period in the future remains the same: the current estimate of Thus a stationary model 0. dose not account for time effects. To demonstrate the of using a possible “damage” stationary model to produce forecasts for a trended time series assume an m-period moving average was used to forecast a linear trend time series. Let MAt(m) represent the moving average value as of time The average age of information used in calculating ‘t’. the moving average is: Average Age = = (m-1)/2. Then, t-(m-1)/2 [0+1+2+…+(m-1)]/m can be interpreted as the point in time where the moving average is centered, and a forecast for period t+p as of for the trended time series should be period‘t’ MAt(m)+1[(m-1)/2+p]. This demonstrates the possible error if using only MAt(m) to forecast a trended time series. 1.1 Test for the Trend of a Time Series Daniel’s The is designed specifically to identify trend. The test is built on the Daniel’s test calculation of differences between the time and the ranks of the time series values for each point; specifically if there is a trend, than the ranks of the ranked data should form a series similar to the time periods themselves. The following graph demonstrates the situation in a “perfect world”: For the case shown above the difference between and the rank Rt assigned to Yt (that ‘t’ is (t Rt)) is always zero. Since the time series is random, its behavior is not – ‘perfect’ and therefore sometimes the differences are positive, other times negative, and of course zeros too. Observe the following example:
period in the future remains the same: the current estimate of 0. Thus a stationary model dose not account for time effects. To demonstrate the possible “damage” of using a stationary model to produce forecasts for a trended time series assume an m-period moving average was used to forecast a linear trend time series. Let MAt(m) represent the moving average value as of time ‘t’. The average age of information used in calculating the moving average is: Average Age = [0+1+2+…+(m-1)]/m = (m-1)/2. Then, t-(m-1)/2 can be interpreted as the point in time where the moving average is centered, and a forecast for period t+p as of period‘t’ for the trended time series should be MAt(m)+1[(m-1)/2+p]. This demonstrates the possible error if using only MAt(m) to forecast a trended time series. 1.1 Daniel’s Test for the Trend of a Time Series The Daniel’s test is designed specifically to identify trend. The test is built on the calculation of differences between the time and the ranks of the time series values for each point; specifically if there is a trend, than the ranks of the ranked data should form a series similar to the time periods themselves. The following graph demonstrates the situation in a “perfect world”: For the case shown above the difference between ‘t’ and the rank Rt assigned to Yt (that is (t – Rt)) is always zero. Since the time series is random, its behavior is not ‘perfect’ and therefore sometimes the differences are positive, other times negative, and of course zeros too. Observe the following example:
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