Additive Model

Description:

• The additive model is expressed as $Y=T+S+R$

Additive model forecasting with Seasonal Variation:

• When random is neglectible, $Y=T_{M{A}_n}+S$
  • where T is the MA of time frame $n$
    • which mean some S will also be missing
• Then seasonal variation of data point $Y_i,S_{Y_i}=Y_i-T_i$
  • ie, de-trended series
  • ex:

Time	y	T(MA_4)	$S$
1	92
2	115
3	104	103.2	0.8
4	131	103.6	27.4
5	74	104.8	-30.8
6	94	105.4	-11.4
…	…	…	…

• Then for every group of data point, $G_{Y_i}=\{Y_{i+an}\}$, there is a group of detrended series, $G_{S_i}=\{S_{i+an}\}$ (except the datapoints that dont have MA)
  • ex: for MA-4, the first group, $G_{Y_1}=\{Y_1,Y_5,Y_9,...\}$ has detrended series $G_{S_1}=\{S_5,S_9,S_1\mathrm{3...}\}$
• Find the average detrended value for each of the group $G_S$, call it ${\bar{G}}_S$
• Minus ${\bar{G}}_S$ with an adjustment value (sum/n) to have the sum of all group’s detrended value to 0
  • The sum is random component if not 0
• Forecast the future MA:
  • Find the long term Trend of MA, $T_{MA}=(M{A}_{max}-M{A}_{min})/M{A}_n$
    • ie. trend of trend = Max MA - Min MA and divided by number of MA
  • Assume the long term trend hold, $M{A}_{i+1}=M{A}_i+T_{MA}$
• Then the next $Y,Y_j^{\ast }=M{A}_j+{\bar{G}}_{S_i}$ where $Y_j^{\ast }\in G_{Y_i}$
  • The next Y is the MA of that data point plus the seasonal variant of that group
  • as same as $Y=T+S$

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