small updates
Per.Andreas.Brodtkorb
13 years ago
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'''
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pychip.py
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chris.michalski@gmail.com
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20090818
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Piecewise cubic Hermite interpolation (monotonic...) in Python
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References:
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Wikipedia: Monotone cubic interpolation
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Cubic Hermite spline
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A cubic Hermite spline is a third degree spline with each polynomial of the spline
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in Hermite form. The Hermite form consists of two control points and two control
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tangents for each polynomial. Each interpolation is performed on one sub-interval
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at a time (piece-wise). A monotone cubic interpolation is a variant of cubic
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interpolation that preserves monotonicity of the data to be interpolated (in other
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words, it controls overshoot). Monotonicity is preserved by linear interpolation
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but not by cubic interpolation.
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Use:
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There are two separate calls, the first call, pchip_slopes(), computes the slopes that
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the interpolator needs. If there are a large number of points to compute,
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it is more efficient to compute the slopes once, rather than for every point
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being evaluated. The second call, pchip_eval(), takes the slopes computed by
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pchip_slopes() along with X, Y, and a vector of desired "xnew"s and computes a vector
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of "ynew"s. If only a handful of points is needed, pchip() is a third function
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which combines a call to pchip_slopes() followed by pchip_eval().
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'''
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import warnings
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import numpy as np
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from matplotlib import pyplot as plt
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from interpolate import slopes2, slopes, stineman_interp
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from scipy.interpolate import PiecewisePolynomial
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#=========================================================
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def pchip(x, y, xnew):
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# Compute the slopes used by the piecewise cubic Hermite interpolator
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m = pchip_slopes(x, y)
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# Use these slopes (along with the Hermite basis function) to interpolate
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ynew = pchip_eval(x, y, m, xnew)
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return ynew
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#=========================================================
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def x_is_okay(x,xvec):
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# Make sure "x" and "xvec" satisfy the conditions for
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# running the pchip interpolator
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n = len(x)
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m = len(xvec)
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# Make sure "x" is in sorted order (brute force, but works...)
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xx = x.copy()
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xx.sort()
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total_matches = (xx == x).sum()
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if total_matches != n:
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warnings.warn( "x values weren't in sorted order --- aborting")
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return False
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# Make sure 'x' doesn't have any repeated values
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delta = x[1:] - x[:-1]
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if (delta == 0.0).any():
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warnings.warn( "x values weren't monotonic--- aborting")
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return False
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# Check for in-range xvec values (beyond upper edge)
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check = xvec > x[-1]
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if check.any():
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print "*" * 50
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print "x_is_okay()"
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print "Certain 'xvec' values are beyond the upper end of 'x'"
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print "x_max = ", x[-1]
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indices = np.compress(check, range(m))
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print "out-of-range xvec's = ", xvec[indices]
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print "out-of-range xvec indices = ", indices
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return False
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# Second - check for in-range xvec values (beyond lower edge)
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check = xvec< x[0]
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if check.any():
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print "*" * 50
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print "x_is_okay()"
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print "Certain 'xvec' values are beyond the lower end of 'x'"
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print "x_min = ", x[0]
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indices = np.compress(check, range(m))
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print "out-of-range xvec's = ", xvec[indices]
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print "out-of-range xvec indices = ", indices
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return False
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return True
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#=========================================================
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def pchip_eval(x, y, m, xvec):
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'''
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Evaluate the piecewise cubic Hermite interpolant with monoticity preserved
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x = array containing the x-data
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y = array containing the y-data
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m = slopes at each (x,y) point [computed to preserve monotonicity]
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xnew = new "x" value where the interpolation is desired
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x must be sorted low to high... (no repeats)
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y can have repeated values
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This works with either a scalar or vector of "xvec"
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'''
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############################
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# Make sure there aren't problems with the input data
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############################
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if not x_is_okay(x, xvec):
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print "pchip_eval2() - ill formed 'x' vector!!!!!!!!!!!!!"
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# Cause a hard crash...
