(phCRSSKrSSKJr SSKrSSKJr SSKJrJ r J r J r J r J r SSKJr SSKJr SSKJr SS KJr SS KJr SS KJr /S QrS rS&SjrSrSr"SS5rS&SjrSr Sr!Sr"Sr#Sr$Sr%Sr&S'Sjr'Sr(S(Sjr)S)SS.S jjr*S!r+S"r,S#r-S*S$jr.S%r/g)+N)prod)normalize_axis_index)get_lapack_funcs LinAlgErrorcholesky_bandedcho_solve_bandedsolve solve_banded)minimize_scalar)_dierckx) _fitpack_impl) csr_array)poch) combinations)BSplinemake_interp_splinemake_lsq_splinemake_smoothing_splinec[R"U[R5(a[R$[R$)z>Return np.complex128 for complex dtypes, np.float64 otherwise.)np issubdtypecomplexfloating complex128float64dtypes N/var/www/html/venv/lib/python3.13/site-packages/scipy/interpolate/_bsplines.py _get_dtypers- }}UB..//}}zzc[R"U5n[UR5nUR USS9nU(a4[R "U5R 5(d [S5eU$)zwConvert the input into a C contiguous float array. NB: Upcasts half- and single-precision floats to double precision. F)copyz$Array must not contain infs or nans.)rascontiguousarrayrrastypeisfiniteall ValueError)x check_finitedtyps r_as_float_arrayr+s^ QA agg D E"ABKKN..00?@@ Hr c US:Xag[R"[SUS-5Vs/sH oCX U-- PM sn5$s snf)z Dual polynomial of the B-spline B_{j,k,t} - polynomial which is associated with B_{j,k,t}: $p_{j,k}(y) = (y - t_{j+1})(y - t_{j+2})...(y - t_{j+k})$ rr )rrrange)jktyis r _dual_polyr3*sA  Av 77E!QUO=n, ...) spline coefficients k : int B-spline degree extrapolate : bool or 'periodic', optional whether to extrapolate beyond the base interval, ``t[k] .. t[n]``, or to return nans. If True, extrapolates the first and last polynomial pieces of b-spline functions active on the base interval. If 'periodic', periodic extrapolation is used. Default is True. axis : int, optional Interpolation axis. Default is zero. Attributes ---------- t : ndarray knot vector c : ndarray spline coefficients k : int spline degree extrapolate : bool If True, extrapolates the first and last polynomial pieces of b-spline functions active on the base interval. axis : int Interpolation axis. tck : tuple A read-only equivalent of ``(self.t, self.c, self.k)`` Methods ------- __call__ basis_element derivative antiderivative integrate insert_knot construct_fast design_matrix from_power_basis Notes ----- B-spline basis elements are defined via .. math:: B_{i, 0}(x) = 1, \textrm{if $t_i \le x < t_{i+1}$, otherwise $0$,} B_{i, k}(x) = \frac{x - t_i}{t_{i+k} - t_i} B_{i, k-1}(x) + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1, k-1}(x) **Implementation details** - At least ``k+1`` coefficients are required for a spline of degree `k`, so that ``n >= k+1``. Additional coefficients, ``c[j]`` with ``j > n``, are ignored. - B-spline basis elements of degree `k` form a partition of unity on the *base interval*, ``t[k] <= x <= t[n]``. Examples -------- Translating the recursive definition of B-splines into Python code, we have: >>> def B(x, k, i, t): ... if k == 0: ... return 1.0 if t[i] <= x < t[i+1] else 0.0 ... if t[i+k] == t[i]: ... c1 = 0.0 ... else: ... c1 = (x - t[i])/(t[i+k] - t[i]) * B(x, k-1, i, t) ... if t[i+k+1] == t[i+1]: ... c2 = 0.0 ... else: ... c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * B(x, k-1, i+1, t) ... return c1 + c2 >>> def bspline(x, t, c, k): ... n = len(t) - k - 1 ... assert (n >= k+1) and (len(c) >= n) ... return sum(c[i] * B(x, k, i, t) for i in range(n)) Note that this is an inefficient (if straightforward) way to evaluate B-splines --- this spline class does it in an equivalent, but much more efficient way. Here we construct a quadratic spline function on the base interval ``2 <= x <= 4`` and compare with the naive way of evaluating the spline: >>> from scipy.interpolate import BSpline >>> k = 2 >>> t = [0, 1, 2, 3, 4, 5, 6] >>> c = [-1, 2, 0, -1] >>> spl = BSpline(t, c, k) >>> spl(2.5) array(1.375) >>> bspline(2.5, t, c, k) 1.375 Note that outside of the base interval results differ. This is because `BSpline` extrapolates the first and last polynomial pieces of B-spline functions active on the base interval. >>> import matplotlib.pyplot as plt >>> import numpy as np >>> fig, ax = plt.subplots() >>> xx = np.linspace(1.5, 4.5, 50) >>> ax.plot(xx, [bspline(x, t, c ,k) for x in xx], 'r-', lw=3, label='naive') >>> ax.plot(xx, spl(xx), 'b-', lw=4, alpha=0.7, label='BSpline') >>> ax.grid(True) >>> ax.legend(loc='best') >>> plt.show() References ---------- .. [1] Tom Lyche and Knut Morken, Spline methods, http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v05/undervisningsmateriale/ .. [2] Carl de Boor, A practical guide to splines, Springer, 2001. cb>[TU]5 [R"U5Ul[ R "U5Ul[ R"U[ RS9Ul US:XaX@l O[U5Ul URRSUR- S- n[XPRR5nXPlUS:wa'[ R""URUS5UlUS:a [%S5eURRS:wa [%S5eX`RS-:a[%SSU-S-U4-5e[ R&"UR5S:R)5(a [%S 5e[+[ R,"URX6S-55S:a [%S 5e[ R."UR5R15(d [%S 5eURRS:a [%S 5eURRSU:a [%S 5e[3URR45n[ R"URUS9Ulg)Nrperiodicrr Spline order cannot be negative.z$Knot vector must be one-dimensional.z$Need at least %d knots for degree %dz(Knots must be in a non-decreasing order.z!Need at least two internal knots.z#Knots should not have nans or infs.z,Coefficients must be at least 1-dimensional.z0Knots, coefficients and degree are inconsistent.)super__init__operatorindexr/rasarraycr#rr0 extrapolateboolshaperndimaxismoveaxisr'diffanyr6uniquer%r&rr) selfr0rGr/rHrLndt __class__s rrCBSpline.