Performing Triple Exponential Smoothing

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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       140

Approach

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