"""
Spatial Error Models with regimes module
"""
__author__ = "Luc Anselin luc.anselin@asu.edu, Pedro V. Amaral pedro.amaral@asu.edu"
import numpy as np
import multiprocessing as mp
from . import regimes as REGI
from . import user_output as USER
from . import summary_output as SUMMARY
from libpysal.weights.spatial_lag import lag_spatial
from .ols import BaseOLS
from .twosls import BaseTSLS
from .error_sp import BaseGM_Error, BaseGM_Endog_Error, _momentsGM_Error
from .utils import set_endog, iter_msg, sp_att, set_warn
from .utils import optim_moments, get_spFilter, get_lags
from .utils import spdot, RegressionPropsY
from .sputils import sphstack
[docs]class GM_Error_Regimes(RegressionPropsY, REGI.Regimes_Frame):
"""
GMM method for a spatial error model with regimes, with results and diagnostics;
based on Kelejian and Prucha (1998, 1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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
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.
regime_lag_sep: boolean
Always False, kept for consistency, ignored.
vm : boolean
If True, include variance-covariance matrix in summary
results
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_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
e_filtered : array
nx1 array of spatially filtered 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)
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
vm : array
Variance covariance matrix (kxk)
sig2 : float
Sigma squared used in computations
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)
z_stat : list of tuples
z 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)
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_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)
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, 'all'
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
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import libpysal
>>> import numpy as np
>>> from libpysal.examples import load_example
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.
>>> nat = load_example('Natregimes')
>>> db = libpysal.io.open(nat.get_path("natregimes.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 = np.array([db.by_col(y_var)]).reshape(3085,1)
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)
Since we want to run a spatial error model, we need to specify
the spatial weights matrix that includes the spatial configuration of the
observations. To do that, we can open an already existing gal file or
create a new one. In this case, we will create one from ``NAT.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(nat.get_path("natregimes.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 all set with the preliminaries, we are good to run the model. In this
case, we will need the variables and the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> from spreg import GM_Error_Regimes
>>> model = GM_Error_Regimes(y, x, regimes, w=w, name_y=y_var, name_x=x_var, name_regimes=r_var, name_ds='NAT.dbf')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas). Alternatively, we can have a summary of the
output by typing: model.summary
>>> print(model.name_x)
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', 'lambda']
>>> np.around(model.betas, decimals=6)
array([[0.074807],
[0.786107],
[0.538849],
[5.103756],
[1.196009],
[0.600533],
[0.364103]])
>>> np.around(model.std_err, decimals=6)
array([0.379864, 0.152316, 0.051942, 0.471285, 0.19867 , 0.057252])
>>> np.around(model.z_stat, decimals=6)
array([[ 0.196932, 0.843881],
[ 5.161042, 0. ],
[10.37397 , 0. ],
[10.829455, 0. ],
[ 6.02007 , 0. ],
[10.489215, 0. ]])
>>> np.around(model.sig2, decimals=6)
28.172732
"""
[docs] def __init__(self, y, x, regimes, w,
vm=False, name_y=None, name_x=None, name_w=None,
constant_regi='many', cols2regi='all', regime_err_sep=False,
regime_lag_sep=False,
cores=False, name_ds=None, name_regimes=None):
n = USER.check_arrays(y, x)
y = USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
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_ds = USER.set_name_ds(name_ds)
self.name_y = USER.set_name_y(name_y)
self.name_w = USER.set_name_w(name_w, w)
self.name_regimes = USER.set_name_ds(name_regimes)
self.n = n
self.y = y
cols2regi = REGI.check_cols2regi(constant_regi, cols2regi, x_constant)
self.regimes_set = REGI._get_regimes_set(regimes)
self.regimes = regimes
USER.check_regimes(self.regimes_set, self.n, x_constant.shape[1])
self.regime_err_sep = regime_err_sep
if regime_err_sep == True:
if set(cols2regi) == set([True]):
self._error_regimes_multi(y, x_constant, regimes, w, cores,
cols2regi, vm, name_x)
else:
raise Exception("All coefficients must vary accross regimes if regime_err_sep = True.")