__init__s "A%%arzz: * $* #K0D  FFLLOdff $q (#D&&++6 19 [[q1DF q5?@ @ 66;;! CD D vvz>CcAgq\*+ + GGDFFOa  $ $ & &GH H ryyA#' (1 ,@A A{{466"&&((BC C 66;;?KL L 66<<?Q OP P  %%%dffB7r cr[RU5nXUsUlUlUlXFlXVlU$)zConstruct a spline without making checks. Accepts same parameters as the regular constructor. Input arrays `t` and `c` must of correct shape and dtype. )object__new__r0rGr/rHrL)clsr0rGr/rHrLrQs rconstruct_fastBSpline.construct_fasts6~~c"!"q&  r cHURURUR4$)z@Equivalent to ``(self.t, self.c, self.k)`` (read-only). )r0rGr/rQs rtck BSpline.tck svvtvvtvv%%r c[U5S- n[U5n[RUSS- 4U-XSS-4U-4n[R"U5nSXC'UR XX25$)aReturn a B-spline basis element ``B(x | t[0], ..., t[k+1])``. Parameters ---------- t : ndarray, shape (k+2,) internal knots extrapolate : bool or 'periodic', optional whether to extrapolate beyond the base interval, ``t[0] .. t[k+1]``, or to return nans. If 'periodic', periodic extrapolation is used. Default is True. Returns ------- basis_element : callable A callable representing a B-spline basis element for the knot vector `t`. Notes ----- The degree of the B-spline, `k`, is inferred from the length of `t` as ``len(t)-2``. The knot vector is constructed by appending and prepending ``k+1`` elements to internal knots `t`. Examples -------- Construct a cubic B-spline: >>> import numpy as np >>> from scipy.interpolate import BSpline >>> b = BSpline.basis_element([0, 1, 2, 3, 4]) >>> k = b.k >>> b.t[k:-k] array([ 0., 1., 2., 3., 4.]) >>> k 3 Construct a quadratic B-spline on ``[0, 1, 1, 2]``, and compare to its explicit form: >>> t = [0, 1, 1, 2] >>> b = BSpline.basis_element(t) >>> def f(x): ... return np.where(x < 1, x*x, (2. - x)**2) >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0, 2, 51) >>> ax.plot(x, b(x), 'g', lw=3) >>> ax.plot(x, f(x), 'r', lw=8, alpha=0.4) >>> ax.grid(True) >>> plt.show() rArr ?)r6r+rr_ zeros_likerZ)rYr0rHr/rGs r basis_elementBSpline.basis_elementssr FQJ A  EE1Q46)a-rU1WJN2 3 MM! !!!77r c [US5n[US5nUS:wa [U5nUS:a [S5eURS:wd$[R "USSUSS:5(a[SUS 35e[ U5S U-S -:a[S US 35eUS:Xa+URU- S- nX#XU- X%X#- --nS nOKU(dD[U5X#:d$[U5X"RSU- S- :a[S US 35eURSnXSS--nU[R"[R5R:a[RnO[Rn[R"XU[R "U5U5upn UR#5nU R$U:waU R'U5n [R("XS-5R+SUS-5n U [R,"US-US9-n U R#5n [R,"SUS-US--US-US9n [/XU 4URSURSU- S- 4S9$)aL Returns a design matrix as a CSR format sparse array. Parameters ---------- x : array_like, shape (n,) Points to evaluate the spline at. t : array_like, shape (nt,) Sorted 1D array of knots. k : int B-spline degree. extrapolate : bool or 'periodic', optional Whether to extrapolate based on the first and last intervals or raise an error. If 'periodic', periodic extrapolation is used. Default is False. .. versionadded:: 1.10.0 Returns ------- design_matrix : `csr_array` object Sparse matrix in CSR format where each row contains all the basis elements of the input row (first row = basis elements of x[0], ..., last row = basis elements x[-1]). Examples -------- Construct a design matrix for a B-spline >>> from scipy.interpolate import make_interp_spline, BSpline >>> import numpy as np >>> x = np.linspace(0, np.pi * 2, 4) >>> y = np.sin(x) >>> k = 3 >>> bspl = make_interp_spline(x, y, k=k) >>> design_matrix = bspl.design_matrix(x, bspl.t, k) >>> design_matrix.toarray() [[1. , 0. , 0. , 0. ], [0.2962963 , 0.44444444, 0.22222222, 0.03703704], [0.03703704, 0.22222222, 0.44444444, 0.2962963 ], [0. , 0. , 0. , 1. ]] Construct a design matrix for some vector of knots >>> k = 2 >>> t = [-1, 0, 1, 2, 3, 4, 5, 6] >>> x = [1, 2, 3, 4] >>> design_matrix = BSpline.design_matrix(x, t, k).toarray() >>> design_matrix [[0.5, 0.5, 0. , 0. , 0. ], [0. , 0.5, 0.5, 0. , 0. ], [0. , 0. , 0.5, 0.5, 0. ], [0. , 0. , 0. , 0.5, 0.5]] This result is equivalent to the one created in the sparse format >>> c = np.eye(len(t) - k - 1) >>> design_matrix_gh = BSpline(t, c, k)(x) >>> np.allclose(design_matrix, design_matrix_gh, atol=1e-14) True Notes ----- .. versionadded:: 1.8.0 In each row of the design matrix all the basis elements are evaluated at the certain point (first row - x[0], ..., last row - x[-1]). `nt` is a length of the vector of knots: as far as there are `nt - k - 1` basis elements, `nt` should be not less than `2 * k + 2` to have at least `k + 1` basis element. Out of bounds `x` raises a ValueError. Tr?rr@r Nraz2Expect t to be a 1-D sorted array_like, but got t=.rAzLength t is not enough for k=FOut of bounds w/ x = rrJ)r+rIr'rKrrOr6sizeminmaxrJiinfoint32int64r data_matrix ones_likeravelrr$repeatreshapearanger) rYr(r0r/rHrRnnz int_dtypedataoffsets_indicesindptrs r design_matrixBSpline.design_matrixSsnX At $ At $ * ${+K q5?