In [1]:
import math
from collections import defaultdict

class MarkovChainOrder(object):
''' Simple higher-order Markov chain, specifically for DNA
sequences.  User provides training data (DNA strings).  '''

def __init__(self, examples, order=1):
''' Initialize the model given collection of example DNA
strings. '''
self.order = order
self.mers = defaultdict(int)
self.longestMer = longestMer = order + 1
for ex in examples:
# count up all k-mers of length 'longestMer' or shorter
for i in range(len(ex) - longestMer + 1):
for j in range(longestMer, -1, -1):
self.mers[ex[i:i+j]] += 1

def condProb(self, obs, given):
''' Return conditional probability of seeing nucleotide "obs"
given we just saw nucleotide string "given".  Length of
"given" can't exceed model order.  Return None if "given"
was never observed. '''
assert len(given) <= self.order
ngiven = self.mers[given]
if ngiven == 0: return None
return float(self.mers[given + obs]) / self.mers[given]

def jointProb(self, s):
''' Return joint probability of observing string s '''
cum = 1.0
for i in range(self.longestMer-1, len(s)):
obs, given = s[i], s[i-self.longestMer+1:i]
p = self.condProb(obs, given)
if p is not None:
cum *= p
# include the smaller k-mers at the very beginning
for j in range(self.longestMer-2, -1, -1):
obs, given = s[j], s[:j]
p = self.condProb(obs, given)
if p is not None:
cum *= p
return cum

def jointProbL(self, s):
''' Return log2 of joint probability of observing string s '''
cum = 0.0
for i in range(self.longestMer-1, len(s)):
obs, given = s[i], s[i-self.longestMer+1:i]
p = self.condProb(obs, given)
if p is not None:
cum += math.log(p, 2)
# include the smaller k-mers at the very beginning
for j in range(self.longestMer-2, -1, -1):
obs, given = s[j], s[:j]
p = self.condProb(obs, given)
if p is not None:
cum += math.log(p, 2)
return cum

In [2]:
mc1 = MarkovChainOrder(['AC' * 10], 1)

In [3]:
mc1.condProb('A', 'C') # should be 1; C always followed by A

Out[3]:
1.0
In [4]:
mc1.condProb('C', 'A') # should be 1; A always followed by C

Out[4]:
1.0
In [5]:
mc1.condProb('G', 'A') # should be 0; A occurs but is never followed by G

Out[5]:
0.0
In [6]:
mc2 = MarkovChainOrder(['AC' * 10], 2)

In [7]:
mc2.condProb('A', 'AC') # AC always followed by A

Out[7]:
1.0
In [8]:
mc2.condProb('C', 'CA') # CA always followed by C

Out[8]:
1.0
In [9]:
mc2.condProb('C', 'AA') is None # because AA doesn't occur

Out[9]:
True
In [10]:
mc3 = MarkovChainOrder(['AAA1AAA2AAA2AAA3AAA3AAA3'], 3)

In [11]:
mc3.condProb('2', 'AAA') # 1/3

Out[11]:
0.3333333333333333
In [12]:
mc3.condProb('3', 'AAA') # 1/2

Out[12]:
0.5
In [13]:
p1 = mc3.condProb('A', '')
p1

Out[13]:
0.7619047619047619
In [14]:
p2 = mc3.condProb('A', 'A')
p2

Out[14]:
0.6875
In [15]:
p3 = mc3.condProb('A', 'AA')
p3

Out[15]:
0.5454545454545454
In [16]:
p4 = mc3.condProb('1', 'AAA')
p4

Out[16]:
0.16666666666666666
In [17]:
p1 * p2 * p3 * p4, mc3.jointProb('AAA1') # should be equal

Out[17]:
(0.0476190476190476, 0.04761904761904761)
In [18]:
import math
math.log(mc3.jointProb('AAA1'), 2), mc3.jointProbL('AAA1') # should be equal

Out[18]:
(-4.392317422778761, -4.392317422778761)