M/PhdZddlmZmZddlmZddlZddlZ ddl m Z ddl m Z ddlmZdZd Zd/d Zd ZdZ d0dZd1dZ d2dZd3dZGddeeZGddeZGddeZGddeZGdd eZGd!d"eZGd#d$eeZGd%d&eZ Gd'd(eZ!Gd)d*eZ"Gd+d,eZ#Gd-d.eZ$dS)4z Spline and other smoother classes for Generalized Additive Models Author: Luca Puggini Author: Josef Perktold Created on Fri Jun 5 16:32:00 2015 )ABCMetaabstractmethod)with_metaclassN)dmatrix)_get_all_sorted_knots)transf_constraintsc|dz }|}|}tj|||}|S)N)minmaxnplinspace)xdfn_knotsx_minx_maxknotss \/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/gam/smooth_basis.py_equally_spaced_knotsrs=1fG EEGGE EEGGE Kug . .E Lctj|}tjfd|dD}||jdS)Nc@g|]}tjd|zS)d)r percentile).0probrs r z&_R_compat_quantile..#s?@@@ $M!S4Z88@@@rC)order)r asarrayravelreshapeshape)rprobs quantiless` r_R_compat_quantiler' s{ Ju  E @@@@(- # (>(>@@@AAI   U[  4 44rallTc ddlm}n#t$rtdwxYwtjtj|t }|jdksJ|t|}tj|}|jdkr|j ddkr |dddf}|jdksJtj |tj |ks*tj |tj |krtddt|z }t||dzz |z }|dvr*tj|j d|ft }|} |d vr*tj|j d|ft } | } |d vr*tj|j d|ft } | } t!|D]v} tj||zf} d| | |z<| }|dvr|||| |f|dd|f<|d vr|||| |fd | dd|f<|d vr|||| |fd | dd|f<w|d kr|| | fS| S) Nr)splevz#spline functionality requires scipydtyper zksome data points fall outside the outermost knots, and I'm not sure how to handle them. (Patches accepted!))r(r)r(r-)r(r )derr()scipy.interpolater* ImportErrorr atleast_1dr!floatndimsortintr$r r NotImplementedErrorlenemptyrangezeros)rrdegreederivinclude_interceptr*k_constn_basesbasisret der1_basis der2_basisicoefsiis r_eval_bspline_basisrG+sA+++++++ AAA?@@@A M"*U%888 9 9E :???? JJLLL [[F aAv{{qwqzQ aaadG 6Q;;;;  vayy26%==  BF1IIu $=$=!#HII I #'(((G%jjFQJ''1G !'!*g.e<<< Xqwqz735AAA  Xqwqz735AAA  7^^ H H'G+-..a'k  J   5UE6$:;;E!!!R%L J   %a%)?Q G G GJqqq"u  J   %a%)?Q G G GJqqq"u  ~~j*,, s #c |dz}||z }tj|}tj|}tjdd|dzdd}t ||}tj||g|z|f} | |||fS)Nr-rr )r r r rr' concatenate) rrr;r n_inner_knots lower_bound upper_boundknot_quantiles inner_knots all_knotss rcompute_all_knotsrQns QJEJM&))K&))K[A}q'899!B$?N$Q77Kk :U B + -..I k; ;;rcbt|||\}}}}t|||\}}}|||fS)z make a spline basis for x )rQrG)rrr;rP_r@ der_basisrCs rmake_bsplines_basisrUzsC+1b&99Iq!Q#6q)V#L#L E9j )Z ''rquantilec ||S|}|} |dkrtd|dt||krtd|d||td|dz} ||| z } | dkrtd|d |d || z |=t || kr)td|d |d | d t |d n|dkr2t jdd| dzdd} t|| } nS|dkr>t jdd| dzdd}||| |z zz} | d| dz }ntd||} |t j|}|t j|}||krtd|d|dt j| } | j dkrtdt j | |kr td| | |kd|dt j | |kr td| | |kd|d|dkrRt j d| dz|z}| d|dddz }| d|z}t j || |f}nt j ||g| z| f}| |S)a7knots for use in B-splines There are two main options for the knot placement - quantile spacing with multiplicity of boundary knots - equal spacing extended to boundary or exterior knots The first corresponds to splines as used by patsy. the second is the knot spacing for P-Splines. Nrz#degree must be greater than 0 (not )zdegree must be an integer (not zmust specify either df or knotsr-zdf=z is too small for degree=z ; must be >= z with degree=z implies z knots, but z knots were providedrWr rIequalzincorrect option for spacingzlower_bound > upper_bound (z > zknots must be 1 dimensionalzsome knot values (z) fall below lower bound (z) fall above upper bound ()r r ValueErrorr5r7r rr'r!r3anyarangerJr4)rrrr;spacingrLrMrPrrr rKrNrOgrid diff_knotsdiffs lower_knots upper_knotss rget_knots_bsplinesrds EEGGE EEGGE zzj"FF%&& & 6{{fj"FF%&& &  zem:;;; QJE ~U 1  * "FFF!