section 4.2.4 MTMAMΒΆ
This is another example from section 4.2.4 of Molecular Evolution: A Statistical Approach, page 109.
It computes the log likelihood for the empirical MTMAM model of molecular evolution for an evolutionary tree of primates.
The conditional expected number of transitions per edge divided by the conditional expected exit rate from each state can be used to iteratively update the parameters corresponding to edge-specific rate scaling factors, reaching the -14,558.59 log likelihood after six iterations. This is an expectation maximization.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | from __future__ import print_function, division, absolute_import
import copy
import json
import numpy as np
from numpy.testing import assert_equal
import jsonctmctree.interface
s_mtmam = """\
32
2 4
11 0 864
0 186 0 0
0 246 8 49 0
0 0 0 569 0 274
78 18 47 79 0 0 22
8 232 458 11 305 550 22 0
75 0 19 0 41 0 0 0 0
21 6 0 0 27 20 0 0 26 232
0 50 408 0 0 242 215 0 0 6 4
76 0 21 0 0 22 0 0 0 378 609 59
0 0 6 5 7 0 0 0 0 57 246 0 11
53 9 33 2 0 51 0 0 53 5 43 18 0 17
342 3 446 16 347 30 21 112 20 0 74 65 47 90 202
681 0 110 0 114 0 4 0 1 360 34 50 691 8 78 614
5 16 6 0 65 0 0 0 0 0 12 0 13 0 7 17 0
0 0 156 0 530 54 0 1 1525 16 25 67 0 682 8 107 0 14
398 0 0 10 0 33 20 5 0 2220 100 0 832 6 0 0 237 0 0\
"""
s_distn = """
0.0692 0.0184 0.0400 0.0186 0.0065 0.0238 0.0236 0.0557 0.0277 0.0905
0.1675 0.0221 0.0561 0.0611 0.0536 0.0725 0.0870 0.0293 0.0340 0.0428
"""
s_aas = 'ARNDCQEGHILKMFPSTWYV'
def main():
nstates = len(s_aas)
assert_equal(nstates, 20)
d = {a : i for i, a in enumerate(s_aas)}
distn = [float(x) for x in s_distn.strip().split()]
assert_equal(len(distn), nstates)
lines = s_mtmam.splitlines()
assert_equal(len(lines), nstates-1)
rate_matrix = np.zeros((nstates, nstates), dtype=int)
for i, line in enumerate(lines):
row_index = i + 1
row = [int(x) for x in line.strip().split()]
assert_equal(len(row), row_index)
rate_matrix[row_index, :row_index] = row
rate_matrix = np.multiply(rate_matrix + rate_matrix.T, distn)
exit_rates = rate_matrix.sum(axis=1)
# This is a partial scene, missing the root distribution,
# the process definition, and the observed data.
scene = {
"node_count" : 12,
"process_count" : 1,
"state_space_shape" : [20],
"tree" : {
"row_nodes" : [
0, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11],
"column_nodes" : [
8, 1, 2, 7, 9, 3, 10, 6, 11, 4, 5],
"edge_rate_scaling_factors" : [
0.001, 0.001, 0.001, 0.001, 0.001, 0.001,
0.001, 0.001, 0.001, 0.001, 0.001],
"edge_processes" : [
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
}
}
# Add the root distribution.
scene['root_prior'] = {
"states" : [[i] for i in range(nstates)],
"probabilities" : distn
}
# Add the process definition.
triples = []
for i in range(nstates):
for j in range(nstates):
r = rate_matrix[i, j]
if i != j and r:
triples.append((i, j, r))
row_states, col_states, rates = zip(*triples)
scene['process_definitions'] = [{
"row_states" : [[s] for s in row_states],
"column_states" : [[s] for s in col_states],
"transition_rates" : rates
}]
# Add the observed data.
sequences = []
with open('mtCDNApri.aa') as fin:
lines = fin.readlines()
header = lines[0]
for line in lines[1:]:
name, sequence = line.strip().split()
sequences.append([d[x] for x in sequence])
columns = [list(x) for x in zip(*sequences)]
nsites = len(columns)
