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State-Space Methods and Kalman Filtering

The Argus project is built around a state-space formulation of stochastic processes, making it well-suited for applications like pulsar timing and gravitational wave detection.

What Are State-Space Models?

A state-space model represents a dynamical system using a pair of equations:

  1. State Equation (Evolution/Transition) \[ x_{t} = F x_{t-1} + w_{t} \]
  2. Describes how the hidden state evolves over time.

  3. \( F \) is the transition matrix, and \( w_t \) is process noise.

  4. Observation Equation \[ y_{t} = H x_{t} + v_{t} \]

  5. Describes how the observed data \( y_t \) relates to the hidden state.
  6. \( H \) is the observation matrix, and \( v_t \) is observation noise.

State-space models are powerful for time series where you want to estimate latent (unobserved) variables over time in a principled way.

Kalman Filter

The Kalman filter is a recursive algorithm that estimates the state of a linear dynamical system from noisy observations. It is optimal under the assumption of linear-Gaussian models.

Key Steps

  1. Predict
  2. Estimate the next state and its uncertainty based on the current estimate.
  3. Update
  4. Incorporate the new observation to refine the prediction.

This makes the Kalman filter especially useful in domains like signal processing, control systems, and astrophysics.

Why Use This for PTAs?

Pulsar timing arrays (PTAs) observe timing residuals from pulsars over long periods. The underlying signals (e.g., gravitational waves) are:

  • Weak
  • Stochastic
  • Embedded in noisy observations

By framing PTA analysis as a latent stochastic process in a state-space form, we can:

  • Efficiently model signal evolution over time
  • Incorporate uncertainty and measurement noise
  • Leverage fast recursive estimation with Kalman filtering

Extensions

Argus can be extended to handle:

  • Nonlinear systems (e.g., Extended or Unscented Kalman Filters)
  • Correlated noise models
  • Time-varying dynamics

For a deeper mathematical treatment, see the detailed Kalman filter mathematics and the publications linked in the main documentation.