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return #STOP_pchip_eval2
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# Find the indices "k" such that x[k] < xvec < x[k+1]
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k = np.searchsorted(x[1:-1], xvec)
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# Create the Hermite coefficients
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h = x[k+1] - x[k]
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t = (xvec - x[k]) / h[k]
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# Hermite basis functions
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h00 = (2 * t**3) - (3 * t**2) + 1
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h10 = t**3 - (2 * t**2) + t
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h01 = (-2* t**3) + (3 * t**2)
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h11 = t**3 - t**2
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# Compute the interpolated value of "y"
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ynew = h00*y[k] + h10*h*m[k] + h01*y[k+1] + h11*h*m[k+1]
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return ynew
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#=========================================================
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def pchip_slopes(x,y, method='secant', tension=0, monotone=True):
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'''
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Return estimated slopes y'(x)
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Parameters
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----------
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x, y : array-like
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array containing the x- and y-data, respectively.
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x must be sorted low to high... (no repeats) while
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y can have repeated values.
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method : string
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defining method of estimation for yp. Valid options are:
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'secant' average secants
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yp = 0.5*((y[k+1]-y[k])/(x[k+1]-x[k]) + (y[k]-y[k-1])/(x[k]-x[k-1]))
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'Catmull-Rom' yp = (y[k+1]-y[k-1])/(x[k+1]-x[k-1])
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'Cardinal' yp = (1-tension) * (y[k+1]-y[k-1])/(x[k+1]-x[k-1])
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tension : real scalar between 0 and 1.
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tension parameter used in Cardinal method
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monotone : bool
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If True modifies yp to preserve monoticity
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x input conditioning is assumed but not checked
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'''
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n = len(x)
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# Compute the slopes of the secant lines between successive points
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delta = (y[1:] - y[:-1]) / (x[1:] - x[:-1])
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# Initialize the tangents at every points as the average of the secants
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m = np.zeros(n, dtype='d')
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# At the endpoints - use one-sided differences
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m[0] = delta[0]
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m[n-1] = delta[-1]
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method = method.lower()
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if method.startswith('secant'):
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# In the middle - use the average of the secants
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m[1:-1] = (delta[:-1] + delta[1:]) / 2.0
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else: # Cardinal or Catmull-Rom method
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m[1:-1] = (y[2:] - y[:-2]) / (x[2:] - x[:-2])
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if method.startswith('cardinal'):
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m = (1-tension) * m
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if monotone:
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# Special case: intervals where y[k] == y[k+1]
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# Setting these slopes to zero guarantees the spline connecting
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# these points will be flat which preserves monotonicity
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ii, = (delta == 0.0).nonzero()
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m[ii] = 0.0
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m[ii+1] = 0.0
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alpha = m[:-1]/delta
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beta = m[1:]/delta
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dist = alpha**2 + beta**2
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tau = 3.0 / np.sqrt(dist)
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# To prevent overshoot or undershoot, restrict the position vector
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# (alpha, beta) to a circle of radius 3. If (alpha**2 + beta**2)>9,
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# then set m[k] = tau[k]alpha[k]delta[k] and m[k+1] = tau[k]beta[b]delta[k]
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# where tau = 3/sqrt(alpha**2 + beta**2).
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# Find the indices that need adjustment
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indices_to_fix, = (dist > 9.0).nonzero()
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for ii in indices_to_fix:
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m[ii] = tau[ii] * alpha[ii] * delta[ii]
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m[ii+1] = tau[ii] * beta[ii] * delta[ii]
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return m
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def _edge_case(m0, d1):
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return np.where((d1==0) | (m0==0), 0.0, 1.0/(1.0/m0+1.0/d1))
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def pchip_slopes2(x, y):
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# Determine the derivatives at the points y_k, d_k, by using
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# PCHIP algorithm is:
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# We choose the derivatives at the point x_k by
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# Let m_k be the slope of the kth segment (between k and k+1)
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# If m_k=0 or m_{k-1}=0 or sgn(m_k) != sgn(m_{k-1}) then d_k == 0
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# else use weighted harmonic mean:
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# w_1 = 2h_k + h_{k-1}, w_2 = h_k + 2h_{k-1}
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# 1/d_k = 1/(w_1 + w_2)*(w_1 / m_k + w_2 / m_{k-1})
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# where h_k is the spacing between x_k and x_{k+1}
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hk = x[1:] - x[:-1]
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mk = (y[1:] - y[:-1]) / hk
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smk = np.sign(mk)
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condition = ((smk[1:] != smk[:-1]) | (mk[1:]==0) | (mk[:-1]==0))
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w1 = 2*hk[1:] + hk[:-1]
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w2 = hk[1:] + 2*hk[:-1]
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whmean = 1.0/(w1+w2)*(w1/mk[1:] + w2/mk[:-1])
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dk = np.zeros_like(y)
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dk[1:-1][condition] = 0.0
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dk[1:-1][~condition] = 1.0/whmean[~condition]
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# For end-points choose d_0 so that 1/d_0 = 1/m_0 + 1/d_1 unless
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# one of d_1 or m_0 is 0, then choose d_0 = 0
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dk[0] = _edge_case(mk[0],dk[1])
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dk[-1] = _edge_case(mk[-1],dk[-2])
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return dk
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class StinemanInterp(PiecewisePolynomial):
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def __init__(self, x, y, yp=None, method='parabola'):
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if yp is None:
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yp = slopes2(x, y, method)
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super(StinemanInterp,self).__init__(x, zip(y,yp))
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def CubicHermiteSpline2(x, y, xnew):
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'''
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Piecewise Cubic Hermite Interpolation using Catmull-Rom
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method for computing the slopes.
