InterpolatedUnivariateSplinewithUnits¶

class interpolated_coordinates.utils.spline.InterpolatedUnivariateSplinewithUnits(x: Quantity, y: Quantity, w: ndarray | None = None, bbox: BBoxType = [None, None], k: int = 3, ext: int = 0, check_finite: bool = False, *, x_unit: UnitLikeType | None = None, y_unit: UnitLikeType | None = None)[source]¶

Bases: UnivariateSplinewithUnits, InterpolatedUnivariateSpline

1-D interpolating spline for a given set of data points, with units.

Fits a spline y = spl(x) of degree k to the provided x, y data. Spline function passes through all provided points. Equivalent to UnivariateSpline with s=0.

Parameters:

x(N,) Quantity array_like

Input dimension of data points – must be strictly increasing

y(N,) Quantity array_like

input dimension of data points

w(N,) Quantity array_like, optional

Weights for spline fitting. Must be positive. If None (default), weights are all equal.

bbox(2,) Quantity array_like, optional

2-sequence specifying the boundary of the approximation interval. If None (default), bbox=[x[0], x[-1]].

kint, optional

Degree of the smoothing spline. Must be 1 <= k <= 5.

extint or str, optional

Controls the extrapolation mode for elements not in the interval defined by the knot sequence.

if ext=0 or ‘extrapolate’, return the extrapolated value.
if ext=1 or ‘zeros’, return 0
if ext=2 or ‘raise’, raise a ValueError
if ext=3 of ‘const’, return the boundary value.

The default value is 0.

check_finitebool, 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 or non-sensical results) if the inputs do contain infinities or NaNs. Default is False.

x_unit, y_unitunit-like or None, optional keyword-only

The Unit of x/y (if not None), and to which x/y will be converted before the value is used in the underlying interpolation machinery. If x/y does not have units (e.g. is an ndarray) this cannot not be None.

See also

UnivariateSpline: Superclass – allows knots to be selected by a smoothing condition
LSQUnivariateSpline: spline for which knots are user-selected
splrep: An older, non object-oriented wrapping of FITPACK
splev, sproot, splint, spalde
BivariateSpline: A similar class for two-dimensional spline interpolation

Notes

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

Examples

>>> import numpy as np
>>> import astropy.units as u
>>> from interpolated_coordinates.utils import InterpolatedUnivariateSplinewithUnits
>>> x = np.linspace(-3, 3, 50) * u.s
>>> y = 8 * u.m / (x.value**2 + 4)
>>> spl = InterpolatedUnivariateSplinewithUnits(x, y)

(Source code, png, hires.png, pdf)

../_images/interpolated_coordinates-utils-spline-InterpolatedUnivariateSplinewithUnits-1.png

Notice that the spl(x) interpolates y:

>>> spl.get_residual()
<Quantity 0. m>

Attributes Summary

`x_unit`	`Unit` of the independent data.
`y_unit`	`Unit` of the dependent data.

Methods Summary

`__call__`(x[, nu, ext])	Evaluate spline (or its nu-th derivative) at positions x.
`antiderivative`([n])	Construct a new spline representing this spline's antiderivative.
`derivative`([n])	Construct a new spline representing the derivative of this spline.
`derivatives`(x)	Return all derivatives of the spline at the point x.
`get_coeffs`()	Return spline coefficients.
`get_knots`()	Return positions of interior knots of the spline.
`get_residual`()	Return weighted sum of squared residuals of spline approximation.
`integral`(a, b)	Return definite integral of the spline between two given points.
`roots`()	Return the zeros of the spline.
`set_smoothing_factor`(s)	Continue spline computation with the given smoothing factor s and with the knots found at the last call.
`validate_input`(x, y, w, bbox, k, s, ext, ...)

Attributes Documentation

x_unit¶: Unit of the independent data.

y_unit¶: Unit of the dependent data.

Methods Documentation

__call__(x: ndarray, nu: int = 0, ext: int | None = None) → Quantity¶

Evaluate spline (or its nu-th derivative) at positions x.

