>hmRSrSSKrSSKJr SSKJr SSKJ r SSK J r SSK r Sr \ R"\ \5 SSjrSrS rSS jrSS jrSS jr"S S5r"SS\5rg)a Bspines and smoothing splines. General references: Craven, P. and Wahba, G. (1978) "Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation." Numerische Mathematik, 31(4), 377-403. Hastie, Tibshirani and Friedman (2001). "The Elements of Statistical Learning." Springer-Verlag. 536 pages. Hutchison, M. and Hoog, F. "Smoothing noisy data with spline functions." Numerische Mathematik, 47(1), 99-106. N) solveh_bandedgolden) _hbsplinez The bspline code is technology preview and requires significant work on the public API and documentation. The API will likely change in the future cURSnURSnSnU(d[U5H{n[R"XS- U- US9XtU-2XtU-24nXh- nU(aUS:aXhR- nMNU(dMWUS:dM_XhR 5R- nM} U$[U5Hqn[R"XUS9SU2SU24nXh- nU(aUS:aXhR- nMDU(dMMUS:dMUXhR 5R- nMs URnU$)a Take an upper or lower triangular banded matrix and return a numpy array. INPUTS: a -- a matrix in upper or lower triangular banded matrix lower -- is the matrix upper or lower triangular? symmetric -- if True, return the original result plus its transpose hermitian -- if True (and symmetric False), return the original result plus its conjugate transposed r)k)shaperangenpdiagT conjugate) alower symmetric hermitiannr_aj_bs nC:\Users\julio\OneDrive\Documentos\Trabajo\Ideas Frescas\venv\Lib\site-packages\statsmodels/sandbox/bspline.py _band2arrayr"s!  A  A B qAQ3q5A&qA#wqA#w7B HBQUdd q1ulln&&& " IqA"1Q3qs7+B HBQUdd q1ulln&&& TT Ic [R"URUR5nURup#[ URS5H.nXS- U- XC24XSX4- 24'XS- U- SU24XX4- S24'M0 U$)z Convert upper triangular banded matrix to lower banded form. INPUTS: ub -- an upper triangular banded matrix OUTPUTS: lb lb -- a lower triangular banded matrix with same entries as ub rrNr zerosr dtyper )ublbnrowncolis r _upper2lowerr%Hs "((BHH %BJD 288A; 1fQhqvo.QZ<!VAXac\*dfY;  Irc [R"URUR5nURup#[ URS5H.nXSX4- 24XS- U- XC24'XX4- S24XS- U- SU24'M0 U$)z Convert lower triangular banded matrix to upper banded form. INPUTS: lb -- a lower triangular banded matrix OUTPUTS: ub ub -- an upper triangular banded matrix with same entries as lb rrNr)r!r r"r#r$s r _lower2upperr'[s "((BHH %BJD 288A;  1df:.6!8AF? k?6!8AaC<  IrcU(aUSR5nOUSR5nU(aX U- 4$[U5nU[X2- 54$)a Take a banded triangular matrix and return its diagonal and the unit matrix: the banded triangular matrix with 1's on the diagonal, i.e. each row is divided by the corresponding entry on the diagonal. INPUTS: tb -- a lower triangular banded matrix lower -- if True, then tb is assumed to be lower triangular banded, in which case return value is also lower triangular banded. OUTPUTS: d, b d -- diagonal entries of tb b -- unit matrix: if lower is False, b is upper triangular banded and its rows of have been divided by d, else lower is True, b is lower triangular banded and its columns have been divieed by d. r)copyr%r')tbrdlnums r_triangle2unitr.nsP& qEJJL rFKKM 6{B,tx(((rcU(a6[X-SS9nUSR5SUSSR5--$[X-SS9nUSR5SUSSR5--$)a/ Compute the trace(ab) for two upper or banded real symmetric matrices stored either in either upper or lower form. INPUTS: a, b -- two banded real symmetric matrices (either lower or upper) lower -- if True, a and b are assumed to be the lower half OUTPUTS: trace trace -- trace(ab) rrrNr)) _zero_tribandsum)rbrts r_trace_symbandedr6sn !