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pywafo/wafo/interpolate.py

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#!/usr/bin/env python
from __future__ import absolute_import, division
import numpy as np
import scipy.signal
# import scipy.sparse.linalg # @UnusedImport
import scipy.sparse as sparse
from numpy import ones, zeros, prod, sin, diff, pi, inf, vstack, linspace
from scipy.interpolate import BPoly, interp1d
from wafo import polynomial as pl
__all__ = [
'PPform', 'savitzky_golay', 'savitzky_golay_piecewise', 'sgolay2d',
'SmoothSpline', 'pchip_slopes', 'slopes', 'stineman_interp', 'Pchip',
'StinemanInterp', 'CubicHermiteSpline']
def savitzky_golay(y, window_size, order, deriv=0):
"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter.
The Savitzky-Golay filter removes high frequency noise from data.
It has the advantage of preserving the original shape and
features of the signal better than other types of filtering
approaches, such as moving averages techhniques.
Parameters
----------
y : array_like, shape (N,)
the values of the time history of the signal.
window_size : int
the length of the window. Must be an odd integer number.
order : int
the order of the polynomial used in the filtering.
Must be less then `window_size` - 1.
deriv: int
order of the derivative to compute (default = 0 means only smoothing)
Returns
-------
ys : ndarray, shape (N)
the smoothed signal (or it's n-th derivative).
Notes
-----
The Savitzky-Golay is a type of low-pass filter, particularly
suited for smoothing noisy data. The test_doctstrings idea behind this
approach is to make for each point a least-square fit with a
polynomial of high order over a odd-sized window centered at
the point.
Examples
--------
>>> t = np.linspace(-4, 4, 500)
>>> noise = np.random.normal(0, 0.05, t.shape)
>>> noise = 0.4*np.sin(100*t)
>>> y = np.exp( -t**2 ) + noise
>>> ysg = savitzky_golay(y, window_size=31, order=4)
>>> np.allclose(ysg[:10],
... [-0.00127789, -0.02390299, -0.04444364, -0.01738837, 0.00585856,
... -0.01675704, -0.03140276, 0.00010455, 0.02099063, -0.00380031])
True
import matplotlib.pyplot as plt
h=plt.plot(t, y, label='Noisy signal')
h=plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
h=plt.plot(t, ysg, 'r', label='Filtered signal')
h=plt.legend()
plt.show()
References
----------
.. [1] A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of
Data by Simplified Least Squares Procedures. Analytical
Chemistry, 1964, 36 (8), pp 1627-1639.
.. [2] Numerical Recipes 3rd Edition: The Art of Scientific Computing
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery
Cambridge University Press ISBN-13: 9780521880688
"""
try:
window_size = np.abs(np.int(window_size))
order = np.abs(np.int(order))
except ValueError:
raise ValueError("window_size and order have to be of type int")
if window_size % 2 != 1 or window_size < 1:
raise TypeError("window_size size must be a positive odd number")
if window_size < order + 2:
raise TypeError("window_size is too small for the polynomials order")
order_range = range(order + 1)
half_window = (window_size - 1) // 2
# precompute coefficients
b = np.mat([[k ** i for i in order_range]
for k in range(-half_window, half_window + 1)])
m = np.linalg.pinv(b).A[deriv]
# pad the signal at the extremes with
# values taken from the signal itself
firstvals = y[0] - np.abs(y[1:half_window + 1][::-1] - y[0])
lastvals = y[-1] + np.abs(y[-half_window - 1:-1][::-1] - y[-1])
y = np.concatenate((firstvals, y, lastvals))
return np.convolve(m, y, mode='valid')
def savitzky_golay_piecewise(xvals, data, kernel=11, order=4):
'''
One of the most popular applications of S-G filter, apart from smoothing
UV-VIS and IR spectra, is smoothing of curves obtained in electroanalytical
experiments. In cyclic voltammetry, voltage (being the abcissa) changes
like a triangle wave. And in the signal there are cusps at the turning
points (at switching potentials) which should never be smoothed.
In this case, Savitzky-Golay smoothing should be
done piecewise, ie. separately on pieces monotonic in x
Example
-------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> n = 1e3
>>> x = np.linspace(0, 25, n)
>>> y = np.round(sin(x))
>>> sig2 = linspace(0,0.5,50)
# As an example, this figure shows the effect of an additive noise with a
# variance of 0.2 (original signal (black), noisy signal (red) and filtered
# signal (blue dots)).
>>> noise = np.sqrt(0.2)*np.random.randn(*x.shape)
>>> noise = np.sqrt(0.2)*np.sin(1000*x)
>>> yn = y + noise
>>> yr = savitzky_golay_piecewise(x, yn, kernel=11, order=4)
>>> np.allclose(yr[:10],
... [-0.02708216, -0.04295155, -0.08522043, -0.13995016, -0.1908162 ,
... -0.22938387, -0.26932722, -0.30614865, -0.33942134, -0.3687596 ])
True
h=plt.plot(x, yn, 'r', x, y, 'k', x, yr, 'b.')
