Time-series Forecasting - Exponential Smoothing

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1 Time-series Forecasting - Exponential Smoothing Dr. Josif Grabocka ISMLL, University of Hildesheim Business Analytics Business Analytics 1 / 26

2 Motivation Extreme recentivism: The Naive forecast considers only the last value: ŷ T +h T = y T Extreme nostalgia: The Average method equally considers all previous values: ŷ T +h T = 1 T T t=1 y t A middle way is needed: How to capture the influence of previous values while prioritizing recent measurements? Business Analytics 2 / 26

3 Exponential Smoothing An exponentially-weighted average for 0 < α 1: ŷ T +1 T = αy T + α(1 α)y T 1 + α(1 α) 2 y T The weights decrease exponentially for past measurements: Observation α = 0.2 α = 0.4 α = 0.6 α = 0.8 y T y T y T y T y T The recentivism is controlled by α, where T α(1 α) T t 1, α (0, 1] Business Analytics 3 / 26 t=1

4 Example - Oil Production Oil production in Saudi Arabia ( ). Hyndman et al Business Analytics 4 / 26

5 Weighted Average Form Forecast of time T + 1 is a weighted average of observation y T and forecast ŷ T T 1 : ŷ T +1 T = αy T + (1 α)ŷ T T 1 ŷ T T 1 = αy T 1 + (1 α)ŷ T 1 T 2... ŷ 3 2 = αy 2 + (1 α)ŷ 2 1 ŷ 2 1 = αy 1 + (1 α)l 0 Business Analytics 5 / 26

6 Component and Error Correction Forms The only component influencing forecast ŷ t t 1 is the level l t : Forecast Equation Smoothing Equation ŷ t+1 t = l t l t = αy t + (1 α)l t 1 Similarly we can derive the error (e t ) correction form: l t = l t 1 + α(y t l t 1 ) = l t 1 + α(y t ŷ t t 1 ) = l t 1 + αe t Business Analytics 6 / 26

7 Holt s Linear Trend Method Tackle trend in ŷ t+1 h by decomposing out a slope/trend term b t : Forecast Equation ŷ t+h t = l t + hb t Level Equation l t = αy t + (1 α)(l t 1 + b t 1 ) Trend Equation b t = β (l t l t 1 ) + (1 β )b t 1 0 < α 1, 0 < β 1 l t as a waited average of current observation y t and estimation ŷ t t 1 = l t 1 + b t 1 b t as a weighted average of prediction differences l t l t 1 and previous trend b t 1 Business Analytics 7 / 26

8 Holt s Linear Trend Method - Error Correction Rewrite by taking l t and b t on the left side: l t = l t 1 + b t 1 + αe t b t = b t 1 + αβ e t Where the isolated error is: e t = y t (l t 1 + b t 1 ) = y t ŷ t t 1 Set l 0 = y 1 and b 0 = y 2 y 1 when making predictions Business Analytics 8 / 26

9 Exercise Given a series y 1 = 2, y 2 = 3, y 3 = 3, y 4 = 2, predict ŷ 5 4 =? using Holt s Linear Trend with α = 0.5, β = 0.5 In fact, it was later observed that y 5 = 2. l 0 = 2, b 0 = 5 l 1 = 0.5( ) = 0.5, b 1 = 0.5( ) = 3.75 l 2 = 0.5( ) = 3.625, b 2 = 0.5( ) = l 3 = 0.5( ) = , b 3 = 0.5( ) = l 4 = 0.5( ) = , b 4 = 0.5( ) = ŷ 5 4 = = Business Analytics 9 / 26

10 Example - Livestock, sheep in Asia Comparison of different forecasting approaches using non-seasonal exponential smoothing. Hyndman et al Business Analytics 10 / 26

11 Exponential Trend Method Replace an additive model (constant slope) with a multiplicative one (constant growth): Forecast Equation ŷ t+h t = l t bt h Level Equation l t = αy t + (1 α)(l t 1 b t 1 ) Trend Equation b t = β l t l t 1 + (1 β )b t 1 b t represents a multiplicative growth rate in relative terms l t l t 1 Forecasts project a constant growth rate rather than a constant slope (as in the additive model) Business Analytics 11 / 26

12 Exponential Trend Method - Error Correction Rewrite by taking l t and b t on the left side: l t = l t 1 b t 1 + αe t b t = b t 1 + αβ e t l t 1 Where the isolated error is: e t = y t (l t 1 b t 1 ) = y t ŷ t t 1 Set l 0 = y 1 and b 0 = y 2 /y 1 when making predictions Business Analytics 12 / 26

