Decomposition

Decomposition of time series

a statistical task that deconstructs a time series into several components
- each one representing one of the underlying categories of patterns

Decomposition based on rates of change

seeks to construct, from an observed time series
- a number of component series (that could be used to reconstruct the original by additions or multiplications)
  - where each of these has a certain characteristic or type of behavior

time series decomposes into:

\(T_t\) the trend component at time \(t\)
- reflects the long-term progression of the series (secular variation)
  - a trend exists when there is a persistent increasing or decreasing direction in the data
\(C_t\) the cyclical component at time \(t\)
- reflects repeated but non-periodic fluctuations
  - durations of the fluctuations depend on the nature of the time series
\(S_t\) seasonal component at time \(t\)
- reflecting seasonal variation
  - a seasonal pattern exists when a time series is influenced by the seasonal factors
\(I_t\) the irregular component or noise at time \(t\)
- describes random, irregular influences
  - it represents the residuals or remainder of the time series after the other components have been removed
    
    Additive \(y_t = T_t + C_t + S_t + I_t\)
    
    additive models is used when the variations around the trend do not vary with the level of the time series
    
    whereas multiplicative models is appropriate if the trend is proportional to the level of the time series
    
    Multiplicative \(y_t = T_t \times C_t \times S_t \times I_t\)
    
    Sometimes the trend and cyclical components are grouped into
    - called trend-cycle component
      
      \(y_t = S_t + T_t + R_t\) \(S_t\) seasonal component, \(T_t\) trend-cycle component, \(R_t\) remainder component

\(M(X_t)=\sum_{k=-p}^{+f}\theta_k X_{t+k}\)

the value of time \(t\) of the series is therefore replaced by a weighted average of \(p\) "past" values of the series, the current value,
- and \(f\) "future" values of the series
the quantity \(p+f+1\) is called the moving average is said to be centred
- if in addition \(\theta_{-k}=\theta_k\) for any \(k\)
  - the moving average \(M\) is said to be symmetric
one of the simplest moving averages is the symmetric moving average of order… \(P=2p+1\) where all the weights are equal to \(\frac{1}{P}\)

cons

not resistant and might be deeply impacted by outliers
smoothing of the ends of the series cannot be done except with asymmetric moving averages which introduce phase-shifts and delays in the detection of turning points

additive \(X_t = TC_t + S_t + I_t\)

result of a long tradition of non-parametric smoothing based on moving averages
- which are weighted averages of a moving span of time series
in the X-11 method, symmetric moving averages play an important role
- they do not introduce any phase-shift in the smoothed series
but, to avoid losing information at the series ends
- they are supplemented by adhoc asymmetric moving averages
  - or applied on the series extended by forecasts

Henderson Moving Averages

pros

relatively robust to outliers and level shifts
sophisticated methods for handling trading day variation, holiday effects, end effects of known predictors
completly automated choices for trend and seasonal changes
widely tested on economic data over a long period of time

cons

Extension: X-12-ARIMA X-13-ARIMA

Seasonal and Trend decomposition using Loess