else:
x_constant = sphstack(np.ones((x_constant.shape[0], 1)), x_constant)
name_x = USER.set_name_x(name_x, x_constant)
self.x, self.name_x = REGI.Regimes_Frame.__init__(self, x_constant,
regimes, constant_regi=None, cols2regi=cols2regi, names=name_x)
ols = BaseOLS(y=y, x=self.x)
self.k = ols.x.shape[1]
moments = _momentsGM_Error(w, ols.u)
lambda1 = optim_moments(moments)
xs = get_spFilter(w, lambda1, x_constant)
ys = get_spFilter(w, lambda1, y)
xs = REGI.Regimes_Frame.__init__(self, xs,
regimes, constant_regi=None, cols2regi=cols2regi)[0]
ols2 = BaseOLS(y=ys, x=xs)
# Output
self.predy = spdot(self.x, ols2.betas)
self.u = y - self.predy
self.betas = np.vstack((ols2.betas, np.array([[lambda1]])))
self.sig2 = ols2.sig2n
self.e_filtered = self.u - lambda1 * lag_spatial(w, self.u)
self.vm = self.sig2 * ols2.xtxi
self.title = "SPATIALLY WEIGHTED LEAST SQUARES - REGIMES"
self.name_x.append('lambda')
self.kf += 1
self.chow = REGI.Chow(self)
self._cache = {}
SUMMARY.GM_Error(reg=self, w=w, vm=vm, regimes=True)
def _error_regimes_multi(self, y, x, regimes, w, cores,
cols2regi, vm, name_x):
regi_ids = dict(
(r, list(np.where(np.array(regimes) == r)[0])) for r in self.regimes_set)
results_p = {}
"""
for r in self.regimes_set:
if system() == 'Windows':
results_p[r] = _work_error(*(y,x,regi_ids,r,w,self.name_ds,self.name_y,name_x+['lambda'],self.name_w,self.name_regimes))
is_win = True
else:
pool = mp.Pool(cores)
results_p[r] = pool.apply_async(_work_error,args=(y,x,regi_ids,r,w,self.name_ds,self.name_y,name_x+['lambda'],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_error, args=(
y, x_constant, regi_ids, r, w, self.name_ds, self.name_y, name_x + ['lambda'], self.name_w, self.name_regimes, ))
else:
results_p[r] = _work_error(
*(y, x_constant, regi_ids, r, w, self.name_ds, self.name_y, name_x + ['lambda'], self.name_w, self.name_regimes))
self.kryd = 0
self.kr = len(cols2regi)
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), 1), float)
self.u = np.zeros((self.n, 1), float)
self.predy = np.zeros((self.n, 1), float)
self.e_filtered = 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 + 1)):((counter + 1) * (self.kr + 1)), ] = results[r].betas
self.u[regi_ids[r], ] = results[r].u
self.predy[regi_ids[r], ] = results[r].predy
self.e_filtered[regi_ids[r], ] = results[r].e_filtered
self.name_y += results[r].name_y
self.name_x += results[r].name_x
counter += 1
self.chow = REGI.Chow(self)
self.multi = results
SUMMARY.GM_Error_multi(
reg=self, multireg=self.multi, vm=vm, regimes=True)
[docs]class GM_Endog_Error_Regimes(RegressionPropsY, REGI.Regimes_Frame):
'''
GMM method for a spatial error model with regimes and endogenous variables, with
results and diagnostics; based on Kelejian and Prucha (1998,
1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
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
constant_regi: string
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.
regime_lag_sep: boolean
Always False, kept for consistency, ignored.
