def _hessian_finite_difference(self,
params,
approx_centered=False,
**kwargs):
params = np.array(params, ndmin=1)
warnings.warn(
'Calculation of the Hessian using finite differences'
' is usually subject to substantial approximation'
' errors.',
PrecisionWarning,
stacklevel=3,
)
if not approx_centered:
epsilon = _get_epsilon(params, 3, None, len(params))
else:
epsilon = _get_epsilon(params, 4, None, len(params)) / 2
hessian = approx_fprime(params,
self._score_finite_difference,
epsilon=epsilon,
kwargs=kwargs,
centered=approx_centered)
# TODO: changed this to nobs_effective, has to be changed when merging
# with statespace mlemodel
return hessian / (self.nobs_effective)
def approx_fprime(x, f, epsilon=None, args=(), kwargs=None, centered=True):
"""
Gradient of function, or Jacobian if function fun returns 1d array
Parameters
----------
x : array
parameters at which the derivative is evaluated
fun : function
`fun(*((x,)+args), **kwargs)` returning either one value or 1d array
epsilon : float, optional
Stepsize, if None, optimal stepsize is used. This is _EPS**(1/2)*x for
`centered` == False and _EPS**(1/3)*x for `centered` == True.
args : tuple
Tuple of additional arguments for function `fun`.
kwargs : dict
Dictionary of additional keyword arguments for function `fun`.
centered : bool
Whether central difference should be returned. If not, does forward
differencing.
Returns
-------
grad : array
gradient or Jacobian
Notes
-----
If fun returns a 1d array, it returns a Jacobian. If a 2d array is returned
by fun (e.g., with a value for each observation), it returns a 3d array
with the Jacobian of each observation with shape xk x nobs x xk. I.e.,
the Jacobian of the first observation would be [:, 0, :]
"""
kwargs = {} if kwargs is None else kwargs
x = np.atleast_1d(x).ravel()
n = len(x)
f0 = f(*(x, ) + args, **kwargs)
dim = np.atleast_1d(f0).shape # it could be a scalar
grad = np.zeros((n, ) + dim, float)
ei = np.zeros(np.shape(x), float)
if not centered:
epsilon = _get_epsilon(x, 2, epsilon, n)
for k in range(n):
ei[k] = epsilon[k]
grad[k, :] = (f(*(x + ei, ) + args, **kwargs) - f0) / epsilon[k]
ei[k] = 0.0
else:
epsilon = _get_epsilon(x, 3, epsilon, n) / 2.
for k in range(n):
ei[k] = epsilon[k]
grad[k, :] = (f(*(x + ei, ) + args, **kwargs) -
f(*(x - ei, ) + args, **kwargs)) / (2 * epsilon[k])
ei[k] = 0.0
# return grad
# return grad.T
# grad = grad.squeeze()
axes = list(range(grad.ndim))
axes[:2] = axes[1::-1]
return np.transpose(grad, axes=axes)
def _score_complex_step(self, params, **kwargs):
# the default epsilon can be too small
# inversion_method = INVERT_UNIVARIATE | SOLVE_LU
epsilon = _get_epsilon(params, 2., None, len(params))
kwargs['transformed'] = True
kwargs['complex_step'] = True
return approx_fprime_cs(params, self.loglike, epsilon=epsilon,
kwargs=kwargs)
def approx_jacobian(x,func,epsilon,*args):
'''Approximate the Jacobian matrix of callable function func
Parameters:
* x - The state vector at which the Jacobian matrix is desired
* func - A vector-valued function of the form f(x,*args)
* epsilon - The step size used to determine the partial derivatives. Set to None to select
the optimal step size.
* *args - Additional arguments passed to func
Returns:
An array of dimensions (lenf, lenx) where lenf is the length
of the outputs of func, and lenx is the number of inputs of func.
Notes:
The approximation is done using fourth order central difference method.
'''
if np.shape(x) == ():
n = 1
x = np.asarray([x])
else:
n = len(x)
method = 'FirstOrderCentralDifference'
method = 'FourthOrderCentralDifference'
x0 = np.asarray(x)
f0 = func(x0, *args)
if method == 'FirstOrderCentralDifference':
jac = np.zeros([len(x0),len(f0)])
df1 = np.zeros([len(x0),len(f0)])
df2 = np.zeros([len(x0),len(f0)])
dx = np.zeros(len(x0))
for i in range(len(x0)):
dx[i] = epsilon
df1[i] = func(*((x0+dx/2,)+args))
df2[i] = func(*((x0-dx/2,)+args))
jac[i] = (df1[i] - df2[i])/epsilon
dx[i] = 0.0
if method == 'FourthOrderCentralDifference':
epsilon = nd._get_epsilon(x,3,epsilon,n)/2.