#] 2 2 455 5  5zz]** j$&BB$1MM3u::::"?@@@+  " "[A}q/@AA!B$GN,Q??KK   ;q!]Q%677"=D$%%-"88K$Q+a.8JJ;<< <  fQii fQii [  j'KK677 7*[))K!6777 vkK'((*j' k(ABBB'KK)** * vkK'((*j' k(ABBB'KK)** * ' !UQY''*4!!nuTTrT{2 !"o- NKk#JKK N[+$>$F$/$122  NN rc$|dz}tj|}tj||z }tj|}|dddf|z}tj|dddf|z|dgf}|S)zmadd points to each subinterval defined by knots inserts k_points between each two consecutive knots r-NrI)r uniquer]diffrJr")rk_pointsdxidxkdxrs r_get_integration_pointsrls !|H Ie  E )H   (C '%..C QQQW B ssDy)B.5577%)EFFA HrFr c ddlm}n#t$r ddlm}YnwxYw|j}|t ||}n|}||||}||dddddf|dddddfz|d} | S)z Approximate integral of cross product of second derivative of smoother This uses scipy.integrate simps to compute an approximation to the integral of the smoother derivative cross-product at knots plus k_points in between knots. r)simpson)simpsN)rh)r< skip_ctransf)raxis)scipy.integraternr0rorrl transform) smootherrhintegration_pointsrpr<rnrrd2covd2s r get_covder2rxs5+++++++ 555444444445 NE! #EH = = =    AU  F FB GBqqq!!!TzNR4 ^3qq A A AE Ls  c|rd}nd}t|}tj||dz|z f}tj||dz|z f}tj||dz|z f}t||dzD]A}||z|dd||z f<|||dz zz|dd||z f<||dz z||dz zz|dd||z f<B|||fS)zj given a vector x returns poly=(1, x, x^2, ..., x^degree) and its first and second derivative rr-r$Nr )r7r r:r9) rr; interceptstartnobsr@rTrCrDs rmake_poly_basisr~s  q66D HD&1*u"45 6 6 6Efqj5&89:::IvzE'9 :;;;J 5&1* % %>>1faaaUl"#aAEl"2 !!!QY,#$A;q1u#= 111a%i<  )Z ''rc0eZdZdZddZedZdS)UnivariateGamSmootherz+Base Class for single smooth component Nrc|_|_|_t|dc__}|dkr%|dddddf}|,t|tst|}|_ ntdsd_ |\_ ___j j }|d |d|_ |d |d|_|d |d|_|d8|j|d|_j jd_fdt+jD_dS)Nr-centerrctransfr rVcDg|]}jdzt|zS)_s) variable_namestr)rrDselfs rrz2UnivariateGamSmoother.__init__..Ts<:::,t3c!ff<:::r)r constraintsrr7r} k_variables!_smooth_basis_for_single_variablemean isinstancerrrhasattrr@rTrCcov_der2dotTr$ dim_basisr9 col_names)rrrrbase4rs` r__init__zUnivariateGamSmoother.__init__3s&*&)!ffa# 4#6688 ( " "(--**473K  ":k3+G+G "(55G"DLL4++ $# EJB DNDOT] < #lGQx#"1X\\'22 Qx#!&qg!6!6Qx#"'(,,w"7"7Qx# ' eAh 7 7 ; ;G D D )!,::::#(#8#8:::rcdSNrs rrz7UnivariateGamSmoother._smooth_basis_for_single_variableWsrNr)__name__ __module__ __qualname____doc__rrrrrrrr0sN":":":":H^rrc,eZdZdZ dfd ZdZxZS)UnivariateGenericSmootherz$Generic single smooth component rc||_||_||_||_t ||dSN)r)r@rTrCrsuperr)rrr@rTrCrr __class__s rrz"UnivariateGenericSmoother.__init___sB "$   -88888rc6|j|j|j|jfSr)r@rTrCrrs rrz;UnivariateGenericSmoother._smooth_basis_for_single_variablehsz4>4?DMIIrrrrrrrr __classcell__rs@rrr\sb #999999JJJJJJJrrc*eZdZdZdfd ZdZxZS)UnivariatePolynomialSmootherz'polynomial single smooth component rc\||_t||dSr)r;rr)rrr;rrs rrz%UnivariatePolynomialSmoother.__init__os-  -88888rctj|j|jf}tj|j|jf}tj|j|jf}t |jD]\}|dz}|j|z|dd|f<||j|dz zz|dd|f<|dkr||dz z|j|dz zz|dd|f<Sd|dd|f<]tj|j|}||||fS)zv given a vector x returns poly=(1, x, x^2, ..., x^degree) and its first and second derivative rzr-Nr r)r r:r}r;r9rrr)rr@rTrCrDdgrs rrz>UnivariatePolynomialSmoother._smooth_basis_for_single_variabless 4;7888HDIt{#;<<< XTY $<=== t{## % %AQB&B,E!!!Q$K 46b1f#55IaaadOAvv#%a=46b1f3E#E 111a4  #$ 111a4  6*, 33iX55rrrrs@rrrlsV9999996666666rrc8eZdZdZ d fd ZdZd d ZxZS) UnivariateBSplinesaWB-Spline single smooth component This creates and holds the B-Spline