Interpolation with SciPy and NumPy

Interpolation with Python

November 20, 2009

Wednesday, December 2, 2009

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Interpolation


Contrast with

Curve-fitting Extrapolation


2-D Interpolation


Curve interpolation and scattered samples


Interpolation basic mathematics

Given data points (xi , fi )

f (x) = fj φ(x − xj )
j
φ(xi − xj ) = δij
f (xi ) = fi


Interpolation
scipy.interpolate — General purpose Interpolation

•1D Interpolating Class
• Constructs callable function from data points and
desired spline interpolation order.
• Function takes vector of inputs and returns interpolated
value using the spline.
•1D and 2D spline interpolation (FITPACK)
• Smoothing splines up to order 5
• Parametric splines

10


1D Spline Interpolation
>>> from scipy.interpolate import interp1d

interp1d(x, y, kind='linear', axis=-1, copy=True,
bounds_error=True, fill_value=numpy.nan)

Returns a function that uses interpolation to find the
value of new points.

• x – 1d array of increasing real values which cannot
contain duplicates
• y – Nd array of real values whose length along the
interpolation axis must be len(x)
• kind – kind of interpolation (e.g. 'linear',
'nearest', 'quadratic', 'cubic'). Can also be an
integer n>1 which returns interpolating spline (with
minimum sum-of-squares discontinuity in nth
derivative).
• axis – axis of y along which to interpolate
• copy – make internal copies of x and y
• bounds_error – raise error for out-of-bounds
• fill_value – if bounds_error is False, then use this
value to fill in out-of-bounds.
11


# demo/interpolate/spline.py
from scipy.interpolate import interp1d
from pylab import plot, axis, legend
from numpy import linspace

# sample values
x = linspace(0,2*pi,6)
y = sin(x)

# Create a spline class for interpolation.
# kind=5 sets to 5th degree spline.
# kind='nearest' -> zeroth older hold.
# kind='linear' -> linear interpolation
# kind=n -> use an nth order spline
spline_fit = interp1d(x,y,kind=5)
xx = linspace(0,2*pi, 50)
yy = spline_fit(xx)

# display the results.
plot(xx, sin(xx), 'r-', x, y, 'ro', xx, yy, 'b--', linewidth=2)
axis('tight')
legend(['actual sin', 'original samples', 'interpolated curve'])

12


>>> from scipy.interpolate import interp2d

interp2d(x, y, z, kind='linear')

Returns a function, f, that uses interpolation to find the value of new points: z_new = f(x_new,
y_new)

x – 1d or 2d array
y – 1d or 2d array
z – 1d or 2d array representing function evaluated at x
and y
kind – kind of interpolation: 'linear', 'quadratic', or
'cubic'

The shape of x, y, and z must be the same.

Resulting function is evaluated
at cross product of new inputs.

13


EXAMPLE

>>> from scipy.interpolate import
... interp2d
>>> from numpy import hypot, mgrid,
... linspace
>>> from scipy.special import j0
>>> x,y = mgrid[-5:6,-5:6]
>>> z = j0(hypot(x,y))
>>> newfunc = interp2d(x, y, z,
... kind='cubic')
>>> xx = linspace(-5,5,100)
>>> yy = xx
# xx and yy are 1-d
# result is evaluated on the
# cross product
>>> zz = newfunc(xx,yy)
>>> from enthought.mayavi import mlab
>>> mlab.surf(x,y,z)
>>> x2, y2 = mgrid[-5:5:100j,
... -5:5:100j]
>>> mlab.surf(x2,y2,zz)

14


What else is in SciPy?
• Interpolate: Fitpack interface
• splrep : represent 1-d data as spline
• splprep : represent N-d parametric curve as spline
• splev : evaluate spline at new data points
• splint : evaluate integral of spline
• splalde : evaluate all derivatives of spline
• bisplrep : reprsent 2-d surface as spline
• bisplev : evaluate 2-d spline at new data points
• Additional class-based interface:
–UnivariateSpline
–InterpolatedUnivariateSpline
–LSQUnivariateSpline
–BivariateSpline
–SmoothBivariateSpline


• Interpolate: General spline interface (no fitpack)
• splmake : create a general spline representation from data
• spleval : evaluate spline at new data
• spline : front end to splmake and spleval (all in one)
• spltopp : take spline reprentation and return piece-wise
polynomial representation of a spline
• ppform : piecewise polynomial representation of spline
• PiecewisePolynomial : class to generate arbitrary piecewise
polynomial interpolator given data + derivatives
• lagrange : Lagrange polynomial interpolator
• BarycentricInterpolator : polynomial interpolation class
• KroghInterpolator : polynomial interpolation class that allows
setting derivatives as well


• Signal: Fast B-spline implementations (equally-spaced)
• bspline : B-spline basis functions of order n
• gauss_spline : Gaussian approximation to the B-spline
• qspline1d : quadratic B-spline coefficients from 1-d data
• cspline1d : cubic B-spline coefficients from 1-d data
• qspline1d_eval : evaluate quadratic spline
• cspline1d_eval : evaluat cubic spline
• qspline2d : quadratic B-spline coefficients from 2-d data
• cspline2d : cubic B-spline coefficients from 2-d data
• spline_filter : spline filter using from coefficients

• resample : sinc-interpolation using a Fourier method


• ndimage: N-d spline interpolation
• map_coordinates : N-d interpolation
• spline_filter : repeated interpolation from spline coefficients

• interpolate : Radial Basis Functions
• Rbf : N-d interpolation using radial basis functions


Things I’d like to see

• Monotonic Splines
• PCHIP
• Finishing out the splmake, spleval interface
• Improving a simplified interface to all the
tools
• An interpnd


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