Scenario
We wish to create a triple exponential smoothing (TES) forecast 2 periods into the future with the following quarterly sales values ( M = seasons = 4):
Quarter Value
1 40
2 46
3 65
4 130
5 50
6 60
7 70
8 140Approach
Equations
We will do this manually in order to illustrate the technique.
Level:
Lₜ=α(Yₜ/(Sₜ -ₘ)+(1-α)(Lₜ-₁+Tₜ-₁)
Trend:
Tₜ=Β(Lₜ-Lₜ-₁)+(1-Β)Tₜ-₁
Seasonality
Sₜ=γ(Yₜ/Lₜ)+(1-γ)Sₜ -ₘ
Forecast
Fₜ = Lₜ + Tₜ
Start Point
S₁:0.57
S₂:0.65
S₃:0.93
S₄:1.85
Month Value Level Trend Seasons 1 40 40 / ((40+46+65+130)/4)=0.57
2 46 46 / ((40+46+65+130)/4)=0.65
3 65 65 / ((40+46+65+130)/4)=0.93
4 130 130 / ((40+46+65+130)/4)=1.85
M = 4
α: 0.1
Β: 0.2
γ: 0.3
Month Value Level Trend Seasons
1 40 0.57
2 46 0.65
3 65 0.93
4 130 1.85 5 50 50/0.57=87.72 87.72-130/1.85=17.45
6 60
7 70
8 140
L₅ = 87.72
T₅ = 17.45
S₅ = 0.3*(50/87.72)+0.7*0.57=0.57
L₆ = 0.1(60/0.65)+0.9(87.72+17.45)=103.88
T₆ = 0.2(103.88–87.72)+0.8*17.45=17.19
S₆ = 0.3*60/103.88 + 0.7*0.65=0.63
L₇ = 0.1(70/0.93)+0.9(103.88+17.19)=116.49
T₇ = 0.2(116.49–103.88)+0.8*17.19=16.27
S₇ = 0.3*70/116.49 + 0.7*0.93=0.83
L₈ = 0.1(140/1.85)+0.9(116.49+16.27)=127.05
T₈ = 0.2(127.05–116.49)+0.8*16.27=15.13
S₈ = 0.3*140/127.05 + 0.7*1.85=1.63
Forecast
Fₜ₊ₖ =( L₈ + k*T₈)*S₈-ₘ₊ₖ
F₉ =(127.05+1*15.13)*0.57 = 81.04
F₁₀ =(127.05+2*15.13)*0.63 = 99.11
