def winner_distance(r1, r2, reverse=False):
""" Asymmetrical winner distance.
This distance is the rank of the winner of r1 in r2, normalized by the number of candidates.
(rank(r1 winner)) - 1 / (n - 1)
Assuming no ties.
Args:
r1: 1D vector representing a judge.
r2: 1D vector representing a judge.
reverse: If True, lower is better.
"""
r1, r2 = np.array(r1), np.array(r2)
if reverse:
w1 = np.argmin(r1) # r1 winner
else:
w1 = np.argmax(r1) # r1 winner
return (rk.rank(r2)[w1] - 1) / (len(r2) - 1)
def kendall_w(matrix, axis=0, ties=False):
""" Kendall's W coefficient of concordance.
See https://en.wikipedia.org/wiki/Kendall%27s_W for more information.
Args:
matrix: Preference matrix.
axis: Axis of judges.
ties: If True, apply the correction for ties
"""
if ties:
return kendall_w_ties(matrix, axis=axis)
matrix = rk.rank(matrix, axis=1 - axis) # compute on ranks
m = matrix.shape[axis] # judges
n = matrix.shape[1 - axis] # candidates
denominator = m**2 * (n**3 - n)
rating_sums = np.sum(matrix, axis=axis)
S = n * np.var(rating_sums)
return 12 * S / denominator
def kendall_w_ties(matrix, axis=0):
""" Kendall's W coefficient of concordance with correction for ties.
The goal of this correction is to avoid having a lower score in the presence of ties in the rankings.
Args:
matrix: Preference matrix.
axis: Axis of judges.
"""
if axis == 1:
matrix = matrix.T
m = matrix.shape[0] # judges
n = matrix.shape[1] # candidates
matrix = rk.rank(matrix, axis=1) # compute on ranks
T = [] # correction factors, one by judge
for j in range(m):
_, counts = np.unique(matrix[j], return_counts=True) # tied groups
correction = np.sum([(t**3 - t) for t in counts])
T.append(correction)
denominator = m**2 * n * (n**2 - 1) - m * np.sum(T)
sum = np.sum([r**2 for r in np.sum(matrix, axis=0)])
numerator = 12 * sum - 3 * m**2 * n * (n + 1)**2
return numerator / denominator