make_splrep#
- scipy.interpolate.make_splrep(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None)[source]#
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 degreekon the intervalxb <= x <= xe.- Parameters:
- x, yarray_like, shape (m,)
The data points defining a curve
y = f(x).- warray_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 be1/d. Default isnp.ones(m).- xb, xefloat, optional
The interval to fit. If None, these default to
x[0]andx[-1], respectively.- kint, 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.- sfloat, 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))wheremis the number of datapoints in x, y, and w. Default iss = 0.0, i.e. interpolation.- tarray_like, optional
The spline knots. If None (default), the knots will be constructed automatically. There must be at least
2*k + 2and at mostm + k + 1knots.- nestint, optional
The target length of the knot vector. Should be between
2*(k + 1)(the minimum number of knots for a degree-kspline), andm + 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 tom + k + 1knots).
- Returns:
- spla
BSplineinstance 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.
- spla
See also
generate_knotsis used under the hood for generating the knots
make_splprepthe analog of this routine for parametric curves
make_interp_splineconstruct an interpolating spline (
s = 0)make_lsq_splineconstruct the least-squares spline given the knot vector
splrepa FITPACK analog of this routine
Notes
This routine constructs the smoothing spline function, \(g(x)\), to minimize the sum of jumps, \(D_j\), of the
k-th derivative at the internal knots (\(x_b < t_i < x_e\)), where\[D_i = g^{(k)}(t_i + 0) - g^{(k)}(t_i - 0)\]Specifically, the routine constructs the spline function \(g(x)\) which minimizes
\[\sum_i | D_i |^2 \to \mathrm{min}\]provided that
\[\sum_{j=1}^m (w_j \times (g(x_j) - y_j))^2 \leqslant s ,\]where \(s > 0\) 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 \(g(x_j)\) from the data \(y_j\) (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, seegenerate_knotsfor 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\[\sum w_j \times (g(x_j) - y_j)^2 \to \mathrm{min}\]for a spline function \(g(x)\) with a _fixed_ knot vector
t.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.