prob.py 43 KB
 Mikael Boden committed Jul 13, 2016 1 2 ``````''' Module for classes and functions that are representing and processing basic probabilities. `````` Mikael Boden committed Aug 22, 2018 3 ``````Also includes Markov chain and hidden Markov model. `````` Mikael Boden committed Jul 13, 2016 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 ``````Uses and depends on "Alphabet" that is used to define discrete random variables. ''' import random from sym import * import math ################################################################################################# # Generic utility functions ################################################################################################# def _getMeTuple(alphas, str): """ Handy function that resolves what entries that are being referred to in the case of written wildcards etc. Example y = _getMeTuple([DNA_Alphabet, Protein_Alphabet], '*R') gives y = (None, 'R') alphas: the alphabets str: the string that specifies entries (may include '*' and '-' signifying any symbol) """ assert len(str) == len(alphas), "Entry invalid" if not type(str) is tuple: list = [] for ndx in range(len(alphas)): if str[ndx] == '*' or str[ndx] == '-': list.append(None) else: list.append(str[ndx]) return tuple(list) else: return str ################################################################################################# # Distrib class ################################################################################################# class Distrib(): """ A class for a discrete probability distribution, defined over a specified "Alphabet" TODO: Fix pseudo counts Exclude from counts, specify in constructor, include only when computing probabilities by standard formula (n_a + pseudo_a * N^(1/2)) / (N + N^(1/2)) Exclude from filesaves, include with filereads (optional) """ def __init__(self, alpha, pseudo = 0.0): """ Construct a new distribution for a specified alphabet, using an optional pseudo-count. alpha: alphabet pseudo: either a single "count" that applies to all symbols, OR a distribution/dictionary with counts. """ self.pseudo = pseudo or 0.0 self.alpha = alpha self.cnt = [0.0 for _ in alpha] try: # assume pseudo is a dictionary or a Distrib itself self.tot = 0 symndx = 0 for sym in alpha: cnt = float(pseudo[sym]) self.cnt[symndx] = cnt self.tot = self.tot + cnt symndx += 1 except TypeError: # assume pseudo is a single count for each symbol self.cnt = [float(self.pseudo) for _ in alpha] self.tot = float(self.pseudo) * len(alpha) # track total counts (for efficiency) def observe(self, sym, cntme = 1.0): """ Make an observation of a symbol sym: symbol that is being observed cntme: number/weight of observation (default is 1) """ ndx = self.alpha.symbols.index(sym) self.cnt[ndx] = self.cnt[ndx] + cntme self.tot = self.tot + cntme return def reset(self): """ Re-set the counts of this distribution. Pseudo-counts are re-applied. """ try: self.tot = 0 symndx = 0 for sym in self.alpha: # assume it is a Distribution cnt = float(self.pseudo[sym]) self.cnt[symndx] = cnt self.tot = self.tot + cnt symndx += 1 except TypeError: # assume pseudo is a single count for each symbol self.cnt = [float(self.pseudo) for _ in self.alpha] self.tot = float(self.pseudo) * len(self.alpha) # track total counts (for efficiency) def reduce(self, new_alpha): """ Create new distribution from self, using (smaller) alphabet new_alpha. """ d = Distrib(new_alpha, self.pseudo) for sym in new_alpha: d.observe(sym, self.cnt[self.alpha.index(sym)]) return d def count(self, sym = None): """ Return the absolute count(s) of the distribution or the count for a specified symbol. """ if sym != None: ndx = self.alpha.symbols.index(sym) return self.cnt[ndx] else: d = {} index = 0 for a in self.alpha: d[a] = self.cnt[index] index += 1 return d def add(self, distrib): """ Add the counts for the provided distribution to the present. """ for i in range(len(self.cnt)): cnt = distrib.count(self.alpha[i]) self.cnt[i] += cnt self.tot += cnt def subtract(self, distrib): """ Subtract the counts for the provided distribution from the present. """ for i in range(len(self.cnt)): cnt = distrib.count(self.alpha[i]) self.cnt[i] -= cnt self.tot -= cnt def getSymbols(self): return self.alpha.symbols def __getitem__(self, sym): """ Retrieve the probability of a symbol (ascertained by counts incl pseudo-counts) """ if self.tot > 0.0: return self.count(sym) / self.tot else: return 1.0 / len(self.alpha) # uniform def prob(self, sym = None): """ Retrieve the probability of a symbol OR the probabilities of all symbols (listed in order of the alphabet index). """ if sym != None: return self.