"""
Spatial diagnostics module
"""
__author__ = "Luc Anselin luc.anselin@asu.edu, Daniel Arribas-Bel darribas@asu.edu"
from .utils import spdot
#from scipy.stats.stats import chisqprob
from scipy import stats
#stats.chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)
chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)
from scipy.stats import norm
import numpy as np
import numpy.linalg as la
__all__ = ['LMtests', 'MoranRes', 'AKtest']
[docs]class LMtests:
"""
Lagrange Multiplier tests. Implemented as presented in :cite:`Anselin1996a`
Attributes
----------
ols : OLS
OLS regression object
w : W
Spatial weights instance
tests : list
Lists of strings with the tests desired to be performed.
Values may be:
* 'all': runs all the options (default)
* 'lme': LM error test
* 'rlme': Robust LM error test
* 'lml' : LM lag test
* 'rlml': Robust LM lag test
Parameters
----------
lme : tuple
(Only if 'lme' or 'all' was in tests). Pair of statistic and
p-value for the LM error test.
lml : tuple
(Only if 'lml' or 'all' was in tests). Pair of statistic and
p-value for the LM lag test.
rlme : tuple
(Only if 'rlme' or 'all' was in tests). Pair of statistic
and p-value for the Robust LM error test.
rlml : tuple
(Only if 'rlml' or 'all' was in tests). Pair of statistic
and p-value for the Robust LM lag test.
sarma : tuple
(Only if 'rlml' or 'all' was in tests). Pair of statistic
and p-value for the SARMA test.
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> from spreg import OLS
>>> import spreg
Open the csv file to access the data for analysis
>>> csv = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
Pull out from the csv the files we need ('HOVAL' as dependent as well as
'INC' and 'CRIME' as independent) and directly transform them into nx1 and
nx2 arrays, respectively
>>> y = np.array([csv.by_col('HOVAL')]).T
>>> x = np.array([csv.by_col('INC'), csv.by_col('CRIME')]).T
Create the weights object from existing .gal file
>>> w = libpysal.io.open(libpysal.examples.get_path('columbus.gal'), 'r').read()
Row-standardize the weight object (not required although desirable in some
cases)
>>> w.transform='r'
Run an OLS regression
>>> ols = OLS(y, x)
Run all the LM tests in the residuals. These diagnostics test for the
presence of remaining spatial autocorrelation in the residuals of an OLS
model and give indication about the type of spatial model. There are five
types: presence of a spatial lag model (simple and robust version),
presence of a spatial error model (simple and robust version) and joint presence
of both a spatial lag as well as a spatial error model.
>>> lms = spreg.LMtests(ols, w)
LM error test:
>>> print(round(lms.lme[0],4), round(lms.lme[1],4))
3.0971 0.0784
LM lag test:
>>> print(round(lms.lml[0],4), round(lms.lml[1],4))
0.9816 0.3218
Robust LM error test:
>>> print(round(lms.rlme[0],4), round(lms.rlme[1],4))
3.2092 0.0732
Robust LM lag test:
>>> print(round(lms.rlml[0],4), round(lms.rlml[1],4))
1.0936 0.2957
LM SARMA test:
>>> print(round(lms.sarma[0],4), round(lms.sarma[1],4))
4.1907 0.123
"""
[docs] def __init__(self, ols, w, tests=['all']):
cache = spDcache(ols, w)
if tests == ['all']:
tests = ['lme', 'lml', 'rlme', 'rlml', 'sarma']
if 'lme' in tests:
self.lme = lmErr(ols, w, cache)
if 'lml' in tests:
self.lml = lmLag(ols, w, cache)
if 'rlme' in tests:
self.rlme = rlmErr(ols, w, cache)
if 'rlml' in tests:
self.rlml = rlmLag(ols, w, cache)
if 'sarma' in tests:
self.sarma = lmSarma(ols, w, cache)
[docs]class MoranRes:
"""
Moran's I for spatial autocorrelation in residuals from OLS regression
Parameters
----------
ols : OLS
OLS regression object
w : W
Spatial weights instance
z : boolean
If set to True computes attributes eI, vI and zI. Due to computational burden of vI, defaults to False.
