import numpy as np
from scipy.stats import t, norm
from scipy import optimize
from scikits.statsmodels.tools.tools import recipr
from scikits.statsmodels.stats.contrast import ContrastResults
from scikits.statsmodels.tools.decorators import (resettable_cache,
cache_readonly)
[docs]class Model(object):
"""
A (predictive) statistical model. The class Model itself is not to be used.
Model lays out the methods expected of any subclass.
Parameters
----------
endog : array-like
Endogenous response variable.
exog : array-like
Exogenous design.
Notes
-----
`endog` and `exog` are references to any data provided. So if the data is
already stored in numpy arrays and it is changed then `endog` and `exog`
will change as well.
"""
_results = None
def __init__(self, endog, exog=None):
endog = np.asarray(endog)
endog = np.squeeze(endog) # for consistent outputs if endog is (n,1)
## # not sure if we want type conversion, needs tests with integers
## if np.issubdtype(endog.dtype, int):
## endog = endog.astype(float)
## if np.issubdtype(exog.dtype, int):
## endog = exog.astype(float)
if not exog is None:
exog = np.asarray(exog)
if exog.ndim == 1:
exog = exog[:,None]
if exog.ndim != 2:
raise ValueError("exog is not 1d or 2d")
if endog.shape[0] != exog.shape[0]:
raise ValueError("endog and exog matrices are not aligned.")
if np.any(exog.var(0) == 0):
# assumes one constant in first or last position
# avoid exception if more than one constant
const_idx = np.where(exog.var(0) == 0)[0][0].item()
if const_idx == exog.shape[1] - 1:
exog_names = ['x%d' % i for i in range(1,exog.shape[1])]
exog_names += ['const']
else:
exog_names = ['x%d' % i for i in range(exog.shape[1])]
exog_names[const_idx] = 'const'
self.exog_names = exog_names
else:
self.exog_names = ['x%d' % i for i in range(exog.shape[1])]
if endog.ndim == 1 or endog.shape[1] == 1:
self.endog_names = ['y']
else: # for VAR
self.endog_names = ['y%d' % (i+1) for i in range(endog.shape[1])]
self.endog = endog
self.exog = exog
[docs] def fit(self):
"""
Fit a model to data.
"""
raise NotImplementedError
[docs] def predict(self, exog, params=None):
"""
After a model has been fit predict returns the fitted values. If
the model has not been fit, then fit is called.
"""
raise NotImplementedError
[docs]class LikelihoodModel(Model):
"""
Likelihood model is a subclass of Model.
"""
def __init__(self, endog, exog=None):
super(LikelihoodModel, self).__init__(endog, exog)
self.initialize()
[docs] def initialize(self):
"""
Initialize (possibly re-initialize) a Model instance. For
instance, the design matrix of a linear model may change
and some things must be recomputed.
"""
pass
#TODO: if the intent is to re-initialize the model with new data then
# this method needs to take inputs...
[docs] def loglike(self, params):
"""
Log-likelihood of model.
"""
raise NotImplementedError
[docs] def score(self, params):
"""
Score vector of model.
The gradient of logL with respect to each parameter.
"""
raise NotImplementedError
[docs] def hessian(self, params):
"""
The Hessian matrix of the model
"""
raise NotImplementedError
[docs] def fit(self, start_params=None, method='newton', maxiter=100,
full_output=1,disp=1, fargs=(), callback=None, retall=0, **kwargs):
"""
Fit method for likelihood based models
Parameters
----------
start_params : array-like, optional
Initial guess of the solution for the loglikelihood maximization.
The default is an array of zeros.
method : str {'newton','nm','bfgs','powell','cg', or 'ncg'}
Method can be 'newton' for Newton-Raphson, 'nm' for Nelder-Mead,
'bfgs' for Broyden-Fletcher-Goldfarb-Shanno, 'powell' for modified
Powell's method, 'cg' for conjugate gradient, or 'ncg' for Newton-
conjugate gradient. `method` determines which solver from
scipy.optimize is used. The explicit arguments in `fit` are passed
to the solver. Each solver has several optional arguments that are
not the same across solvers. See the notes section below (or
scipy.optimize) for the available arguments.
maxiter : int
The maximum number of iterations to perform.
full_output : bool
Set to True to have all available output in the Results object's
mle_retvals attribute. The output is dependent on the solver.
See LikelihoodModelResults notes section for more information.
disp : bool
Set to True to print convergence messages.
fargs : tuple
Extra arguments passed to the likelihood function, i.e.,
loglike(x,*args)
callback : callable callback(xk)
Called after each iteration, as callback(xk), where xk is the
current parameter vector.
retall : bool
Set to True to return list of solutions at each iteration.
Available in Results object's mle_retvals attribute.
Notes
-----
Optional arguments for the solvers (available in Results.mle_settings):
'newton'
tol : float
Relative error in params acceptable for convergence.
'nm' -- Nelder Mead
xtol : float
Relative error in params acceptable for convergence
ftol : float
Relative error in loglike(params) acceptable for
convergence
maxfun : int
Maximum number of function evaluations to make.
'bfgs'
gtol : float
Stop when norm of gradient is less than gtol.
norm : float
Order of norm (np.Inf is max, -np.Inf is min)
epsilon
If fprime is approximated, use this value for the step
size. Only relevant if LikelihoodModel.score is None.
