Time Series Decomposition

3 Types of Patterns:
Trend

Trend exists when there is a long-term [on average] increase or decrease in the data. It does not have to be linear. Sometimes trend change direction when it might go from an increasing trend to a decreasing trend.

Cycle

cyclic pattern exists when data exhibit rises and falls that are not of fixed period. The duration of these fluctuations is usually of at least 1.5-2 years

Seasonal parts

Seasonal pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week). Seasonality is always of a fixed and known period.

Differences:
- seasonal pattern constant length; cyclic pattern variable length
- average length of cycle longer than length of seasonal pattern
- magnitude of cycle more variable than magnitude of seasonal pattern
Often Trend and Cycle components are combined and simply called Trend
To make Decomposition easier it may be necessary to adjust or transform the series

Transformations and adjustments

In general, there are four kind of adjustments, namely, Calendar Adjustment, Population Adjustment, Inflation Adjustment and Transformation, used to obtain a simpler time series
Simplify the patterns in the historical data by removing known sources of variation
Make the pattern more consistent across the whole data set
Simpler patterns are usually easier to model and lead to more accurate forecasts.

Calendar Adjustments are often called calendar effects, are the adjustment to remove these effects before analysis

Population adjustments are often use to remove the effect of population variation.

Inflation adjustments are often used for the data that are affected by the value of money

Mathematical Transformations

Mathematical transformations are often used for the data shows variation that increases or decreases with the level of the series, then a transformation can be useful. Most commonly used transformation is log transformation, $w_t=log(y_t)$ (note: changes in a log value are relative (or percentage) changes on the original scale). Another example is power transformations (flexible family of transformations introduced by Box and Cox 1964, Box-Cox transformation)

Log Transformation, $log(y_t)$

Used when all $Y_t>0$. Let $E[Y_t]=\mu_t$ and $\sqrt{V(Y_t)}=\mu_t \sigma$ then $E[log(Y_t)]=log(\mu_t)$ and $V(log(Y_t))=\sigma^2$

$\implies log(Y_t) \approx log(\mu_t) + \frac{Y_t-\mu_t}{\mu_t}$

if the standard deviation of the series is proportional to the level of the series, then transforming to logarithms will produce a series with approximately constant variance over time

Box- transformations

The aim of the Box-Cox transformations is to ensure the usual assumptions for Linear Model hold $y \sim N(X\beta,\sigma^2I_n)$

\[\begin{equation} w_t = \begin{cases} \log(y_t) & \text{if $\lambda=0$}; \\ (y_t^\lambda-1)/\lambda & \text{otherwise}. \end{cases} \end{equation}\]

There are family of box-cox transformation proposed in the literature (click here to read more)

Time Series Components

Additive Decomposition

$y_{t} = S_{t} + T_{t} + R_t$

It is the most appropriate if the magnitude of the seasonal fluctuations, or the variation around the trend-cycle, does not vary with the level of the time series

If not,

Multiplicative Decomposition

$y_{t} = S_{t} \times T_{t} \times R_t$

equivalent

$log(y_{t}) = log(S_{t}) + log(T_{t}) + log(R_t)$

Just to show an example, let’s use STL decomposition and get the first 6 observations

# A dable: 6 x 7 [1M]
# Key:     .model [1]
# :        Employed = trend + season_year + remainder
  .model    Month Employed  trend season_year remainder season_adjust
  <chr>     <mth>    <dbl>  <dbl>       <dbl>     <dbl>         <dbl>
1 stl    1990 Jan   13256. 13288.       -33.0     0.836        13289.
2 stl    1990 Feb   12966. 13269.      -258.    -44.6          13224.
3 stl    1990 Mar   12938. 13250.      -290.    -22.1          13228.
4 stl    1990 Apr   13012. 13231.      -220.      1.05         13232.
5 stl    1990 May   13108. 13211.      -114.     11.3          13223.
6 stl    1990 Jun   13183. 13192.       -24.3    15.5          13207.