@ @ 66Q;"&&123B00 &&'S+, , q6AEAI A+=)> >4QCq9: : GGAJq5k "((#'' 'II$//aa+Vqzz| ==I %nnY/G))GqS)11"ac:BIIac;;--/1q1uQ/QiH F #771:qwwqzA~12  r c Uc URn[R"U5nURURpT[R "UR 5[RS9nUS:XaURRUR- S- nURURXRUR- URUURUR- --nSn[R"[U5[URRSS54URRS9nUR!5 URR#[$5nURRS:XaMURRR&S:Xa)UR)URRSS5n[*R,"URUR)URSS 5URXX7R#[$55 UR)X@RRSS-5nUR.S:waW[1[3UR55n XXPR.-U SU-XUR.-S-n UR5U 5nU$) a  Evaluate a spline function. Parameters ---------- x : array_like points to evaluate the spline at. nu : int, optional derivative to evaluate (default is 0). extrapolate : bool or 'periodic', optional whether to extrapolate based on the first and last intervals or return nans. If 'periodic', periodic extrapolation is used. Default is `self.extrapolate`. Returns ------- y : array_like Shape is determined by replacing the interpolation axis in the coefficient array with the shape of `x`. Nrr?r FrGrrAra)rHrrFrJrKr#rsrr0rkr/emptyr6rrGr_ensure_c_contiguousviewfloatkindrur evaluate_splinerLr5r- transpose) rQr(nurHx_shapex_ndimrRoutccls r__call__BSpline.__call__s,  **K JJqM''166  "** = * $ dff$q(Atvv!ffTVVn"49=:H"IIAKhhATVV\\!"%5 67tvv||L !!#VV[[  66;;!  1 1S 8DFFLLOQ/B  BHHQK)D"ffa[((5/ Kkk'FFLL$445 99>U388_%A )*QwZ7!499>> from scipy.interpolate import BSpline >>> b = BSpline.basis_element([0, 1, 2]) >>> b.integrate(0, 1) array(0.5) If the integration limits are outside of the base interval, the result is controlled by the `extrapolate` parameter >>> b.integrate(-1, 1) array(0.0) >>> b.integrate(-1, 1, extrapolate=False) array(0.5) >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.grid(True) >>> ax.axvline(0, c='r', lw=5, alpha=0.5) # base interval >>> ax.axvline(2, c='r', lw=5, alpha=0.5) >>> xx = [-1, 1, 2] >>> ax.plot(xx, b(xx)) >>> plt.show() Nr rar?rArrF)rHrr0rkr/rmrlrGrKrsplintr^rrFrrrJrr6rcrrdivmodrr rru)rQabrHsignrRintegralrrGrtacakatsteperiodinterval n_periodsleftr(s r integrateBSpline.integratees`  **K !!# 5qD FFKK$&& 1 $ * $[Avvdff~&AAvvay!Avv{{a)//dhh?zz(/22hh4 QR 012$&&,,G FF [3q6 ! 6a2%!''!"+"5667A"--tvvq&&.A1E  * $VVDFF^TVVAYWFuH$Xv6OIt1}JJxrzz:((ZZ R-H&(!Qs<q6CF?I%88QTVV\\!"-=(>$?*.&&,,8"f&&ADAwJJvRZZ8((ZZ R-H&(!Qs<FSVO+JJwbjj9((ZZ R-H&(!Qs<FSVO+JJFRK0 C((ZZ R-H&(!Qs<FSVO+ A64A  $ $RBHHQK)D"$aK >1vAH --r c SSKJn [X5(d[S[ U5S35eUR nUR nUR RSS- nURSnUS:Xa [XF5nO8US:XdUS:Xa [XF5nO US :Xa [XF5nO[S U35eURSXv-S-- n [R"Xy-UR RS 9n [US-5Hn [US - 5HNn X==[!US-U *5X[U 4-[R""S Xk- 5-[%XXLX5-- ss'MP [US - Xy-5HTn X==[!US-U *5X[US - 4-[R""S Xk- 5-[%XXGS - X5-- ss'MV M UR'XXaR(UR*5$)am Construct a polynomial in the B-spline basis from a piecewise polynomial in the power basis. For now, accepts ``CubicSpline`` instances only. Parameters ---------- pp : CubicSpline A piecewise polynomial in the power basis, as created by ``CubicSpline`` bc_type : string, optional Boundary condition type as in ``CubicSpline``: one of the ``not-a-knot``, ``natural``, ``clamped``, or ``periodic``. Necessary for construction an instance of ``BSpline`` class. Default is ``not-a-knot``. Returns ------- b : BSpline object A new instance representing the initial polynomial in the B-spline basis. Notes ----- .. versionadded:: 1.8.0 Accepts only ``CubicSpline`` instances for now. The algorithm follows from differentiation the Marsden's identity [1]: each of coefficients of spline interpolation function in the B-spline basis is computed as follows: .. math:: c_j = \sum_{m=0}^{k} \frac{(k-m)!}{k!} c_{m,i} (-1)^{k-m} D^m p_{j,k}(x_i) :math:`c_{m, i}` - a coefficient of CubicSpline, :math:`D^m p_{j, k}(x_i)` - an m-th defivative of a dual polynomial in :math:`x_i`. ``k`` always equals 3 for now. First ``n - 2`` coefficients are computed in :math:`x_i = x_j`, e.g. .. math:: c_1 = \sum_{m=0}^{k} \frac{(k-1)!}{k!} c_{m,1} D^m p_{j,3}(x_1) Last ``nod + 2`` coefficients are computed in ``x[-2]``, ``nod`` - number of derivatives at the ends. For example, consider :math:`x = [0, 1, 2, 3, 4]`, :math:`y = [1, 1, 1, 1, 1]` and bc_type = ``natural`` The coefficients of CubicSpline in the power basis: :math:`[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, 1, 1, 1]]` The knot vector: :math:`t = [0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4]` In this case .. math:: c_j = \frac{0!}{k!} c_{3, i} k! = c_{3, i} = 1,~j = 0, ..., 6 References ---------- .. [1] Tom Lyche and Knut Morken, Spline Methods, 2005, Section 3.1.2 r ) CubicSplinez3Only CubicSpline objects are accepted for now. Got z instead.r not-a-knotnaturalclampedr?Unknown boundary condition: rrAra)_cubicr isinstanceNotImplementedErrortyper(rGrJ _not_a_knot_augknt_periodic_knots TypeErrorrrrr-rpowerr;rZrHrL)rYppbc_typerr(coefr/rRr0nodrGmr2r.s rfrom_power_basisBSpline.from_power_basissX ("**%)66:2hZy'JK K DDtt DDJJqMA  GGAJ l "A!A  !W %9 A  "%A:7)DE EggajAEAI& HHQWBDDJJ /q1uA1q5\QUQB$!t*4((2qu-.)!a;<<"1q5!'*QUQB$!a%x.8((2qu-.)!a%!?@@+ !!!