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dip.py
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"""
Module for computing The Hartigans' dip statistic
The dip statistic measures unimodality of a sample from a random process.
See:
Hartigan, J. A.; Hartigan, P. M. The Dip Test of Unimodality. The Annals
of Statistics 13 (1985), no. 1, 70--84. doi:10.1214/aos/1176346577.
http://projecteuclid.org/euclid.aos/1176346577.
"""
import numpy as np
import collections
def _gcm_(cdf, idxs):
work_cdf = cdf
work_idxs = idxs
gcm = [work_cdf[0]]
touchpoints = [0]
while len(work_cdf) > 1:
distances = work_idxs[1:] - work_idxs[0]
slopes = (work_cdf[1:] - work_cdf[0]) / distances
minslope = slopes.min()
minslope_idx = np.where(slopes == minslope)[0][0] + 1
gcm.extend(work_cdf[0] + distances[:minslope_idx] * minslope)
touchpoints.append(touchpoints[-1] + minslope_idx)
work_cdf = work_cdf[minslope_idx:]
work_idxs = work_idxs[minslope_idx:]
return np.array(np.array(gcm)),np.array(touchpoints)
def _lcm_(cdf, idxs):
g,t = _gcm_(1-cdf[::-1], idxs.max() - idxs[::-1])
return 1-g[::-1], len(cdf) - 1 - t[::-1]
def _touch_diffs_(part1, part2, touchpoints):
diff = np.abs((part2[touchpoints] - part1[touchpoints]))
return diff.max(), diff
def dip(histogram=None, idxs=None):
"""
Compute the Hartigans' dip statistic either for a histogram of
samples (with equidistant bins) or for a set of samples.
"""
if idxs is None:
idxs = np.arange(len(histogram))
elif histogram is None:
h = collections.Counter(idxs)
idxs = np.msort(h.keys())
histogram = np.array([h[i] for i in idxs])
else:
if len(histogram) != len(idxs):
raise ValueError("Need exactly as many indices as histogram bins.")
if len(idxs) != len(set(idxs)):
raise ValueError("idxs must be unique if histogram is given.")
if not np.array_equal(np.msort(idxs), idxs):
idxs_s = np.argsort(idxs)
idx = np.asarray(idxs)[idxs_s]
histogram = np.asarray(histogram)[idxs_s]
cdf = np.cumsum(histogram, dtype=float)
cdf /= cdf[-1]
work_idxs = idxs
work_histogram = np.asarray(histogram, dtype=float) / np.sum(histogram)
work_cdf = cdf
D = 0
left = [0]
right = [1]
while True:
left_part, left_touchpoints = _gcm_(work_cdf - work_histogram, work_idxs)
right_part, right_touchpoints = _lcm_(work_cdf, work_idxs)
d_left, left_diffs = _touch_diffs_(left_part, right_part, left_touchpoints)
d_right, right_diffs = _touch_diffs_(left_part, right_part, right_touchpoints)
if d_right > d_left:
xr = right_touchpoints[d_right == right_diffs][-1]
xl = left_touchpoints[left_touchpoints <= xr][-1]
d = d_right
else:
xl = left_touchpoints[d_left == left_diffs][0]
xr = right_touchpoints[right_touchpoints >= xl][0]
d = d_left
left_diff = np.abs(left_part[:xl+1] - work_cdf[:xl+1]).max()
right_diff = np.abs(right_part[xr:] - work_cdf[xr:] + work_histogram[xr:]).max()
if d <= D or xr == 0 or xl == len(work_cdf):
the_dip = max(np.abs(cdf[:len(left)] - left).max(), np.abs(cdf[-len(right)-1:-1] - right).max())
return the_dip/2, (cdf, idxs, left, left_part, right, right_part)
else:
D = max(D, left_diff, right_diff)
work_cdf = work_cdf[xl:xr+1]
work_idxs = work_idxs[xl:xr+1]
work_histogram = work_histogram[xl:xr+1]
left[len(left):] = left_part[1:xl+1]
right[:0] = right_part[xr:-1]
def plot_dip(histogram=None, idxs=None):
from matplotlib import pyplot as plt
d,(cdf,idxs,left,left_part,right,right_part) = dip(histogram,idxs)
plt.plot(idxs[:len(left)], left, color='red')
plt.plot(idxs[len(left)-1:len(left)+len(left_part) - 1], left_part, color='gray')
plt.plot(idxs[-len(right):], right, color='blue')
plt.plot(idxs[len(cdf) - len(right) + 1 - len(right_part):len(cdf) - len(right) + 1], right_part, color='gray')
the_dip = max(np.abs(cdf[:len(left)] - left).max(), np.abs(cdf[-len(right)-1:-1] - right).max())
l_dip_idxs = np.abs(cdf[:len(left)] - left) == the_dip
r_dip_idxs = np.abs(cdf[-len(right)-1:-1] - right) == the_dip
print(the_dip/2, d)
plt.vlines(x=idxs[:len(left)][l_dip_idxs], ymin=cdf[:len(left)][l_dip_idxs], ymax = cdf[:len(left)][l_dip_idxs] - the_dip)
plt.vlines(x=idxs[-len(right):][r_dip_idxs], ymin=cdf[-len(right)-1:-1][r_dip_idxs], ymax = cdf[-len(right)-1:][r_dip_idxs] + the_dip)
plt.plot(np.repeat(idxs,2)[1:], np.repeat(cdf,2)[:-1], color='black')
plt.scatter(idxs, cdf)
plt.show()
def crit_points(random_function, quantiles, sample_size, n_samples):
"""
Compute the quantiles for the dip statistic for n_samples
samples of size sample_size from the random process given by
random_function.
Parameters:
random_function : a paramter-free function which returns random values.
quantiles : a sequence of values between 0 and 1
sample_size : the size of the samples to draw from random_function
n_samples : the number of samples for which to compute dips
Returns: a list such that the i'th value is the greatest dip observed
such that the fraction of dips less than or equal to that value is less
than the i'th value from quantiles.
"""
data = [[random_function() for _ in range(sample_size)] for __ in range(n_samples)]
dips = np.array([dip(idxs=samples)[0] for samples in data])
return np.percentile(dips, [p * 100 for p in quantiles])