The trend column (containing the trend-cycle $T_t$) follows the overall movement of the series, ignoring any seasonality and random fluctuations

Three additive components

And the Seasonally adjusted data (for additive, $Y_t-S_t$ and for multiplicative $Y_t/S_t$)

Seasonal Plots

Data plotted against the individual “seasons” in which the data were observed. (In this example a “season” is a month.)
Like a time plot except that the data from each season are overlapped
Enables the underlying seasonal pattern to be seen more clearly, and also allows any substantial departures from the seasonal pattern to be easily identified

Multiple seasonal periods

Data: Half-hourly electricity demand for Victoria, Australia

# A tsibble: 6 x 5 [30m] <Australia/Melbourne>
  Time                Demand Temperature Date       Holiday
  <dttm>               <dbl>       <dbl> <date>     <lgl>  
1 2012-01-01 00:00:00  4383.        21.4 2012-01-01 TRUE   
2 2012-01-01 00:30:00  4263.        21.0 2012-01-01 TRUE   
3 2012-01-01 01:00:00  4049.        20.7 2012-01-01 TRUE   
4 2012-01-01 01:30:00  3878.        20.6 2012-01-01 TRUE   
5 2012-01-01 02:00:00  4036.        20.4 2012-01-01 TRUE   
6 2012-01-01 02:30:00  3866.        20.2 2012-01-01 TRUE

Sub-series Plot

Data for each season collected together in time plot as separate time series
Enables the underlying seasonal pattern to be seen clearly, and changes in seasonality over time to be visualized

Moving averages

The first step in a classical decomposition is to use a moving average method to estimate the trend-cycle
A moving average of order $m$ can be written as:

$\hat{T}_{t} = \frac{1}{m} \sum_{j=-k}^k y_{t+j}$

where, $m=2k+1$

The estimate of the trend-cycle at time $t$ is obtained by averaging values of the time series within $k$ periods of $t$
The average eliminates some of the randomness in the data
It is called an $m-MA$
Here is the 12 Observations of Turkish Export and $5-MA$ of the series

# A tsibble: 12 x 3 [1Y]
    Year Exports `5-MA`
   <dbl>   <dbl>  <dbl>
 1  1960    2.06  NA   
 2  1961    5.12  NA   
 3  1962    5.60   4.29
 4  1963    4.18   4.79
 5  1964    4.47   4.58
 6  1965    4.56   4.28
 7  1966    4.09   4.18
 8  1967    4.11   4.01
 9  1968    3.68   3.98
10  1969    3.60   4.23
11  1970    4.43   4.61
12  1971    5.32   5.28

It is possible to apply a moving average to a moving average
One reason for doing this is to make an even-order moving average symmetric

# A tsibble: 10 x 4 [1Q]
   Quarter  Beer `4-MA` `2x4-MA`
     <qtr> <dbl>  <dbl>    <dbl>
 1 1992 Q1   443    NA       NA 
 2 1992 Q2   410   451.      NA 
 3 1992 Q3   420   449.     450 
 4 1992 Q4   532   452.     450.
 5 1993 Q1   433   449      450.
 6 1993 Q2   421   444      446.
 7 1993 Q3   410   448      446 
 8 1993 Q4   512   438      443 
 9 1994 Q1   449   441.     440.
10 1994 Q2   381   446      444.

$2\times4-MA$

\[\begin{align*} \hat{T}_{t} &= \frac{1}{2}\Big[ \frac{1}{4} (y_{t-2}+y_{t-1}+y_{t}+y_{t+1}) + \frac{1}{4} (y_{t-1}+y_{t}+y_{t+1}+y_{t+2})\Big] \\ &= \frac{1}{8}y_{t-2}+\frac14y_{t-1} + \frac14y_{t}+\frac14y_{t+1}+\frac18y_{t+2}. \end{align*}\]

Similarly for Monthly data (12 obs. per year), $2 \times 4-MA$ obtained US retail employment

Classical decomposition

Additive decomposition

$y_{t} = T_{t} + S_{t} + R_t$

Step 1: compute the trend-cycle component, $\hat{T_t}$

Step 2: Calculate the detrended series, $Y_t - \hat{T_t}$

Step 3: To estimate the seasonal component for each season, simply average the detrended values for that season, $\hat{S_t}$

Step 4: The remainder component is calculated by subtracting the estimated seasonal and trend-cycle components

$\hat{R_t}=Y_t - \hat{T_t} - \hat{S_t}$