jac = np.zeros([len(x0),len(f0)])
df1 = np.zeros([len(x0),len(f0)])
df2 = np.zeros([len(x0),len(f0)])
df3 = np.zeros([len(x0),len(f0)])
df4 = np.zeros([len(x0),len(f0)])
dx = np.zeros(len(x0))
for i in range(len(x0)):
dx[i] = epsilon[i]
df1[i] = -func(*((x0+2*dx,)+args))
df2[i] = 8*func(*((x0+dx,)+args))
df3[i] = -8*func(*((x0-dx,)+args))
df4[i] = func(*((x0-2*dx,)+args))
jac[i] = (df1[i]+df2[i] + df3[i] + df4[i])/(12*dx[i])
dx[i] = 0.0
return jac.transpose()
def approx_jacobian(x,func,epsilon,*args):
'''Approximate the Jacobian matrix of callable function func
Parameters:
* x - The state vector at which the Jacobian matrix is desired
* func - A vector-valued function of the form f(x,*args)
* epsilon - The step size used to determine the partial derivatives. Set to None to select
the optimal step size.
* *args - Additional arguments passed to func
Returns:
An array of dimensions (lenf, lenx) where lenf is the length
of the outputs of func, and lenx is the number of inputs of func.
Notes:
The approximation is done using fourth order central difference method.
'''
if np.shape(x) == ():
n = 1
x = np.asarray([x])
else:
n = len(x)
method = 'FirstOrderCentralDifference'
method = 'FourthOrderCentralDifference'
x0 = np.asarray(x)
f0 = func(x0, *args)
if method == 'FirstOrderCentralDifference':
jac = np.zeros([len(x0),len(f0)])
df1 = np.zeros([len(x0),len(f0)])
df2 = np.zeros([len(x0),len(f0)])
dx = np.zeros(len(x0))
for i in range(len(x0)):
dx[i] = epsilon
df1[i] = func(*((x0+dx/2,)+args))
df2[i] = func(*((x0-dx/2,)+args))
jac[i] = (df1[i] - df2[i])/epsilon
dx[i] = 0.0
if method == 'FourthOrderCentralDifference':
epsilon = nd._get_epsilon(x,3,epsilon,n)/2.
jac = np.zeros([len(x0),len(f0)])
df1 = np.zeros([len(x0),len(f0)])
df2 = np.zeros([len(x0),len(f0)])
df3 = np.zeros([len(x0),len(f0)])
df4 = np.zeros([len(x0),len(f0)])
dx = np.zeros(len(x0))
for i in range(len(x0)):
dx[i] = epsilon[i]
df1[i] = -func(*((x0+2*dx,)+args))
df2[i] = 8*func(*((x0+dx,)+args))
df3[i] = -8*func(*((x0-dx,)+args))
df4[i] = func(*((x0-2*dx,)+args))
jac[i] = (df1[i]+df2[i] + df3[i] + df4[i])/(12*dx[i])
dx[i] = 0.0
return jac.transpose()
def arma_scoreobs(endog,
ar_params=None,
ma_params=None,
sigma2=1,
prefix=None):
"""
Compute the score per observation (gradient of the loglikelihood function)
Parameters
----------
endog : ndarray
The observed time-series process.
ar_params : ndarray, optional
Autoregressive coefficients, not including the zero lag.
ma_params : ndarray, optional
Moving average coefficients, not including the zero lag, where the sign
convention assumes the coefficients are part of the lag polynomial on
the right-hand-side of the ARMA definition (i.e. they have the same
sign from the usual econometrics convention in which the coefficients
are on the right-hand-side of the ARMA definition).
sigma2 : ndarray, optional
The ARMA innovation variance. Default is 1.
prefix : str, optional
The BLAS prefix associated with the datatype. Default is to find the
best datatype based on given input. This argument is typically only
used internally.
Returns
----------
scoreobs : array
Score per observation, evaluated at the given parameters.
Notes
-----
This is a numerical approximation, calculated using first-order complex
step differentiation on the `arma_loglike` method.
"""
ar_params = [] if ar_params is None else ar_params
ma_params = [] if ma_params is None else ma_params
p = len(ar_params)
q = len(ma_params)
def func(params):
return arma_loglikeobs(endog, params[:p], params[p:p + q],
params[p + q:])
params0 = np.r_[ar_params, ma_params, sigma2]
epsilon = _get_epsilon(params0, 2., None, len(params0))
return approx_fprime_cs(params0, func, epsilon)
def _hessian_complex_step(self, params, **kwargs):
"""
Hessian matrix computed by second-order complex-step differentiation
on the `loglike` function.
