"""
Functions that are general enough to use for any model fitting. The idea is
to untie these from LikelihoodModel so that they may be re-used generally.
"""
from __future__ import print_function
import numpy as np
from scipy import optimize
def _check_method(method, methods):
if method not in methods:
message = "Unknown fit method %s" % method
raise ValueError(message)
[docs]class Optimizer(object):
def _fit(self, objective, gradient, start_params, fargs, kwargs,
hessian=None, method='newton', maxiter=100, full_output=True,
disp=True, callback=None, retall=False):
"""
Fit function for any model with an objective function.
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','ncg','basinhopping',
'minimize'}
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, 'ncg' for Newton-
conjugate gradient, 'basinhopping' for global basin-hopping
solver, if available or a generic 'minimize' which is a wrapper for
scipy.optimize.minimize. `method` determines which solver from
scipy.optimize is used. The explicit arguments in `fit` are passed
to the solver, with the exception of the basin-hopping 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 and for the list of explicit arguments that the
basin-hopping solver supports..
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.
Returns
-------
xopt : array
The solution to the objective function
retvals : dict, None
If `full_output` is True then this is a dictionary which holds
information returned from the solver used. If it is False, this is
None.
optim_settings : dict
A dictionary that contains the parameters passed to the solver.
Notes
-----
The 'basinhopping' solver ignores `maxiter`, `retall`, `full_output`
explicit arguments.
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.
'lbfgs'
m : int
The maximum number of variable metric corrections used to
define the limited memory matrix. (The limited memory BFGS
method does not store the full hessian but uses this many
terms in an approximation to it.)
pgtol : float
The iteration will stop when
``max{|proj g_i | i = 1, ..., n} <= pgtol`` where pg_i is
the i-th component of the projected gradient.
factr : float
The iteration stops when
``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
where eps is the machine precision, which is automatically
generated by the code. Typical values for factr are: 1e12
for low accuracy; 1e7 for moderate accuracy; 10.0 for
extremely high accuracy. See Notes for relationship to
ftol, which is exposed (instead of factr) by the
scipy.optimize.minimize interface to L-BFGS-B.
maxfun : int
Maximum number of iterations.
epsilon : float
Step size used when approx_grad is True, for numerically
calculating the gradient
approx_grad : bool
Whether to approximate the gradient numerically (in which
case func returns only the function value).
'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.
'basinhopping'
niter : integer
The number of basin hopping iterations.
niter_success : integer
Stop the run if the global minimum candidate remains the
same for this number of iterations.
T : float
The "temperature" parameter for the accept or reject
criterion. Higher "temperatures" mean that larger jumps
in function value will be accepted. For best results
`T` should be comparable to the separation (in function
value) between local minima.
stepsize : float
Initial step size for use in the random displacement.
interval : integer
The interval for how often to update the `stepsize`.
minimizer : dict
Extra keyword arguments to be passed to the minimizer
`scipy.optimize.minimize()`, for example 'method' - the
minimization method (e.g. 'L-BFGS-B'), or 'tol' - the
tolerance for termination. Other arguments are mapped from
explicit argument of `fit`:
- `args` <- `fargs`
- `jac` <- `score`
- `hess` <- `hess`
'minimize'
min_method : str, optional
Name of minimization method to use.
Any method specific arguments can be passed directly.
For a list of methods and their arguments, see
documentation of `scipy.optimize.minimize`.
If no method is specified, then BFGS is used.
