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 – array of partial second derivatives, Hessian
Return type:

ndarray

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