__getitem__(sym) elif self.tot > 0: return [ s / self.tot for s in self.cnt ] else: return [ 1.0 / len(self.alpha) for _ in self.cnt ] def __iter__(self): return self.alpha def __str__(self): """ Return a readable representation of the distribution """ str = '< ' for s in self.alpha: str += (s + ("=%4.2f " % self[s])) return str + ' >' def swap(self, sym1, sym2): """ Swap the entries for specified symbols. Useful for reverse complement etc. Note that changes are made to the current instance. Use swapxcopy if you want to leave this instance intact. """ sym1ndx = self.alpha.index(sym1) sym2ndx = self.alpha.index(sym2) tmpcnt = self.cnt[sym1ndx] self.cnt[sym1ndx] = self.cnt[sym2ndx] self.cnt[sym2ndx] = tmpcnt def swapxcopy(self, sym1, sym2): """ Create a new instance with swapped entries for specified symbols. Useful for reverse complement etc. Note that changes are NOT made to the current instance. Use swap if you want to modify this instance. """ newdist = Distrib(self.alpha, self.count()) newdist.swap(sym1, sym2) return newdist def writeDistrib(self, filename = None): """ Write the distribution to a file or string. Note that the total number of counts is also saved, e.g. * 1000 """ str = '' for s in self.alpha: str += (s + ("\t%f\n" % self[s])) str += "*\t%d\n" % self.tot if filename != None: fh = open(filename, 'w') fh.write(str) fh.close() return str def generate(self): """ Generate and return a symbol from the distribution using assigned probabilities. """ alpha = self.alpha p = random.random() # get a random value between 0 and 1 q = 0.0 for sym in alpha: # pick a symbol with a frequency proportional to its probability q = q + self[sym] if p < q: return sym return alpha[len(alpha)] def getmax(self): """ Generate the symbol with the largest probability. """ maxprob = 0.0 maxsym = None for sym in self.alpha: if self[sym] > maxprob or maxprob == 0.0: maxsym = sym maxprob = self[sym] return maxsym def getsort(self): """ Return the list of symbols, in order of their probability. """ symlist = [sym for (sym, _) in self.getProbsort()] return symlist def getProbsort(self): """ Return the list of symbol-probability pairs, in order of their probability. """ s = [(sym, self.prob(sym)) for sym in self.alpha] ss = sorted(s, key=lambda y: y[1], reverse=True) return ss def divergence(self, distrib2): """ Calculate the Kullback-Leibler divergence between two discrete distributions. Note that when self.prob(x) is 0, the divergence for x is 0. When distrib2.prob(x) is 0, it is replaced by 0.0001. """ assert self.alpha == distrib2.alpha sum = 0.0 base = len(self.alpha) for sym in self.alpha: if self[sym] > 0: if distrib2[sym] > 0: sum += math.log(self[sym] / distrib2[sym]) * self[sym] else: sum += math.log(self[sym] / 0.0001) * self[sym] return sum def entropy(self): """ Calculate the information (Shannon) entropy of the distribution. Note that the base is the size of the alphabet, so maximum entropy is by definition 1. Also note that if the probability is exactly zero, it is replaced by a small value to avoid numerical issues with the logarithm. """ sum = 0.0 base = len(self.alpha) for sym in self.alpha: p = self.__getitem__(sym) if p == 0: p = 0.000001 sum += p * math.log(p, base) return -sum def writeDistribs(distribs, filename): """ Write a list/set of distributions to a single file. """ str = '' k = 0 for d in distribs: str += "[%d]\n%s" % (k, d.writeDistrib()) k += 1 fh = open(filename, 'w') fh.write(str) fh.close() def _readDistrib(linelist): """ Extract distribution from a pre-processed list if strings. """ symstr = '' d = {} for line in linelist: line = line.strip() if len(line) == 0 or line.startswith('#'): continue sections = line.split() sym, value = sections[0:2] if len(sym) == 1: if sym != '*': symstr += sym else: raise RuntimeError("Invalid symbol in distribution: " + sym) try: d[sym] = float(value) except ValueError: raise RuntimeError("Invalid value in distribution for symbol " + sym + ": " + value) if len(d) == 0: return None alpha = Alphabet(symstr) `````` Mikael Boden committed Feb 14, 2017 280 `````` if '*' in list(d.keys()): # tot provided `````` Mikael Boden committed Jul 13, 2016 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 `````` for sym in d: if sym != '*': d[sym] = d[sym] * d['*'] distrib = Distrib(alpha, d) return distrib def readDistribs(filename): """ Load a list of distributions from file. Note that if a row contains '* ' then it is assumed that each probability associated with the specific distribution is based on counts. """ fh = open(filename) string = fh.read() distlist = [] linelist = [] for line in string.splitlines(): line = line.strip() if line.startswith('['): if len(linelist) != 0: distlist.append(_readDistrib(linelist)) linelist = [] elif len(line) == 0 or line.startswith('#'): pass # comment or blank line --> ignore else: linelist.append(line) # end for-loop, reading the file if len(linelist) != 0: distlist.append(_readDistrib(linelist)) fh.close() return distlist def readDistrib(filename): """ Load a distribution from file. Note that if a row contains '* ' then it is assumed that each probability is based on counts. """ dlist = readDistribs(filename) if len(dlist) > 0: # if at least one distribution was in the file... return dlist[0] # return the first import re def _readMultiCount(linelist, format = 'JASPAR'): ncol = 0 symcount = {} if format == 'JASPAR2010': for line in linelist: line = line.strip() if len(line) > 0: name = line.split()[0] counts = [] for txt in re.findall(r'\w+', line): try: y = float(txt) counts.append(y) except ValueError: pass # ignore non-numeric entries if len(counts) != ncol and ncol != 0: raise RuntimeError('Invalid row in file: ' + line) ncol = len(counts) if len(name) == 1: # proper symbol symcount[name] = counts `````` Mikael Boden committed Feb 14, 2017 341 `````` alpha = Alphabet(''.join(list(symcount.keys()))) `````` Mikael Boden committed Jul 13, 2016 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 `````` distribs = [] for col in range(ncol): d = dict([(sym, symcount[sym][col]) for sym in symcount]) distribs.append(Distrib(alpha, d)) elif format == 'JASPAR': alpha_str = 'ACGT' alpha = Alphabet(alpha_str) cnt = 0 for sym in alpha_str: line = linelist[cnt].strip() counts = [] for txt in re.findall(r'\w+', line): try: y = float(txt) counts.append(y) except ValueError: pass # ignore non-numeric entries if len(counts) != ncol and ncol != 0: raise RuntimeError('Invalid row in file: ' + line) ncol = len(counts) symcount[sym] = counts cnt += 1 distribs = [] for col in range(ncol): d = dict([(sym, symcount[sym][col]) for sym in symcount]) distribs.append(Distrib(alpha, d)) else: raise RuntimeError('Unsupported format: ' + format) return distribs def readMultiCounts(filename, format = 'JASPAR'): """ Read a file of raw counts for multiple distributions over the same set of symbols for (possibly) multiple (named) entries. filename: name of file format: format of file, default is 'JASPAR' exemplified below >MA0001.1 SEP4 0 3 79 40 66 48 65 11 65 0 94 75 4 3 1 2 5 2 3 3 1 0 3 4 1 0 5 3 28 88 2 19 11 50 29 47 22 81 1 6 returns a dictionary of Distrib's, key:ed by entry name (e.g. MA001.1) """ fh = open(filename) linelist = [] entryname = '' entries = {} for row in fh: row = row.strip() if len(row) < 1: continue if row.startswith('>'): if len(linelist) > 0: entries[entryname] = _readMultiCount(linelist, format=format) linelist = [] entryname = row[1:].split()[0] else: linelist.append(row) if len(linelist) > 0: entries[entryname] = _readMultiCount(linelist, format=format) fh.close() return entries def readMultiCount(filename, format = 'JASPAR'): """ Read a file of raw counts for multiple distributions over the same set of symbols. filename: name of file format: format of file, default is 'JASPAR' exemplified below 0 3 79 40 66 48 65 11 65 0 94 75 4 3 1 2 5 2 3 3 1 0 3 4 1 0 5 3 28 88 2 19 11 50 29 47 22 81 1 6 returns a list of Distrib's """ d = readMultiCounts(filename, format=format) if len(d) > 0: `````` Mikael Boden committed Feb 14, 2017 415 `````` return list(d.values())[0] `````` Mikael Boden committed Jul 13, 2016 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 `````` ################################################################################################# # Joint class ################################################################################################# class Joint(object): """ A joint probability class. The JP is represented as a distribution over n-tuples where n is the number of variables. Variables can be for any defined alphabet. The size of each alphabet determine the number of entries in the table (with probs that add up to 1.0) """ def __init__(self, alphas): """ A distribution of n-tuples. alphas: Alphabet(s) over which the distribution is defined """ if type(alphas) is Alphabet: self.alphas = tuple( [alphas] ) elif type(alphas) is tuple: self.alphas = alphas else: self.alphas = tuple( alphas ) self.store = TupleStore(self.alphas) self.totalCnt = 0 def getN(self): """ Retrieve the number of distributions/random variables. """ return len(self.alphas) def __iter__(self): return self.store.