table 1 by Minin and SuchardΒΆ

This example reproduces Table 1 of “Counting labeled transitions in continuous-time Markov models of evolution” by Minin and Suchard, up to sampling error.

In the paper they sampled nucleotide sequences of length 1000 and computed labeled transition count expectations conditional on those sequences, but here I’ve computed the expectations in the limit as the sequence length increases. The discrepancies in the reported conditional expectations seem to be within the sampling error.

Original table published by Minin and Suchard:

k Transitions Transversions
  Prior iid Post. CP Post. Prior iid Post. CP Post.
1 233.3 333.7 313.0 466.7 342.3 316.4
2 350.0 395.6 373.6 350.0 281.1 257.3
4 466.7 455.4 433.0 233.3 244.8 221.4

Reconstructed table from the output of the Python script below:

k Transitions Transversions
  Prior iid Post. CP Post. Prior iid Post. CP Post.
1 233.3 332.6 316.3 466.7 343.2 333.2
2 350.0 399.5 381.7 350.0 274.8 267.2
4 466.7 464.5 446.3 233.3 232.8 226.3 
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"""
This is really a K80 model because the nucleotide distribution is uniform.

Reproduce an example from Minin and Suchard
"Counting labeled transitions in continuous-time Markov models of evolution."

"""
from __future__ import print_function, division

import itertools
import copy

import numpy as np
from numpy.testing import assert_allclose

from jsonctmctree.interface import process_json_in


_template = """\
+------+-------------------------------+-------------------------------+
|  k   |           Transitions         |            Transversions      |
+------+--------+-----------+----------+--------+-----------+----------+
|      | Prior  | iid Post. | CP Post. | Prior  | iid Post. | CP Post. | 
+======+========+===========+==========+========+===========+==========+
|   1  | {0000} | {0001}    | {0002}   | {0003} | {0004}    | {0005}   |
+------+--------+-----------+----------+--------+-----------+----------+
|   2  | {0006} | {0007}    | {0008}   | {0009} | {0010}    | {0011}   |
+------+--------+-----------+----------+--------+-----------+----------+
|   4  | {0012} | {0013}    | {0014}   | {0015} | {0016}    | {0017}   |
+------+--------+-----------+----------+--------+-----------+----------+\
"""



def gen_K80():
    # Use the nucleotide order from the Minin and Suchard paper.
    # A, G, C, T
    transitions = ((0, 1), (1, 0), (2, 3), (3, 2))
    for i in range(4):
        for j in range(4):
            if i != j:
                if (i, j) in transitions:
                    ts, tv = 1, 0
                else:
                    ts, tv = 0, 1
                yield i, j, ts, tv


def get_K80_process_definition(kappa):
    exit_rates = np.zeros(4)
    row_states = []
    column_states = []
    unnormalized_rates = []
    for i, j, ts, tv in gen_K80():
        row_states.append([i])
        column_states.append([j])
        rate = kappa * ts + tv
        exit_rates[i] += rate
        unnormalized_rates.append(rate)

    # In the K80 model all exit rates are equal,
    # and each is equal to the expected rate.
    expected_rate = exit_rates.mean()
    assert_allclose(exit_rates, expected_rate)
    transition_rates = [r / expected_rate for r in unnormalized_rates]

    return dict(
            row_states = row_states,
            column_states = column_states,
            transition_rates = transition_rates)


def get_observation_reduction(likelihoods):
    npatterns = len(likelihoods)
    return dict(
            observation_indices = range(npatterns),
            weights = likelihoods)


def get_ts_reduction():
    return dict(
            row_states = [[i] for i, j, ts, tv in gen_K80() if ts],
            column_states = [[j] for i, j, ts, tv in gen_K80() if ts],
            weights = [1 for i, j, ts, tv in gen_K80() if ts])


def get_tv_reduction():
    return dict(
            row_states = [[i] for i, j, ts, tv in gen_K80() if tv],
            column_states = [[j] for i, j, ts, tv in gen_K80() if tv],
            weights = [1 for i, j, ts, tv in gen_K80() if tv])


def run_partitioned_analysis(scene, partitioned_likelihoods, kappa):

    # Copy the scene because we are going to do some surgery.
    scene = copy.deepcopy(scene)
    scene['tree'] = get_analysis_tree()
    analysis_process = get_K80_process_definition(kappa)
    scene['process_definitions'] = [analysis_process]

    # Request nucleotide transition count expectations.
    ts_requests = []
    for likelihoods in partitioned_likelihoods:
        ts_request = dict(
                property = 'WSNTRAN',
                observation_reduction = get_observation_reduction(likelihoods),
                transition_reduction = get_ts_reduction())
        ts_requests.append(ts_request)

    # Request nucleotide transversion count expectations.
    tv_requests = []
    for likelihoods in partitioned_likelihoods:
        tv_request = dict(
                property = 'WSNTRAN',
                observation_reduction = get_observation_reduction(likelihoods),
                transition_reduction = get_tv_reduction())
        tv_requests.append(tv_request)

    # Run the analysis.
    j_in = dict(
            scene = scene,
            requests = ts_requests + tv_requests)
    j_out = process_json_in(j_in)
    ts_responses = j_out['responses'][:3]
    tv_responses = j_out['responses'][3:]

    partitioned_ts = np.mean(ts_responses)
    partitioned_tv = np.mean(tv_responses)

    return partitioned_ts, partitioned_tv


def run_analysis(scene, likelihoods, kappa):

