|
|
|
@ -18,9 +18,13 @@ from numpy.ma.core import ones, zeros, prod, sin
|
|
|
|
from numpy import diff, pi, inf #@UnresolvedImport
|
|
|
|
from numpy import diff, pi, inf #@UnresolvedImport
|
|
|
|
from numpy.lib.shape_base import vstack
|
|
|
|
from numpy.lib.shape_base import vstack
|
|
|
|
from numpy.lib.function_base import linspace
|
|
|
|
from numpy.lib.function_base import linspace
|
|
|
|
|
|
|
|
from scipy.interpolate import PiecewisePolynomial
|
|
|
|
|
|
|
|
|
|
|
|
import polynomial as pl
|
|
|
|
import polynomial as pl
|
|
|
|
|
|
|
|
|
|
|
|
__all__ = ['PPform', 'savitzky_golay', 'savitzky_golay_piecewise', 'sgolay2d','SmoothSpline', 'stineman_interp']
|
|
|
|
|
|
|
|
|
|
|
|
__all__ = ['PPform', 'savitzky_golay', 'savitzky_golay_piecewise', 'sgolay2d','SmoothSpline',
|
|
|
|
|
|
|
|
'slopes','pchip_slopes','slopes2','stineman_interp', 'Pchip','StinemanInterp', 'CubicHermiteSpline']
|
|
|
|
|
|
|
|
|
|
|
|
def savitzky_golay(y, window_size, order, deriv=0):
|
|
|
|
def savitzky_golay(y, window_size, order, deriv=0):
|
|
|
|
r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter.
|
|
|
|
r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter.
|
|
|
|
@ -531,13 +535,11 @@ class SmoothSpline(PPform):
|
|
|
|
di = di.T
|
|
|
|
di = di.T
|
|
|
|
ci = ci.T
|
|
|
|
ci = ci.T
|
|
|
|
ai = ai.T
|
|
|
|
ai = ai.T
|
|
|
|
#end
|
|
|
|
|
|
|
|
if not any(di):
|
|
|
|
if not any(di):
|
|
|
|
if not any(ci):
|
|
|
|
if not any(ci):
|
|
|
|
coefs = vstack([bi.ravel(), ai.ravel()])
|
|
|
|
coefs = vstack([bi.ravel(), ai.ravel()])
|
|
|
|
else:
|
|
|
|
else:
|
|
|
|
coefs = vstack([ci.ravel(), bi.ravel(), ai.ravel()])
|
|
|
|
coefs = vstack([ci.ravel(), bi.ravel(), ai.ravel()])
|
|
|
|
#end
|
|
|
|
|
|
|
|
else:
|
|
|
|
else:
|
|
|
|
coefs = vstack([di.ravel(), ci.ravel(), bi.ravel(), ai.ravel()])
|
|
|
|
coefs = vstack([di.ravel(), ci.ravel(), bi.ravel(), ai.ravel()])
|
|
|
|
|
|
|
|
|
|
|
|
@ -566,9 +568,126 @@ class SmoothSpline(PPform):
|
|
|
|
ddydx = diff(dydx, axis=0)
|
|
|
|
ddydx = diff(dydx, axis=0)
|
|
|
|
sp.linalg.use_solver(useUmfpack=True)
|
|
|
|
sp.linalg.use_solver(useUmfpack=True)
|
|
|
|
u = 2 * sp.linalg.spsolve((QQ + QQ.T), ddydx)
|
|
|
|
u = 2 * sp.linalg.spsolve((QQ + QQ.T), ddydx)
|
|
|
|
#faster than u=QQ\(Q' * yi);
|
|
|
|
|
|
|
|
return u.reshape(n - 2, -1), p
|
|
|
|
return u.reshape(n - 2, -1), p
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def _edge_case(m0, d1):
|
|
|
|
|
|
|
|
