Contents

State space models (SSM)¶

Is just an hidden markov model, except the hidden states are continuous:

\[z_t = g(u_t, z_{t-1}, \epsilon_t)\]
\[y_t = h(z_t, u_t, \delta_t)\]

  • \(z_t\) is the hidden state

  • \(u_t\) is an optional input or control signal

  • \(y_t\) is the observation

  • \(g\) is the transition model

  • \(h\) is the observation model

  • \(\epsilon_t\) is the system noise at time t

  • \(\delta_t\) is the observation noise at time t.

Our goal is to recursively estimate the belief state:

\[p(z_t| y_{1:t}, u_{1:t}, \theta)\]

Linear-Gaussian SSM (LG-SSM) (Linear dynamical system (LDS))¶

\[\begin{split} z_t = A_tz_{t-1} + B_t u_t + \epsilon_t \\ y_t = C_t z_t + D_tu_t + \delta_t \\ \epsilon_t \sim N(0, Q_t) \\ \delta_t \sim N(0, R_t) \end{split}\]

  1. The transition model is a linear function

  2. The observation model is a linear function

  3. The system noise is Gaussian:

  4. The observation noise is Gaussian

If all the parameters \(\theta_t = (A_t, B_t, C_t, D_t, Q_t, R_t)\) are independent of time, the model is called stationary.

LG-SSM supports exact inference. The initial belief state is a Gaussian:

\[p(z_1) = N(\mu_{1|0}, \Sigma_{1|0})\]

All subsequent belief states are also Gaussian:

\[p(z_t| y_{1:t}) = N(\mu_{t|t}, \Sigma_{t|t})\]

Kalman filtering algorithm can compute this quantities efficiently.

Application of SSM¶

  1. Object tracking

  2. Robotic slam (Essentially this is the base for robotic vacuums)

  3. Time-series forecasting: Here we can show that the popular ARMA model can be viewed as a form of SSM.

Non-Linear and non-Gaussian SSM¶

A lot of models are not linear, or the noise is non-Gaussian. Hence the posterior is no longer Gaussian. So we have to approximate \(p(Y)\)

Transfer learning and domain adaptation Long term dependencies in RNN