forked from sympy/sympy
-
Notifications
You must be signed in to change notification settings - Fork 0
/
crv.py
311 lines (261 loc) · 11.2 KB
/
crv.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
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
"""
Continuous Random Variables Module
See Also
========
sympy.stats.crv_types
sympy.stats.rv
sympy.stats.frv
"""
from sympy.stats.rv import (RandomDomain, SingleDomain, ConditionalDomain,
ProductDomain, PSpace, SinglePSpace, random_symbols, ProductPSpace)
from sympy.functions.special.delta_functions import DiracDelta
from sympy import (S, Interval, symbols, Dummy, FiniteSet, Mul, Tuple,
Integral, And, Or, Piecewise, solve, cacheit, integrate, oo, Lambda)
from sympy.solvers.inequalities import reduce_poly_inequalities
from sympy.polys.polyerrors import PolynomialError
import random
class ContinuousDomain(RandomDomain):
"""
A domain with continuous support
Represented using symbols and Intervals.
"""
is_Continuous = True
def as_boolean(self):
raise NotImplementedError("Not Implemented for generic Domains")
class SingleContinuousDomain(ContinuousDomain, SingleDomain):
"""
A univariate domain with continuous support
Represented using a single symbol and interval.
"""
def __new__(cls, symbol, set):
assert symbol.is_Symbol
symbols = FiniteSet(symbol)
return RandomDomain.__new__(cls, symbols, set)
def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
assert frozenset(variables) == frozenset(self.symbols)
# assumes only intervals
evaluate = kwargs.pop('evaluate', True)
if evaluate:
return integrate(expr, (self.symbol, self.set), **kwargs)
else:
return Integral(expr, (self.symbol, self.set), **kwargs)
def as_boolean(self):
return self.set.as_relational(self.symbol)
class ProductContinuousDomain(ProductDomain, ContinuousDomain):
"""
A collection of independent domains with continuous support
"""
def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
for domain in self.domains:
domain_vars = frozenset(variables) & frozenset(domain.symbols)
if domain_vars:
expr = domain.integrate(expr, domain_vars, **kwargs)
return expr
def as_boolean(self):
return And(*[domain.as_boolean() for domain in self.domains])
class ConditionalContinuousDomain(ContinuousDomain, ConditionalDomain):
"""
A domain with continuous support that has been further restricted by a
condition such as x > 3
"""
def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
# Extract the full integral
fullintgrl = self.fulldomain.integrate(expr, variables, evaluate=False)
# separate into integrand and limits
integrand, limits = fullintgrl.function, list(fullintgrl.limits)
conditions = [self.condition]
while conditions:
cond = conditions.pop()
if cond.is_Boolean:
if isinstance(cond, And):
conditions.extend(cond.args)
elif isinstance(cond, Or):
raise NotImplementedError("Or not implemented here")
elif cond.is_Relational:
if cond.is_Equality:
# Add the appropriate Delta to the integrand
integrand *= DiracDelta(cond.lhs-cond.rhs)
else:
symbols = cond.free_symbols & set(self.symbols)
if len(symbols)!=1: # Can't handle x > y
raise NotImplementedError(
"Multivariate Inequalities not yet implemented")
# Can handle x > 0
symbol = symbols.pop()
# Find the limit with x, such as (x, -oo, oo)
for i, limit in enumerate(limits):
if limit[0]==symbol:
# Make condition into an Interval like [0, oo]
cintvl = reduce_poly_inequalities_wrap(cond, symbol)
# Make limit into an Interval like [-oo, oo]
lintvl = Interval(limit[1], limit[2])
# Intersect them to get [0, oo]
intvl = cintvl.intersect(lintvl)
# Put back into limits list
limits[i] = (symbol, intvl.left, intvl.right)
else:
raise TypeError(
"Condition %s is not a relational or Boolean"%cond)
evaluate = kwargs.pop('evaluate', True)
if evaluate:
return integrate(integrand, *limits, **kwargs)
return Integral(integrand, *limits, **kwargs)
def as_boolean(self):
return And(self.fulldomain.as_boolean(), self.condition)
@property
def set(self):
if len(self.symbols) == 1:
return (self.fulldomain.set & reduce_poly_inequalities_wrap(
self.condition, tuple(self.symbols)[0]))
else:
raise NotImplementedError(
"Set of Conditional Domain not Implemented")
class ContinuousPSpace(PSpace):
"""
A Continuous Probability Space
Represents the likelihood of an event space defined over a continuum.
Represented with a set of symbols and a probability density function.