>>277CCr c XRUR:d XRUR*S- :a[SUS35eUS::a[SU<S35eURR5nURR5n[ U5H)n[ XX@RURS:H5up4M+ URX4URURUR5$)aInsert a new knot at `x` of multiplicity `m`. Given the knots and coefficients of a B-spline representation, create a new B-spline with a knot inserted `m` times at point `x`. Parameters ---------- x : float The position of the new knot m : int, optional The number of times to insert the given knot (its multiplicity). Default is 1. Returns ------- spl : BSpline object A new BSpline object with the new knot inserted. Notes ----- Based on algorithms from [1]_ and [2]_. In case of a periodic spline (``self.extrapolate == "periodic"``) there must be either at least k interior knots t(j) satisfying ``t(k+1)>> import numpy as np >>> from scipy.interpolate import BSpline, make_interp_spline >>> x = np.linspace(0, 10, 5) >>> y = np.sin(x) >>> spl = make_interp_spline(x, y, k=3) >>> spl.t array([ 0., 0., 0., 0., 5., 10., 10., 10., 10.]) Insert a single knot >>> spl_1 = spl.insert_knot(3) >>> spl_1.t array([ 0., 0., 0., 0., 3., 5., 10., 10., 10., 10.]) Insert a multiple knot >>> spl_2 = spl.insert_knot(8, m=3) >>> spl_2.t array([ 0., 0., 0., 0., 5., 8., 8., 8., 10., 10., 10., 10.]) r zCannot insert a knot at rhrz`m` must be positive, got m = r?) r0r/r'r"rGr-_insertrHrZrL)rQr(rttrr{s r insert_knotBSpline.insert_knotTsJ vvdff~ VVTVVGAI%6!67s!<= = 6>!qAB B VV[[] VV[[]qAQB0@0@J0NOFB""246643C3CTYYOOr )rLrGrHr/r0)Tr)TFrN)r N)r)__name__ __module__ __qualname____firstlineno____doc__rC classmethodrZpropertyr^rer~rrrrrrr__static_attributes__ __classcell__)rTs@rrrEsIV-8^  && =8=8~D D L:x #38&3PB.HhDhDTOPOPr rc[R"X[U5US5nUS:a[SUS35eXXU-S-:XaUS-nU(a3US-SU-::a'US-URSSU-- :a [S5e[ R USUS-XUS-S4nURSS-4URSS-n[ R"XrRS 9nX%S2S 4XS-S2S 4'[XUU- S 5H7n XU - XiU-S-Xi- - n XU S 4-S U - X)S- S 4--XS 4'M9 USXS- S-2S 4USXS- S-2S 4'U(aURSn X- S- n U SU-- S- n XlXc- nX\U- :aXlU- U U- USU&XX-2S 4USU2S 4'USU-S- ::a&XcS-US-U-U-XkU- S&USU2S 4XX-2S 4'Xh4$) zInsert a single knot at `xval`.FrzCannot insert the knot at rhr rAzNot enough internal knots.Nr.rarb) r find_intervalrr'rJrrcrrr-)xvalr0rGr/r?rrnewshaperr2facrRnkn2kTs rrrsx$%%aE$KEBH!|5dV1=>> {a1 q())A  qLAaC hlaggaj1Q36F&F9: : q(1*~txz{^3 4B Q 17712;.H ('' *BY^,Bz{C 8aZ ,e|Q3q5 BE 121c6]b3hA#s( %;;c6 - !(,q.#!56B Q HHQK UQY!A#gk FRUN Av Qr]Q&BrF[#-.Brr3wK qs1u A#ac!e}q(BstH#%bqb#g;B37{C 6Mr c[R"U5nUS-S:XaUS-S-nUR5nOUS-nUSSUSS-S- nX2U*n[RUS4US--X0S4US--4nU$)aFGiven data x, construct the knot vector w/ not-a-knot BC. cf de Boor, XIII(12). For even k, it's a bit ad hoc: Greville sites + omit 2nd and 2nd-to-last data points, a la not-a-knot. This seems to match what Dierckx does, too: https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L63-L80 rAr Nrar)rrFr"rc)r(r/k2r0s rrrs 1 A1uz!e\ FFH !V qrUQsV^q  bS A qtgqsmQ2!A#./A Hr cH[RUS4U-XS4U-4$)zBConstruct a knot vector appropriate for the order-k interpolation.rra)rrc)r(r/s rrrs* 55!A$A"xz) **r c[U[5(aPUS:XaS[R"U54/nU$US:XaS[R"U54/nU$[ SU35eU$)Nrr rrAzUnknown boundary condition : )rstrrrr')deriv target_shapes r_convert_string_aliasesr su% I ,/01E L i ,/01E LAbXq$A 1f A Hr c[R"U5n[U5nUS-S:Xa*[R"U5nUSS===USSS- -sss&[R"U5n[R"USU--5nX%X*&[ SU5H=nXQU- XFUS- -*S- - XQU- S- 'XQ*U-S- XFUS- --XQ*U-'M? U$)z# returns vector of nodes on circle rArr raN)rr"r6rNrr-)r(r/xcrRdxr0r2s rrrys B BA1uz WWR[ 1b RWq[ B QUAaH 1a[Qx"AE{^a%7"88a%!) b1fqjMBAE{O3"q&  Hr c [[RXU45upnURn[R"XC-S- XC-S- 45n[ US- 5Hln[ R"X#USX6S-5nXVSUS-24==U- ss'[ R"X#USXC-S- US-5SSnXVU*S24==U-ss'Mn [ U5HTnXnXU:XaUn O[R"X(5S- n [ R"X#X5nXuXc-S- X- U S-24'MV [RS/US- -U4n [XZ5n U $)a Returns a solution of a system for B-spline interpolation with periodic boundary conditions. First ``k - 1`` rows of matrix are conditions of periodicity (continuity of ``k - 1`` derivatives at the boundary points). Last ``n`` rows are interpolation conditions. RHS is ``k - 1`` zeros and ``n`` ordinates in this case. Parameters ---------- x : 1-D array, shape (n,) Values of x - coordinate of a given set of points. y : 1-D array, shape (n,) Values of y - coordinate of a given set of points. t : 1-D array, shape(n+2*k,) Vector of knots. k : int The maximum degree of spline Returns ------- c : 1-D array, shape (n+k-1,) B-spline coefficients Notes ----- ``t`` is supposed to be taken on circle. r rNra) maprrFrkrr-r evaluate_all_bspl searchsortedrcr ) r(r1r0r/rRmatrr2bbrrrrGs r_make_interp_per_full_matrrsc<"**qQi(GA! A 88QUQY * +D1q5\  ' 'adA1u = !a%ZB  ' 'aeQUQYA Fs K V  1Xt Q4<D??1+a/D ' 'd 9)+QUQYtAv %& qcQUmQA dA Hr c[R"X[U5US5n[URS5HOn Xin [R "XX(U 5n [US-5Hn X- U -n XX4U-U-U -U - U 4'M MQ g)aFill in the entries of the colocation matrix corresponding to known derivatives at `xval`. The colocation matrix is in the banded storage, as prepared by _coloc. No error checking. Parameters ---------- t : ndarray, shape (nt + k + 1,) knots k : integer B-spline order xval : float The value at which to evaluate the derivatives at. ab : ndarray, shape(2*kl + ku + 1, nt), Fortran order B-spline colocation matrix. This argument is modified *in-place*. kl : integer Number of lower diagonals of ab. ku : integer Number of upper diagonals of ab. deriv_ords : 1D ndarray