scene['observed_data'] = {
"nodes" : [0, 1, 2, 3, 4, 5, 6],
"variables" : [0, 0, 0, 0, 0, 0, 0],
"iid_observations" : columns
}
# Define some requests.
# These include the log likelihood,
# some dwell time expectations, and some transition count expectations.
requests = [
{"property" : "SNNLOGL"},
{
"property" : "SDWDWEL",
"state_reduction" : {
"states" : [[i] for i in range(nstates)],
"weights" : exit_rates.tolist()
}
},
{
"property" : "SDNTRAN",
"transition_reduction" : {
"row_states" : [[s] for s in row_states],
"column_states" : [[s] for s in col_states],
"weights" : [1] * len(triples)
}
}]
# Request some stuff.
j_in = {
"scene" : scene,
"requests" : requests
}
arr = []
j_out = None
for i in range(7):
if j_out is None:
j_out = jsonctmctree.interface.process_json_in(j_in)
else:
dwells = j_out['responses'][1]
transitions = j_out['responses'][2]
scaling_factors = [t/d for t, d in zip(transitions, dwells)]
j_in['scene']['tree']['edge_rate_scaling_factors'] = scaling_factors
j_out = jsonctmctree.interface.process_json_in(j_in)
arr.append(copy.deepcopy(j_out))
print(json.dumps(arr, indent=4))
main()
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 | [
{
"status": "feasible",
"responses": [
-16504.121032993055,
[
390460.1059915314,
390424.5844230256,
391224.60535462154,
390824.8875002111,
390697.27993762447,
390043.24810754554,
390554.30746285967,
390649.92327320843,
389776.7310744418,
389136.96053040074,
388944.61400367785
],
[
95.79212073378258,
61.93135540104879,
60.24470946801712,
72.95137332382043,
74.8324931106707,
114.68492421659182,
131.4201313568659,
230.46271166866788,
201.85718287482823,
113.2558893397393,
102.61130105420631
]
]
},
{
"status": "feasible",
"responses": [
-14615.45177721429,
[
390329.090286233,
390266.26908660866,
391066.39607547666,
390611.3625268762,
390524.70785371325,
389931.06101992645,
390351.1126739523,
390499.7650051904,
389459.55536128406,
388892.806748215,
388701.9705687931
],
[
73.70766635159143,
38.18459705173582,
35.316752969600415,
43.078659444371354,
37.97967486227084,
89.13839402137754,
93.82513950950059,
217.02806583421844,
179.5893615119534,
87.72147690595901,
70.28055492514683
]
]
},
{
"status": "feasible",
"responses": [
-14559.544636017676,
[
390328.61872730206,
390254.32603035594,
391054.73439382075,
390600.5686739526,
390552.6731726354,
389956.3666851394,
390395.35882293235,
390516.58402257576,
389456.21133322164,
388870.1013831647,
388680.18994273257
],
[
74.47857833147525,
37.87478016778081,
34.5555552841812,
40.995137533150285,
33.042365140020976,
90.80469992743372,
87.98117910994463,
217.92673877299362,
179.02334833050713,
87.9519282620357,
66.95304311276284
]
]
},
{
"status": "feasible",
"responses": [
-14558.679492389878,
[
390327.18704844266,
390252.11766812106,
391052.5501532332,
390596.7203876089,
390561.4667511602,
389966.79685595026,
390416.48503409344,
390528.2576211807,
389464.82133761985,
388867.6363936431,
388677.6635125758
],
[
74.9180216155701,
37.89988987007537,
34.522729385014436,
40.66327688510903,
31.70563683058224,
92.13453901806906,
86.60930481714882,
218.5168970329599,
179.50084156422488,
88.46335584529685,
66.27857440209739
]
]
},
{
"status": "feasible",
"responses": [
-14558.604011930442,
[
390326.3472349087,
390251.4306210645,
391051.87252850796,
390595.1125301453,
390564.2062571244,
389970.5063065846,
390424.09742512205,
390532.5646433043,
389468.47017160396,
388867.2068578355,
388677.19180607935
],
[
75.04424767171336,
37.90244282411175,
34.522795173905514,
40.58652860788022,
31.259257663704524,
92.63978474704587,
86.23039258851394,
218.6998496012418,
179.73012202094094,
88.63464270632278,
66.09143469841817
]
]
},
{
"status": "feasible",
"responses": [
-14558.59572822224,
[
390326.0002153936,
390251.20633083815,
391051.6521278921,
390594.5170990737,
390565.1258302591,
389971.81857915595,
390426.77312118816,
390534.05067966715,
389469.7970711172,
388867.1093589269,
388677.08085152676
],
[
75.0808895141623,
37.90229913040323,
34.523694320534304,
40.56723880147044,
31.10348951003328,
92.81602807303793,
86.12107550833387,