'''
turnpoint = 0
last = len(xvals)
if xvals[1] > xvals[0]: # x is increasing?
for i in range(1, last): # yes
if xvals[i] < xvals[i - 1]: # search where x starts to fall
turnpoint = i
break
else: # no, x is decreasing
for i in range(1, last): # search where it starts to rise
if xvals[i] > xvals[i - 1]:
turnpoint = i
break
if turnpoint == 0: # no change in direction of x
return savitzky_golay(data, kernel, order)
else:
# smooth the first piece
firstpart = savitzky_golay(data[0:turnpoint], kernel, order)
# recursively smooth the rest
rest = savitzky_golay_piecewise(
xvals[turnpoint:], data[turnpoint:], kernel, order)
return np.concatenate((firstpart, rest))
def sgolay2d(z, window_size, order, derivative=None):
"""
Savitsky - Golay filters can also be used to smooth two dimensional data
affected by noise. The algorithm is exactly the same as for the one
dimensional case, only the math is a bit more tricky. The basic algorithm
is as follow: for each point of the two dimensional matrix extract a sub
- matrix, centered at that point and with a size equal to an odd number
"window_size". for this sub - matrix compute a least - square fit of a
polynomial surface, defined as
p(x, y) = a0 + a1 * x + a2 * y + a3 * x2 + a4 * y2 + a5 * x * y + ... .
Note that x and y are equal to zero at the central point.
replace the initial central point with the value computed with the fit.
Note that because the fit coefficients are linear with respect to the data
spacing, they can pre - computed for efficiency. Moreover, it is important
to appropriately pad the borders of the data, with a mirror image of the
data itself, so that the evaluation of the fit at the borders of the data
can happen smoothly.
Here is the code for two dimensional filtering.
Example
-------
# create some sample twoD data
>>> x = np.linspace(-3,3,100)
>>> y = np.linspace(-3,3,100)
>>> X, Y = np.meshgrid(x,y)
>>> Z = np.exp( -(X**2+Y**2))
# add noise
>>> noise = np.random.normal( 0, 0.2, Z.shape )
>>> noise = np.sqrt(0.2) * np.sin(100*X)*np.sin(100*Y)
>>> Zn = Z + noise
# filter it
>>> Zf = sgolay2d( Zn, window_size=29, order=4)
>>> np.allclose(Zf[:3,:5],
... [[ 0.29304073, 0.29749652, 0.29007645, 0.2695685 , 0.23541966],
... [ 0.29749652, 0.29819304, 0.28766723, 0.26524542, 0.23081572],
... [ 0.29007645, 0.28766723, 0.27483445, 0.25141198, 0.21769662]])
True
# do some plotting
import matplotlib.pyplot as plt
h=plt.matshow(Z)
h=plt.matshow(Zn)
h=plt.matshow(Zf)
"""
# number of terms in the polynomial expression
n_terms = (order + 1) * (order + 2) / 2.0
if window_size % 2 == 0:
raise ValueError('window_size must be odd')
if window_size ** 2 < n_terms:
raise ValueError('order is too high for the window size')
