statsmodels.tools.numdiff.approx_hess3

statsmodels.tools.numdiff.approx_hess3(x, f, epsilon=None, args=(), kwargs={})

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**(1/4)*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 (9) in Ridout. Computes the Hessian as:

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]))

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

This is an alias for approx_hess3