"""
# the default epsilon can be too small
epsilon = _get_epsilon(params, 3., None, len(params))
kwargs['transformed'] = True
kwargs['complex_step'] = True
hessian = approx_hess_cs(
params, self.loglike, epsilon=epsilon, kwargs=kwargs)
# TODO: changed this to nobs_effective, has to be changed when merging
# with statespace mlemodel
return hessian / (self.nobs_effective)
def arma_scoreobs(endog, ar_params=None, ma_params=None, sigma2=1,
prefix=None):
"""
Compute the score per observation (gradient of the loglikelihood function)
Parameters
----------
endog : ndarray
The observed time-series process.
ar_params : ndarray, optional
Autoregressive coefficients, not including the zero lag.
ma_params : ndarray, optional
Moving average coefficients, not including the zero lag, where the sign
convention assumes the coefficients are part of the lag polynomial on
the right-hand-side of the ARMA definition (i.e. they have the same
sign from the usual econometrics convention in which the coefficients
are on the right-hand-side of the ARMA definition).
sigma2 : ndarray, optional
The ARMA innovation variance. Default is 1.
prefix : str, optional
The BLAS prefix associated with the datatype. Default is to find the
best datatype based on given input. This argument is typically only
used internally.
Returns
----------
scoreobs : array
Score per observation, evaluated at the given parameters.
Notes
-----
This is a numerical approximation, calculated using first-order complex
step differentiation on the `arma_loglike` method.
"""
ar_params = [] if ar_params is None else ar_params
ma_params = [] if ma_params is None else ma_params
p = len(ar_params)
q = len(ma_params)
def func(params):
return arma_loglikeobs(endog, params[:p], params[p:p + q],
params[p + q:])
params0 = np.r_[ar_params, ma_params, sigma2]
epsilon = _get_epsilon(params0, 2., None, len(params0))
return approx_fprime_cs(params0, func, epsilon)
def approx_fprime_cs(x, f, epsilon=None, args=(), kwargs=None):
'''
Calculate gradient or Jacobian with complex step derivative approximation
Parameters
----------
x : array
parameters at which the derivative is evaluated
f : function
`f(*((x,)+args), **kwargs)` returning either one value or 1d array
epsilon : float, optional
Stepsize, if None, optimal stepsize is used. Optimal step-size is
EPS*x. See note.
args : tuple
Tuple of additional arguments for function `f`.
kwargs : dict
Dictionary of additional keyword arguments for function `f`.
Returns
-------
partials : ndarray
array of partial derivatives, Gradient or Jacobian
Notes
-----
The complex-step derivative has truncation error O(epsilon**2), so
truncation error can be eliminated by choosing epsilon to be very small.
The complex-step derivative avoids the problem of round-off error with
small epsilon because there is no subtraction.
'''
# From Guilherme P. de Freitas, numpy mailing list
# May 04 2010 thread "Improvement of performance"
# http://mail.scipy.org/pipermail/numpy-discussion/2010-May/050250.html
kwargs = {} if kwargs is None else kwargs
x = np.atleast_1d(x).ravel()
n = len(x)
epsilon = _get_epsilon(x, 1, epsilon, n)
increments = np.identity(n) * 1j * epsilon
# TODO: see if this can be vectorized, but usually dim is small
partials = [
f(x + ih, *args, **kwargs).imag / epsilon[i]
for i, ih in enumerate(increments)
]
axes = list(range(partials[0].ndim + 1))
axes[:2] = axes[1::-1]
return np.transpose(partials, axes=axes)
def _approx_hess1_backward(x, f, epsilon=None, args=(), kwargs=None):
n = len(x)
epsilon = -np.abs(_get_epsilon(x, 3, epsilon, n))
return approx_hess1(x, f, epsilon, args, kwargs, centered=False)
def observed_information_matrix(self, params, **kwargs):
"""
Observed information matrix
Parameters
----------
params : array_like, optional
Array of parameters at which to evaluate the loglikelihood
function.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
Notes
-----
This method is from Harvey (1989), which shows that the information
matrix only depends on terms from the gradient. This implementation is
partially analytic and partially numeric approximation, therefore,
because it uses the analytic formula for the information matrix, with
numerically computed elements of the gradient.
References
----------
Harvey, Andrew C. 1990.
Forecasting, Structural Time Series Models and the Kalman Filter.
Cambridge University Press.
