Spaces:
No application file
No application file
File size: 26,511 Bytes
b7731cd |
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 |
# Copyright 2001 Brad Chapman. All rights reserved.
#
# This file is part of the Biopython distribution and governed by your
# choice of the "Biopython License Agreement" or the "BSD 3-Clause License".
# Please see the LICENSE file that should have been included as part of this
# package.
"""Deal with representations of Markov Models."""
# standard modules
import copy
import math
import random
from collections import defaultdict
from Bio.Seq import Seq
def _gen_random_array(n):
"""Return an array of n random numbers summing to 1.0 (PRIVATE)."""
randArray = [random.random() for _ in range(n)]
total = sum(randArray)
return [x / total for x in randArray]
def _calculate_emissions(emission_probs):
"""Calculate which symbols can be emitted in each state (PRIVATE)."""
# loop over all of the state-symbol duples, mapping states to
# lists of emitted symbols
emissions = defaultdict(list)
for state, symbol in emission_probs:
emissions[state].append(symbol)
return emissions
def _calculate_from_transitions(trans_probs):
"""Calculate which 'from transitions' are allowed for each state (PRIVATE).
This looks through all of the trans_probs, and uses this dictionary
to determine allowed transitions. It converts this information into
a dictionary, whose keys are source states and whose values are
lists of destination states reachable from the source state via a
transition.
"""
transitions = defaultdict(list)
for from_state, to_state in trans_probs:
transitions[from_state].append(to_state)
return transitions
def _calculate_to_transitions(trans_probs):
"""Calculate which 'to transitions' are allowed for each state (PRIVATE).
This looks through all of the trans_probs, and uses this dictionary
to determine allowed transitions. It converts this information into
a dictionary, whose keys are destination states and whose values are
lists of source states from which the destination is reachable via a
transition.
"""
transitions = defaultdict(list)
for from_state, to_state in trans_probs:
transitions[to_state].append(from_state)
return transitions
class MarkovModelBuilder:
"""Interface to build up a Markov Model.
This class is designed to try to separate the task of specifying the
Markov Model from the actual model itself. This is in hopes of making
the actual Markov Model classes smaller.
So, this builder class should be used to create Markov models instead
of trying to initiate a Markov Model directly.
"""
# the default pseudo counts to use
DEFAULT_PSEUDO = 1
def __init__(self, state_alphabet, emission_alphabet):
"""Initialize a builder to create Markov Models.
Arguments:
- state_alphabet -- An iterable (e.g., tuple or list) containing
all of the letters that can appear in the states
- emission_alphabet -- An iterable (e.g., tuple or list) containing
all of the letters for states that can be emitted by the HMM.
"""
self._state_alphabet = tuple(state_alphabet)
self._emission_alphabet = tuple(emission_alphabet)
# probabilities for the initial state, initialized by calling
# set_initial_probabilities (required)
self.initial_prob = {}
# the probabilities for transitions and emissions
# by default we have no transitions and all possible emissions
self.transition_prob = {}
self.emission_prob = self._all_blank(state_alphabet, emission_alphabet)
# the default pseudocounts for transition and emission counting
self.transition_pseudo = {}
self.emission_pseudo = self._all_pseudo(state_alphabet, emission_alphabet)
def _all_blank(self, first_alphabet, second_alphabet):
"""Return a dictionary with all counts set to zero (PRIVATE).
This uses the letters in the first and second alphabet to create
a dictionary with keys of two tuples organized as
(letter of first alphabet, letter of second alphabet). The values
are all set to 0.
"""
all_blank = {}
for first_state in first_alphabet:
for second_state in second_alphabet:
all_blank[(first_state, second_state)] = 0
return all_blank
def _all_pseudo(self, first_alphabet, second_alphabet):
"""Return a dictionary with all counts set to a default value (PRIVATE).
This takes the letters in first alphabet and second alphabet and
creates a dictionary with keys of two tuples organized as:
(letter of first alphabet, letter of second alphabet). The values
are all set to the value of the class attribute DEFAULT_PSEUDO.