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'''
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# Non-montonic cubic Hermite spline interpolator using
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# Catmul-Rom method for computing slopes...
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m = pchip_slopes(x, y, method='catmull', monotone=False)
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# Use these slopes (along with the Hermite basis function) to interpolate
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ynew = pchip_eval(x, y, m, xnew)
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return ynew
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#==============================================================
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def main():
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############################################################
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# Sine wave test
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############################################################
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# Create a example vector containing a sine wave.
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x = np.arange(30.0)/10.
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y = np.sin(x)
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# Interpolate the data above to the grid defined by "xvec"
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xvec = np.arange(250.)/100.
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# Initialize the interpolator slopes
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m = pchip_slopes(x,y)
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m1 = slopes(x, y)
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m2 = pchip_slopes(x,y,method='catmul',monotone=False)
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m3 = pchip_slopes2(x, y)
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# Call the monotonic piece-wise Hermite cubic interpolator
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yvec = pchip_eval(x, y, m, xvec)
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yvec1 = pchip_eval(x, y, m1, xvec)
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yvec2 = pchip_eval(x, y, m2, xvec)
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yvec3 = pchip_eval(x, y, m3, xvec)
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plt.figure(1)
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plt.plot(x,y, 'ro')
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plt.title("pchip() Sin test code")
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# Plot the interpolated points
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plt.plot(xvec, yvec, xvec, yvec1, xvec, yvec2,xvec, yvec3, )
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plt.legend(['true','m0','m1','m2','m3'])
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# Step function test...
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plt.figure(2)
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plt.title("pchip() step function test")
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# Create a step function (will demonstrate monotonicity)
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x = np.arange(7.0) - 3.0
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y = np.array([-1.0, -1,-1,0,1,1,1])
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# Interpolate using monotonic piecewise Hermite cubic spline
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xvec = np.arange(599.)/100. - 3.0
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# Create the pchip slopes
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m = pchip_slopes(x,y)
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m1 = slopes(x, y)
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m2 = pchip_slopes(x,y,method='catmul',monotone=False)
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m3 = pchip_slopes2(x, y)
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# Interpolate...
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yvec = pchip_eval(x, y, m, xvec)
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# Call the Scipy cubic spline interpolator
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from scipy.interpolate import interpolate
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function = interpolate.interp1d(x, y, kind='cubic')
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yvec2 = function(xvec)
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# Non-montonic cubic Hermite spline interpolator using
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# Catmul-Rom method for computing slopes...
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yvec3 = CubicHermiteSpline(x,y)(xvec)
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yvec4 = StinemanInterp(x, y)(xvec)
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#yvec4 = stineman_interp(xvec, x, y, m)
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yvec5 = pchip_eval(x, y, m3, xvec)
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# Plot the results
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plt.plot(x, y, 'ro')
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plt.plot(xvec, yvec, 'b')
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plt.plot(xvec, yvec2, 'k')
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plt.plot(xvec, yvec3, 'g')
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plt.plot(xvec, yvec4, 'm')
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#plt.plot(xvec, yvec5, 'y')
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plt.xlabel("X")
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plt.ylabel("Y")
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plt.title("Comparing pypchip() vs. Scipy interp1d() vs. non-monotonic CHS")
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legends = ["Data", "pypchip()", "interp1d","CHS", 'SI']
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plt.legend(legends, loc="upper left")
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plt.ioff()
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plt.show()
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if __name__ == '__main__':
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###################################################################
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main()
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