Decomposition

• Z is a time-varying variable depending on random factors

• times seires is the set of time-ordered valued by measuring Z at some time points

statistical task that deconstructs a time series into several components:

• trend

• seasonal component

• cyclic component

• irrrgular component/ noise

• long-term increase or decrease in the level of time series

• systematic change in a time series, that does not appear to be periodic

when a time series is influenced by seasonal factors

• occurs over a fixed and known period (quarter of a year, month, day or week)

data shows rises and falls that are not of fixed period

• duration of these fluctuation is of at least 2 years

• random, irregular influences

• represents the residuals or remainder of the time series after the other commponents have been removed

Global trend:

• these are trends that apply to the whole time series

• easier to identify

• f(t) = f(t, beta)

  • t - time

  • beta - estimated parameters

Local trends

• trends that apply to parts or subsequences of a time series

• choice of the trend function f(t)

  • should be based on data and logicaly justified

  • must fit well the time series data

• estimating the trend-function parameters beta

  • model fit is as good as possible

  • e.g. via least squares method

• Additive decomposition

• Multiplicative decomposition

Tt - Trend

St - Seasonality

It - Irregular component

variations around the trend does not vary with the level of time series

• unclear, how well the deterministic part of the decomposition can be used for forecasting

• it is assumed, that the character of seasonality will not chnage over time

Seasonal trend decomposition based on LOESS

Seasonal trend decomposition based on REGRESSION