"""
all_counts = {}
for first_state in first_alphabet:
for second_state in second_alphabet:
all_counts[(first_state, second_state)] = self.DEFAULT_PSEUDO
return all_counts
def get_markov_model(self):
"""Return the markov model corresponding with the current parameters.
Each markov model returned by a call to this function is unique
(ie. they don't influence each other).
"""
# user must set initial probabilities
if not self.initial_prob:
raise Exception(
"set_initial_probabilities must be called to "
"fully initialize the Markov model"
)
initial_prob = copy.deepcopy(self.initial_prob)
transition_prob = copy.deepcopy(self.transition_prob)
emission_prob = copy.deepcopy(self.emission_prob)
transition_pseudo = copy.deepcopy(self.transition_pseudo)
emission_pseudo = copy.deepcopy(self.emission_pseudo)
return HiddenMarkovModel(
self._state_alphabet,
self._emission_alphabet,
initial_prob,
transition_prob,
emission_prob,
transition_pseudo,
emission_pseudo,
)
def set_initial_probabilities(self, initial_prob):
"""Set initial state probabilities.
initial_prob is a dictionary mapping states to probabilities.
Suppose, for example, that the state alphabet is ('A', 'B'). Call
set_initial_prob({'A': 1}) to guarantee that the initial
state will be 'A'. Call set_initial_prob({'A': 0.5, 'B': 0.5})
to make each initial state equally probable.
This method must now be called in order to use the Markov model
because the calculation of initial probabilities has changed
incompatibly; the previous calculation was incorrect.
If initial probabilities are set for all states, then they should add up
to 1. Otherwise the sum should be <= 1. The residual probability is
divided up evenly between all the states for which the initial
probability has not been set. For example, calling
set_initial_prob({}) results in P('A') = 0.5 and P('B') = 0.5,
for the above example.
"""
self.initial_prob = copy.copy(initial_prob)
# ensure that all referenced states are valid
for state in initial_prob:
if state not in self._state_alphabet:
raise ValueError(
f"State {state} was not found in the sequence alphabet"
)
# distribute the residual probability, if any
num_states_not_set = len(self._state_alphabet) - len(self.initial_prob)
if num_states_not_set < 0:
raise Exception("Initial probabilities can't exceed # of states")
prob_sum = sum(self.initial_prob.values())
if prob_sum > 1.0:
raise Exception("Total initial probability cannot exceed 1.0")
if num_states_not_set > 0:
prob = (1.0 - prob_sum) / num_states_not_set
for state in self._state_alphabet:
if state not in self.initial_prob:
self.initial_prob[state] = prob
def set_equal_probabilities(self):
"""Reset all probabilities to be an average value.
Resets the values of all initial probabilities and all allowed
transitions and all allowed emissions to be equal to 1 divided by the
number of possible elements.
This is useful if you just want to initialize a Markov Model to
starting values (ie. if you have no prior notions of what the
probabilities should be -- or if you are just feeling too lazy
to calculate them :-).
Warning 1 -- this will reset all currently set probabilities.
Warning 2 -- This just sets all probabilities for transitions and
emissions to total up to 1, so it doesn't ensure that the sum of
each set of transitions adds up to 1.
"""
# set initial state probabilities
new_initial_prob = 1.0 / len(self.transition_prob)
for state in self._state_alphabet:
self.initial_prob[state] = new_initial_prob
# set the transitions
new_trans_prob = 1.0 / len(self.transition_prob)
for key in self.transition_prob:
self.transition_prob[key] = new_trans_prob
# set the emissions
new_emission_prob = 1.0 / len(self.emission_prob)
for key in self.emission_prob:
self.emission_prob[key] = new_emission_prob
def set_random_initial_probabilities(self):
"""Set all initial state probabilities to a randomly generated distribution.
Returns the dictionary containing the initial probabilities.
"""
initial_freqs = _gen_random_array(len(self._state_alphabet))
for state in self._state_alphabet:
self.initial_prob[state] = initial_freqs.pop()
return self.initial_prob
def set_random_transition_probabilities(self):
"""Set all allowed transition probabilities to a randomly generated distribution.
