M/Phh;dZddlZddlmZmZejejZ dZ dZ ddidfdZ ddidfd Z ddifd Zddifd Zddifd Zed ddddee ddidfdZed ddddee ddidfdZedddddee ddifdZeZexjdz c_dS)aynumerical 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 does not use loop over observations. * numerical precision will vary and depend on the choice of stepsizes N)Appender Substitutiona 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 c|Htd|z ztjtjtj|dz}nqtj|r*tj|}||n3tj|}|j|jkrtdtj|S)Ng?g?z6If h is not a scalar it must have the same shape as x.) EPSnpmaximumabsasarrayisscalaremptyfillshape ValueError)xsepsilonnhs Y/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/tools/numdiff.py _get_epsilonr^s "q&MBJrvbjmm'<' 1 A A aT$Y "6 " "B ;1a!,,QsUHtO///"4;1a!,,r1QsUHTM-f--QsUHTM-f--.23c'; Krct}td|tj|dzz}fdt |D}tj|jS)a Calculate gradient or Jacobian with complex step derivative approximation Parameters ---------- x : ndarray parameters at which the derivative is evaluated f : function `f(*((x,)+args), **kwargs)` returning either one value or 1d array epsilon : float, optional Stepsize, if None, optimal stepsize is used. Optimal step-size is EPS*x. See note. args : tuple Tuple of additional arguments for function `f`. kwargs : dict Dictionary of additional keyword arguments for function `f`. Returns ------- partials : ndarray array of partial derivatives, Gradient or Jacobian Notes ----- The complex-step derivative has truncation error O(epsilon**2), so truncation error can be eliminated by choosing epsilon to be very small. The complex-step derivative avoids the problem of round-off error with small epsilon because there is no subtraction. r?cPg|]"\}}|zgRij|z #Sr)imag).0iihr(rr'r)rs r z$approx_fprime_cs..sW444Ar!B$((((((- :444r)rrridentity enumeratearrayr%)rr'rr(r)r incrementspartialss````` rapprox_fprime_csrAsB AA1a!,,GQ"$w.J44444444&z22444H 8H   rctj|}|jd}t|d||}d|z}|||zg|Ri|j|z }tj|S)a Calculate gradient for scalar parameter with complex step derivatives. This assumes that the function ``f`` is vectorized for a scalar parameter. The function value ``f(x)`` has then the same shape as the input ``x``. The derivative returned by this function also has the same shape as ``x``. Parameters ---------- x : ndarray Parameters at which the derivative is evaluated. f : function `f(*((x,)+args), **kwargs)` returning either one value or 1d array. epsilon : float, optional Stepsize, if None, optimal stepsize is used. Optimal step-size is EPS*x. See note. args : tuple Tuple of additional arguments for function `f`. kwargs : dict Dictionary of additional keyword arguments for function `f`. Returns ------- partials : ndarray Array of derivatives, gradient evaluated for parameters ``x``. Notes ----- The complex-step derivative has truncation error O(epsilon**2), so truncation error can be eliminated by choosing epsilon to be very small. The complex-step derivative avoids the problem of round-off error with small epsilon because there is no