%q )txxzA!" O++ !%q )uyy{Q3B---rcURup#U(a[U5H nSXX4- S24'M U$[U5H nSXSU24'M U$)z Explicitly zero out unused elements of a real symmetric banded matrix. INPUTS: a -- a real symmetric banded matrix (either upper or lower hald) lower -- if True, a is assumed to be the lower half Nr)r r )rrr"r#r$s rr2r2sZJD tA A$&lO HtAA1fI Hrcd\rSrSrSrS SjrSrSr\"\\5r Sr S Sjr SS jr S S jr S rg)BSplineaE Bsplines of a given order and specified knots. Implementation is based on description in Chapter 5 of Hastie, Tibshirani and Friedman (2001). "The Elements of Statistical Learning." Springer-Verlag. 536 pages. INPUTS: knots -- a sorted array of knots with knots[0] the lower boundary, knots[1] the upper boundary and knots[1:-1] the internal knots. order -- order of the Bspline, default is 4 which yields cubic splines M -- number of additional boundary knots, if None it defaults to order coef -- an optional array of real-valued coefficients for the Bspline of shape (knots.shape + 2 * (M - 1) - order,). x -- an optional set of x values at which to evaluate the Bspline to avoid extra evaluation in the __call__ method Nc0[R"[R"[R"U555nURS:wa [ S5eX lUc UR nX0l[R"US/URS- -XS/URS- -/5Ul URSS- Ul UcR[R"URSUR--UR - [R5UlOg[R"U5UlURRURSUR--UR - :wa [ S5eUbXPlgg)Nrzexpecting 1d array for knotsrr)r1z,coefficients of Bspline have incorrect shape)r squeezeuniqueasarrayndim ValueErrormMhstacktaur Krfloat64coefx)selfknotsorderrCrHrIs r__init__BSpline.__init__s+ 299RZZ%678 ::?;< < 9A99uQxj$&&(3U2YKPQLDI 4(DIyy466AJ#6#?@ !OPP =F rcPXlURUR5UlgN)_xbasis_basisx)rJrIs r_setx BSpline._setxszz$''* rcUR$rP)rQrJs r_getx BSpline._getxs wwrcU(dURRnO4USn[R"UR U55Rn[R "[R "X R55$)a Evaluate the BSpline at a given point, yielding a matrix B and return B * self.coef INPUTS: args -- optional arguments. If None, it returns self._basisx, the BSpline evaluated at the x values passed in __init__. Otherwise, return the BSpline evaluated at the first argument args[0]. OUTPUTS: y y -- value of Bspline at specified x values BUGS: If self has no attribute x, an exception will be raised because self has no attribute _basisx. r)rSrr r?rRr=dotrH)rJargsr4rIs r__call__BSpline.__call__sU, AQA 4::a=)++Azz"&&II.//rc [R"U[R5nURnUS:XaSUl[R"USS94UlX R RSS- :a1[ R"XR URX2US-5nO/[R"UR[R5$X R RSUR- :Xa9[R"[R"XR S5SU5nXElU$)aQ Evaluate a particular basis element of the BSpline, or its derivative. INPUTS: x -- x values at which to evaluate the basis element i -- which element of the BSpline to return d -- the order of derivative OUTPUTS: y y -- value of d-th derivative of the i-th basis element of the BSpline at specified x values rraxisrr)) r r?rGr productrErevaluaterBrwhereequal)rJrIr$r,_shapevs r basis_elementBSpline.basis_elements JJq"** % R<AG::f!,. xx~~a 1$ $""1hhacBA88AGGRZZ0 0 "TVV+ +!XXb\2Aq9Arc [R"U5nURnUS:XaSUl[R"USS94UlUc&URRSUR - nUcSn[ X@RRSUR - 5n[SU5n[R"U5nURS:Xa7[R"XRUR [U5X45nO{URSS:wa [S5eSn[URS5H?nXbSU4[R"XRUR USU4X45-- nMA XC- 4U-UlX@RRSUR - :Xa?