half_size = window_size // 2
# exponents of the polynomial.
# p(x,y) = a0 + a1*x + a2*y + a3*x^2 + a4*y^2 + a5*x*y + ...
# this line gives a list of two item tuple. Each tuple contains
# the exponents of the k-th term. First element of tuple is for x
# second element for y.
# Ex. exps = [(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...]
exps = [(k - n, n) for k in range(order + 1) for n in range(k + 1)]
# coordinates of points
ind = np.arange(-half_size, half_size + 1, dtype=np.float)
dx = np.repeat(ind, window_size)
dy = np.tile(ind, [window_size, 1]).reshape(window_size ** 2,)
# build matrix of system of equation
A = np.empty((window_size ** 2, len(exps)))
for i, exp in enumerate(exps):
A[:, i] = (dx ** exp[0]) * (dy ** exp[1])
# pad input array with appropriate values at the four borders
new_shape = z.shape[0] + 2 * half_size, z.shape[1] + 2 * half_size
Z = np.zeros((new_shape))
# top band
band = z[0, :]
Z[:half_size, half_size:-half_size] = band - \
np.abs(np.flipud(z[1:half_size + 1, :]) - band)
# bottom band
band = z[-1, :]
Z[-half_size:, half_size:-half_size] = band + \
np.abs(np.flipud(z[-half_size - 1:-1, :]) - band)
# left band
band = np.tile(z[:, 0].reshape(-1, 1), [1, half_size])
Z[half_size:-half_size, :half_size] = band - \
np.abs(np.fliplr(z[:, 1:half_size + 1]) - band)
# right band
band = np.tile(z[:, -1].reshape(-1, 1), [1, half_size])
Z[half_size:-half_size, -half_size:] = band + \
np.abs(np.fliplr(z[:, -half_size - 1:-1]) - band)
# central band
Z[half_size:-half_size, half_size:-half_size] = z
# top left corner
band = z[0, 0]
Z[:half_size, :half_size] = band - \
np.abs(
np.flipud(np.fliplr(z[1:half_size + 1, 1:half_size + 1])) - band)
# bottom right corner
band = z[-1, -1]
Z[-half_size:, -half_size:] = band + \
np.abs(np.flipud(np.fliplr(z[-half_size - 1:-1, -half_size - 1:-1])) -
band)
# top right corner
band = Z[half_size, -half_size:]
Z[:half_size, -half_size:] = band - \
np.abs(
np.flipud(Z[half_size + 1:2 * half_size + 1, -half_size:]) - band)
# bottom left corner
band = Z[-half_size:, half_size].reshape(-1, 1)
Z[-half_size:, :half_size] = band - \
np.abs(
np.fliplr(Z[-half_size:, half_size + 1:2 * half_size + 1]) - band)
# solve system and convolve
if derivative is None:
m = np.linalg.pinv(A)[0].reshape((window_size, -1))
return scipy.signal.fftconvolve(Z, m, mode='valid')
elif derivative == 'col':
c = np.linalg.pinv(A)[1].reshape((window_size, -1))
return scipy.signal.fftconvolve(Z, -c, mode='valid')
elif derivative == 'row':
r = np.linalg.pinv(A)[2].reshape((window_size, -1))
return scipy.signal.fftconvolve(Z, -r, mode='valid')
elif derivative == 'both':
c = np.linalg.pinv(A)[1].reshape((window_size, -1))
r = np.linalg.pinv(A)[2].reshape((window_size, -1))
return (scipy.signal.fftconvolve(Z, -r, mode='valid'),
scipy.signal.fftconvolve(Z, -c, mode='valid'))
class PPform(object):
"""The ppform of the piecewise polynomials
is given in terms of coefficients and breaks.
The polynomial in the ith interval is
x_{i} <= x < x_{i+1}
S_i = sum(coefs[m,i]*(x-breaks[i])^(k-m), m=0..k)
where k is the degree of the polynomial.
Example
-------
>>> import matplotlib.pyplot as plt
>>> coef = np.array([[1,1]]) # unit step function
>>> coef = np.array([[1,1],[0,1]]) # linear from 0 to 2
>>> coef = np.array([[1,1],[1,1],[0,2]]) # linear from 0 to 2
>>> breaks = [0,1,2]
>>> self = PPform(coef, breaks)
>>> x = linspace(-1, 3, 21)
>>> y = self(x)
>>> np.allclose(y, [ 0. , 0. , 0. , 0. , 0. , 0. , 0.24, 0.56,
... 0.96, 1.44, 2. , 2.24, 2.56, 2.96, 3.44, 4. , 0. , 0. ,
... 0. , 0. , 0. ])
True
h=plt.plot(x, y)
"""
def __init__(self, coeffs, breaks, fill=0.0, sort=False, a=None, b=None):
if sort:
self.breaks = np.sort(breaks)
else:
self.breaks = np.asarray(breaks)
if a is None:
a = self.breaks[0]
if b is None:
b = self.breaks[-1]
self.coeffs = np.asarray(coeffs)
self.order = self.coeffs.shape[0]
self.fill = fill
self.a = a
self.b = b
def __call__(self, xnew):
saveshape = np.shape(xnew)
xnew = np.ravel(xnew)
res = np.empty_like(xnew)
mask = (self.a <= xnew) & (xnew <= self.b)
res[~mask] = self.fill
xx = xnew.compress(mask)
indxs = np.searchsorted(self.breaks[:-1], xx) - 1
indxs = indxs.clip(0, len(self.breaks))
pp = self.coeffs
dx = xx - self.breaks.take(indxs)
v = pp[0, indxs]
for i in range(1, self.order):
v = dx * v + pp[i, indxs]
values = v
res[mask] = values
res.shape = saveshape
return res
def linear_extrapolate(self, output=True):
'''
Return 1D PPform which extrapolate linearly outside its basic interval
'''
max_order = 2
if self.order <= max_order:
if output:
return self
else:
return
breaks = self.breaks.copy()
coefs = self.coeffs.copy()
# pieces = len(breaks) - 1
# Add new breaks beyond each end
breaks2add = breaks[[0, -1]] + np.array([-1, 1])
newbreaks = np.hstack([breaks2add[0], breaks, breaks2add[1]])
dx = newbreaks[[0, -2]] - breaks[[0, -2]]
dx = dx.ravel()
# Get coefficients for the new last polynomial piece (a_n)
# by just relocate the previous last polynomial and
# then set all terms of order > maxOrder to zero
a_nn = coefs[:, -1]
dxN = dx[-1]
