"""numerical differentiation function, gradient, Jacobian, and Hessian
Author : josef-pkt
License : BSD
Notes
-----
These are simple forward differentiation, so that we have them available
without dependencies.
* Jacobian should be faster than numdifftools because it does not use loop over
observations.
* numerical precision will vary and depend on the choice of stepsizes
"""
# TODO:
# * some cleanup
# * check numerical accuracy (and bugs) with numdifftools and analytical
# derivatives
# - linear least squares case: (hess - 2*X'X) is 1e-8 or so
# - gradient and Hessian agree with numdifftools when evaluated away from
# minimum
# - forward gradient, Jacobian evaluated at minimum is inaccurate, centered
# (+/- epsilon) is ok
# * dot product of Jacobian is different from Hessian, either wrong example or
# a bug (unlikely), or a real difference
#
#
# What are the conditions that Jacobian dotproduct and Hessian are the same?
#
# See also:
#
# BHHH: Greene p481 17.4.6, MLE Jacobian = d loglike / d beta , where loglike
# is vector for each observation
# see also example 17.4 when J'J is very different from Hessian
# also does it hold only at the minimum, what's relationship to covariance
# of Jacobian matrix
# http://projects.scipy.org/scipy/ticket/1157
# https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm
# objective: sum((y-f(beta,x)**2), Jacobian = d f/d beta
# and not d objective/d beta as in MLE Greene
# similar: http://crsouza.blogspot.com/2009/11/neural-network-learning-by-levenberg_18.html#hessian
#
# in example: if J = d x*beta / d beta then J'J == X'X
# similar to https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm
import numpy as np
from statsmodels.compat.pandas import Appender, Substitution
# NOTE: we only do double precision internally so far
EPS = np.finfo(float).eps
_hessian_docs = """
Calculate Hessian with finite difference derivative approximation
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/%(scale)s)*x.
args : tuple
Arguments for function `f`.
kwargs : dict
Keyword arguments for function `f`.
%(extra_params)s
Returns
-------
hess : ndarray
array of partial second derivatives, Hessian
%(extra_returns)s
Notes
-----
Equation (%(equation_number)s) in Ridout. Computes the Hessian as::
%(equation)s
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
"""
def _get_epsilon(x, s, epsilon, n):
if epsilon is None:
h = EPS**(1. / s) * np.maximum(np.abs(x), 0.1)
else:
if np.isscalar(epsilon):
h = np.empty(n)
h.fill(epsilon)
else: # pragma : no cover
h = np.asarray(epsilon)
if h.shape != x.shape:
raise ValueError("If h is not a scalar it must have the same"
" shape as x.")
return h
[docs]def approx_fprime(x, f, epsilon=None, args=(), kwargs={}, centered=False):
'''
Gradient of function, or Jacobian if function f returns 1d array
Parameters
----------
x : ndarray
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. 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 `f`.
kwargs : dict
Dictionary of additional keyword arguments for function `f`.
centered : bool
Whether central difference should be returned. If not, does forward
differencing.
Returns
-------
grad : ndarray
gradient or Jacobian
Notes
-----
If f returns a 1d array, it returns a Jacobian. If a 2d array is returned
by f (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, :]
'''
n = len(x)
f0 = f(*((x,)+args), **kwargs)
dim = np.atleast_1d(f0).shape # it could be a scalar
grad = np.zeros((n,) + dim, np.promote_types(float, x.dtype))
ei = np.zeros((n,), 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.squeeze().T
def _approx_fprime_scalar(x, f, epsilon=None, args=(), kwargs={},
centered=False):
'''
Gradient of function vectorized for scalar parameter.
This assumes that the function ``f`` is vectorized for a scalar parameter.
The function value ``f(x)`` has then the same shape as the input ``x``.
The derivative returned by this function also has the same shape as ``x``.
Parameters
----------
x : ndarray
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. 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 `f`.
kwargs : dict
Dictionary of additional keyword arguments for function `f`.
centered : bool
Whether central difference should be returned. If not, does forward
differencing.
Returns
-------
grad : ndarray
Array of derivatives, gradient evaluated at parameters ``x``.
'''
x = np.asarray(x)
n = 1
f0 = f(*((x,)+args), **kwargs)
if not centered:
eps = _get_epsilon(x, 2, epsilon, n)
grad = (f(*((x+eps,) + args), **kwargs) - f0) / eps
else:
eps = _get_epsilon(x, 3, epsilon, n) / 2.
grad = (f(*((x+eps,)+args), **kwargs) -
f(*((x-eps,)+args), **kwargs)) / (2 * eps)
return grad
[docs]def approx_fprime_cs(x, f, epsilon=None, args=(), kwargs={}):
'''
Calculate gradient or Jacobian with complex step derivative approximation
Parameters
----------
x : ndarray
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
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)]
return np.array(partials).T
def _approx_fprime_cs_scalar(x, f, epsilon=None, args=(), kwargs={}):
'''
Calculate gradient for scalar parameter with complex step derivatives.
