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The “past history" data can be smoothed in many ways. We consider two straightforward averaging methods, namely the mean and simple moving averages. In both cases the objective is to make use of past data to develop a forecasting system for future periods.
The mean
The method of simple averages is simply to take the average of all observed data as the forecast. So,
Ft+1 = 1/t ∑_(i=1)^t▒Yi
When a new observation, Yt+1, becomes available, the forecast for time t+2 is the new mean including the previously observed data plus this new observation:
Ft+2 = 1/(t+1) ∑_(i=1)^(t+1)▒Yi
Moving averages
One way to modify the influence of past data on the mean-as-a-forecast is to specify at the outset just how many past observations will be included in a mean. The term “moving average" is used to describe this procedure because as each new observation becomes
Available, a new average can be computed by dropping the oldest observation and including the newest one. This moving average will then be the forecast for the next period. Note that the number of data points in each average remains constant and includes the most recent observations. A moving average forecast of order k, or MA(k), is given by
Ft+1 = 1/k ∑_(i=t-k+1)^t▒Yi
Compared with the simple mean (of all past data) the moving average of order k has the following characteristics:
It deals only with the latest k periods of known data,
The number of data points in each average does not change astime goes on.
But it also has the following disadvantages:
It requires more storage because all of the k latest observationsmust be stored, not just the average,
It cannot handle trend or seasonality very well, although it cando better than the total mean.
MA(1) : That is, a moving average of order 1-the last known data point (Yt) is taken as the forecast for the next period (Ft+1 = Yt). An example of this is “the forecast of tomorrow's closing price of IBM stock is today's closing price." This was called the naive forecast (NF1)
MA(n): In this case, the mean of all observations is used as a forecast.So this is equivalent to the mean forecast method.
Note that use of a small value for k will allow the moving averageto follow the pattern, but these MA forecasts will nevertheless trailthe pattern, lagging behind by one or more periods. In general, thelarger the order of the moving average-that is, the number of data
points used for each average, the greater the smoothing effect.Algebraically, the moving average can be written as follows:
Ft+1 = (Yt + Yt-1 + … + Yt-k+1)/k ,
Ft+2= (Yt+1 + Yt + … + Yt-k+2)/k
Comparing Ft+1 and Ft+2, it can be seen that Ft+2 requires droppingthe value Yt-k+1 and adding the value Yt+1 as it becomes available,so that another way to write Ft+2 is
Ft+2= Ft+1+ 1/k(Yt+1 - Yt-k+1)
It can be seen from above equation that each new forecast (Ft+2) is simply an adjustment of the immediately preceding forecast (Ft+1). This adjustment is (1/k)th of the difference between Yt+1 and Yt-k+1.
Clearly if k is a big number, this adjustment is small, so that moving averages of high order provide forecasts that do not change very much. In summary, an MA(k) forecasting system will require k data points to be stored at any one time. If k is small (say 4), then the
storage requirements are not severe although for many thousands of time series (say for inventories involving thousands of stock keeping units) this can be a problem. In practice, however, the technique of moving averages as a forecasting procedure is not used often because the methods of exponential smoothing are generally superior.
Exponential smoothing methods
The averaging method gives equal weight to observations. There is another method which applies an unequal weight to the past data and weight typically decays in an exponential manner from the most recent to the most recent to the most distant data point, the methods are known as exponential smoothingmethods. This method does not smooth data in sense of estimating a trend-cycle; they are taking a weighted average of past observations using weights that decay smoothly.
Single exponential smoothing
We wish to forecast the next value of our time series Ytwhich is yet to be observed. Forecast is denoted by Ft. Whenthe observation Ytbecomes available, the forecast error is found tobe Yt- Ft. The method of single exponential forecasting takes theforecast for the previous period and adjusts it using the forecast error.That is, the forecast for the next period is
Ft+1 = Ft + α(Yt -Ft) , where αis a constant between 0 and 1.
It can be seen that the new forecast is simply the old forecast plus an adjustment for the error that occurred in the last forecast. When α has a value close to 1, the new forecast will include a substantial adjustment for the error in the previous forecast. Conversely, when
α is close to 0, the new forecast will include very little adjustment. Thus, the effect of a large or small α is completely analogous (in an opposite direction) to the effect of including a small or a large number of observations when computing a moving average.