a_n = pl.polyreloc(a_nn, -dxN) # Relocate last polynomial
# set to zero all terms of order > maxOrder
a_n[0:self.order - max_order] = 0
# Get the coefficients for the new first piece (a_1)
# by first setting all terms of order > maxOrder to zero and then
# relocate the polynomial.
# Set to zero all terms of order > maxOrder, i.e., not using them
a_11 = coefs[self.order - max_order::, 0]
dx1 = dx[0]
a_1 = pl.polyreloc(a_11, -dx1) # Relocate first polynomial
a_1 = np.hstack([zeros(self.order - max_order), a_1])
newcoefs = np.hstack([a_1.reshape(-1, 1), coefs, a_n.reshape(-1, 1)])
if output:
return PPform(newcoefs, newbreaks, a=-inf, b=inf)
else:
self.coeffs = newcoefs
self.breaks = newbreaks
self.a = -inf
self.b = inf
def derivative(self):
"""
Return first derivative of the piecewise polynomial
"""
cof = pl.polyder(self.coeffs)
brks = self.breaks.copy()
return PPform(cof, brks, fill=self.fill)
def integrate(self):
"""
Return the indefinite integral of the piecewise polynomial
"""
cof = pl.polyint(self.coeffs)
pieces = len(self.breaks) - 1
if 1 < pieces:
# evaluate each integrated polynomial at the right endpoint of its
# interval
xs = diff(self.breaks[:-1, ...], axis=0)
index = np.arange(pieces - 1)
vv = xs * cof[0, index]
k = self.order
for i in range(1, k):
vv = xs * (vv + cof[i, index])
cof[-1] = np.hstack((0, vv)).cumsum()
return PPform(cof, self.breaks, fill=self.fill)
# def fromspline(self, xk, cvals, order, fill=0.0):
# N = len(xk) - 1
# sivals = np.empty((order + 1, N), dtype=float)
# for m in range(order, -1, -1):
# fact = spec.gamma(m + 1)
# res = _fitpack._bspleval(xk[:-1], xk, cvals, order, m)
# res /= fact
# sivals[order - m, :] = res
# return self(sivals, xk, fill=fill)
class SmoothSpline(PPform):
"""
Cubic Smoothing Spline.
Parameters
----------
x : array-like
x-coordinates of data. (vector)
y : array-like
y-coordinates of data. (vector or matrix)
p : real scalar
smoothing parameter between 0 and 1:
0 -> LS-straight line
1 -> cubic spline interpolant
lin_extrap : bool
if False regular smoothing spline
if True a smoothing spline with a constraint on the ends to
ensure linear extrapolation outside the range of the data (default)
var : array-like
variance of each y(i) (default 1)
Returns
-------
pp : ppform
If xx is not given, return self-form of the spline.
Given the approximate values
y(i) = g(x(i))+e(i)
of some smooth function, g, where e(i) is the error. SMOOTH tries to
recover g from y by constructing a function, f, which minimizes
p * sum (Y(i) - f(X(i)))^2/d2(i) + (1-p) * int (f'')^2
Example
-------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 1, 21)
>>> noise = 1e-1*np.random.randn(x.size)
>>> noise = np.array(
... [-0.03298601, -0.08164429, -0.06845745, -0.20718593, 0.08666282,
... 0.04702094, 0.08208645, -0.1017021 , -0.03031708, 0.22871709,
... -0.10302486, -0.17724316, -0.05885157, -0.03875947, -0.1102984 ,
... -0.05542001, -0.12717549, 0.14337697, -0.02637848, -0.10353976,
... -0.0618834 ])
>>> y = np.exp(x) + noise
>>> pp9 = SmoothSpline(x, y, p=.9)
>>> pp99 = SmoothSpline(x, y, p=.99, var=0.01)
>>> y99 = pp99(x); y9 = pp9(x)
>>> np.allclose(y9,
... [ 0.8754795 , 0.95285289, 1.03033239, 1.10803792, 1.18606854,
... 1.26443234, 1.34321265, 1.42258227, 1.5027733 , 1.58394785,
... 1.66625727, 1.74998243, 1.8353173 , 1.92227431, 2.01076693,
... 2.10064087, 2.19164551, 2.28346334, 2.37573696, 2.46825194,
... 2.56087699])
True
>>> np.allclose(y99,
... [ 0.95227461, 0.97317995, 1.01159244, 1.08726908, 1.21260587,
... 1.31545644, 1.37829108, 1.42719649, 1.51308685, 1.59669367,
... 1.61486217, 1.64481078, 1.72970022, 1.83208819, 1.93312796,
... 2.05164767, 2.19326122, 2.34608425, 2.45023567, 2.5357288 ,
... 2.6357401 ])
True
h=plt.plot(x,y, x,pp99(x),'g', x,pp9(x),'k', x,np.exp(x),'r')
See also
--------
lc2tr, dat2tr
References
----------
Carl de Boor (1978)
'Practical Guide to Splines'
Springer Verlag
Uses EqXIV.6--9, self 239
"""
def __init__(self, xx, yy, p=None, lin_extrap=True, var=1):
coefs, brks = self._compute_coefs(xx, yy, p, var)
super(SmoothSpline, self).__init__(coefs, brks)
if lin_extrap:
self.linear_extrapolate(output=False)
def _compute_coefs(self, xx, yy, p=None, var=1):
x, y = np.atleast_1d(xx, yy)
x = x.ravel()
dx = np.diff(x)
must_sort = (dx < 0).any()
if must_sort:
ind = x.argsort()
x = x[ind]
y = y[..., ind]
dx = np.diff(x)
n = len(x)
# ndy = y.ndim
szy = y.shape
nd = prod(szy[:-1])
ny = szy[-1]
if n < 2:
raise ValueError('There must be >=2 data points.')
