-
Notifications
You must be signed in to change notification settings - Fork 0
/
variance.py
241 lines (194 loc) · 9.47 KB
/
variance.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
from math import floor, ceil
from pandas import DataFrame
EPS = .00001
def _releq(a, b):
assert abs(a - b) < EPS
def _releql(l1, l2):
assert len(l1) == len(l2)
for a, b in zip(l1, l2):
_releq(a, b)
def hypergeometric(k, m, n):
return k * m/n * (1 - m/n) * (n-k) / (n-1)
def count_mi_ni(feature_column, stratum_column):
if not min(stratum_column) == 0:
raise ValueError(f"The minimum stratum entry must be 0, not {min(stratum_column)}.")
ell = max(stratum_column) + 1
if not ell >= 1:
raise ValueError(f"There must be at least one stratum, but the maximum stratum number found is {ell - 1}.")
if not len(set(stratum_column)) == ell:
raise ValueError(f"Strata must should be numbered 0, …, l-1, and all strata should be non-empty. However, "
f"a there are the following strata numbers: {set(stratum_column)}.")
ni = [0] * ell
mi = [0] * ell
for stratum, feature in zip(stratum_column, feature_column):
assert feature == 0 or feature == 1
ni[stratum] += 1
if feature == 1:
mi[stratum] += 1
return mi, ni
def get_layout(ni, k):
"""Given strata sizes `n_i` and panel size `k`, compute the `rho_i` and `g_i` for block rounding."""
assert k > 0
ell = len(ni)
n = sum(ni)
gi = [0] * ell
rhoi = [0] * ell
count = 0
for stratum in range(ell):
if ni[stratum] * k < n:
raise ValueError(f"All n_i must be at least n/k, but n_{stratum} = {ni[stratum]} < {n / k} = n/k.")
gi[stratum] = (count + ni[stratum]) * k // n - (count * k + n - 1) // n
_releq(gi[stratum], floor((count + ni[stratum]) / (n/k)) - ceil(count / (n/k)))
assert gi[stratum] >= 0
rhoi[stratum] = (((count + ni[stratum]) * k) % n) / n
_releq(rhoi[stratum], (count + ni[stratum]) / (n/k) - floor((count + ni[stratum]) / (n/k)))
assert 0 <= rhoi[stratum] < 1
assert rhoi[stratum] == 0 or rhoi[stratum] > EPS
count += ni[stratum]
assert rhoi[ell - 1] == 0
assert sum(gi) + sum(1 for rho in rhoi if rho != 0) == k
return rhoi, gi
def expvar(ni, mi, rhoi, gi):
"""Compute the expected value (over rounding) of the variance (over selection inside of strata) of the
representation of the hidden feature.
"""
ell = len(ni)
if ell != len(mi) or ell != len(rhoi) or ell != len(gi):
raise ValueError("The lengths of arguments ni, mi, rhoi, and gi must coincide.")
ev = 0
for stratum in range(ell):
if stratum == ell - 1:
rho = 0
else:
rho = rhoi[stratum]
if stratum == 0 or rhoi[stratum - 1] == 0:
lam = 0
else:
lam = 1 - rhoi[stratum - 1]
gx = gi[stratum]
mx = mi[stratum]
nx = ni[stratum]
ev += (1 - lam) * (1 - rho) * hypergeometric(gx, mx, nx)
ev += ((1 - lam) * rho + lam * (1 - rho)) * hypergeometric(gx + 1, mx, nx)
ev += lam * rho * hypergeometric(gx + 2, mx, nx)
return ev
def varexp(ni, mi, rhoi):
"""Compute the variance (over rounding) of the expected value (over selection inside of strata) of the
representation of the hidden feature.
"""
ell = len(ni)
ve = 0
for stratum in range(ell - 1):
ve += (mi[stratum]/ni[stratum] - mi[stratum+1]/ni[stratum+1])**2 * rhoi[stratum] * (1 - rhoi[stratum])
return ve
def compute_variance(data, hidden_feature, strata, k, partial=None):
"""Compute the variance in representation of a feature, given a stratification.
Args:
data (DataFrame): Table containing an index column and a column with the value of `hidden_feature` as its name.
hidden_feature (string): The column name to use as the hidden feature. `data[hidden_feature]` should be a binary
column with only entries 0 and 1.
strata (DataFrame): Table containing an index column referring to the same agents as `data` and a column
"stratum" describing the stratification. Entries should be between 0 and (number of
strata - 1), each stratum is expected to be non-empty.
partial: if "varexp" or "expvar", return only corresponding summand of variance
"""
assert len(data) == len(strata)
combination = data.join(strata, how="inner")
assert len(combination) == len(data)
assert len(strata) != 0
mi, ni = count_mi_ni(combination[hidden_feature], combination["stratum"])
rhoi, gi = get_layout(ni, k)
if partial == "varexp":
return varexp(ni, mi, rhoi)
elif partial == "expvar":
return expvar(ni, mi, rhoi, gi)
else:
return expvar(ni, mi, rhoi, gi) + varexp(ni, mi, rhoi)
def test_var():
"""Run usage examples and verify that variance computation functions work as expected."""
