"""
Ordinary Least Squares regression with regimes.
"""
__author__ = "Luc Anselin luc.anselin@asu.edu, Pedro V. Amaral pedro.amaral@asu.edu, Daniel Arribas-Bel darribas@asu.edu"
from . import regimes as REGI
from . import user_output as USER
from .ols import BaseOLS
from .utils import set_warn, spbroadcast, RegressionProps_basic, RegressionPropsY, spdot
from .robust import hac_multi
from . import summary_output as SUMMARY
import numpy as np
import multiprocessing as mp
from platform import system
import scipy.sparse as SP
import copy as COPY
[docs]class OLS_Regimes(BaseOLS, REGI.Regimes_Frame, RegressionPropsY):
"""
Ordinary least squares with results and diagnostics.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
regimes : list
List of n values with the mapping of each
observation to a regime. Assumed to be aligned with 'x'.
w : pysal W object
Spatial weights object (required if running spatial
diagnostics)
robust : string
If 'white', then a White consistent estimator of the
variance-covariance matrix is given. If 'hac', then a
HAC consistent estimator of the variance-covariance
matrix is given. Default set to None.
gwk : pysal W object
Kernel spatial weights needed for HAC estimation. Note:
matrix must have ones along the main diagonal.
sig2n_k : boolean
If True, then use n-k to estimate sigma^2. If False, use n.
nonspat_diag : boolean
If True, then compute non-spatial diagnostics on
the regression.
spat_diag : boolean
If True, then compute Lagrange multiplier tests (requires
w). Note: see moran for further tests.
moran : boolean
If True, compute Moran's I on the residuals. Note:
requires spat_diag=True.
white_test : boolean
If True, compute White's specification robust test.
(requires nonspat_diag=True)
vm : boolean
If True, include variance-covariance matrix in summary
results
constant_regi: string, optional
Switcher controlling the constant term setup. It may take
the following values:
* 'one': a vector of ones is appended to x and held constant across regimes
* 'many': a vector of ones is appended to x and considered different per regime (default)
cols2regi : list, 'all'
Argument indicating whether each
column of x should be considered as different per regime
or held constant across regimes (False).
If a list, k booleans indicating for each variable the
option (True if one per regime, False to be held constant).
If 'all' (default), all the variables vary by regime.
regime_err_sep : boolean
If True, a separate regression is run for each regime.
cores : boolean
Specifies if multiprocessing is to be used
Default: no multiprocessing, cores = False
Note: Multiprocessing may not work on all platforms.
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_w : string
Name of weights matrix for use in output
name_gwk : string
Name of kernel weights matrix for use in output
name_ds : string
Name of dataset for use in output
name_regimes : string
Name of regime variable for use in the output
Attributes
----------
summary : string
Summary of regression results and diagnostics (note: use in
conjunction with the print command)
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
robust : string
Adjustment for robust standard errors
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
r2 : float
R squared
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
ar2 : float
Adjusted R squared
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
utu : float
Sum of squared residuals
sig2 : float
Sigma squared used in computations
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
sig2ML : float
Sigma squared (maximum likelihood)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
f_stat : tuple
Statistic (float), p-value (float)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
logll : float
Log likelihood
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
aic : float
Akaike information criterion
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
schwarz : float
Schwarz information criterion
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
std_err : array
1xk array of standard errors of the betas
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
t_stat : list of tuples
t statistic; each tuple contains the pair (statistic,
p-value), where each is a float
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
mulColli : float
Multicollinearity condition number
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
jarque_bera : dictionary
'jb': Jarque-Bera statistic (float); 'pvalue': p-value
(float); 'df': degrees of freedom (int)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
breusch_pagan : dictionary
'bp': Breusch-Pagan statistic (float); 'pvalue': p-value
(float); 'df': degrees of freedom (int)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
koenker_bassett: dictionary
'kb': Koenker-Bassett statistic (float); 'pvalue': p-value (float);
'df': degrees of freedom (int). Only available in dictionary
'multi' when multiple regressions (see 'multi' below for details).
white : dictionary
'wh': White statistic (float); 'pvalue': p-value (float);
'df': degrees of freedom (int).