Attributes
----------
I : float
Moran's I statistic
eI : float
Moran's I expectation
vI : float
Moran's I variance
zI : float
Moran's I standardized value
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> from spreg import OLS
>>> import spreg
Open the csv file to access the data for analysis
>>> csv = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
Pull out from the csv the files we need ('HOVAL' as dependent as well as
'INC' and 'CRIME' as independent) and directly transform them into nx1 and
nx2 arrays, respectively
>>> y = np.array([csv.by_col('HOVAL')]).T
>>> x = np.array([csv.by_col('INC'), csv.by_col('CRIME')]).T
Create the weights object from existing .gal file
>>> w = libpysal.io.open(libpysal.examples.get_path('columbus.gal'), 'r').read()
Row-standardize the weight object (not required although desirable in some
cases)
>>> w.transform='r'
Run an OLS regression
>>> ols = OLS(y, x)
Run Moran's I test for residual spatial autocorrelation in an OLS model.
This computes the traditional statistic applying a correction in the
expectation and variance to account for the fact it comes from residuals
instead of an independent variable
>>> m = spreg.MoranRes(ols, w, z=True)
Value of the Moran's I statistic:
>>> print(round(m.I,4))
0.1713
Value of the Moran's I expectation:
>>> print(round(m.eI,4))
-0.0345
Value of the Moran's I variance:
>>> print(round(m.vI,4))
0.0081
Value of the Moran's I standardized value. This is
distributed as a standard Normal(0, 1)
>>> print(round(m.zI,4))
2.2827
P-value of the standardized Moran's I value (z):
>>> print(round(m.p_norm,4))
0.0224
"""
[docs] def __init__(self, ols, w, z=False):
cache = spDcache(ols, w)
self.I = get_mI(ols, w, cache)
if z:
self.eI = get_eI(ols, w, cache)
self.vI = get_vI(ols, w, self.eI, cache)
self.zI, self.p_norm = get_zI(self.I, self.eI, self.vI)
[docs]class AKtest:
"""
Moran's I test of spatial autocorrelation for IV estimation.
Implemented following the original reference :cite:`Anselin1997`
Parameters
----------
iv : TSLS
Regression object from TSLS class
w : W
Spatial weights instance
case : string
Flag for special cases (default to 'nosp'):
* 'nosp': Only NO spatial end. reg.
* 'gen': General case (spatial lag + end. reg.)
Attributes
----------
mi : float
Moran's I statistic for IV residuals
ak : float
Square of corrected Moran's I for residuals
:math:`ak = \dfrac{N \times I^*}{\phi^2}`.
Note: if case='nosp' then it simplifies to the LMerror
p : float
P-value of the test
Examples
--------
We first need to import the needed modules. Numpy is needed to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis. The TSLS is required to run the model on
which we will perform the tests.
>>> import numpy as np
>>> import libpysal
>>> from spreg import TSLS, GM_Lag, AKtest
Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open().
This is the DBF associated with the Columbus shapefile. Note that
libpysal.io.open() also reads data in CSV format; since the actual class
requires data to be passed in as numpy arrays, the user can read their
data in using any method.
>>> db = libpysal.io.open(libpysal.examples.get_path("columbus.dbf"),'r')
Before being able to apply the diagnostics, we have to run a model and,
for that, we need the input variables. Extract the CRIME column (crime
rates) from the DBF file and make it the dependent variable for the
regression. Note that PySAL requires this to be an numpy array of shape
(n, 1) as opposed to the also common shape of (n, ) that other packages
accept.