Parameters:

xndarray or Quantity array_like

A 1-D array of points at which to return the value of the smoothed spline or its derivatives. Note: x can be unordered but the evaluation is more efficient if x is (partially) ordered.

nuint, optional

The order of derivative of the spline to compute.

extint, optional

Controls the value returned for elements of x not in the interval defined by the knot sequence.

if ext=0 or ‘extrapolate’, return the extrapolated value.
if ext=1 or ‘zeros’, return 0
if ext=2 or ‘raise’, raise a ValueError
if ext=3 or ‘const’, return the boundary value.

The default value is 0, passed from the initialization of UnivariateSpline.

Returns:

yQuantity array_like: Evaluated spline with units y_unit. Same shape as x.

antiderivative(n: int = 1) → USwUType¶

Construct a new spline representing this spline’s antiderivative.

Parameters:

nint, optional: Order of antiderivative to evaluate. Default: 1

Returns:

splineUnivariateSplinewithUnits: Spline of order k2=k+n representing the antiderivative of this spline.

See also

splder, antiderivative

Examples

This can be used for finding maxima of a curve:

>>> from interpolated_coordinates.utils import UnivariateSplinewithUnits
>>> x = np.linspace(0, 10, 70) * u.s
>>> y = np.sin(x.value) * u.m
>>> spl = UnivariateSplinewithUnits(x, y, k=4, s=0)

Now, differentiate the spline and find the zeros of the derivative. (NB: sproot only works for order 3 splines, so we fit an order 4 spline):

>>> spl.derivative().roots() / np.pi  
<Quantity [0.50000001, 1.5       , 2.49999998] s>

This agrees well with roots \(\\pi/2 + n\\pi\) of \(\\cos(x) = \\sin'(x)\).

derivatives(x: Quantity) → ndarray¶

Return all derivatives of the spline at the point x.

Parameters:

xQuantity: The point to evaluate the derivatives at.

Returns:

derndarray of Quantity object, shape(k+1,): Derivatives of the orders 0 to k.

Examples

>>> from scipy.interpolate import UnivariateSpline
>>> x = np.linspace(0, 3, 11)
>>> y = x**2
>>> spl = UnivariateSpline(x, y)
>>> np.round(spl.derivatives(1.5), 2)  
array([2.25, 3.  , 2.  , 0.  ])

get_coeffs() → Quantity¶: Return spline coefficients.

get_knots() → Quantity¶

Return positions of interior knots of the spline.

Internally, the knot vector contains 2*k additional boundary knots. Has units of x position

get_residual() → Quantity¶

Return weighted sum of squared residuals of spline approximation.

This is equivalent to::: sum((w[i] * (y[i]-spl(x[i])))**2, axis=0)

integral(a: Quantity, b: Quantity) → Quantity¶

Return definite integral of the spline between two given points.

Parameters:

aQuantity: Lower limit of integration.
bQuantity: Upper limit of integration.

Returns:

integralfloat: The value of the definite integral of the spline between limits.

Examples

>>> from scipy.interpolate import UnivariateSpline
>>> x = np.linspace(0, 3, 11)
>>> y = x**2
>>> spl = UnivariateSpline(x, y)
>>> spl.integral(0, 3)
9.0

which agrees with \(\int x^2 dx = x^3 / 3\) between the limits of 0 and 3.

A caveat is that this routine assumes the spline to be zero outside of the data limits:

>>> spl.integral(-1, 4)
9.0

>>> spl.integral(-1, 0)
0.0

roots() → Quantity¶

Return the zeros of the spline.

Restriction: only cubic splines are supported by fitpack.

set_smoothing_factor(s)¶

Continue spline computation with the given smoothing factor s and with the knots found at the last call.

This routine modifies the spline in place.

validate_input(x: ndarray | Quantity, y: ndarray | Quantity, w: ndarray | Quantity, bbox: list[Quantity | None], k: int, s: float, ext: int | str, check_finite: float) → tuple[ndarray, ndarray, ndarray, list[Quantity], int | str]¶

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InterpolatedUnivariateSplinewithUnits¶