Sign up ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

In much of the literature I'm studying it's one of those terms that occurs frequently yet without a rigorous definition to be found. Specifically, I am told:

For time-indexed random variables (RVs) $\{X_t\}$, the additive decomposition model is given as

$$X_t = {ll}(X_{t-1}, X_{t-2}, \ldots) + {fc}(X_{t-1}, X_{t-2}, \ldots, \varepsilon_t, \varepsilon_{t-1}, \ldots)$$


  • $ll$ is the long-term level, which is a stochastic process and can be visualised as a smoothed version of $\{X_t\}$, not to be confused with trends which are deterministic patterns
  • $fc$ is the fluctuation component which represents changes in local level, assumed stationary and with zero mean level
  • $\{\varepsilon_t\}$ are innovations, and are IID mean-zero RVs

But what is the difference in meaning between trend vs. long-term level vs. local level vs. mean level?

Additionally, aren't the fluctutation component and innovations modelling the same thing, which is the noise associated with each observation? So why complicate things by including both?

share|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.