"""
#TODO: generalize the regularization stuff
# Extract kwargs specific to fit_regularized calling fit
extra_fit_funcs = kwargs.setdefault('extra_fit_funcs', dict())
methods = ['newton', 'nm', 'bfgs', 'lbfgs', 'powell', 'cg', 'ncg',
'basinhopping', 'minimize']
methods += extra_fit_funcs.keys()
method = method.lower()
_check_method(method, methods)
fit_funcs = {
'newton': _fit_newton,
'nm': _fit_nm, # Nelder-Mead
'bfgs': _fit_bfgs,
'lbfgs': _fit_lbfgs,
'cg': _fit_cg,
'ncg': _fit_ncg,
'powell': _fit_powell,
'basinhopping': _fit_basinhopping,
'minimize': _fit_minimize # wrapper for scipy.optimize.minimize
}
#NOTE: fit_regularized checks the methods for these but it should be
# moved up probably
if extra_fit_funcs:
fit_funcs.update(extra_fit_funcs)
func = fit_funcs[method]
xopt, retvals = func(objective, gradient, start_params, fargs, kwargs,
disp=disp, maxiter=maxiter, callback=callback,
retall=retall, full_output=full_output,
hess=hessian)
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)
# set as attributes or return?
return xopt, retvals, optim_settings
def _fit_constrained(self, params):
"""
TODO: how to add constraints?
Something like
sm.add_constraint(Model, func)
or
model_instance.add_constraint(func)
model_instance.add_constraint("x1 + x2 = 2")
result = model_instance.fit()
"""
raise NotImplementedError
def _fit_regularized(self, params):
# TODO: code won't necessarily be general here. 3 options.
# 1) setup for scipy.optimize.fmin_sqlsqp
# 2) setup for cvxopt
# 3) setup for openopt
raise NotImplementedError
########################################
# Helper functions to fit
def _fit_minimize(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
kwargs.setdefault('min_method', 'BFGS')
# prepare options dict for minimize
filter_opts = ['extra_fit_funcs', 'niter', 'min_method', 'tol']
options = dict((k,v) for k,v in kwargs.items() if k not in filter_opts)
options['disp'] = disp
options['maxiter'] = maxiter
# Use Hessian/Jacobian only if they're required by the method
no_hess = ['Nelder-Mead', 'Powell', 'CG', 'BFGS', 'COBYLA', 'SLSQP']
no_jac = ['Nelder-Mead', 'Powell', 'COBYLA']
if kwargs['min_method'] in no_hess: hess = None
if kwargs['min_method'] in no_jac: score = None
res = optimize.minimize(f, start_params, args=fargs, method=kwargs['min_method'],
jac=score, hess=hess, callback=callback, options=options)
xopt = res.x
retvals = None
if full_output:
nit = getattr(res, 'nit', np.nan) # scipy 0.14 compat
retvals = {'fopt': res.fun, 'iterations': nit,
'fcalls': res.nfev, 'warnflag': res.status,
'converged': res.success}
if retall:
retvals.update({'allvecs': res.values()})
return xopt, retvals
[docs]def _fit_newton(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None, ridge_factor=1e-10):
tol = kwargs.setdefault('tol', 1e-8)
iterations = 0
oldparams = np.inf
newparams = np.asarray(start_params)
if retall:
history = [oldparams, newparams]
while (iterations < maxiter and np.any(np.abs(newparams -
oldparams) > tol)):
H = np.asarray(hess(newparams))
# regularize Hessian, not clear what ridge factor should be
# keyword option with absolute default 1e-10, see #1847
if not np.all(ridge_factor == 0):
H[np.diag_indices(H.shape[0])] += ridge_factor
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:
xopt = newparams
retvals = None
return xopt, retvals
[docs]def _fit_bfgs(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
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})
else:
xopt = retvals
retvals = None
return xopt, retvals
[docs]def _fit_lbfgs(f, score, start_params, fargs, kwargs, disp=True, maxiter=100,
callback=None, retall=False, full_output=True, hess=None):
"""
Fit model using L-BFGS algorithm
Parameters
----------
f : function
Returns negative log likelihood given parameters.
score : function
Returns gradient of negative log likelihood with respect to params.
Notes
-----
Within the mle part of statsmodels, the log likelihood function and
its gradient with respect to the parameters do not have notationally
consistent sign.