[R"[R"XRS5SUS5US'U$) a8 Evaluate the basis of the BSpline or its derivative. If lower or upper is specified, then only the [lower:upper] elements of the basis are returned. INPUTS: x -- x values at which to evaluate the basis element i -- which element of the BSpline to return d -- the order of derivative lower -- optional lower limit of the set of basis elements upper -- optional upper limit of the set of basis elements OUTPUTS: y y -- value of d-th derivative of the basis elements of the BSpline at specified x values r`rarrbr1if d is not an integer, expecting a jx2 array with first row indicating order of derivative, second row coefficient in front.rr))r r?r rdrErBminmaxrreintrAr rfrg)rJrIr,rupperrhrir$s rrR BSpline.basis2s& JJqM R<AG::f!,. =HHNN1%.E =EE88>>!,tvv56Au  JJqM 77b=""1hhAMAwwqzQ "DEEA1771:&qsVi00HHdffa!fe[[[';.6) HHNN1%. .HHRXXa"61R5AAbErcz[R"U5n[R"U5RS:XaE[R "UR UR[U5[U55Ul O[R"U5nURSS:wa [S5eURS:XaSUlSUl [URS5Hn[URS5HonU=RUSU4USU4-[R "UR UR[USU45[USU455-- sl Mq M URRUl Xl [R"UR5$)a Compute Gram inner product matrix, storing it in lower triangular banded form. The (i,j) entry is G_ij = integral b_i^(d) b_j^(d) where b_i are the basis elements of the BSpline and (d) is the d-th derivative. If d is a matrix then, it is assumed to specify a differential operator as follows: the first row represents the order of derivative with the second row the coefficient corresponding to that order. For instance: [[2, 3], [3, 1]] represents 3 * f^(2) + 1 * f^(3). INPUTS: d -- which derivative to apply to each basis element, if d is a matrix, it is assumed to specify a differential operator as above OUTPUTS: gram gram -- the matrix of inner products of (derivatives) of the BSpline elements r`rr1rm)r1)r1rr)r r=r?r rgramrErBrpgrAr rr, nan_to_num)rJr,r$rs rrt BSpline.gramcsUB JJqM ::a=  " $^^DHHdffc!fc!fEDF 1 AwwqzQ "DEEww$DF1771:&qwwqz*AFFa!fa!fny~~dhhPSTUVWXYVYTZP[]`abcdefcfag]h/iiiF+'}}TVV$$r) rFrCrSrQrHr,rurBrErI)NNNr)rNN)__name__ __module__ __qualname____firstlineno____doc__rMrTrXpropertyrIr]rjrRrt__static_attributes__r`rrr:r:s9:0+ A0:>/b2%rr:cr\rSrSrSrSrSrSrSrSSjr SS jr S r S r S r S rSSjrSSjrSrg)SmoothingSplineig>@ target_dfMbP?TNc SnUc'URnURR5nOURU5nUS:XaSnURUR:wa [ S5eX@R :a UR nUbX0lOSUl[R"UR5nXg-n[R"S[R"[R"US5SS 9- 5nUSS2U4nXnURSUl [R"[R"XgU-55n URSUlU(d[R"XfR"5Ul['UR(SSS 9n [*R,"UR$XJ--U 5SS uUlol[3UR0UR$RS5UlA O[R4"UR(R[R65UlUR(Rup[9U 5HIn[9[3XU- 55H+nXnXnU--R;5UR$X4'M- MK SU RS4U lX@l[?UR$X@R(--U SS 9uUl Ul[R"UR.5UlXR-[R"UR.U5- Ul!X@lA AAg) a Fit the smoothing spline to a set of (x,y) pairs. INPUTS: y -- response variable x -- if None, uses self.x weights -- optional array of weights pen -- constant in front of Gram matrix OUTPUTS: None The smoothing spline is determined by self.coef, subsequent calls of __call__ will be the smoothing spline. ALGORITHM: Formally, this solves a minimization: fhat = ARGMIN_f SUM_i=1^n (y_i-f(x_i))^2 + pen * int f^(2)^2 int is integral. pen is lambda (from Hastie) See Chapter 5 of Hastie, Tibshirani and Friedman (2001). "The Elements of Statistical Learning." Springer-Verlag. 536 pages. for more details. TODO: Should add arbitrary derivative penalty instead of just second derivative. TNr8FzXx and y shape do not agree, by default x are the Bspline's internal knotsg?rrrb)rrr0)"rQrSr*rRr rApenmaxweightsr sqrt flatnonzeroallrgdf_totalr=r[NrbtbrruLlstsqrHrankrnrrGr r3penrcholresid)rJyrIrrbandedbt_wmaskbty_g_nbandnbasisr$r s rfitSmoothingSpline.fitsB 9A""$BAB "9F 77agg ./ / ++ ++C  "LDL WWT\\ " ~~a"&&"aq"AAB $Z G  jjF+,vvb$$'DHTVV1=B&'ggdhh.?