vm : boolean
If True, include variance-covariance matrix in summary
results
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_yend : list of strings
Names of endogenous variables for use in output
name_q : list of strings
Names of instruments for use in output
name_w : string
Name of 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
e_filtered : array
nx1 array of spatially filtered 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)
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
z : array
nxk array of variables (combination of x and yend)
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)
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
sig2 : float
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
Sigma squared used in computations
std_err : array
1xk array of standard errors of the betas
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
z_stat : list of tuples
z 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)
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_yend : list of strings
Names of endogenous variables for use in output
name_z : list of strings
Names of exogenous and endogenous variables for use in
output
name_q : list of strings
Names of external instruments
name_h : list of strings
Names of all instruments used in ouput
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
name_regimes : string
Name of regimes variable for use in output
title : string
Name of the regression method used
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 : ['one', 'many']
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 (default).
cols2regi : list, 'all'
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
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import libpysal
>>> import numpy as np
>>> from libpysal.examples import load_example
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.
>>> nat = load_example('Natregimes')
>>> db = libpysal.io.open(nat.get_path("natregimes.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 = np.array([db.by_col(y_var)]).reshape(3085,1)
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
For the endogenous models, we add the endogenous variable RD90 (resource deprivation)
and we decide to instrument for it with FP89 (families below poverty):
>>> yd_var = ['RD90']
>>> yend = np.array([db.by_col(name) for name in yd_var]).T
>>> q_var = ['FP89']
>>> q = np.array([db.by_col(name) for name in q_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)
Since we want to run a spatial error model, we need to specify the spatial
weights matrix that 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 ``NAT.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(nat.get_path("natregimes.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 all set with the preliminaries, we are good to run the model. In this
case, we will need the variables (exogenous and endogenous), the
instruments and the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> from spreg import GM_Endog_Error_Regimes
>>> model = GM_Endog_Error_Regimes(y, x, yend, q, regimes, w=w, name_y=y_var, name_x=x_var, name_yend=yd_var, name_q=q_var, name_regimes=r_var, name_ds='NAT.dbf')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas). Also, this regression uses a two stage least
squares estimation method that accounts for the endogeneity created by the
endogenous variables included. Alternatively, we can have a summary of the
output by typing: model.summary
>>> print(model.name_z)
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', '0_RD90', '1_RD90', 'lambda']
>>> np.around(model.betas, decimals=5)
array([[ 3.59718],
[ 1.0652 ],
[ 0.15822],
[ 9.19754],
[ 1.88082],
[-0.24878],
[ 2.46161],
[ 3.57943],
[ 0.25564]])
>>> np.around(model.std_err, decimals=6)
array([0.522633, 0.137555, 0.063054, 0.473654, 0.18335 , 0.072786,
0.300711, 0.240413])
'''
[docs] def __init__(self, y, x, yend, q, regimes, w, cores=False,
vm=False, constant_regi='many', cols2regi='all',
regime_err_sep=False, regime_lag_sep=False, name_y=None,
name_x=None, name_yend=None, name_q=None, name_w=None,
name_ds=None, name_regimes=None, summ=True, add_lag=False):
n = USER.check_arrays(y, x, yend, q)
y = USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
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.constant_regi = constant_regi
self.cols2regi = cols2regi
self.name_ds = USER.set_name_ds(name_ds)
self.name_regimes = USER.set_name_ds(name_regimes)
self.name_w = USER.set_name_w(name_w, w)
self.n = n
self.y = y
if summ:
name_yend = USER.set_name_yend(name_yend, yend)
self.name_y = USER.set_name_y(name_y)
name_q = USER.set_name_q(name_q, q)
self.name_x_r = USER.set_name_x(name_x, x_constant) + name_yend
cols2regi = REGI.check_cols2regi(
constant_regi, cols2regi, x_constant, yend=yend)
self.regimes_set = REGI._get_regimes_set(regimes)
self.regimes = regimes
USER.check_regimes(self.regimes_set, self.n, x_constant.shape[1])
self.regime_err_sep = regime_err_sep
if regime_err_sep == True:
if set(cols2regi) == set([True]):
self._endog_error_regimes_multi(y, x_constant, regimes, w, yend, q, cores,
cols2regi, vm, name_x, name_yend, name_q, add_lag)
else:
raise Exception("All coefficients must vary accross regimes if regime_err_sep = True.")
