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scipy.interpolate.

# make_splrep

scipy.interpolate.make_splrep(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None)

  Create a smoothing B-spline function with bounded error, minimizing derivative jumps.

  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 LSQ (least-squares) constraint:

          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

  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, 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

  \[\sum w_j \times (g(x_j) - y_j)^2 \to \mathrm{min}\]

  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.