SciPy

scipy.interpolate.splprep

scipy.interpolate.splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, full_output=0, nest=None, per=0, quiet=1)[source]

Find the B-spline representation of an N-dimensional curve.

Given a list of N rank-1 arrays, x, which represent a curve in N-dimensional space parametrized by u, find a smooth approximating spline curve g(u). Uses the FORTRAN routine parcur from FITPACK.

Parameters :

x : array_like

A list of sample vector arrays representing the curve.

w : array_like

Strictly positive rank-1 array of weights the same length as x[0]. 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 ones(len(x[0])).

u : array_like, optional

An array of parameter values. If not given, these values are calculated automatically as M = len(x[0]), where

v[0] = 0

v[i] = v[i-1] + distance(x[i], x[i-1])

u[i] = v[i] / v[M-1]

ub, ue : int, optional

The end-points of the parameters interval. Defaults to u[0] and u[-1].

k : int, optional

Degree of the spline. Cubic splines are recommended. Even values of k should be avoided especially with a small s-value. 1 <= k <= 5, default is 3.

task : int, optional

If task==0 (default), find t and c for a given smoothing factor, s. If task==1, find t and c for another value of the smoothing factor, s. There must have been a previous call with task=0 or task=1 for the same set of data. If task=-1 find the weighted least square spline for a given set of knots, t.

s : float, optional

A smoothing condition. The amount of smoothness is determined by satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s, where g(x) is the smoothed interpolation of (x,y). 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 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 data points in x, y, and w.

t : int, optional

The knots needed for task=-1.

full_output : int, optional

If non-zero, then return optional outputs.

nest : int, optional

An over-estimate of the total number of knots of the spline to help in determining the storage space. By default nest=m/2. Always large enough is nest=m+k+1.

per : int, optional

If non-zero, data points are considered periodic with period x[m-1] - x[0] and a smooth periodic spline approximation is returned. Values of y[m-1] and w[m-1] are not used.

quiet : int, optional

Non-zero to suppress messages.

Returns :

tck : tuple

A tuple (t,c,k) containing the vector of knots, the B-spline coefficients, and the degree of the spline.

u : array

An array of the values of the parameter.

fp : float

The weighted sum of squared residuals of the spline approximation.

ier : int

An integer flag about splrep success. Success is indicated if ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.

msg : str

A message corresponding to the integer flag, ier.

Notes

See splev for evaluation of the spline and its derivatives.

References

[R55]P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing”, 20 (1982) 171-184.
[R56]P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines”, report tw55, Dept. Computer Science, K.U.Leuven, 1981.
[R57]P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.