'cg'
gtol : float
Stop when norm of gradient is less than gtol.
norm : float
Order of norm (np.Inf is max, -np.Inf is min)
epsilon : float
If fprime is approximated, use this value for the step
size. Can be scalar or vector. Only relevant if
Likelihoodmodel.score is None.
'ncg'
fhess_p : callable f'(x,*args)
Function which computes the Hessian of f times an arbitrary
vector, p. Should only be supplied if
LikelihoodModel.hessian is None.
avextol : float
Stop when the average relative error in the minimizer
falls below this amount.
epsilon : float or ndarray
If fhess is approximated, use this value for the step size.
Only relevant if Likelihoodmodel.hessian is None.
'powell'
xtol : float
Line-search error tolerance
ftol : float
Relative error in loglike(params) for acceptable for
convergence.
maxfun : int
Maximum number of function evaluations to make.
start_direc : ndarray
Initial direction set.
"""
Hinv = None #JP error if full_output=0, Hinv not defined
methods = ['newton', 'nm', 'bfgs', 'powell', 'cg', 'ncg']
if start_params is None:
if hasattr(self, 'start_params'):
start_params = self.start_params
elif self.exog is not None:
start_params = [0]*self.exog.shape[1] # fails for shape (K,)?
else:
raise ValueError("If exog is None, then start_params should be "
"specified")
#print 'repr(start_params)', repr(start_params)
if method.lower() not in methods:
raise ValueError, "Unknown fit method %s" % method
method = method.lower()
#TODO: separate args from nonarg taking score and hessian, ie.,
# user-supplied and numerically evaluated
# estimate frprime doesn't take args in most (any?) of the optimize function
f = lambda params, *args: -self.loglike(params, *args)
score = lambda params: -self.score(params)
try:
hess = lambda params: -self.hessian(params)
except:
hess = None
if method == 'newton':
tol = kwargs.setdefault('tol', 1e-8)
score = lambda params: self.score(params) #JP: redundant ?
hess = lambda params: self.hessian(params) #JP: redundant ?
iterations = 0
oldparams = np.inf
newparams = np.asarray(start_params)
if retall:
history = [oldparams, newparams]
while (iterations < maxiter and np.all(np.abs(newparams -
oldparams) > tol)):
H = hess(newparams)
oldparams = newparams
newparams = oldparams - np.dot(np.linalg.inv(H),
score(oldparams))
if retall:
history.append(newparams)
if callback is not None:
callback(newparams)
iterations += 1
fval = f(newparams, *fargs) # this is the negative likelihood
if iterations == maxiter:
warnflag = 1
if disp:
print "Warning: Maximum number of iterations has been \
exceeded."
print " Current function value: %f" % fval
print " Iterations: %d" % iterations
else:
warnflag = 0
if disp:
print "Optimization terminated successfully."
print " Current function value: %f" % fval
print " Iterations %d" % iterations
if full_output:
xopt, fopt, niter, gopt, hopt = (newparams, f(newparams, *fargs),
iterations, score(newparams), hess(newparams))
converged = not warnflag
retvals = {'fopt' : fopt, 'iterations' : niter, 'score' : gopt,
'Hessian' : hopt, 'warnflag' : warnflag,
'converged' : converged}
if retall:
retvals.update({'allvecs' : history})
else:
retvals = newparams
elif method == 'nm': # Nelder-Mead
xtol = kwargs.setdefault('xtol', 0.0001)
ftol = kwargs.setdefault('ftol', 0.0001)
maxfun = kwargs.setdefault('maxfun', None)
retvals = optimize.fmin(f, start_params, args=fargs, xtol=xtol,
ftol=ftol, maxiter=maxiter, maxfun=maxfun,
full_output=full_output, disp=disp, retall=retall,
callback=callback)
if full_output:
if not retall:
xopt, fopt, niter, fcalls, warnflag = retvals
else:
xopt, fopt, niter, fcalls, warnflag, allvecs = retvals
converged = not warnflag
retvals = {'fopt' : fopt, 'iterations' : niter,
'fcalls' : fcalls, 'warnflag' : warnflag,
'converged' : converged}
if retall:
retvals.update({'allvecs' : allvecs})
elif method == 'bfgs':
gtol = kwargs.setdefault('gtol', 1.0000000000000001e-05)
norm = kwargs.setdefault('norm', np.Inf)
epsilon = kwargs.setdefault('epsilon', 1.4901161193847656e-08)
retvals = optimize.fmin_bfgs(f, start_params, score, args=fargs,
gtol=gtol, norm=norm, epsilon=epsilon,
maxiter=maxiter, full_output=full_output,
disp=disp, retall=retall, callback=callback)
if full_output:
if not retall:
xopt, fopt, gopt, Hinv, fcalls, gcalls, warnflag = retvals
else:
xopt, fopt, gopt, Hinv, fcalls, gcalls, warnflag, allvecs =\
retvals
converged = not warnflag
retvals = {'fopt' : fopt, 'gopt' : gopt, 'Hinv' : Hinv,
'fcalls' : fcalls, 'gcalls' : gcalls, 'warnflag' :
