# Time Series Decomposition: Classical Method

Last Update: June 1, 2022

Time Series Decomposition: Classical Method is used to estimate time series trend-cycle, seasonal and remainder components.

As example, we can delimit univariate time series $y_{t}$ into training range $y_{t(a)}$ for model fitting and testing range $y_{t(b)}$ for model forecasting.

Then, we can do training range univariate time series classical additive seasonal decomposition by moving averages trend-cycle, seasonal and remainder components estimation. Notice that we have to evaluate whether time series classical additive or multiplicative seasonal decomposition is needed.

Next, we estimate trend-cycle component with formulas $\hat{t}_{t(a)}=\frac{1}{2}\left&space;(&space;sma_{m,t(a)}\left&space;(&space;y_{t(a)}&space;\right&space;)+sma_{m,t(a)-1}\left&space;(&space;y_{t(a)}&space;\right&space;)&space;\right&space;)\;(1)$ when seasonal period $m$ is an even number and $\hat{t}_{t(a)}=sma_{m,t(a)}\left&space;(&space;y_{t(a)}&space;\right&space;)\;(2)$ when seasonal period $m$ is an odd number. As example, with monthly data, $m=12$. Training range trend-cycle component estimated values $\hat{t}_{t(a)}$ are the two periods averages of univariate time series centered seasonal simple moving averages when seasonal period $m$ is an even number and the univariate time series centered seasonal simple moving averages when seasonal period $m$ is an odd number. Training range univariate time series centered seasonal simple moving averages $sma_{m,t(a)}\left&space;(&space;y_{t(a)}&space;\right&space;)$ are the $y_{t(a)}$ rolling averages with formulas $sma_{m,t(a)}\left&space;(&space;y_{t(a)}&space;\right&space;)=\frac{1}{m}\sum_{j=-k+1}^{k}y_{t(a)+j}\;(3)$ where $k=\frac{m}{2}\;(4)$ when seasonal period $m$ is an even number and $sma_{m,t(a)}\left&space;(&space;y_{t(a)}&space;\right&space;)=\frac{1}{m}\sum_{j=-k}^{k}y_{t(a)+j}\;(5)$ where $k=\frac{m-1}{2}\;(6)$ when seasonal period $m$ is an odd number. After that we estimate detrended univariate time series values with formula $\hat{d}_{t(a)}=y_{t(a)}-\hat{t}_{t(a)}\;(7)$.

Later, we estimate seasonal component with formula $\hat{s}_{t(a)}=\hat{si}_{t(a)}+\frac{1}{m}\left&space;(&space;0-\sum_{t(a)=1}^{m}\hat{si}_{t(a)}&space;\right&space;)\;(8)$. Training range seasonal component estimated values $\hat{s}_{t(a)}$ are the estimated seasonal index values $\hat{si}_{t(a)}$ adjusted so that they add to zero. As example, with monthly data, we estimate December seasonal index with formula $\hat{si}_{d(a)}=\frac{1}{k}\sum_{j=1}^{k}\hat{d}_{d(a),j}\;(9)$ where $k$ is the number of December detrended univariate time series values within training range. Training range December seasonal index value $\hat{si}_{d(a)}$ is the average of all estimated December detrended univariate time series values.

Then we estimate remainder component with formula $\hat{r}_{t(a)}=y_{t(a)}-\hat{t}_{t(a)}-\hat{s}_{t(a)}\;(10)$. Training range remainder component estimated values $\hat{r}_{t(a)}$ are the $y_{t(a)}$ values minus estimated trend-cycle component values $\hat{t}_{t(a)}$ minus estimated seasonal component values $\hat{s}_{t(a)}$.

Below, we find example of training range univariate time series classical additive seasonal decomposition by moving averages using airline passengers data [1]. Training range as first ten years and testing range as last two years of data.

References

Data Description: Monthly international airline passenger numbers in thousands from 1949 to 1960.

Original Source: Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1976). “Time Series Analysis, Forecasting and Control”. Third Edition. Holden-Day. Series G.

Source: datasets R Package AirPassengers Object. R Core Team (2021). “R: A language and environment for statistical computing”. R Foundation for Statistical Computing, Vienna, Austria.

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