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
lm_error : tuple
Lagrange multiplier test for spatial error model; tuple
contains the pair (statistic, p-value), where each is a
float. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
lm_lag : tuple
Lagrange multiplier test for spatial lag model; tuple
contains the pair (statistic, p-value), where each is a
float. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
rlm_error : tuple
Robust lagrange multiplier test for spatial error model;
tuple contains the pair (statistic, p-value), where each
is a float. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
rlm_lag : tuple
Robust lagrange multiplier test for spatial lag model;
tuple contains the pair (statistic, p-value), where each
is a float. Only available in dictionary 'multi' when
multiple regressions (see 'multi' below for details)
lm_sarma : tuple
Lagrange multiplier test for spatial SARMA model; tuple
contains the pair (statistic, p-value), where each is a
float. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
moran_res : tuple
Moran's I for the residuals; tuple containing the triple
(Moran's I, standardized Moran's I, p-value)
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_w : string
Name of weights matrix for use in output
name_gwk : string
Name of kernel weights matrix for use in output
name_ds : string
Name of dataset for use in output
name_regimes : string
Name of regime variable for use in the output
title : string
Name of the regression method used.
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
sig2n : float
Sigma squared (computed with n in the denominator)
sig2n_k : float
Sigma squared (computed with n-k in the denominator)
xtx : float
:math:`X'X`. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
xtxi : float
:math:`(X'X)^{-1}`. Only available in dictionary 'multi' when multiple
regressions (see 'multi' below for details)
regimes : list
List of n values with the mapping of each observation to
a regime. Assumed to be aligned with 'x'.
constant_regi : string
Ignored if regimes=False. Constant option for regimes.
Switcher controlling the constant term setup. It may take
the following values:
* 'one': a vector of ones is appended to x and held constant across regimes.
* 'many': a vector of ones is appended to x and considered different per regime.
cols2regi : list
Ignored if regimes=False. Argument indicating whether each
column of x should be considered as different per regime
or held constant across regimes (False).
If a list, k booleans indicating for each variable the
option (True if one per regime, False to be held constant).
If 'all', all the variables vary by regime.
regime_err_sep: boolean
If True, a separate regression is run for each regime.
kr : int
Number of variables/columns to be "regimized" or subject
to change by regime. These will result in one parameter
estimate by regime for each variable (i.e. nr parameters per
variable)
kf : int
Number of variables/columns to be considered fixed or
global across regimes and hence only obtain one parameter
estimate.
nr : int
Number of different regimes in the 'regimes' list.
multi : dictionary
Only available when multiple regressions are estimated,
i.e. when regime_err_sep=True and no variable is fixed
across regimes.
Contains all attributes of each individual regression.
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> from spreg import OLS_Regimes
Open data on NCOVR US County Homicides (3085 areas) using libpysal.io.open().
This is the DBF associated with the NAT 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("NAT.dbf"),'r')
Extract the HR90 column (homicide rates in 1990) 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_var = 'HR90'
>>> y = db.by_col(y_var)
>>> y = np.array(y).reshape(len(y), 1)
Extract UE90 (unemployment rate) and PS90 (population structure) vectors from
the DBF to be used as independent variables in the regression. Other variables
can be inserted by adding their names to x_var, such as x_var = ['Var1','Var2','...]
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.
>>> x_var = ['PS90','UE90']
>>> x = np.array([db.by_col(name) for name in x_var]).T
The different regimes in this data are given according to the North and
South dummy (SOUTH).