return np.where((d1==0) | (m0==0), 0.0, 1.0/(1.0/m0+1.0/d1))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def pchip_slopes(x, y):
|
|
|
|
|
|
|
|
# Determine the derivatives at the points y_k, d_k, by using
|
|
|
|
|
|
|
|
# PCHIP algorithm is:
|
|
|
|
|
|
|
|
# We choose the derivatives at the point x_k by
|
|
|
|
|
|
|
|
# Let m_k be the slope of the kth segment (between k and k+1)
|
|
|
|
|
|
|
|
# If m_k=0 or m_{k-1}=0 or sgn(m_k) != sgn(m_{k-1}) then d_k == 0
|
|
|
|
|
|
|
|
# else use weighted harmonic mean:
|
|
|
|
|
|
|
|
# w_1 = 2h_k + h_{k-1}, w_2 = h_k + 2h_{k-1}
|
|
|
|
|
|
|
|
# 1/d_k = 1/(w_1 + w_2)*(w_1 / m_k + w_2 / m_{k-1})
|
|
|
|
|
|
|
|
# where h_k is the spacing between x_k and x_{k+1}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
hk = x[1:] - x[:-1]
|
|
|
|
|
|
|
|
mk = (y[1:] - y[:-1]) / hk
|
|
|
|
|
|
|
|
smk = np.sign(mk)
|
|
|
|
|
|
|
|
condition = ((smk[1:] != smk[:-1]) | (mk[1:]==0) | (mk[:-1]==0))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
w1 = 2*hk[1:] + hk[:-1]
|
|
|
|
|
|
|
|
w2 = hk[1:] + 2*hk[:-1]
|
|
|
|
|
|
|
|
whmean = 1.0/(w1+w2)*(w1/mk[1:] + w2/mk[:-1])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
dk = np.zeros_like(y)
|
|
|
|
|
|
|
|
dk[1:-1][condition] = 0.0
|
|
|
|
|
|
|
|
dk[1:-1][~condition] = 1.0/whmean[~condition]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# For end-points choose d_0 so that 1/d_0 = 1/m_0 + 1/d_1 unless
|
|
|
|
|
|
|
|
# one of d_1 or m_0 is 0, then choose d_0 = 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
dk[0] = _edge_case(mk[0],dk[1])
|
|
|
|
|
|
|
|
dk[-1] = _edge_case(mk[-1],dk[-2])
|
|
|
|
|
|
|
|
return dk
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def slopes2(x,y, method='parabola', tension=0, monotone=True):
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
Return estimated slopes y'(x)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
|
|
----------
|
|
|
|
|
|
|
|
x, y : array-like
|
|
|
|
|
|
|
|
array containing the x- and y-data, respectively.
|
|
|
|
|
|
|
|
x must be sorted low to high... (no repeats) while
|
|
|
|
|
|
|
|
y can have repeated values.
|
|
|
|
|
|
|
|
method : string
|
|
|
|
|
|
|
|
defining method of estimation for yp. Valid options are:
|
|
|
|
|
|
|
|
'Catmull-Rom' yp = (y[k+1]-y[k-1])/(x[k+1]-x[k-1])
|
|
|
|
|
|
|
|
'Cardinal' yp = (1-tension) * (y[k+1]-y[k-1])/(x[k+1]-x[k-1])
|
|
|
|
|
|
|
|
'parabola'
|
|
|
|
|
|
|
|
'secant' average secants
|
|
|
|
|
|
|
|
yp = 0.5*((y[k+1]-y[k])/(x[k+1]-x[k]) + (y[k]-y[k-1])/(x[k]-x[k-1]))
|
|
|
|
|
|
|
|
tension : real scalar between 0 and 1.
