├── .gitignore ├── setup.py ├── README.md ├── LICENSE └── pykov.py /.gitignore: -------------------------------------------------------------------------------- 1 | __pycache__ 2 | *.pyc -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python 2 | 3 | from distutils.core import setup 4 | 5 | setup( 6 | name='pykov', 7 | version='1.1', 8 | description='Pykov is a tiny Python module on finite regular Markov chains.', 9 | author='Riccardo Scalco', 10 | author_email='riccardo.scalco@gmail.com', 11 | url='https://github.com/riccardoscalco/Pykov', 12 | py_modules=['pykov'], 13 | install_requires=['scipy', 'numpy', 'six'], 14 | ) 15 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | 2 | Welcome to Pykov docs 3 | ==================== 4 | 5 | Pykov is a tiny Python module on *finite regular Markov chains*. 6 | 7 | You can define a Markov chain from scratch or read it from a text file according specific format. Pykov is versatile, being it able to manipulate the chain, inserting and removing nodes, and to calculate various kind of quantities, like the steady state distribution, mean first passage times, random walks, absorbing times, and so on. 8 | 9 | Pykov is licensed under the terms of the **GNU General Public License** as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. 10 | 11 | --------------- 12 | 13 | Installation 14 | ------------- 15 | Pykov can be installed/upgraded via [`pip`](http://pip.readthedocs.org/en/latest/#) 16 | ```sh 17 | $ pip install git+git://github.com/riccardoscalco/Pykov@master #both Python2 and Python3 18 | $ pip install --upgrade git+git://github.com/riccardoscalco/Pykov@master 19 | ``` 20 | Note that Pykov depends on **numpy** and **scipy**. 21 | 22 | ----------------- 23 | 24 | Getting started 25 | ------------------ 26 | 27 | Open your favourite Python shell and import pykov: 28 | ```python 29 | >>> import pykov 30 | >>> pykov.__version__ 31 | ``` 32 | 33 | ###Vector class 34 | The **Vector class** inherits from python `collections.OrderedDict`, which means it has the same behaviors and features of python ordered dictionaries, with few exceptions. The states and the corresponding probabilities are the keys and the values of the dictionary, respectively. 35 | 36 | collections.OrderedDict is used instead of the default dict because from Python 3.3 onwards, dict is non-deterministic for security reasons (see this [stackoverflow question](http://stackoverflow.com/questions/14956313/dictionary-ordering-non-deterministic-in-python3)). For programs that needs determinism (such as simulations), an OrderedDict should be passed. But for programs where determinism is not an issue, dict can be used. 37 | 38 | Definition of a `pykov.Vector()`: 39 | ```python 40 | >>> p = pykov.Vector() 41 | ``` 42 | You can *get* and *set* states in many ways: 43 | ```python 44 | >>> p['A'] = .2 45 | >>> p 46 | {'A': 0.2} 47 | 48 | >>> # Non-deterministic example 49 | >>> p = pykov.Vector({'A':.3, 'B':.7}) 50 | >>> p 51 | {'A':0.3, 'B':0.7} 52 | 53 | >>> # Deterministic example 54 | >>> data = collections.OrderedDict((('A', .3), ('B', .7))) 55 | >>> p = pykov.Vector(data) 56 | >>> p 57 | {'A':0.3, 'B':0.7} 58 | 59 | >>> pykov.Vector(A=.3, B=.6, C=.1) 60 | {'A':0.3, 'B':0.6, 'C':0.1} 61 | ``` 62 | States not belonging to the vector have zero probability, moreover states with zero probability are not shown: 63 | ```python 64 | >>> q = pykov.Vector(C=.4, B=.6) 65 | >>> q['C'] 66 | 0.4 67 | >>> q['Z'] 68 | 0.0 69 | >>> 'Z' in q 70 | False 71 | 72 | >>> q['Z']=.2 73 | >>> q 74 | {'C': 0.4, 'B': 0.6, 'Z': 0.2} 75 | >>> 'Z' in q 76 | True 77 | >>> q['Z']=0 78 | >>> q 79 | {'C': 0.4, 'B': 0.6} 80 | >>> 'Z' in q 81 | False 82 | ``` 83 | 84 | #### Vector operations 85 | ##### **Sum** 86 | A `pykov.Vector()` instance can be added or substracted to another `pykov.Vector()` instance: 87 | ```python 88 | >>> p = pykov.Vector(A=.3, B=.7) 89 | >>> q = pykov.Vector(C=.5, B=.5) 90 | >>> p + q 91 | {'A': 0.3, 'C': 0.5, 'B': 1.2} 92 | >>> p - q 93 | {'A': 0.3, 'C': -0.5, 'B': 0.2} 94 | >>> q - p 95 | {'A': -0.3, 'C': 0.5, 'B': -0.2} 96 | ``` 97 | 98 | ##### **Product** 99 | A `pykov.Vector()` instance can be multiplied by a scalar. The *dot product* with another `pykov.Vector()` or `pykov.Matrix()` instance is also supported. 100 | ```python 101 | >>> p = pykov.Vector(A=.3, B=.7) 102 | >>> p * 3 103 | {'A': 0.9, 'B': 2.1} 104 | >>> 3 * p 105 | {'A': 0.9, 'B': 2.1} 106 | >>> q = pykov.Vector(C=.5, B=.5) 107 | >>> p * q 108 | 0.35 109 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 110 | >>> p * T 111 | {'A': 0.91, 'B': 0.09} 112 | >>> T * p 113 | {'A': 0.42, 'B': 0.3} 114 | ``` 115 | 116 | #### Vector methods 117 | 118 | ##### **sum()** 119 | Sum the probability values. 120 | ```python 121 | >>> p = pykov.Vector(A=.3, B=.7) 122 | >>> p.sum() 123 | 1.0 124 | ``` 125 | 126 | ##### **sort(reverse=False)** 127 | Return a list of tuples `(state, probability)` sorted according the probability values. 128 | ```python 129 | >>> p = pykov.Vector({'A':.3, 'B':.1, 'C':.6}) 130 | >>> p.sort() 131 | [('B', 0.1), ('A', 0.3), ('C', 0.6)] 132 | >>> p.sort(reverse=True) 133 | [('C', 0.6), ('A', 0.3), ('B', 0.1)] 134 | ``` 135 | ##### **choose(random_func=None)** 136 | Choose a state at random, according to its probability. 137 | ```python 138 | >>> p = pykov.Vector(A=.3, B=.7) 139 | >>> p.choose() 140 | 'B' 141 | >>> p.choose() 142 | 'B' 143 | >>> p.choose() 144 | 'A' 145 | ``` 146 | 147 | Optionally, if you need to supply your own random number generator, you can pass a function that takes two inputs (the min and max) and Pykov will use that function. 148 | ```python 149 | >>> def FakeRandom(min, max): return 0.01 150 | >>> p = pykov.Vector(A=.05, B=.4, C=.4, D=.15) 151 | >>> p.choose(FakeRandom) 152 | 'A' 153 | ``` 154 | 155 | 156 | ##### **normalize()** 157 | Normalize the `pykov.Vector`, after normalization the probabilities sum to 1. 158 | ```python 159 | >>> p = pykov.Vector({'A':3, 'B':1, 'C':6}) 160 | >>> p.sum() 161 | 10.0 162 | >>> p.normalize() 163 | >>> p 164 | {'A': 0.3, 'C': 0.6, 'B': 0.1} 165 | >>> p.sum() 166 | 1.0 167 | ``` 168 | 169 | ##### **copy()** 170 | Return a shallow copy. 171 | ```python 172 | >>> p = pykov.Vector(A=.3, B=.7) 173 | >>> q = p.copy() 174 | >>> p['C'] = 1. 175 | >>> q 176 | {'A': 0.3, 'B': 0.7} 177 | ``` 178 | 179 | ##### **entropy()** 180 | Return the Shannon entropy, defined as $H(p) = \sum_i p_i \ln p_i$. 181 | ```python 182 | >>> p = pykov.Vector(A=.3, B=.7) 183 | >>> p.entropy() 184 | 0.6108643020548935 185 | ``` 186 | For further details, have a look at *Khinchin A. I., Mathematical Foundations of Information Theory Dover, 1957*. 187 | 188 | ##### **dist(q)** 189 | Return the distance to another `pykov.Vector`, defined as $d(p,q) = \sum_i | p_i - q_i |$. 190 | ```python 191 | >>> p = pykov.Vector(A=.3, B=.7) 192 | >>> q = pykov.Vector(C=.5, B=.5) 193 | >>> p.dist(q) 194 | 1.0 195 | ``` 196 | 197 | ##### **relative_entropy(q)** 198 | Return the *Kullback-Leibler* distance, defined as $d(p,q) = \sum_i p_i \ln (p_i/q_i)$. 199 | ```python 200 | >>> p = pykov.Vector(A=.3, B=.7) 201 | >>> q = pykov.Vector(A=.4, B=.6) 202 | >>> p.relative_entropy(q) #d(p,q) 203 | 0.02160085414354654 204 | >>> q.relative_entropy(p) #d(q,p) 205 | 0.022582421084357485 206 | ``` 207 | Note that the Kullback-Leibler distance is not symmetric. 208 | 209 | ------------ 210 | 211 | ### Matrix class 212 | The `pykov.Matrix()` class inherits from python collections.OrderedDict. Similar to the default dict, OrderedDict `keys` are `tuple` of states, OrderedDict `values` are the matrix entries. Indexes do not need to be `int`, they can be `string`, as the states of a `pykov.Vector()`. 213 | 214 | Definition of `pykov.Matrix()`: 215 | ```python 216 | >>> T = pykov.Matrix() 217 | ``` 218 | You can *get* and *set* items in many ways: 219 | ```python 220 | >>> T = pykov.Matrix() 221 | >>> T[('A','B')] = .3 222 | >>> T 223 | {('A', 'B'): 0.3} 224 | >>> T['A','A'] = .7 225 | >>> T 226 | {('A', 'B'): 0.3, ('A', 'A'): 0.7} 227 | 228 | >>> # Non-deterministic example 229 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 230 | >>> T[('A','B')] 231 | 0.3 232 | >>> T['A','B'] 233 | 0.3 234 | 235 | >>> # deterministic example 236 | >>> data = collections.OrderedDict(( 237 | (('A','B'), .3), 238 | (('A','A'), .7), 239 | (('B','A'), 1.))) 