__iter__() def reset(self): """ Re-set the counts of this joint distribution. Pseudo-counts are re-applied. """ for entry in self.store: self.store[entry] = None self.totalCnt = 0 def observe(self, key, cnt = 1): """ Make an observation of a tuple/key key: tuple that is being observed cnt: number/weight of observation (default is 1) """ key = _getMeTuple(self.alphas, key) if not None in key: score = self.store[key] if (score == None): score = 0 self.totalCnt += cnt self.store[key] = score + cnt else: # there are wildcards in the key allkeys = [mykey for mykey in self.store.getAll(key)] mycnt = float(cnt)/float(len(allkeys)) self.totalCnt += cnt for mykey in allkeys: score = self.store[mykey] if (score == None): score = 0 self.store[mykey] = score + mycnt return def count(self, key): """ Return the absolute count that is used for the joint probability table. """ key = _getMeTuple(self.alphas, key) score = self.store[key] if (score == None): score = 0.0 for match in self.store.getAll(key): y = self.store[match] if y != None: score += y return score def __getitem__(self, key): """ Determine and return the probability of a specified expression of the n-tuple which can involve "wildcards" Note that no assumptions are made regarding independence. """ key = _getMeTuple(self.alphas, key) score = self.store[key] if (score == None): score = 0.0 for match in self.store.getAll(key): y = self.store[match] if y != None: score += y if self.totalCnt == 0: return 0.0 return float(score) / float(self.totalCnt) def __str__(self): """ Return a textual representation of the JP. """ str = '< ' if self.totalCnt == 0.0: return str + 'None >' for s in self.store: if self[s] == None: y = 0.0 else: y = self[s] str += (''.join(s) + ("=%4.2f " % y)) return str + ' >' def items(self, sort = False): """ In a dictionary-like way return all entries as a list of 2-tuples (key, prob). If sort is True, entries are sorted in descending order of probability. Note that this function should NOT be used for big (>5 variables) tables.""" if self.totalCnt == 0.0: return [] ret = [] for s in self.store: if self[s] != None: ret.append((s, self[s])) if sort: return sorted(ret, key=lambda v: v[1], reverse=True) return ret class IndepJoint(Joint): def __init__(self, alphas, pseudo = 0.0): """ A distribution of n-tuples. All positions are assumed to be independent. alphas: Alphabet(s) over which the distribution is defined """ self.pseudo = pseudo if type(alphas) is Alphabet: self.alphas = tuple( [alphas] ) elif type(alphas) is tuple: self.alphas = alphas else: self.alphas = tuple( alphas ) self.store = [Distrib(alpha, pseudo) for alpha in self.alphas] def getN(self): """ Retrieve the number of distributions/random variables. """ return len(self.alphas) def __iter__(self): return TupleStore(self.alphas).__iter__() def reset(self): """ Re-set the counts of each distribution. Pseudo-counts are re-applied. """ self.store = [Distrib(alpha, self.pseudo) for alpha in self.alphas] def observe(self, key, cnt = 1, countGaps = True): """ Make an observation of a tuple/key key: tuple that is being observed cnt: number/weight of observation (default is 1) """ assert len(key) == len(self.store), "Number of symbols must agree with the number of positions" for i in range(len(self.store)): subkey = key[i] if subkey == '-' and countGaps == False: continue if subkey == '*' or subkey == '-': for sym in self.alphas[i]: score = self.store[i][sym] if (score == None): score = 0 self.store[i].observe(sym, float(cnt)/float(len(self.alphas[i]))) else: score = self.store[i][subkey] if (score == None): score = 0 self.store[i].observe(subkey, cnt) def __getitem__(self, key): """ Determine and return the probability of a specified expression of the n-tuple which can involve "wildcards" Note that variables are assumed to be independent. """ assert len(key) == len(self.store), "Number of symbols must agree with the number of positions" prob = 1.0 for i in range(len(self.store)): mykey = key[i] if mykey == '*' or mykey == '-': pass # same as multiplying with 1.0 (all symbols possible) else: prob *= self.store[i][mykey] return prob def get(self, sym, pos): """ Retrieve the probability of a specific symbol at a specified position. """ mystore = self.store[pos] return mystore[sym] def getColumn(self, column, count = False): """ Retrieve all the probabilities (or counts) for a specified position. Returns values as a dictionary, with symbol as key.""" d = {} for a in self.alphas[column]: if count: # absolute count d[a] = self.store[column].count(a) else: # probability d[a] = self.store[column][a] return d def getRow(self, sym, count = False): """ Retrieve the probabilities (or counts) for a specific symbol over all columns/positions. Returns a list of values in the order of the variables/alphabets supplied to the constructor. """ d = [] for store in self.store: if count: # absolute count d.append(store.count(sym)) else: # probability d.append(store[sym]) return d def getMatrix(self, count = False): """ Retrieve the full matrix of probabilities (or counts) """ d = {} for a in self.alphas[0]: d[a] = self.getRow(a, count) return d def displayMatrix(self, count = False): """ Pretty-print