elif (dx <= 0).any():
raise ValueError('Two consecutive values in x can not be equal.')
elif n != ny:
raise ValueError('x and y must have the same length.')
dydx = np.diff(y) / dx
if (n == 2): # % straight line
coefs = np.vstack([dydx.ravel(), y[0, :]])
else:
dx1 = 1. / dx
D = sparse.spdiags(var * ones(n), 0, n, n) # The variance
u, p = self._compute_u(p, D, dydx, dx, dx1, n)
dx1.shape = (n - 1, -1)
dx.shape = (n - 1, -1)
zrs = zeros(nd)
if p < 1:
# faster than yi-6*(1-p)*Q*u
Qu = D * diff(vstack([zrs, diff(vstack([zrs, u, zrs]),
axis=0) * dx1, zrs]), axis=0)
ai = (y - (6 * (1 - p) * Qu).T).T
else:
ai = y.reshape(n, -1)
# The piecewise polynominals are written as
# fi=ai+bi*(x-xi)+ci*(x-xi)^2+di*(x-xi)^3
# where the derivatives in the knots according to Carl de Boor are:
# ddfi = 6*p*[0;u] = 2*ci;
# dddfi = 2*diff([ci;0])./dx = 6*di;
# dfi = diff(ai)./dx-(ci+di.*dx).*dx = bi;
ci = np.vstack([zrs, 3 * p * u])
di = (diff(vstack([ci, zrs]), axis=0) * dx1 / 3)
bi = (diff(ai, axis=0) * dx1 - (ci + di * dx) * dx)
ai = ai[:n - 1, ...]
if nd > 1:
di = di.T
ci = ci.T
ai = ai.T
if not any(di):
if not any(ci):
coefs = vstack([bi.ravel(), ai.ravel()])
else:
coefs = vstack([ci.ravel(), bi.ravel(), ai.ravel()])
else:
coefs = vstack(
[di.ravel(), ci.ravel(), bi.ravel(), ai.ravel()])
return coefs, x
def _compute_u(self, p, D, dydx, dx, dx1, n):
if p is None or p != 0:
data = [dx[1:n - 1], 2 * (dx[:n - 2] + dx[1:n - 1]), dx[:n - 2]]
R = sparse.spdiags(data, [-1, 0, 1], n - 2, n - 2)
if p is None or p < 1:
Q = sparse.spdiags(
[dx1[:n - 2], -(dx1[:n - 2] + dx1[1:n - 1]), dx1[1:n - 1]],
[0, -1, -2], n, n - 2)
QDQ = (Q.T * D * Q)
if p is None or p < 0:
# Estimate p
p = 1. / \
(1. + QDQ.diagonal().sum() /
(100. * R.diagonal().sum() ** 2))
if p == 0:
QQ = 6 * QDQ
else:
QQ = (6 * (1 - p)) * (QDQ) + p * R
else:
QQ = R
# Make sure it uses symmetric matrix solver
ddydx = diff(dydx, axis=0)
# sp.linalg.use_solver(useUmfpack=True)
u = 2 * sparse.linalg.spsolve((QQ + QQ.T), ddydx) # @UndefinedVariable
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 slopes(x, y, method='parabola', tension=0, monotone=False):
'''
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('p'): # 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('s'): # 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('car'): # 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 stineman_interp(xi, x, y, yp=None):
"""
Given data vectors *x* and *y*, the slope vector *yp* and a new
abscissa vector *xi*, the function :func:`stineman_interp` uses
Stineman interpolation to calculate a vector *yi* corresponding to
*xi*.
