tools.numdiff.approx_hess2()

statsmodels.tools.numdiff.approx_hess2

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