matrix """ `````` Mikael Boden committed Feb 14, 2017 631 `````` print((" \t%s" % (''.join("\t%5d" % (i + 1) for i in range(len(self.alphas)))))) `````` Mikael Boden committed Jul 13, 2016 632 633 `````` for a in self.alphas[0]: if count: `````` Mikael Boden committed Feb 14, 2017 634 `````` print(("%s\t%s" % (a, ''.join("\t%5d" % (y) for y in self.getRow(a, True))))) `````` Mikael Boden committed Jul 13, 2016 635 `````` else: `````` Mikael Boden committed Feb 14, 2017 636 `````` print(("%s\t%s" % (a, ''.join("\t%5.3f" % (y) for y in self.getRow(a))))) `````` Mikael Boden committed Jul 13, 2016 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 `````` def __str__(self): """ Text representation of the table. Note that size is an issue so big tables will not be retrieved and displayed. """ if self.alphas > 5: return '< ... too large to process ... >' tstore = TupleStore(self.alphas) str = '< ' for key in tstore: p = 1.0 for i in range(len(self.store)): value = self.store[i][key[i]] if value != None and value != 0.0: p *= value else: p = 0; break; str += (''.join(key) + ("=%4.2f " % p)) return str + ' >' def items(self, sort = False): """ In a dictionary-like way return all entries as a list of 2-tuples (key, prob). If sort is True, entries are sorted in descending order of probability. Note that this function should NOT be used for big (>5 variables) tables.""" tstore = TupleStore(self.alphas) ret = [] for key in tstore: p = 1.0 for i in range(len(self.store)): value = self.store[i][key[i]] if value != None and value != 0.0: p *= value else: p = 0; break; if p > 0.0: ret.append((key, p)) if sort: return sorted(ret, key=lambda v: v[1], reverse=True) return ret `````` Mikael Boden committed Aug 22, 2018 678 679 680 681 ``````################################################################################################# # Naive Bayes' classifier ################################################################################################# `````` Mikael Boden committed Jul 13, 2016 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 ``````class NaiveBayes(): """ NaiveBayes implements a classifier: a model defined over a class variable and conditional on a list of discrete feature variables. Note that feature variables are assumed to be independent. """ def __init__(self, inputs, output, pseudo_input = 0.0, pseudo_output = 0.0): """ Initialise a classifier. inputs: list of alphabets that define the values that input variables can take. output: alphabet that defines the possible values the output variable takes pseudo_input: pseudo-count used for each input variable (default is 0.0) pseudo_output: pseudo-count used for the output variable (default is 0.0) """ if type(inputs) is Alphabet: self.inputs = tuple( [inputs] ) elif type(inputs) is tuple: self.inputs = inputs else: self.inputs = tuple( inputs ) self.condprobs = {} # store conditional probabilities as a dictionary (class is key) for outsym in output: # GIVEN the class # for each input variable initialise a conditional probability self.condprobs[outsym] = [ Distrib(input, pseudo_input) for input in self.inputs ] self.classprob = Distrib(output, pseudo_output) # the class prior def observe(self, inpseq, outsym): """ Record an observation of an input sequence of feature values that belongs to a class. inpseq: sequence/list of feature values, e.g. 'ATG' outsym: the class assigned to these feature values. """ condprob = self.condprobs[outsym] for i in range(len(inpseq)): condprob[i].observe(inpseq[i]) self.classprob.observe(outsym) def __getitem__(self, key): """ Determine and return the class probability GIVEN a specified n-tuple of feature values The class probability is given as an instance of Distrib. """ out = Distrib(self.classprob.alpha) for outsym in self.classprob.getSymbols(): condprob = self.condprobs[outsym] prob = self.classprob[outsym] for i in range(len(key)): prob *= condprob[i][key[i]] or 0.0 out.observe(outsym, prob) return out `````` Mikael Boden committed Aug 16, 2018 725 `````` `````` Mikael Boden committed Aug 22, 2018 726 727 728 729 ``````################################################################################################# # Markov chain ################################################################################################# `````` Mikael Boden committed Aug 16, 2018 730 ``````class MarkovChain(): `````` Mikael Boden committed Aug 22, 2018 731 `````` """ Markov Chain in a simple form; supports higher-orders and can determine (joint) probability of sequence. `````` Mikael Boden committed Aug 16, 2018 732 733 `````` """ `````` Mikael Boden committed Aug 22, 2018 734 735 736 737 738 739 740 `````` def __init__(self, alpha, order = 1, startsym = '^', endsym = '\$'): """ Construct a new Markov chain based on a given alphabet of characters. alpha: alphabet of allowed characters and states order: the number of states to include in memory (default is 1) startsym: the symbol to mark the first character in the internal sequence, and the first state endsym: the symbol to mark the termination of the internal sequence (and the last state) """ `````` Mikael Boden committed Aug 16, 2018 741 `````` self.order = order `````` Mikael Boden committed Aug 22, 2018 742 743 744 `````` self.startsym = startsym self.endsym = endsym self.alpha = getTerminatedAlphabet(alpha, self.startsym, self.endsym) `````` Mikael Boden committed Aug 16, 2018 745 746 747 `````` self.transit = TupleStore([self.alpha for _ in range(order)]) # transition probs, i.e. given key (prev state/s) what is the prob of current state def _getpairs(self, term_seq): `````` Mikael Boden committed Aug 22, 2018 748 `````` """ Return a tuple of all (tuple) Markov pairs from a sequence. Used internally. """ `````` Mikael Boden committed Aug 16, 2018 749 750 751 752 753 754 755 756 `````` ret = [] for i in range(len(term_seq) - self.order): past = tuple(term_seq[i:i + self.order]) present = term_seq[i + self.order] ret.append(tuple([past, present])) return tuple(ret) def observe(self, wholeseq): `````` Mikael Boden committed Aug 22, 2018 757 758 759 760 `````` """ Set parameters of Markov chain by counting transitions, as observed in the sequence. wholeseq: the sequence not including the termination symbols. """ myseq = _terminate(wholeseq, self.order, self.startsym, self.endsym) `````` Mikael Boden committed Aug 16, 2018 761 762 763 764 765 766 767 768 `````` for (past, present) in self._getpairs(myseq): d = self.transit[past] if not d: # no distrib d = Distrib(self.alpha) self.transit[past] = d d.observe(present) def __getitem__(self, wholeseq): `````` Mikael Boden committed Aug 22, 2018 769 770 771 772 773 `````` """ Determine the log probability of a given sequence. wholeseq: the sequence not including the termination symbols. returns the joint probability """ myseq = _terminate(wholeseq, self.order, self.startsym, self.endsym) `````` Mikael Boden committed Aug 16, 2018 774 775 776 777 778 779 780 781 782 `````` logp = 0 for (past, present) in self._getpairs(myseq): d = self.transit[past] if not d: return None p = d[present] if p == 0: return None logp += math.log(p) `````` Mikael Boden committed Aug 22, 2018 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 `````` return logp def _terminate(unterm_seq, order = 1, startsym = '^', endsym = '\$'): """ Terminate sequence with start and end symbols """ term_seq = [startsym for _ in range(order)] term_seq.extend(unterm_seq) term_seq.append(endsym) return term_seq def getTerminatedAlphabet(alpha, startsym = '^', endsym = '\$'): """ Amend the given alphabet with termination symbols """ return Alphabet(alpha.symbols + tuple([startsym, endsym])) ################################################################################################# # Hidden Markov model (HMM) ################################################################################################# class HMM(): """ Basic, first-order HMM. Has functionality to set up HMM, and query it with Viterbi and Forward algorithms.""" def __init__(self, states, symbols, startstate = '^', endstate = '\$'): """ Construct HMM with states and symbols, here given as strings of characters. > cpg_hmm = prob.HMM('HL','ACGT') """ if isinstance(states, str): states = Alphabet(states) self.mystates = getTerminatedAlphabet(states, startstate, endstate) if isinstance(symbols, str): symbols = Alphabet(symbols) self.mysymbols = getTerminatedAlphabet(symbols, startstate, endstate) self.a = dict() # transition probabilities self.e = dict() # emission probabilities self.startsym = startstate self.endsym = endstate def transition(self, fromstate, distrib): """ Add a transition to the HMM, determining with the probability of transitions, e.g. > cpg_hmm.transition('^',{'^':0,'\$':0,'H':0.5,'L':0.5}) > cpg_hmm.transition('H',{'^':0,'\$':0.001,'H':0.5,'L':0.5}) > cpg_hmm.transition('L',{'^':0,'\$':0.001,'H':0.4,'L':0.6}) > cpg_hmm.transition('\$',{'^':1,'\$':0,'H':0,'L':0}) """ if not isinstance(distrib, Distrib): distrib = Distrib(self.mystates, distrib) self.a[fromstate] = distrib def emission(self, state, distrib): """ Add an emission probability to the HMM, e.g. > cpg_hmm.emission('^',{'^':1,'\$':0,'A':0,'C':0,'G':0,'T':0}) > cpg_hmm.emission('H',{'^':0,'\$':0,'A':0.2,'C':0.3,'G':0.3,'T':0.2}) > cpg_hmm.emission('L',{'^':0,'\$':0,'A':0.3,'C':0.2,'G':0.2,'T':0.3}) > cpg_hmm.emission('\$',{'^':0,'\$':1,'A':0,'C':0,'G':0,'T':0}) """ if not isinstance(distrib, Distrib): distrib = Distrib(self.mysymbols, distrib) self.e[state] = distrib