Orders of derivatives known at xval offset : integer, optional Skip this many rows of the matrix ab. Frr N)r rrr-rJr)r0r/rabklku deriv_ordsoffsetrrowrwrkrclmns r_handle_lhs_derivativesrs:  ! !!d Q >DZ%%a() _((t2>qsA8aS9n [$R'X:TUS9$T=nn[R<"S"U-U-S-U4[R@S#S$9n[BRD"XTURFU 5 U S:a[IUTUSUUUU 5 US:a[IUTUSUUUUUU- S%9 [KURSS5n[RL"UU4UR"S9nU S:aU ROSU5USU &UROSU5UU UU- &US:aUROSU5UUU- S&U(a[Q[RRUU45unn[US&UU45unU"UUUUS'S'S(9unnn nUS:a [WS)5eUS:a[S*U*-5e[R"U ROU4URSS-55n [$R'X:TUS9$![an [SU35U eSn A ff=f)+avCompute the (coefficients of) interpolating B-spline. Parameters ---------- x : array_like, shape (n,) Abscissas. y : array_like, shape (n, ...) Ordinates. k : int, optional B-spline degree. Default is cubic, ``k = 3``. t : array_like, shape (nt + k + 1,), optional. Knots. The number of knots needs to agree with the number of data points and the number of derivatives at the edges. Specifically, ``nt - n`` must equal ``len(deriv_l) + len(deriv_r)``. bc_type : 2-tuple or None Boundary conditions. Default is None, which means choosing the boundary conditions automatically. Otherwise, it must be a length-two tuple where the first element (``deriv_l``) sets the boundary conditions at ``x[0]`` and the second element (``deriv_r``) sets the boundary conditions at ``x[-1]``. Each of these must be an iterable of pairs ``(order, value)`` which gives the values of derivatives of specified orders at the given edge of the interpolation interval. Alternatively, the following string aliases are recognized: * ``"clamped"``: The first derivatives at the ends are zero. This is equivalent to ``bc_type=([(1, 0.0)], [(1, 0.0)])``. * ``"natural"``: The second derivatives at ends are zero. This is equivalent to ``bc_type=([(2, 0.0)], [(2, 0.0)])``. * ``"not-a-knot"`` (default): The first and second segments are the same polynomial. This is equivalent to having ``bc_type=None``. * ``"periodic"``: The values and the first ``k-1`` derivatives at the ends are equivalent. axis : int, optional Interpolation axis. Default is 0. check_finite : bool, optional Whether to check that the input arrays contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- b : a BSpline object of the degree ``k`` and with knots ``t``. See Also -------- BSpline : base class representing the B-spline objects CubicSpline : a cubic spline in the polynomial basis make_lsq_spline : a similar factory function for spline fitting UnivariateSpline : a wrapper over FITPACK spline fitting routines splrep : a wrapper over FITPACK spline fitting routines Examples -------- Use cubic interpolation on Chebyshev nodes: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> def cheb_nodes(N): ... jj = 2.*np.arange(N) + 1 ... x = np.cos(np.pi * jj / 2 / N)[::-1] ... return x >>> x = cheb_nodes(20) >>> y = np.sqrt(1 - x**2) >>> from scipy.interpolate import BSpline, make_interp_spline >>> b = make_interp_spline(x, y) >>> np.allclose(b(x), y) True Note that the default is a cubic spline with a not-a-knot boundary condition >>> b.k 3 Here we use a 'natural' spline, with zero 2nd derivatives at edges: >>> l, r = [(2, 0.0)], [(2, 0.0)] >>> b_n = make_interp_spline(x, y, bc_type=(l, r)) # or, bc_type="natural" >>> np.allclose(b_n(x), y) True >>> x0, x1 = x[0], x[-1] >>> np.allclose([b_n(x0, 2), b_n(x1, 2)], [0, 0]) True Interpolation of parametric curves is also supported. As an example, we compute a discretization of a snail curve in polar coordinates >>> phi = np.linspace(0, 2.*np.pi, 40) >>> r = 0.3 + np.cos(phi) >>> x, y = r*np.cos(phi), r*np.sin(phi) # convert to Cartesian coordinates Build an interpolating curve, parameterizing it by the angle >>> spl = make_interp_spline(phi, np.c_[x, y]) Evaluate the interpolant on a finer grid (note that we transpose the result to unpack it into a pair of x- and y-arrays) >>> phi_new = np.linspace(0, 2.*np.pi, 100) >>> x_new, y_new = spl(phi_new).T Plot the result >>> plt.plot(x, y, 'o') >>> plt.plot(x_new, y_new, '-') >>> plt.show() Build a B-spline curve with 2 dimensional y >>> x = np.linspace(0, 2*np.pi, 10) >>> y = np.array([np.sin(x), np.cos(x)]) Periodic condition is satisfied because y coordinates of points on the ends are equivalent >>> ax = plt.axes(projection='3d') >>> xx = np.linspace(0, 2*np.pi, 100) >>> bspl = make_interp_spline(x, y, k=5, bc_type='periodic', axis=1) >>> ax.plot3D(xx, *bspl(xx)) >>> ax.scatter3D(x, *y, color='red') >>> plt.show() Nrr?NNrrragV瞯<)atolzAFirst and last points does not match while periodic case expected Shapes of x  and y  are incompatibler zExpect x to not have duplicates1Expect x to be a 1D strictly increasing sequence.c3(# UHoSLv M g7fr).0r{s r %make_interp_spline..s<&;}&;sz6Too much info for k=0: t and bc_type can only be None.rrLz0Too much info for k=1: bc_type can only be None.zOFor periodic case t is constructed automatically and can not be passed manuallyExpect non-negative k.z'Expect t to be a 1-D sorted array_like.zGot %d knots, need at least %d.rirhc3N># UHnSUs=:*=(a T:*Os v M g7frr&r'r2r/s rr(r)11LqqA{{{{L"%zBad boundary conditions at c3N># UHnSUs=:*=(a T:*Os v M g7frr&r-s rr(r)4r.r/zAThe number of derivatives at boundaries does not match: expected z, got +rArr)r )gbsvT) overwrite_ab overwrite_bzColocation matrix is singular.z0illegal value in %d-th argument of internal gbsv),rrrr'rrFrrKr+rMallcloserkrJrOrcr#rrrrZrDrErrrrrrrr&rrrr rrrrrrurasarray_chkfiniterr)r(r1r/r0rrLr)deriv_lderiv_rrrG deriv_l_ords deriv_l_valsnleft deriv_r_ords deriv_r_valsnrightrRrr r rrrhsr2lupivinfos ` rrrSsH'\1W 5J% GS ! !" N& G 1 A ff -D(A(A AQA*R[[1qu5%I23 3vv<yy@QRSS vvaeq"vo:;;vv{bffQqrUQsV^,,LMM Av %%aAD%99 Av!)GOOP P EE!A$R5. ! JJqM  *QWW*= >%%aAD%99qA*!