ó áp7]c @shdZddlmZddlmZddlZejƒjZ dZ d„Z ddie d„Zddid„Zdd id „Zdd!ie d „Ze ed d d dddddddƒe_dd"ie d„Ze ed d d dddddddƒe_dd#id„Ze ed dd dddddddƒe_eZejd7_dS($sxnumerical differentiation function, gradient, Jacobian, and Hessian Author : josef-pkt License : BSD Notes ----- These are simple forward differentiation, so that we have them available without dependencies. * Jacobian should be faster than numdifftools because it doesn't use loop over observations. * numerical precision will vary and depend on the choice of stepsizes iÿÿÿÿ(tprint_function(trangeNs 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/%(scale)s)*x. args : tuple Arguments for function `f`. kwargs : dict Keyword arguments for function `f`. %(extra_params)s Returns ------- hess : ndarray array of partial second derivatives, Hessian %(extra_returns)s Notes ----- Equation (%(equation_number)s) in Ridout. Computes the Hessian as:: %(equation)s 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 cCs˜|dkr6td|tjtj|ƒdƒ}n^tj|ƒrdtj|ƒ}|j|ƒn0tj|ƒ}|j |j kr”t dƒ‚n|S(Ngð?gš™™™™™¹?s6If h is not a scalar it must have the same shape as x.( tNonetEPStnptmaximumtabstisscalartemptytfilltasarraytshapet ValueError(txtstepsilontnth((s8lib/python2.7/site-packages/statsmodels/tools/numdiff.pyt _get_epsilon^s *c Cst|ƒ}||f||Ž}tj|ƒj}tj|f|tjt|jƒƒ} tj|ftƒ} |sôt|d||ƒ}xt |ƒD]T} || | | <||| f||Ž||| | | dd…ft|ƒ}t|d||ƒ}tj|ƒ}tj||ƒ}t|ƒ}xét|ƒD]Û} xÒt| |ƒD]Á} ||d|| dd…f|| dd…ff||Ž||d|| dd…f|| dd…ff||Žjd|| | f|| | f<|| | f|| | fs8  ( 7*+