Untitled Quiz

Double Exponential Smoothing (Holt's Method)

• Used to forecast time series with a linear trend and no seasonality.
• Also known as Holt's trend-corrected or second-order exponential smoothing.
• Introduces a term to account for the trend in the time series.
• Capable of capturing increases or decreases in linear trends.

Steps for Double Exponential Smoothing

• For t = 0, S₀ = Y₀.
• For t > 0, S_t = aY_t + (1 - a)(S_t-1 + b_t-1)
• b_t = β(S_t - S_t-1) + (1 - β)b_t-1
• b_t is the best estimate of the trend at time t.
• 0 < a < 1 is the smoothing parameter for data.
• 0 < β < 1 is the trend smoothing factor.

Triple Exponential Smoothing (Holt-Winters Method)

• Used for forecasting time series with both linear trends and seasonal patterns.
• Also called Holt-Winters method or third-order exponential smoothing.
• Introduces two terms to account for both trend and seasonality in the time series.
• Can capture increases or decreases in linear trends and seasonal patterns.

Involved Notations

• S_t: smoothed statistic
• a: smoothing parameter for data, 0 < a < 1.
• b_t: best estimate of the trend at time t.
• β: trend smoothing factor, 0 < β < 1.
• c_t: sequence of seasonal correction factors at time t.
• γ: seasonal change smoothing factor, 0 < γ < 1.
• L: length of the seasonal cycle (e.g., 12 for monthly data)
• N: number of seasonal cycles (e.g., 10 for 10 year monthly data)

Seasonality Types

• Additive Seasonality: The seasonal effect is added to the trend, and the seasonal effect is roughly constant over time.
  • Example: Sales of a product increase by a fixed amount every December due to holiday shopping.
• Multiplicative Seasonality: The seasonal effect is multiplied to the trend, resulting in larger seasonal fluctuations when the time series is at a higher level.
  • Example: Sales in December might double compared to other months.

Steps for Multiplicative Seasonality

• S₀ = Y₀
• S_t = a(Y_t/C_t-L) + (1-a)(S_t-1 + b_t-1)
• b_t= β(S_t - S_t-1) + (1 - β)b_t-1
• C_t = γ(Y_t/S_t) + (1 - γ)C_t-L

Steps for Additive Seasonality

• S₀ = Y₀
• S_t = aY_t + (1-a)(S_t-1 + b_t-1)
• b_t = β(S_t - S_t-1) + (1 - β)b_t-1
• C_t = γ(Y_t-S_t-b_t) + (1 - γ)C_t-L

Numerical Example

• Data on monthly air passengers used as an example.

Holt-Winters Filtering Summary

• Use for exponential smoothing with trend & without seasonal component.
  • Example: HoltWinters(x = AirPassengers, gamma =F)
  • Smoothing parameters in example: alpha=1; beta= 0.0032185; gamma=FALSE
• Use for exponential smoothing with trend & additive seasonal component.
  • Example: HoltWinters(x = log(AirPassengers))
  • Smoothing parameters in example: alpha=0.3266015; beta=0.005744138; gamma=0.8206654
• Used to filter and forecast data from a time series. (plots provided)