#-. .  y ?w*$#Aq)1%1 A(A1u122vv{bffQqrUQsV^,,BCCvv Q:&&!&&1*q.123 3 !qt 2A201566*$Q1a66&gwwqr{;G!4W!=L   q !E%gwwqr{;G!4W!=L    "F 1L1 1 16qtfA>?? 1L1 1 16qugQ?@@ A !aB Av,,.qD6waxIJ J vv{ HHbUQWWQR[( 6%%aAD%99 KB 1R4"9q="%RZZs CB OOA!RTT5) qy1adBB E z1aeRR')&y 2AGGABK H ((B> 1C qy"**2x8FU YYr84Cb6k z(00X>BKLb**RI6C YS 2EDBB)-4ABQ ax:;; KteSTT QYYuqwwqr{':;>> import numpy as np >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> x = np.linspace(-3, 3, 50) >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50) Now fit a smoothing cubic spline with a pre-defined internal knots. Here we make the knot vector (k+1)-regular by adding boundary knots: >>> from scipy.interpolate import make_lsq_spline, BSpline >>> t = [-1, 0, 1] >>> k = 3 >>> t = np.r_[(x[0],)*(k+1), ... t, ... (x[-1],)*(k+1)] >>> spl = make_lsq_spline(x, y, t, k) For comparison, we also construct an interpolating spline for the same set of data: >>> from scipy.interpolate import make_interp_spline >>> spl_i = make_interp_spline(x, y) Plot both: >>> xs = np.linspace(-3, 3, 100) >>> plt.plot(x, y, 'ro', ms=5) >>> plt.plot(xs, spl(xs), 'g-', lw=3, label='LSQ spline') >>> plt.plot(xs, spl_i(xs), 'b-', lw=3, alpha=0.7, label='interp spline') >>> plt.legend(loc='best') >>> plt.show() **NaN handling**: If the input arrays contain ``nan`` values, the result is not useful since the underlying spline fitting routines cannot deal with ``nan``. A workaround is to use zero weights for not-a-number data points: >>> y[8] = np.nan >>> w = np.isnan(y) >>> y[w] = 0. >>> tck = make_lsq_spline(x, y, t, w=~w) Notice the need to replace a ``nan`` by a numerical value (precise value does not matter as long as the corresponding weight is zero.) Nrr zExpect x to be a 1-D sequence.zNeed more x points.r+raz1Expect t to be a 1D strictly increasing sequence.r!r"r#rirhz and w znorm-eqr$rCz,Expect x to be a 1D non-decreasing sequence.rGrATrrr)r3lowerr))r4r)zUnknown method =r*)r+rrrrDrErrKrMr'rJrOrkrrrrrurrrr _norm_eq_lsqrcomplexrr _lsq_solve_qrr#rrZ)r(r1r0r/wrLr)rDrR was_complexyyrrFrr? cho_decomprGr{s rrris8` (A(A(A} A , LLOqA ff -D AQAvv{9::wwqzAaC.//1u122vv{bffQqrUQsV^a/00LMMvv<yy@QRSS1uqTaQB%i01101566vv<yy@QRSS rvvaeafn&9::LMM ~#aeafnq011GHH  QA77<<3&K Bqvv{ ZZ A &BHHQRL!H B !B77<<3&K Bqvv{ ZZ A &BHHQRL!H B !B  XXqsAhbjj <hh}BJJ7aA  ddC ) ((7#Ckk1$,-%Rd%2>@ j0#4*6 8 4qQ/1a wA,F:Q/00 1$$%A QA  ! !!Q ! 55r cURS:XdeXSS2S4-n[R"XX45upgn[R"XgX5 [R"XhU5n XeU 4$)zSolve for the LSQ spline coeffs given x, y and knots. `y` is always 2D: for 1D data, the shape is ``(m, 1)``. `w` is always 1D: one weight value per `x` value. rAN)rKr rq qr_reducefpback) r(r1r0r/rJy_wrr ncrGs rrIrI:sc 66Q;; 4j.C((q4MAr q"*s#A 19r c^^^^^ ^ ^ SnSm U U4Sjm TRSnU"TUT5m U"TUT5mUUU U U4Sjn[USU4SS9nUR(a UR$[ S UR 35e) a Returns an optimal regularization parameter from the GCV criteria [1]. Parameters ---------- X : array, shape (5, n) 5 bands of the design matrix ``X`` stored in LAPACK banded storage. wE : array, shape (5, n) 5 bands of the penalty matrix :math:`W^{-1} E` stored in LAPACK banded storage. y : array, shape (n,) Ordinates. w : array, shape (n,) Vector of weights. Returns ------- lam : float An optimal from the GCV criteria point of view regularization parameter. Notes ----- No checks are performed. References ---------- .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models for observational data, Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics, 1990, pp. 45-65. :doi:`10.1137/1.9781611970128` c [R"U5nUS==U-ss'[S5H9nX4SU- S24==USSU--ss'USU-SSU- 24==USU-S-ss'M; URSn[R"SU45n[U5HKn[[ XT- S55H-n[ XS2U4USSU- 2XG-4-5Xg*S- XG-4'M/ MM U$) a Assuming that the product :math:`X^T W Y` is symmetric and both ``X`` and ``Y`` are 5-banded, compute the unique bands of the product. Parameters ---------- X : array, shape (5, n) 5 bands of the matrix ``X`` stored in LAPACK banded storage. w : array, shape (n,) Array of weights Y : array, shape (5, n) 5 bands of the matrix ``Y`` stored in LAPACK banded storage. Returns ------- res : array, shape (4, n) The result of the product :math:`X^T Y` stored in the banded way. Notes ----- As far as the matrices ``X`` and ``Y`` are 5-banded, their product :math:`X^T W Y` is 7-banded. It is also symmetric, so we can store only unique diagonals. rANrrar )rr"r-rJrrlsum)XrJYW_Yr2rRr9r.s rcompute_banded_symmetric_XT_W_YG_compute_optimal_gcv_parameter..compute_banded_symmetric_XT_W_Yrs6ggaj A! qA 1q56 Naaj (N AwQw 1QUV9 ,  GGAJhh1vqA3qsA;'#&qQx#dqsdAEk2B'B#CBqD!