else:
x_constant = sphstack(np.ones((x_constant.shape[0], 1)), x_constant)
name_x = USER.set_name_x(name_x, x_constant)
q, name_q = REGI.Regimes_Frame.__init__(self, q,
regimes, constant_regi=None, cols2regi='all', names=name_q)
x, name_x = REGI.Regimes_Frame.__init__(self, x_constant,
regimes, constant_regi=None, cols2regi=cols2regi,
names=name_x)
yend2, name_yend = REGI.Regimes_Frame.__init__(self, yend,
regimes, constant_regi=None,
cols2regi=cols2regi, yend=True, names=name_yend)
tsls = BaseTSLS(y=y, x=x, yend=yend2, q=q)
self.k = tsls.z.shape[1]
self.x = tsls.x
self.yend, self.z = tsls.yend, tsls.z
moments = _momentsGM_Error(w, tsls.u)
lambda1 = optim_moments(moments)
xs = get_spFilter(w, lambda1, x_constant)
xs = REGI.Regimes_Frame.__init__(self, xs,
regimes, constant_regi=None, cols2regi=cols2regi)[0]
ys = get_spFilter(w, lambda1, y)
yend_s = get_spFilter(w, lambda1, yend)
yend_s = REGI.Regimes_Frame.__init__(self, yend_s,
regimes, constant_regi=None, cols2regi=cols2regi,
yend=True)[0]
tsls2 = BaseTSLS(ys, xs, yend_s, h=tsls.h)
# Output
self.betas = np.vstack((tsls2.betas, np.array([[lambda1]])))
self.predy = spdot(tsls.z, tsls2.betas)
self.u = y - self.predy
self.sig2 = float(np.dot(tsls2.u.T, tsls2.u)) / self.n
self.e_filtered = self.u - lambda1 * lag_spatial(w, self.u)
self.vm = self.sig2 * tsls2.varb
self.name_x = USER.set_name_x(name_x, x_constant, constant=True)
self.name_yend = USER.set_name_yend(name_yend, yend)
self.name_z = self.name_x + self.name_yend
self.name_z.append('lambda')
self.name_q = USER.set_name_q(name_q, q)
self.name_h = USER.set_name_h(self.name_x, self.name_q)
self.kf += 1
self.chow = REGI.Chow(self)
self._cache = {}
if summ:
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES - REGIMES"
SUMMARY.GM_Endog_Error(reg=self, w=w, vm=vm, regimes=True)
def _endog_error_regimes_multi(self, y, x, regimes, w, yend, q, cores,
cols2regi, vm, name_x, name_yend, name_q, add_lag):
regi_ids = dict(
(r, list(np.where(np.array(regimes) == r)[0])) for r in self.regimes_set)
if add_lag != False:
self.cols2regi += [True]
cols2regi += [True]
self.predy_e = np.zeros((self.n, 1), float)
self.e_pred = np.zeros((self.n, 1), float)
results_p = {}
"""
for r in self.regimes_set:
if system() == 'Windows':
results_p[r] = _work_endog_error(*(y,x,yend,q,regi_ids,r,w,self.name_ds,self.name_y,name_x,name_yend,name_q,self.name_w,self.name_regimes,add_lag))
is_win = True
else:
pool = mp.Pool(cores)
results_p[r] = pool.apply_async(_work_endog_error,args=(y,x,yend,q,regi_ids,r,w,self.name_ds,self.name_y,name_x,name_yend,name_q,self.name_w,self.name_regimes,add_lag, ))
is_win = False
"""
x_constant,name_x = REGI.check_const_regi(self,x,name_x,regi_ids)
self.name_x_r = name_x + name_yend
for r in self.regimes_set:
if cores:
pool = mp.Pool(None)
results_p[r] = pool.apply_async(_work_endog_error, args=(
y, x_constant, yend, q, regi_ids, r, w, self.name_ds, self.name_y, name_x, name_yend, name_q, self.name_w, self.name_regimes, add_lag, ))
else:
results_p[r] = _work_endog_error(
*(y, x_constant, yend, q, regi_ids, r, w, self.name_ds, self.name_y, name_x, name_yend, name_q, self.name_w, self.name_regimes, add_lag))
self.kryd, self.kf = 0, 0
self.kr = len(cols2regi)
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), 1), float)
self.u = np.zeros((self.n, 1), float)
self.predy = np.zeros((self.n, 1), float)
self.e_filtered = 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, self.name_yend, self.name_q, self.name_z, self.name_h = [
], [], [], [], [], []
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 + 1)):((counter + 1) * (self.kr + 1)), ] = results[r].betas
self.u[regi_ids[r], ] = results[r].u
self.predy[regi_ids[r], ] = results[r].predy
self.e_filtered[regi_ids[r], ] = results[r].e_filtered
self.name_y += results[r].name_y
self.name_x += results[r].name_x
self.name_yend += results[r].name_yend
self.name_q += results[r].name_q
self.name_z += results[r].name_z
self.name_h += results[r].name_h
if add_lag != False:
self.predy_e[regi_ids[r], ] = results[r].predy_e
self.e_pred[regi_ids[r], ] = results[r].e_pred
counter += 1
self.chow = REGI.Chow(self)
self.multi = results
if add_lag != False:
