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:

xarray_like: value at which function derivative is evaluated
ffunction: function of one array f(x, *args, **kwargs)
epsilonfloat or array_like, optional: Stepsize used, if None, then stepsize is automatically chosen according to EPS**(1/3)*x.
argstuple: Arguments for function f.
kwargsdict: Keyword arguments for function f.
return_gradbool: Whether or not to also return the gradient

Returns:

hessndarray: array of partial second derivatives, Hessian
gradndarray: 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