163 lines
5.0 KiB
Python
163 lines
5.0 KiB
Python
"""
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Sample a 1D function to given tolerance by adaptive subdivision.
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The result of sampling is a set of points that, if plotted, produces a smooth
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curve with also sharp features of the function resolved.
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This routine is useful in computing functions that are expensive to compute,
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and have sharp features — it makes more sense to adaptively dedicate more
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sampling points for the sharp features than the smooth parts.
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Source: http://central.scipy.org/item/53/1/adaptive-sampling-of-1d-functions
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License: Creative Commons Zero (almost public domain) http://scpyce.org/cc0
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(Slightly) modified by Phyks (Lucas Verney).
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"""
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import numpy as np
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def sample_function(func, points, tol=0.05, min_points=16, max_level=16,
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sample_transform=None):
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"""
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Sample a 1D function to given tolerance by adaptive subdivision.
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The result of sampling is a set of points that, if plotted,
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produces a smooth curve with also sharp features of the function
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resolved.
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Parameters
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----------
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func : callable
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Function func(x) of a single argument. It is assumed to be vectorized.
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points : array-like, 1D
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Initial points to sample, sorted in ascending order.
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These will determine also the bounds of sampling.
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tol : float, optional
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Tolerance to sample to. The condition is roughly that the total
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length of the curve on the (x, y) plane is computed up to this
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tolerance.
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min_point : int, optional
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Minimum number of points to sample.
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max_level : int, optional
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Maximum subdivision depth.
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sample_transform : callable, optional
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Function w = g(x, y). The x-samples are generated so that w
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is sampled.
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Returns
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-------
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x : ndarray
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X-coordinates
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y : ndarray
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Corresponding values of func(x)
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Notes
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-----
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This routine is useful in computing functions that are expensive
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to compute, and have sharp features --- it makes more sense to
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adaptively dedicate more sampling points for the sharp features
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than the smooth parts.
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Examples
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--------
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>>> def func(x):
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... '''Function with a sharp peak on a smooth background'''
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... a = 0.001
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... return x + a**2/(a**2 + x**2)
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...
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>>> x, y = sample_function(func, [-1, 1], tol=1e-3)
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>>> import matplotlib.pyplot as plt
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>>> xx = np.linspace(-1, 1, 12000)
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>>> plt.plot(xx, func(xx), '-', x, y, '.')
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>>> plt.show()
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"""
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x, y = _sample_function(func, points, values=None, mask=None, depth=0,
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tol=tol,
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min_points=min_points, max_level=max_level,
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sample_transform=sample_transform)
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return (x, y[0])
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def _sample_function(func, points, values=None, mask=None, tol=0.05,
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depth=0, min_points=16, max_level=16,
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sample_transform=None):
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points = np.asarray(points)
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if values is None:
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values = np.atleast_2d(func(points))
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if mask is None:
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mask = Ellipsis
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if depth > max_level:
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# recursion limit
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return points, values
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x_a = points[:-1][mask]
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x_b = points[1:][mask]
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x_c = .5*(x_a + x_b)
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y_c = np.atleast_2d(func(x_c))
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x_2 = np.r_[points, x_c]
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y_2 = np.r_['-1', values, y_c]
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j = np.argsort(x_2)
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x_2 = x_2[..., j]
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y_2 = y_2[..., j]
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# -- Determine the intervals at which refinement is necessary
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if len(x_2) < min_points:
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mask = np.ones([len(x_2)-1], dtype=bool)
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else:
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# represent the data as a path in N dimensions (scaled to unit box)
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if sample_transform is not None:
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y_2_val = sample_transform(x_2, y_2)
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else:
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y_2_val = y_2
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p = np.r_['0',
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x_2[None, :],
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y_2_val.real.reshape(-1, y_2_val.shape[-1]),
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y_2_val.imag.reshape(-1, y_2_val.shape[-1])]
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sz = (p.shape[0]-1) // 2
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xscale = x_2.ptp(axis=-1)
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yscale = abs(y_2_val.ptp(axis=-1)).ravel()
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p[0] /= xscale
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p[1:sz+1] /= yscale[:, None]
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p[sz+1:] /= yscale[:, None]
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# compute the length of each line segment in the path
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dp = np.diff(p, axis=-1)
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s = np.sqrt((dp**2).sum(axis=0))
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s_tot = s.sum()
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# compute the angle between consecutive line segments
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dp /= s
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dcos = np.arccos(np.clip((dp[:, 1:] * dp[:, :-1]).sum(axis=0), -1, 1))
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# determine where to subdivide: the condition is roughly that
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# the total length of the path (in the scaled data) is computed
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# to accuracy `tol`
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dp_piece = dcos * .5*(s[1:] + s[:-1])
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mask = (dp_piece > tol * s_tot)
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mask = np.r_[mask, False]
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mask[1:] |= mask[:-1].copy()
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# -- Refine, if necessary
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if mask.any():
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return _sample_function(func, x_2, y_2, mask, tol=tol, depth=depth+1,
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min_points=min_points, max_level=max_level,
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sample_transform=sample_transform)
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else:
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return x_2, y_2
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