sojourn times.
Notes
-----
Refer to :cite:`Rey2014a` Equation (13)-(16) for details. Ties are
resolved by assigning distinct ranks, corresponding to the order
that the values occur in each cross section.
Examples
--------
US nominal per capita income 48 states 81 years 1929-2009
>>> from giddy.markov import GeoRank_Markov
>>> import libpysal as ps
>>> import numpy as np
>>> f = ps.io.open(ps.examples.get_path("usjoin.csv"))
>>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)]).transpose()
>>> m = GeoRank_Markov(pci)
>>> m.transitions
array([[38., 0., 8., ..., 0., 0., 0.],
[ 0., 15., 0., ..., 0., 1., 0.],
[ 6., 0., 44., ..., 5., 0., 0.],
...,
[ 2., 0., 5., ..., 34., 0., 0.],
[ 0., 0., 0., ..., 0., 18., 2.],
[ 0., 0., 0., ..., 0., 3., 14.]])
>>> m.p
array([[0.475 , 0. , 0.1 , ..., 0. , 0. , 0. ],
[0. , 0.1875, 0. , ..., 0. , 0.0125, 0. ],
[0.075 , 0. , 0.55 , ..., 0.0625, 0. , 0. ],
...,
[0.025 , 0. , 0.0625, ..., 0.425 , 0. , 0. ],
[0. , 0. , 0. , ..., 0. , 0.225 , 0.025 ],
[0. , 0. , 0. , ..., 0. , 0.0375, 0.175 ]])
>>> m.fmpt
array([[ 48. , 63.35532038, 92.75274652, ..., 82.47515731,
71.01114491, 68.65737127],
[108.25928005, 48. , 127.99032986, ..., 92.03098299,
63.36652935, 61.82733039],
[ 76.96801786, 64.7713783 , 48. , ..., 73.84595169,
72.24682723, 69.77497173],
...,
[ 93.3107474 , 62.47670463, 105.80634118, ..., 48. ,
69.30121319, 67.08838421],
[113.65278078, 61.1987031 , 133.57991745, ..., 96.0103924 ,
48. , 56.74165107],
[114.71894813, 63.4019776 , 134.73381719, ..., 97.287895 ,
61.45565054, 48. ]])
>>> m.sojourn_time
array([ 1.9047619 , 1.23076923, 2.22222222, 1.73913043, 1.15942029,
3.80952381, 1.70212766, 1.25 , 1.31147541, 1.11111111,
1.73913043, 1.37931034, 1.17647059, 1.21212121, 1.33333333,
1.37931034, 1.09589041, 2.10526316, 2. , 1.45454545,
1.26984127, 26.66666667, 1.19402985, 1.23076923, 1.09589041,
1.56862745, 1.26984127, 2.42424242, 1.50943396, 2. ,
1.29032258, 1.09589041, 1.6 , 1.42857143, 1.25 ,
1.45454545, 1.29032258, 1.6 , 1.17647059, 1.56862745,
1.25 , 1.37931034, 1.45454545, 1.42857143, 1.29032258,
1.73913043, 1.29032258, 1.21212121])
"""
def __init__(self, y):
y = np.asarray(y)
n = y.shape[0]
# resolve ties: All values are given a distinct rank, corresponding
# to the order that the values occur in each cross section.
ranks = np.array([rankdata(col, method='ordinal') for col in y.T]).T
geo_ranks = np.argsort(ranks, axis=0) + 1
grm = Markov(geo_ranks)
self.p = grm.p
self.transitions = grm.transitions
@property
def steady_state(self):
if not hasattr(self, '_steady_state'):
self._steady_state = STEADY_STATE(self.p)
return self._steady_state
@property
def fmpt(self):
if not hasattr(self, '_fmpt'):
self._fmpt = fmpt(self.p)
return self._fmpt
@property
def sojourn_time(self):
if not hasattr(self, '_st'):
self._st = sojourn_time(self.p)
return self._st
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