basis function for one component. Parameters ---------- x : ndarray, 1-D underlying explanatory variable for smooth terms. df : int number of basis functions or degrees of freedom degree : int degree of the spline include_intercept : bool If False, then the basis functions are transformed so that they do not include a constant. This avoids perfect collinearity if a constant or several components are included in the model. constraints : {None, str, array} Constraints are used to transform the basis functions to satisfy those constraints. `constraints = 'center'` applies a linear transform to remove the constant and center the basis functions. variable_name : {None, str} The name for the underlying explanatory variable, x, used in for creating the column and parameter names for the basis functions. covder2_kwds : {None, dict} options for computing the penalty matrix from the second derivative of the spline. knot_kwds : {None, list[dict]} option for the knot selection. By default knots are selected in the same way as in patsy, however the number of knots is independent of keeping or removing the constant. Interior knot selection is based on quantiles of the data and is the same in patsy and mgcv. Boundary points are at the limits of the data range. The available options use with `get_knots_bsplines` are - knots : None or array interior knots - spacing : 'quantile' or 'equal' - lower_bound : None or float location of lower boundary knots, all boundary knots are at the same point - upper_bound : None or float location of upper boundary knots, all boundary knots are at the same point - all_knots : None or array If all knots are provided, then those will be taken as given and all other options will be ignored. rVFNrc ||_||_||_t|f||d||_||ni|_t |||dS)N)r;rrr)r;rr=rdr covder2_kwdsrr) rrrr;r=rrr knot_kwdsrs rrzUnivariateBSplines.__init__s !2'M&RMM9MM -9-E\\"$   ;m      rct|j|j|j|j\}}}t |fddi|j}||||fS)N)r=rpT)rGrrr;r=rxr)rr@rTrCrs rrz4UnivariateBSplines._smooth_basis_for_single_variablesk': FDJ "4(6(6(6$y* t44$4!%!244iX55rrc||j}t||j|j||j}t |dd}||s||j}|S)acreate the spline basis for new observations The main use of this stateful transformation is for prediction using the same specification of the spline basis. Parameters ---------- x_new : ndarray observations of the underlying explanatory variable deriv : int which derivative of the spline basis to compute This is an options for internal computation. skip_ctransf : bool whether to skip the constraint transform This is an options for internal computation. Returns ------- basis : ndarray design matrix for the spline basis for given ``x_new`` N)r<r=r)rrGrr;r=getattrrr)rx_newr<rpexogrs rrszUnivariateBSplines.transformsp. =FE"5$*dk).595KMMM $ 400  | 88DL))D r)rVFNrN)rF)rrrrrrrsrrs@rrrsw11d;@14"        6 6 6!!!!!!!!rrcPeZdZdZ d fd ZddZdZd Zdd Zd Z d Z xZ S)UnivariateCubicSplinesziCubic Spline single smooth component Cubic splines as described in the wood's book in chapter 3 Ndomainrcd|_||_||_||dx|_}t |||_t|||dS)NrVT initializer) r;rtransform_data_methodtransform_datarrrrr)rrrrrsrrs rrzUnivariateCubicSplines.__init__s} %."