    # Copy the scene because we are going to do some surgery.
    scene = copy.deepcopy(scene)

    # Change the edge rate scaling factors.
    scene['tree'] = get_analysis_tree()

    # Define the model to be used for the analysis.
    analysis_process = get_K80_process_definition(kappa)
    scene['process_definitions'] = [analysis_process]

    # Request nucleotide transition count expectations.
    ts_transition_request = dict(
            property = 'WSNTRAN',
            observation_reduction = get_observation_reduction(likelihoods),
            transition_reduction = get_ts_reduction())

    # Request nucleotide transversion count expectations.
    tv_transition_request = dict(
            property = 'WSNTRAN',
            observation_reduction = get_observation_reduction(likelihoods),
            transition_reduction = get_tv_reduction())

    # Define the requests.
    requests = [ts_transition_request, tv_transition_request]

    # Run the analysis.
    j_in = dict(
            scene = scene,
            requests = requests)
    j_out = process_json_in(j_in)
    posterior_ts, posterior_tv = j_out['responses']

    # Re-run the analysis without any observations.
    scene['observed_data'] = dict(
            nodes = [],
            variables = [],
            iid_observations = [[] for p in likelihoods])
    j_in = dict(
            scene = scene,
            requests = requests)
    j_out = process_json_in(j_in)
    prior_ts, prior_tv = j_out['responses']

    return prior_ts, prior_tv, posterior_ts, posterior_tv


def get_analysis_tree():
    return dict(
            row_nodes = [0, 0, 1, 1],
            column_nodes = [2, 1, 3, 4],
            edge_rate_scaling_factors = [0.28, 0.21, 0.12, 0.09],
            edge_processes = [0, 0, 0, 0])


def get_simulation_tree(rate_scaling_factor):
    # Define the tree used for simulation in the Minin and Suchard paper.
    # In our case, we will use this to compute log likelihoods for all possible
    # alignment site patterns and to subsequently use these likelihoods
    # as site-specific weights.
    row_nodes = [0, 0, 1, 1]
    column_nodes = [2, 1, 3, 4]
    rates = np.array([0.3, 0.2, 0.1, 0.1]) * rate_scaling_factor
    return dict(
            row_nodes = row_nodes,
            column_nodes = column_nodes,
            edge_rate_scaling_factors = rates.tolist(),
            edge_processes = [0, 0, 0, 0])


def get_pattern_likelihoods(scene, rate_scaling_factor):
    scene = copy.deepcopy(scene)
    scene['tree'] = get_simulation_tree(rate_scaling_factor)

    # Define the request for per-pattern log likelihoods.
    log_likelihoods_request = {'property' : 'DNNLOGL'}

    # Get the per-pattern log likelihoods.
    j_in = dict(
            scene = scene,
            requests = [log_likelihoods_request])
    j_out = process_json_in(j_in)
    pattern_log_likelihoods = j_out['responses'][0]
    pattern_likelihoods = np.exp(pattern_log_likelihoods)

    # The sum of likelihoods over all patterns should be 1.
    assert_allclose(pattern_likelihoods.sum(), 1)

    # Return the pattern likelihoods.
    return pattern_likelihoods.tolist()


def main():

    # The root prior for every process in this project is uniform.
    root_prior = dict(
            states = [[0], [1], [2], [3]],
            probabilities = [0.25, 0.25, 0.25, 0.25])

    # Define the simulation process.
    simulation_process = get_K80_process_definition(4)

    # Define all possible site patterns.
    all_site_patterns = list(itertools.product(range(4), repeat=3))

    # Observation data consisting of all site patterns.
    observed_all_site_patterns = dict(
            nodes = [2, 3, 4],
            variables = [0, 0, 0],
            iid_observations = all_site_patterns)

    # Define the scene associated with the simulation.
    scene = dict(
            node_count = 5,
            process_count = 1,
            state_space_shape = [4],
            tree = None,
            root_prior = root_prior,
            process_definitions = [simulation_process],
            observed_data = observed_all_site_patterns)

    pattern_likelihoods = get_pattern_likelihoods(scene, 1)
    partitioned_pattern_likelihoods = []
    position_rates = np.array([1.5, 1.0, 3.0])
    position_rates = position_rates / position_rates.mean()
    for rate_scaling_factor in position_rates:
         likelihoods = get_pattern_likelihoods(scene, rate_scaling_factor)
         partitioned_pattern_likelihoods.append(likelihoods)

    # Run an analysis for each of a few values of kappa.
    arr = []
    for kappa in 1, 2, 4:
        prior_ts, prior_tv, post_ts, post_tv = run_analysis(
                scene, pattern_likelihoods, kappa)
        partitioned_ts, partitioned_tv = run_partitioned_analysis(
                scene, partitioned_pattern_likelihoods, kappa)
        arr.extend((
                prior_ts, post_ts, partitioned_ts,
                prior_tv, post_tv, partitioned_tv))

    # Print the table using the template.
    s = ['{:>6.1f}'.format(x * 1000) for x in arr]
    rst_table_string = _template.format(*s)
    print(rst_table_string)


main()