|
|
|
|
|
|
|
|
tension parameter used in Cardinal method
|
|
|
|
|
|
|
|
monotone : bool
|
|
|
|
|
|
|
|
If True modifies yp to preserve monoticity
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Returns
|
|
|
|
|
|
|
|
-------
|
|
|
|
|
|
|
|
yp : ndarray
|
|
|
|
|
|
|
|
estimated slope
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
References:
|
|
|
|
|
|
|
|
-----------
|
|
|
|
|
|
|
|
Wikipedia: Monotone cubic interpolation
|
|
|
|
|
|
|
|
Cubic Hermite spline
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
x = np.asarray(x, np.float_)
|
|
|
|
|
|
|
|
y = np.asarray(y, np.float_)
|
|
|
|
|
|
|
|
yp = np.zeros(y.shape, np.float_)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
dx = x[1:] - x[:-1]
|
|
|
|
|
|
|
|
# Compute the slopes of the secant lines between successive points
|
|
|
|
|
|
|
|
dydx = (y[1:] - y[:-1]) / dx
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
method = method.lower()
|
|
|
|
|
|
|
|
if method.startswith('parabola'):
|
|
|
|
|
|
|
|
yp[1:-1] = (dydx[:-1] * dx[1:] + dydx[1:] * dx[:-1]) / (dx[1:] + dx[:-1])
|
|
|
|
|
|
|
|
yp[0] = 2.0 * dydx[0] - yp[1]
|
|
|
|
|
|
|
|
yp[-1] = 2.0 * dydx[-1] - yp[-2]
|
|
|
|
|
|
|
|
else:
|
|
|
|
|
|
|
|
# At the endpoints - use one-sided differences
|
|
|
|
|
|
|
|
yp[0] = dydx[0]
|
|
|
|
|
|
|
|
yp[-1] = dydx[-1]
|
|
|
|
|
|
|
|
if method.startswith('secant'):
|
|
|
|
|
|
|
|
# In the middle - use the average of the secants
|
|
|
|
|
|
|
|
yp[1:-1] = (dydx[:-1] + dydx[1:]) / 2.0
|
|
|
|
|
|
|
|
else: # Cardinal or Catmull-Rom method
|
|
|
|
|
|
|
|
yp[1:-1] = (y[2:] - y[:-2]) / (x[2:] - x[:-2])
|
|
|
|
|
|
|
|
if method.startswith('cardinal'):
|
|
|
|
|
|
|
|
yp = (1-tension) * yp
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
if monotone:
|
|
|
|
|
|
|
|
# Special case: intervals where y[k] == y[k+1]
|
|
|
|
|
|
|
|
# Setting these slopes to zero guarantees the spline connecting
|
|
|
|
|
|
|
|
# these points will be flat which preserves monotonicity
|
|
|
|
|
|
|
|
ii, = (dydx == 0.0).nonzero()
|
|
|
|
|
|
|
|
yp[ii] = 0.0
|
|
|
|
|
|
|
|
yp[ii+1] = 0.0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
alpha = yp[:-1]/dydx
|
|
|
|
|
|
|
|
beta = yp[1:]/dydx
|
|
|
|
|
|
|
|
dist = alpha**2 + beta**2
|
|
|
|
|
|
|
|
tau = 3.0 / np.sqrt(dist)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# To prevent overshoot or undershoot, restrict the position vector
|
|
|
|
|
|
|
|
# (alpha, beta) to a circle of radius 3. If (alpha**2 + beta**2)>9,
|
|
|
|
|
|
|
|
# then set m[k] = tau[k]alpha[k]delta[k] and m[k+1] = tau[k]beta[b]delta[k]
|
|
|
|
|
|
|
|
# where tau = 3/sqrt(alpha**2 + beta**2).
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# Find the indices that need adjustment
|
|
|
|
|
|
|
|
indices_to_fix, = (dist > 9.0).nonzero()
|
|
|
|
|
|
|
|
for ii in indices_to_fix:
|
|
|
|
|
|
|
|
yp[ii] = tau[ii] * alpha[ii] * dydx[ii]
|
|
|
|
|
|
|
|
yp[ii+1] = tau[ii] * beta[ii] * dydx[ii]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
return yp
|
|
|
|
|
|
|
|
|
|
|
|
def slopes(x, y):
|
|
|
|
def slopes(x, y):
|
|
|
|
"""
|
|
|
|
"""
|
|
|
|
@ -610,7 +729,118 @@ def slopes(x, y):
|
|
|
|
yp[-1] = 2.0 * dy[-1] / dx[-1] - yp[-2]
|
|
|
|
yp[-1] = 2.0 * dy[-1] / dx[-1] - yp[-2]
|
|
|
|
return yp
|
|
|
|
return yp
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class StinemanInterp(object):
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
Returns the values of an interpolating function that runs through a set of points according to the algorithm of Stineman (1980).
