""" Replicate FITPACK's logic for constructing smoothing spline functions and curves. Currently provides analogs of splrep and splprep python routines, i.e. curfit.f and parcur.f routines (the drivers are fpcurf.f and fppara.f, respectively) The Fortran sources are from https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/ .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing", 20 (1982) 171-184. :doi:`10.1016/0146-664X(82)90043-0`. .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on Numerical Analysis, Oxford University Press, 1993. .. [3] P. Dierckx, "An algorithm for smoothing, differentiation and integration of experimental data using spline functions", Journal of Computational and Applied Mathematics, vol. I, no 3, p. 165 (1975). https://doi.org/10.1016/0771-050X(75)90034-0 """ import warnings import operator import numpy as np from ._bsplines import ( _not_a_knot, make_interp_spline, BSpline, fpcheck, _lsq_solve_qr ) from . import _dierckx # type: ignore[attr-defined] # cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc # c part 1: determination of the number of knots and their position c # c ************************************************************** c # # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L31 # Hardcoded in curfit.f TOL = 0.001 MAXIT = 20 def _get_residuals(x, y, t, k, w): # FITPACK has (w*(spl(x)-y))**2; make_lsq_spline has w*(spl(x)-y)**2 w2 = w**2 # inline the relevant part of # >>> spl = make_lsq_spline(x, y, w=w2, t=t, k=k) # NB: # 1. y is assumed to be 2D here. For 1D case (parametric=False), # the call must have been preceded by y = y[:, None] (cf _validate_inputs) # 2. We always sum the squares across axis=1: # * For 1D (parametric=False), the last dimension has size one, # so the summation is a no-op. # * For 2D (parametric=True), the summation is actually how the # 'residuals' are defined, see Eq. (42) in Dierckx1982 # (the reference is in the docstring of `class F`) below. _, _, c = _lsq_solve_qr(x, y, t, k, w) c = np.ascontiguousarray(c) spl = BSpline(t, c, k) return _compute_residuals(w2, spl(x), y) def _compute_residuals(w2, splx, y): delta = ((splx - y)**2).sum(axis=1) return w2 * delta def add_knot(x, t, k, residuals): """Add a new knot. (Approximately) replicate FITPACK's logic: 1. split the `x` array into knot intervals, ``t(j+k) <= x(i) <= t(j+k+1)`` 2. find the interval with the maximum sum of residuals 3. insert a new knot into the middle of that interval. NB: a new knot is in fact an `x` value at the middle of the interval. So *the knots are a subset of `x`*. This routine is an analog of https://github.com/scipy/scipy/blob/v1.11.4/scipy/interpolate/fitpack/fpcurf.f#L190-L215 (cf _split function) and https://github.com/scipy/scipy/blob/v1.11.4/scipy/interpolate/fitpack/fpknot.f """ new_knot = _dierckx.fpknot(x, t, k, residuals) idx_t = np.searchsorted(t, new_knot) t_new = np.r_[t[:idx_t], new_knot, t[idx_t:]] return t_new def _validate_inputs(x, y, w, k, s, xb, xe, parametric): """Common input validations for generate_knots and make_splrep. """ x = np.asarray(x, dtype=float) y = np.asarray(y, dtype=float) if w is None: w = np.ones_like(x, dtype=float) else: w = np.asarray(w, dtype=float) if w.ndim != 1: raise ValueError(f"{w.ndim = } not implemented yet.") if (w < 0).any(): raise ValueError("Weights must be non-negative") if y.ndim == 0 or y.ndim > 2: raise ValueError(f"{y.ndim = } not supported (must be 1 or 2.)") parametric = bool(parametric) if parametric: if y.ndim != 2: raise ValueError(f"{y.ndim = } != 2 not supported with {parametric =}.") else: if y.ndim != 1: raise ValueError(f"{y.ndim = } != 1 not supported with {parametric =}.") # all _impl functions expect y.ndim = 2 y = y[:, None] if w.shape[0] != x.shape[0]: raise ValueError(f"Weights is incompatible: {w.shape =} != {x.shape}.") if