218.75005119403716,
179.81468114217932,
88.68620712347064,
66.03581896848449
]
]
},
{
"status": "feasible",
"responses": [
-14558.594786234356,
[
390325.86895244475,
390251.13295395265,
391051.58020462503,
390594.30511277943,
390565.4422909194,
389972.28152996185,
390427.7080287603,
390534.5573468209,
389470.2646910455,
388867.08596201823,
388677.0534854288
],
[
75.09203425335421,
37.90219174533584,
34.52400997815937,
40.56224880903427,
31.048668496080666,
92.87676799816758,
86.08922785361307,
218.76314629525257,
179.84374455303276,
88.70164218119001,
66.01901840002965
]
]
}
]
|
And using the generic edge rate EM in the ‘extras’ module:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | from __future__ import print_function, division, absolute_import
import numpy as np
from numpy.testing import assert_equal
from jsonctmctree.interface import process_json_in
from jsonctmctree.extras import optimize_em
s_mtmam = """\
32
2 4
11 0 864
0 186 0 0
0 246 8 49 0
0 0 0 569 0 274
78 18 47 79 0 0 22
8 232 458 11 305 550 22 0
75 0 19 0 41 0 0 0 0
21 6 0 0 27 20 0 0 26 232
0 50 408 0 0 242 215 0 0 6 4
76 0 21 0 0 22 0 0 0 378 609 59
0 0 6 5 7 0 0 0 0 57 246 0 11
53 9 33 2 0 51 0 0 53 5 43 18 0 17
342 3 446 16 347 30 21 112 20 0 74 65 47 90 202
681 0 110 0 114 0 4 0 1 360 34 50 691 8 78 614
5 16 6 0 65 0 0 0 0 0 12 0 13 0 7 17 0
0 0 156 0 530 54 0 1 1525 16 25 67 0 682 8 107 0 14
398 0 0 10 0 33 20 5 0 2220 100 0 832 6 0 0 237 0 0\
"""
s_distn = """
0.0692 0.0184 0.0400 0.0186 0.0065 0.0238 0.0236 0.0557 0.0277 0.0905
0.1675 0.0221 0.0561 0.0611 0.0536 0.0725 0.0870 0.0293 0.0340 0.0428
"""
s_aas = 'ARNDCQEGHILKMFPSTWYV'
def main():
nstates = len(s_aas)
assert_equal(nstates, 20)
d = {a : i for i, a in enumerate(s_aas)}
distn = [float(x) for x in s_distn.strip().split()]
assert_equal(len(distn), nstates)
lines = s_mtmam.splitlines()
assert_equal(len(lines), nstates-1)
rate_matrix = np.zeros((nstates, nstates), dtype=int)
for i, line in enumerate(lines):
row_index = i + 1
row = [int(x) for x in line.strip().split()]
assert_equal(len(row), row_index)
rate_matrix[row_index, :row_index] = row
rate_matrix = np.multiply(rate_matrix + rate_matrix.T, distn)
exit_rates = rate_matrix.sum(axis=1)
# This is a partial scene, missing the root distribution,
# the process definition, and the observed data.
scene = {
"node_count" : 12,
"process_count" : 1,
"state_space_shape" : [20],
"tree" : {
"row_nodes" : [
0, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11],
"column_nodes" : [
8, 1, 2, 7, 9, 3, 10, 6, 11, 4, 5],
"edge_rate_scaling_factors" : [
0.001, 0.001, 0.001, 0.001, 0.001, 0.001,
0.001, 0.001, 0.001, 0.001, 0.001],
"edge_processes" : [
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
}
}
# Add the root distribution.
scene['root_prior'] = {
"states" : [[i] for i in range(nstates)],
"probabilities" : distn
}
# Add the process definition.
triples = []
for i in range(nstates):
for j in range(nstates):
r = rate_matrix[i, j]
if i != j and r:
triples.append((i, j, r))
row_states, col_states, rates = zip(*triples)
scene['process_definitions'] = [{
"row_states" : [[s] for s in row_states],
"column_states" : [[s] for s in col_states],
"transition_rates" : rates
}]
# Add the observed data.
sequences = []
with open('mtCDNApri.aa') as fin:
lines = fin.readlines()
header = lines[0]
for line in lines[1:]:
name, sequence = line.strip().split()
sequences.append([d[x] for x in sequence])
columns = [list(x) for x in zip(*sequences)]
nsites = len(columns)
scene['observed_data'] = {
"nodes" : [0, 1, 2, 3, 4, 5, 6],
"variables" : [0, 0, 0, 0, 0, 0, 0],
"iid_observations" : columns
}
# Update the edge rates according to a few iterations of EM.
observation_reduction = None
em_iterations = 6
edge_rates = optimize_em(scene, observation_reduction, em_iterations)
scene['tree']['edge_rate_scaling_factors'] = edge_rates
# Report the log likelihood for the updated edge rates.
j_in = dict(
scene = scene,
requests = [dict(property = 'SNNLOGL')])
ll = process_json_in(j_in)['responses'][0]
print(ll)
main()
|
-14558.5947862