Double Exponential Smoothing (Holt's Method) PDF

Double Exponential Smoothing (Holt’s Method) Used for forecasting the time series when the data has a linear trend and no seasonal pattern. Also called Holt’s trend corrected or second-order exponential smoothing. Introduce a term to take care of trend present in the time series. Cap...

Double Exponential Smoothing (Holt’s Method) Used for forecasting the time series when the data has a linear trend and no seasonal pattern. Also called Holt’s trend corrected or second-order exponential smoothing. Introduce a term to take care of trend present in the time series. Capable of capturing increase or decrease in linear trend. 1  Steps For 𝑡 = 0, 𝑠0 = 𝑦0. For 𝑡 > 0, 𝑠𝑡 = 𝛼𝑦𝑡 + (1 − 𝛼)(𝑠𝑡−1 + 𝑏𝑡−1 ) 𝑏𝑡 = 𝛽 𝑠𝑡 − 𝑠𝑡−1 + (1 − 𝛽)𝑏𝑡−1 Where, 𝑏𝑡 is the best estimate of trend at time t. 0 < 𝛽 < 1 is the trend smoothing factor. 2  Triple Exponential Smoothing (Holt Winter’s Method) Used for forecasting the time series when the data has a linear trend and a seasonal pattern also. Also called Holt Winter’s method or third-order exponential smoothing. Introduce two terms to take care of trend and seasonality present in the time series. Capable of capturing increase or decrease in linear trend and seasonal patterns. 3  Involved Notations 𝑠𝑡 : smoothed statistic 𝛼: smoothing parameter of data. 0 < 𝛼 < 1. 𝑏𝑡 : best estimate of trend at time t. 𝛽: trend smoothing factor. 0 < 𝛽 < 1. 𝑐𝑡 : sequence of seasonal correction factor at time t. 𝛾: seasonal change smoothing factor. 0 < 𝛾 < 1. 4  Involved Notations Further, Let L denote the length of the cycle of seasonal change. If we have a monthly data having seasonality of period 12, 𝐿 = 12. Let N denote the number of cycles. If we have monthly data with seasonality of period 12 for 10 years, 𝑁 = 10. Two cases of seasonality: Multiplicative and Additive. 5  Additive Seasonality 𝑌𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝑒𝑡 The seasonal effect is added to the trend, and the seasonal effect is roughly constant over time. E.g., imagine sales of a product over the year. In an additive model, the sales might increase by a fixed number (e.g., 100 units) every December due to holiday shopping, regardless of the general sales level throughout the year. 6  Multiplicative Seasonality 𝑌𝑡 = 𝑇𝑡 × 𝑆𝑡 × 𝑒𝑡 The seasonal effect is multiplied to the trend, resulting in larger seasonal fluctuations when the time series is at a higher level. E.g., imagine sales of a product over the year. In a multiplicative model, the sales in December might double as compared to other months. 7 Steps for Multiplicative Seasonality 𝑠0 = 𝑦0 𝑦𝑡 𝑠𝑡 = 𝛼 + (1 − 𝛼)(𝑠𝑡−1 + 𝑏𝑡−1 ) 𝑐𝑡−𝐿 𝑏𝑡 = 𝛽 𝑠𝑡 − 𝑠𝑡−1 + (1 − 𝛽)𝑏𝑡−1 𝑦𝑡 𝑐𝑡 = 𝛾 + (1 − 𝛾)𝑐𝑡−𝐿 𝑠𝑡 8  Steps for Additive Seasonality 𝑠0 = 𝑦0 𝑠𝑡 = 𝛼 + 𝑦𝑡 − 𝑐𝑡−𝐿 + (1 − 𝛼)(𝑠𝑡−1 + 𝑏𝑡−1 ) 𝑏𝑡 = 𝛽 𝑠𝑡 − 𝑠𝑡−1 + (1 − 𝛽)𝑏𝑡−1 𝑐𝑡 = 𝛾(𝑦𝑡 − 𝑠𝑡−1 − 𝑏𝑡−1 ) + (1 − 𝛾)𝑐𝑡−𝐿 9  Numerical Example Data on monthly air passengers. 10 Holt’s Filtering on the Data 11 Holt’s Filtering on the Data 12 Forecasts from the Holt’s Method 13 Holt Winter’s Filtering on the Data 14 Holt Winter’s Filtering on the Data 15 Forecasts from the Holt Winter’s Method 16