Activity29

Introduction to State-space Models

1. Core Concept: The Hidden Layer of Time Series

Why State-Space? ARIMA models directly relate past observations to present values. State-space models add a hidden layer that evolves independently:

Simple Weather Example

  • Observed: Daily temperature (\(y_t\))
  • Hidden State: Atmospheric pressure system (\(x_t\))
  • Relationship:
    Pressure evolves: \(x_t = 0.9x_{t-1} + \text{weather\_disturbance}\)
    We observe: \(y_t = x_t + \text{thermometer\_error}\)

Model Equations

State Transition:

\[x_t = \phi x_{t-1} + w_t \quad (\text{Memory factor } \phi, \text{ system noise } w_t)\]

Observation Process:

\[y_t = x_t + v_t \quad (\text{Measurement noise } v_t)\]

2. Retail Inventory Analysis

Interpretation:

  • Level: Long-term employment capacity
  • Trend: Damped growth pattern (Ad = additive damped)
  • Noise: Irregular fluctuations (\(v_t\) in our equations)

State Transition Equations:

  • Level:

    \[ l_t = l_{t-1} + \phi b_{t-1} + \alpha \epsilon_t \quad \Rightarrow \quad l_t = l_{t-1} + 0.98b_{t-1} + 0.9999\epsilon_t \]

  • Damped Trend:

    \[ b_t = \phi b_{t-1} + \beta \epsilon_t \quad \Rightarrow \quad b_t = 0.98b_{t-1} + 0.0162\epsilon_t \]

Observation Equation:

\[ y_t = l_{t-1} + \phi b_{t-1} + \epsilon_t \quad \Rightarrow \quad y_t = l_{t-1} + 0.98b_{t-1} + \epsilon_t \]

3. Activity: Employment Dynamics

Step 1 - Model Setup

Step 2 - Extract States

Step 3 - Equation Translation

Based on the ETS(A,Ad,A) model output with parameters:

  • α = 0.6488 (level smoothing)
  • β = 0.1251 (trend smoothing)
  • φ = 0.9794 (damping factor)
  • γ = 0.3298 (seasonal smoothing)

State Transition Equations

  1. Level:
    \(l_t = l_{t-1} + \phi b_{t-1} + \alpha \epsilon_t\)
    \(\Rightarrow l_t = l_{t-1} + 0.9794b_{t-1} + 0.6488\epsilon_t\)

  2. Damped Trend:
    \(b_t = \phi b_{t-1} + \beta \epsilon_t'\)
    \(\Rightarrow b_t = 0.9794b_{t-1} + 0.1251\epsilon_t'\)

  3. Seasonal Component:
    \(s_t = s_{t-m} + \gamma \epsilon_t''\)
    \(\Rightarrow s_t = s_{t-12} + 0.3298\epsilon_t''\) (monthly seasonality: \(m = 12\))

Observation Equation

\(y_t = l_{t-1} + \phi b_{t-1} + s_{t-m} + v_t\)
\(\Rightarrow y_t = l_{t-1} + 0.9794b_{t-1} + s_{t-12} + v_t\)

4. Activity

Problem 1: Fit damped trend model to US_employment

Problem 2: Perform a residual diagnostics vs ARIMA