SUMMARY.GM_Combo_multi(
reg=self, multireg=self.multi, vm=vm, regimes=True)
else:
SUMMARY.GM_Endog_Error_multi(
reg=self, multireg=self.multi, vm=vm, regimes=True)
[docs]class GM_Combo_Regimes(GM_Endog_Error_Regimes, REGI.Regimes_Frame):
"""
GMM method for a spatial lag and error model with regimes and endogenous
variables, with results and diagnostics; based on Kelejian and Prucha (1998,
1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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'.
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
w : pysal W object
Spatial weights object (always needed)
constant_regi: string
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.
regime_lag_sep: boolean
If True, the spatial parameter for spatial lag is also
computed according to different regimes. If False (default),
the spatial parameter is fixed accross regimes.
w_lags : integer
Orders of W to include as instruments for the spatially
lagged dependent variable. For example, w_lags=1, then
instruments are WX; if w_lags=2, then WX, WWX; and so on.
lag_q : boolean
If True, then include spatial lags of the additional
instruments (q).
vm : boolean
If True, include variance-covariance matrix in summary
results
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_yend : list of strings
Names of endogenous variables for use in output
name_q : list of strings
Names of instruments for use in output
name_w : string
Name of 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
e_filtered : array
nx1 array of spatially filtered residuals
e_pred : array
nx1 array of residuals (using reduced form)
predy : array
nx1 array of predicted y values
predy_e : array
nx1 array of predicted y values (using reduced form)
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)
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
z : array
nxk array of variables (combination of x and yend)
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)
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
pr2_e : float
Pseudo R squared (squared correlation between y and ypred_e
(using reduced form))
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
sig2 : float
Sigma squared used in computations (based on filtered
residuals)
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)
z_stat : list of tuples
z 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)
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_yend : list of strings
Names of endogenous variables for use in output
name_z : list of strings
Names of exogenous and endogenous variables for use in
output
name_q : list of strings
Names of external instruments
name_h : list of strings
Names of all instruments used in ouput
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
name_regimes : string
Name of regimes variable for use in output
title : string
Name of the regression method used
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 (default).
cols2regi : list, 'all'
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.
regime_lag_sep: boolean
If True, the spatial parameter for spatial lag is also
computed according to different regimes. If False (default),
the spatial parameter is fixed accross regimes.
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
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import numpy as np
>>> import libpysal
>>> from libpysal.examples import load_example
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.
>>> nat = load_example('Natregimes')
>>> db = libpysal.io.open(nat.get_path("natregimes.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 = np.array([db.by_col(y_var)]).reshape(3085,1)
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)
Since we want to run a spatial lag model, we need to specify
the spatial weights matrix that includes the spatial configuration of the
observations. To do that, we can open an already existing gal file or
create a new one. In this case, we will create one from ``NAT.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(nat.get_path("natregimes.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'
The Combo class runs an SARAR model, that is a spatial lag+error model.