This assumes that the function ``f`` is vectorized for a scalar parameter.
The function value ``f(x)`` has then the same shape as the input ``x``.
The derivative returned by this function also has the same shape as ``x``.
Parameters
----------
x : ndarray
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 derivatives, gradient evaluated for parameters ``x``.
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
x = np.asarray(x)
n = x.shape[-1]
epsilon = _get_epsilon(x, 1, epsilon, n)
eps = 1j * epsilon
partials = f(x + eps, *args, **kwargs).imag / epsilon
return np.array(partials)
[docs]def approx_hess_cs(x, f, epsilon=None, args=(), kwargs={}):
'''Calculate Hessian with complex-step derivative approximation
Parameters
----------
x : array_like
value at which function derivative is evaluated
f : function
function of one array f(x)
epsilon : float
stepsize, if None, then stepsize is automatically chosen
Returns
-------
hess : ndarray
array of partial second derivatives, Hessian
Notes
-----
based on equation 10 in
M. S. RIDOUT: Statistical Applications of the Complex-step Method
of Numerical Differentiation, University of Kent, Canterbury, Kent, U.K.
The stepsize is the same for the complex and the finite difference part.
'''
# TODO: might want to consider lowering the step for pure derivatives
n = len(x)
h = _get_epsilon(x, 3, epsilon, n)
ee = np.diag(h)
hess = np.outer(h, h)
n = len(x)
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(*((x + 1j*ee[i, :] + ee[j, :],) + args), **kwargs)
- f(*((x + 1j*ee[i, :] - ee[j, :],)+args),
**kwargs)).imag/2./hess[i, j]
hess[j, i] = hess[i, j]
return hess
[docs]@Substitution(
scale="3",
extra_params="""return_grad : bool
Whether or not to also return the gradient
""",
extra_returns="""grad : nparray
Gradient if return_grad == True
""",
equation_number="7",
equation="""1/(d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j])))
"""
)
@Appender(_hessian_docs)
def approx_hess1(x, f, epsilon=None, args=(), kwargs={}, return_grad=False):
n = len(x)
h = _get_epsilon(x, 3, epsilon, n)
ee = np.diag(h)
f0 = f(*((x,)+args), **kwargs)
# Compute forward step
g = np.zeros(n)
for i in range(n):
g[i] = f(*((x+ee[i, :],)+args), **kwargs)
hess = np.outer(h, h) # this is now epsilon**2
# Compute "double" forward step
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(*((x + ee[i, :] + ee[j, :],) + args), **kwargs) -
g[i] - g[j] + f0)/hess[i, j]
hess[j, i] = hess[i, j]
if return_grad:
grad = (g - f0)/h
return hess, grad
else:
return hess
[docs]@Substitution(
scale="3",
extra_params="""return_grad : bool
Whether or not to also return the gradient
""",
extra_returns="""grad : ndarray
Gradient if return_grad == True
""",
equation_number="8",
equation="""1/(2*d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j])) -
(f(x + d[k]*e[k]) - f(x)) +
(f(x - d[j]*e[j] - d[k]*e[k]) - f(x + d[j]*e[j])) -
(f(x - d[k]*e[k]) - f(x)))
"""
)
@Appender(_hessian_docs)
def approx_hess2(x, f, epsilon=None, args=(), kwargs={}, return_grad=False):
#
n = len(x)
# NOTE: ridout suggesting using eps**(1/4)*theta
h = _get_epsilon(x, 3, epsilon, n)
ee = np.diag(h)
f0 = f(*((x,)+args), **kwargs)
# Compute forward step
g = np.zeros(n)
gg = np.zeros(n)
for i in range(n):
g[i] = f(*((x+ee[i, :],)+args), **kwargs)
gg[i] = f(*((x-ee[i, :],)+args), **kwargs)
hess = np.outer(h, h) # this is now epsilon**2
# Compute "double" forward step
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(*((x + ee[i, :] + ee[j, :],) + args), **kwargs) -
g[i] - g[j] + f0 +
f(*((x - ee[i, :] - ee[j, :],) + args), **kwargs) -
gg[i] - gg[j] + f0)/(2 * hess[i, j])
hess[j, i] = hess[i, j]
if return_grad:
grad = (g - f0)/h
return hess, grad
else:
return hess
[docs]@Substitution(
scale="4",
extra_params="",
extra_returns="",
equation_number="9",
equation="""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]))"""
)
@Appender(_hessian_docs)
def approx_hess3(x, f, epsilon=None, args=(), kwargs={}):
n = len(x)
h = _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
approx_hess = approx_hess3
approx_hess.__doc__ += "\n This is an alias for approx_hess3"