Time Series Prediction

• Task:
  • discrete time, equidistant timesteps
  • Approximate teacher signal
  • Input signal : $u(t{)}_{t\in \mathcal{T}}$
  • Desired output : $(y(t){)}_{t\in \mathcal{T}}=u(t+h){)}_{t\in \mathcal{T}}$
  • Dynamical system state : $x(t)\in {\mathbb{R}}^n$
  • Temporal evolution governed by
    • State update map
      • Governs how $x(t)$ develops over time
      • $x(t+1)=f(x(t),u(t+1))$
    • Observation function
      • What output can be observed when in state $x(t)$
      • $y(t)=g(x(t))$
• Short range
  • PDE , ODE
• Long range
  • HMM
  • Recurrent