rhoi, gi = get_layout([2, 2], 3)
_releql(rhoi, [0.5, 0])
_releql(gi, [1, 1])
rhoi, gi = get_layout([4, 5, 6], 5)
_releql(rhoi, [1/3, 0, 0])
assert rhoi[1] == 0
assert rhoi[2] == 0
_releql(gi, [1, 1, 2])
_releq(expvar([2, 2], [1, 1], [0, 0], [1, 1]), 0.5)
_releq(expvar([3, 6, 3], [0, 3, 3], [.5, .5, 0], [1, 2, 1]), 17/40)
_releq(varexp([3, 6, 9, 6], [2, 4, 6, 4], [.5, .5, 0, 0]), 0)
_releq(compute_variance(DataFrame({"m": [1, 1]}), "m", DataFrame({"stratum": [0, 0]}), 1), 0)
_releq(compute_variance(DataFrame({"m": [0, 0, 1, 1]}), "m", DataFrame({"stratum": [0, 0, 1, 1]}), 2), 0)
_releq(compute_variance(DataFrame({"m": [0, 0, 1, 1]}), "m", DataFrame({"stratum": [0, 1, 0, 1]}), 2), 0.5)
def _stratify_successively(data, features, min_size, prestratification=None):
if prestratification is None:
prestratification = [list(data.index)]
if len(features) == 0:
return prestratification
f = features[0]
new_stratification = []
for stratum in prestratification:
if len(stratum) < 2 * min_size:
new_stratification.append(stratum)
else:
mini = int(data[f].min())
maxi = int(data[f].max())
lower = mini - 1
remaining = len(stratum)
for upper in range(mini, maxi):
new_index = [i for i in stratum if lower < data.loc[i, f] <= upper]
if len(new_index) >= min_size and remaining - len(new_index) >= min_size:
new_stratification.append(new_index)
lower = upper
remaining -= len(new_index)
new_index = [i for i in stratum if not data.loc[i, f] <= lower]
assert len(new_index) == remaining
new_stratification.append(new_index)
return _stratify_successively(data, features[1:], min_size, new_stratification)
def successive_stratification(data, features, min_size):
"""Stratify by successively refining based on a sequence of features, up to a minimum size.
Args:
data (DataFrame): Table with index column and columns named like the elements of `features`.
features (list of string): Column names. Stratify the agents by the entry in data[features[0]], then subdivide
based on data[features[1]] and so forth, until doing so would create strata with size
less than `min_size`.
min_size (int): Minimum size for strata.
Returns:
DataFrame
Table with index column matching `data` and column "stratum" with stratum number.
"""
strat = _stratify_successively(data, features, min_size)
index = data.index
positions = {}
for pos, i in enumerate(index):
positions[i] = pos
stratum = [None] * len(data)
for stratum_name, elements in enumerate(strat):
for x in elements:
stratum[positions[x]] = stratum_name
assert all(x is not None for x in stratum)
return DataFrame({"stratum": stratum}, index=index)
def test_successive_stratification():
d = DataFrame({"a": [0, 0, 0, 1, 1, 1, 2, 2, 2], "b": [0, 1, 2, 0, 1, 2, 0, 1, 2]})
assert _stratify_successively(d, [], 3) == [[0, 1, 2, 3, 4, 5, 6, 7, 8]]
assert _stratify_successively(d, ["a"], 3) == [[0, 1, 2], [3, 4, 5], [6, 7, 8]]
assert _stratify_successively(d, ["a"], 1) == [[0, 1, 2], [3, 4, 5], [6, 7, 8]]
assert _stratify_successively(d, ["b"], 3) == [[0, 3, 6], [1, 4, 7], [2, 5, 8]]
assert _stratify_successively(d, ["a"], 4) == [[0, 1, 2, 3, 4, 5, 6, 7, 8]]
assert _stratify_successively(d, ["a","b"], 1) == [[0], [1], [2], [3], [4], [5], [6], [7], [8]]
d = DataFrame({"a": [0, 0, 0, 1, 1, 1, 2, 2, 2], "b": [0, 1, 2, 0, 1, 2, 0, 1, 2]}, index=[7, 8, 9, 10, 11, 12, 13, 15, 16])
assert _stratify_successively(d, [], 3) == [[7, 8, 9, 10, 11, 12, 13, 15, 16]]
d = DataFrame({"a": [0, 0, 0, 0, 0, 1]})
assert _stratify_successively(d, ["a"], 2) == [[0, 1, 2, 3, 4, 5]]
d = DataFrame({"a": [0, 0, 0, 0, 1, 2], "b": [1, 2, 1, 2, 1, 2]})
assert _stratify_successively(d, ["a"], 2) == [[0, 1, 2, 3], [4, 5]]
assert _stratify_successively(d, ["a", "b"], 2) == [[0, 2], [1, 3], [4, 5]]
def equivalent_panel_size(variance, n, m, k):
"""Compute k' such that Var(U_M^{n,k'} / k') = Var(X_M^{n,k} / k).
To be exact, the resulting k' is a fractional interpolation of the variance
formula k * m/n * (1 - m/n) * (n-k)/(n-1).
Argument `variance` is Var(X_M^{n,k}), not normalized.
"""
assert m > 0
assert m < n
return (k**2 * m * n * (n-m)) / (variance * n**2 * (n-1) + k**2 * m * (n-m))