# 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 model
```

This 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.
```