Interpolation with SciPy and NumPy

Interpolation with Python November 20, 2009 Wednesday, December 2, 2009

Enthought Python Distribution (EPD) http://www.enthought.com/products/epd.php Wednesday, December 2, 2009

Enthought Training Courses Dec. 7 - 9 Dec. 10 - 11 Python Basics, NumPy, SciPy, Matplotlib, Traits, TraitsUI, Chaco… Wednesday, December 2, 2009

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Interpolation Wednesday, December 2, 2009

Contrast with Curve-fitting Extrapolation Wednesday, December 2, 2009

2-D Interpolation Wednesday, December 2, 2009

Curve interpolation and scattered samples Wednesday, December 2, 2009

Interpolation basic mathematics Given data points (xi , fi ) f (x) = fj φ(x − xj ) j φ(xi − xj ) = δij f (xi ) = fi Wednesday, December 2, 2009

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 Wednesday, December 2, 2009

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 Wednesday, December 2, 2009

1D Spline Interpolation # 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 Wednesday, December 2, 2009

2D Spline Interpolation >>> 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 Wednesday, December 2, 2009

2D Spline Interpolation 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 Wednesday, December 2, 2009

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 Wednesday, December 2, 2009

What else is in SciPy? • 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 Wednesday, December 2, 2009

What else is in SciPy? • 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 Wednesday, December 2, 2009

What else is in SciPy? • 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 Wednesday, December 2, 2009

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 Wednesday, December 2, 2009

Scientific Python Classes http://www.enthought.com/training Dec 7-11 Feb 22-26 Python for Scientists and Engineers (3 days) Advanced Modules (2 days) May 17-21 Take both together to receive a discount Aug 23-27 Wednesday, December 2, 2009

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Interpolation with SciPy and NumPy

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