Simple Hidden Markov Model (HMM)
Start the Julia REPL. Press ] to enter the Pkg mode.
Then execute the following line:
Pkg> add SimpleHMM
By now SimpleHMM is all installed and ready to use!
In your Julia code, load the package with:
using SimpleHMM# Load the example model
model_path = joinpath(dirname(pathof(SimpleHMM)), "../data", "example_model.json")
model = HMM_from_json(model_path)# Sequence length = 100
_, emitted_seq = emit(model, 100)
# Print the sequence:
println(emitted_seq)[3, 3, 3, 3, 3, 3, 4, 5, 3, 4, 2, 2, 1, 3, 1, 3, 2, 4, 4, 5, 3, 4, 2, 3, 3, 3, 4, 2, 1, 2, 2, 2, 1, 2, 3, 3, 3, 4, 2, 4, 3, 3, 3, 5, 2, 3, 3, 4, 2, 4, 4, 3, 4, 5, 4, 3, 3, 4, 4, 5, 4, 3, 3, 4, 4, 4, 3, 5, 2, 2, 2, 1, 4, 2, 4, 3, 4, 4, 3, 4, 3, 2, 4, 4, 4, 2, 3, 2, 5, 2, 4, 2, 3, 3, 1, 2, 4, 5, 3, 4]
We can also calculate the log-likelihood of this particular sequence being observed:
log_likelihood(model, emitted_seq)-139.14822148811922
First, we will initialize an HMM as the start point of inference:
# Initialize a HMM with random parameters
# The size of the hidden state space is 2
# The size of the observed state space is 5
initial_model = init_random_HMM(2, 5)
# Check the emission probability matrix:
display(initial_model.emission_matrix)2×5 Array{Float64,2}:
0.273413 0.197861 0.117856 0.14479 0.26608
0.127547 0.211586 0.0694058 0.366105 0.225356
Then, use the previously generated observed sequence to train HMM
# The new parameters are stored in the "new model"
new_model = baum_welch(initial_model, emitted_seq)
# Examine the trained emission probabilities:
display(new_model.emission_matrix)2×5 Array{Float64,2}:
0.285027 0.545211 0.146414 0.0224742 0.000874392
3.35735e-11 0.133287 0.391617 0.373998 0.101098
Infer the hidden states from an observed sequence
# Let's use the trained model to infer the hidden states
hidden_seq = viterbi(new_model, emitted_seq)
# check it out
println(hidden_seq)[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1]
SimpleHMM also supports models with continuous emission probabilities. The emission probability conforms a Gaussian mixture model and the number of mixtures can be adjusted.
# Initialized a random HMM with continuous emission
# The size of hidden states space is 2
# The number of mixtures for the Gaussian mixture model is 2
continuous_model = init_random_HMM(2, 2, "Gaussian")
# The emit an observed sequence or infer the parameters/hidden states
# just like we did to the discrete modelThis Julia package is the product of the Genomic Sequence Analysis module, by Dr Aylwyn Scally, as part of the Cambridge MPhil in Computational Biology programme.
[1]L. R. Rabiner, “A tutorial on hidden Markov models and selected applications in speech recognition,” Proceedings of the IEEE, vol. 77, no. 2, pp. 257–286, Feb. 1989, doi: 10.1109/5.18626.
[2]A. N. of Loc Nguyen, “Continuous Observation Hidden Markov Model,” Revista Kasmera, vol. 44, pp. 65–149, Jun. 2016.