Returns the dictionary containing the transition probabilities.
"""
if not self.transition_prob:
raise Exception(
"No transitions have been allowed yet. "
"Allow some or all transitions by calling "
"allow_transition or allow_all_transitions first."
)
transitions_from = _calculate_from_transitions(self.transition_prob)
for from_state in transitions_from:
freqs = _gen_random_array(len(transitions_from[from_state]))
for to_state in transitions_from[from_state]:
self.transition_prob[(from_state, to_state)] = freqs.pop()
return self.transition_prob
def set_random_emission_probabilities(self):
"""Set all allowed emission probabilities to a randomly generated distribution.
Returns the dictionary containing the emission probabilities.
"""
if not self.emission_prob:
raise Exception(
"No emissions have been allowed yet. Allow some or all emissions."
)
emissions = _calculate_emissions(self.emission_prob)
for state in emissions:
freqs = _gen_random_array(len(emissions[state]))
for symbol in emissions[state]:
self.emission_prob[(state, symbol)] = freqs.pop()
return self.emission_prob
def set_random_probabilities(self):
"""Set all probabilities to randomly generated numbers.
Resets probabilities of all initial states, transitions, and
emissions to random values.
"""
self.set_random_initial_probabilities()
self.set_random_transition_probabilities()
self.set_random_emission_probabilities()
# --- functions to deal with the transitions in the sequence
def allow_all_transitions(self):
"""Create transitions between all states.
By default all transitions within the alphabet are disallowed;
this is a convenience function to change this to allow all
possible transitions.
"""
# first get all probabilities and pseudo counts set
# to the default values
all_probs = self._all_blank(self._state_alphabet, self._state_alphabet)
all_pseudo = self._all_pseudo(self._state_alphabet, self._state_alphabet)
# now set any probabilities and pseudo counts that
# were previously set
for set_key in self.transition_prob:
all_probs[set_key] = self.transition_prob[set_key]
for set_key in self.transition_pseudo:
all_pseudo[set_key] = self.transition_pseudo[set_key]
# finally reinitialize the transition probs and pseudo counts
self.transition_prob = all_probs
self.transition_pseudo = all_pseudo
def allow_transition(
self, from_state, to_state, probability=None, pseudocount=None
):
"""Set a transition as being possible between the two states.
probability and pseudocount are optional arguments
specifying the probabilities and pseudo counts for the transition.
If these are not supplied, then the values are set to the
default values.
Raises:
KeyError -- if the two states already have an allowed transition.
"""
# check the sanity of adding these states
for state in [from_state, to_state]:
if state not in self._state_alphabet:
raise ValueError(
f"State {state} was not found in the sequence alphabet"
)
# ensure that the states are not already set
if (from_state, to_state) not in self.transition_prob and (
from_state,
to_state,
) not in self.transition_pseudo:
# set the initial probability
if probability is None:
probability = 0
self.transition_prob[(from_state, to_state)] = probability
# set the initial pseudocounts
if pseudocount is None:
pseudocount = self.DEFAULT_PSEUDO
self.transition_pseudo[(from_state, to_state)] = pseudocount
else:
raise KeyError(
f"Transition from {from_state} to {to_state} is already allowed."
)
def destroy_transition(self, from_state, to_state):
"""Restrict transitions between the two states.
Raises:
KeyError if the transition is not currently allowed.
"""
try:
del self.transition_prob[(from_state, to_state)]
del self.transition_pseudo[(from_state, to_state)]
except KeyError:
raise KeyError(
f"Transition from {from_state} to {to_state} is already disallowed."
)
def set_transition_score(self, from_state, to_state, probability):
"""Set the probability of a transition between two states.
Raises:
KeyError if the transition is not allowed.
"""
if (from_state, to_state) in self.transition_prob:
self.transition_prob[(from_state, to_state)] = probability
else:
raise KeyError(
f"Transition from {from_state} to {to_state} is not allowed."
)
def set_transition_pseudocount(self, from_state, to_state, count):
"""Set the default pseudocount for a transition.