def joint(self, symseq, stateseq): """ Determine the joint probability of the sequence and the given path. :param symseq: sequence of characters :param stateseq: sequence of states :return: the probability """ X = _terminate(symseq, 1, self.startsym, self.endsym) P = _terminate(stateseq, 1, self.startsym, self.endsym) p = 1 for i in range(len(X) - 1): p = p * self.e[P[i]][X[i]] * self.a[P[i]][P[i + 1]] return p def viterbi(self, symseq, V = dict(), trace = dict()): """ Determine the Viterbi path (the most probable sequence of states) given a sequence of symbols :param symseq: sequence of symbols :param V: the Viterbi dynamic programming variable as a matrix (optional; pass an empty dict if you need it) :param trace: the traceback (optional; pass an empty dict if you need it) :return: the Viterbi path as a string of characters > X = 'GGCACTGAA' # sequence of characters > states = cpg_hmm.viterbi(X) > print(states) """ `````` Mikael Boden committed Aug 22, 2018 867 `````` X = _terminate(symseq, 1, self.startsym, self.endsym) # put start and end symbols on sequence `````` Mikael Boden committed Aug 22, 2018 868 869 `````` # Initialise state scores for each index in X for state in self.mystates: `````` Mikael Boden committed Aug 22, 2018 870 871 `````` # Fill in emission probabilities in V for each index of X # (only the first position is really needed) `````` Mikael Boden committed Aug 22, 2018 872 873 `````` V[state] = [self.e[state][x] for x in X] trace[state] = [] `````` Mikael Boden committed Aug 22, 2018 874 `````` # Next loop through the sequence `````` Mikael Boden committed Aug 22, 2018 875 `````` for j in range(len(X) - 1): `````` Mikael Boden committed Aug 22, 2018 876 877 `````` i = j + 1 # sequence index that we're processing start with 1, not 0 for tostate in self.mystates: # check each state for i = 1, ... `````` Mikael Boden committed Aug 22, 2018 878 `````` tracemax = 0 `````` Mikael Boden committed Aug 22, 2018 879 `````` beststate = None # the state v with max[Vv(i-1) * t(v,u)] `````` Mikael Boden committed Aug 22, 2018 880 `````` for fromstate in self.mystates: `````` Mikael Boden committed Aug 22, 2018 881 `````` # determine the best score propagated forward from previous state `````` Mikael Boden committed Aug 22, 2018 882 883 884 885 `````` score = V[fromstate][i - 1] * self.a[fromstate][tostate] if score > tracemax: beststate = fromstate tracemax = score `````` Mikael Boden committed Aug 22, 2018 886 `````` # record the transition that will appear in the traceback `````` Mikael Boden committed Aug 22, 2018 887 `````` trace[tostate].append(beststate) `````` Mikael Boden committed Aug 22, 2018 888 `````` # finalise the dynamic programming score for current i in state u `````` Mikael Boden committed Aug 22, 2018 889 `````` V[tostate][i] = self.e[tostate][X[i]] * tracemax `````` Mikael Boden committed Aug 22, 2018 890 `````` # finally, assemble the string that describes the most probable path `````` Mikael Boden committed Aug 22, 2018 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 `````` ret = '' traced = '\$' for j in range(len(X)): i = len(X) - 2 - j traced = trace[traced][i] if j > 0 and j < len(X) - 2: ret = traced + ret return ret def forward(self, symseq, F = dict()): """ Determine the probability of the sequence, summing over all possible state paths :param symseq: sequence of symbols :param F: the Forward dynamic programming variable as a matrix (optional; pass an empty dict if you need it) :return: the probability > X = 'GGCACTGAA' # sequence of characters > prob = cpg_hmm.forward(X) > print(prob) """ X = _terminate(symseq, 1, self.startsym, self.endsym) # Initialise state scores for each index in X for state in self.mystates: # Fill in emission probabilities for each index in X F[state] = [self.e[state][x] for x in X] for j in range(len(X) - 1): i = j + 1 # sequence index that we're processing for tostate in self.mystates: mysum = 0 for fromstate in self.mystates: mysum += F[fromstate][i - 1] * self.a[fromstate][tostate] F[tostate][i] = self.e[tostate][X[i]] * mysum traced = '\$' return F[traced][len(X) - 1] def writeHTML(self, X, Viterbi, Trace = None, filename = None): """ Generate HTML that displays a DP matrix from Viterbi (or Forward) algorithms. > from IPython.core.display import HTML > X = 'GGCACTGAA' # sequence of characters > V = dict() > T = dict() > cpg_hmm.viterbi(X, V, T) > HTML(cpg_hmm.writeHTML(X, V, T)) """ html = '''\nHMM dynamic programming matrix\n