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
|
|
---------
|
|
|
|
|
|
|
|
x,y : array-like
|
|
|
|
|
|
|
|
coordinates of points defining the interpolating function.
|
|
|
|
|
|
|
|
yp : array-like
|
|
|
|
|
|
|
|
slopes of the interpolating function at x. Optional: only given if they are known, else the argument is not used.
|
|
|
|
|
|
|
|
method : string
|
|
|
|
|
|
|
|
method for computing the slope at the given points if the slope is not known. With method=
|
|
|
|
|
|
|
|
"parabola" calculates the slopes from a parabola through every three points.
|
|
|
|
|
|
|
|
Notes
|
|
|
|
|
|
|
|
-----
|
|
|
|
|
|
|
|
The interpolation method is described in an article by Russell W. Stineman (1980)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
According to Stineman, the interpolation procedure has "the following properties:
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
If values of the ordinates of the specified points change monotonically, and the slopes of the line segments joining
|
|
|
|
|
|
|
|
the points change monotonically, then the interpolating curve and its slope will change monotonically.
|
|
|
|
|
|
|
|
If the slopes of the line segments joining the specified points change monotonically, then the slopes of the interpolating
|
|
|
|
|
|
|
|
curve will change monotonically. Suppose that the conditions in (1) or (2) are satisfied by a set of points, but a small
|
|
|
|
|
|
|
|
change in the ordinate or slope at one of the points will result conditions (1) or (2) being not longer satisfied. Then
|
|
|
|
|
|
|
|
making this small change in the ordinate or slope at a point will cause no more than a small change in the interpolating
|
|
|
|
|
|
|
|
curve." The method is based on rational interpolation with specially chosen rational functions to satisfy the above three
|
|
|
|
|
|
|
|
conditions.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Slopes computed at the given points with the methods provided by the `StinemanInterp' function satisfy Stineman's requirements.
|
|
|
|
|
|
|
|
The original method suggested by Stineman (method="scaledstineman", the default, and "stineman") result in lower slopes near
|
|
|
|
|
|
|
|
abrupt steps or spikes in the point sequence, and therefore a smaller tendency for overshooting. The method based on a second
|
|
|
|
|
|
|
|
degree polynomial (method="parabola") provides better approximation to smooth functions, but it results in in higher slopes
|
|
|
|
|
|
|
|
near abrupt steps or spikes and can lead to some overshooting where Stineman's method does not. Both methods lead to much
|
|
|
|
|
|
|
|
less tendency for `spurious' oscillations than traditional interplation methods based on polynomials, such as splines
|
|
|
|
|
|
|
|
(see the examples section).
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Stineman states that "The complete assurance that the procedure will never generate `wild' points makes it attractive as a
|
|
|
|
|
|
|
|
general purpose procedure".
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
This interpolation method has been implemented in Matlab and R in addition to Python.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Examples
|
|
|
|
|
|
|
|
--------
|
|
|
|
|
|
|
|
>>> import wafo.interpolate as wi
|
|
|
|
|
|
|
|
>>> x = linspace(0,2*pi,20)
|
|
|
|
|
|
|
|
>>> y = sin(x); yp = cos(x)
|
|
|
|
|
|
|
|
>>> xi = linspace(0,2*pi,40);
|
|
|
|
|
|
|
|
>>> yi = wi.StinemanInterp(x,y)(xi)
|
|
|
|
|
|
|
|
>>> yi1 = wi.CubicHermiteSpline(x,y, yp)(xi)
|
|
|
|
|
|
|
|
>>> yi2 = wi.Pchip(x,y, method='parabola')(xi)
|
|
|
|
|
|
|
|
>>> plt.subplot(211)
|
|
|
|
|
|
|
|
>>> plt.plot(x,y,'o',xi,yi,'r', xi,yi1, 'g', xi,yi1, 'b')
|
|
|
|
|
