bundles / scipy latest / scipy / interpolate / _bsplines / make_lsq_spline

function

`scipy.interpolate._bsplines:make_lsq_spline`

source: /scipy/interpolate/_bsplines.py :1694

Signature

   def     make_lsq_spline (    x    ,    y    ,    t    ,    k    =  3   ,    w    =  None   ,    axis    =  0   ,    check_finite    =  True   ,  * ,     method    =  qr     )

Summary

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

Extended Summary

The result is a linear combination

S (x) = j \sum c_{j} B_{j} (x; t)

of the B-spline basis elements, $B_{j} (x; t)$ , which minimizes

j \sum (w_{j} \times (S (x_{j}) - y_{j}))^{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.

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 dev-arrayapi 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.)

Aliases

scipy.interpolate.make_lsq_spline