240 | >>> T = pykov.Matrix(data) 241 | >>> T[('A','B')] 242 | 0.3 243 | >>> T['A','B'] 244 | 0.3 245 | ``` 246 | 247 | Items not belonging to the matrix have value equal to zero, moreover items with value equal to zero are not shown: 248 | ```python 249 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 250 | >>> T['B','B'] 251 | 0.0 252 | 253 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 254 | >>> T 255 | {('A', 'B'): 0.3, ('A', 'A'): 0.7, ('B', 'A'): 1.0} 256 | >>> T['A','A'] = 0 257 | >>> T 258 | {('A', 'B'): 0.3, ('B', 'A'): 1.0} 259 | ``` 260 | 261 | #### Matrix operations 262 | 263 | ##### **Sum** 264 | A `pykov.Matrix()` instance can be added or substracted to another `pykov.Matrix()` instance. 265 | ```python 266 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 267 | >>> I = pykov.Matrix({('A','A'):1, ('B','B'):1}) 268 | >>> T + I 269 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 1.7, ('B', 'B'): 1.0} 270 | >>> T - I 271 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): -0.3, ('B', 'B'): -1} 272 | ``` 273 | 274 | ##### **Product** 275 | A `pykov.Matrix()` instance can be multiplied by a scalar, the dot product with a `pykov.Vector()` or another `pykov.Matrix()` instance is also supported. 276 | ```python 277 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 278 | >>> T * 3 279 | {('B', 'A'): 3.0, ('A', 'B'): 0.9, ('A', 'A'): 2.1} 280 | 281 | >>> p = pykov.Vector(A=.3, B=.7) 282 | >>> T * p 283 | {'A': 0.42, 'B': 0.3} 284 | 285 | >>> W = pykov.Matrix({('N', 'M'): 0.5, ('M', 'N'): 0.7, 286 | ('M', 'M'): 0.3, ('O', 'N'): 0.5, 287 | ('O', 'O'): 0.5, ('N', 'O'): 0.5}) 288 | >>> W * W 289 | {('N', 'M'): 0.15, ('M', 'N'): 0.21, ('M', 'O'): 0.35, 290 | ('M', 'M'): 0.44, ('O', 'M'): 0.25, ('O', 'N'): 0.25, 291 | ('O', 'O'): 0.5, ('N', 'O'): 0.25, ('N', 'N'): 0.6} 292 | ``` 293 | 294 | #### Matrix methods 295 | 296 | ##### **states()** 297 | Return the `set` of states. 298 | ```python 299 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 300 | >>> T.states() 301 | {'B', 'A'} 302 | ``` 303 | States without ingoing or outgoing transitions are removed from the set of states. 304 | ```python 305 | >>> T['A','C']=1 306 | >>> T.states() 307 | {'A', 'C', 'B'} 308 | >>> T['A','C']=0 309 | >>> T.states() 310 | {'A', 'B'} 311 | ``` 312 | 313 | ##### **pred(key=None)** 314 | Return the precedessors of a state (if not indicated, of all states). In matrix notation, return the column of the indicated state. 315 | ```python 316 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 317 | >>> T.pred() 318 | {'A': {'A': 0.7, 'B': 1.0}, 'B': {'A': 0.3}} 319 | >>> T.pred('A') 320 | {'A': 0.7, 'B': 1.0} 321 | ``` 322 | 323 | ##### **succ(key=None)** 324 | Return the successors of a state (if not indicated, of all states). In matrix notation, return the row of the indicated state. 325 | ```python 326 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 327 | >>> T.succ() 328 | {'A': {'A': 0.7, 'B': 0.3}, 'B': {'A': 1.0}} 329 | >>> T.succ('A') 330 | {'A': 0.7, 'B': 0.3} 331 | ``` 332 | 333 | ##### **copy()** 334 | Return a shallow copy. 335 | ```python 336 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 337 | >>> W = T.copy() 338 | >>> T[('B','B')] = 1. 339 | >>> W 340 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 341 | ``` 342 | 343 | ##### **remove(states)** 344 | Return a shallow copy of the matrix without the indicated states. 345 | All the links where the states appear are deleted, so that the result will not be in general a stochastic matrix. 346 | ```python 347 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 348 | >>> T.remove(['B']) 349 | {('A', 'A'): 0.7} 350 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1., 351 | ('C','D'): .5, ('D','C'): 1., ('C','B'): .5}) 352 | >>> T.remove(['A','B']) 353 | {('C', 'D'): 0.5, ('D', 'C'): 1.0} 354 | ``` 355 | 356 | ##### **stochastic()** 357 | Change the `pykov.Matrix()` instance in a right [stochastic matrix](http://en.wikipedia.org/wiki/Stochastic_matrix). 358 | Set the sum of every row equal to one, raise `PykovError` if not possible. 359 | ```python 360 | >>> T = pykov.Matrix({('A','B'): 3, ('A','A'): 7, ('B','A'): .2}) 361 | >>> T.stochastic() 362 | >>> T 363 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 364 | 365 | >>> T[('A','C')]=1 366 | >>> T.stochastic() 367 | pykov.PykovError: 'Zero links from node C' 368 | ``` 369 | 370 | ##### **transpose()** 371 | Return the transpose of the `pykov.Matrix()` instance. 372 | ```python 373 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 374 | >>> T.transpose() 375 | {('B', 'A'): 0.3, ('A', 'B'): 1.0, ('A', 'A'): 0.7} 376 | >>> T 377 | {('A', 'B'): 0.3, ('A', 'A'): 0.7, ('B', 'A'): 1.0} 378 | ``` 379 | 380 | ##### **eye()** 381 | Return the [Identity matrix](http://en.wikipedia.org/wiki/Identity_matrix). 382 | ```python 383 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 384 | >>> T.eye() 385 | {('A', 'A'): 1., ('B', 'B'): 1.} 386 | >>> type(T.eye()) 387 | 388 | ``` 389 | 390 | ##### **ones()** 391 | Return a `pykov.Vector()` instance with entries equal to 1. 392 | ```python 393 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 394 | >>> T.ones() 395 | {'A': 1.0, 'B': 1.0} 396 | >>> type(T.ones()) 397 | 398 | ``` 399 | 400 | ##### **trace()** 401 | Return the matrix [trace](http://en.wikipedia.org/wiki/Trace_%28linear_algebra%29). 402 | ```python 403 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 404 | >>> T.trace() 405 | 0.7 406 | ``` 407 | 408 | --------------- 409 | 410 | ### Chain class 411 | 412 | The `pykov.Chain` class inherits from `pykov.Matrix` class. 413 | The OrderedDict `key` is a tuple of states, the OrderedDict `value` is the transition 414 | probability to go from the first state to the second state, in other words 415 | pykov describes the transitions of a Markov chain with a *right* stochastic matrix. 416 | 417 | #### Chain methods 418 | 419 | ##### **adjacency()** 420 | Return the [adjacency matrix](http://en.wikipedia.org/wiki/Adjacency_matrix). 421 | ```python 422 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 423 | >>> T.adjacency() 424 | {('B', 'A'): 1, ('A', 'B'): 1, ('A', 'A'): 1} 425 | >>> type(T.adjacency()) 426 | 427 | ``` 428 | 429 | ##### **pow(p, n)** 430 | Find the probability distribution after `n` steps, starting from an initial `pykov.Vector()` `p`. 431 | ```python 432 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 433 | >>> p = pykov.Vector(A=1) 434 | >>> T.pow(p,3) 435 | {'A': 0.7629999999999999, 'B': 0.23699999999999996} 436 | >>> p * T * T * T #not efficient 437 | {'A': 0.7629999999999999, 'B': 0.23699999999999996} 438 | ``` 439 | 440 | ##### **move(state)** 441 | Do one step from the indicated `state` to one of its successors, chosen at random according to the transition probability. Return the new state. 442 | ```python 443 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 444 | >>> T.move('A') 445 | 'B' 446 | ``` 447 | 448 | Optionally, if you need to supply your own random number generator, you can pass a function that takes two inputs (the min and max) and Pykov will use that function. 449 | ```python 450 | >>> def FakeRandom(min, max): return 0.01 451 | >>> T.move('A', FakeRandom) 452 | 'B' 453 | ``` 454 | 455 | ##### **walk(steps, start=None, stop=None)** 456 | Return a random walk of n `steps`, starting and stopping at the indicated states. If not indicated, then the starting state is chosen according to the steady state probability. If the stopping state is reached before to do n steps, then the walker stops. 457 | ```python 458 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 459 | >>> T.walk(10) 460 | ['B', 'A', 'B', 'A', 'A', 'B', 'A', 'A', 'A', 'B', 'A'] 461 | >>> T.walk(10,'B','B') 462 | ['B', 'A', 'A', 'A', 'A', 'A', 'B'] 463 | ``` 464 | 465 | ##### **walk_probability(walk)** 466 | Return the *logarithm* of the **walk** probability (see `walk()` method). Impossible walks have probability equal to zero. 467 | ```python 468 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 469 | >>> T.walk_probability(['A','A','B','A','A']) 470 | -1.917322692203401 471 | >>> probability = math.exp(-1.917322692203401) 472 | >>> probability 473 | 0.147 474 | 475 | >>> p = T.walk_probability(['A','B','B','B','A']) 476 | >>> math.exp(p) 477 | 0.0 478 | ``` 479 | 480 | 481 | ##### **steady()** 482 | Return the steady state, i.e. the equilibrium distribution of the chain. 483 | ```python 484 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 485 | >>> T.steady() 486 | {'A': 0.7692307692307676, 'B': 0.23076923076923028} 487 | ``` 488 | Since Pykov describes the chain with a right stochatic matrix, 489 | the steady state $x$ satisfies at the condition $p=pT$ 490 | and it is calculated with the *inverse iteration method* 491 | $Q^t x = e$, where $Q = I - T$ and $e = (0,0,...,1)$. 