4.7.9.1.7. statsmodels.tools.numdiff.approx_hess_cs¶

statsmodels.tools.numdiff.approx_hess_cs(x, f, epsilon=None, args=(), kwargs={})[source]¶

Calculate Hessian with complex-step derivative approximation 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.

Returns:

hess : ndarray

array of partial second derivatives, Hessian

Notes

Equation (10) in Ridout. Computes the Hessian as:

1/(2*d_j*d_k) * imag(f(x + i*d[j]*e[j] + d[k]*e[k]) -
               f(x + i*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