scipy.interpolate.

make_splprep

scipy.interpolate.make_splprep(x, *, w=None, u=None, ub=None, ue=None, k=3, s=0, t=None, nest=None)

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:

xarray_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.

warray_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).

uarray_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, uefloat, optional

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

kint, 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

sfloat, 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.

tarray_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.

nestint, 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:

spla 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.

undarray

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 \(m\) data points in \(D\) dimensions, \(\vec{x}_j\), with \(j=1, ..., m\) and \(\vec{x}_j = (x_{j; 1}, ..., x_{j; D})\), this routine constructs the parametric spline curve \(g_a(u)\) with \(a=1, ..., D\), to minimize the sum of jumps, \(D_{i; a}\), of the k-th derivative at the internal knots (\(u_b < t_i < u_e\)), where

\[D_{i; a} = g_a^{(k)}(t_i + 0) - g_a^{(k)}(t_i - 0)\]

Specifically, the routine constructs the spline function \(g(u)\) which minimizes

\[\sum_i \sum_{a=1}^D | D_{i; a} |^2 \to \mathrm{min}\]

provided that

\[\sum_{j=1}^m \sum_{a=1}^D (w_j \times (g_a(u_j) - x_{j; a}))^2 \leqslant s\]

where \(u_j\) is the value of the parameter corresponding to the data point \((x_{j; 1}, ..., x_{j; D})\), and \(s > 0\) 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 \(g(u_j)\) from the data \(x_j\) (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.

Added in version 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.