((t(<<<*1b11   ;m      rFc|j}||S|dur|dkr5|d|_|d|_nFt |t r"|d|_|d|_d|_ntd|j|jz |_|jdkr||jz |jz }|Std)NTrrr-z-transform should be None, 'domain' or a tuplezincorrect transform_data_method) rr domain_lowr domain_upprtupler[ domain_diff)rrrtms rrz%UnivariateCubicSplines.transform_data s  ' :H   X~~"#%%(("#%%((B&& /"$Q%"$Q%-5** ".///#@D   % 1 1T_$(88AH>?? ?rcx|ddddf}|jdksB|ddddfd|_|ddddfxx|jzcc<n$t j|jd|_|ddddf}|jdks>t jdt j t j |dz }nt j |jd}|jdkr |dd}||_ |dd|fS)NrInoner-r)rqzno-const) _splines_xrr transf_meanr r:r$ _splines_sdiagr abseyer)rr@srs rrz8UnivariateCubicSplines._smooth_basis_for_single_variable%s8!!!!!SbS&)6))$QQQU|0033D  !!!QRR%LLLD, ,LLLL!x A77D  OO  crc3B3h '6))garve}}1 = = ==>>GGfU[^,,G  z ) )abbkG dD!##rc|dz dzdz |dz dzdz zdz }tj||z dz dzdtj||z dz dzzz dzdz }||z S)Ng?r gUUUUUU?gݝ?g8@)r r)rrzp1p2s r_rkzUnivariateCubicSplines._rk=s5yQ'QY1,||j}t|jdz}|jd}t j||f}||dddf<t |D]?\}}t |jD]%\}}|||} | |||dzf<&@|S)Nr rrzr-)rr7rr$r ones enumerater) rr n_columnsr}r@rDxijxkjs_ijs rrz!UnivariateCubicSplines._splines_xDs 9A OOa' wqztY/000aaad q\\ ' 'EAr#DJ// ' '3xxC(("&aQh ' rc t|jdz}tj||f}t |jD]@\}}t |jD]&\}}|||||dz|dzf<'A|S)Nr rz)r7rr r:rr)rqrrDx1rx2s rrz!UnivariateCubicSplines._splines_sRs  OOa  HAq6 " " "tz** 3 3EAr"4:.. 3 32"&((2r"2"2!a%Q, 3rc||d}||}|ddddfxx|jzcc<|j||j}|S)NFrr-)rrrrrrrrs rrsz UnivariateCubicSplines.transformZsp##Ee#<<u%% QQQU t'' < #88DL))D r)Nrr)Fr) rrrrrrrrrrrsrrs@rrrs ;C"       @@@@0$$$0    rrc<eZdZdZd fd ZdZdZdZdZxZ S) UnivariateCubicCyclicSplinesacyclic cubic regression spline single smooth component This creates and holds the Cyclic CubicSpline basis function for one component. Parameters ---------- x : ndarray, 1-D underlying explanatory variable for smooth terms. df : int number of basis functions or degrees of freedom degree : int degree of the spline include_intercept : bool If False, then the basis functions are transformed so that they do not include a constant. This avoids perfect collinearity if a constant or several components are included in the model. constraints : {None, str, array} Constraints are used to transform the basis functions to satisfy those constraints. `constraints = 'center'` applies a linear transform to remove the constant and center the basis functions. variable_name : None or str The name for the underlying explanatory variable, x, used in for creating the column and parameter names for the basis functions. Nrcd|_||_||_t|||_t |||dS)NrVr)r;rrrrrr)rrrrrrs rrz%UnivariateCubicCyclicSplines.__init__~s[ *1b11   ;m      rc,tdt|jzdzd|ji}|j|_|jdz dz}t |j|ddd}||\}}|||}|dd|fS)Nz cc(x, df=z) - 1rr r-)rKrOrLrM)rrrr design_infor _get_b_and_d_get_s)rr@rKrPbdrs rrz>UnivariateCubicCyclicSplines._smooth_basis_for_single_variables c$'ll2W.s8'I'I'I+,(+SVV|'I'I'Irc3$K|] }|jV dSr)r@rrts r z/AdditiveGamSmoother.__init__..s8$?$?#+%-N$?$?$?$?$?$?rcg|] }|j Sr)rrs rrz0AdditiveGamSmoother.__init__..s1!A!A!A%-"*!2!A!A!ArrFT) rpd DataFramecolumnstolistSeriesnamer r!r3copyrr7r$r}rboolr=variable_namesr9_make_smoothers_list smoothershstacklistr@rpenalty_matricesrextendmaskarrayappend) rrrr=kwargs data_namesrt last_columnr s rrzAdditiveGamSmoother.__init__sh a & & ))++JJ 29 % % &JJJ JqMM 6Q;;VVXXDFFFA;DFLLDF&*fl# 4# ' . . 7&7%84;K%KD " "%6D "  !%&0##'I'I05d6F0G0G'I'I'I###1D 2244Yt$?$?/3~$?$?$? ? ?@@ )!,!A!A15!A!A!A 6 6H N ! !