In this case we will run a simple version of that, where we have the
spatial effects as well as exogenous variables. Since it is a spatial
model, we have to pass in the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> from spreg import GM_Combo_Regimes
>>> model = GM_Combo_Regimes(y, x, regimes, w=w, name_y=y_var, name_x=x_var, name_regimes=r_var, name_ds='NAT')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas). Also, this regression uses a two stage least
squares estimation method that accounts for the endogeneity created by the
spatial lag of the dependent variable. We can have a summary of the
output by typing: model.summary
Alternatively, we can check the betas:
>>> print(model.name_z)
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', '_Global_W_HR90', 'lambda']
>>> print(np.around(model.betas,4))
[[ 1.4607]
[ 0.958 ]
[ 0.5658]
[ 9.113 ]
[ 1.1338]
[ 0.6517]
[-0.4583]
[ 0.6136]]
And lambda:
>>> print('lambda: ', np.around(model.betas[-1], 4))
lambda: [0.6136]
This class also allows the user to run a spatial lag+error model with the
extra feature of including non-spatial endogenous regressors. This means
that, in addition to the spatial lag and error, we consider some of the
variables on the right-hand side of the equation as endogenous and we
instrument for this. In this case we consider RD90 (resource deprivation)
as an endogenous regressor. We use FP89 (families below poverty)
for this and hence put it in the instruments parameter, 'q'.
>>> yd_var = ['RD90']
>>> yd = np.array([db.by_col(name) for name in yd_var]).T
>>> q_var = ['FP89']
>>> q = np.array([db.by_col(name) for name in q_var]).T
And then we can run and explore the model analogously to the previous combo:
>>> model = GM_Combo_Regimes(y, x, regimes, yd, q, w=w, name_y=y_var, name_x=x_var, name_yend=yd_var, name_q=q_var, name_regimes=r_var, name_ds='NAT')
>>> print(model.name_z)
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', '0_RD90', '1_RD90', '_Global_W_HR90', 'lambda']
>>> print(model.betas)
[[ 3.41963782]
[ 1.04065841]
[ 0.16634393]
[ 8.86544628]
[ 1.85120528]
[-0.24908469]
[ 2.43014046]
[ 3.61645481]
[ 0.03308671]
[ 0.18684992]]
>>> print(np.sqrt(model.vm.diagonal()))
[0.53067577 0.13271426 0.06058025 0.76406411 0.17969783 0.07167421
0.28943121 0.25308326 0.06126529]
>>> print('lambda: ', np.around(model.betas[-1], 4))
lambda: [0.1868]
"""
[docs] def __init__(self, y, x, regimes, yend=None, q=None,
w=None, w_lags=1, lag_q=True, cores=False,
constant_regi='many', cols2regi='all',
regime_err_sep=False, regime_lag_sep=False,
vm=False, name_y=None, name_x=None,
name_yend=None, name_q=None,
name_w=None, name_ds=None, name_regimes=None):
n = USER.check_arrays(y, x)
y = USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
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_y = USER.set_name_y(name_y)
name_yend = USER.set_name_yend(name_yend, yend)
name_q = USER.set_name_q(name_q, q)
name_q.extend(
USER.set_name_q_sp(name_x, w_lags, name_q, lag_q, force_all=True))
cols2regi = REGI.check_cols2regi(
constant_regi, cols2regi, x_constant, yend=yend, add_cons=False)
self.regimes_set = REGI._get_regimes_set(regimes)
self.regimes = regimes
USER.check_regimes(self.regimes_set, n, x_constant.shape[1])
self.regime_err_sep = regime_err_sep
self.regime_lag_sep = regime_lag_sep
if regime_lag_sep == True:
if regime_err_sep == False:
raise Exception("For spatial combo models, if spatial lag is set by regimes (regime_lag_sep=True), spatial error must also be set by regimes (regime_err_sep=True).")