%K ( r c ^U4Sjn[U5n[SS5H"nX#*US- S24==USSU*S-24-ss'M$ SUSS-- nUS==US-ss'URSm[R"ST4S 9n[TS- SS5H1n[[ S TU- S- 5SS5H nU"X6X$U5 M M3 S /T-US 'U$) aY Inverse 3 central bands of matrix :math:`A=U^T D^{-1} U` assuming that ``U`` is a unit upper triangular banded matrix using an algorithm proposed in [1]. Parameters ---------- A : array, shape (4, n) Matrix to inverse, stored in LAPACK banded storage. Returns ------- B : array, shape (4, n) 3 unique bands of the symmetric matrix that is an inverse to ``A``. The first row is filled with zeros. Notes ----- The algorithm is based on the cholesky decomposition and, therefore, in case matrix ``A`` is close to not positive defined, the function raises LinalgError. Both matrices ``A`` and ``B`` are stored in LAPACK banded storage. References ---------- .. [1] M. F. Hutchinson and F. R. de Hoog, "Smoothing noisy data with spline functions," Numerische Mathematik, vol. 47, no. 1, pp. 99-106, 1985. :doi:`10.1007/BF01389878` cl>[ST U- S- 5nSnUS:Xa@[SUS-5HnXbU*S- X-4XG*S- X-4--nM! XcU- nXdSU4'g[SUS-5H:n[Xq- 5nU[Xq5-n XbU*S- X-4XH*S- X-4--nM< XdU*S- X-4'g)Nrr rra)rlr-abs) r2r.rDBrngrng_sumr/rindrRs rfind_b_inv_elemN_compute_optimal_gcv_parameter..compute_b_inv..find_b_inv_elemsaQ#CGAvq#'*A!a/!BFAEM2BBBG+Q4""a%q#'*Aqu:Dc!i-C!a/!EAIsz4I2JJJG+$+1"q&!%- r rArWr NrarbrVrjrr`r)rr-rJrrrl)rrgrr2rbrcr.rRs @r compute_b_inv5_compute_optimal_gcv_parameter..compute_b_invsD +" A q!A b!A#$hK1R1"Q$Y< 'K !B%!O "2 GGAJ HHAq6 "q1ub"%A3q!a%!),b"5aA.6&tax!r c 2>URSn[SXU--T5n[R"U5nX6-n[ U5HHn [ [ SX- S-5[ SU S-55Hn Xy==XU S-U - 4- ss'M MJ [RRX-5S-U- n X U--n T"U 5n X-nUSS===S-sss&S[[U55U- - S-nX- nU$![a [S 5ef=f) aW Computes the generalized cross-validation criteria [1]. Parameters ---------- lam : float, (:math:`\lambda \geq 0`) Regularization parameter. X : array, shape (5, n) Matrix is stored in LAPACK banded storage. XtWX : array, shape (4, n) Product :math:`X^T W X` stored in LAPACK banded storage. wE : array, shape (5, n) Matrix :math:`W^{-1} E` stored in LAPACK banded storage. XtE : array, shape (4, n) Product :math:`X^T E` stored in LAPACK banded storage. Returns ------- res : float Value of the GCV criteria with the regularization parameter :math:`\lambda`. Notes ----- Criteria is computed from the formula (1.3.2) [3]: .. math: GCV(\lambda) = \dfrac{1}{n} \sum\limits_{k = 1}^{n} \dfrac{ \left( y_k - f_{\lambda}(x_k) \right)^2}{\left( 1 - \Tr{A}/n\right)^2}$. The criteria is discussed in section 1.3 [3]. The numerator is computed using (2.2.4) [3] and the denominator is computed using an algorithm from [2] (see in the ``compute_b_inv`` function). References ---------- .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models for observational data, Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics, 1990, pp. 45-65. :doi:`10.1137/1.9781611970128` .. [2] M. F. Hutchinson and F. R. de Hoog, "Smoothing noisy data with spline functions," Numerische Mathematik, vol. 47, no. 1, pp. 99-106, 1985. :doi:`10.1007/BF01389878` .. [3] E. Zemlyanoy, "Generalized cross-validation smoothing splines", BSc thesis, 2022. Might be available (in Russian) `here `_ r rArArrrWrANraz#Seems like the problem is ill-posed) rJr rrr-rmrllinalgnormrXrr')lamrYXtWXwEXtErRrGr9tmpr2r.numerlhsb_bandedtrdenomrir1s r_gcv,_compute_optimal_gcv_parameter.._gcvs)j GGAJ 2Xq 1hhqkfqA3q!%!),c!QUm<#Ql++= sy)1,q03Y D$S)HB sGqLGSW))A-E m  DBC C Ds 7DDr c>T"UTTTT5$rr&)rorYrrrpryrqs rfun+_compute_optimal_gcv_parameter..fun3sCD"c**r rBounded)boundsrDz,Unable to find minimum of the GCV function: )rJr successr(r'message) rYrqr1rJr\rRr|gcv_estrrrpryris ``` @@@@r_compute_optimal_gcv_parameterrOsF'RAFN`  A *1a 3D )!Q 3C++c1a&CGyy "")//!24 55r cURSn[R"U5n[U5H1nSn[U5HnXS:wdM X@UX- -nM SU- X#'M3 U$)a Returns the coefficients of the divided difference. Parameters ---------- x : array, shape (n,) Array which is used for the computation of divided difference. Returns ------- res : array_like, shape (n,) Coefficients of the divided difference. Notes ----- Vector ``x`` should have unique elements, otherwise an error division by zero might be raised. No checks are performed. rrb)rJrrr-)r(rRr9r2rr/s r_coeff_of_divided_diffr=sj,  A ((1+C 1X qAvtad{#b  Jr c [R"U[S9n[R"U[S9n[USSUSS- S:*5(a [ S5eUR S:wd0UR S:wd UR SUR S:wa [ S5eUc [R"[U55nO4[R"U5n[US:*5(a [ S5e[RUS/S -XS/S -4nUR SnUS ::a [ S 5e[RXS 5n[R"S U45n[SS 5H0nXhUS - 2S S 24[R"US - 5XxSS24'M2 USUS'USUS-SUS-- US-USUS-4USSS24'USUS- US-US4US SS24'USUSUS - US-4USSS24'USUS-SUS-US- US - US-4USSS24'USUS'[R"S U45n [USS 5USS - U SS2S4'[USS 5USS - U SS2S4'[SUS- 5H8n X S-X S- - [X S- U S -5-X*S- U S -- U SS2U 4'M: [USS5*USS- U SS2S4'[US S5US S- U SS2S4'U