To avoid computational problems, it is helpful to be able to
set a 'default' pseudocount to start with for estimating
transition and emission probabilities (see p62 in Durbin et al
for more discussion on this. By default, all transitions have
a pseudocount of 1.
Raises:
KeyError if the transition is not allowed.
"""
if (from_state, to_state) in self.transition_pseudo:
self.transition_pseudo[(from_state, to_state)] = count
else:
raise KeyError(
f"Transition from {from_state} to {to_state} is not allowed."
)
# --- functions to deal with emissions from the sequence
def set_emission_score(self, seq_state, emission_state, probability):
"""Set the probability of a emission from a particular state.
Raises:
KeyError if the emission from the given state is not allowed.
"""
if (seq_state, emission_state) in self.emission_prob:
self.emission_prob[(seq_state, emission_state)] = probability
else:
raise KeyError(
f"Emission of {emission_state} from {seq_state} is not allowed."
)
def set_emission_pseudocount(self, seq_state, emission_state, count):
"""Set the default pseudocount for an emission.
To avoid computational problems, it is helpful to be able to
set a 'default' pseudocount to start with for estimating
transition and emission probabilities (see p62 in Durbin et al
for more discussion on this. By default, all emissions have
a pseudocount of 1.
Raises:
KeyError if the emission from the given state is not allowed.
"""
if (seq_state, emission_state) in self.emission_pseudo:
self.emission_pseudo[(seq_state, emission_state)] = count
else:
raise KeyError(
f"Emission of {emission_state} from {seq_state} is not allowed."
)
class HiddenMarkovModel:
"""Represent a hidden markov model that can be used for state estimation."""
def __init__(
self,
state_alphabet,
emission_alphabet,
initial_prob,
transition_prob,
emission_prob,
transition_pseudo,
emission_pseudo,
):
"""Initialize a Markov Model.
Note: You should use the MarkovModelBuilder class instead of
initiating this class directly.
Arguments:
- state_alphabet -- A tuple containing all of the letters that can
appear in the states.
- emission_alphabet -- A tuple containing all of the letters for
states that can be emitted by the HMM.
- initial_prob - A dictionary of initial probabilities for all states.
- transition_prob -- A dictionary of transition probabilities for all
possible transitions in the sequence.
- emission_prob -- A dictionary of emission probabilities for all
possible emissions from the sequence states.
- transition_pseudo -- Pseudo-counts to be used for the transitions,
when counting for purposes of estimating transition probabilities.
- emission_pseudo -- Pseudo-counts to be used for the emissions,
when counting for purposes of estimating emission probabilities.
"""
self.state_alphabet = state_alphabet
self.emission_alphabet = emission_alphabet
self.initial_prob = initial_prob
self._transition_pseudo = transition_pseudo
self._emission_pseudo = emission_pseudo
self.transition_prob = transition_prob
self.emission_prob = emission_prob
# a dictionary of the possible transitions from each state
# each key is a source state, mapped to a list of the destination states
# that are reachable from the source state via a transition
self._transitions_from = _calculate_from_transitions(self.transition_prob)
# a dictionary of the possible transitions to each state
# each key is a destination state, mapped to a list of source states
# from which the destination is reachable via a transition
self._transitions_to = _calculate_to_transitions(self.transition_prob)
def get_blank_transitions(self):
"""Get the default transitions for the model.
Returns a dictionary of all of the default transitions between any
two letters in the sequence alphabet. The dictionary is structured
with keys as (letter1, letter2) and values as the starting number
of transitions.
"""
return self._transition_pseudo
def get_blank_emissions(self):
"""Get the starting default emmissions for each sequence.
This returns a dictionary of the default emmissions for each
letter. The dictionary is structured with keys as
(seq_letter, emmission_letter) and values as the starting number
of emmissions.
"""
return self._emission_pseudo
def transitions_from(self, state_letter):
"""Get all destination states which can transition from source state_letter.
This returns all letters which the given state_letter can transition
to, i.e. all the destination states reachable from state_letter.
An empty list is returned if state_letter has no outgoing transitions.
"""
if state_letter in self._transitions_from:
return self._transitions_from[state_letter]
else:
return []
def transitions_to(self, state_letter):
"""Get all source states which can transition to destination state_letter.
