Factorial Hidden Markov Model (FHMM)

FHMM represents the aggregate power signal as the sum of multiple independent hidden Markov chains, each corresponding to a single appliance. This additive structure allows the model to naturally decompose total consumption into constituent loads.

• Each chain evolves independently according to its own transition matrix
• The observed aggregate is a linear combination of state-dependent emissions
• Enables simultaneous tracking of multiple appliances without combinatorial state explosion

Each appliance is modeled with a discrete state space and a probabilistic transition matrix governing movement between operational modes. This captures realistic appliance behavior patterns.

• States represent operational modes: OFF, ON-low, ON-high, standby
• Transition probabilities encode typical usage durations and duty cycles
• Prior knowledge about appliance cycling can be encoded directly into the model

FHMM is a fully generative model that defines a joint probability distribution over hidden appliance states and observed aggregate readings. This enables principled inference and uncertainty quantification.

• Can generate synthetic aggregate data by sampling from the learned distribution
• Provides posterior probabilities over appliance states rather than hard classifications
• Supports Bayesian treatment of model parameters for robust estimation

Exact inference in FHMM is computationally intractable due to the exponential state space. Practical implementations rely on approximate techniques to estimate appliance states.

• Gibbs sampling iteratively samples each chain conditioned on all others
• Variational inference approximates the true posterior with a tractable distribution
• Mean-field approximations decouple chains for efficient message passing
• Inference complexity scales linearly with the number of appliances

The observation model linking hidden states to aggregate power can accommodate various noise assumptions to match real-world measurement characteristics.

• Gaussian emissions model additive sensor noise around the mean aggregate
• Poisson emissions handle count-based or low-magnitude power readings
• Can incorporate appliance-specific emission variances
• Supports modeling of both active and reactive power observations

FHMM parameters are typically learned using the Baum-Welch algorithm, a specialized form of Expectation-Maximization that iteratively refines transition and emission probabilities.

• E-step: Estimate posterior state probabilities given current parameters
• M-step: Update transition matrices and emission parameters to maximize likelihood
• Converges to a local optimum of the data likelihood
• Requires careful initialization to avoid poor local minima

The core mechanism inside each chain of the FHMM. It defines the probabilistic rules governing how an appliance moves between discrete operational states.

• State Definitions: Finite states like 'OFF', 'ON-Low', 'ON-High'.
• Transition Matrix: A square matrix where entry (i, j) is the probability of moving from state i to state j.
• Markov Property: The next state depends only on the current state, not the history.

The overarching application domain for FHMMs. NILM uses a single-point sensor to deduce individual appliance loads.

• Cost Efficiency: Eliminates the need for per-appliance sub-metering hardware.
• Signal Decomposition: The FHMM mathematically separates the additive aggregate signal.
• Use Case: Providing consumers with itemized energy bills to drive behavioral efficiency.

The feature engineering process required to parameterize the FHMM. It isolates unique electrical fingerprints.

• Steady-State Signatures: Differences in real and reactive power draw during stable operation.
• Transient Signatures: The brief, high-amplitude inrush current spike when a motor starts.
• Harmonic Content: Non-linear loads (like electronics) inject specific frequency multiples into the current waveform.

A pre-processing step that segments the aggregate signal before FHMM inference. It identifies significant state changes.

• Edge Detection: Algorithms like the Generalized Likelihood Ratio (GLR) test detect step-changes in power.
• Clustering: Grouping similar events to define the observable symbols for the Hidden Markov chains.
• FHMM Integration: Events trigger the inference engine to re-estimate the most likely combination of appliance states.

A canonical public benchmark used to train and validate FHMM-based disaggregators. It provides ground-truth data.

• Structure: Contains both low-frequency aggregate data and high-frequency circuit-level data from real homes.
• FHMM Training: Used to learn the transition matrices and emission probabilities for common appliances.
• Standardization: Allows researchers to compare FHMM performance against other architectures like deep neural networks.

An alternative signal processing approach that, unlike FHMM, does not require a pre-defined appliance model.

• Statistical Independence: Assumes source signals are statistically independent to separate the mixture.
• FHMM Advantage: FHMMs incorporate strong prior knowledge of appliance physics (finite states), making them more robust to overlapping loads.
• Hybrid Systems: Modern systems often use event detection to initialize a Blind Source Separation matrix.