x.shape[0] != y.shape[0]: raise ValueError(f"Data is incompatible: {x.shape = } and {y.shape = }.") if x.ndim != 1 or (x[1:] < x[:-1]).any(): raise ValueError("Expect `x` to be an ordered 1D sequence.") k = operator.index(k) if s < 0: raise ValueError(f"`s` must be non-negative. Got {s = }") if xb is None: xb = min(x) if xe is None: xe = max(x) return x, y, w, k, s, xb, xe def generate_knots(x, y, *, w=None, xb=None, xe=None, k=3, s=0, nest=None): """Replicate FITPACK's constructing the knot vector. Parameters ---------- x, y : array_like The data points defining the curve ``y = f(x)``. w : array_like, optional Weights. xb : float, optional The boundary of the approximation interval. If None (default), is set to ``x[0]``. xe : float, optional The boundary of the approximation interval. If None (default), is set to ``x[-1]``. k : int, optional The spline degree. Default is cubic, ``k = 3``. s : float, optional The smoothing factor. Default is ``s = 0``. nest : int, optional Stop when at least this many knots are placed. Yields ------ t : ndarray Knot vectors with an increasing number of knots. The generator is finite: it stops when the smoothing critetion is satisfied, or when then number of knots exceeds the maximum value: the user-provided `nest` or `x.size + k + 1` --- which is the knot vector for the interpolating spline. Examples -------- Generate some noisy data and fit a sequence of LSQ splines: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.interpolate import make_lsq_spline, generate_knots >>> rng = np.random.default_rng(12345) >>> x = np.linspace(-3, 3, 50) >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(size=50) >>> knots = list(generate_knots(x, y, s=1e-10)) >>> for t in knots[::3]: ... spl = make_lsq_spline(x, y, t) ... xs = xs = np.linspace(-3, 3, 201) ... plt.plot(xs, spl(xs), '-', label=f'n = {len(t)}', lw=3, alpha=0.7) >>> plt.plot(x, y, 'o', label='data') >>> plt.plot(xs, np.exp(-xs**2), '--') >>> plt.legend() Note that increasing the number of knots make the result follow the data more and more closely. Also note that a step of the generator may add multiple knots: >>> [len(t) for t in knots] [8, 9, 10, 12, 16, 24, 40, 48, 52, 54] Notes ----- The routine generates successive knots vectors of increasing length, starting from ``2*(k+1)`` to ``len(x) + k + 1``, trying to make knots more dense in the regions where the deviation of the LSQ spline from data is large. When the maximum number of knots, ``len(x) + k + 1`` is reached (this happens when ``s`` is small and ``nest`` is large), the generator stops, and the last output is the knots for the interpolation with the not-a-knot boundary condition. Knots are located at data sites, unless ``k`` is even and the number of knots is ``len(x) + k + 1``. In that case, the last output of the generator has internal knots at Greville sites, ``(x[1:] + x[:-1]) / 2``. .. versionadded:: 1.15.0 """ if s == 0: if nest is not None or w is not None: raise ValueError("s == 0 is interpolation only") t = _not_a_knot(x, k) yield t return x, y, w, k, s, xb, xe = _validate_inputs( x, y, w, k, s, xb, xe, parametric=np.ndim(y) == 2 ) yield from _generate_knots_impl(x, y, w=w, xb=xb, xe=xe, k=k, s=s, nest=nest) def _generate_knots_impl(x, y, *, w=None, xb=None, xe=None, k=3, s=0, nest=None): acc = s * TOL m = x.size # the number of data points if nest is None: # the max number of knots. This is set in _fitpack_impl.py line 274 # and fitpack.pyf line 198 nest = max(m + k + 1, 2*k + 3) else: if nest < 2*(k + 1): raise ValueError(f"`nest` too small: {nest = } < 2*(k+1) = {2*(k+1)}.") nmin = 2*(k + 1) # the number of knots for an LSQ polynomial approximation nmax = m + k + 1 # the number of knots for the spline interpolation # start from no internal knots t = np.asarray([xb]*(k+1) + [xe]*(k+1), dtype=float) n = t.shape[0] fp = 0.0 fpold = 