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This forecasting technique in its basic form (simple exponential smoothing) should only be used for non seasonal time series showing no systematic trend, both of these chosen series visibly exhibit at least one of these phenomenon, but it is not fair to say it is of little use as these effects can be measured and removed to produce a stationary series. Given a non seasonal time series with no systematic trend (which is what I hope to reduce these series to), it is considered natural to take as an estimate of Xn+1 a weighted sum of the past observations, X(T,1) = c0XT + c1XT-1+ …. It also seems sensible to give more weight to recent observations and less weight to observations further back in the past. A set of weights which fit this description are geometric weights, which decrease by a constant ratio. In order that the weights sum to 1 we take ci = α ( 1 – α) i where α is a constant such that 0 < α < 1 so that the above equation becomes X(T,1) = α XT + α ( 1 - α) XT-1+….+ = α XT + ( 1 – α)( α XT-1 + α ( 1 – α)XT-2 + ...+ = α XT + ( 1 - α)X( T - 1,1) If we set X(1,1) = X1 we can recursively compute forecasts, from the most recent observation and the most recent prediction, this drastically reduces the amount of arithmetic involved in a forecast and the equation can easily be updated to make future forecasts. Several things are worth noting with regard to simple exponential smoothing. Firstly the value of X(T,1) represents an average of the series to date. Second the function X(T,t) is a linear combination of all past observations, it can be shown Xt = mu + α Σi(Z j ) + Z t j < t This infinite MA process is non stationary but the first differences X t+1 – X t form a first order MA process so that X t is an ARIMA(0,1,1). (Note this is assuming 0 < α < 1, since we do not generally make this restriction for ARIMA(0,1,1) it is not always true that ARIMA(0,1,1) model is an exponential smoothing). The speed at which remote values are dampened out depends on the value of α . Below is a graph showing the decreasing values of the weights obtained by running SES on the seasonal differenced production data and the sales data, who have a values of 0.628797 0.256208 respectively. Finally from the fact that the larger the value of α the faster past responses are dampened out from the smoothing we may derive a rule of thumb for determining the value of α. When the magnitude of the variation in the series is quite large we would like to average out the random effects quickly. This being the case we would select a small value of α so that the smoothed value of X(T,1) will reflect X(T-1,1) to a greater extent than the last observation. In practice the value of a falls in the interval 0.6 and is determined by means of simulation with the purpose of minimizing the variance. Example: The Dairy Production series is clearly non stationary, when an optimal simple exponential smoothing is run on it an α value of 1.5734 is obtained, this is clearly outside the range of exponential smoothing and the graph below shows the weight on past observations is oscillating to zero rather than decreasing exponentially to zero. Taking 1st differences and then 12th differences from the Milk Sales series and then running SES an α value of 0.0000103 was obtained this tells us that the series is very random and the method will predict close to the mean of the transformed series as can be seen in the graph. Holt-Winters Exponential smoothing may easily be generalised to deal with time series with trend and seasonal variation. The resulting procedure is Holt-Winters exponential smoothing or HWES for short. In order to perform HWES is implemented firstly by introducing trend and seasonal parameters which are also updated by exponential smoothing. Let L t, Tt, It denote the local level, trend, and seasonal index at time t. Let α, γ, δ, denote the three smoothing parameters for updating the level, trend and seasonal index respectively, these should all lie in the interval (0,1). If the seasonal variation is multiplicative then the updating equations are Lt = α (Xt/It-12)+(1- α)(Lt-1 + Tt-1) Tt = γ (Lt-Lt-1) +(1-γ)Tt-1 It= δ (Xt/Lt) +(1-δ)It-12 while for additive seasonal variation we have Lt= α (Xt-It-12)+(1- α)(Lt-1+ Tt-1) Tt= γ (Lt-Lt-1) +(1-γ)Tt-1 It = δ (Xt-Lt) +(1-δ)It-12 And the respective forecasts from time t are then X(T,k) = (Lt+ kTt)It-12+k X(T,k) = Lt+ kTt+ It-12+k The Holt Winters additive may also be described in terms of ARIMA(0,1,p+1)(0,1,0)p. As the seasonality in the Production series is of order 12 and MINITAB only computes upto MA5 it is not possible to compute an ARIMA(0,1,12+1)(0,1,0)12. No such ARIMA series exists for multiplicive Holt Winters. |
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