Anastasia Giachanou, Tina Shahedi
Machine Learning with Python - Utrecht Summer School
Welcome to the Hidden Markov Model practical!
This practice is built on Emmeke Aarts's practical workshop on "Extracting personalised latent dynamics using a multilevel hidden Markov model".
In this practical, we introduce a dataset and use it to fit a two-state hidden Markov model (HMM) (so not multilevel).
Explore the documentation provided in libraries like hmmlearn or pomegranate, which are used for building and analyzing hidden Markov models. The complete documentation for both can be accessed in bellow:
For this practical, we will use several Python libraries. The hmmlearn library is a popular choice for building Hidden Markov Models in Python. We start by ensuring that it is installed in our environment.
!pip install hmmlearn
!pip install plotly
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# Importing necessary libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import hmmlearn
import plotly.graph_objects as go
from hmmlearn import hmm
from scipy.stats import norm
We will be working with an open access dataset from Rowland and Wenzel (2020), which is part of a study involving 125 undergraduate students from the University of Mainz in Germany. These students completed a 40-day ambulatory assessment six times a day, reporting on their affective experiences such as happiness, excitement, relaxation, satisfaction, anger, anxiety, depression, and sadness. These affective states were quantified using a visual analog slider, ranging from 0 to 100.
Before the data collection, participants were randomly assigned to either a group receiving weekly mindfulness treatment during the study or a control group. We will be working with the specific dataset here. This datase has been cleaned and is provided by Haslbeck, Ryan, & Dablander (2023) and can be found in their OSF repository.
1. Load the dataset emotion_data.csv using the function pd.read_csv(). Give the dataset the name emotion_data. Then, inspect the first few rows of the data.
The dataset contains multiple affective states recorded over time for multiple participants. Each row in the dataset represents a unique measurement instance To fit a hidden Markov model, we need to understand the structure of the data, including the variables and their relationships.
Ee can see the summary here:
| No | Name | Label | Options |
|---|---|---|---|
| 1 | subjno | individual code | 1-164 |
| 2 | dayno | study days | 1-40 |
| 3 | beep | daily signals | 1-6 |
| 4 | group | condition allocation | 1 = control, 2 = training |
| 5 | emo1_m | happy | 0-100 |
| 6 | emo2_m | excited | 0-100 |
| 7 | emo3_m | relaxed | 0-100 |
| 8 | emo4_m | satisfied | 0-100 |
| 9 | emo5_m | angry | 0-100 |
| 10 | emo6_m | anxious | 0-100 |
| 11 | emo7_m | depressed | 0-100 |
| 12 | emo8_m | sad | 0-100 |
Next, we need to preprocess the data to make it suitable for fitting a hidden Markov model. First we are gonna check for missing values across the columns by applying the .isnull().sum() function to each column in the dataset:
# Count of NaN values in each column
NaN_counts = emotion_data.isnull().sum()
NaN_counts
subj_id 0 dayno 0 beep 0 group 0 happy 8430 excited 8430 relaxed 8430 satisfied 8430 angry 8430 anxious 8430 depressed 8430 sad 8430 dtype: int64
For this case, we will fill the NaN value by mean.
# Remove rows with any NaN values
emotion_data_clean = emotion_data.fillna(emotion_data.mean())
Now, we will create a plot to visualize the hidden states over time for each subject in the dataset.
2. Visualize time sequences for individual subjects separately to analyze emotional responses over time. Start by transforming your dataset from wide to long format using pandas.melt(), which prepares it for detailed analysis. Select only five emotions: 'happy', 'excited', 'relaxed', 'angry', 'depressed'.
3. Now, create multi-faceted line plots for each subject using seaborn.FacetGrid, ensuring clear visualization by selecting a manageable subset of subjects, such as the first four.
In this section, you will fit a 2-state hidden Markov model. For this, you need a DataFrame in which only the affective variables you want to use in the model are included, plus the subject ID as the first column. Please feel free to use a subset of the provided affective variables, but do make sure to select at least two affective variables.
4. Create a dataset emotion_mHMM which has the subject ID variable in the first column, followed by only the affective variables you want to use in the hidden Markov model.
