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scipy.interpolate.splrep

scipy.interpolate.splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None, full_output=0, per=0, quiet=1)

Find the B-spline representation of 1-D curve.

Given the set of data points (x[i], y[i]) determine a smooth spline approximation of degree k on the interval xb <= x <= xe. The coefficients, c, and the knot points, t, are returned. Uses the FORTRAN routine curfit from FITPACK.

Parameters :

x, y : array_like

The data points defining a curve y = f(x).

w : array_like

Strictly positive rank-1 array of weights 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 ones(len(x)).

xb, xe : float

The interval to fit. If None, these default to x[0] and x[-1] respectively.

k : int

The order of the spline fit. It is recommended to use cubic splines. Even order splines should be avoided especially with small s values. 1 <= k <= 5

task : {1, 0, -1}

If task==0 find t and c for a given smoothing factor, s.

If task==1 find t and c for another value of the smoothing factor, s. There must have been a previous call with task=0 or task=1 for the same set of data (t will be stored an used internally)

If task=-1 find the weighted least square spline for a given set of knots, t. These should be interior knots as knots on the ends will be added automatically.

s : float

A smoothing condition. The amount of smoothness is determined by satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s where g(x) is the smoothed interpolation of (x,y). The user can use s to control the tradeoff between closeness 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 : s=m-sqrt(2*m) if weights are supplied. s = 0.0 (interpolating) if no weights are supplied.

t : int

The knots needed for task=-1. If given then task is automatically set to -1.

full_output : bool

If non-zero, then return optional outputs.

per : bool

If non-zero, data points are considered periodic with period x[m-1] - x[0] and a smooth periodic spline approximation is returned. Values of y[m-1] and w[m-1] are not used.

quiet : bool

Non-zero to suppress messages.

Returns :

tck : tuple

(t,c,k) a tuple containing the vector of knots, the B-spline coefficients, and the degree of the spline.

fp : array, optional

The weighted sum of squared residuals of the spline approximation.

ier : int, optional

An integer flag about splrep success. Success is indicated if ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.

msg : str, optional

A message corresponding to the integer flag, ier.

See also

UnivariateSpline, BivariateSpline, splprep, splev, sproot, spalde, splint, bisplrep, bisplev

Notes

See splev for evaluation of the spline and its derivatives.

References

Based on algorithms described in [1], [2], [3], and [4]:

[R22]

P. Dierckx, “An algorithm for smoothing, differentiation and integration of experimental data using spline functions”, J.Comp.Appl.Maths 1 (1975) 165-184.

[R23]

P. Dierckx, “A fast algorithm for smoothing data on a rectangular grid while using spline functions”, SIAM J.Numer.Anal. 19 (1982) 1286-1304.

[R24]

P. Dierckx, “An improved algorithm for curve fitting with spline functions”, report tw54, Dept. Computer Science,K.U. Leuven, 1981.

[R25]

P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.

Examples

>>> x = linspace(0, 10, 10)
>>> y = sin(x)
>>> tck = splrep(x, y)
>>> x2 = linspace(0, 10, 200)
>>> y2 = splev(x2, tck)
>>> plot(x, y, 'o', x2, y2)