0.0 # c main loop for the different sets of knots. m is a safe upper bound # c for the number of trials. for _ in range(m): yield t # construct the LSQ spline with this set of knots fpold = fp residuals = _get_residuals(x, y, t, k, w=w) fp = residuals.sum() fpms = fp - s # c test whether the approximation sinf(x) is an acceptable solution. # c if f(p=inf) < s accept the choice of knots. if (abs(fpms) < acc) or (fpms < 0): return # ### c increase the number of knots. ### # c determine the number of knots nplus we are going to add. if n == nmin: # the first iteration nplus = 1 else: delta = fpold - fp npl1 = int(nplus * fpms / delta) if delta > acc else nplus*2 nplus = min(nplus*2, max(npl1, nplus//2, 1)) # actually add knots for j in range(nplus): t = add_knot(x, t, k, residuals) # check if we have enough knots already n = t.shape[0] # c if n = nmax, sinf(x) is an interpolating spline. # c if n=nmax we locate the knots as for interpolation. if n >= nmax: t = _not_a_knot(x, k) yield t return # c if n=nest we cannot increase the number of knots because of # c the storage capacity limitation. if n >= nest: yield t return # recompute if needed if j < nplus - 1: residuals = _get_residuals(x, y, t, k, w=w) # this should never be reached return # cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc # c part 2: determination of the smoothing spline sp(x). c # c *************************************************** c # c we have determined the number of knots and their position. c # c we now compute the b-spline coefficients of the smoothing spline c # c sp(x). the observation matrix a is extended by the rows of matrix c # c b expressing that the kth derivative discontinuities of sp(x) at c # c the interior knots t(k+2),...t(n-k-1) must be zero. the corres- c # c ponding weights of these additional rows are set to 1/p. c # c iteratively we then have to determine the value of p such that c # c f(p)=sum((w(i)*(y(i)-sp(x(i))))**2) be = s. we already know that c # c the least-squares kth degree polynomial corresponds to p=0, and c # c that the least-squares spline corresponds to p=infinity. the c # c iteration process which is proposed here, makes use of rational c # c interpolation. since f(p) is a convex and strictly decreasing c # c function of p, it can be approximated by a rational function c # c r(p) = (u*p+v)/(p+w). three values of p(p1,p2,p3) with correspond- c # c ing values of f(p) (f1=f(p1)-s,f2=f(p2)-s,f3=f(p3)-s) are used c # c to calculate the new value of p such that r(p)=s. convergence is c # c guaranteed by taking f1>0 and f3<0. c # cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc def prodd(t, i, j, k): res = 1.0 for s in range(k+2): if i + s != j: res *= (t[j] - t[i+s]) return res def disc(t, k): """Discontinuity matrix: jumps of k-th derivatives of b-splines at internal knots. See Eqs. (9)-(10) of Ref. [1], or, equivalently, Eq. (3.43) of Ref. [2]. This routine assumes internal knots are all simple (have multiplicity =1). Parameters ---------- t : ndarray, 1D, shape(n,) Knots. k : int The spline degree Returns ------- disc : ndarray, shape(n-2*k-1, k+2) The jumps of the k-th derivatives of b-splines at internal knots, ``t[k+1], ...., t[n-k-1]``. offset : ndarray, shape(2-2*k-1,) Offsets nc : int Notes ----- The normalization here follows FITPACK: (https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpdisc.f#L36) The k-th derivative jumps are multiplied by a factor:: (delta / nrint)**k / k! where ``delta`` is the length of the interval spanned by internal knots, and ``nrint`` is one less the number of internal knots (i.e., the number of subintervals between them). References ---------- .. [1] Paul Dierckx, Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing, vol. 20, p. 171 (1982). :doi:`10.1016/0146-664X(82)90043-0` .. [2] Tom Lyche and Knut Morken, Spline methods, http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v05/undervisningsmateriale/ """ n = t.shape[0] # the length of the base interval spanned by internal knots & the number # of subintervas between these internal knots delta = t[n - k - 1] - t[k] nrint = n - 2*k - 1 matr = np.empty((nrint - 1, k + 2), dtype=float) for jj in range(nrint - 1): j = jj + k + 1 for ii in range(k + 2): i = jj + ii matr[jj, ii] = (t[i + k + 1] - t[i]) / prodd(t, i, j, k) # NB: equivalent to # row = [(t[i + k + 1] - t[i]) / prodd(t, i, j, k) for i in range(j-k-1, j+1)] # assert (matr[j-k-1, :] == row).all() # follow FITPACK matr *= (delta/ nrint)**k # make it packed offset = np.array([i for i in range(nrint-1)], dtype=np.int64) nc = n - k - 1 return matr, offset, nc class F: """ The r.h.s. of ``f(p) = s``. Given scalar `p`, we solve the system of equations in the LSQ sense: | A | @ | c | = | y | | B / p | | 0 | | 0 | where `A` is the matrix of b-splines and `b` is the discontinuity matrix (the jumps of the k-th derivatives of b-spline basis elements at knots). Since we do that repeatedly while minimizing over `p`, we QR-factorize `A` only once and update the QR factorization only of the `B` rows of the augmented matrix |A, B/p|. The system of equations is Eq. (15) Ref. [1]_, the strategy and implementation follows that of FITPACK, see specific links below. References ---------- [1] P. Dierckx, Algorithms for Smoothing Data with Periodic and Parametric Splines, COMPUTER GRAPHICS AND IMAGE PROCESSING vol. 20, pp 171-184 (1982.) https://doi.org/10.1016/0146-664X(82)90043-0 """ def __init__(self, x, y, t, k, s, w=None, *, R=None, Y=None): self.x = x self.y = y self.t = t self.k = k w = np.ones_like(x, dtype=float) if w is None else w if w.ndim != 1: raise ValueError(f"{w.ndim = } != 1.") self.w = w self.s = s if y.ndim != 2: raise ValueError(f"F: expected y.ndim == 2, got {y.ndim = } instead.") # ### precompute what we can ### # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L250 # c evaluate the discontinuity jump of the kth derivative of the # c b-splines at the knots t(l),l=k+2,...n-k-1 and store in b. b, b_offset, b_nc = disc(t, k) # the QR factorization of the data matrix, if not provided # NB: otherwise, must be consistent with x,y & s, but this is not checked if R is None and Y is None: R, Y, _ = _lsq_solve_qr(x, y, t, k, w) # prepare to combine R and the discontinuity matrix (AB); also r.h.s. (YY) # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L269 # c the rows of matrix b with weight 1/p are rotated into the # c triangularised observation matrix a which is stored in g. nc = t.shape[0] - k - 1 nz = k + 1 if R.shape[1] != nz: raise ValueError(f"Internal error: {R.shape[1] =} != {k+1 =}.") # r.h.s. of the augmented system z = np.zeros((b.shape[0], Y.shape[1]), dtype=float) self.YY = np.r_[Y[:nc], z] # l.h.s. of the augmented system AA = np.zeros((nc + b.shape[0], self.k+2), dtype=float) AA[:nc, :nz] = R[:nc, :] # AA[nc:, :] = b.a / p # done in __call__(self, p) self.AA = AA self.offset = np.r_[np.arange(nc, dtype=np.int64), b_offset] self.nc = nc self.b = b def __call__(self, p): # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L279 # c the row of matrix b is rotated into triangle by givens transformation # copy the precomputed matrices over for in-place work # R = PackedMatrix(self.AB.a.copy(), self.AB.offset.copy(), nc) AB = self.AA.copy() offset = self.offset.copy() nc = self.nc AB[nc:, :] = self.b / p QY = self.YY.copy() # heavy lifting happens here, in-place _dierckx.qr_reduce(AB, offset, nc, QY, startrow=nc) # solve for the coefficients c = _dierckx.fpback(AB, nc, QY) spl = BSpline(self.t, c, self.k) residuals = _compute_residuals(self.w**2, spl(self.x), self.y) fp = residuals.sum() self.spl = spl # store it return fp - self.s def fprati(p1, f1, p2, f2, p3, f3): """The root of r(p) = (u*p + v) / (p + w) given three points and values, (p1, f2), (p2, f2) and (p3, f3). The FITPACK analog adjusts the bounds, and we do not https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fprati.f NB: FITPACK uses p < 0 to encode p=infinity. We just use the infinity itself. Since the bracket is ``p1 <= p2 <= p3``, ``p3`` can be infinite (in fact, this is what the minimizer starts with, ``p3=inf``). """ h1 = f1 * (f2 - f3) h2 = f2 * (f3 - f1) h3 = f3 * (f1 - f2) if p3 == np.inf: return -(p2*h1 + p1*h2) / h3 return -(p1*p2*h3 + p2*p3*h1 + p1*p3*h2) / (p1*h1 + p2*h2 + p3*h3) class Bunch: def __init__(self, **kwargs): self.__dict__.update(**kwargs) _iermesg = { 2: """error. a theoretically impossible result was found during the iteration process for finding a smoothing spline with fp = s. probably causes : s too small. there is an approximation returned but the corresponding weighted sum of squared residuals does not satisfy the condition abs(fp-s)/s < tol. """, 3: """error. the maximal number of iterations maxit (set to 20 by the program) allowed for finding a smoothing spline with fp=s has been reached. probably causes : s too small there is an approximation returned but the corresponding weighted sum of squared residuals does not satisfy the condition abs(fp-s)/s < tol. """ } def root_rati(f, p0, bracket, acc): """Solve `f(p) = 0` using a rational function approximation. In a nutshell, since the function f(p) is known to be monotonically decreasing, we - maintain the bracket (p1, f1), (p2, f2) and (p3, f3) - at each iteration step, approximate f(p) by a rational function r(p) = (u*p + v) / (p + w) and make a step to p_new to the root of f(p): r(p_new) = 0. The coefficients u, v and w are found from the bracket values p1..3 and f1...3 The algorithm and implementation follows https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L229 and https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fppara.f#L290 Note that the latter is for parametric splines and the former is for 1D spline functions. The minimization is indentical though [modulo a summation over the dimensions in the computation of f(p)], so we reuse the minimizer for both d=1 and d>1. """ # Magic values from # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L27 con1 = 0.1 con9 = 0.9 con4 = 0.04 # bracketing flags (follow FITPACK) # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fppara.f#L365 ich1, ich3 = 0, 0 (p1, f1), (p3, f3) = bracket p = p0 for it in range(MAXIT): p2, f2 = p, f(p) # c test whether the approximation sp(x) is an acceptable solution. if abs(f2) < acc: ier, converged = 0, True break # c carry out one more step of the iteration process. if ich3 == 0: if f2 - f3 <= acc: # c our initial choice of p is too large. p3 = p2 f3 = f2 p = p*con4 if p <= p1: p = p1*con9 + p2*con1 continue else: if f2 < 0: ich3 = 1 if ich1 == 0: if f1 - f2 <= acc: # c our initial choice of p is too small p1 = p2 f1 = f2 p = p/con4 if p3 != np.inf and p <= p3: p = p2*con1 + p3*con9 continue else: if f2 > 0: ich1 = 1 # c test whether the iteration process proceeds as theoretically expected. # [f(p) should be monotonically decreasing] if f1 <= f2 or f2 <= f3: ier, converged = 2, False break # actually make the iteration step p = fprati(p1, f1, p2, f2, p3, f3) # c adjust the value of p1,f1,p3 and f3 such that f1 > 0 and f3 < 0. if f2 < 0: p3, f3 = p2, f2 else: p1, f1 = p2, f2 else: # not converged in MAXIT iterations ier, converged = 3, False if ier != 0: warnings.warn(RuntimeWarning(_iermesg[ier]), stacklevel=2) return Bunch(converged=converged, root=p, iterations=it, ier=ier) def _make_splrep_impl(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None): """Shared infra for make_splrep and make_splprep. """ acc = s * TOL m = x.size # the number of data points if nest is None: # the