>>> y = np.array(db.by_col("CRIME"))
>>> y = np.reshape(y, (49,1))
Extract INC (income) vector from the DBF to be used as
independent variables in the regression. Note that PySAL requires this to
be an nxj numpy array, where j is the number of independent variables (not
including a constant). By default this model adds a vector of ones to the
independent variables passed in, but this can be overridden by passing
constant=False.
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
In this case, we consider HOVAL (home value) as an endogenous regressor,
so we acknowledge that by reading it in a different category.
>>> yd = []
>>> yd.append(db.by_col("HOVAL"))
>>> yd = np.array(yd).T
In order to properly account for the endogeneity, we have to pass in the
instruments. Let us consider DISCBD (distance to the CBD) is a good one:
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
Now we are good to run the model. It is an easy one line task.
>>> reg = TSLS(y, X, yd, q=q)
Now we are concerned with whether our non-spatial model presents spatial
autocorrelation in the residuals. To assess this possibility, we can run
the Anselin-Kelejian test, which is a version of the classical LM error
test adapted for the case of residuals from an instrumental variables (IV)
regression. First we need an extra object, the weights matrix, which
includes the spatial configuration of the observations
into the error component of the model. To do that, we can open an already
existing gal file or create a new one. In this case, we will create one
from ``columbus.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
Unless there is a good reason not to do it, the weights have to be
row-standardized so every row of the matrix sums to one. Among other
things, this allows to interpret the spatial lag of a variable as the
average value of the neighboring observations. In PySAL, this can be
easily performed in the following way:
>>> w.transform = 'r'
We are good to run the test. It is a very simple task:
>>> ak = AKtest(reg, w)
And explore the information obtained:
>>> print('AK test: %f\tP-value: %f'%(ak.ak, ak.p))
AK test: 4.642895 P-value: 0.031182
The test also accomodates the case when the residuals come from an IV
regression that includes a spatial lag of the dependent variable. The only
requirement needed is to modify the ``case`` parameter when we call
``AKtest``. First, let us run a spatial lag model:
>>> reg_lag = GM_Lag(y, X, yd, q=q, w=w)
And now we can run the AK test and obtain similar information as in the
non-spatial model.
>>> ak_sp = AKtest(reg, w, case='gen')
>>> print('AK test: %f\tP-value: %f'%(ak_sp.ak, ak_sp.p))
AK test: 1.157593 P-value: 0.281965
"""
[docs] def __init__(self, iv, w, case='nosp'):
if case == 'gen':
cache = spDcache(iv, w)
self.mi, self.ak, self.p = akTest(iv, w, cache)
elif case == 'nosp':
cache = spDcache(iv, w)
self.mi = get_mI(iv, w, cache)
self.ak, self.p = lmErr(iv, w, cache)
else:
print("""\n
Fix the optional argument 'case' to match the requirements:
* 'gen': General case (spatial lag + end. reg.)
* 'nosp': No spatial end. reg.
\n""")
class spDcache:
"""
Helper class to compute reusable pieces in the spatial diagnostics module
...
