# Double Exponential Smoothing

Double Exponential Smoothing

The Double Exponential Smoothing is a little complicated in computation than the Single Exponential Smoothing but more efficient, as it takes into account the short comings of the Single Exponential Smoothing on a Time Series with Trend.

It possess a trend Smoothing factor (β) where  0 < β < 1, also it has b(t), which represents the vest estimate for the trend and computed using the Formula

b(t) = β(S(t) - S(t-1)) + (1-β) b(t-1)
Where
0< β <1

The introduction of the Trend Smoothing Factor (β) and the best Trend estimate b(t) brings us a slight modification to the Simple Exponential Smoothing Model.

The Simple Exponential Smoothing Model is given as
S(t) = αY(t) +(1- α)S(t-1)
Where 0<α<1

By as we introduce the best estimate for trend and the trend Smoothing factor ( β),

S(t) = αY(t) + (1- α)(S(t-1)-b(t-1))
Where
0<α<1
0< β <1

The S(t)  will then be substituted into b(t), then b(t)  into the Model for the Smoothened series
F(t+m) = S(t) + mb(t)

Where (m>0)
Computations for the initial values b(1), F(1) and S(1)
S(1)=Y(1)
For b(0) we could set
b(1)=(Y(n)-Y(0))/n
for n>0
Or Set

b(1)=Y(1)-Y(0)

Then apply the Model from t>2
Its can also be set with

b(1)=(1/3)[((y(2)-y(1))+(y(3)-y(2))+(y(4)-y(3))]

For F(0)

There is no estimate for F(0). Hence we start computations at F(1) with the Formula
F(1)= S(0) + b(0) from which other values are computed

Brown's Linear Exponential Smoothing (LES) or Brown's Double Exponential Smoothing

S'(0)=Y(0)
S"(0)=Y(0)
S'(t) = αy(t) +(1- α)S(t-1)
S"(t) = αS'(t) +(1- α)S"(t-1)

Then

F(t+m) = α(t) + mb(t)

Where
α(t) =2S'(t)-S"(t)
b(t)=(α/(1- α)) (S'(t)-S"(t))

α(t) is the Smoothing factor and b(t) the best trend estimate

Click on the topics Below for more lectures on the topic

⏩   Brown's Linear Exponential Smoothing (LES) or Brown's Double Exponential Smoothing