warnflag, 'converged' : converged}
if retall:
retvals.update({'allvecs' : allvecs})
elif method == 'ncg':
fhess_p = kwargs.setdefault('fhess_p', None)
avextol = kwargs.setdefault('avextol', 1.0000000000000001e-05)
epsilon = kwargs.setdefault('epsilon', 1.4901161193847656e-08)
retvals = optimize.fmin_ncg(f, start_params, score, fhess_p=fhess_p,
fhess=hess, args=fargs, avextol=avextol,
epsilon=epsilon, maxiter=maxiter,
full_output=full_output, disp=disp, retall=retall,
callback=callback)
if full_output:
if not retall:
xopt, fopt, fcalls, gcalls, hcalls, warnflag = retvals
else:
xopt, fopt, fcalls, gcalls, hcalls, warnflag, allvecs =\
retvals
converged = not warnflag
retvals = {'fopt' : fopt, 'fcalls' : fcalls, 'gcalls' : gcalls,
'hcalls' : hcalls, 'warnflag' : warnflag,
'converged' : converged}
if retall:
retvals.update({'allvecs' : allvecs})
elif method == 'cg':
gtol = kwargs.setdefault('gtol', 1.0000000000000001e-05)
norm = kwargs.setdefault('norm', np.Inf)
epsilon = kwargs.setdefault('epsilon', 1.4901161193847656e-08)
retvals = optimize.fmin_cg(f, start_params, score,
gtol=gtol, norm=norm,
epsilon=epsilon, maxiter=maxiter,
full_output=full_output, disp=disp, retall=retall,
callback=callback)
if full_output:
if not retall:
xopt, fopt, fcalls, gcalls, warnflag = retvals
else:
xopt, fopt, fcalls, gcalls, warnflag, allvecs = retvals
converged = not warnflag
retvals = {'fopt' : fopt, 'fcalls' : fcalls, 'gcalls' : gcalls,
'warnflag' : warnflag, 'converged' : converged}
if retall:
retvals.update({'allvecs' : allvecs})
elif method == 'powell':
xtol = kwargs.setdefault('xtol', 0.0001)
ftol = kwargs.setdefault('ftol', 0.0001)
maxfun = kwargs.setdefault('maxfun', None)
start_direc = kwargs.setdefault('start_direc', None)
retvals = optimize.fmin_powell(f, start_params, args=fargs,
xtol=xtol, ftol=ftol, maxiter=maxiter,
maxfun=maxfun, full_output=full_output, disp=disp,
retall=retall, callback=callback, direc=start_direc)
if full_output:
if not retall:
xopt, fopt, direc, niter, fcalls, warnflag = retvals
else:
xopt, fopt, direc, niter, fcalls, warnflag, allvecs =\
retvals
converged = not warnflag
retvals = {'fopt' : fopt, 'direc' : direc, 'iterations' : niter,
'fcalls' : fcalls, 'warnflag' : warnflag, 'converged' :
converged}
if retall:
retvals.update({'allvecs' : allvecs})
if not full_output:
xopt = retvals
#NOTE: better just to use the Analytic Hessian here, as approximation isn't
# great
# if method == 'bfgs' and full_output:
# Hinv = retvals.setdefault('Hinv', 0)
elif method == 'newton' and full_output:
Hinv = np.linalg.inv(-hopt)
else:
try:
Hinv = np.linalg.inv(-1*self.hessian(xopt))
except:
#might want custom warning ResultsWarning? NumericalWarning?
from warnings import warn
warndoc= 'Inverting hessian failed, no bse or cov_params \
available'
warn(warndoc, Warning)
Hinv = None
#TODO: add Hessian approximation and change the above if needed
mlefit = LikelihoodModelResults(self, xopt, Hinv, scale=1.)
#TODO: hardcode scale?
if isinstance(retvals, dict):
mlefit.mle_retvals = retvals
optim_settings = {'optimizer' : method, 'start_params' : start_params,
'maxiter' : maxiter, 'full_output' : full_output, 'disp' : disp,
'fargs' : fargs, 'callback' : callback, 'retall' : retall}
optim_settings.update(kwargs)
mlefit.mle_settings = optim_settings
self._results = mlefit
return mlefit
#TODO: the below is unfinished
[docs]class GenericLikelihoodModel(LikelihoodModel):
"""
Allows the fitting of any likelihood function via maximum likelihood.
A subclass needs to specify at least the log-likelihood
If the log-likelihood is specified for each observation, then results that
require the Jacobian will be available. (The other case is not tested yet.)
Notes
-----
Optimization methods that require only a likelihood function are 'nm' and
'powell'
Optimization methods that require a likelihood function and a
score/gradient are 'bfgs', 'cg', and 'ncg'. A function to compute the
Hessian is optional for 'ncg'.
Optimization method that require a likelihood function, a score/gradient,
and a Hessian is 'newton'
If they are not overwritten by a subclass, then numerical gradient, Jacobian
and Hessian of the log-likelihood are caclulated by numerical forward
differentiation. This might results in some cases in precision problems, and
the Hessian might not be positive definite. Even if the Hessian is not
positive definite the covariance matrix of the parameter estimates based on
the outer product of the Jacobian might still be valid.
Examples
--------
see also subclasses in directory miscmodels
import scikits.statsmodels.api as sm
data = sm.datasets.spector.load()
data.exog = sm.add_constant(data.exog)
# in this dir
from model import GenericLikelihoodModel
probit_mod = sm.Probit(data.endog, data.exog)
probit_res = probit_mod.fit()
loglike = probit_mod.loglike
score = probit_mod.score
mod = GenericLikelihoodModel(data.endog, data.exog, loglike, score)
res = mod.fit(method="nm", maxiter = 500)
import numpy as np
np.allclose(res.params, probit_res.params)
"""
def __init__(self, endog, exog=None, loglike=None, score=None,
hessian=None):
# let them be none in case user wants to use inheritance
if loglike:
self.loglike = loglike
if score:
self.score = score
if hessian:
self.hessian = hessian
self.confint_dist = norm
if not exog is None: #this won't work for ru2nmnl, maybe np.ndim of a dict?
#try:
self.nparams = self.df_model = exog.shape[1] if np.ndim(exog)==2 else 1
super(GenericLikelihoodModel, self).__init__(endog, exog)
#this is redundant and not used when subclassing
[docs] def initialize(self):
if not self.score: # right now score is not optional
from scikits.statsmodels.sandbox.regression.numdiff import approx_fprime1
self.score = approx_fprime1
if not self.hessian:
pass
else: # can use approx_hess_p if we have a gradient
if not self.hessian:
pass
[docs] def expandparams(self, params):
'''
expand to full parameter array when some parameters are fixed
Parameters
----------
params : array
reduced parameter array
Returns
-------
paramsfull : array
expanded parameter array where fixed parameters are included
Notes
-----
Calling this requires that self.fixed_params and self.fixed_paramsmask
are defined.
*developer notes:*
This can be used in the log-likelihood to ...
this could also be replaced by a more general parameter
transformation.
'''
paramsfull = self.fixed_params.copy()
paramsfull[self.fixed_paramsmask] = params
return paramsfull
[docs] def reduceparams(self, params):
return params[self.fixed_paramsmask]
[docs] def loglike(self, params):
return self.loglikeobs(params).sum(0)
[docs] def nloglike(self, params):
return -self.loglikeobs(params).sum(0)
[docs] def loglikeobs(self, params):
return -self.nloglikeobs(params)
[docs] def score(self, params):
'''
Gradient of log-likelihood evaluated at params
'''
from scikits.statsmodels.sandbox.regression.numdiff import approx_fprime1
return approx_fprime1(params, self.loglike, epsilon=1e-4).ravel()
[docs] def jac(self, params, **kwds):
'''
Jacobian/Gradient of log-likelihood evaluated at params for each
observation.
'''
kwds.setdefault('epsilon', 1e-4)
from scikits.statsmodels.sandbox.regression.numdiff import approx_fprime1
return approx_fprime1(params, self.loglikeobs, **kwds)
[docs] def hessian(self, params):
'''
Hessian of log-likelihood evaluated at params
'''
from scikits.statsmodels.sandbox.regression.numdiff import approx_hess
return approx_hess(params, self.loglike)[0] #need options for hess (epsilon)
[docs] def fit(self, start_params=None, method='nm', maxiter=500, full_output=1,
disp=1, callback=None, retall=0, **kwargs):
"""
Fit the model using maximum likelihood.
The rest of the docstring is from
scikits.statsmodels.LikelihoodModel.fit
"""
if start_params is None:
if hasattr(self, 'start_params'):
start_params = self.start_params
else:
start_params = 0.1 * np.ones(self.nparams)
mlefit = super(GenericLikelihoodModel, self).fit(start_params=start_params,
method=method, maxiter=maxiter, full_output=full_output,
disp=disp, callback=callback, **kwargs)
genericmlefit = GenericLikelihoodModelResults(self, mlefit)
return genericmlefit
#fit.__doc__ += LikelihoodModel.fit.__doc__
#------------------------------
#TODO: the following have been moved to the result mixin class
# check if anything is still using them from here
@cache_readonly
def jacv(self):
if not hasattr(self, '_results'):
raise ValueError('need to call fit first')
return self.jac(self._results.params)
@cache_readonly
def hessv(self):
if not hasattr(self, '_results'):
raise ValueError('need to call fit first')
return self.hessian(self._results.params)
# the following could be moved to results
@cache_readonly
def covjac(self):
'''
covariance of parameters based on loglike outer product of jacobian
'''
## if not hasattr(self, '_results'):
## raise ValueError('need to call fit first')
## #self.fit()
## self.jacv = jacv = self.jac(self._results.params)
jacv = self.jacv
return np.linalg.inv(np.dot(jacv.T, jacv))
@cache_readonly
def covjhj(self):
jacv = self.jacv
## hessv = self.hessv
## hessinv = np.linalg.inv(hessv)
## self.hessinv = hessinv
hessinv = self._results.cov_params()
return np.dot(hessinv, np.dot(np.dot(jacv.T, jacv), hessinv))
@cache_readonly
def bsejhj(self):
return np.sqrt(np.diag(self.covjhj))
@cache_readonly
def bsejac(self):
return np.sqrt(np.diag(self.covjac))
[docs]class Results(object):
"""
Class to contain model results
Parameters
----------
model : class instance
the previously specified model instance
params : array
parameter estimates from the fit model
"""
def __init__(self, model, params, **kwd):
self.__dict__.update(kwd)
self.initialize(model, params, **kwd)
[docs] def initialize(self, model, params, **kwd):
self.params = params
self.model = model
#TODO: public method?