Parameters
----------
reg : OLS_dev, TSLS_dev, STSLS_dev
Instance from a regression class
w : W
Spatial weights instance
Attributes
----------
j : array
1x1 array with the result from:
:math:`J = \dfrac{1}{[(WX\beta)' M (WX\beta) + T \sigma^2]}`
wu : array
nx1 array with spatial lag of the residuals
utwuDs : array
1x1 array with the result from:
:math:`utwuDs = \dfrac{u' W u}{\tilde{\sigma^2}}`
utwyDs : array
1x1 array with the result from:
:math:`utwyDs = \dfrac{u' W y}{\tilde{\sigma^2}}`
t : array
1x1 array with the result from :
:math:` T = tr[(W' + W) W]`
trA : float
Trace of A as in Cliff & Ord (1981)
"""
def __init__(self, reg, w):
self.reg = reg
self.w = w
self._cache = {}
@property
def j(self):
if 'j' not in self._cache:
wxb = self.w.sparse * self.reg.predy
wxb2 = np.dot(wxb.T, wxb)
xwxb = spdot(self.reg.x.T, wxb)
num1 = wxb2 - np.dot(xwxb.T, np.dot(self.reg.xtxi, xwxb))
num = num1 + (self.t * self.reg.sig2n)
den = self.reg.n * self.reg.sig2n
self._cache['j'] = num / den
return self._cache['j']
@property
def wu(self):
if 'wu' not in self._cache:
self._cache['wu'] = self.w.sparse * self.reg.u
return self._cache['wu']
@property
def utwuDs(self):
if 'utwuDs' not in self._cache:
res = np.dot(self.reg.u.T, self.wu) / self.reg.sig2n
self._cache['utwuDs'] = res
return self._cache['utwuDs']
@property
def utwyDs(self):
if 'utwyDs' not in self._cache:
res = np.dot(self.reg.u.T, self.w.sparse * self.reg.y)
self._cache['utwyDs'] = res / self.reg.sig2n
return self._cache['utwyDs']
@property
def t(self):
if 't' not in self._cache:
prod = (self.w.sparse.T + self.w.sparse) * self.w.sparse
self._cache['t'] = np.sum(prod.diagonal())
return self._cache['t']
@property
def trA(self):
if 'trA' not in self._cache:
xtwx = spdot(self.reg.x.T, spdot(self.w.sparse, self.reg.x))
mw = np.dot(self.reg.xtxi, xtwx)
self._cache['trA'] = np.sum(mw.diagonal())
return self._cache['trA']
@property
def AB(self):
"""
Computes A and B matrices as in Cliff-Ord 1981, p. 203
"""
if 'AB' not in self._cache:
U = (self.w.sparse + self.w.sparse.T) / 2.
z = spdot(U, self.reg.x, array_out=False)
c1 = spdot(self.reg.x.T, z, array_out=False)
c2 = spdot(z.T, z, array_out=False)
G = self.reg.xtxi
A = spdot(G, c1)
B = spdot(G, c2)
self._cache['AB'] = [A, B]
return self._cache['AB']
def lmErr(reg, w, spDcache):
"""
LM error test. Implemented as presented in eq. (9) of Anselin et al.
(1996) :cite:`Anselin1996a`.
Attributes
----------
reg : OLS_dev, TSLS_dev, STSLS_dev
Instance from a regression class
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
lme : tuple
Pair of statistic and p-value for the LM error test.
"""
lm = spDcache.utwuDs ** 2 / spDcache.t
pval = chisqprob(lm, 1)
return (lm[0][0], pval[0][0])
def lmLag(ols, w, spDcache):
"""
LM lag test. Implemented as presented in eq. (13) of Anselin et al.
(1996) :cite:`Anselin1996a`.
Attributes
----------
ols : OLS_dev
Instance from an OLS_dev regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
lml : tuple
Pair of statistic and p-value for the LM lag test.
"""
lm = spDcache.utwyDs ** 2 / (ols.n * spDcache.j)
pval = chisqprob(lm, 1)
return (lm[0][0], pval[0][0])
def rlmErr(ols, w, spDcache):
"""
Robust LM error test. Implemented as presented in eq. (8) of Anselin et
al. (1996) :cite:`Anselin1996a`.
NOTE: eq. (8) has an errata, the power -1 in the denominator
should be inside the square bracket.
Attributes
----------
ols : OLS_dev
Instance from an OLS_dev regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
rlme : tuple
Pair of statistic and p-value for the Robust LM error test.
"""
nj = ols.n * spDcache.j
num = (spDcache.utwuDs - (spDcache.t * spDcache.utwyDs) / nj) ** 2
den = spDcache.t * (1. - (spDcache.t / nj))
lm = num / den
pval = chisqprob(lm, 1)
return (lm[0][0], pval[0][0])
def rlmLag(ols, w, spDcache):
"""
Robust LM lag test. Implemented as presented in eq. (12) of Anselin et al.
