HMM vs Headache: Predicting Headache Occurrences Using Hidden Markov Models

Overview

This repository contains our analysis on predicting future occurrences of headaches using Hidden Markov Models (HMMs). The dataset comprises a binary sequence of 296 days, indicating the presence (1) or absence (0) of a headache.

Authors

• Valeria Insogna
• Guglielmo Padula

University

Universit`a degli Studi di Trieste

Dataset

The dataset represents a binary sequence:

• 1: Day with headache
• 0: Day without headache

Objective

To use HMMs and other statistical models to predict the likelihood of experiencing headaches in the future.

Analysis:

Autocorrelation: The first three lags are statistically significant, leading to the decision to model the data as a time series using short memory HMM models.

General Model Structure:

• ( Z_t ): Markov process (possibly with memory) parametrized by ( \theta ).
• ( X_t | Z_t = z_t ): Follows a Bernoulli distribution governed by a function ( g ).

Inference: Used Black Box Expectation Maximization algorithm (via Pyro library).

Performance Measures:

• BIC and autocorrelation distance in the training data for internal model selection.
• BIC in test data and the autocorrelation distance for the best average model.

Models Explored:

Model 1 - Wiener Process:

• ( Z_t ) is modeled as a Wiener process.
• Performance: BIC=1277, Autocorrelation distance=0.12.

Model 2 - Stagional ARIMA:

• ( Z_t ) is modeled as sARIMA(0, 0, 0) × (0, 1, 0)3.
• Performance: BIC=1023, Autocorrelation distance=0.08.

Model 3 - Bayesian Normal Mixture:

• Combines various distributions including Beta, Normal, and InverseGamma.
• Performance: BIC=662, Autocorrelation distance=0.12.

Model 4 - Discrete Time Markov Chain (DTMC):

• ( Z_t ) follows a discrete transition given by matrix ( A ).
• Performance: BIC=288, Autocorrelation distance=0.006.

Model 5 - Mixture of Categorical Variables:

• Combines several distributions like Dirichlet and Beta.
• Performance: BIC=252, Autocorrelation distance=0.04.

Key Findings

• The DTMC Model (Model 4) was found to be the most optimal in predicting headache occurrences.
• The analysis suggests a cyclical pattern: a three-day high likelihood of headaches followed by a three-day low likelihood.

Tools and Libraries Used

• Pyro Library for model implementation and inference
• Other common data science libraries like Pandas, NumPy, and Matplotlib