[docs]class LikelihoodModelResults(Results):
"""
Class to contain results from likelihood models
Parameters
-----------
model : LikelihoodModel instance or subclass instance
LikelihoodModelResults holds a reference to the model that is fit.
params : 1d array_like
parameter estimates from estimated model
normalized_cov_params : 2d array
Normalized (before scaling) covariance of params. (dot(X.T,X))**-1
scale : float
For (some subset of models) scale will typically be the
mean square error from the estimated model (sigma^2)
Returns
-------
**Attributes**
mle_retvals : dict
Contains the values returned from the chosen optimization method if
full_output is True during the fit. Available only if the model
is fit by maximum likelihood. See notes below for the output from
the different methods.
mle_settings : dict
Contains the arguments passed to the chosen optimization method.
Available if the model is fit by maximum likelihood. See
LikelihoodModel.fit for more information.
model : model instance
LikelihoodResults contains a reference to the model that is fit.
params : ndarray
The parameters estimated for the model.
scale : float
The scaling factor of the model given during instantiation.
tvalues : array
The t-values of the standard errors.
Notes
--------
The covariance of params is given by scale times normalized_cov_params.
Return values by solver if full_ouput is True during fit:
'newton'
fopt : float
The value of the (negative) loglikelihood at its
minimum.
iterations : int
Number of iterations performed.
score : ndarray
The score vector at the optimum.
Hessian : ndarray
The Hessian at the optimum.
warnflag : int
1 if maxiter is exceeded. 0 if successful convergence.
converged : bool
True: converged. False: did not converge.
allvecs : list
List of solutions at each iteration.
'nm'
fopt : float
The value of the (negative) loglikelihood at its
minimum.
iterations : int
Number of iterations performed.
warnflag : int
1: Maximum number of function evaluations made.
2: Maximum number of iterations reached.
converged : bool
True: converged. False: did not converge.
allvecs : list
List of solutions at each iteration.
'bfgs'
fopt : float
Value of the (negative) loglikelihood at its minimum.
gopt : float
Value of gradient at minimum, which should be near 0.
Hinv : ndarray
value of the inverse Hessian matrix at minimum. Note
that this is just an approximation and will often be
different from the value of the analytic Hessian.
fcalls : int
Number of calls to loglike.
gcalls : int
Number of calls to gradient/score.
warnflag : int
1: Maximum number of iterations exceeded. 2: Gradient
and/or function calls are not changing.
converged : bool
True: converged. False: did not converge.
allvecs : list
Results at each iteration.
'powell'
fopt : float
Value of the (negative) loglikelihood at its minimum.
direc : ndarray
Current direction set.
iterations : int
Number of iterations performed.
fcalls : int
Number of calls to loglike.
warnflag : int
1: Maximum number of function evaluations. 2: Maximum number
of iterations.
converged : bool
True : converged. False: did not converge.
allvecs : list
Results at each iteration.
'cg'
fopt : float
Value of the (negative) loglikelihood at its minimum.
fcalls : int
Number of calls to loglike.
gcalls : int
Number of calls to gradient/score.
warnflag : int
1: Maximum number of iterations exceeded. 2: Gradient and/
or function calls not changing.
converged : bool
True: converged. False: did not converge.
allvecs : list
Results at each iteration.
'ncg'
fopt : float
Value of the (negative) loglikelihood at its minimum.
fcalls : int
Number of calls to loglike.
gcalls : int
Number of calls to gradient/score.
hcalls : int
Number of calls to hessian.
warnflag : int
1: Maximum number of iterations exceeded.
converged : bool
True: converged. False: did not converge.
allvecs : list
Results at each iteration.
"""
def __init__(self, model, params, normalized_cov_params=None, scale=1.):
super(LikelihoodModelResults, self).__init__(model, params)
self.normalized_cov_params = normalized_cov_params
self.scale = scale
[docs] def normalized_cov_params(self):
raise NotImplementedError
#JP: add methods that are valid generically higher up in class hierarchy
@cache_readonly
def llf(self):
return self.model.loglike(self.params)
@cache_readonly
def bse(self):
return np.sqrt(np.diag(self.cov_params()))
[docs] def t(self, column=None):
"""
deprecated: Return the t-statistic for a given parameter estimate.
FutureWarning: use attribute tvalues instead, t will be removed
in the next release
Parameters
----------
column : array-like
The columns for which you would like the t-value.
Note that this uses Python's indexing conventions.
See also
---------
Use t_test for more complicated t-statistics.
Examples
--------
>>> import scikits.statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> results = sm.OLS(data.endog, data.exog).fit()
>>> results.tvalues
array([ 0.17737603, -1.06951632, -4.13642736, -4.82198531, -0.22605114,
4.01588981, -3.91080292])
>>> results.tvalues[[1,2,4]]
array([-1.06951632, -4.13642736, -0.22605114])
>>> import numpy as np
>>> results.tvalues[np.array([1,2,4]]
array([-1.06951632, -4.13642736, -0.22605114])
"""
import warnings
warnings.warn("`t` will be removed in the next release, use attribute"
"`tvalues` instead", FutureWarning)
if self.normalized_cov_params is None:
raise ValueError('need covariance of parameters for computing T '
'statistics')
if column is None:
column = range(self.params.shape[0])
column = np.asarray(column)
_params = self.params[column]
_cov = self.cov_params(column=column)
if _cov.ndim == 2:
_cov = np.diag(_cov)
_t = _params * recipr(np.sqrt(_cov))
# repicr drops precision for MNLogit?
_t = _params / np.sqrt(_cov)
return _t
@cache_readonly
def tvalues(self):
"""
Return the t-statistic for a given parameter estimate.