13 Example - Air Passengers (Hyndman et al. 2014) Notice the over-forecast of the linear and exponential trend methods. The damped trend approach addresses the issue... Business Analytics 13 / 26

14 Damped Additive Trend Model Include a damping/prohibitory parameter 0 < φ < 1: Forecast Equation ŷ t+h t = l t + (φ + φ φ h )b t Level Equation l t = αy t + (1 α)(l t 1 + φb t 1 ) Trend Equation b t = β (l t l t 1 ) + (1 β )φb t 1 φ = 1 leads Holt s additive linear trend model Error correction form: l t = l t 1 + φb t 1 + αe t b t = φb t 1 + αβ e t Business Analytics 14 / 26

15 Example - Livestock, sheep in Asia Level l t and trend b t computed from a Livestock, sheep time series. Hyndman et al Business Analytics 15 / 26

16 Damped Multiplicative Trend Model Include a damping/prohibitory parameter 0 < φ < 1 to the exponential trend: Forecast Equation ŷ t+h t = l t b (φ+φ2 + +φ h ) t Level Equation l t = αy t + (1 α)(l t 1 b φ t 1 ) Trend Equation b t = β l t l t 1 + (1 β )b φ t 1 φ = 1 leads to exponential trend model Error correction form: l t = l t 1 b φ t 1 + αe t b t = b φ t 1 + αβ e t l t 1 Business Analytics 16 / 26

17 Example - Livestock, sheep in Asia Hyndman et al Business Analytics 17 / 26

18 Exercise (2) Given a series y 1 = 2, y 2 = 3, y 3 = 3, y 4 = 2, predict ŷ 5 4 =? using: Damped Additive Trend Model with α = 0.5, β = 0.5, φ = 0.5 In fact, it was later observed that y 5 = 2. Solve it at home! Business Analytics 18 / 26

19 Setting parameters How to set α, β, b 0, l 0, φ? Search for combinations that minimize the Sum of Squared Errors: T T SSE = et 2 = (y t ŷ t t 1 ) 2 t=1 t=1 Business Analytics 19 / 26

20 Holt-Winters Additive Seasonal Method Forecast ŷ t+h t = l t + hb t + s t m+h + m Level l t = α(y t s t m ) + (1 α)(l t 1 + b t 1 ) Trend Seasonality b t = β (l t l t 1 ) + (1 β )b t 1 s t = γ(y t l t 1 b t 1 ) + (1 γ)s t m h + m = (h 1) mod m + 1 ensures estimates come from final year l t is the weighted average between seasonally-adjusted observation and non-seasonal forecast b t same as Holt s linear model s t is the weighted average between current seasonal value and the last season s (m periods ago) Business Analytics 20 / 26

21 Holt-Winters Additive Seasonal Method (2) With error correction forms: l t = l t 1 + b t 1 + αe t b t = b t 1 + αβ e t s t = s t m + γe t and residual errors: e t = y t (l t 1 + b t + s t m ) = y t ŷ t t 1 Business Analytics 21 / 26

22 Holt-Winters Multiplicative Seasonal Method Forecast Level Trend Seasonality ŷ t+h t = (l t + hb t )s t m+h + m l t = α y t + (1 α)(l t 1 + b t 1 ) s t m b t = β (l t l t 1 ) + (1 β )b t 1 y t s t = γ + (1 γ)s t m l t 1 + b t 1 Business Analytics 22 / 26

23 Holt-Winters Multiplicative Seasonal Method (2) With an error correction form: l t = l t 1 + b t 1 + α e t s t m b t = b t 1 + αβ e t s t m e t s t = s t m + γ l t 1 + b t 1 With residual errors: e t = y t (l t 1 + b t )s t m Business Analytics 23 / 26

24 Exercise (3) Given a series y 1 = 2, y 2 = 3, y 3 = 3, y 4 = 2, predict ŷ 5 4 =? using: Holt-Winters Additive Seasonal Model with α = 0.5, β = 0.5, γ = 0.5, m = 2 In fact, it was later observed that y 5 = 2. t l b s ŷ Business Analytics 24 / 26

25 Example - International Visitors in Australia Hyndman et al Business Analytics 25 / 26

26 Example - International Visitors in Australia (2) Decomposing l t, b t and s t. Hyndman et al Business Analytics 26 / 26