"""
# Use unconstrained optimization by default.
bounds = kwargs.setdefault('bounds', [(None, None)] * len(start_params))
kwargs.setdefault('iprint', 0)
# Pass the following keyword argument names through to fmin_l_bfgs_b
# if they are present in kwargs, otherwise use the fmin_l_bfgs_b
# default values.
names = ('m', 'pgtol', 'factr', 'maxfun', 'epsilon', 'approx_grad')
extra_kwargs = dict((x, kwargs[x]) for x in names if x in kwargs)
# Extract values for the options related to the gradient.
approx_grad = kwargs.get('approx_grad', False)
loglike_and_score = kwargs.get('loglike_and_score', None)
epsilon = kwargs.get('epsilon', None)
# The approx_grad flag has superpowers nullifying the score function arg.
if approx_grad:
score = None
# Choose among three options for dealing with the gradient (the gradient
# of a log likelihood function with respect to its parameters
# is more specifically called the score in statistics terminology).
# The first option is to use the finite-differences
# approximation that is built into the fmin_l_bfgs_b optimizer.
# The second option is to use the provided score function.
# The third option is to use the score component of a provided
# function that simultaneously evaluates the log likelihood and score.
if epsilon and not approx_grad:
raise ValueError('a finite-differences epsilon was provided '
'even though we are not using approx_grad')
if approx_grad and loglike_and_score:
raise ValueError('gradient approximation was requested '
'even though an analytic loglike_and_score function '
'was given')
if loglike_and_score:
func = lambda p, *a : tuple(-x for x in loglike_and_score(p, *a))
elif score:
func = f
extra_kwargs['fprime'] = score
elif approx_grad:
func = f
retvals = optimize.fmin_l_bfgs_b(func, start_params, maxiter=maxiter,
callback=callback, args=fargs,
bounds=bounds, disp=disp,
**extra_kwargs)
if full_output:
xopt, fopt, d = retvals
# The warnflag is
# 0 if converged
# 1 if too many function evaluations or too many iterations
# 2 if stopped for another reason, given in d['task']
warnflag = d['warnflag']
converged = (warnflag == 0)
gopt = d['grad']
fcalls = d['funcalls']
iterations = d['nit']
retvals = {'fopt': fopt, 'gopt': gopt, 'fcalls': fcalls,
'warnflag': warnflag, 'converged': converged,
'iterations': iterations}
else:
xopt = retvals[0]
retvals = None
return xopt, retvals
[docs]def _fit_nm(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
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})
else:
xopt = retvals
retvals = None
return xopt, retvals
[docs]def _fit_cg(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
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})
else:
xopt = retvals
retvals = None
return xopt, retvals
[docs]def _fit_ncg(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
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})
else:
xopt = retvals
retvals = None
return xopt, retvals
[docs]def _fit_powell(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
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})
else:
xopt = retvals
retvals = None
return xopt, retvals
[docs]def _fit_basinhopping(f, score, start_params, fargs, kwargs, disp=True,
maxiter=100, callback=None, retall=False,
full_output=True, hess=None):
from copy import copy
kwargs = copy(kwargs)
niter = kwargs.setdefault('niter', 100)
niter_success = kwargs.setdefault('niter_success', None)
T = kwargs.setdefault('T', 1.0)
stepsize = kwargs.setdefault('stepsize', 0.5)
interval = kwargs.setdefault('interval', 50)
minimizer_kwargs = kwargs.get('minimizer', {})
minimizer_kwargs['args'] = fargs
minimizer_kwargs['jac'] = score
method = minimizer_kwargs.get('method', None)
if method and method != 'L-BFGS-B': # l_bfgs_b doesn't take a hessian
minimizer_kwargs['hess'] = hess
retvals = optimize.basinhopping(f, start_params,
minimizer_kwargs=minimizer_kwargs,
niter=niter, niter_success=niter_success,
T=T, stepsize=stepsize, disp=disp,
callback=callback, interval=interval)
if full_output:
xopt, fopt, niter, fcalls = map(lambda x : getattr(retvals, x),
['x', 'fun', 'nit', 'nfev'])
converged = 'completed successfully' in retvals.message[0]
retvals = {'fopt': fopt, 'iterations': niter,
'fcalls': fcalls, 'converged': converged}
else:
xopt = retvals.x
retvals = None
return xopt, retvals
