statsmodels.tools.numdiff.approx_hess2¶
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statsmodels.tools.numdiff.
approx_hess2
(x, f, epsilon=None, args=(), kwargs={}, return_grad=False)[source]¶ 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**(1/3)*x.
args : tuple
Arguments for function f.
kwargs : dict
Keyword arguments for function f.
return_grad : bool
Whether or not to also return the gradient
Returns: hess : ndarray
array of partial second derivatives, Hessian
grad : nparray
Gradient if return_grad == True
Notes
Equation (8) in Ridout. Computes the Hessian as:
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)))
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