Summary of local level model and local linear trend model to time series data

Overview

On this article, I’ll leave the summary about local level model and local linear trend model.
The both models are for time series analysis. Those are too simple to adapt for real data as they are. But those are very fundamental in many cases and by adding some other factors, those can become practical. So, here, I’ll leave rough memos about those.
As a text book, I’m using the following book. This article responds to chapters two and three.

The local level model

The local level model can simply be expressed as followings.
$y_t = u_t + e_t$ …①
$u_{t+1} = u_t + ξ_t$ …②
$e_t \sim Normal(0, σ_e^2)$
$ξ_t \sim Normal(0, σ_ξ^2)$

• $y_t$ : observed value at time t
• $u_t$ : unobserved level at time t
• $e_t$ : observation disturbance of time t
• $ξ_t$ : level disturbance of time t

Equation ① is called observation or measurement equation. Equation ② is called state equation.
If you compare this with classical linear regression model, the role of $u_t$ is equivalent to intercept. I’ll explain this points later on the phase of the local linear trend model.
About the example of local level model, please check the article below.

• Local level model to time series data on Stan

The local linear trend model

As the same manner as local level model, local linear trend model can also be expressed in the form of equation.
$y_t = u_t + e_t$ …③
$u_{t+1} = u_t + v_t + ξ_t$ …④
$v_{t+1} = v_t + ζ_t$ …⑤
$e_t \sim Normal(0, σ_e^2)$
$ξ_t \sim Normal(0, σ_ξ^2)$
$ζ_t \sim Normal(0, σ_ζ^2)$

Equation ③ is observation or measurement equation. Equations ④ and ⑤ are state equations. When we compare with local level model, we can notice that one more variable, $v_t$ was added. This $v_t$ is called slope, because it is equivalent to the classical regression model’s slope. The role of $v_t$ is not intuitive. So, by fixing $ξ_t$ and $ζ_t$ to zero, let’s re-express the equation.
The value $v_t$ doesn’t change from the value $v_1$. So we can express the equations ③ ~⑤ as the following one equation.
$y_t = u_1 + v_1 \times g_t + e_t$
Here, $g_t = t - 1$. $e_t$ just follows the normal distribution. So only changeable variable is $g_t$. To make the comparison with classical linear model easier, see the following corresponding model.
$y_t = a + b \times x_t + e_t$
We can understand by this that $v_t$ is equivalent to the slope and $u_t$ to the intercept.
About the example of local linear trend model, please check the article below.

• Local Linear Trend Model for time series analysis on Stan