|
|
|
>>> plt.subplot(212)
|
|
|
|
|
|
|
|
>>> plt.plot(xi,np.abs(sin(xi)-yi), 'r', xi, np.abs(sin(xi)-yi1), 'g', xi, np.abs(sin(xi)-yi2), 'b')
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
References
|
|
|
|
|
|
|
|
----------
|
|
|
|
|
|
|
|
Stineman, R. W. A Consistently Well Behaved Method of Interpolation. Creative Computing (1980), volume 6, number 7, p. 54-57.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
See Also
|
|
|
|
|
|
|
|
--------
|
|
|
|
|
|
|
|
slopes, Pchip
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
def __init__(self, x,y,yp=None,method='parabola'):
|
|
|
|
|
|
|
|
if yp is None:
|
|
|
|
|
|
|
|
yp = slopes2(x, y, method)
|
|
|
|
|
|
|
|
self.x = np.asarray(x, np.float_)
|
|
|
|
|
|
|
|
self.y = np.asarray(y, np.float_)
|
|
|
|
|
|
|
|
self.yp = np.asarray(yp, np.float_)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def __call__(self, xi):
|
|
|
|
|
|
|
|
xi = np.asarray(xi, np.float_)
|
|
|
|
|
|
|
|
x = self.x
|
|
|
|
|
|
|
|
y = self.y
|
|
|
|
|
|
|
|
yp = self.yp
|
|
|
|
|
|
|
|
# calculate linear slopes
|
|
|
|
|
|
|
|
dx = x[1:] - x[:-1]
|
|
|
|
|
|
|
|
dy = y[1:] - y[:-1]
|
|
|
|
|
|
|
|
s = dy / dx #note length of s is N-1 so last element is #N-2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# find the segment each xi is in
|
|
|
|
|
|
|
|
# this line actually is the key to the efficiency of this implementation
|
|
|
|
|
|
|
|
idx = np.searchsorted(x[1:-1], xi)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# now we have generally: x[idx[j]] <= xi[j] <= x[idx[j]+1]
|
|
|
|
|
|
|
|
# except at the boundaries, where it may be that xi[j] < x[0] or xi[j] > x[-1]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# the y-values that would come out from a linear interpolation:
|
|
|
|
|
|
|
|
sidx = s.take(idx)
|
|
|
|
|
|
|
|
xidx = x.take(idx)
|
|
|
|
|
|
|
|
yidx = y.take(idx)
|
|
|
|
|
|
|
|
xidxp1 = x.take(idx + 1)
|
|
|
|
|
|
|
|
yo = yidx + sidx * (xi - xidx)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# the difference that comes when using the slopes given in yp
|
|
|
|
|
|
|
|
dy1 = (yp.take(idx) - sidx) * (xi - xidx) # using the yp slope of the left point
|
|
|
|
|
|
|
|
dy2 = (yp.take(idx + 1) - sidx) * (xi - xidxp1) # using the yp slope of the right point
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
dy1dy2 = dy1 * dy2
|
|
|
|
|
|
|
|
# The following is optimized for Python. The solution actually
|
|
|
|
|
|
|
|
# does more calculations than necessary but exploiting the power
|
|
|
|
|
|
|
|
# of numpy, this is far more efficient than coding a loop by hand
|
|
|
|
|
|
|
|
# in Python
|
|
|
|
|
|
|
|
dy1mdy2 = np.where(dy1dy2,dy1-dy2,np.inf)
|
|
|
|
|
|
|
|
dy1pdy2 = np.where(dy1dy2,dy1+dy2,np.inf)
|
|
|
|
|
|
|
|
yi = yo + dy1dy2 * np.choose(np.array(np.sign(dy1dy2), np.int32) + 1,
|
|
|
|
|
|
|
|
((2 * xi - xidx - xidxp1) / ((dy1mdy2) * (xidxp1 - xidx)),
|
|
|
|
|
|
|
|
0.0,
|
|
|
|
|
|
|
|
1 / (dy1pdy2)))
|
|
|
|
|
|
|
|
return yi
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def stineman_interp(xi, x, y, yp=None):
|
|
|
|
def stineman_interp(xi, x, y, yp=None):
|
|
|
|
"""
|
|
|
|
"""
|
|
|
|
Given data vectors *x* and *y*, the slope vector *yp* and a new
|
|
|
|
Given data vectors *x* and *y*, the slope vector *yp* and a new
|
|
|
|
@ -703,6 +933,80 @@ def stineman_interp(xi, x, y, yp=None):
|
|
|
|
1 / (dy1pdy2)))
|
|
|
|
1 / (dy1pdy2)))
|
|
|
|
return yi
|
|
|
|
return yi
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class CubicHermiteSpline(PiecewisePolynomial):
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
Piecewise Cubic Hermite Interpolation using Catmull-Rom
|
|
|
|
|
|
|
|
method for computing the slopes.
