Dj8=dZddlZddlmZmZej ejZ dZ dZ ddidfdZ ddidfd Z ddifd Zddifd Zddifd Zed ddddee ddidfdZed ddddee ddidfdZedddddee ddifdZeZexjdz c_y)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|Jtd|z ztjtjtj|dz}nutj |r'tj |}|j|n9tj|}|j|jk7r tdtj|S)Ng?g?z6If h is not a scalar it must have the same shape as x.) EPSnpmaximumabsasarrayisscalaremptyfillshape ValueError)xsepsilonnhs ]C:\Crop_Prediction\Backend\crop-ai-system\venv\Lib\site-packages\statsmodels\tools\numdiff.py _get_epsilonr^s "q&MBJJrvvbjjm'} || | | <||| zf|zi|||| z f|zi|z d|| zz | | ddf<d| | <@|dk(r | jS| jjS)aR Gradient of function, or Jacobian if function f returns 1d array 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. This is EPS**(1/2)*x for `centered` == False and EPS**(1/3)*x for `centered` == True. args : tuple Tuple of additional arguments for function `f`. kwargs : dict Dictionary of additional keyword arguments for function `f`. centered : bool Whether central difference should be returned. If not, does forward differencing. Returns ------- grad : ndarray gradient or Jacobian Notes ----- If f returns a 1d array, it returns a Jacobian. If a 2d array is returned by f (e.g., with a value for each observation), it returns a 3d array with the Jacobian of each observation with shape xk x nobs x xk. I.e., the Jacobian of the first observation would be [:, 0, :] Ng@) lenr atleast_1drzeros promote_typesfloatdtyperrangeTsqueeze) rfrargskwargscenteredrf0dimgradeiks r approx_fprimer0msB AA aT$Y "6 "B --  ! !C 88QD3J 0 0 @ AD 1$ B q!Wa0q AAJBqEqtgn882=wqzIDAJBqE  q!Wa025q AAJBqEqtgdl6v6qtgdl6v679:WQZIDAJBqE    Avvv ||~rctj|}d}||f|zi|}|s%t|d||}|||zf|zi||z |z } | St|d||dz }|||zf|zi||||z f|zi|z d|zz } | S)a' Gradient of function vectorized for scalar parameter. 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. This is EPS**(1/2)*x for `centered` == False and EPS**(1/3)*x for `centered` == True. args : tuple Tuple of additional arguments for function `f`. kwargs : dict Dictionary of additional keyword arguments for function `f`. centered : bool Whether central difference should be returned. If not, does forward differencing. Returns ------- grad : ndarray Array of derivatives, gradient evaluated at parameters ``x``. rrrr)rr r) rr'rr(r)r*rr+epsr-s r_approx_fprime_scalarr3s> 1 A A aT$Y "6 "B 1a!,QsUHtO//"4; K 1a!,r1QsUHTM-f-QsUHTM-f-.23c'; Krc (t|}t|d||}tj|dz|z}t |Dcgc]$\}}|||zg|i|j ||z &} }}tj | jScc}}w)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?)rrridentity enumerateimagarrayr%) rr'rr(r)r incrementsiihpartialss rapprox_fprime_csr>sB AA1a!,GQ"$w.J'z24Ar!B$(((-- :4H4 88H   4s)Bctj|}|jd}t|d||}d|z}|||zg|i|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 rrr8r9)rr'rr(r)rr2r=s r_approx_fprime_cs_scalarrAsjJ 1 A  A1a!,G w,CS*4*6*//'9H 88H 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$r&r8 rr'rr(r)rreehessr;js rapprox_hess_csrI0s(4 AAQ7A&A B 88Aq>D AA 1X$q! $Aa"R1X+o1a402T9EfER1a4[2ad8!; =d B( &(()-b115ad<DAJ adDAJ  $$ Kr3zFreturn_grad : bool Whether or not to also return the gradient z7grad : nparray Gradient if return_grad == True 7zB1/(d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j]))) )scale extra_params extra_returnsequation_numberequationc t|}t|d||}tj|}||f|zi|} tj|} t |D]} |||| ddfzf|zi|| | <tj ||} t |D][} t | |D]J} |||| ddfz|| ddfzf|zi|| | z | | z | z| | | fz | | | f<| | | f| | | f<L]|r | | z |z }| |fS| S)NrrrrrCr r$rD)rr'rr(r) return_gradrrrFr+gr;rGrHr-s r approx_hess1rU]s\ AAQ7A&A B aT$Y "6 "B  A 1X2AbAhJ=%1&1!2 88Aq>D 1X$q! $Aq2ad8|bAh684?KFKA$!"1&(*+,0AJ7DAJadDAJ $$ 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]4} |||| ddfzf|zi|| | <|||| ddfz f|zi|| | <6tj ||} 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)NrrrR)rr'rr(r)rSrrrFr+rTggr;rGrHr-s r approx_hess2rYs$ AAQ7A&A B aT$Y "6 "B  A !B 1X3AbAhJ=%1&1!Qr!Q$xZM$&26213 88Aq>D 1X$q! $Aq2ad8|bAh684?KFKA$!"1&(*+q2ad8|bAh684?KFKLQ% #%Q%(+--0141:~?DAJadDAJ  $$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 4t|}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<|S)Ng@)rrrrCrDr$r&rEs r approx_hess3r_su AAQ7A&A B 88Aq>D 1X $q! $Aa"QT(lR1X-/$6B6BBq!tH r!Q$x/1D8DVDER1X1a402T9EfE1r!Q$x<"QT(24t;GGHItAqDzM #DAJadDAJ $ $ Krz& This is an alias for approx_hess3)__doc__numpyrstatsmodels.compat.pandasrrfinfor"r2r _hessian_docsrr0r3r>rArIrUrYr_ approx_hessrrrrfsq Z<bhhuo& R !%2b57 t)-2b#(+\$(b) X,0b,^"&Br*Z   -#"RU 2  -#"RU < F  -#"R & @@r