max number of knots. This is set in _fitpack_impl.py line 274 # and fitpack.pyf line 198 nest = max(m + k + 1, 2*k + 3) else: if nest < 2*(k + 1): raise ValueError(f"`nest` too small: {nest = } < 2*(k+1) = {2*(k+1)}.") if t is not None: raise ValueError("Either supply `t` or `nest`.") if t is None: gen = _generate_knots_impl(x, y, w=w, k=k, s=s, xb=xb, xe=xe, nest=nest) t = list(gen)[-1] else: fpcheck(x, t, k) if t.shape[0] == 2 * (k + 1): # nothing to optimize _, _, c = _lsq_solve_qr(x, y, t, k, w) return BSpline(t, c, k) ### solve ### # c initial value for p. # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L253 R, Y, _ = _lsq_solve_qr(x, y, t, k, w) nc = t.shape[0] -k -1 p = nc / R[:, 0].sum() # ### bespoke solver #### # initial conditions # f(p=inf) : LSQ spline with knots t (XXX: reuse R, c) residuals = _get_residuals(x, y, t, k, w=w) fp = residuals.sum() fpinf = fp - s # f(p=0): LSQ spline without internal knots residuals = _get_residuals(x, y, np.array([xb]*(k+1) + [xe]*(k+1)), k, w) fp0 = residuals.sum() fp0 = fp0 - s # solve bracket = (0, fp0), (np.inf, fpinf) f = F(x, y, t, k=k, s=s, w=w, R=R, Y=Y) _ = root_rati(f, p, bracket, acc) # solve ALTERNATIVE: is roughly equivalent, gives slightly different results # starting from scratch, that would have probably been tolerable; # backwards compatibility dictates that we replicate the FITPACK minimizer though. # f = F(x, y, t, k=k, s=s, w=w, R=R, Y=Y) # from scipy.optimize import root_scalar # res_ = root_scalar(f, x0=p, rtol=acc) # assert res_.converged # f.spl is the spline corresponding to the found `p` value return f.spl def make_splrep(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None): r"""Find the B-spline representation of a 1D function. Given the set of data points ``(x[i], y[i])``, determine a smooth spline approximation of degree ``k`` on the interval ``xb <= x <= xe``. Parameters ---------- x, y : array_like, shape (m,) The data points defining a curve ``y = f(x)``. w : array_like, shape (m,), optional Strictly positive 1D array of weights, of the same length as `x` and `y`. The weights are used in computing the weighted least-squares spline fit. If the errors in the y values have standard-deviation given by the vector ``d``, then `w` should be ``1/d``. Default is ``np.ones(m)``. xb, xe : float, optional The interval to fit. If None, these default to ``x[0]`` and ``x[-1]``, respectively. k : int, optional The degree of the spline fit. It is recommended to use cubic splines, ``k=3``, which is the default. Even values of `k` should be avoided, especially with small `s` values. s : float, optional The smoothing condition. The amount of smoothness is determined by satisfying the conditions:: sum((w * (g(x) - y))**2 ) <= s where ``g(x)`` is the smoothed fit to ``(x, y)``. The user can use `s` to control the tradeoff between closeness to data and smoothness of fit. Larger `s` means more smoothing while smaller values of `s` indicate less smoothing. Recommended values of `s` depend on the weights, `w`. If the weights represent the inverse of the standard deviation of `y`, then a good `s` value should be found in the range ``(m-sqrt(2*m), m+sqrt(2*m))`` where ``m`` is the number of datapoints in `x`, `y`, and `w`. Default is ``s = 0.0``, i.e. interpolation. t : array_like, optional The spline knots. If None (default), the knots will be constructed automatically. There must be at least ``2*k + 2`` and at most ``m + k + 1`` knots. nest : int, optional The target length of the knot vector. Should be between ``2*(k + 1)`` (the minimum number of knots for a degree-``k`` spline), and ``m + k + 1`` (the number of knots of the interpolating spline). The actual number of knots returned by this routine may be slightly larger than `nest`. Default is None (no limit, add up to ``m + k + 1`` knots). Returns ------- spl : a `BSpline` instance For `s=0`, ``spl(x) == y``. For non-zero values of `s` the `spl` represents the smoothed approximation to `(x, y)`, generally with fewer knots. See Also -------- generate_knots : is used under the hood for generating the knots make_splprep : the analog of this routine for parametric curves make_interp_spline : construct an interpolating spline (``s = 0``) make_lsq_spline : construct the least-squares spline given the knot vector splrep : a FITPACK analog of this routine References ---------- .