Here's an example that generates a coarse sine curve, then
interpolates over a finer abscissa::
x = linspace(0,2*pi,20); y = sin(x); yp = cos(x)
xi = linspace(0,2*pi,40);
yi = stineman_interp(xi,x,y,yp);
plot(x,y,'o',xi,yi)
The interpolation method is described in the article A
CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russell
W. Stineman. The article appeared in the July 1980 issue of
Creative Computing with a note from the editor stating that while
they were:
not an academic journal but once in a while something serious
and original comes in adding that this was
"apparently a real solution" to a well known problem.
For *yp* = *None*, the routine automatically determines the slopes
using the :func:`slopes` routine.
*x* is assumed to be sorted in increasing order.
For values ``xi[j] < x[0]`` or ``xi[j] > x[-1]``, the routine
tries an extrapolation. The relevance of the data obtained from
this, of course, is questionable...
Original implementation by Halldor Bjornsson, Icelandic
Meteorolocial Office, March 2006 halldor at vedur.is
Completely reworked and optimized for Python by Norbert Nemec,
Institute of Theoretical Physics, University or Regensburg, April
2006 Norbert.Nemec at physik.uni-regensburg.de
"""
# Cast key variables as float.
x = np.asarray(x, np.float_)
y = np.asarray(y, np.float_)
assert x.shape == y.shape
# N = len(y)
if yp is None:
yp = slopes(x, y)
else:
yp = np.asarray(yp, np.float_)
xi = np.asarray(xi, np.float_)
# yi = np.zeros(xi.shape, np.float_)
# 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
# using the yp slope of the left point
dy1 = (yp.take(idx) - sidx) * (xi - xidx)
# using the yp slope of the right point
dy2 = (yp.take(idx + 1) - sidx) * (xi - xidxp1)
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
class StinemanInterp(object):
'''
Returns 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 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
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0,2*pi,20)
>>> y = np.sin(x); yp = np.cos(x)
>>> xi = np.linspace(0,2*pi,40);
>>> yi = wi.StinemanInterp(x,y)(xi)
>>> np.allclose(yi[:10],
... [ 0., 0.16258231, 0.31681338, 0.46390886, 0.60091421,
... 0.7206556 , 0.82314953, 0.90304148, 0.96059538, 0.99241945])
True
>>> yi1 = wi.CubicHermiteSpline(x,y, yp)(xi)
>>> yi2 = wi.Pchip(x,y, method='parabola')(xi)
h=plt.subplot(211)
h=plt.plot(x,y,'o',xi,yi,'r', xi,yi1, 'g', xi,yi1, 'b')
h=plt.subplot(212)
h=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', monotone=False):
if yp is None:
yp = slopes(x, y, method, monotone=monotone)
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
# using the yp slope of the left point
dy1 = (yp.take(idx) - sidx) * (xi - xidx)
# using the yp slope of the right point
dy2 = (yp.take(idx + 1) - sidx) * (xi - xidxp1)
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
class StinemanInterp2(BPoly):
def __init__(self, x, y, yp=None, method='parabola', monotone=False):
if yp is None:
yp = slopes(x, y, method, monotone=monotone)
yyp = [z for z in zip(y, yp)]
bpoly = BPoly.from_derivatives(x, yyp)
super(StinemanInterp2, self).__init__(bpoly.c, x)
class CubicHermiteSpline(BPoly):
'''
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 = slopes(x, y, method, monotone=False)
yyp = [z for z in zip(y, yp)]
bpoly = BPoly.from_derivatives(x, yyp, orders=3)
super(CubicHermiteSpline, self).__init__(bpoly.c, x)
# super(CubicHermiteSpline, self).__init__(x, yyp, orders=3)
class Pchip(BPoly):
"""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.
yp : array
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.
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
>>> n = 20.