\n' html += '\n' for state in Viterbi: html += '\n' % str(state) html += '\n' # process each sequence symbol X = _terminate(X, 1, self.startsym, self.endsym) for row in range(len(X)): html += '\n' html += '\n' % (row, str(X[row])) for state in Viterbi: if Trace and row > 0: html += '\n' % (str(state),X[row],self.e[state][X[row]],str(state),Viterbi[state][row],Trace[state][row - 1],self.a[Trace[state][row - 1]][state] if Trace[state][row - 1] != None else 0) else: html += '\n' % (str(state),X[row],self.e[state][X[row]],str(state),Viterbi[state][row]) html += '\n' html += '\n' html += '' if filename: fh = open(filename, 'w') fh.write(html) fh.close() return html `````` Mikael Boden committed Jan 07, 2019 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 `````` ################################################################################################# # The univariate Gaussian density function. ################################################################################################# class Gaussian(): mu = None # mean sigma = None # standard deviation sigmaSquared = None # variance def ROOT_2PI(self): return (2 * math.pi) ** 2 def LOG_ROOT_2PI(self): return 0.5 * (math.log(2) + math.log(math.pi)) def __init__(self, mean, variance): """ Creates a univariate Gaussian distribution with the given fixed mean and variance. """ self.mu = mean self.sigmaSquared = variance; self.sigma = math.sqrt(variance); self.normConst = self.sigma * self.ROOT_2PI(); self.logNormConst = (0.5 * math.log(variance)) + self.LOG_ROOT_2PI(); def __str__(self): return "<" + "%5.3f" % self.mu + u"\u00B1%5.3f" % self.sigma + ">" def getDensity(self, x): """ Returns the density of this Gaussian distribution at the given value. """ return (math.exp(-math.pow((x - self.mu), 2) / (2 * self.sigmaSquared)) / self.normConst); def __getitem__(self, value): """ Get the probability density of a specified value for this Gaussian """ return self.getDensity(value) def getMean(self): return self.mu def getVariance(self): return self.sigmaSquared def getLogDensity(self, x): """ Returns the natural log of the density of this Gaussian distribution at the given value. """ return (-math.pow((x - self.mu), 2) / (2 * self.sigmaSquared)) - self.logNormConst; def sample(self): """ Returns a value sampled from this Gaussian distribution. The implementation uses the Box - Muller transformation [G.E.P.Box and M.E.Muller(1958) "A note on the generation of random normal deviates". Ann.Math.Stat 29: 610 - 611]. """ U = random.random() # get a random value between 0 and 1 V = random.random() # get a random value between 0 and 1 return (self.mu + (self.sigma * math.sin(2 * math.pi * V) * math.sqrt((-2 * math.log(U))))) def estimate(samples, count = None): """ Create a density based on the specified samples. Optionally, provide an iterable with the corresponding counts/weights. """ mean = 0 if count == None: for i in range(len(samples)): mean += samples[i] / len(samples) diff = 0 for i in range(len(samples)): diff += (mean - samples[i]) * (mean - samples[i]); if (diff == 0): return None return Gaussian(mean, diff / len(samples)) elif len(count) == len(samples): totcnt = 0 for i in range(len(samples)): mean += samples[i] * count[i] totcnt += count[i] mean /= totcnt diff = 0 for i in range(len(samples)): diff += (mean - samples[i]) * (mean - samples[i]) * count[i]; if (diff == 0): return None return Gaussian(mean, diff / totcnt) ################################################################################################# # The univariate Poisson distribution ################################################################################################# class Poisson(): def __init__(self, LAMBDA): """ * Define a Poisson distribution * @ param lambda the average number of events per interval """ self.LAMBDA = LAMBDA def p(self, k): """ * The probability mass function * @param k the number of events in the interval * @return the probability of k events """ return math.exp(k * math.log(self.LAMBDA) - self.LAMBDA - lgamma(k + 1)) def cdf(self, k): """ * The cumulative probability function. * The implementation calls the PMF for all values i from 0 to floor(k) * @param k the number of events in the interval * @return the cumulative probability of k events * https://en.wikipedia.org/wiki/Poisson_distribution """ sum = 0 for i in range(k + 1): sum += self.p(i) if (sum >= 1.0): # happens only due to poor numerical precision return 1.0 return sum def lgamma(x): """ * Returns an approximation of the log of the Gamma function of x. Laczos * Approximation Reference: Numerical Recipes in C * http://www.library.cornell.edu/nr/cbookcpdf.html """ cof = [ 76.18009172947146, -86.50532032941677, 24.01409824083091, -1.231739572450155, 0.1208650973866179e-2, -0.5395239384953e-5 ] y = x tmp = x + 5.5 tmp -= ((x + 0.5) * math.log(tmp)) ser = 1.000000000190015 for j in range(len(cof)): y += 1 ser += (cof[j] / y) return (-tmp + math.log(2.5066282746310005 * ser / x))``````