Exponential Smoothing: Overview, Questions, Preparation

For statistics, exponential smoothing plays an important role. It is a comprehensive principle for smoothing the time series using the exponential window functions. It is used as one of the window functions applied in signal processing to smoothen the data.

Exponential Smoothing
Exponential smoothing is a technique applied to smoothing the time series by using the exponential window functions. 

The formula for exponential smoothing is

s_t = αx_t + (1-α)* s_t-1 , when t > 0

where α = smoothing factor and 0         s_t = smoothed statistic
    s_t-1 = previous smoothed statistic
     t = time period

If the smoothing factor increases, the level of smoothing will decrease. There is no official number for α. The value of α nearer to 1 would provide a lesser level of smoothing and are less responsive to change.   

Methods to Estimate Exponential Smoothing

There are three different methods to determine exponential smoothing.

Simple exponential smoothing- When the data displays no periodic trend and no seasonal pattern, then the simple exponential smoothing is applied.

    s_t = αx_t + (1-α)* s_t-1

_{Double exponential smoothing- When the data displays a linear trend but shows no seasonal pattern, then the double exponential smoothing is applied. This is also called Holt's trend corrected.}

s_t = αx_t + (1-α)* (s_t-1 + b_t-1)
          β_t = β(s_t + s_t-1) + (1 - β)b_t-1

where bt = best estimate of the trend at time ‘ t.’
     β  = trend smoothing factor where 0

Triple exponential smoothing- When the data displays both the linear trend and seasonal patterns, then triple exponential smoothing is used. This is also known as Holt-Winters.
        s_t = α*(x_t/c_t-L) + (1- α)*(s_t-1 + b_t-1)
         bt    = β(s_t + s_t-1 ) + (1 - β)b_t-1
           c_t = γ* (x_t/s_t ) + (1- γ)* c_t-L 

where  c_t = sequence of seasonal error-free factor at time t
       γ = seasonal variation smoothing factor where 0

In the chapter ‘Statistics,’ detail regarding dispersion, dispersion measures, range, mean deviation, variance, standard deviation, analysis of frequency distributions, and exponential smoothing are explained.

1. The mean and SD of 100 observations were calculated as 40  and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and SD?

Solution.
     n = 100, x= 40, 

x = 1/n _i=1∑ⁿ x_i
    = nx = 100 x 40 = 4000

α_x² = 1/n _i∑x_i² - (x)²

       = 162601

Corrected ∑x_i = 4000 - 50 + 40 = 3990
Corrected ∑x_i² = 162601 - (50)2 + (40)2 = 161701

Corrected mean = 3990/100 = 39.9
Corrected SD = (161701/100) - (39.9)² = 5

2. The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14. Find the other two observations?

Solution.
= Mean = 2+4+10+12+14+x+y/7
= 8 = 56 + x + y/7
= x + y = 14 — (1)

Variance = 1/n _i=1∑⁷(x_i - x)²
= 16 = 1/7 [ (-6)² + (-4)² + (2)² + (4)² + (6)² + x² + y² - 2 * 8(x + y) + 2 * (8)²]
= x² + y² = 100 — (2)

From (1)
= x² + y² + 2xy = 196 — (3)

From (2) and (3),

= 2xy = 196 - 100
= 2xy = 96 — (4)

Subtracting (4) and (2), 

= x² + y² - 2xy = 100 - 96
= x - y = 2

x = 8 and y = 6  when x - y = 2
x = 6 and y = 8  when x - y = -2

3. Find the mean and variance of 6, 7, 10, 12, 13, 4, 8, 12

Solution.

Mean = (6+7+10+12+13+4+8+12)/8
Mean = 9

Variance = 1/n _i=1∑⁸(x_i - x)²
Variance = 1/8 * 74 
Variance = 9.25

Q: What is exponential smoothing?

Q: What is the formula of exponential smoothing?

Q: What is simple exponential smoothing?

Q: What is Double exponential smoothing?

Q: What is Triple exponential smoothing?