&Ea&J #DIq)DIItxx~~a'89DIxx bjj9DH FFLLME6]s5(34A%'UR!W_$9$9$;DHHQSM5#399Q<(CIH#014VV2<14A$? DItyJJtyy) %tyy"(==    rcURS:XaG[US5(aURXX0RS9 gUR XX0R S9 gURS:XaUR XUS9 gg)NrrrIrr)rIrdf optimize_gcv)rIr)methodhasattrrr fit_target_dfrfit_optimize_gcv)rJrrIrs rsmoothSmoothingSpline.smooth sj ;;+ %tU##hh?""17~~"N [[N *  ! !!' ! :+rczURS-R5nXRUR5- - $)z Generalized cross-validation score of current fit. Craven, P. and Wahba, G. "Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation." Numerische Mathematik, 31(4), 377-403. r1)rr3rtrace)rJ norm_resids rgcvSmoothingSpline.gcvs2jj!m((* ]]TZZ\9::rc<URUR5- $)zy Residual degrees of freedom in the fit. self.N - self.trace() where self.N is the number of observations of last fit. )rrrWs rdf_residSmoothingSpline.df_resid"svv $$rc"UR5$)z< How many degrees of freedom used in the fit? self.trace() )rrWs rdf_fitSmoothingSpline.df_fit-s zz|rcURS:aD[R"URR 55n[ XR SS9nU$UR$)zb Trace of the smoothing matrix S(pen) TODO: addin a reference to Wahba, and whoever else I used. rrr0)rrinvbandrr*r6rr)rJ_invbandtrs rrSmoothingSpline.trace5sI 88a< (()9:H!(HHA>BI99 rcXU=(d URnURSUR- n[US5(awUR XX@R S9 UR 5n [R"X- 5U- U:agX:aUR SUR -pvO SUR pvSXg--n UR XXJS9 UR 5n X:aU SU -pvOXjpvX`R:a [S5e[R"X- 5U- U:agMq) a Fit smoothing spline with approximately df degrees of freedom used in the fit, i.e. so that self.trace() is approximately df. Uses binary search strategy. In general, df must be greater than the dimension of the null space of the Gram inner product. For cubic smoothing splines, this means that df > 2. INPUTS: y -- response variable x -- if None, uses self.x df -- target degrees of freedom weights -- optional array of weights tol -- (relative) tolerance for convergence apen -- lower bound of penalty for binary search bpen -- upper bound of penalty for binary search OUTPUTS: None The smoothing spline is determined by self.coef, subsequent calls of __call__ will be the smoothing spline. rrrNr1r8g?zPpenalty too large, try setting penmax higher or decreasing df) rr rBrrrrr fabsrrA) rJrrIrrtolapenbpenolddfcurdfcurpens rrSmoothingSpline.fit_target_dfCs 4 !4>> TVV# 4   HHQW((H ;JJLEwwuz"R'#-z!XXq488|ddDK(F HHQWH 9JJLEz#QZd!d{{" ",--wwuz"R'#-rc*^U4Sjn[XaU4XTS9ng)ar Fit smoothing spline trying to optimize GCV. Try to find a bracketing interval for scipy.optimize.golden based on bracket. It is probably best to use target_df instead, as it is sometimes difficult to find a bracketing interval. INPUTS: y -- response variable x -- if None, uses self.x df -- target degrees of freedom weights -- optional array of weights tol -- (relative) tolerance for convergence brack -- an initial guess at the bracketing interval OUTPUTS: None The smoothing spline is determined by self.coef, subsequent calls of __call__ will be the smoothing spline. cp>TRX[R"U5S9 TR5nU$)N)rIr)rr expr)rrrIrrJs r_gcv.SmoothingSpline.fit_optimize_gcv.._gcvs+ HHQH - AHr)r\brackrNr)rJrrIrrrrrs` rr SmoothingSpline.fit_optimize_gcvzs0  4e5 :r) rrrrHrrrrr)NNr8)NN)NNNrrr)NNr)i)rzr{r|r}rrr default_penoptimizerrrrrrrrrr`rrrrs^ F FIKHaF; ; % CJ#*5n=D(;rr)rFFry)r~numpyr numpy.linalglinalgr scipy.linalgrscipy.optimizermodelsrwarnings_msgwarn FutureWarningrr%r'r.r6r2r:rr`rrrsq"&! dM"#L&&)>., &_%_%B@;g@;r