"""
return self.params / self.bse
[docs] def cov_params(self, r_matrix=None, column=None, scale=None, other=None):
"""
Returns the variance/covariance matrix.
The variance/covariance matrix can be of a linear contrast
of the estimates of params or all params multiplied by scale which
will usually be an estimate of sigma^2. Scale is assumed to be
a scalar.
Parameters
----------
r_matrix : array-like
Can be 1d, or 2d. Can be used alone or with other.
column : array-like, optional
Must be used on its own. Can be 0d or 1d see below.
scale : float, optional
Can be specified or not. Default is None, which means that
the scale argument is taken from the model.
other : array-like, optional
Can be used when r_matrix is specified.
Returns
-------
(The below are assumed to be in matrix notation.)
cov : ndarray
If no argument is specified returns the covariance matrix of a model
(scale)*(X.T X)^(-1)
If contrast is specified it pre and post-multiplies as follows
(scale) * r_matrix (X.T X)^(-1) r_matrix.T
If contrast and other are specified returns
(scale) * r_matrix (X.T X)^(-1) other.T
If column is specified returns
(scale) * (X.T X)^(-1)[column,column] if column is 0d
OR
(scale) * (X.T X)^(-1)[column][:,column] if column is 1d
"""
if self.normalized_cov_params is None:
raise ValueError('need covariance of parameters for computing '
'(unnormalized) covariances')
if column is not None and (r_matrix is not None or other is not None):
raise ValueError('Column should be specified without other '
'arguments.')
if other is not None and r_matrix is None:
raise ValueError('other can only be specified with r_matrix')
if scale is None:
scale = self.scale
if column is not None:
column = np.asarray(column)
if column.shape == ():
return self.normalized_cov_params[column, column] * scale
else:
return self.normalized_cov_params[column][:,column] * scale
elif r_matrix is not None:
r_matrix = np.asarray(r_matrix)
if r_matrix.shape == ():
raise ValueError("r_matrix should be 1d or 2d")
if other is None:
other = r_matrix
else:
other = np.asarray(other)
tmp = np.dot(r_matrix, np.dot(self.normalized_cov_params,
np.transpose(other)))
return tmp * scale
if r_matrix is None and column is None:
return self.normalized_cov_params * scale
#TODO: make sure this works as needed for GLMs
[docs] def t_test(self, r_matrix, q_matrix=None, scale=None):
"""
Compute a tcontrast/t-test for a row vector array of the form Rb = q
where R is r_matrix, b = the parameter vector, and q is q_matrix.
Parameters
----------
r_matrix : array-like
A length p row vector specifying the linear restrictions.
q_matrix : array-like or scalar, optional
Either a scalar or a length p row vector.
scale : float, optional
An optional `scale` to use. Default is the scale specified
by the model fit.
Examples
--------
>>> import numpy as np
>>> import scikits.statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> results = sm.OLS(data.endog, data.exog).fit()
>>> r = np.zeros_like(results.params)
>>> r[4:6] = [1,-1]
>>> print r
[ 0. 0. 0. 0. 1. -1. 0.]
r tests that the coefficients on the 5th and 6th independent
variable are the same.
>>>T_Test = results.t_test(r)
>>>print T_test
<T contrast: effect=-1829.2025687192481, sd=455.39079425193762,
t=-4.0167754636411717, p=0.0015163772380899498, df_denom=9>
>>> T_test.effect
-1829.2025687192481
>>> T_test.sd
455.39079425193762
>>> T_test.t
-4.0167754636411717
>>> T_test.p
0.0015163772380899498
See also
---------
t : method to get simpler t values
f_test : for f tests
"""
r_matrix = np.atleast_2d(np.asarray(r_matrix))
num_ttests = r_matrix.shape[0]
num_params = r_matrix.shape[1]
if self.normalized_cov_params is None:
raise ValueError('Need covariance of parameters for computing '
'T statistics')
if num_params != self.params.shape[0]:
raise ValueError, 'r_matrix and params are not aligned'
if q_matrix is None:
q_matrix = np.zeros(num_ttests)
else:
q_matrix = np.asarray(q_matrix)
if q_matrix.size > 1:
if q_matrix.shape[0] != num_ttests:
raise ValueError("r_matrix and q_matrix must have the same "
"number of rows")
_t = _sd = None
_effect = np.dot(r_matrix, self.params)
if num_ttests > 1:
_sd = np.sqrt(np.diag(self.cov_params(r_matrix=r_matrix)))
else:
_sd = np.sqrt(self.cov_params(r_matrix=r_matrix))
_t = (_effect-q_matrix) * recipr(_sd)
return ContrastResults(effect=_effect, t=_t, sd=_sd,
df_denom=self.model.df_resid)
#TODO: untested for GLMs?
[docs] def f_test(self, r_matrix, q_matrix=None, scale=1.0, invcov=None):
"""
Compute an Fcontrast/F-test for a contrast matrix.
Here, matrix `r_matrix` is assumed to be non-singular. More precisely,
r_matrix (pX pX.T) r_matrix.T
is assumed invertible. Here, pX is the generalized inverse of the
design matrix of the model. There can be problems in non-OLS models
where the rank of the covariance of the noise is not full.
