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forecasting1_2PerPage

forecasting1_2PerPage - FORECASTING Constant mean model...

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FORECASTING March 26, 2007 Constant mean model: introduction Suppose demand for a product follows the (very) simple model x t = μ + ± t Here I x t = demand for time period t I μ is the constant expected demand I ± 1 , ± 2 , . . . are independent with mean 0 I the best forecast of a future value of x t is μ I we want to estimate μ and update the estimate as each new x t is observed
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Constant mean model: forecasts I Let b x n ( ` ) be the ` -step ahead forecast at time period n I Stated differently, b x n ( ` ) is the forecast at time n of demand at time n + ` I Let b μ n = x 1 + ··· + x n n I Then b x n ( ` ) = b μ n , for all ` Constant mean model: updating b μ n I In this simple model, μ does not change, but our estimate of μ does I 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 )
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Slowly changing mean model: introduction I Now suppose that x t = μ n + ± t where μ n is slowly changing I The forecast is the same as for the constant mean model: b x n ( ` ) = b μ n , for all ` I What changes is the way b μ n is updated I We need for b μ n to track μ n Slowly changing mean: updating I For a constant mean, the update is b μ n + 1 = b μ n + 1 n + 1 ( x n + 1 - b μ n ) I 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 + α x n + 1 for a constant α I α is adjusted depending on how fast μ n is changing I 0 < α < 1 I faster changes in μ necessitate larger α
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Exponential weighting b μ n + 1 = ( 1 - α ) b μ n + α x n + 1 = ( 1 - α ) { b μ n - 1 ( 1 - α ) + α x n } + α x n + 1 = ( 1 - α ) 2 b μ n - 1 + ( 1 - α ) α x n + α x n + 1 = ( 1 - α ) 3 b μ n - 2 + ( 1 - α ) 2 α x n - 1 + ( 1 - α ) α x n + α x 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 I 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 ² ± x n + 1 + ( 1 - α ) x n + ( 1 - α ) 2 x n - 1 + ··· + ( 1 - α ) n x 1 ² 1 + ( 1 - α ) + ··· + ( 1 - α ) n
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Exponential weights: examples 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 n weight = ( 1 -α29 n α =0.4 α =0.2 α =0.4 Forecasting with trends and seasonality: example Time air passengers 1950 1952 1954 1956 1958 1960 100 200 300 400 500 600
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Holt method: forecasting with trend Holt’s forecasting method uses a linear trend b x n ( ` ) = b μ n + b β n ( ` - n ) I b μ n is called the level I 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 ) + α x n + 1 or, equivalently, b μ n + 1 = b μ n + ( 1 - α ) b β n + α ( x n + 1 - b μ n ) Compare with previous updating equation (for the no-trend model): b μ n + 1 = b μ n + α ( x n + 1 - b μ n ) = ( 1 - α ) b μ n + α x n + 1
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