>>> xvec = np.arange(n)/10. - 1.0
>>> yvec = wi.Pchip(x, y)(xvec)
>>> np.allclose(yvec, [-1. , -0.981, -0.928, -0.847, -0.744, -0.625,
... -0.496, -0.363, -0.232, -0.109, 0. , 0.109, 0.232, 0.363,
... 0.496, 0.625, 0.744, 0.847, 0.928, 0.981])
True
# Call the Scipy cubic spline interpolator
>>> from scipy.interpolate import interpolate
>>> function = interpolate.interp1d(x, y, kind='cubic')
>>> yvec1 = function(xvec)
>>> np.allclose(yvec1, [-1.00000000e+00, -9.41911765e-01, -8.70588235e-01,
... -7.87500000e-01, -6.94117647e-01, -5.91911765e-01,
... -4.82352941e-01, -3.66911765e-01, -2.47058824e-01,
... -1.24264706e-01, 2.49800181e-16, 1.24264706e-01,
... 2.47058824e-01, 3.66911765e-01, 4.82352941e-01,
... 5.91911765e-01, 6.94117647e-01, 7.87500000e-01,
... 8.70588235e-01, 9.41911765e-01])
True
# 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)
>>> np.allclose(yvec2, [-1., -0.9405, -0.864 , -0.7735, -0.672 , -0.5625,
... -0.448 , -0.3315, -0.216 , -0.1045, 0. , 0.1045, 0.216 ,
... 0.3315, 0.448 , 0.5625, 0.672 , 0.7735, 0.864 , 0.9405])
True
>>> np.allclose(yvec3, [-1. , -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3,
... -0.2, -0.1, 0. , 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9])
True
# Plot the results
import matplotlib.pyplot as plt
h=plt.plot(x, y, 'ro')
h=plt.plot(xvec, yvec, 'b')
h=plt.plot(xvec, yvec1, 'k')
h=plt.plot(xvec, yvec2, 'g')
h=plt.plot(xvec, yvec3, 'm')
h=plt.title("pchip() step function test")
h=plt.xlabel("X")
h=plt.ylabel("Y")
txt = "Comparing pypchip() vs. Scipy interp1d() vs. non-monotonic CHS"
h=plt.title(txt)
legends = ["Data", "pypchip()", "interp1d","CHS", 'SI']
h=plt.legend(legends, loc="upper left")
plt.show()
"""
def __init__(self, x, y, yp=None, method='secant'):
if yp is None:
yp = slopes(x, y, method=method, monotone=True)
yyp = [z for z in zip(y, yp)]
bpoly = BPoly.from_derivatives(x, yyp, orders=3)
super(Pchip, self).__init__(bpoly.c, x)
# super(Pchip, self).__init__(x, yyp, orders=3)
def interp3(x, y, z, v, xi, yi, zi, method='cubic'):
"""Interpolation on 3-D. x, y, xi, yi should be 1-D
and z.shape == (len(x), len(y), len(z))"""
q = (x, y, z)
qi = (xi, yi, zi)
for j in range(3):
pp = interp1d(q[j], v, axis=j, kind=method)
v = pp(qi[j])
return v
def somefunc(x, y, z):
return x**2 + y**2 - z**2 + x*y*z
def test_interp3():
# some input data
x = np.linspace(0, 1, 5)
y = np.linspace(0, 2, 6)
z = np.linspace(0, 3, 7)
v = somefunc(x[:, None, None], y[None, :, None], z[None, None, :])
# interpolate
xi = np.linspace(0, 1, 45)
yi = np.linspace(0, 2, 46)
zi = np.linspace(0, 3, 47)
vi = interp3(x, y, z, v, xi, yi, zi)
import matplotlib.pyplot as plt
X, Y = np.meshgrid(xi, yi)
plt.figure(1)
plt.subplot(1, 2, 1)
plt.pcolor(X, Y, vi[:, :, 12].T)
plt.title('interpolated')
plt.subplot(1, 2, 2)
plt.pcolor(X, Y, somefunc(xi[:, None], yi[None, :], zi[12]).T)
plt.title('exact')
plt.show('hold')
def test_smoothing_spline():
x = linspace(0, 2 * pi + pi / 4, 20)
y = sin(x) # + np.random.randn(x.size)
pp = SmoothSpline(x, y, p=1)
x1 = linspace(-1, 2 * pi + pi / 4 + 1, 20)
y1 = pp(x1)
pp1 = pp.derivative()
pp0 = pp1.integrate()
dy1 = pp1(x1)
y01 = pp0(x1)
# dy = y-y1
import matplotlib.pyplot as plt
plt.plot(x, y, x1, y1, '.', x1, dy1, 'ro', x1, y01, 'r-')
plt.show('hold')
pass
# tck = interpolate.splrep(x, y, s=len(x))
def compare_methods():
#
# Sine wave test
#
fun = np.sin
# Create a example vector containing a sine wave.
x = np.arange(30.0) / 10.
y = fun(x)
# Interpolate the data above to the grid defined by "xvec"
xvec = np.arange(250.) / 100.
# Initialize the interpolator slopes
# Create the pchip slopes
m = slopes(x, y, method='parabola', monotone=True)
m1 = slopes(x, y, method='parabola', monotone=False)
m2 = slopes(x, y, method='catmul', monotone=False)
m3 = pchip_slopes(x, y)
# Call the monotonic piece-wise Hermite cubic interpolator
yvec = Pchip(x, y, m)(xvec)
yvec1 = Pchip(x, y, m1)(xvec)
yvec2 = Pchip(x, y, m2)(xvec)
yvec3 = Pchip(x, y, m3)(xvec)
import matplotlib.pyplot as plt
plt.figure()
plt.plot(x, y, 'ro', xvec, fun(xvec), 'r')
plt.title("pchip() Sin test code")
# Plot the interpolated points
plt.plot(xvec, yvec, xvec, yvec1, xvec, yvec2, 'g.', xvec, yvec3)
plt.legend(
['true', 'true', 'parbola_monoton', 'parabola', 'catmul', 'pchip'],
frameon=False, loc=0)
plt.ioff()
plt.show()
def demo_monoticity():