|
|
|
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
def __init__(self, x, y, yp=None, method='Catmull-Rom'):
|
|
|
|
|
|
|
|
if yp is None:
|
|
|
|
|
|
|
|
yp = slopes2(x, y, method, monotone=False)
|
|
|
|
|
|
|
|
super(CubicHermiteSpline, self).__init__(x, zip(y,yp), orders=3)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class Pchip(PiecewisePolynomial):
|
|
|
|
|
|
|
|
"""PCHIP 1-d monotonic cubic interpolation
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Description
|
|
|
|
|
|
|
|
-----------
|
|
|
|
|
|
|
|
x and y are arrays of values used to approximate some function f:
|
|
|
|
|
|
|
|
y = f(x)
|
|
|
|
|
|
|
|
This class factory function returns a callable class whose __call__ method
|
|
|
|
|
|
|
|
uses monotonic cubic, interpolation to find the value of new points.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
|
|
----------
|
|
|
|
|
|
|
|
x : array
|
|
|
|
|
|
|
|
A 1D array of monotonically increasing real values. x cannot
|
|
|
|
|
|
|
|
include duplicate values (otherwise f is overspecified)
|
|
|
|
|
|
|
|
y : array
|
|
|
|
|
|
|
|
A 1-D array of real values. y's length along the interpolation
|
|
|
|
|
|
|
|
axis must be equal to the length of x.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Assumes x is sorted in monotonic order (e.g. x[1] > x[0])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Example
|
|
|
|
|
|
|
|
-------
|
|
|
|
|
|
|
|
>>> import wafo.interpolate as wi
|
|
|
|
|
|
|
|
# Create a step function (will demonstrate monotonicity)
|
|
|
|
|
|
|
|
>>> x = np.arange(7.0) - 3.0
|
|
|
|
|
|
|
|
>>> y = np.array([-1.0, -1,-1,0,1,1,1])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# Interpolate using monotonic piecewise Hermite cubic spline
|
|
|
|
|
|
|
|
>>> xvec = np.arange(599.)/100. - 3.0
|
|
|
|
|
|
|
|
>>> yvec = wi.Pchip(x, y)(xvec)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# Call the Scipy cubic spline interpolator
|
|
|
|
|
|
|
|
>>> from scipy.interpolate import interpolate
|
|
|
|
|
|
|
|
>>> function = interpolate.interp1d(x, y, kind='cubic')
|
|
|
|
|
|
|
|
>>> yvec1 = function(xvec)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# Non-montonic cubic Hermite spline interpolator using
|
|
|
|
|
|
|
|
# Catmul-Rom method for computing slopes...
|
|
|
|
|
|
|
|
>>> yvec2 = wi.CubicHermiteSpline(x,y)(xvec)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
>>> yvec3 = wi.StinemanInterp(x, y)(xvec)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
# Plot the results
|
|
|
|
|
|
|
|
>>> plt.plot(x, y, 'ro')
|
|
|
|
|
|
|
|
>>> plt.plot(xvec, yvec, 'b')
|
|
|
|
|
|
|
|
>>> plt.plot(xvec, yvec1, 'k')
|
|
|
|
|
|
|
|
>>> plt.plot(xvec, yvec2, 'g')
|
|
|
|
|
|
|
|
>>> plt.plot(xvec, yvec3, 'm')
|
|
|
|
|
|
|
|
>>> plt.title("pchip() step function test")
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
>>> plt.xlabel("X")
|
|
|
|
|
|
|
|
>>> plt.ylabel("Y")
|
|
|
|
|
|
|
|
>>> plt.title("Comparing pypchip() vs. Scipy interp1d() vs. non-monotonic CHS")
|
|
|
|
|
|
|
|
>>> legends = ["Data", "pypchip()", "interp1d","CHS", 'SI']
|
|
|
|
|
|
|
|
>>> plt.legend(legends, loc="upper left")
|
|
|
|
|
|
|
|
>>> plt.show()
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
"""
|
|
|
|
|
|
|
|
def __init__(self, x, y, yp=None, method='secant'):
|
|
|
|
|
|
|
|
if yp is None:
|
|
|
|
|
|
|
|
yp = slopes2(x, y, method=method, monotone=True)
|
|
|
|
|
|
|
|
super(Pchip, self).__init__(x, zip(y,yp), orders=3)
|
|
|
|
|
|
|
|
|
|
|
|
def test_smoothing_spline():
|
|
|
|
def test_smoothing_spline():
|
|
|
|
x = linspace(0, 2 * pi + pi / 4, 20)
|
|
|
|
x = linspace(0, 2 * pi + pi / 4, 20)
|
|
|
|
y = sin(x) #+ np.random.randn(x.size)
|
|
|
|
y = sin(x) #+ np.random.randn(x.size)
|
|
|
|
|
per.andreas.brodtkorb
13 years ago