Bayesian estimation of discrete system states from high-dimensional data
Probabilistic state estimation from complex measurement data using dimensionality reduction, mixture models, and sequential Bayesian inference.
Project Overview
This project focused on estimating discrete system states from high-dimensional measurement data, where the original 56-dimensional measurement vector required dimensionality reduction before probabilistic interpretation.
The method combined PCA-based feature extraction with Gaussian mixture models in the reduced principal-component space and a Hidden Markov Model for sequential Bayesian state estimation, enabling robust inference of system states from noisy and complex measurement patterns.
Context: Academic research · TU Graz
Role: Role: Method development, probabilistic modeling, dimensionality reduction, Bayesian inference, and implementation of the state-estimation workflow.
Methods: Bayesian Inference · Hidden Markov Models · Gaussian Mixture Models · PCA · State Estimation · High-Dimensional Data · Probabilistic Modeling
Problem
In many technical and scientific systems, the relevant system state is not directly observable. Instead, it has to be inferred from high-dimensional measurement data that may contain noise, redundant information, and complex correlations between variables.
In this project, each observation consisted of a 56-dimensional measurement vector. Direct interpretation of the original measurement space was difficult, both because of the dimensionality of the data and because the system state was only indirectly reflected in the measured variables.
The key challenge was therefore to transform the high-dimensional measurements into a compact representation and to estimate the underlying discrete system state in a probabilistic and temporally consistent way.
Key challenges
- High-dimensional measurement vectors
- Indirect observation of discrete system states
- Noise and overlap between state-dependent measurement patterns
- Need for probabilistic rather than purely deterministic classification
- Temporal consistency of state estimates over time
Approach
The analysis workflow combined dimensionality reduction, probabilistic observation modeling, and sequential Bayesian inference.
1. Dimensionality reduction
Principal component analysis was used to transform the original 56-dimensional measurement vector into a lower-dimensional representation. The first principal components captured the dominant patterns in the data and provided a compact feature space for subsequent probabilistic modeling.
2. Probabilistic observation model
A Gaussian mixture model was fitted in the reduced principal-component space to describe the probability of observing a given measurement pattern under each discrete system state. This provided a likelihood model linking the observed data to the hidden state.
3. Sequential state estimation
A Hidden Markov Model was used to incorporate temporal information and probabilistic state transitions. This allowed the state estimate at each time point to depend not only on the current measurement but also on the estimated state sequence over time.
4. Posterior inference
The final output of the workflow was a posterior probability for each possible system state over time. This enabled uncertainty-aware state estimation instead of a single hard classification without confidence information.
Figures
Figure 1: Workflow for Bayesian estimation of discrete system states from high-dimensional measurement data. The original 56-dimensional measurement vector is transformed using principal component analysis. A Gaussian mixture model defines the likelihood in the reduced principal-component space, while a Hidden Markov Model incorporates temporal state transitions. The resulting posterior probabilities provide an uncertainty-aware estimate of the discrete system state over time.
Figure 2: Example of time-resolved Bayesian state estimation. The upper panel shows the temporal evolution of the first principal component derived from the high-dimensional measurement vector. The lower panel shows the corresponding posterior probabilities of the two discrete system states estimated by the Hidden Markov Model, illustrating how measurement dynamics are translated into probabilistic state estimates over time.
Figure 1: Workflow for Bayesian estimation of discrete system states from high-dimensional measurement data. The original 56-dimensional measurement vector is transformed using principal component analysis. A Gaussian mixture model defines the likelihood in the reduced principal-component space, while a Hidden Markov Model incorporates temporal state transitions. The resulting posterior probabilities provide an uncertainty-aware estimate of the discrete system state over time.
Figure 2: Example of time-resolved Bayesian state estimation. The upper panel shows the temporal evolution of the first principal component derived from the high-dimensional measurement vector. The lower panel shows the corresponding posterior probabilities of the two discrete system states estimated by the Hidden Markov Model, illustrating how measurement dynamics are translated into probabilistic state estimates over time.
Outcome
The project demonstrated how high-dimensional measurement data can be transformed into a compact probabilistic representation and used for sequential estimation of discrete system states.
By combining PCA, Gaussian mixture modeling, and Hidden Markov Models, the workflow provided a structured way to connect complex measurement patterns with interpretable state estimates. Instead of producing only a deterministic classification, the method yielded posterior state probabilities, making uncertainty in the state estimate explicit.
The project illustrates a general analytical approach that is relevant for systems where hidden or discrete states need to be inferred from indirect, noisy, or high-dimensional measurements.
- Compact representation of 56-dimensional measurement data using PCA
- Probabilistic observation model in principal-component space
- Sequential Bayesian state estimation using a Hidden Markov Model
- Posterior probabilities for two discrete system states over time
- Interpretable workflow for high-dimensional measurement-based inference
Related Services
Modeling & Inference →
Probabilistic modeling, state estimation, Hidden Markov Models, and Bayesian inference for hidden or indirectly observed quantities.
Statistical Analysis →
Evaluation of model outputs, posterior probabilities, classification behavior, uncertainty, and performance of state-estimation workflows.
Automated Workflows →
Implementation of reproducible workflows for dimensionality reduction, model fitting, inference, visualization, and repeated analysis.
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