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# lecture1 - 1 Introduction to Forecasting A time series is...

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1 Introduction to Forecasting A time series is sequence of random variables Y t , t = 1 , 2 , 3 . . . indexed by time t . The index t will typically refer to some standard unit of time , e.g., seconds, hours, days, weeks, months, years. The historical data is a time-ordered sample of observations Y 1 , Y 2 , . . ., Y n obtained from the time series. Examples. Fig. 1.1-1.4. Figure 1.1: Time JJ 1960 1965 1970 1975 1980 0 5 10 15 In this course we will be interested in constructing models to forecast (or to predict) future values of Y t , i.e. values of Y t for t beyond the end of the data set. Typically we will use the historical data (or some appropriate subset of it) to build our forecasting models. 1

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Figure 1.2: A time series giving the monthly totals of accidental deaths in the USA. The values for the Frst six months of 1979 are 7798, 7406, 8363, 8460, 9217, 9316. Time USAccDeaths 1973 1974 1975 1976 1977 1978 1979 7000 8000 9000 10000 11000 How do we compare two forecasting models to decide which one is better? We will look at various ways of choosing between models as the course progresses, but the most obvious answer is to see which one better at predicting. Suppose we have used the historical data Y 1 , . . ., Y n to build two forecasting models ˆ Y t and ¯ Y t and we now obtain future observation Y n +1 , . . ., Y n + k . Note that the di±erence Y t ˆ Y t is the forecast (or prediction) error at time t for Model 1. ²or each model compute the Sum of Squares of Forecast Errors (SOS): 1. SOS 1 = n + k t = n +1 ( Y t ˆ Y t ) 2 2. SOS 2 = n + k t = n +1 ( Y t ¯ Y t ) 2 2
If SOS 1 is substantially smaller than SOS 2 then we would choose the Frst model. One obvious drawback to the above method is that it requires you to wait for future observations in order to compare models. A way around this to take the historical data Y 1 . . ., Y n and split it into a training set Y 1 , . . ., Y k and a testing set Y k +1 , . . ., Y n , where ( n k ) k , i.e.most of the data goes into the training set. ±orecasting models are then built using only the training set. The testing set is used as a set of future observations. The SOS is

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lecture1 - 1 Introduction to Forecasting A time series is...

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