"""
# Setup
n = len(params)
epsilon = _get_epsilon(params, 1, None, n)
increments = np.identity(n) * 1j * epsilon
# Get values at the params themselves
self.update(params)
res = self.ssm.filter(**kwargs)
dtype = self.ssm.dtype
# Save this for inversion later
inv_forecasts_error_cov = res.forecasts_error_cov.copy()
# Compute partial derivatives
partials_forecasts_error = (
np.zeros((self.k_endog, self.nobs, n))
)
partials_forecasts_error_cov = (
np.zeros((self.k_endog, self.k_endog, self.nobs, n))
)
for i, ih in enumerate(increments):
self.update(params + ih)
res = self.ssm.filter(**kwargs)
partials_forecasts_error[:, :, i] = (
res.forecasts_error.imag / epsilon[i]
)
partials_forecasts_error_cov[:, :, :, i] = (
res.forecasts_error_cov.imag / epsilon[i]
)
# Compute the information matrix
tmp = np.zeros((self.k_endog, self.k_endog, self.nobs, n), dtype=dtype)
information_matrix = np.zeros((n, n), dtype=dtype)
for t in range(self.ssm.loglikelihood_burn, self.nobs):
inv_forecasts_error_cov[:, :, t] = (
np.linalg.inv(inv_forecasts_error_cov[:, :, t])
)
for i in range(n):
tmp[:, :, t, i] = np.dot(
inv_forecasts_error_cov[:, :, t],
partials_forecasts_error_cov[:, :, t, i]
)
for i in range(n):
for j in range(n):
information_matrix[i, j] += (
0.5 * np.trace(np.dot(tmp[:, :, t, i],
tmp[:, :, t, j]))
)
information_matrix[i, j] += np.inner(
partials_forecasts_error[:, t, i],
np.dot(inv_forecasts_error_cov[:,:,t],
partials_forecasts_error[:, t, j])
)
return information_matrix / (self.nobs - self.ssm.loglikelihood_burn)
def _approx_fprime_backward(x, f, epsilon=None, args=(), kwargs=None):
x = np.atleast_1d(x).ravel()
n = len(x)
epsilon = -np.abs(_get_epsilon(x, 2, epsilon, n))
return approx_fprime(x, f, epsilon, args, kwargs, centered=False)
def _approx_hess1_backward(x, f, epsilon=None, args=(), kwargs=None):
n = len(x)
kwargs = {} if kwargs is None else kwargs
epsilon = -np.abs(_get_epsilon(x, 3, epsilon, n))
return approx_hess1(x, f, epsilon, args, kwargs)
def observed_information_matrix(self, params, **kwargs):
"""
Observed information matrix
Parameters
----------
params : array_like, optional
Array of parameters at which to evaluate the loglikelihood
function.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
Notes
-----
This method is from Harvey (1989), which shows that the information
matrix only depends on terms from the gradient. This implementation is
partially analytic and partially numeric approximation, therefore,
because it uses the analytic formula for the information matrix, with
numerically computed elements of the gradient.
References
----------
Harvey, Andrew C. 1990.
Forecasting, Structural Time Series Models and the Kalman Filter.
Cambridge University Press.
"""
# Setup
n = len(params)
epsilon = _get_epsilon(params, 1, None, n)
increments = np.identity(n) * 1j * epsilon
kwargs['results'] = FilterResults
# Get values at the params themselves
self.update(params)
res = self.filter(**kwargs)
dtype = self.dtype
# Save this for inversion later
inv_forecasts_error_cov = res.forecasts_error_cov.copy()
# Compute partial derivatives
partials_forecasts_error = (np.zeros((self.k_endog, self.nobs, n)))
partials_forecasts_error_cov = (np.zeros(
(self.k_endog, self.k_endog, self.nobs, n)))
for i, ih in enumerate(increments):
self.update(params + ih)
res = self.filter(**kwargs)
partials_forecasts_error[:, :, i] = (res.forecasts_error.imag /
epsilon[i])
partials_forecasts_error_cov[:, :, :,
i] = (res.forecasts_error_cov.imag /
epsilon[i])
# Compute the information matrix
tmp = np.zeros((self.k_endog, self.k_endog, self.nobs, n), dtype=dtype)
information_matrix = np.zeros((n, n), dtype=dtype)
for t in range(self.loglikelihood_burn, self.nobs):
inv_forecasts_error_cov[:, :, t] = (np.linalg.inv(
inv_forecasts_error_cov[:, :, t]))
for i in range(n):
tmp[:, :, t,
i] = np.dot(inv_forecasts_error_cov[:, :, t],
partials_forecasts_error_cov[:, :, t, i])
for i in range(n):
for j in range(n):
information_matrix[i, j] += (
0.5 *
np.trace(np.dot(tmp[:, :, t, i], tmp[:, :, t, j])))
information_matrix[i, j] += np.inner(
partials_forecasts_error[:, t, i],
np.dot(inv_forecasts_error_cov[:, :, t],
partials_forecasts_error[:, t, j]))
return information_matrix / (self.nobs - self.loglikelihood_burn)
def approx_hess(x, f, epsilon=None, args=(), kwargs={}):
"""
Parameters
----------
x : array_like
value at which function derivative is evaluated
f : function
function of one array f(x, `*args`, `**kwargs`)
epsilon : float or array-like, optional
Stepsize used, if None, then stepsize is automatically chosen
according to EPS**(1/4)*x.