"""
is_Continuous = True
def integrate(self, expr, rvs=None, **kwargs):
if rvs == None:
rvs = self.values
else:
rvs = frozenset(rvs)
expr = expr.subs(dict((rv, rv.symbol) for rv in rvs))
domain_symbols = frozenset(rv.symbol for rv in rvs)
return self.domain.integrate(self.density * expr,
domain_symbols, **kwargs)
def compute_density(self, expr, **kwargs):
# Common case Density(X) where X in self.values
if expr in self.values:
# Marginalize all other random symbols out of the density
density = self.domain.integrate(self.density, set(rs.symbol
for rs in self.values - frozenset((expr,))), **kwargs)
return Lambda(expr.symbol, density)
z = Dummy('z', real=True, bounded=True)
return Lambda(z, self.integrate(DiracDelta(expr - z), **kwargs))
def compute_cdf(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise ValueError("CDF not well defined on multivariate expressions")
d = self.compute_density(expr, **kwargs)
x, z = symbols('x, z', real=True, bounded=True, cls=Dummy)
left_bound = self.domain.set.start
# CDF is integral of PDF from left bound to z
cdf = integrate(d(x), (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((0, z<left_bound), (cdf, True))
return Lambda(z, cdf)
def P(self, condition, **kwargs):
evaluate = kwargs.get("evaluate", True)
z = Dummy('z', real=True, bounded=True)
# Univariate case can be handled by where
try:
domain = self.where(condition)
rv = [rv for rv in self.values if rv.symbol == domain.symbol][0]
# Integrate out all other random variables
pdf = self.compute_density(rv, **kwargs)
# Integrate out the last variable over the special domain
if evaluate:
return integrate(pdf(z), (z, domain.set), **kwargs)
else:
return Integral(pdf(z), (z, domain.set), **kwargs)
# Other cases can be turned into univariate case
# by computing a density handled by density computation
except NotImplementedError:
expr = condition.lhs - condition.rhs
density = self.compute_density(expr, **kwargs)
# Turn problem into univariate case
space = SingleContinuousPSpace(z, density(z))
return space.P(condition.__class__(space.value, 0))
def where(self, condition):
rvs = frozenset(random_symbols(condition))
if not (len(rvs)==1 and rvs.issubset(self.values)):
raise NotImplementedError(
"Multiple continuous random variables not supported")
rv = tuple(rvs)[0]
interval = reduce_poly_inequalities_wrap(condition, rv)
interval = interval.intersect(self.domain.set)
return SingleContinuousDomain(rv.symbol, interval)
def conditional_space(self, condition, normalize=True, **kwargs):
condition = condition.subs(dict((rv,rv.symbol) for rv in self.values))
domain = ConditionalContinuousDomain(self.domain, condition)
density = self.density
if normalize:
density = density / domain.integrate(density, **kwargs)
return ContinuousPSpace(domain, density)
class SingleContinuousPSpace(ContinuousPSpace, SinglePSpace):
"""
A continuous probability space over a single univariate domain
This class is commonly implemented by the various ContinuousRV types
such as Normal, Exponential, Uniform, etc....
"""
_count = 0
_name = 'x'
def __new__(cls, symbol, density, set=Interval(-oo, oo)):
assert symbol.is_Symbol
domain = SingleContinuousDomain(symbol, set)
obj = ContinuousPSpace.__new__(cls, domain, density)
obj._cdf = None
return obj
@cacheit
def _inverse_cdf_expression(self):
"""
Inverse of the CDF
See Also
========
compute_cdf
sample
"""
d = self.compute_cdf(self.value)
x, z = symbols('x, z', real=True, positive=True, cls=Dummy)
# Invert CDF
try:
inverse_cdf = solve(d(x)-z, x)
except NotImplementedError:
raise NotImplementedError("Could not invert CDF")
if len(inverse_cdf) != 1:
raise NotImplementedError("Could not invert CDF")
return Lambda(z, inverse_cdf[0])
def sample(self):
"""
Internal sample method
Returns dictionary mapping RandomSymbol to realization value.
"""
icdf = self._inverse_cdf_expression()
return {self.value: icdf(random.uniform(0,1))}
class ProductContinuousPSpace(ProductPSpace, ContinuousPSpace):
"""
A collection of independent continuous probability spaces
"""
@property
def density(self):
return Mul(*[space.density for space in self.spaces])
def _reduce_inequalities(conditions, var, **kwargs):
try:
return reduce_poly_inequalities(conditions, var, **kwargs)
except PolynomialError:
raise ValueError("Reduction of condition failed %s\n"%conditions[0])
def reduce_poly_inequalities_wrap(condition, var):
if condition.is_Relational:
return _reduce_inequalities([[condition]], var, relational=False)
if condition.__class__ is Or:
return _reduce_inequalities([list(condition.args)],
var, relational=False)
if condition.__class__ is And:
intervals = [_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args]
I = intervals[0]
for i in intervals:
I = I.intersect(i)
return I