>>> r_var = 'SOUTH'
>>> regimes = db.by_col(r_var)
We can now run the regression and then have a summary of the output
by typing: olsr.summary
Alternatively, we can just check the betas and standard errors of the
parameters:
>>> olsr = OLS_Regimes(y, x, regimes, nonspat_diag=False, name_y=y_var, name_x=['PS90','UE90'], name_regimes=r_var, name_ds='NAT')
>>> olsr.betas
array([[0.39642899],
[0.65583299],
[0.48703937],
[5.59835 ],
[1.16210453],
[0.53163886]])
>>> np.sqrt(olsr.vm.diagonal())
array([0.24816345, 0.09662678, 0.03628629, 0.46894564, 0.21667395,
0.05945651])
>>> olsr.cols2regi
'all'
"""
[docs] def __init__(self, y, x, regimes,
w=None, robust=None, gwk=None, sig2n_k=True,
nonspat_diag=True, spat_diag=False, moran=False, white_test=False,
vm=False, constant_regi='many', cols2regi='all',
regime_err_sep=True, cores=False,
name_y=None, name_x=None, name_regimes=None,
name_w=None, name_gwk=None, name_ds=None):
n = USER.check_arrays(y, x)
y = USER.check_y(y, n)
USER.check_weights(w, y)
USER.check_robust(robust, gwk)
USER.check_spat_diag(spat_diag, w)
x_constant,name_x,warn = USER.check_constant(x,name_x,just_rem=True)
set_warn(self,warn)
name_x = USER.set_name_x(name_x, x_constant, constant=True)
self.name_x_r = USER.set_name_x(name_x, x_constant)
self.constant_regi = constant_regi
self.cols2regi = cols2regi
self.name_w = USER.set_name_w(name_w, w)
self.name_gwk = USER.set_name_w(name_gwk, gwk)
self.name_ds = USER.set_name_ds(name_ds)
self.name_y = USER.set_name_y(name_y)
self.name_regimes = USER.set_name_ds(name_regimes)
self.n = n
cols2regi = REGI.check_cols2regi(
constant_regi, cols2regi, x_constant, add_cons=False)
self.regimes_set = REGI._get_regimes_set(regimes)
self.regimes = regimes
USER.check_regimes(self.regimes_set, self.n, x_constant.shape[1])
if regime_err_sep == True and robust == 'hac':
set_warn(
self, "Error by regimes is incompatible with HAC estimation. Hence, error by regimes has been disabled for this model.")
regime_err_sep = False
self.regime_err_sep = regime_err_sep
if regime_err_sep == True and set(cols2regi) == set([True]) and constant_regi == 'many':
self.y = y
regi_ids = dict(
(r, list(np.where(np.array(regimes) == r)[0])) for r in self.regimes_set)
self._ols_regimes_multi(x_constant, w, regi_ids, cores,
gwk, sig2n_k, robust, nonspat_diag, spat_diag, vm, name_x, moran, white_test)
else:
x, self.name_x = REGI.Regimes_Frame.__init__(self, x_constant,
regimes, constant_regi, cols2regi, name_x)
BaseOLS.__init__(
self, y=y, x=x, robust=robust, gwk=gwk, sig2n_k=sig2n_k)
if regime_err_sep == True and robust == None:
y2, x2 = REGI._get_weighted_var(
regimes, self.regimes_set, sig2n_k, self.u, y, x)
ols2 = BaseOLS(y=y2, x=x2, sig2n_k=sig2n_k)
RegressionProps_basic(self, betas=ols2.betas, vm=ols2.vm)
self.title = "ORDINARY LEAST SQUARES - REGIMES (Group-wise heteroskedasticity)"
nonspat_diag = None
set_warn(
self, "Residuals treated as homoskedastic for the purpose of diagnostics.")
else:
self.title = "ORDINARY LEAST SQUARES - REGIMES"
self.robust = USER.set_robust(robust)
self.chow = REGI.Chow(self)
SUMMARY.OLS(reg=self, vm=vm, w=w, nonspat_diag=nonspat_diag,
spat_diag=spat_diag, moran=moran, white_test=white_test, regimes=True)
def _ols_regimes_multi(self, x, w, regi_ids, cores,
gwk, sig2n_k, robust, nonspat_diag, spat_diag, vm, name_x, moran, white_test):
results_p = {}
"""
for r in self.regimes_set:
if system() == 'Windows':
is_win = True
results_p[r] = _work(*(self.y,x,w,regi_ids,r,robust,sig2n_k,self.name_ds,self.name_y,name_x,self.name_w,self.name_regimes))
else:
pool = mp.Pool(cores)
results_p[r] = pool.apply_async(_work,args=(self.y,x,w,regi_ids,r,robust,sig2n_k,self.name_ds,self.name_y,name_x,self.name_w,self.name_regimes))
