forecasting1_2PerPage - FORECASTING March 26, 2007 Constant...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 )
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 22

forecasting1_2PerPage - FORECASTING March 26, 2007 Constant...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online