492 | Moreover, the Markov chain is assumed to be ergodic, i.e. the transition matrix 493 | must be irreducible and acyclic. 494 | You can easily test such properties by means of 495 | [NetworkX](http://networkx.github.io/), let's see how: 496 | ```python 497 | >>> import networkx as nx 498 | 499 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 500 | >>> G = nx.DiGraph(list(T.keys())) 501 | >>> nx.is_strongly_connected(G) # is irreducible 502 | True 503 | >>> nx.is_aperiodic(G) 504 | True 505 | 506 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','B'): 1.}) 507 | >>> G = nx.DiGraph(list(T.keys())) 508 | >>> nx.is_strongly_connected(G) # is irreducible 509 | False 510 | >>> nx.is_aperiodic(G) 511 | True 512 | 513 | >>> T = pykov.Chain({('A','B'): 1, ('B','C'): 1., ('C','A'): 1.}) 514 | >>> G = nx.DiGraph(list(T.keys())) 515 | >>> nx.is_strongly_connected(G) 516 | True 517 | >>> nx.is_aperiodic(G) 518 | False 519 | ``` 520 | 521 | Often, Markov chains created from raw data are not irreducibles. 522 | In such cases, the Markov chain may be defined by means of the 523 | largest strongly connected component of the associated graph. 524 | Strongly connected components can be found with NetworkX: 525 | ```python 526 | >>> nx.strongly_connected_components(G) 527 | ``` 528 | 529 | For further details on the inverse iteration method, 530 | have a look at *W. Stewart, Introduction to the Numerical Solution of 531 | Markov Chains, Princeton University Press, Chichester, West Sussex, 1994*. 532 | 533 | ##### **mixing_time(cutoff=0.25, jump=1, p=None)** 534 | Return the [mixing time](http://en.wikipedia.org/wiki/Markov_chain_mixing_time), defined as the number of steps needed to have $|pT^n - \pi| \lt 0.25$, where $\pi$ is the steady state $\pi = \pi T$. 535 | 536 | If the initial distribution `p` is not indicated, then the iteration starts from the less probable state of the steady distribution. The parameter `jump` controls the iteration step, for example with `jump=2` n has values 2,4,6,8,.. 537 | ```python 538 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 539 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 540 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 541 | >>> T = pykov.Chain(d) 542 | >>> T.mixing_time() 543 | 2 544 | ``` 545 | 546 | ##### **entropy(p=None, norm=False)** 547 | Return the Chain entropy, defined as $H = \sum_i \pi_i H_i$, where $H_i=\sum_j T_{ij}\ln T_{ij}$. 548 | If `p` is not `None`, then the entropy is calculated with the indicated probability `pykov.Vector()`. 549 | ```python 550 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 551 | >>> T.entropy() 552 | 0.46989561696530169 553 | ``` 554 | With **norm=True** entropy belongs to [0,1]. 555 | ```python 556 | >>> T.entropy(norm=True) 557 | 0.33895603665233132 558 | ``` 559 | For further details, have a look at *Khinchin A. I., Mathematical Foundations of Information Theory, Dover, 1957*. 560 | 561 | ##### **mfpt_to(state)** 562 | Return the Mean First Passage Times of every state *to* the indicated `state`. 563 | ```python 564 | >>> d = {('R', 'N'): 0.25, ('R', 'S'): 0.25, ('S', 'R'): 0.25, 565 | ('R', 'R'): 0.5, ('N', 'S'): 0.5, ('S', 'S'): 0.5, 566 | ('S', 'N'): 0.25, ('N', 'R'): 0.5, ('N', 'N'): 0.0} 567 | >>> T = pykov.Chain(d) 568 | >>> T.mfpt_to('R') 569 | {'S': 3.333333333333333, 'N': 2.666666666666667} #mfpt from 'S' to 'R' is 3.33 570 | ``` 571 | See also *Kemeny J. G. and Snell, J. L., Finite Markov Chains. Springer-Verlag: New York, 1976*. 572 | 573 | ##### **absorbing_time(transient_set)** 574 | Mean number of steps needed to leave the `transient_set`, return the `pykov.Vector()` `tau` where `tau[i]` is the mean number of steps needed to leave the transient set starting from state `i`. The parameter `transient_set` is a subset of states (iterable). 575 | ```python 576 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 577 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 578 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 579 | >>> T = pykov.Chain(d) 580 | >>> p = pykov.Vector({'N':.3, 'S':.7}) 581 | >>> tau = T.absorbing_time(p.keys()) 582 | >>> tau 583 | {'S': 3.333333333333333, 'N': 2.6666666666666665} 584 | ``` 585 | In other words, the mean number of steps in order to leave states `'S'` and `'N'` starting from `'S'` is 3.33. 586 | It is sufficient to calculate `p * tau` in order to weight the mean times according an initial distribution `p`. 587 | ```python 588 | >>> p * tau 589 | 3.1333333333333329 590 | ``` 591 | In order to better understand the meaning of the method, the calculation of the previous example can be approximated by means of many random walkers: 592 | ```python 593 | >>> numpy.mean([len(T.walk(10000000,"S","R"))-1 for i in range(1000000)]) 594 | 3.3326020000000001 595 | >>> numpy.mean([len(T.walk(10000000,"N","R"))-1 for i in range(1000000)]) 596 | 2.6665549999999998 597 | ``` 598 | 599 | ##### **absorbing_tour(p, transient_set=None)** 600 | Return a `pykov.Vector()` `v`, where `v[i]` is the mean time the process is in the transient state `i` before leaving the transient set. 601 | 602 | Note that `v.sum()` is equal to `p * tau` (see `absorbing_time()` method). 603 | If not specified, the transient set is defined as the set of states in vector `p`. 604 | ```python 605 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 606 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 607 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 608 | >>> T = pykov.Chain(d) 609 | >>> p = pykov.Vector({'N':.3, 'S':.7}) 610 | >>> T.absorbing_tour(p) 611 | {'S': 2.2666666666666666, 'N': 0.8666666666666669} 612 | ``` 613 | 614 | ##### **fundamental_matrix()** 615 | Return the fundamental matrix, have a look at *Kemeny J. G. and Snell J. L., Finite Markov Chains. Springer-Verlag: New York, 1976* for further details. 616 | ```python 617 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 618 | >>> T.fundamental_matrix() 619 | {('B', 'A'): 0.17751479289940991, ('A', 'B'): 0.053254437869822958, 620 | ('A', 'A'): 0.94674556213017902, ('B', 'B'): 0.82248520710059214} 621 | ``` 622 | Note that the fundamental matrix is not sparse. 623 | 624 | ##### **kemeny_constant()** 625 | Return the Kemeny constant of the transition matrix. 626 | ```python 627 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 628 | >>> T.kemeny_constant() 629 | 1.7692307692307712 630 | ``` 631 | 632 | ---------------------------- 633 | 634 | Utilities 635 | ---------- 636 | Pykov comes with an utility useful to create a `pykov.Chain()` from a text file, let say file `/mypath/mat`, which contains the transition matrix defined with the following format: 637 | ```python 638 | A A .7 639 | A B .3 640 | B A 1 641 | ``` 642 | The `pykov.Chain()` instance is created with the command: 643 | ```python 644 | >>> P = pykov.readmat('/mypath/mat') 645 | >>> P 646 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 647 | ``` 648 | ----------------------------------- 649 | > Docs written with [StackEdit](https://stackedit.io/). 650 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /pykov.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | 3 | # PyKov is Python package for the creation, manipulation and study of Markov 4 | # Chains. 5 | # Copyright (C) 2014 Riccardo Scalco 6 | # 7 | # This program is free software: you can redistribute it and/or modify 8 | # it under the terms of the GNU General Public License as published by 9 | # the Free Software Foundation, either version 3 of the License, or 10 | # (at your option) any later version. 11 | # 12 | # This program is distributed in the hope that it will be useful, 13 | # but WITHOUT ANY WARRANTY; without even the implied warranty of 14 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 | # GNU General Public License for more details. 16 | # 17 | # You should have received a copy of the GNU General Public License 18 | # along with this program. If not, see . 19 | 20 | # Email: riccardo.scalco@gmail.com 21 | 22 | """Pykov documentation. 23 | 24 | .. module:: A Python module for finite Markov chains. 25 | :platform: Unix, Windows, Mac 26 | 27 | .. moduleauthor:: 28 | Riccardo Scalco 29 | 30 | """ 31 | import random 32 | import math 33 | import six 34 | import numpy 35 | import sys 36 | 37 | from collections import OrderedDict 38 | 39 | import scipy.sparse as ss 40 | import scipy.sparse.linalg as ssl 41 | 42 | if sys.version_info < (2, 6): 43 | from sets import Set 44 | else: 45 | Set = set 46 | 47 | __date__ = 'March 2015' 48 | 49 | __version__ = 1.1 50 | 51 | __license__ = 'GNU General Public License Version 3' 52 | 53 | __authors__ = 'Riccardo Scalco' 54 | 55 | __many_thanks_to__ = 'Sandra Steiner, Nicky Van Foreest, Adel Qalieh' 56 | 57 | 58 | def _del_cache(fn): 59 | """ 60 | Delete cache. 61 | """ 62 | def wrapper(*args, **kwargs): 63 | self = args[0] 64 | try: 65 | del(self._states) 66 | except AttributeError: 67 | pass 68 | try: 69 | del(self._succ) 70 | except AttributeError: 71 | pass 72 | try: 73 | del(self._pred) 74 | except AttributeError: 75 | pass 76 | try: 77 | del(self._steady) 78 | except AttributeError: 79 | pass 80 | try: 81 | del(self._guess) 82 | except AttributeError: 83 | pass 84 | try: 85 | del(self._fundamental_matrix) 86 | except AttributeError: 87 | pass 88 | return fn(*args, **kwargs) 89 | return wrapper 90 | 91 | 92 | class PykovError(Exception): 93 | 94 | """ 95 | Exception definition form Pykov Errors. 96 | """ 97 | 98 | def __init__(self, value): 99 | self.value = value 100 | 101 | def __str__(self): 102 | return repr(self.value) 103 | 104 | 105 | class Vector(OrderedDict): 106 | 107 | """ 108 | """ 109 | 110 | def __init__(self, data=None, **kwargs): 111 | """ 112 | >>> pykov.Vector({'A':.3, 'B':.7}) 113 | {'A':.3, 'B':.7} 114 | >>> pykov.Vector(A=.3, B=.7) 115 | {'A':.3, 'B':.7} 116 | """ 117 | OrderedDict.__init__(self) 118 | 119 | if data: 120 | self.update([item for item in six.iteritems(data) 121 | if abs(item[1]) > numpy.finfo(numpy.float).eps]) 122 | if len(kwargs): 123 | self.update([item for item in six.iteritems(kwargs) 124 | if abs(item[1]) > numpy.finfo(numpy.float).eps]) 125 | 126 | def __getitem__(self, key): 127 | """ 128 | >>> q = pykov.Vector(C=.4, B=.6) 129 | >>> q['C'] 130 | 0.4 131 | >>> q['Z'] 132 | 0.0 133 | """ 134 | try: 135 | return OrderedDict.__getitem__(self, key) 136 | except KeyError: 137 | return 0.0 138 | 139 | def __setitem__(self, key, value): 140 | """ 141 | >>> q = pykov.Vector(C=.4, B=.6) 142 | >>> q['Z']=.2 143 | >>> q 144 | {'C': 0.4, 'B': 0.6, 'Z': 0.2} 145 | >>> q['Z']=0 146 | >>> q 147 | {'C': 0.4, 'B': 0.6} 148 | """ 149 | if abs(value) > numpy.finfo(numpy.float).eps: 150 | OrderedDict.__setitem__(self, key, value) 151 | elif key in self: 152 | del(self[key]) 153 | 154 | def __mul__(self, M): 155 | """ 156 | >>> p = pykov.Vector(A=.3, B=.7) 157 | >>> p * 3 158 | {'A': 0.9, 'B': 2.1} 159 | >>> q = pykov.Vector(C=.5, B=.5) 160 | >>> p * q 161 | 0.35 162 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 163 | >>> p * T 164 | {'A': 0.91, 'B': 0.09} 165 | >>> T * p 166 | {'A': 0.42, 'B': 0.3} 167 | """ 168 | if isinstance(M, int) or isinstance(M, float): 169 | return self.__rmul__(M) 170 | if isinstance(M, Matrix): 171 | e2p, p2e = M._el2pos_() 172 | x = self._toarray(e2p) 173 | A = M._dok_(e2p).tocsr().transpose() 174 | y = A.dot(x) 175 | result = Vector() 176 | result._fromarray(y, e2p) 177 | return result 178 | elif isinstance(M, Vector): 179 | result = 0 180 | for state, value in six.iteritems(self): 181 | result += value * M[state] 182 | return result 183 | else: 184 | raise TypeError('unsupported operand type(s) for *:' + 185 | ' \'Vector\' and ' + repr(type(M))[7:-1]) 186 | 187 | def __rmul__(self, M): 188 | """ 189 | >>> p = pykov.Vector(A=.3, B=.7) 190 | >>> 3 * p 191 | {'A': 0.9, 'B': 2.1} 192 | """ 193 | if isinstance(M, int) or isinstance(M, float): 194 | result = Vector() 195 | for state, value in six.iteritems(self): 196 | result[state] = value * M 197 | return result 198 | else: 199 | raise TypeError('unsupported operand type(s) for *: ' + 200 | repr(type(M))[7:-1] + ' and \'Vector\'') 201 | 202 | def __add__(self, v): 203 | """ 204 | >>> p = pykov.Vector(A=.3, B=.7) 205 | >>> q = pykov.Vector(C=.5, B=.5) 206 | >>> p + q 207 | {'A': 0.3, 'C': 0.5, 'B': 1.2} 208 | """ 209 | if isinstance(v, Vector): 210 | result = Vector() 211 | for state in set(six.iterkeys(self)) | set(v.keys()): 212 | result[state] = self[state] + v[state] 213 | return result 214 | else: 215 | raise TypeError('unsupported operand type(s) for +:' + 216 | ' \'Vector\' and ' + repr(type(v))[7:-1]) 217 | 218 | def __sub__(self, v): 219 | """ 220 | >>> p = pykov.Vector(A=.3, B=.7) 221 | >>> q = pykov.Vector(C=.5, B=.5) 222 | >>> p - q 223 | {'A': 0.3, 'C': -0.5, 'B': 0.2} 224 | >>> q - p 225 | {'A': -0.3, 'C': 0.5, 'B': -0.2} 226 | """ 227 | if isinstance(v, Vector): 228 | result = Vector() 229 | for state in set(six.iterkeys(self)) | set(v.keys()): 230 | result[state] = self[state] - v[state] 231 | return result 232 | else: 233 | raise TypeError('unsupported operand type(s) for -:' + 234 | ' \'Vector\' and ' + repr(type(v))[7:-1]) 235 | 236 | def _toarray(self, el2pos): 237 | """ 238 | >>> p = pykov.Vector(A=.3, B=.7) 239 | >>> el2pos = {'A': 1, 'B': 0} 240 | >>> v = p._toarray(el2pos) 241 | >>> v 242 | array([ 0.7, 0.3]) 243 | """ 244 | p = numpy.zeros(len(el2pos)) 245 | for key, value in six.iteritems(self): 246 | p[el2pos[key]] = value 247 | return p 248 | 249 | def _fromarray(self, arr, el2pos): 250 | """ 251 | >>> p = pykov.Vector() 252 | >>> el2pos = {'A': 1, 'B': 0} 253 | >>> v = numpy.array([ 0.7, 0.3]) 254 | >>> p._fromarray(v,el2pos) 255 | >>> p 256 | {'A': 0.3, 'B': 0.7} 257 | """ 258 | for elem, pos in el2pos.items(): 259 | self[elem] = arr[pos] 260 | return None 261 | 262 | def sort(self, reverse=False): 263 | """ 264 | List of (state,probability) sorted according the probability. 265 | 266 | >>> p = pykov.Vector({'A':.3, 'B':.1, 'C':.6}) 267 | >>> p.sort() 268 | [('B', 0.1), ('A', 0.3), ('C', 0.6)] 269 | >>> p.sort(reverse=True) 270 | [('C', 0.6), ('A', 0.3), ('B', 0.1)] 271 | """ 272 | res = list(six.iteritems(self)) 273 | res.sort(key=lambda lst: lst[1], reverse=reverse) 274 | return res 275 | 276 | def normalize(self): 277 | """ 278 | Normalize the vector so that the entries sum is 1. 279 | 280 | >>> p = pykov.Vector({'A':3, 'B':1, 'C':6}) 281 | >>> p.normalize() 282 | >>> p 283 | {'A': 0.3, 'C': 0.6, 'B': 0.1} 284 | """ 285 | s = self.sum() 286 | for k in six.iterkeys(self): 287 | self[k] = self[k] / s 288 | 289 | def choose(self, random_func = None): 290 | """ 291 | Choose a state according to its probability. 292 | 293 | >>> p = pykov.Vector(A=.3, B=.7) 294 | >>> p.choose() 295 | 'B' 296 | 297 | Optionally, a function that generates a random number can be supplied. 298 | >>> def FakeRandom(min, max): return 0.01 299 | >>> p = pykov.Vector(A=.05, B=.4, C=.4, D=.15) 300 | >>> p.choose(FakeRandom) 301 | 'A' 302 | 303 | .. seealso:: 304 | 305 | `Kevin Parks recipe `_ 306 | """ 307 | if random_func is None: 308 | random_func = random.uniform 309 | n = random_func(0, 1) 310 | for state, prob in six.iteritems(self): 311 | if n < prob: 312 | break 313 | n = n - prob 314 | return state 315 | 316 | def entropy(self): 317 | """ 318 | Return the entropy. 319 | 320 | .. math:: 321 | 322 | H(p) = \sum_i p_i \ln p_i 323 | 324 | .. seealso:: 325 | 326 | Khinchin, A. I. 327 | Mathematical Foundations of Information Theory 328 | Dover, 1957. 329 | 330 | >>> p = pykov.Vector(A=.3, B=.7) 331 | >>> p.entropy() 332 | 0.6108643020548935 333 | """ 334 | return -sum([v * math.log(v) for v in self.values()]) 335 | 336 | def relative_entropy(self, p): 337 | """ 338 | Return the Kullback-Leibler distance. 339 | 340 | .. math:: 341 | 342 | d(q,p) = \sum_i q_i \ln (q_i/p_i) 343 | 344 | .. note:: 345 | 346 | The Kullback-Leibler distance is not symmetric. 347 | 348 | >>> p = pykov.Vector(A=.3, B=.7) 349 | >>> q = pykov.Vector(A=.4, B=.6) 350 | >>> p.relative_entropy(q) 351 | 0.02160085414354654 352 | >>> q.relative_entropy(p) 353 | 0.022582421084357485 354 | """ 355 | states = set(six.iterkeys(self)) & set(p.keys()) 356 | return sum([self[s] * math.log(self[s] / p[s]) for s in states]) 357 | 358 | def copy(self): 359 | """ 360 | Return a shallow copy. 361 | 362 | >>> p = pykov.Vector(A=.3, B=.7) 363 | >>> q = p.copy() 364 | >>> p['C'] = 1. 365 | >>> q 366 | {'A': 0.3, 'B': 0.7} 367 | """ 368 | return Vector(self) 369 | 370 | def sum(self): 371 | """ 372 | Sum the values. 373 | 374 | >>> p = pykov.Vector(A=.3, B=.7) 375 | >>> p.sum() 376 | 1.0 377 | """ 378 | return float(sum(self.values())) 379 | 380 | def dist(self, v): 381 | """ 382 | Return the distance between the two probability vectors. 383 | 384 | .. math:: 385 | 386 | d(q,p) = \sum_i |q_i - p_i| 387 | 388 | >>> p = pykov.Vector(A=.3, B=.7) 389 | >>> q = pykov.Vector(C=.5, B=.5) 390 | >>> q.dist(p) 391 | 1.0 392 | """ 393 | if isinstance(v, Vector): 394 | result = 0 395 | for state in set(six.iterkeys(self)) | set(v.keys()): 396 | result += abs(v[state] - self[state]) 397 | return result 398 | 399 | 400 | class Matrix(OrderedDict): 401 | 402 | """ 403 | """ 404 | 405 | def __init__(self, data=None): 406 | """ 407 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 408 | """ 409 | OrderedDict.__init__(self) 410 | 411 | if data: 412 | self.update([item for item in six.iteritems(data) 413 | if abs(item[1]) > numpy.finfo(numpy.float).eps]) 414 | 415 | def __getitem__(self, *args): 416 | """ 417 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 418 | >>> T[('A','B')] 419 | 0.3 420 | >>> T['A','B'] 421 | 0.3 422 | >>> 423 | 0.0 424 | """ 425 | try: 426 | return OrderedDict.__getitem__(self, args[0]) 427 | except KeyError: 428 | return 0.0 429 | 430 | @_del_cache 431 | def __setitem__(self, key, value): 432 | """ 433 | >>> T = pykov.Matrix() 434 | >>> T[('A','B')] = .3 435 | >>> T 436 | {('A', 'B'): 0.3} 437 | >>> T['A','A'] = .7 438 | >>> T 439 | {('A', 'B'): 0.3, ('A', 'A'): 0.7} 440 | >>> T['B','B'] = 0 441 | >>> T 442 | {('A', 'B'): 0.3, ('A', 'A'): 0.7} 443 | >>> T['A','A'] = 0 444 | >>> T 445 | {('A', 'B'): 0.3} 446 | 447 | >>> T = pykov.Matrix({('A','B'): 3, ('A','A'): 7, ('B','A'): .1}) 448 | >>> T.states() 449 | {'A', 'B'} 450 | >>> T['A','C']=1 451 | >>> T.states() 452 | {'A', 'B', 'C'} 453 | >>> T['A','C']=0 454 | >>> T.states() 455 | {'A', 'B'} 456 | """ 457 | if abs(value) > numpy.finfo(numpy.float).eps: 458 | OrderedDict.__setitem__(self, key, value) 459 | elif key in self: 460 | del(self[key]) 461 | 462 | @_del_cache 463 | def __delitem__(self, key): 464 | """ 465 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 466 | >>> del(T['B', 'A']) 467 | >>> T 468 | {('A', 'B'): 0.3, ('A', 'A'): 0.7} 469 | """ 470 | OrderedDict.__delitem__(self, key) 471 | 472 | @_del_cache 473 | def pop(self, key): 474 | """ 475 | Remove specified key and return the corresponding value. 476 | See: help(OrderedDict.pop) 477 | 478 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 479 | >>> T.pop(('A','B')) 480 | 0.3 481 | >>> T 482 | {('B', 'A'): 1.0, ('A', 'A'): 0.7} 483 | """ 484 | return OrderedDict.pop(self, key) 485 | 486 | @_del_cache 487 | def popitem(self): 488 | """ 489 | Remove and return some (key, value) pair as a 2-tuple. 490 | See: help(OrderedDict.popitem) 491 | 492 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 493 | >>> T.popitem() 494 | (('B', 'A'), 1.0) 495 | >>> T 496 | {('A', 'B'): 0.3, ('A', 'A'): 0.7} 497 | """ 498 | return OrderedDict.popitem(self) 499 | 500 | @_del_cache 501 | def clear(self): 502 | """ 503 | Remove all keys. 504 | See: help(OrderedDict.clear) 505 | 506 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 507 | >>> T.clear() 508 | >>> T 509 | {} 510 | """ 511 | OrderedDict.clear(self) 512 | 513 | @_del_cache 514 | def update(self, other): 515 | """ 516 | Update with keys and their values present in other. 517 | See: help(OrderedDict.update) 518 | 519 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 520 | >>> d = {('B', 'C'):2} 521 | >>> T.update(d) 522 | >>> T 523 | {('B', 'A'): 1.0, ('B', 'C'): 2, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 524 | """ 525 | OrderedDict.update(self, other) 526 | 527 | @_del_cache 528 | def setdefault(self, k, *args): 529 | """ 530 | See: help(OrderedDict.setdefault) 531 | 532 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 533 | >>> T.setdefault(('A','A'),1) 534 | 0.7 535 | >>> T 536 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 537 | >>> T.setdefault(('A','C'),1) 538 | 1 539 | >>> T 540 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7, ('A', 'C'): 1} 541 | """ 542 | return OrderedDict.setdefault(self, k, *args) 543 | 544 | def copy(self): 545 | """ 546 | Return a shallow copy. 547 | 548 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 549 | >>> W = T.copy() 550 | >>> T[('B','B')] = 1. 551 | >>> W 552 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 553 | """ 554 | return Matrix(self) 555 | 556 | def __reduce__(self): 557 | """Return state information for pickling""" 558 | # Required because we changed the OrderedDict.__init__ signature 559 | return (self.__class__, (), None, None, six.iteritems(self)) 560 | 561 | def _dok_(self, el2pos, method=''): 562 | """ 563 | """ 564 | m = len(el2pos) 565 | S = ss.dok_matrix((m, m)) 566 | if method == '': 567 | for k, v in six.iteritems(self): 568 | i = el2pos[k[0]] 569 | j = el2pos[k[1]] 570 | S[i, j] = float(v) 571 | elif method == 'transpose': 572 | for k, v in six.iteritems(self): 573 | i = el2pos[k[0]] 574 | j = el2pos[k[1]] 575 | S[j, i] = float(v) 576 | return S 577 | 578 | def _from_dok_(self, mat, pos2el): 579 | """ 580 | """ 581 | for ii, val in mat.items(): 582 | self[pos2el[ii[0]], pos2el[ii[1]]] = val 583 | return None 584 | 585 | def _numpy_mat(self, el2pos): 586 | """ 587 | Return a numpy.matrix object from a dictionary. 588 | 589 | -- Parameters -- 590 | t_ij : the OrderedDict, values must be real numbers, keys should be tuples of 591 | two strings. 592 | el2pos : see _map() 593 | """ 594 | m = len(el2pos) 595 | T = numpy.matrix(numpy.zeros((m, m))) 596 | for k, v in six.iteritems(self): 597 | T[el2pos[k[0]], el2pos[k[1]]] = v 598 | return T 599 | 600 | def _from_numpy_mat(self, T, pos2el): 601 | """ 602 | Return a dictionary from a numpy.matrix object. 603 | 604 | -- Parameters -- 605 | T : the numpy.matrix. 606 | pos2el : see _map() 607 | """ 608 | for i in range(len(T)): 609 | for j in range(len(T)): 610 | if T[i, j]: 611 | self[(pos2el[i], pos2el[j])] = T[i, j] 612 | return None 613 | 614 | def _el2pos_(self): 615 | """ 616 | """ 617 | el2pos = {} 618 | pos2el = {} 619 | for pos, element in enumerate(list(self.states())): 620 | el2pos[element] = pos 621 | pos2el[pos] = element 622 | return el2pos, pos2el 623 | 624 | def stochastic(self): 625 | """ 626 | Make a right stochastic matrix. 627 | 628 | Set the sum of every row equal to one, 629 | raise ``PykovError`` if it is not possible. 630 | 631 | >>> T = pykov.Matrix({('A','B'): 3, ('A','A'): 7, ('B','A'): .2}) 632 | >>> T.stochastic() 633 | >>> T 634 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 635 | >>> T[('A','C')]=1 636 | >>> T.stochastic() 637 | pykov.PykovError: 'Zero links from node C' 638 | """ 639 | s = {} 640 | for k, v in self.succ().items(): 641 | summ = float(sum(v.values())) 642 | if summ: 643 | s[k] = summ 644 | else: 645 | raise PykovError('Zero links from state ' + k) 646 | for k in six.iterkeys(self): 647 | self[k] = self[k] / s[k[0]] 648 | 649 | def pred(self, key=None): 650 | """ 651 | Return the precedessors of a state (if not indicated, of all states). 652 | In Matrix notation: return the coloum of the indicated state. 653 | 654 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 655 | >>> T.pred() 656 | {'A': {'A': 0.7, 'B': 1.0}, 'B': {'A': 0.3}} 657 | >>> T.pred('A') 658 | {'A': 0.7, 'B': 1.0} 659 | """ 660 | try: 661 | if key is not None: 662 | return self._pred[key] 663 | else: 664 | return self._pred 665 | except AttributeError: 666 | self._pred = OrderedDict([(state, Vector()) for state in self.states()]) 667 | for link, probability in six.iteritems(self): 668 | self._pred[link[1]][link[0]] = probability 669 | if key is not None: 670 | return self._pred[key] 671 | else: 672 | return self._pred 673 | 674 | def succ(self, key=None): 675 | """ 676 | Return the successors of a state (if not indicated, of all states). 677 | In Matrix notation: return the row of the indicated state. 678 | 679 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 680 | >>> T.succ() 681 | {'A': {'A': 0.7, 'B': 0.3}, 'B': {'A': 1.0}} 682 | >>> T.succ('A') 683 | {'A': 0.7, 'B': 0.3} 684 | """ 685 | try: 686 | if key is not None: 687 | return self._succ[key] 688 | else: 689 | return self._succ 690 | except AttributeError: 691 | self._succ = OrderedDict([(state, Vector()) for state in self.states()]) 692 | for link, probability in six.iteritems(self): 693 | self._succ[link[0]][link[1]] = probability 694 | if key is not None: 695 | return self._succ[key] 696 | else: 697 | return self._succ 698 | 699 | def remove(self, states): 700 | """ 701 | Return a copy of the Chain, without the indicated states. 702 | 703 | .. warning:: 704 | 705 | All the links where the states appear are deleted, so that the result 706 | will not be in general a stochastic matrix. 707 | .. 708 | 709 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 710 | >>> T.remove(['B']) 711 | {('A', 'A'): 0.7} 712 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1., 713 | ('C','D'): .5, ('D','C'): 1., ('C','B'): .5}) 714 | >>> T.remove(['A','B']) 715 | {('C', 'D'): 0.5, ('D', 'C'): 1.0} 716 | """ 717 | return Matrix(OrderedDict([(key, value) for key, value in six.iteritems(self) if 718 | key[0] not in states and key[1] not in states])) 719 | 720 | def states(self): 721 | """ 722 | Return the set of states. 723 | 724 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 725 | >>> T.states() 726 | {'A', 'B'} 727 | """ 728 | try: 729 | return self._states 730 | except AttributeError: 731 | self._states = set() 732 | for link in six.iterkeys(self): 733 | self._states.add(link[0]) 734 | self._states.add(link[1]) 735 | return self._states 736 | 737 | def __pow__(self, n): 738 | """ 739 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 740 | >>> T**2 741 | {('A', 'B'): 0.21, ('B', 'A'): 0.70, ('A', 'A'): 0.79, ('B', 'B'): 0.30} 742 | >>> T**0 743 | {('A', 'A'): 1.0, ('B', 'B'): 1.0} 744 | """ 745 | el2pos, pos2el = self._el2pos_() 746 | P = self._numpy_mat(el2pos) 747 | P = P**n 748 | res = Matrix() 749 | res._from_numpy_mat(P, pos2el) 750 | return res 751 | 752 | def pow(self, n): 753 | return self.__pow__(n) 754 | 755 | def __mul__(self, v): 756 | """ 757 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 758 | >>> T * 3 759 | {('B', 'A'): 3.0, ('A', 'B'): 0.9, ('A', 'A'): 2.1} 760 | >>> p = pykov.Vector(A=.3, B=.7) 761 | >>> T * p 762 | {'A': 0.42, 'B': 0.3} 763 | >>> W = pykov.Matrix({('N', 'M'): 0.5, ('M', 'N'): 0.7, 764 | ('M', 'M'): 0.3, ('O', 'N'): 0.5, 765 | ('O', 'O'): 0.5, ('N', 'O'): 0.5}) 766 | >>> W * W 767 | {('N', 'M'): 0.15, ('M', 'N'): 0.21, ('M', 'O'): 0.35, 768 | ('M', 'M'): 0.44, ('O', 'M'): 0.25, ('O', 'N'): 0.25, 769 | ('O', 'O'): 0.5, ('N', 'O'): 0.25, ('N', 'N'): 0.6} 770 | """ 771 | if isinstance(v, Vector): 772 | e2p, p2e = self._el2pos_() 773 | x = v._toarray(e2p) 774 | M = self._dok_(e2p).tocsr() 775 | y = M.dot(x) 776 | result = Vector() 777 | result._fromarray(y, e2p) 778 | return result 779 | elif isinstance(v, Matrix): 780 | e2p, p2e = self._el2pos_() 781 | M = self._dok_(e2p).tocsr() 782 | N = v._dok_(e2p).tocsr() 783 | C = M.dot(N).todok() 784 | if 'Chain' in repr(self.__class__): 785 | res = Chain() 786 | elif 'Matrix' in repr(self.__class__): 787 | res = Matrix() 788 | res._from_dok_(C, p2e) 789 | return res 790 | elif isinstance(v, int) or isinstance(v, float): 791 | return Matrix(OrderedDict([(key, value * v) for key, value in 792 | six.iteritems(self)])) 793 | else: 794 | raise TypeError('unsupported operand type(s) for *:' + 795 | ' \'Matrix\' and ' + repr(type(v))[7:-1]) 796 | 797 | def __rmul__(self, v): 798 | """ 799 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 800 | >>> 3 * T 801 | {('B', 'A'): 3.0, ('A', 'B'): 0.9, ('A', 'A'): 2.1} 802 | """ 803 | if isinstance(v, int) or isinstance(v, float): 804 | return Matrix(OrderedDict([(key, value * v) for key, value in 805 | six.iteritems(self)])) 806 | else: 807 | raise TypeError('unsupported operand type(s) for *:' + 808 | ' \'Matrix\' and ' + repr(type(v))[7:-1]) 809 | 810 | def __add__(self, M): 811 | """ 812 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 813 | >>> I = pykov.Matrix({('A','A'):1, ('B','B'):1}) 814 | >>> T + I 815 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 1.7, ('B', 'B'): 1.0} 816 | """ 817 | if isinstance(M, Matrix): 818 | result = Matrix() 819 | for link in set(six.iterkeys(self)) | set(M.keys()): 820 | result[link] = self[link] + M[link] 821 | return result 822 | else: 823 | raise TypeError('unsupported operand type(s) for +:' + 824 | ' \'Matrix\' and ' + repr(type(M))[7:-1]) 825 | 826 | def __sub__(self, M): 827 | """ 828 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 829 | >>> I = pykov.Matrix({('A','A'):1, ('B','B'):1}) 830 | >>> T - I 831 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): -0.3, ('B', 'B'): -1} 832 | """ 833 | if isinstance(M, Matrix): 834 | result = Matrix() 835 | for link in set(six.iterkeys(self)) | set(M.keys()): 836 | result[link] = self[link] - M[link] 837 | return result 838 | else: 839 | raise TypeError('unsupported operand type(s) for -:' + 840 | ' \'Matrix\' and ' + repr(type(M))[7:-1]) 841 | 842 | def trace(self): 843 | """ 844 | Return the Matrix trace. 845 | 846 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 847 | >>> T.trace() 848 | 0.7 849 | """ 850 | return sum([self[k, k] for k in self.states()]) 851 | 852 | def eye(self): 853 | """ 854 | Return the Identity Matrix. 855 | 856 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 857 | >>> T.eye() 858 | {('A', 'A'): 1., ('B', 'B'): 1.} 859 | """ 860 | return Matrix(OrderedDict([((state, state), 1.) for state in self.states()])) 861 | 862 | def ones(self): 863 | """ 864 | Return a ``Vector`` instance with entries equal to one. 865 | 866 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 867 | >>> T.ones() 868 | {'A': 1.0, 'B': 1.0} 869 | """ 870 | return Vector(OrderedDict([(state, 1.) for state in self.states()])) 871 | 872 | def transpose(self): 873 | """ 874 | Return the transpose Matrix. 875 | 876 | >>> T = pykov.Matrix({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 877 | >>> T.transpose() 878 | {('B', 'A'): 0.3, ('A', 'B'): 1.0, ('A', 'A'): 0.7} 879 | """ 880 | return Matrix(OrderedDict([((key[1], key[0]), value) for key, value in 881 | six.iteritems(self)])) 882 | 883 | def _UMPFPACKSolve(self, b, x=None, method='UMFPACK_A'): 884 | """ 885 | UMFPACK ( U nsymmetric M ulti F Rontal PACK age) 886 | 887 | Parameters 888 | ---------- 889 | method: 890 | "UMFPACK_A" : \mathbf{A} x = b (default) 891 | "UMFPACK_At" : \mathbf{A}^T x = b 892 | 893 | References 894 | ---------- 895 | A column pre-ordering strategy for the unsymmetric-pattern multifrontal 896 | method, T. A. Davis, ACM Transactions on Mathematical Software, vol 30, 897 | no. 2, June 2004, pp. 165-195. 898 | """ 899 | e2p, p2e = self._el2pos_() 900 | if method == "UMFPACK_At": 901 | A = self._dok_(e2p).tocsr().transpose() 902 | else: 903 | A = self._dok_(e2p).tocsr() 904 | bb = b._toarray(e2p) 905 | x = ssl.spsolve(A, bb, use_umfpack=True) 906 | res = Vector() 907 | res._fromarray(x, e2p) 908 | return res 909 | 910 | 911 | class Chain(Matrix): 912 | 913 | """ 914 | """ 915 | 916 | def move(self, state, random_func = None): 917 | """ 918 | Do one step from the indicated state, and return the final state. 919 | 920 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 921 | >>> T.move('A') 922 | 'B' 923 | 924 | Optionally, a function that generates a random number can be supplied. 925 | >>> def FakeRandom(min, max): return 0.01 926 | >>> T.move('A', FakeRandom) 927 | 'B' 928 | 929 | """ 930 | return self.succ(state).choose(random_func) 931 | 932 | def pow(self, p, n): 933 | """ 934 | Find the probability distribution after n steps, starting from an 935 | initial ``Vector``. 936 | 937 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 938 | >>> p = pykov.Vector(A=1) 939 | >>> T.pow(p,3) 940 | {'A': 0.7629999999999999, 'B': 0.23699999999999996} 941 | >>> p * T * T * T 942 | {'A': 0.7629999999999999, 'B': 0.23699999999999996} 943 | """ 944 | return p * self**n 945 | 946 | def steady(self): 947 | """ 948 | With the assumption of ergodicity, return the steady state. 949 | 950 | .. note:: 951 | 952 | Inverse iteration method (P is the Markov chain) 953 | 954 | .. math:: 955 | 956 | Q = \mathbf{I} - P 957 | 958 | Q^T x = e 959 | 960 | e = (0,0,\dots,0,1) 961 | .. 962 | .. 963 | 964 | 965 | .. seealso:: 966 | 967 | W. Stewart: Introduction to the Numerical Solution of Markov Chains, 968 | Princeton University Press, Chichester, West Sussex, 1994. 969 | 970 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 971 | >>> T.steady() 972 | {'A': 0.7692307692307676, 'B': 0.23076923076923028} 973 | """ 974 | try: 975 | return self._steady 976 | except AttributeError: 977 | e2p, p2e = self._el2pos_() 978 | m = len(e2p) 979 | P = self._dok_(e2p).tocsr() 980 | Q = ss.eye(m, format='csr') - P 981 | e = numpy.zeros(m) 982 | e[-1] = 1. 983 | Q = Q.transpose() 984 | # not elegant singular matrix error 985 | Q[0, 0] = Q[0, 0] + _machineEpsilon() 986 | x = ssl.spsolve(Q, e, use_umfpack=True) 987 | x = x/sum(x) 988 | res = Vector() 989 | res._fromarray(x, e2p) 990 | self._steady = res 991 | return res 992 | 993 | def entropy(self, p=None, norm=False): 994 | """ 995 | Return the ``Chain`` entropy, calculated with the indicated probability 996 | Vector (the steady state by default). 997 | 998 | .. math:: 999 | 1000 | H_i = \sum_j P_{ij} \ln P_{ij} 1001 | 1002 | H = \sum \pi_i H_i 1003 | 1004 | .. seealso:: 1005 | 1006 | Khinchin, A. I. 1007 | Mathematical Foundations of Information Theory 1008 | Dover, 1957. 1009 | 1010 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1011 | >>> T.entropy() 1012 | 0.46989561696530169 1013 | 1014 | With normalization entropy belongs to [0,1] 1015 | 1016 | >>> T.entropy(norm=True) 1017 | 0.33895603665233132 1018 | 1019 | """ 1020 | if not p: 1021 | p = self.steady() 1022 | H = 0. 1023 | for state in self.states(): 1024 | H += p[state] * sum([v * math.log(v) for v in 1025 | self.succ(state).values()]) 1026 | if norm: 1027 | n = len(self.states()) 1028 | return -H / (n * math.log(n)) 1029 | return -H 1030 | 1031 | def mfpt_to(self, state): 1032 | """ 1033 | Return the Mean First Passage Times of every state to the indicated 1034 | state. 1035 | 1036 | .. seealso:: 1037 | 1038 | Kemeny J. G.; Snell, J. L. 1039 | Finite Markov Chains. 1040 | Springer-Verlag: New York, 1976. 1041 | 1042 | >>> d = {('R', 'N'): 0.25, ('R', 'S'): 0.25, ('S', 'R'): 0.25, 1043 | ('R', 'R'): 0.5, ('N', 'S'): 0.5, ('S', 'S'): 0.5, 1044 | ('S', 'N'): 0.25, ('N', 'R'): 0.5, ('N', 'N'): 0.0} 1045 | >>> T = pykov.Chain(d) 1046 | >>> T.mfpt_to('R') 1047 | {'S': 3.333333333333333, 'N': 2.666666666666667} 1048 | """ 1049 | if len(self.states()) == 2: 1050 | self.states().remove(state) 1051 | other = self.states().pop() 1052 | self.states().add(state) 1053 | self.states().add(other) 1054 | return Vector({other: 1. / self[other, state]}) 1055 | T = self.remove([state]) 1056 | T = T.eye() - T 1057 | return T._UMPFPACKSolve(T.ones()) 1058 | 1059 | def adjacency(self): 1060 | """ 1061 | Return the adjacency matrix. 1062 | 1063 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1064 | >>> T.adjacency() 1065 | {('B', 'A'): 1, ('A', 'B'): 1, ('A', 'A'): 1} 1066 | """ 1067 | return Matrix(OrderedDict.fromkeys(self, 1)) 1068 | 1069 | def walk(self, steps, start=None, stop=None): 1070 | """ 1071 | Return a random walk of n steps, starting and stopping at the 1072 | indicated states. 1073 | 1074 | .. note:: 1075 | 1076 | If not indicated or is `None`, then the starting state is chosen 1077 | according to its steady probability. 1078 | If the stopping state is not `None`, the random walk stops early if 1079 | the stopping state is reached. 1080 | 1081 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1082 | >>> T.walk(10) 1083 | ['B', 'A', 'B', 'A', 'A', 'B', 'A', 'A', 'A', 'B', 'A'] 1084 | >>> T.walk(10,'B','B') 1085 | ['B', 'A', 'A', 'A', 'A', 'A', 'B'] 1086 | """ 1087 | if start is None: 1088 | steady = self.steady() 1089 | if len(steady) != 0: 1090 | start = steady.choose() 1091 | else: 1092 | # There is no steady state, so choose a state uniformly at 1093 | # random. 1094 | start = random.sample(self.states(), 1)[0] 1095 | if stop is None: 1096 | result = [start] 1097 | for i in range(steps): 1098 | result.append(self.move(result[-1])) 1099 | return result 1100 | else: 1101 | result = [start] 1102 | for i in range(steps): 1103 | result.append(self.move(result[-1])) 1104 | if result[-1] == stop: 1105 | return result 1106 | return result 1107 | 1108 | def walk_probability(self, walk): 1109 | """ 1110 | Given a walk, return the log of its probability. 1111 | 1112 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1113 | >>> T.walk_probability(['A','A','B','A','A']) 1114 | -1.917322692203401 1115 | >>> probability = math.exp(-1.917322692203401) 1116 | 0.147 1117 | >>> p = T.walk_probability(['A','B','B','B','A']) 1118 | >>> math.exp(p) 1119 | 0.0 1120 | """ 1121 | res = 0 1122 | for step in zip(walk[:-1], walk[1:]): 1123 | if not self[step]: 1124 | return -float('Inf') 1125 | res += math.log(self[step]) 1126 | return res 1127 | 1128 | def mixing_time(self, cutoff=.25, jump=1, p=None): 1129 | """ 1130 | Return the mixing time. 1131 | 1132 | If the initial distribution (p) is not indicated, 1133 | then it is set to p={'less probable state':1}. 1134 | 1135 | .. note:: 1136 | 1137 | The mixing time is calculated here as the number of steps (n) needed to 1138 | have 1139 | 1140 | .. math:: 1141 | 1142 | |p(n)-\pi| < 0.25 1143 | 1144 | p(n)=p P^n 1145 | 1146 | \pi=\pi P 1147 | .. 1148 | 1149 | The parameter ``jump`` controls the iteration step, for example with 1150 | ``jump=2`` n has values 2,4,6,8,.. 1151 | .. 1152 | 1153 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 1154 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 1155 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 1156 | >>> T = pykov.Chain(d) 1157 | >>> T.mixing_time() 1158 | 2 1159 | """ 1160 | res = [] 1161 | d = 1 1162 | n = 0 1163 | if not p: 1164 | p = Vector({self.steady().sort()[0][0]: 1}) 1165 | res.append(p.dist(self.steady())) 1166 | while d > cutoff: 1167 | n = n + jump 1168 | p = self.pow(p, jump) 1169 | d = p.dist(self.steady()) 1170 | res.append(d) 1171 | return n 1172 | 1173 | def absorbing_time(self, transient_set): 1174 | """ 1175 | Mean number of steps needed to leave the transient set. 1176 | 1177 | Return the ``Vector tau``, the ``tau[i]`` is the mean number of steps needed 1178 | to leave the transient set starting from state ``i``. The parameter 1179 | ``transient_set`` is a subset of nodes. 1180 | 1181 | .. note:: 1182 | 1183 | If the starting point is a ``Vector p``, then it is sufficient to 1184 | calculate ``p * tau`` in order to weigh the mean times according the 1185 | initial conditions. 1186 | 1187 | 1188 | .. seealso: 1189 | 1190 | Kemeny J. G.; Snell, J. L. 1191 | Finite Markov Chains. 1192 | Springer-Verlag: New York, 1976. 1193 | 1194 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 1195 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 1196 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 1197 | >>> T = pykov.Chain(d) 1198 | >>> p = pykov.Vector({'N':.3, 'S':.7}) 1199 | >>> tau = T.absorbing_time(p.keys()) 1200 | >>> p * tau 1201 | 3.1333333333333329 1202 | """ 1203 | Q = self.remove(self.states() - set(transient_set)) 1204 | K = Q.eye() - Q 1205 | # means 1206 | tau = K._UMPFPACKSolve(K.ones()) 1207 | return tau 1208 | 1209 | def absorbing_tour(self, p, transient_set=None): 1210 | """ 1211 | Return a ``Vector v``, ``v[i]`` is the mean of the total number of times 1212 | the process is in a given transient state ``i`` before to leave the 1213 | transient set. 1214 | 1215 | .. note:: 1216 | ``v.sum()`` is equal to ``p * tau`` (see :meth:`absorbing_time` method). 1217 | 1218 | In not specified, the ``transient set`` is defined 1219 | by means of the ``Vector p``. 1220 | 1221 | .. seealso:: 1222 | 1223 | Kemeny J. G.; Snell, J. L. 1224 | Finite Markov Chains. 1225 | Springer-Verlag: New York, 1976. 1226 | 1227 | >>> d = {('R','R'):1./2, ('R','N'):1./4, ('R','S'):1./4, 1228 | ('N','R'):1./2, ('N','N'):0., ('N','S'):1./2, 1229 | ('S','R'):1./4, ('S','N'):1./4, ('S','S'):1./2} 1230 | >>> T = pykov.Chain(d) 1231 | >>> p = pykov.Vector({'N':.3, 'S':.7}) 1232 | >>> T.absorbing_tour(p) 1233 | {'S': 2.2666666666666666, 'N': 0.8666666666666669} 1234 | """ 1235 | if transient_set: 1236 | Q = self.remove(self.states() - transient_set) 1237 | else: 1238 | Q = self.remove(self.states() - set(p.keys())) 1239 | K = Q.eye() - Q 1240 | return K._UMPFPACKSolve(p, method='UMFPACK_At') 1241 | 1242 | def fundamental_matrix(self): 1243 | """ 1244 | Return the fundamental matrix. 1245 | 1246 | .. seealso:: 1247 | 1248 | Kemeny J. G.; Snell, J. L. 1249 | Finite Markov Chains. 1250 | Springer-Verlag: New York, 1976. 1251 | 1252 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1253 | >>> T.fundamental_matrix() 1254 | {('B', 'A'): 0.17751479289940991, ('A', 'B'): 0.053254437869822958, 1255 | ('A', 'A'): 0.94674556213017902, ('B', 'B'): 0.82248520710059214} 1256 | """ 1257 | try: 1258 | return self._fundamental_matrix 1259 | except AttributeError: 1260 | el2pos, pos2el = self._el2pos_() 1261 | p = self.steady()._toarray(el2pos) 1262 | P = self._numpy_mat(el2pos) 1263 | d = len(p) 1264 | A = numpy.matrix([p for i in range(d)]) 1265 | I = numpy.matrix(numpy.identity(d)) 1266 | E = numpy.matrix(numpy.ones((d, d))) 1267 | D = numpy.zeros((d, d)) 1268 | diag = 1. / p 1269 | for pos, val in enumerate(diag): 1270 | D[pos, pos] = val 1271 | Z = numpy.linalg.inv(I - P + A) 1272 | res = Matrix() 1273 | res._from_numpy_mat(Z, pos2el) 1274 | self._fundamental_matrix = res 1275 | return res 1276 | 1277 | def kemeny_constant(self): 1278 | """ 1279 | Return the Kemeny constant of the transition matrix. 1280 | 1281 | >>> T = pykov.Chain({('A','B'): .3, ('A','A'): .7, ('B','A'): 1.}) 1282 | >>> T.kemeny_constant() 1283 | 1.7692307692307712 1284 | """ 1285 | Z = self.fundamental_matrix() 1286 | return Z.trace() 1287 | 1288 | 1289 | def accessibility_matrix(self): 1290 | """ 1291 | Return the accessibility matrix of the Markov chain. 1292 | 1293 | ..see also: http://www.ssc.wisc.edu/~jmontgom/commclasses.pdf 1294 | """ 1295 | el2pos, pos2el = self._el2pos_() 1296 | Z = self.adjacency() 1297 | I = self.eye() 1298 | n = len(self.states()) 1299 | 1300 | A = (I + Z)**(n-1) 1301 | numpy_A = A._numpy_mat(el2pos) 1302 | numpy_A = numpy_A > 0 1303 | numpy_A = numpy_A.astype(int) 1304 | res = Matrix() 1305 | res._from_numpy_mat(numpy_A, pos2el) 1306 | return res 1307 | 1308 | def is_accessible(self, i, j): 1309 | """ 1310 | Return whether state j is accessible from state i. 1311 | """ 1312 | 1313 | A = self.accessibility_matrix() 1314 | return A.get((i, j)) > 0 1315 | 1316 | def communicates(self, i, j): 1317 | """ 1318 | Return whether states i and j communicate. 1319 | """ 1320 | return self.is_accessible(i, j) and self.is_accessible(j, i) 1321 | 1322 | def communication_classes(self): 1323 | """ 1324 | Return a Set of all communication classes of the Markov chain. 1325 | 1326 | ..see also: http://www.ssc.wisc.edu/~jmontgom/commclasses.pdf 1327 | 1328 | >>> T = pykov.Chain({('A','A'): 1.0, ('B','B'): 1.0}) 1329 | >>> T.communication_classes() 1330 | """ 1331 | el2pos, pos2el = self._el2pos_() 1332 | A = self.accessibility_matrix() 1333 | 1334 | numpy_A = A._numpy_mat(el2pos) 1335 | numpy_A_trans = numpy.transpose(numpy_A) 1336 | numpy_res = numpy.logical_and(numpy_A, numpy_A_trans) 1337 | numpy_res = numpy_res.astype(int) 1338 | 1339 | #remove duplicate rows 1340 | #remaining rows will give communication 1341 | #ref: http://stackoverflow.com/questions/16970982/find-unique-rows-in-numpy-array 1342 | a = numpy_res 1343 | b = numpy.ascontiguousarray(a).view( 1344 | numpy.dtype( 1345 | (numpy.void, a.dtype.itemsize * a.shape[1]) 1346 | ) 1347 | ) 1348 | _, idx = numpy.unique(b, return_index=True) 1349 | 1350 | unique_a = a[idx] 1351 | 1352 | res = Set() 1353 | for row in unique_a: 1354 | #each iteration here is a comm. class 1355 | comm_class = Set() 1356 | number_of_elements = len(A.states()) 1357 | for el in range(number_of_elements): 1358 | if row[0, el] == 1: 1359 | comm_class.add(pos2el[el]) 1360 | res.add(comm_class) 1361 | return res 1362 | 1363 | def readmat(filename): 1364 | """ 1365 | Read an external file and return a Chain. 1366 | 1367 | The file must be of the form: 1368 | 1369 | A A .7 1370 | A B .3 1371 | B A 1 1372 | 1373 | Example 1374 | ------- 1375 | >>> P = pykov.readmat('/mypath/mat') 1376 | >>> P 1377 | {('B', 'A'): 1.0, ('A', 'B'): 0.3, ('A', 'A'): 0.7} 1378 | """ 1379 | with open(filename) as f: 1380 | P = Chain() 1381 | for line in f: 1382 | tmp = line.split() 1383 | P[(tmp[0], tmp[1])] = float(tmp[2]) 1384 | return P 1385 | 1386 | 1387 | def readtrj(filename): 1388 | """ 1389 | In the case the :class:`Chain` instance must be created from a finite chain 1390 | of states, the transition matrix is not fully defined. 1391 | The function defines the transition probabilities as the maximum likelihood 1392 | probabilities calculated along the chain. Having the file ``/mypath/trj`` 1393 | with the following format:: 1394 | 1395 | 1 1396 | 1 1397 | 1 1398 | 2 1399 | 1 1400 | 3 1401 | 1402 | the :class:`Chain` instance defined from that chain is: 1403 | 1404 | >>> t = pykov.readtrj('/mypath/trj') 1405 | >>> t 1406 | (1, 1, 1, 2, 1, 3) 1407 | >>> p, P = maximum_likelihood_probabilities(t,lag_time=1, separator='0') 1408 | >>> p 1409 | {1: 0.6666666666666666, 2: 0.16666666666666666, 3: 0.16666666666666666} 1410 | >>> P 1411 | {(1, 2): 0.25, (1, 3): 0.25, (1, 1): 0.5, (2, 1): 1.0, (3, 3): 1.0} 1412 | >>> type(P) 1413 | 1414 | >>> type(p) 1415 | 1416 | """ 1417 | with open(filename) as f: 1418 | return tuple(line.strip() for line in f) 1419 | 1420 | 1421 | def _writefile(mylist, filename): 1422 | """ 1423 | Export in a file the list. 1424 | 1425 | mylist could be a list of list. 1426 | 1427 | Example 1428 | ------- 1429 | >>> L = [[2,3],[4,5]] 1430 | >>> pykov.writefile(L,'tmp') 1431 | >>> l = [1,2] 1432 | >>> pykov.writefile(l,'tmp') 1433 | """ 1434 | try: 1435 | L = [[str(i) for i in line] for line in mylist] 1436 | except TypeError: 1437 | L = [str(i) for i in mylist] 1438 | with open(filename, mode='w') as f: 1439 | tmp = '\n'.join('\t'.join(x) for x in L) 1440 | f.write(tmp) 1441 | return None 1442 | 1443 | 1444 | def transitions(trj, nsteps=1, lag_time=1, separator='0'): 1445 | """ 1446 | Return the temporal list of transitions observed. 1447 | 1448 | Parameters 1449 | ---------- 1450 | trj : the symbolic trajectory. 1451 | nsteps : number of steps. 1452 | lag_time : step length. 1453 | separator: the special symbol indicating the presence of sub-trajectories. 1454 | 1455 | Example 1456 | ------- 1457 | >>> trj = [1,2,1,0,2,3,1,0,2,3,2,3,1,2,3] 1458 | >>> pykov.transitions(trj,1,1,0) 1459 | [(1, 2), (2, 1), (2, 3), (3, 1), (2, 3), (3, 2), (2, 3), (3, 1), (1, 2), 1460 | (2, 3)] 1461 | >>> pykov.transitions(trj,1,2,0) 1462 | [(1, 1), (2, 1), (2, 2), (3, 3), (2, 1), (3, 2), (1, 3)] 1463 | >>> pykov.transitions(trj,2,2,0) 1464 | [(2, 2, 1), (3, 3, 2), (2, 1, 3)] 1465 | """ 1466 | result = [] 1467 | for pos in range(len(trj) - nsteps * lag_time): 1468 | if separator not in trj[pos:(pos + nsteps * lag_time + 1)]: 1469 | tmp = trj[pos:(pos + nsteps * lag_time + 1):lag_time] 1470 | result.append(tuple(tmp)) 1471 | return result 1472 | 1473 | 1474 | def maximum_likelihood_probabilities(trj, lag_time=1, separator='0'): 1475 | """ 1476 | Return a Chain calculated by means of maximum likelihood probabilities. 1477 | 1478 | Return two objects: 1479 | p : a Vector object, the probability distribution over the nodes. 1480 | T : a Chain object, the Markov chain. 1481 | 1482 | Parameters 1483 | ---------- 1484 | trj : the symbolic trajectory. 1485 | lag_time : number of steps defining a transition. 1486 | separator: the special symbol indicating the presence of sub-trajectories. 1487 | 1488 | Example 1489 | ------- 1490 | >>> t = [1,2,3,2,3,2,1,2,2,3,3,2] 1491 | >>> p, T = pykov.maximum_likelihood_probabilities(t) 1492 | >>> p 1493 | {1: 0.18181818181818182, 2: 0.4545454545454546, 3: 0.36363636363636365} 1494 | >>> T 1495 | {(1, 2): 1.0, (3, 2): 0.7499999999999999, (2, 3): 0.5999999999999999, (3, 1496 | 3): 0.25, (2, 2): 0.19999999999999998, (2, 1): 0.19999999999999998} 1497 | """ 1498 | q_ij = {} 1499 | tr = transitions(trj, nsteps=1, lag_time=lag_time, separator=separator) 1500 | _remove_dead_branch(tr) 1501 | tot = len(tr) 1502 | for step in tr: 1503 | q_ij[step] = q_ij.get(step, 0.) + 1 1504 | for key in q_ij.keys(): 1505 | q_ij[key] = q_ij[key] / tot 1506 | p_i = {} 1507 | for k, v in q_ij.items(): 1508 | p_i[k[0]] = p_i.get(k[0], 0) + v 1509 | t_ij = {} 1510 | for k, v in q_ij.items(): 1511 | t_ij[k] = v / p_i[k[0]] 1512 | T = Chain(t_ij) 1513 | p = Vector(p_i) 1514 | T._guess = Vector(p_i) 1515 | return p, T 1516 | 1517 | 1518 | def _remove_dead_branch(transitions_list): 1519 | """ 1520 | Remove dead branchs inserting a selfloop in every node that has not 1521 | outgoing links. 1522 | 1523 | Example 1524 | ------- 1525 | >>> trj = [1,2,3,1,2,3,2,2,4,3,5] 1526 | >>> tr = pykov.transitions(trj, nsteps=1) 1527 | >>> tr 1528 | [(1, 2), (2, 3), (3, 1), (1, 2), (2, 3), (3, 2), (2, 2), (2, 4), (4, 3), 1529 | (3, 5)] 1530 | >>> pykov._remove_dead_branch(tr) 1531 | >>> tr 1532 | [(1, 2), (2, 3), (3, 1), (1, 2), (2, 3), (3, 2), (2, 2), (2, 4), (4, 3), 1533 | (3, 5), (5, 5)] 1534 | """ 1535 | head_set = set() 1536 | tail_set = set() 1537 | for step in transitions_list: 1538 | head_set.add(step[1]) 1539 | tail_set.add(step[0]) 1540 | for head in head_set: 1541 | if head not in tail_set: 1542 | transitions_list.append((head, head)) 1543 | return None 1544 | 1545 | 1546 | def _machineEpsilon(func=float): 1547 | """ 1548 | should be the same result of: numpy.finfo(numpy.float).eps 1549 | """ 1550 | machine_epsilon = func(1) 1551 | while func(1) + func(machine_epsilon) != func(1): 1552 | machine_epsilon_last = machine_epsilon 1553 | machine_epsilon = func(machine_epsilon) / func(2) 1554 | return machine_epsilon_last 1555 | --------------------------------------------------------------------------------