S-n Uc [!XyX5nOUS:a [ S5e[#SXsU --U5n [RU SUS US -SUS -- -U S-U SUS US - -U S-U SSU SUSUS- -U S-U SSUS-US- US- -U S-4n [R%XLS 5$)a^ Compute the (coefficients of) smoothing cubic spline function using ``lam`` to control the tradeoff between the amount of smoothness of the curve and its proximity to the data. In case ``lam`` is None, using the GCV criteria [1] to find it. A smoothing spline is found as a solution to the regularized weighted linear regression problem: .. math:: \sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2 + \lambda\int\limits_{x_1}^{x_n} (f^{(2)}(u))^2 d u where :math:`f` is a spline function, :math:`w` is a vector of weights and :math:`\lambda` is a regularization parameter. If ``lam`` is None, we use the GCV criteria to find an optimal regularization parameter, otherwise we solve the regularized weighted linear regression problem with given parameter. The parameter controls the tradeoff in the following way: the larger the parameter becomes, the smoother the function gets. Parameters ---------- x : array_like, shape (n,) Abscissas. `n` must be at least 5. y : array_like, shape (n,) Ordinates. `n` must be at least 5. w : array_like, shape (n,), optional Vector of weights. Default is ``np.ones_like(x)``. lam : float, (:math:`\lambda \geq 0`), optional Regularization parameter. If ``lam`` is None, then it is found from the GCV criteria. Default is None. Returns ------- func : a BSpline object. A callable representing a spline in the B-spline basis as a solution of the problem of smoothing splines using the GCV criteria [1] in case ``lam`` is None, otherwise using the given parameter ``lam``. Notes ----- This algorithm is a clean room reimplementation of the algorithm introduced by Woltring in FORTRAN [2]. The original version cannot be used in SciPy source code because of the license issues. The details of the reimplementation are discussed here (available only in Russian) [4]. If the vector of weights ``w`` is None, we assume that all the points are equal in terms of weights, and vector of weights is vector of ones. Note that in weighted residual sum of squares, weights are not squared: :math:`\sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2` while in ``splrep`` the sum is built from the squared weights. In cases when the initial problem is ill-posed (for example, the product :math:`X^T W X` where :math:`X` is a design matrix is not a positive defined matrix) a ValueError is raised. References ---------- .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models for observational data, Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics, 1990, pp. 45-65. :doi:`10.1137/1.9781611970128` .. [2] H. J. Woltring, A Fortran package for generalized, cross-validatory spline smoothing and differentiation, Advances in Engineering Software, vol. 8, no. 2, pp. 104-113, 1986. :doi:`10.1016/0141-1195(86)90098-7` .. [3] T. Hastie, J. Friedman, and R. Tisbshirani, "Smoothing Splines" in The elements of Statistical Learning: Data Mining, Inference, and prediction, New York: Springer, 2017, pp. 241-249. :doi:`10.1007/978-0-387-84858-7` .. [4] E. Zemlyanoy, "Generalized cross-validation smoothing splines", BSc thesis, 2022. ``_ (in Russian) Examples -------- Generate some noisy data >>> import numpy as np >>> np.random.seed(1234) >>> n = 200 >>> def func(x): ... return x**3 + x**2 * np.sin(4 * x) >>> x = np.sort(np.random.random_sample(n) * 4 - 2) >>> y = func(x) + np.random.normal(scale=1.5, size=n) Make a smoothing spline function >>> from scipy.interpolate import make_smoothing_spline >>> spl = make_smoothing_spline(x, y) Plot both >>> import matplotlib.pyplot as plt >>> grid = np.linspace(x[0], x[-1], 400) >>> plt.plot(grid, spl(grid), label='Spline') >>> plt.plot(grid, func(grid), label='Original function') >>> plt.scatter(x, y, marker='.') >>> plt.legend(loc='best') >>> plt.show() rr Nrarz"``x`` should be an ascending arrayz;``x`` and ``y`` should be one dimensional and the same sizezInvalid vector of weightsrrVz)``x`` and ``y`` length must be at least 5rWrArU)rr)r r )r rArl)rr)rUrU)rUr)rara)rrUr`z/Regularization parameter should be non-negativei)rr#rrOr'rKrJonesr6rcrr~rr- diag_indicesrrr rZ) r(r1rJror0rRX_bsplrYr2rqr.rGc_s rrr^s\ Qe,A Qe,A 1QR51Sb6>Q =>>vv{affkQWWQZ1771:%=&' ' y GGCFO   # qAv;;89 9 qtfqj!eWq[()A  AAvDEE " "1 +F !QA 1a[AXq"u_-booa!e.DEQU( TlAdG1!q1Q4x'6$<7t vd|+-Aa!eH1!t ,fTl;Aa!eH!B%!B%-6&>!ABAafI&.0ae)ae#ae+vf~=?AafIf~AeH 1a& B&q!u-"15Bqr1uI&q!u-"15Bqr1uI 1a!e_cFQsVO'=a!AaCj'IIsAaC[!1a4*!BC&11AbcF:BssBwK(2301RS69BssBwK!GB {,QA9 rJKK VQr\1-A qtqtad{Q1X-.15tqtad{#ad*Bxu"" &2.uAbE AbE)AbE12QrU: ; >AA6qe1 MD !A#AbEQs1uX-  A !A %!)C Qw 1c!e_ T Q T IIaf  !8T" " 42:T" " r r)r)rNNrT)rNrTr)0rDmathrnumpyrscipy._lib._utilr scipy.linalgrrrrr r scipy.optimizer r r scipy.sparser scipy.specialr itertoolsr__all__rr+r3r;rrrrrrrrrrrrrrIrrrrr&r rrs1//+"" $  >  ^ P^ PBCT ,+  %S l (< ~(=VYNx>?$(S6lJ6tJ6b*k5\By,@R r