("4 5 5 5 5   # #H8UGdn455DAEDX/+== >%(::K I  T " " " "  # #rcdSrrrs rrz(AdditiveGamSmoother._make_smoothers_list s rcjdkr!jdkrddtjt fdt jD}|S)acreate the spline basis for new observations The main use of this stateful transformation is for prediction using the same specification of the spline basis. Parameters ---------- x_new: ndarray observations of the underlying explanatory variable Returns ------- basis : ndarray design matrix for the spline basis for given ``x_new``. r-rIc3lK|].}j|dd|fV/dSr)rrs)rrDrrs rrz0AdditiveGamSmoother.transform..!sX;;#nQ/99%1+FF;;;;;;r)r3rr#r r r r9rs`` rrszAdditiveGamSmoother.transforms :??t/144MM"a((Ey;;;;;"'(8"9"9;;;;;<< r)NF)rrrrrrrrsrrrrrs]2#2#2#2#h  ^ rrc(eZdZdZfdZdZxZS)GenericSmoothersz9generic class for additive smooth components for GAM c\||_t|ddSN)r)rrr)rrrrs rrzGenericSmoothers.__init__)s-" 400000rc|jSr)rrs rrz%GenericSmoothers._make_smoothers_list-s ~rrrrrrrrrs@rrr&sQ11111rrc*eZdZdZdfd ZdZxZS)PolynomialSmootherz+additive polynomial components for GAM Nc\||_t||dSr)degreesrr)rrrrrs rrzPolynomialSmoother.__init__4s-  >:::::rcg}t|jD]N}t|jdd|f|j||j|}||O|S)N)r;r)r9rrrrrrrrv uv_smoothers rrz'PolynomialSmoother._make_smoothers_list8sy t'(( * *A6qqq!t |A"1!4666K   [ ) ) ) )rrrrs@rrr1sV;;;;;;rrc.eZdZdZ dfd ZdZxZS)BSplinesa9additive smooth components using B-Splines This creates and holds the B-Spline basis function for several components. Parameters ---------- x : array_like, 1-D or 2-D underlying explanatory variable for smooth terms. If 2-dimensional, then observations should be in rows and explanatory variables in columns. df : {int, array_like[int]} number of basis functions or degrees of freedom; should be equal in length to the number of columns of `x`; may be an integer if `x` has one column or is 1-D. degree : {int, array_like[int]} degree(s) of the spline; the same length and type rules apply as to `df` include_intercept : bool If False, then the basis functions are transformed so that they do not include a constant. This avoids perfect collinearity if a constant or several components are included in the model. constraints : {None, str, array} Constraints are used to transform the basis functions to satisfy those constraints. `constraints = 'center'` applies a linear transform to remove the constant and center the basis functions. variable_names : {list[str], None} The names for the underlying explanatory variables, x used in for creating the column and parameter names for the basis functions. If ``x`` is a pandas object, then the names will be taken from it. knot_kwds : None or list of dict option for the knot selection. By default knots are selected in the same way as in patsy, however the number of knots is independent of keeping or removing the constant. Interior knot selection is based on quantiles of the data and is the same in patsy and mgcv. Boundary points are at the limits of the data range. The available options use with `get_knots_bsplines` are - knots : None or array interior knots - spacing : 'quantile' or 'equal' - lower_bound : None or float location of lower boundary knots, all boundary knots are at the same point - upper_bound : None or float location of upper boundary knots, all boundary knots are at the same point - all_knots : None or array If all knots are provided, then those will be taken as given and all other options will be ignored. Attributes ---------- smoothers : list of univariate smooth component instances basis : design matrix, array of spline bases columns for all components penalty_matrices : list of penalty matrices, one for each smooth term dim_basis : number of columns in the basis k_variables : number of smooth components col_names : created names for the basis columns There are additional attributes about the specification of the splines and some attributes mainly for internal use. Notes ----- A constant in the spline basis function can be removed in two different ways. The first is by dropping one basis column and normalizing the remaining columns. This is obtained by the default ``include_intercept=False, constraints=None`` The second option is by using the centering transform which is a linear transformation of all basis functions. As a consequence of the transformation, the B-spline basis functions do not have locally bounded support anymore. This is obtained ``constraints='center'``. In this case ``include_intercept`` will be automatically set to True to avoid dropping an additional column. FNctt|tr"tj|gt|_n||_t|tr"tj|gt|_n||_||_||_|dkrd}t |||dS)Nr+rT)r=r) rr5r rrdfsrrrr) rrrr;r=rrrrs rrzBSplines.__init__s fc " " "8VHC888DLL!DL b#   xC000DHHDH"& ( " " $   /)      rc 4g}t|jD]}|jr |j|ni}t|jdd|ff|j||j||j||j|j |d|}| ||S)N)rr;r=rr) r9rrrrr'rr=rrr)rrr"kwdsr#s rrzBSplines._make_smoothers_lists t'(( * *A(,>4>!$$BD,qqq!t >8A;t|A"&"8"; ,"1!4 >> 9= >>K   [ ) ) ) )r)FNNNrrs@rr%r%CsaOO`9>BF      ,       rr%c.eZdZdZ dfd ZdZxZS) CubicSplineszadditive smooth components using cubic splines as in Wood 2006. Note, these splines do NOT use the same spline basis as ``Cubic Regression Splines``. rrNcz||_||_||_t|||dS)N)rr)r'rrsrr)rrrrrsrrs rrzCubicSplines.__init__sK&"  ;~      rc g}t|jD]Z}t|jdd|f|j||j|j|j|}||[|S)N)rrrsr) r9rrrr'rrsrrr!s rrz!CubicSplines._make_smoothers_lists t'(( * *A0 F111a4LTXa[(,(8&*n*.*=a*@ BBBK   [ ) ) ) )r)rrNrrs@rr+r+s_ ?G $             rr+c*eZdZdZdfd ZdZxZS)CyclicCubicSplinesa,additive smooth components using cyclic cubic regression splines This spline basis is the same as in patsy. Parameters ---------- x : array_like, 1-D or 2-D underlying explanatory variable for smooth terms. If 2-dimensional, then observations should be in rows and explanatory variables in columns. df : int numer of basis functions or degrees of freedom constraints : {None, str, array} Constraints are used to transform the basis functions to satisfy those constraints. variable_names : {list[str], None} The names for the underlying explanatory variables, x used in for creating the column and parameter names for the basis functions. If ``x`` is a pandas object, then the names will be taken from it. Ncj||_||_t||dSr)r'rrr)rrrrrrs rrzCyclicCubicSplines.__init__s5& >:::::rcg}t|jD]T}t|jdd|f|j||j|j|}||U|S)N)rrr)r9rrrr'rrrr!s rrz'CyclicCubicSplines._make_smoothers_lists~ t'(( * *A6qqq!t 8A;D,<"1!4666K   [ ) ) ) )r)NNrrs@rr/r/sV(;;;;;;       rr/)r(T)NNNrVrWNNN)rV)rVNFr )T)%rabcrrstatsmodels.compat.pythonrnumpyr pandasrpatsyrpatsy.mgcv_cubic_splinesrstatsmodels.tools.linalgrrr'rGrQrUrdrlrxr~rrrrrrrrrr%r+r/rrrr9s('''''''444444::::::777777 555@@@@F < < <(((<=7;37\\\\~     :>*+2((((P)))))NN733)))X J J J J J 5 J J J 66666#8666>lllll.lll^fffff2fffRnnnnn#8nnnbOOOOO..11OOOd*,$sssss"sssl&8$$$$$,$$$$$r