(1996) :cite:`Anselin1996a`.
Attributes
----------
ols : OLS_dev
Instance from an OLS_dev regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
rlml : tuple
Pair of statistic and p-value for the Robust LM lag test.
"""
lm = (spDcache.utwyDs - spDcache.utwuDs) ** 2 / \
((ols.n * spDcache.j) - spDcache.t)
pval = chisqprob(lm, 1)
return (lm[0][0], pval[0][0])
def lmSarma(ols, w, spDcache):
"""
LM error test. Implemented as presented in eq. (15) of Anselin et al.
(1996) :cite:`Anselin1996a`.
Attributes
----------
ols : OLS_dev
Instance from an OLS_dev regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
sarma : tuple
Pair of statistic and p-value for the LM sarma test.
"""
first = (spDcache.utwyDs - spDcache.utwuDs) ** 2 / \
(w.n * spDcache.j - spDcache.t)
secnd = spDcache.utwuDs ** 2 / spDcache.t
lm = first + secnd
pval = chisqprob(lm, 2)
return (lm[0][0], pval[0][0])
def get_mI(reg, w, spDcache):
"""
Moran's I statistic of spatial autocorrelation as showed in Cliff & Ord
(1981) :cite:`clifford1981`, p. 201-203
Attributes
----------
reg : OLS_dev, TSLS_dev, STSLS_dev
Instance from a regression class
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Returns
-------
moran : float
Statistic Moran's I test.
"""
mi = (w.n * np.dot(reg.u.T, spDcache.wu)) / (w.s0 * reg.utu)
return mi[0][0]
def get_vI(ols, w, ei, spDcache):
"""
Moran's I variance coded as in :cite:`clifford1981` (p. 201-203) and R's spdep
"""
A = spDcache.AB[0]
trA2 = np.dot(A, A)
trA2 = np.sum(trA2.diagonal())
B = spDcache.AB[1]
trB = np.sum(B.diagonal()) * 4.
vi = (w.n ** 2 / (w.s0 ** 2 * (w.n - ols.k) * (w.n - ols.k + 2.))) * \
(w.s1 + 2. * trA2 - trB -
((2. * (spDcache.trA ** 2)) / (w.n - ols.k)))
return vi
def get_eI(ols, w, spDcache):
"""
Moran's I expectation using matrix M
"""
return - (w.n * spDcache.trA) / (w.s0 * (w.n - ols.k))
def get_zI(I, ei, vi):
"""
Standardized I
Returns two-sided p-values as provided in the GeoDa family
"""
z = abs((I - ei) / np.sqrt(vi))
pval = norm.sf(z) * 2.
return (z, pval)
def akTest(iv, w, spDcache):
"""
Computes AK-test for the general case (end. reg. + sp. lag)
Parameters
----------
iv : STSLS_dev
Instance from spatial 2SLS regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Attributes
----------
mi : float
Moran's I statistic for IV residuals
ak : float
Square of corrected Moran's I for residuals:
:math:`ak = \dfrac{N \times I^*}{\phi^2}`
p : float
P-value of the test
ToDo:
* Code in as Nancy
* Compare both
"""
mi = get_mI(iv, w, spDcache)
# Phi2
etwz = spdot(iv.u.T, spdot(w.sparse, iv.z))
a = np.dot(etwz, np.dot(iv.varb, etwz.T))
s12 = (w.s0 / w.n) ** 2
phi2 = (spDcache.t + (4.0 / iv.sig2n) * a) / (s12 * w.n)
ak = w.n * mi ** 2 / phi2
pval = chisqprob(ak, 1)
return (mi, ak[0][0], pval[0][0])
def _test():
import doctest
doctest.testmod()
if __name__ == '__main__':
_test()