Parameters
----------
r_matrix : array-like
q x p array where q is the number of restrictions to test and
p is the number of regressors in the full model fit.
If q is 1 then f_test(r_matrix).fvalue is equivalent to
the square of t_test(r_matrix).t
q_matrix : array-like
q x 1 array, that represents the sum of each linear restriction.
Default is all zeros for each restriction.
scale : float, optional
Default is 1.0 for no scaling.
invcov : array-like, optional
A qxq matrix to specify an inverse covariance
matrix based on a restrictions matrix.
Examples
--------
>>> import numpy as np
>>> import scikits.statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> results = sm.OLS(data.endog, data.exog).fit()
>>> A = np.identity(len(results.params))
>>> A = A[:-1,:]
This tests that each coefficient is jointly statistically
significantly different from zero.
>>> print results.f_test(A)
<F contrast: F=330.28533923463488, p=4.98403052872e-10, df_denom=9, df_num=6>
Compare this to
>>> results.F
330.2853392346658
>>> results.F_p
4.98403096572e-10
>>> B = np.array(([0,1,-1,0,0,0,0],[0,0,0,0,1,-1,0]))
This tests that the coefficient on the 2nd and 3rd regressors are
equal and jointly that the coefficient on the 5th and 6th regressors
are equal.
>>> print results.f_test(B)
<F contrast: F=9.740461873303655, p=0.00560528853174, df_denom=9, df_num=2>
See also
--------
scikits.statsmodels.contrasts
scikits.statsmodels.model.t_test
"""
r_matrix = np.asarray(r_matrix)
r_matrix = np.atleast_2d(r_matrix)
if self.normalized_cov_params is None:
raise ValueError('need covariance of parameters for computing '
'F statistics')
cparams = np.dot(r_matrix, self.params[:,None])
J = float(r_matrix.shape[0]) # number of restrictions
if q_matrix is None:
q_matrix = np.zeros(J)
else:
q_matrix = np.asarray(q_matrix)
if q_matrix.ndim == 1:
q_matrix = q_matrix[:,None]
if q_matrix.shape[0] != J:
raise ValueError("r_matrix and q_matrix must have the same "
"number of rows")
Rbq = cparams - q_matrix
if invcov is None:
invcov = np.linalg.inv(self.cov_params(r_matrix=r_matrix))
F = np.dot(np.dot(Rbq.T,invcov),Rbq)/J
return ContrastResults(F=F, df_denom=self.model.df_resid,
df_num=invcov.shape[0])
[docs] def conf_int(self, alpha=.05, cols=None, method='default'):
"""
Returns the confidence interval of the fitted parameters.
Parameters
----------
alpha : float, optional
The `alpha` level for the confidence interval.
ie., The default `alpha` = .05 returns a 95% confidence interval.
cols : array-like, optional
`cols` specifies which confidence intervals to return
method : string
Not Implemented Yet
Method to estimate the confidence_interval.
"Default" : uses self.bse which is based on inverse Hessian for MLE
"jhj" :
"jac" :
"boot-bse"
"boot_quant"
"profile"
Returns
--------
conf_int : array
Each row contains [lower, upper] confidence interval
Examples
--------
>>> import scikits.statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> results = sm.OLS(data.endog, data.exog).fit()
>>> results.conf_int()
array([[ -1.77029035e+02, 2.07152780e+02],
[ -1.11581102e-01, 3.99427438e-02],
[ -3.12506664e+00, -9.15392966e-01],
[ -1.51794870e+00, -5.48505034e-01],
[ -5.62517214e-01, 4.60309003e-01],
[ 7.98787515e+02, 2.85951541e+03],
[ -5.49652948e+06, -1.46798779e+06]])
>>> results.conf_int(cols=(1,2))
array([[-0.1115811 , 0.03994274],
[-3.12506664, -0.91539297]])
Notes
-----
The confidence interval is based on Student's t distribution for all
models except RLM and GLM, which uses the standard normal distribution.