add_lag = [w_lags, lag_q]
else:
if regime_err_sep == True:
raise Exception("For spatial combo models, if spatial error is set by regimes (regime_err_sep=True), all coefficients including lambda (regime_lag_sep=True) must be set by regimes.")
cols2regi += [False]
add_lag = False
yend, q = set_endog(y, x_constant, w, yend, q, w_lags, lag_q)
name_yend.append(USER.set_name_yend_sp(self.name_y))
GM_Endog_Error_Regimes.__init__(self, y=y, x=x_constant, yend=yend,
q=q, regimes=regimes, w=w, vm=vm, constant_regi=constant_regi,
cols2regi=cols2regi, regime_err_sep=regime_err_sep, cores=cores,
name_y=self.name_y, name_x=name_x,
name_yend=name_yend, name_q=name_q, name_w=name_w,
name_ds=name_ds, name_regimes=name_regimes, summ=False, add_lag=add_lag)
if regime_err_sep != True:
self.rho = self.betas[-2]
self.predy_e, self.e_pred, warn = sp_att(w, self.y,
self.predy, yend[:, -1].reshape(self.n, 1), self.rho)
set_warn(self, warn)
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES - REGIMES"
SUMMARY.GM_Combo(reg=self, w=w, vm=vm, regimes=True)
def _work_error(y, x, regi_ids, r, w, name_ds, name_y, name_x, name_w, name_regimes):
w_r, warn = REGI.w_regime(w, regi_ids[r], r, transform=True)
y_r = y[regi_ids[r]]
x_r = x[regi_ids[r]]
model = BaseGM_Error(y_r, x_r, w_r.sparse)
set_warn(model, warn)
model.w = w_r
model.title = "SPATIALLY WEIGHTED LEAST SQUARES ESTIMATION - REGIME %s" % r
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
return model
def _work_endog_error(y, x, yend, q, regi_ids, r, w, name_ds, name_y, name_x, name_yend, name_q, name_w, name_regimes, add_lag):
w_r, warn = REGI.w_regime(w, regi_ids[r], r, transform=True)
y_r = y[regi_ids[r]]
x_r = x[regi_ids[r]]
if yend is not None:
yend_r = yend[regi_ids[r]]
q_r = q[regi_ids[r]]
else:
yend_r, q_r = None, None
if add_lag != False:
yend_r, q_r = set_endog(
y_r, x_r[:,1:], w_r, yend_r, q_r, add_lag[0], add_lag[1])
model = BaseGM_Endog_Error(y_r, x_r, yend_r, q_r, w_r.sparse)
set_warn(model, warn)
if add_lag != False:
model.rho = model.betas[-2]
model.predy_e, model.e_pred, warn = sp_att(w_r, model.y,
model.predy, model.yend[:, -1].reshape(model.n, 1), model.rho)
set_warn(model, warn)
model.w = w_r
model.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES - REGIME %s" % r
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_yend = ['%s_%s' % (str(r), i) for i in name_yend]
model.name_z = model.name_x + model.name_yend + ['lambda']
model.name_q = ['%s_%s' % (str(r), i) for i in name_q]
model.name_h = model.name_x + model.name_q
model.name_w = name_w
model.name_regimes = name_regimes
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 libpysal
import numpy as np
dbf = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'), 'r')
y = np.array([dbf.by_col('CRIME')]).T
names_to_extract = ['INC']
x = np.array([dbf.by_col(name) for name in names_to_extract]).T
yd_var = ['HOVAL']
yend = np.array([dbf.by_col(name) for name in yd_var]).T
q_var = ['DISCBD']
q = np.array([dbf.by_col(name) for name in q_var]).T
regimes = regimes = dbf.by_col('NSA')
w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
w.transform = 'r'
model = GM_Error_Regimes(y, x, regimes=regimes, w=w, name_y='crime', name_x=[
'income'], name_regimes='nsa', name_ds='columbus', regime_err_sep=True)
print(model.summary)