5. Set up the first set of model input arguments: m for the number of states, n_dep for the number of dependent variables, and starting values for gamma and the emission distribution.
Next, we convert the data into a format required by hmmlearn
# Convert start_emiss to the format required by hmmlearn
start_means = np.array([emiss[:, 0] for emiss in start_emiss]).T
start_covars = np.array([emiss[:, 1] for emiss in start_emiss]).T ** 2
We also drop the subj_id column from the emotion_mHMM DataFrame to ensure the observations matrix consists only of emotional state data for Hidden Markov Model (HMM) analysis.
observations = emotion_mHMM.drop(columns=['subj_id']).values
# Ensure the shape of observations
print(f"Shape of observations: {observations.shape}")
Shape of observations: (30000, 5)
6. Initialize a Gaussian Hidden Markov Model with a n_component=m smensioned and n_iter=500. Set the model's initial state probabilities, transition matrix, means, and covariances using predefined variables (start_gamma, start_means, start_covars)
Next, we will proceed to fit a Hidden Markov Model to this preprocessed data.
7. Fit a 2-state hidden Markov model. Use the fit method of the HMM model to train it on the prepared observations data. Assign the fitted model to a variable named out_2st_emotion.
8. Determine the total number of unique subjects using .nunique() on subj_id and calculate the average log-likelihood with the model.score() method. Print the number of subjects and the average log-likelihood to summarize key metrics.
9. Find the number of states using the n_components attribute and identify the number of dependent variables using the .shape attribute of the observations array. Print these values to document the model's characteristics.
10. Access and print the transition probability matrix using transmat_, then print the mean and standard deviation of each dependent variable within each state using the means_ and covars_ attributes.
11. To facilitate easy post-processing, save the transition probabilities gamma in an object named gamma. Use the transition matrix transmat_ from the model to extract and save these probabilities. Then, print the gamma to inspect the transition probabilities from one state to another.
12. Now, Save the emission distribution parameters in an object named emiss. Use the means_ and covars_ attributes from the model to extract and save these parameters. Print the emiss to inspect the mean and standard deviation for each dependent variable in each state.
13. Visualizing the transition probabilities can be very helpful, especially when dealing with a large number of states and/or dependent variables.
14. Visualize the transition probabilities gamma saved in the gamma object. Use a heatmap to represent the probabilities of transitioning from one state to another.
15. Visualize the emission distributions to understand the mean values of different dependent variables across states. Prepare the data by creating a DataFrame with the state, mean emission value, and dependent variable name using the emiss dictionary. Use Seaborn's barplot function to create the plot,
16. Compute the density probabilities for each emotion and state. First define the length of the grid and generate a sequence of mood values ranging from 1 to 100. Select only those emotions that are present in the emiss dataset to ensure the accuracy of the density plots. Create a DataFrame emiss_dens with columns for state, emotion (Dep), mood value (X), and probability (Prob).Finally by using the normal distribution parameters calculate the density probabilitiesfor each emotion-state pair
17. Create subplots for each emotion using seaborn.FacetGrid with specified attributes. Plot density probabilities for mood values differentiated by state using sns.lineplot. Add vertical dashed lines at the mean mood values for each state. Set y-axis limits from 0 to 0.05 for consistency, and include a legend and axis labels for mood value and density.
18. Using the predict method from hmmlearn, the fitted model, and the observations emotion_mHMM, obtain the most likely hidden state sequence. Save the state sequences in the object emotion_states_2st, and inspect the object.
In this section we will plot the inferred state sequence over time for a selection of the subjects within the data. To plot the state sequences over time, a new variable denoting the beep moment needs to be added to the matrix containing the inferred states.
19. Now, Create a DataFrame that includes subj_id, state, and beep moment. Filter the DataFrame to include subjects 1 to 10, and map the states to categorical labels ('state 1' and 'state 2'). Use Seaborn's 'Set2' palette for the states and FacetGrid to create horizontal bar plots for each subject, adjusting the figure size and aspect ratio for better readability. Customize the plot with axis labels, titles, and a legend.
End of Practical!