forecasting1_article - 1 Constant mean model Suppose demand...

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1 Constant mean model Constant mean model: introduction Suppose demand for a product follows the (very) simple model x t = μ + ± t Here x t = demand for time period t μ is the constant expected demand ± 1 , ± 2 , . . . are independent with mean 0 the best forecast of a future value of x t is μ we want to estimate μ and update the estimate as each new x t is observed Constant mean model: forecasts Let b x n ( ` ) be the ` -step ahead forecast at time period n Stated differently, b x n ( ` ) is the forecast at time n of demand at time n + ` Let b μ n = x 1 + · · · + x n n Then b x n ( ` ) = b μ n , for all ` Constant mean model: updating b μ n In this simple model, μ does not change, but our estimate of μ does
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Here is a simple way to update b μ n to b μ n +1 b μ n +1 = ( x 1 + · · · + x n ) + x n +1 n + 1 = n n + 1 b μ n + 1 n + 1 x n +1 = b μ n + 1 n + 1 ( x n +1 - b μ n ) 2 Simple exponential smoothing Slowly changing mean model: introduction Now suppose that x t = μ n + ± t where μ n is slowly changing The forecast is the same as for the constant mean model: b x n ( ` ) = b μ n , for all ` What changes is the way b μ n is updated We need for b μ n to track μ n Slowly changing mean: updating For a constant mean, the update is b μ n +1 = b μ n + 1 n + 1 ( x n +1 - b μ n ) For a slowly changing mean, the update is b μ n +1 = b μ n + α ( x n +1 - b μ n ) = (1 - α ) b μ n + αx n +1 for a constant α α is adjusted depending on how fast μ n is changing 2
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0 < α < 1 faster changes in μ necessitate larger α Exponential weighting b μ n +1 = (1 - α ) b μ n + αx n +1 = (1 - α ) { b μ n - 1 (1 - α ) + αx n } + αx n +1 = (1 - α ) 2 b μ n - 1 + (1 - α ) αx n + αx n +1 = (1 - α ) 3 b μ n - 2 + (1 - α ) 2 αx n - 1 + (1 - α ) αx n + αx n +1 α ± x n +1 + (1 - α ) x n + (1 - α ) 2 x n - 1 +(1 - α ) 3 b μ n - 2 + · · · + (1 - α ) n x 1 ² Exponential weighted moving average For previous page: b μ n +1 α x n +1 + (1 - α ) x n + (1 - α ) 2 x n - 1 + ··· + (1 - α ) n x 1 x n +1 + (1 - α ) x n + (1 - α ) 2 x n - 1 + ··· + (1 - α ) n x 1 1 + (1 - α ) + ··· + (1 - α ) n Exponential weights: examples 3
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0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 n weight = ( 1 29 n α =0.4 α =0.2 α =0.4 3 Holt-Winters Forecasting with trends and seasonality: example Time air passengers 1950 1952 1954 1956 1958 1960 100 200 300 400 500 600 4
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3.1 Holt’s nonseasonal model Holt method: forecasting with trend Holt’s forecasting method uses a linear trend b x n ( ` ) = b μ n + b β n ( ` - n ) b μ n is called the level b β n is called the slope Both b μ n and b β n must be updated Holt method: Updating the level In the Holt model, the level b μ n is updated by the equation: b μ n +1 = (1 - α )( b μ n + b β n ) + αx n +1 or, equivalently, b μ n +1 = b μ n + (1 - α ) b β n + α ( x n +1 - b μ n )
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This note was uploaded on 01/09/2009 for the course ORIE 312 taught by Professor D.ruppert,p.jacks during the Spring '08 term at Cornell University (Engineering School).

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forecasting1_article - 1 Constant mean model Suppose demand...

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