subtraction. rr5)rr rrr7r>)rr'rr(r)rr2r@s r_approx_fprime_cs_scalarrDstJ 1 A  A1a!,,G w,CqS*4***6**/'9H 8H  rc t|}t|d||}tj|}tj||}t|}t |D]} t | |D]} tj||d|| ddfzz|| ddfzf|zi|||d|| ddfzz|| ddfz f|zi|z jdz || | fz || | f<|| | f|| | f<|S)aCalculate Hessian with complex-step derivative approximation Parameters ---------- x : array_like value at which function derivative is evaluated f : function function of one array f(x) epsilon : float stepsize, if None, then stepsize is automatically chosen Returns ------- hess : ndarray array of partial second derivatives, Hessian Notes ----- based on equation 10 in M. S. RIDOUT: Statistical Applications of the Complex-step Method of Numerical Differentiation, University of Kent, Canterbury, Kent, U.K. The stepsize is the same for the complex and the finite difference part. rr5Nr)rrrdiagouterr$r&r7 rr'rr(r)rreehessr9js rapprox_hess_csrL0sW4 AAQ7A&&A B 8Aq>>D AA 1XX$$q! $ $Aa"R111X+o1aaa402T9EfEEAR1aaa4[2ad8!; =d B( &((()-b115ad<DAJ adDAJJ  $ Kr3zFreturn_grad : bool Whether or not to also return the gradient z7grad : nparray Gradient if return_grad == True 7zB1/(d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j]))) )scale extra_params extra_returnsequation_numberequationc Pt|}t|d||}tj|}||f|zi|} tj|} t |D]} |||| ddfzf|zi|| | <tj||} t |D]o} t | |D]\} |||| ddfz|| ddfzf|zi|| | z | | z | z| | | fz | | | f<| | | f| | | f<]p|r | | z |z }| |fS| S)NrrrrrFr r$rG)rr'rr(r) return_gradrrrIr+gr9rJrKr-s r approx_hess1rX]s AAQ7A&&A B aT$Y "6 " "B  A 1XX22qAbAAAhJ=%1&11! 8Aq>>D 1XX$$q! $ $A!q2ad8|bAAAh684?KFKKA$!"1&(*+,0AJ7DAJadDAJJ $BzTz rz7grad : ndarray Gradient if return_grad == True 8z1/(2*d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j])) - (f(x + d[k]*e[k]) - f(x)) + (f(x - d[j]*e[j] - d[k]*e[k]) - f(x + d[j]*e[j])) - (f(x - d[k]*e[k]) - f(x))) c .t|}t|d||}tj|}||f|zi|} tj|} tj|} t |D]:} |||| ddfzf|zi|| | <|||| ddfz f|zi|| | <;tj||} t |D]} t | |D]}|||| ddfz||ddfzf|zi|| | z | |z | z|||| ddfz ||ddfz f|zi|z| | z | |z | zd| | |fzz | | |f<| | |f| || f<|r | | z |z }| |fS| S)NrrrU)rr'rr(r)rVrrrIr+rWggr9rJrKr-s r approx_hess2r\s9$ AAQ7A&&A B aT$Y "6 " "B  A !B 1XX33qAbAAAhJ=%1&11!Qr!QQQ$xZM$&26221 8Aq>>D 1XX$$q! $ $A!q2ad8|bAAAh684?KFKKA$!"1&(*+!q2ad8|bAAAh684?KFKKLQ% #%Q%(+--0141:~?DAJadDAJJ  $ BzTz r49a 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]))c t|}t|d||}tj|}tj||}t |D]} t | |D]} tj|||| ddfz|| ddfzf|zi||||| ddfz|| ddfz f|zi|z |||| ddfz || ddfzf|zi||||| ddfz || ddfz f|zi|z z d|| | fzz || | f<|| | f|| | f<Ҍ|S)Ng@)rrrrFrGr$r&rHs r approx_hess3rbs AAQ7A&&A B 8Aq>>D 1XX $ $q! $ $Aa"QT(lR111X-/$6B6BB1Bq!!!tH r!QQQ$x/1D8DVDDEAR111X1aaa402T9EfEEa1r!QQQ$x<"QT(24t;GGGHItAqDzM #DAJadDAJJ $ Krz& This is an alias for approx_hess3)__doc__numpyrstatsmodels.compat.pandasrrfinfor"r2r _hessian_docsrr0r3rArDrLrXr\rb approx_hessrrrrisX  Z<<<<<<<<bhuoo& R   !%2b57 7 7 7 t)-2b#(++++\$(b) ) ) ) X,0b,,,,^"&Br****Z     -#"RU  2  -#"RU < F    -#"R  & @@r