Graph shows that exponential smoothing is equivalent to using the last observation as a forecast. That is, it is the same as NF1, the naïve forecast method
Holt's linear method
Holt (1957) extended single exponential smoothing to linear exponential smoothing to allow forecasting of data with trends. The forecast for Holt's linear exponential smoothing is found using two smoothing constants, α and β (with values between 0 and 1), and three equations:
Lt = αYt + (1- α)(Lt-1 + bt-1) (i)
bt = β(Lt – Lt-1) + (1 – β)bt-1(ii)
Ft+m = Lt + btm (iii)
Here Ltdenotes an estimate of the level of the series at time t and btdenotes an estimate of the slope of the series at time t. Equation (i) adjusts Ltdirectly for the trend of the previous period, bt-1, by adding it to the last smoothed value, Lt-1. This helps to eliminate the lag and brings Ltto the approximate level of the current data value. Equation (ii) then updates the trend, which is expressed as the difference between the last two smoothed values. This is appropriate because if there is a trend in the data, new values should be higher or lower than the previous ones. Since there may be some randomness remaining, the trend is modified by smoothing with βthe trend in the last period (Lt-Lt-1), and adding that to the previous estimate of the trend multiplied by (1-β). Thus, (ii) is similar to the basic form of single smoothing but applies to the updating of the trend. Finally, equation (iii) is used to forecast ahead. The trend, bt, is multiplied by the number of periods ahead to be forecast, m, and added to the base value, Lt.
Holt-Winters' trend and seasonality method
The set of moving average and exponential smoothing methods can deal with almost any type of data as long as such data are non-seasonal. When seasonality does exist, however, these methods are not appropriate on their own.
Holt's method was extended by Winters (1960) to capture seasonality directly. The Holt Winters' method is based on three smoothing equations-one for the level, one for trend, and one for seasonality. It is similar to Holt's method, with one additional equation to deal with seasonality. In fact there are two different Holt-Winters' methods, depending on whether seasonality is modelled in an additive or multiplicative way.
Multiplicative seasonality
The basic equations for Holt-Winters' multiplicative method are as follows:
Level: Lt = αYt/(St-s) + (1- α)(Lt-1 + bt-1) (i)
Trend: bt = β(Lt – Lt-1) + (1 – β)bt-1(ii)
Seasonal: St = γYt/Lt + (1- γ)St-s (iii)
Forecast: Ft+m = (Lt + btm)St-s+m (iv)
where s is the length of seasonality (e.g., number of months or quarters in a year), Lt represents the level of the series, btdenotes the trend, St is the seasonal component, and Ft+mis the forecast for m periods ahead.
Equation (iii) is comparable to a seasonal index that is found as a ratio of the current values of the series, Yt, divided by the current single smoothed value for the series, Lt. If Ytis larger than Lt, the ratio will be greater than 1, while if it is smaller than Lt, the ratio will be less than 1. Important to understanding this method is realizing that Ltis a smoothed (average) value of the series that does not include seasonality (this is the equivalent of saying that the data have been seasonally adjusted). The data values Yt, on the other hand, do contain seasonality. Yt includes randomness. In order to smooth this randomness, equation
(iii) weights the newly computed seasonal factor with γ and the most recent seasonal number corresponding to the same season with (1 - γ). (This prior seasonal factor was computed in period t - s, since s is the length of seasonality.) Equation (ii) is exactly the same as Holt's equation for smoothing the trend. Equation (i) differs slightly from Holt's equation in that the first term is divided by the seasonal number St-S. This is done to deseasonalize (eliminate seasonal fluctuations from) Yt. This adjustment can be illustrated by considering the case when St-s is greater than 1, which occurs when the value in period t - s is greater than average in its seasonality. Dividing Ytby this number greater than 1 gives a value that is smaller than the original value by a percentage just equal to the amount that the seasonality of period t -s was higher than average. The opposite adjustment occurs when the seasonality number is less than 1. The value St-Sis used in these calculations because St cannot be calculated until Lt is known from (i).