args : tuple
Arguments for function `f`.
kwargs : dict
Keyword arguments for function `f`.
Returns
-------
hess : ndarray
array of partial second derivatives, Hessian
Notes
-----
Equation (9) in Ridout. Computes the Hessian as::
1/(4*d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j]
- d[k]*e[k])) -
(f(x - d[j]*e[j] + d[k]*e[k]) - f(x - d[j]*e[j]
- d[k]*e[k]))
where e[j] is a vector with element j == 1 and the rest are zero and
d[i] is epsilon[i].
References
----------:
Ridout, M.S. (2009) Statistical applications of the complex-step method
of numerical differentiation. The American Statistician, 63, 66-74
Copyright
---------
This is an adaptation of the function approx_hess3() in
statsmodels.tools.numdiff. That code is BSD (3 clause) licensed as
follows:
Copyright (C) 2006, Jonathan E. Taylor
All rights reserved.
Copyright (c) 2006-2008 Scipy Developers.
All rights reserved.
Copyright (c) 2009-2012 Statsmodels Developers.
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
a. Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
b. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
c. Neither the name of Statsmodels nor the names of its contributors
may be used to endorse or promote products derived from this software
without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL STATSMODELS OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
DAMAGE.
"""
n = len(x)
h = smnd._get_epsilon(x, 4, epsilon, n)
ee = np.diag(h)
hess = np.outer(h, h)
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(*((x + ee[i, :] + ee[j, :], ) + args), **kwargs) -
f(*((x + ee[i, :] - ee[j, :], ) + args), **kwargs) -
(f(*((x - ee[i, :] + ee[j, :], ) + args), **kwargs) -
f(*((x - ee[i, :] - ee[j, :], ) + args), **kwargs))
) / (4. * hess[i, j])
hess[j, i] = hess[i, j]
return hess
def approx_hess(x, f, epsilon=None, args=(), kwargs={}):
"""
Parameters
----------
x : array_like
value at which function derivative is evaluated
f : function
function of one array f(x, `*args`, `**kwargs`)
epsilon : float or array-like, optional
Stepsize used, if None, then stepsize is automatically chosen
according to EPS**(1/4)*x.
args : tuple
Arguments for function `f`.
kwargs : dict
Keyword arguments for function `f`.
Returns
-------
hess : ndarray
array of partial second derivatives, Hessian
Notes
-----
Equation (9) in Ridout. Computes the Hessian as::
1/(4*d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j]
- d[k]*e[k])) -
(f(x - d[j]*e[j] + d[k]*e[k]) - f(x - d[j]*e[j]
- d[k]*e[k]))
where e[j] is a vector with element j == 1 and the rest are zero and
d[i] is epsilon[i].
References
----------:
Ridout, M.S. (2009) Statistical applications of the complex-step method
of numerical differentiation. The American Statistician, 63, 66-74
Copyright
---------
This is an adaptation of the function approx_hess3() in
statsmodels.tools.numdiff. That code is BSD (3 clause) licensed as
follows:
Copyright (C) 2006, Jonathan E. Taylor
All rights reserved.
Copyright (c) 2006-2008 Scipy Developers.
All rights reserved.
Copyright (c) 2009-2012 Statsmodels Developers.
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
a. Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
b. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
c. Neither the name of Statsmodels nor the names of its contributors
may be used to endorse or promote products derived from this software
without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL STATSMODELS OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
DAMAGE.
"""
n = len(x)
h = smnd._get_epsilon(x, 4, epsilon, n)
ee = np.diag(h)
hess = np.outer(h,h)
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(*((x + ee[i, :] + ee[j, :],) + args), **kwargs)
- f(*((x + ee[i, :] - ee[j, :],) + args), **kwargs)
- (f(*((x - ee[i, :] + ee[j, :],) + args), **kwargs)
- f(*((x - ee[i, :] - ee[j, :],) + args), **kwargs))
)/(4.*hess[i, j])
hess[j, i] = hess[i, j]
return hess