This returns all letters which the given state_letter is reachable
from, i.e. all the source states which can reach state_later
An empty list is returned if state_letter is unreachable.
"""
if state_letter in self._transitions_to:
return self._transitions_to[state_letter]
else:
return []
def viterbi(self, sequence, state_alphabet):
"""Calculate the most probable state path using the Viterbi algorithm.
This implements the Viterbi algorithm (see pgs 55-57 in Durbin et
al for a full explanation -- this is where I took my implementation
ideas from), to allow decoding of the state path, given a sequence
of emissions.
Arguments:
- sequence -- A Seq object with the emission sequence that we
want to decode.
- state_alphabet -- An iterable (e.g., tuple or list) containing
all of the letters that can appear in the states
"""
# calculate logarithms of the initial, transition, and emission probs
log_initial = self._log_transform(self.initial_prob)
log_trans = self._log_transform(self.transition_prob)
log_emission = self._log_transform(self.emission_prob)
viterbi_probs = {}
pred_state_seq = {}
# --- recursion
# loop over the training sequence (i = 1 .. L)
# NOTE: My index numbers are one less than what is given in Durbin
# et al, since we are indexing the sequence going from 0 to
# (Length - 1) not 1 to Length, like in Durbin et al.
for i in range(0, len(sequence)):
# loop over all of the possible i-th states in the state path
for cur_state in state_alphabet:
# e_{l}(x_{i})
emission_part = log_emission[(cur_state, sequence[i])]
max_prob = 0
if i == 0:
# for the first state, use the initial probability rather
# than looking back to previous states
max_prob = log_initial[cur_state]
else:
# loop over all possible (i-1)-th previous states
possible_state_probs = {}
for prev_state in self.transitions_to(cur_state):
# a_{kl}
trans_part = log_trans[(prev_state, cur_state)]
# v_{k}(i - 1)
viterbi_part = viterbi_probs[(prev_state, i - 1)]
cur_prob = viterbi_part + trans_part
possible_state_probs[prev_state] = cur_prob
# calculate the viterbi probability using the max
max_prob = max(possible_state_probs.values())
# v_{k}(i)
viterbi_probs[(cur_state, i)] = emission_part + max_prob
if i > 0:
# get the most likely prev_state leading to cur_state
for state in possible_state_probs:
if possible_state_probs[state] == max_prob:
pred_state_seq[(i - 1, cur_state)] = state
break
# --- termination
# calculate the probability of the state path
# loop over all states
all_probs = {}
for state in state_alphabet:
# v_{k}(L)
all_probs[state] = viterbi_probs[(state, len(sequence) - 1)]
state_path_prob = max(all_probs.values())
# find the last pointer we need to trace back from
last_state = ""
for state in all_probs:
if all_probs[state] == state_path_prob:
last_state = state
assert last_state != "", "Didn't find the last state to trace from!"
# --- traceback
traceback_seq = []
loop_seq = list(range(1, len(sequence)))
loop_seq.reverse()
# last_state is the last state in the most probable state sequence.
# Compute that sequence by walking backwards in time. From the i-th
# state in the sequence, find the (i-1)-th state as the most
# probable state preceding the i-th state.
state = last_state
traceback_seq.append(state)
for i in loop_seq:
state = pred_state_seq[(i - 1, state)]
traceback_seq.append(state)
# put the traceback sequence in the proper orientation
traceback_seq.reverse()
traceback_seq = "".join(traceback_seq)
return Seq(traceback_seq), state_path_prob
def _log_transform(self, probability):
"""Return log transform of the given probability dictionary (PRIVATE).
When calculating the Viterbi equation, add logs of probabilities rather
than multiplying probabilities, to avoid underflow errors. This method
returns a new dictionary with the same keys as the given dictionary
and log-transformed values.
"""
log_prob = copy.copy(probability)
for key in log_prob:
prob = log_prob[key]
if prob > 0:
log_prob[key] = math.log(log_prob[key])
else:
log_prob[key] = -math.inf
return log_prob
|