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scipy.interpolate.

# make_lsq_spline

scipy.interpolate.make_lsq_spline(x, y, t, k=3, w=None, axis=0, check_finite=True, *, method='qr')

  Create a smoothing B-spline satisfying the Least SQuares (LSQ) criterion.

  The result is a linear combination

  \[S(x) = \sum_j c_j B_j(x; t)\]

  of the B-spline basis elements, \(B_j(x; t)\), which minimizes

  \[\sum_{j} \left( w_j \times (S(x_j) - y_j) \right)^2\]
  Parameters:
    x array_like, shape (m,)

      Abscissas.

    y array_like, shape (m, …)

      Ordinates.

    t array_like, shape (n + k + 1,).

      Knots. Knots and data points must satisfy Schoenberg-Whitney conditions.

    k int, optional

      B-spline degree. Default is cubic, `k = 3`.

    w array_like, shape (m,), optional

      Weights for spline fitting. Must be positive. If `None`, then weights are all equal. Default is `None`.

    axis int, optional

      Interpolation axis. Default is zero.

    check_finite bool, optional

      Whether to check that the input arrays contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True.

    method str, optional

      Method for solving the linear LSQ problem. Allowed values are “norm-eq” (Explicitly construct and solve the normal system of equations), and “qr” (Use the QR factorization of the design matrix). Default is “qr”.

  Returns:
    b `BSpline` object

      A `BSpline` object of the degree `k` with knots `t`.

  See also

  `BSpline`

    base class representing the B-spline objects

  `make_interp_spline`

    a similar factory function for interpolating splines

  `LSQUnivariateSpline`

    a FITPACK-based spline fitting routine

  `splrep`

    a FITPACK-based fitting routine

  Notes

  The number of data points must be larger than the spline degree `k`.

  Knots `t` must satisfy the Schoenberg-Whitney conditions, i.e., there must be a subset of data points `x[j]` such that `t[j] < x[j] < t[j+k+1]`, for `j=0, 1,...,n-k-2`.

  Array API Standard Support

  `make_lsq_spline` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ⛔
  PyTorch | ✅ | ⛔
  JAX | ⚠️ no JIT | ⛔
  Dask | ⚠️ computes graph | n/a

  > See Support for the array API standard for more information.

  Examples

  Generate some noisy data:

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> rng = np.random.default_rng()
      >>> x = np.linspace(-3, 3, 50)
      >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50)

  Now fit a smoothing cubic spline with a pre-defined internal knots. Here we make the knot vector (k+1)-regular by adding boundary knots:

      >>> from scipy.interpolate import make_lsq_spline, BSpline
      >>> t = [-1, 0, 1]
      >>> k = 3
      >>> t = np.r_[(x[0],)*(k+1),
      ...           t,
      ...           (x[-1],)*(k+1)]
      >>> spl = make_lsq_spline(x, y, t, k)

  For comparison, we also construct an interpolating spline for the same set of data:

      >>> from scipy.interpolate import make_interp_spline
      >>> spl_i = make_interp_spline(x, y)

  Plot both:

      >>> xs = np.linspace(-3, 3, 100)
      >>> plt.plot(x, y, 'ro', ms=5)
      >>> plt.plot(xs, spl(xs), 'g-', lw=3, label='LSQ spline')
      >>> plt.plot(xs, spl_i(xs), 'b-', lw=3, alpha=0.7, label='interp spline')
      >>> plt.legend(loc='best')
      >>> plt.show()

  NaN handling: If the input arrays contain `nan` values, the result is not useful since the underlying spline fitting routines cannot deal with `nan`. A workaround is to use zero weights for not-a-number data points:

      >>> y[8] = np.nan
      >>> w = np.isnan(y)
      >>> y[w] = 0.
      >>> tck = make_lsq_spline(x, y, t, w=~w)

  Notice the need to replace a `nan` by a numerical value (precise value does not matter as long as the corresponding weight is zero.)