is_win = False
"""
x_constant,name_x = REGI.check_const_regi(self,x,name_x,regi_ids)
self.name_x_r = name_x
for r in self.regimes_set:
if cores:
pool = mp.Pool(None)
results_p[r] = pool.apply_async(_work, args=(
self.y, x_constant, w, regi_ids, r, robust, sig2n_k, self.name_ds, self.name_y, name_x, self.name_w, self.name_regimes))
else:
results_p[r] = _work(*(self.y, x_constant, w, regi_ids, r, robust, sig2n_k,
self.name_ds, self.name_y, name_x, self.name_w, self.name_regimes))
self.kryd = 0
self.kr = x_constant.shape[1]
self.kf = 0
self.nr = len(self.regimes_set)
self.vm = np.zeros((self.nr * self.kr, self.nr * self.kr), float)
self.betas = np.zeros((self.nr * self.kr, 1), float)
self.u = np.zeros((self.n, 1), float)
self.predy = np.zeros((self.n, 1), float)
"""
if not is_win:
pool.close()
pool.join()
"""
if cores:
pool.close()
pool.join()
results = {}
self.name_y, self.name_x = [], []
counter = 0
for r in self.regimes_set:
"""
if is_win:
results[r] = results_p[r]
else:
results[r] = results_p[r].get()
"""
if not cores:
results[r] = results_p[r]
else:
results[r] = results_p[r].get()
self.vm[(counter * self.kr):((counter + 1) * self.kr),
(counter * self.kr):((counter + 1) * self.kr)] = results[r].vm
self.betas[
(counter * self.kr):((counter + 1) * self.kr), ] = results[r].betas
self.u[regi_ids[r], ] = results[r].u
self.predy[regi_ids[r], ] = results[r].predy
self.name_y += results[r].name_y
self.name_x += results[r].name_x
counter += 1
self.multi = results
self.hac_var = x_constant[:,1:]
if robust == 'hac':
hac_multi(self, gwk)
self.chow = REGI.Chow(self)
if spat_diag:
self._get_spat_diag_props(x_constant, sig2n_k)
SUMMARY.OLS_multi(reg=self, multireg=self.multi, vm=vm, nonspat_diag=nonspat_diag,
spat_diag=spat_diag, moran=moran, white_test=white_test, regimes=True, w=w)
def _get_spat_diag_props(self, x, sig2n_k):
self.k = self.kr
self._cache = {}
self.x = REGI.regimeX_setup(
x, self.regimes, [True] * x.shape[1], self.regimes_set)
self.xtx = spdot(self.x.T, self.x)
self.xtxi = np.linalg.inv(self.xtx)
def _work(y, x, w, regi_ids, r, robust, sig2n_k, name_ds, name_y, name_x, name_w, name_regimes):
y_r = y[regi_ids[r]]
x_r = x[regi_ids[r]]
#x_constant,name_x,warn = USER.check_constant(x_r, name_x)
#name_x = USER.set_name_x(name_x, x_constant)
if robust == 'hac':
robust = None
model = BaseOLS(y_r, x_r, robust=robust, sig2n_k=sig2n_k)
model.title = "ORDINARY LEAST SQUARES ESTIMATION - REGIME %s" % r
model.robust = USER.set_robust(robust)
model.name_ds = name_ds
model.name_y = '%s_%s' % (str(r), name_y)
model.name_x = ['%s_%s' % (str(r), i) for i in name_x]
model.name_w = name_w
model.name_regimes = name_regimes
if w:
w_r, warn = REGI.w_regime(w, regi_ids[r], r, transform=True)
set_warn(model, warn)
model.w = w_r
return model
def _test():
import doctest
start_suppress = np.get_printoptions()['suppress']
np.set_printoptions(suppress=True)
doctest.testmod()
np.set_printoptions(suppress=start_suppress)
if __name__ == '__main__':
_test()
import numpy as np
import libpysal
db = libpysal.io.open(libpysal.examples.get_path("NAT.dbf"),'r')
y_var = 'CRIME'
y = np.array([db.by_col(y_var)]).reshape(49, 1)
x_var = ['INC', 'HOVAL']
x = np.array([db.by_col(name) for name in x_var]).T
r_var = 'NSA'
regimes = db.by_col(r_var)
w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
w.transform = 'r'
olsr = OLS_Regimes(y, x, regimes, w=w, constant_regi='many', nonspat_diag=False, spat_diag=False, name_y=y_var, name_x=['INC', 'HOVAL'],
name_ds='columbus', name_regimes=r_var, name_w='columbus.gal', regime_err_sep=True, cols2regi=[True, True], sig2n_k=True, robust='white')
print(olsr.summary)