"""
bse = self.bse
#TODO: simplify structure, DRY
if hasattr(self.model, 'confint_dist'):
dist = self.model.confint_dist
elif self.__class__.__name__ in ['RLMResults','GLMResults','DiscreteResults']:
dist = norm
else:
dist = t
if cols is None and dist == t:
lower = self.params - dist.ppf(1-alpha/2,self.model.df_resid) *\
bse
upper = self.params + dist.ppf(1-alpha/2,self.model.df_resid) *\
bse
elif cols is None and dist == norm:
lower = self.params - dist.ppf(1-alpha/2) * bse
upper = self.params + dist.ppf(1-alpha/2) * bse
elif cols is not None and dist == t:
cols = np.asarray(cols)
lower = self.params[cols] - dist.ppf(1-\
alpha/2,self.model.df_resid) * bse[cols]
upper = self.params[cols] + dist.ppf(1-\
alpha/2,self.model.df_resid) * bse[cols]
elif cols is not None and dist == norm:
cols = np.asarray(cols)
lower = self.params[cols] - dist.ppf(1-alpha/2) * bse[cols]
upper = self.params[cols] + dist.ppf(1-alpha/2) * bse[cols]
return np.asarray(zip(lower,upper))
[docs]class ResultMixin(object):
@cache_readonly
def df_modelwc(self):
#collect different ways of defining the number of parameters, used for aic, bic
if hasattr(self, 'df_model'):
if hasattr(self, 'hasconst'):
hasconst = self.hasconst
else:
hasconst = 1 #default assumption
return self.df_model + hasconst
else:
return self.params.size
@cache_readonly
def aic(self):
return -2 * self.llf + 2 * (self.df_modelwc)
@cache_readonly
def bic(self):
return -2 * self.llf + np.log(self.nobs) * (self.df_modelwc)
@cache_readonly
def jacv(self):
'''cached Jacobian of log-likelihood
'''
return self.model.jac(self.params)
@cache_readonly
def hessv(self):
'''cached Hessian of log-likelihood
'''
return self.model.hessian(self.params)
@cache_readonly
def covjac(self):
'''
covariance of parameters based on outer product of jacobian of
log-likelihood
'''
## if not hasattr(self, '_results'):
## raise ValueError('need to call fit first')
## #self.fit()
## self.jacv = jacv = self.jac(self._results.params)
jacv = self.jacv
return np.linalg.inv(np.dot(jacv.T, jacv))
@cache_readonly
def covjhj(self):
'''covariance of parameters based on HJJH
dot product of Hessian, Jacobian, Jacobian, Hessian of likelihood
name should be covhjh
'''
jacv = self.jacv
## hessv = self.hessv
## hessinv = np.linalg.inv(hessv)
## self.hessinv = hessinv
hessinv = self.cov_params()
return np.dot(hessinv, np.dot(np.dot(jacv.T, jacv), hessinv))
@cache_readonly
def bsejhj(self):
'''standard deviation of parameter estimates based on covHJH
'''
return np.sqrt(np.diag(self.covjhj))
@cache_readonly
def bsejac(self):
'''standard deviation of parameter estimates based on covjac
'''
return np.sqrt(np.diag(self.covjac))
[docs] def bootstrap(self, nrep=100, method='nm', disp=0, store=1):
'''simple bootstrap to get mean and variance of estimator
see notes
Parameters
----------
nrep : int
number of bootstrap replications
method : str
optimization method to use
disp : bool
If true, then optimization prints results
store : bool
If true, then parameter estimates for all bootstrap iterations
are attached in self.bootstrap_results
Returns
-------
mean : array
mean of parameter estimates over bootstrap replications
std : array
standard deviation of parameter estimates over bootstrap replications
Notes
-----
This was mainly written to compare estimators of the standard errors
of the parameter estimates.
It uses independent random sampling from the original endog and exog, and
therefore is only correct if observations are independently distributed.
This will be moved to apply only to models with independently distributed
observations.
'''
results = []
print self.model.__class__
hascloneattr = True if hasattr(self, 'cloneattr') else False
for i in xrange(nrep):
rvsind = np.random.randint(self.nobs-1, size=self.nobs )
#this needs to set startparam and get other defining attributes
#need a clone method on model
fitmod = self.model.__class__(self.endog[rvsind],
self.exog[rvsind,:]
)
if hascloneattr:
for attr in self.model.cloneattr:
setattr(fitmod, attr, getattr(self.model, attr))
fitres = fitmod.fit(method=method, disp=disp)
results.append(fitres.params)
results = np.array(results)
if store:
self.bootstrap_results = results
return results.mean(0), results.std(0), results
[docs] def get_nlfun(self, fun):
#I think this is supposed to get the delta method that is currently
#in miscmodels count (as part of Poisson example)
pass
[docs]class GenericLikelihoodModelResults(LikelihoodModelResults, ResultMixin):
"""
A results class for the discrete dependent variable models.
..Warning :
The following description has not been updated to this version/class.
Where are AIC, BIC, ....? docstring looks like copy from discretemod
Parameters
----------
model : A DiscreteModel instance
mlefit : instance of LikelihoodResults
This contains the numerical optimization results as returned by
LikelihoodModel.fit(), in a superclass of GnericLikelihoodModels
Returns
-------
*Attributes*
Warning most of these are not available yet
aic : float
Akaike information criterion. -2*(`llf` - p) where p is the number
of regressors including the intercept.
bic : float
Bayesian information criterion. -2*`llf` + ln(`nobs`)*p where p is the
number of regressors including the intercept.
bse : array
The standard errors of the coefficients.
df_resid : float
See model definition.
df_model : float
See model definition.
fitted_values : array
Linear predictor XB.
llf : float
Value of the loglikelihood
llnull : float
Value of the constant-only loglikelihood
llr : float
Likelihood ratio chi-squared statistic; -2*(`llnull` - `llf`)
llr_pvalue : float
The chi-squared probability of getting a log-likelihood ratio
statistic greater than llr. llr has a chi-squared distribution
with degrees of freedom `df_model`.
prsquared : float
McFadden's pseudo-R-squared. 1 - (`llf`/`llnull`)
"""
def __init__(self, model, mlefit):
# super(DiscreteResults, self).__init__(model, params,
# np.linalg.inv(-hessian), scale=1.)
self.model = model
#self.df_model = model.df_model
#self.df_resid = model.df_resid
self.endog = model.endog
self.exog = model.exog
self.nobs = model.endog.shape[0]
self._cache = resettable_cache()
self.__dict__.update(mlefit.__dict__)