forecastingLectures_4xgs - Forecasting time series A time...

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FORECASTING April 24, 2011 Forecasting time series A time series , x 1 , x 2 , x 3 , . . . is a data sequence observed over time, for example, demand for parts sales of a product unemployment rate In this segment of the course we study special methods for forecasting time series. the idea is to develop an algorithm to track the time series and to extrapolate into the future Constant mean model: introduction Suppose demand for a product follows the (very) simple model x t = a + ǫ t Here x t = demand for time period t a is the expected demand – which is constant in this simple model ǫ 1 , ǫ 2 , . . . are independent with mean 0 the best forecast of a future value of x t is a we want to estimate a and update the estimate as each new x t is observed Constant mean model: forecasts Let h x n ( ) be the -step ahead forecast at time period n Stated di f erently, h x n ( ) is the forecast at time n of demand at time n + Let h a n = x 1 + · · · + x n n Then, in this simple model, the best forecasts at time n are h x n ( )= h a n , for all > 0
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Constant mean model: updating h a n In this simple model, a does not change, but our estimate of a does Here is a simple way to update h a n to h a n + 1 h a n + 1 = ( x 1 + · · · + x n )+ x n + 1 n + 1 = n n + 1 h a n + 1 n + 1 x n + 1 = h a n + 1 n + 1 ( x n + 1 - h a n ) Advantages of the updating formula The simple updating formula h a n + 1 = h a n + 1 n + 1 ( x n + 1 - h a n ) has several advantages: reduced storage we only store a n computational speed the mean need not be recomputed each time suggests ways to handle a slowly changing mean coming soon Lake Huron level – example with a slowly changing mean Time LakeHuron 1880 1900 1920 1940 1960 576 577 578 579 580 581 582 * * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Slowly changing mean model: introduction Now suppose that x t = a t + ǫ t where a t is slowly changing The forecast is the same as for the constant mean model: h x n ( )= h a n , for all > 0 What changes is the way h a n is updated We need for h a t to track a t
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Slowly changing mean: updating For a constant mean, the update is h a n + 1 = h a n + 1 n + 1 ( x n + 1 - h a n ) For a slowly changing mean, the update is h a n + 1 = h a n + α ( x n + 1 - h a n ) = ( 1 - α ) h a n + α x n + 1 for a constant α α is adjusted depending on how fast a n is changing 0 < α < 1 faster changes in a necessitate larger α Exponential weighting Start with the updating equation and iterate backwards: h a n + 1 =( 1 - α ) h a n + α x n + 1 1 - α ) { h a n - 1 ( 1 - α )+ α x n } + α x n + 1 1 - α ) 2 h a n - 1 +( 1 - α ) α x n + α x n + 1 1 - α ) 3 h a n - 2 1 - α ) 2 α x n - 1 1 - α ) α x n + α x n + 1 α b x n + 1 1 - α ) x n 1 - α ) 2 x n - 1 +( 1 - α ) 3 x n - 2 + · · · 1 - α ) n x 1 B Exponential weighted moving average For previous page and using the summation formula for geometric series (see next page): h a n + 1 α b ( 1 - α ) 0 x n + 1 1 - α ) 1 x n 1 - α ) 2 x n - 1 + · · · 1 - α ) n x 1 B b ( 1 - α ) 0 x n + 1 1 - α ) 1 x n 1 - α ) 2 x n - 1 + · · · 1 - α ) n x 1 B 1 1 - α · · · 1 - α ) n This results shows that h a n + 1 is an exponentially weighted moving average Large values of α mean faster discounting of the past values.
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This note was uploaded on 03/18/2012 for the course ORIE 3120 taught by Professor Jackson during the Spring '09 term at Cornell University (Engineering School).

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forecastingLectures_4xgs - Forecasting time series A time...

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