>h8=SrSSKrSSKJrJr \R "\5Rr Sr Sr SS0S4Sjr SS0S4S jr SS04S jrSS04S jrSS04S jr\"S SSSSS9\"\ 5SS0S4Sj55r\"S SSSSS9\"\ 5SS0S4Sj55r\"SSSSSS9\"\ 5SS04Sj55r\r\=RS- slg)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 cUcM[SU- -[R"[R"[R"U55S5-nO~[R "U5(a([R "U5nURU5 O;[R"U5nURUR:wa [S5e[R"U5$)Ng?g?z6If h is not a scalar it must have the same shape as x.) EPSnpmaximumabsasarrayisscalaremptyfillshape ValueError)xsepsilonnhs lC:\Users\julio\OneDrive\Documentos\Trabajo\Ideas Frescas\venv\Lib\site-packages\statsmodels/tools/numdiff.py _get_epsilonr^s "q&MBJJrvvbjjm' 1 A A aT$Y "6 "B 1a,QUHtO//"4; K 1a,r1QUHTM-f-QUHTM-f-.23c'; Krc *[U5n[USX%5n[R"U5S-U-n[ U5VVs/sH"upxU"X-/UQ70UD6R X'- PM$ n nn[R "U 5R$s snnf)ad Calculate gradient or Jacobian with complex step derivative approximation Parameters ---------- x : ndarray parameters at which the derivative is evaluated f : function `f(*((x,)+args), **kwargs)` returning either one value or 1d array epsilon : float, optional Stepsize, if None, optimal stepsize is used. Optimal step-size is EPS*x. See note. args : tuple Tuple of additional arguments for function `f`. kwargs : dict Dictionary of additional keyword arguments for function `f`. Returns ------- partials : ndarray array of partial derivatives, Gradient or Jacobian Notes ----- The complex-step derivative has truncation error O(epsilon**2), so truncation error can be eliminated by choosing epsilon to be very small. The complex-step derivative avoids the problem of round-off error with small epsilon because there is no subtraction. r?)rrridentity enumerateimagarrayr%) rr'rr(r)r incrementsiihpartialss rapprox_fprime_csr>sB AA1a,GQ"$w.J'z242EA!$(((-- :2 4 88H   4s)Bc[R"U5nURSn[USX%5nSU-nU"X-/UQ70UD6RU- n[R "U5$)aN 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 rrr8r9)rr'rr(r)rr2r=s r_approx_fprime_cs_scalarrAsfJ 1 A  A1a,G w,C*4*6*//'9H 88H rc [U5n[USX%5n[R"U5n[R"Xf5n[U5n[ U5Hn [ X5H|n [R "U"USXySS24--XzSS24-4U-0UD6U"USXySS24--XzSS24- 4U-0UD6- RS- XU 4- 5XU 4'XU 4XU 4'M~ M U$)aWCalculate Hessian with complex-step derivative approximation Parameters ---------- x : array_like value at which function derivative is evaluated f : function function of one array f(x) epsilon : float stepsize, if None, then stepsize is automatically chosen Returns ------- hess : ndarray array of partial second derivatives, Hessian Notes ----- based on equation 10 in M. S. RIDOUT: Statistical Applications of the Complex-step Method of Numerical Differentiation, University of Kent, Canterbury, Kent, U.K. The stepsize is the same for the complex and the finite difference part. rr5Nr)rrrdiagouterr$r&r8 rr'rr(r)rreehessr;js rapprox_hess_csrI0s4 AAQ7&A B 88A>D AA 1XqAa"R1X+oa402T9EfERa4[2d8!; =d B( &(()-b115d<DAJ dDAJ  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 [U5n[USX&5n[R"U5nU"U4U-0UD6n [R"U5n [ U5Hn U"XU SS24-4U-0UD6X'M [R "Xw5n [ U5HUn [ X5HCn U"XU SS24-XSS24-4U-0UD6X- X- U -XU 4- XU 4'XU 4XU 4'ME MW U(a X- U- nX4$U $)NrrrrrCr r$rD)rr'rr(r) return_gradrrrFr+gr;rGrHr-s r approx_hess1rU]s/ AAQ7&A B aT$Y "6 "B  A 1XAAhJ=%1&1 88A>D 1XqAqad8|bAh684?KFK$!"&(*+,0AJ7DAJdDAJ zz 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 [U5n[USX&5n[R"U5nU"U4U-0UD6n [R"U5n [R"U5n [ U5H1n U"XU SS24-4U-0UD6X'U"XU SS24- 4U-0UD6X'M3 [R "Xw5n [ U5Hn [ X5HrnU"XU SS24-XSS24-4U-0UD6X- X- U -U"XU SS24- XSS24- 4U-0UD6-X- X- U -SXU4-- XU4'XU4XU 4'Mt M U(a X- U- nX4$U $)NrrrR)rr'rr(r)rSrrrFr+rTggr;rGrHr-s r approx_hess2rYs$ AAQ7&A B aT$Y "6 "B  A !B 1XAAhJ=%1&1Q!Q$xZM$&262 88A>D 1XqAqad8|bAh684?KFK$!"&(*+qad8|bAh684?KFKL% #%%(+--0141:~?DAJdDAJ zz 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 $[U5n[USX%5n[R"U5n[R"Xf5n[ U5Hn [ X5Hn [R "U"XU SS24-XzSS24-4U-0UD6U"XU SS24-XzSS24- 4U-0UD6- U"XU SS24- XzSS24-4U-0UD6U"XU SS24- XzSS24- 4U-0UD6- - SXU 4-- 5XU 4'XU 4XU 4'M M U$)Ng@)rrrrCrDr$r&rEs r approx_hess3r_sM AAQ7&A B 88A>D 1XqAaQT(lR1X-/$6B6Bq!tH rQ$x/1D8DVDE1Xa402T9EfE1!Q$x<"T(24t;GGHItqDzM #DAJdDAJ Krz& This is an alias for approx_hess3)__doc__numpyrstatsmodels.compat.pandasrrfinfor"r2r _hessian_docsrr0r3r>rArIrUrYr_ approx_hessrrrrfso Z< hhuo& R !%2b57 t)-2b#(+\$(b) X,0b,^"&Br*Z   -#"RU 2  -#"RU < F  -#"R & @@r