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing", 20 (1982) 171-184. .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on Numerical Analysis, Oxford University Press, 1993. Notes ----- This routine constructs the smoothing spline function, :math:`g(x)`, to minimize the sum of jumps, :math:`D_j`, of the ``k``-th derivative at the internal knots (:math:`x_b < t_i < x_e`), where .. math:: D_i = g^{(k)}(t_i + 0) - g^{(k)}(t_i - 0) Specifically, the routine constructs the spline function :math:`g(x)` which minimizes .. math:: \sum_i | D_i |^2 \to \mathrm{min} provided that .. math:: \sum_{j=1}^m (w_j \times (g(x_j) - y_j))^2 \leqslant s , where :math:`s > 0` is the input parameter. In other words, we balance maximizing the smoothness (measured as the jumps of the derivative, the first criterion), and the deviation of :math:`g(x_j)` from the data :math:`y_j` (the second criterion). Note that the summation in the second criterion is over all data points, and in the first criterion it is over the internal spline knots (i.e. those with ``xb < t[i] < xe``). The spline knots are in general a subset of data, see `generate_knots` for details. Also note the difference of this routine to `make_lsq_spline`: the latter routine does not consider smoothness and simply solves a least-squares problem .. math:: \sum w_j \times (g(x_j) - y_j)^2 \to \mathrm{min} for a spline function :math:`g(x)` with a _fixed_ knot vector ``t``. .. versionadded:: 1.15.0 """ if s == 0: if t is not None or w is not None or nest is not None: raise ValueError("s==0 is for interpolation only") return make_interp_spline(x, y, k=k) x, y, w, k, s, xb, xe = _validate_inputs(x, y, w, k, s, xb, xe, parametric=False) spl = _make_splrep_impl(x, y, w=w, xb=xb, xe=xe, k=k, s=s, t=t, nest=nest) # postprocess: squeeze out the last dimension: was added to simplify the internals. spl.c = spl.c[:, 0] return spl def make_splprep(x, *, w=None, u=None, ub=None, ue=None, k=3, s=0, t=None, nest=None): r""" Find a smoothed B-spline representation of a parametric N-D curve. Given a list of N 1D arrays, `x`, which represent a curve in N-dimensional space parametrized by `u`, find a smooth approximating spline curve ``g(u)``. Parameters ---------- x : array_like, shape (m, ndim) Sampled data points representing the curve in ``ndim`` dimensions. The typical use is a list of 1D arrays, each of length ``m``. w : array_like, shape(m,), optional Strictly positive 1D array of weights. The weights are used in computing the weighted least-squares spline fit. If the errors in the `x` values have standard deviation given by the vector d, then `w` should be 1/d. Default is ``np.ones(m)``. u : array_like, optional An array of parameter values for the curve in the parametric form. If not given, these values are calculated automatically, according to:: v[0] = 0 v[i] = v[i-1] + distance(x[i], x[i-1]) u[i] = v[i] / v[-1] ub, ue : float, optional The end-points of the parameters interval. Default to ``u[0]`` and ``u[-1]``. k : int, optional Degree of the spline. Cubic splines, ``k=3``, are recommended. Even values of `k` should be avoided especially with a small ``s`` value. Default is ``k=3`` s : float, optional A smoothing condition. The amount of smoothness is determined by satisfying the conditions:: sum((w * (g(u) - x))**2) <= s, where ``g(u)`` is the smoothed approximation to ``x``. The user can use `s` to control the trade-off between closeness and smoothness of fit. Larger ``s`` means more smoothing while smaller values of ``s`` indicate less smoothing. Recommended values of ``s`` depend on the weights, ``w``. If the weights represent the inverse of the standard deviation of ``x``, then a good ``s`` value should be found in the range ``(m - sqrt(2*m), m + sqrt(2*m))``, where ``m`` is the number of data points in ``x`` and ``w``. t : array_like, optional The spline knots. If None (default), the knots will be constructed automatically. There must be at least ``2*k + 2`` and at most ``m + k + 1`` knots. nest : int, optional The target length of the knot vector. Should be between ``2*(k + 1)`` (the minimum number of knots for a degree-``k`` spline), and ``m + k + 1`` (the number of knots of the interpolating spline). The actual number of knots returned by this routine may be slightly larger than `nest`. Default is None (no limit, add up to ``m + k + 1`` knots). Returns ------- spl : a `BSpline` instance For `s=0`, ``spl(u) == x``. For non-zero values of ``s``, `spl` represents the smoothed approximation to ``x``, generally with fewer knots. u : ndarray The values of the parameters See Also -------- generate_knots : is used under the hood for generating the knots make_splrep : the analog of this routine 1D functions make_interp_spline : construct an interpolating spline (``s = 0``) make_lsq_spline : construct the least-squares spline given the knot vector splprep : a FITPACK analog of this routine Notes ----- Given a set of :math:`m` data points in :math:`D` dimensions, :math:`\vec{x}_j`, with :math:`j=1, ..., m` and :math:`\vec{x}_j = (x_{j; 1}, ..., x_{j; D})`, this routine constructs the parametric spline curve :math:`g_a(u)` with :math:`a=1, ..., D`, to minimize the sum of jumps, :math:`D_{i; a}`, of the ``k``-th derivative at the internal knots (:math:`u_b < t_i < u_e`), where .. math:: D_{i; a} = g_a^{(k)}(t_i + 0) - g_a^{(k)}(t_i - 0) Specifically, the routine constructs the spline function :math:`g(u)` which minimizes .. math:: \sum_i \sum_{a=1}^D | D_{i; a} |^2 \to \mathrm{min} provided that .. math:: \sum_{j=1}^m \sum_{a=1}^D (w_j \times (g_a(u_j) - x_{j; a}))^2 \leqslant s where :math:`u_j` is the value of the parameter corresponding to the data point :math:`(x_{j; 1}, ..., x_{j; D})`, and :math:`s > 0` is the input parameter. In other words, we balance maximizing the smoothness (measured as the jumps of the derivative, the first criterion), and the deviation of :math:`g(u_j)` from the data :math:`x_j` (the second criterion). Note that the summation in the second criterion is over all data points, and in the first criterion it is over the internal spline knots (i.e. those with ``ub < t[i] < ue``). The spline knots are in general a subset of data, see `generate_knots` for details. .. versionadded:: 1.15.0 References ---------- .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing", 20 (1982) 171-184. .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on Numerical Analysis, Oxford University Press, 1993. """ x = np.stack(x, axis=1) # construct the default parametrization of the curve if u is None: dp = (x[1:, :] - x[:-1, :])**2 u = np.sqrt((dp).sum(axis=1)).cumsum() u = np.r_[0, u / u[-1]] if s == 0: if t is not None or w is not None or nest is not None: raise ValueError("s==0 is for interpolation only") return make_interp_spline(u, x.T, k=k, axis=1), u u, x, w, k, s, ub, ue = _validate_inputs(u, x, w, k, s, ub, ue, parametric=True) spl = _make_splrep_impl(u, x, w=w, xb=ub, xe=ue, k=k, s=s, t=t, nest=nest) # posprocess: `axis=1` so that spl(u).shape == np.shape(x) # when `x` is a list of 1D arrays (cf original splPrep) cc = spl.c.T spl1 = BSpline(spl.t, cc, spl.k, axis=1) return spl1, u