# Step function test...
import matplotlib.pyplot as plt
plt.figure(2)
plt.title("pchip() step function test")
# 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
# Create the pchip slopes
m = slopes(x, y, monotone=True)
# m1 = slopes(x, y, monotone=False)
# m2 = slopes(x,y,method='catmul',monotone=False)
m3 = pchip_slopes(x, y)
# Interpolate...
yvec = Pchip(x, y, m)(xvec)
# Call the Scipy cubic spline interpolator
from scipy.interpolate import interpolate as ip
function = ip.interp1d(x, y, kind='cubic')
yvec2 = function(xvec)
# Non-montonic cubic Hermite spline interpolator using
# Catmul-Rom method for computing slopes...
yvec3 = CubicHermiteSpline(x, y)(xvec)
yvec4 = StinemanInterp(x, y)(xvec)
yvec5 = Pchip(x, y, m3)(xvec) # @UnusedVariable
# Plot the results
plt.plot(x, y, 'ro', label='Data')
plt.plot(xvec, yvec, 'b', label='Pchip')
plt.plot(xvec, yvec2, 'k', label='interp1d')
plt.plot(xvec, yvec3, 'g', label='CHS')
plt.plot(xvec, yvec4, 'm', label='Stineman')
# plt.plot(xvec, yvec5, 'yo', label='Pchip2')
plt.xlabel("X")
plt.ylabel("Y")
plt.title("Comparing Pchip() vs. Scipy interp1d() vs. non-monotonic CHS")
# legends = ["Data", "Pchip()", "interp1d","CHS", 'Stineman']
plt.legend(loc="upper left", frameon=False)
plt.ioff()
plt.show()
def test_func():
from scipy import interpolate
import matplotlib.pyplot as plt
import matplotlib
matplotlib.interactive(False)
coef = np.array([[1, 1], [0, 1]]) # linear from 0 to 2
# coef = np.array([[1,1],[1,1],[0,2]]) # linear from 0 to 2
breaks = [0, 1, 2]
pp = PPform(coef, breaks, a=-100, b=100)
x = linspace(-1, 3, 20)
y = pp(x) # @UnusedVariable
x = linspace(0, 2 * pi + pi / 4, 20)
y = sin(x) + np.random.randn(x.size)
tck = interpolate.splrep(x, y, s=len(x)) # @UndefinedVariable
xnew = linspace(0, 2 * pi, 100)
ynew = interpolate.splev(xnew, tck, der=0) # @UndefinedVariable
tck0 = interpolate.splmake( # @UndefinedVariable
xnew, ynew, order=3, kind='smoothest', conds=None)
pp = interpolate.ppform.fromspline(*tck0) # @UndefinedVariable
plt.plot(x, y, "x", xnew, ynew, xnew, sin(xnew), x, y, "b", x, pp(x), 'g')
plt.legend(['Linear', 'Cubic Spline', 'True'])
plt.title('Cubic-spline interpolation')
plt.show()
t = np.arange(0, 1.1, .1)
x = np.sin(2 * np.pi * t)
y = np.cos(2 * np.pi * t)
_tck1, _u = interpolate.splprep([t, y], s=0) # @UndefinedVariable
tck2 = interpolate.splrep(t, y, s=len(t), task=0) # @UndefinedVariable
# interpolate.spl
tck = interpolate.splmake(t, y, order=3, kind='smoothest', conds=None)
self = interpolate.ppform.fromspline(*tck2) # @UndefinedVariable
plt.plot(t, self(t))
plt.show('hold')
pass
def test_pp():
coef = np.array([[1, 1], [0, 0]]) # linear from 0 to 2 @UnusedVariable
# quadratic from 0 to 1 and 1 to 2.
coef = np.array([[1, 1], [1, 1], [0, 2]])
dc = pl.polyder(coef, 1)
c2 = pl.polyint(dc, 1) # @UnusedVariable
breaks = [0, 1, 2]
pp = PPform(coef, breaks)
pp(0.5)
pp(1)
pp(1.5)
dpp = pp.derivative()
import matplotlib.pyplot as plt
x = np.linspace(-1, 3)
plt.plot(x, pp(x), x, dpp(x), '.')
plt.show()
def test_docstrings():
import doctest
print('Testing docstrings in %s' % __file__)
doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE)
if __name__ == '__main__':
# test_func()
test_docstrings()
# test_smoothing_spline()
# compare_methods()
# demo_monoticity()
# test_interp3()
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