Multiple Moving Averages

Xt	: Xt  means the observed data at the current period t
Ft	: Ft means the fitted values (using a forecasting model) at current period t
Data set covering the last n time periods	: X1 X2 X3  ...... Xn - 1 Xn
Past Data for n time periods	: Xt - n + 1 ...... Xt - 2 Xt - 1 Xt
Future Forecasts required for m periods	: Ft + 1 Ft + 2 ..... Ft + m
Fitted Values using a forecasting model	: Ft - n + 1 ......  Ft - 2 Ft - 1 Ft
Fitting Errors	: (Xt - n + 1- - Ft - n + 1) ...... (Xt - 1 - Ft - 1) (Xt - Ft)
Forecasting Errors (when Xt + 1 .... Xt + m   become available)	: (Xt + 1 - Ft + 1) , (Xt + 2 - Ft + 2) ..... (Xt + m - Ft + m)

Once a forecasting model is selected, we fit the model to the known data (by your choice of parameters and initializing procedures) and obtain the fitted values. We than calculate the fitted errors, and as new new observations become available we examine the forecasting errors. Moving averages are mostly recursive in nature - moving through the past data period by period, as opposed to using all the past data in one fitting such as in least squares fitting. Figure 2.1 shows a 6 stages procedure that is used to appraise any forecasting methods.

 

Figure 2.1 Procedure for appraising any forecasting method

The Single Moving Average of the latest n observations has a few drawbacks in that it gives equal weightage to the observation data; and also it can not handle trend and seasonality very well, although it can forecast better than the Mean. Double Moving Average (or moving averages of moving averages) ends up with unequally weighted averages, and can be used within a method called linear moving averages. The method using Double Moving Averages applies an unequal weights to the past data, and because the weights typically decay in an exponential manner from the most recent to the most distant data point, the method is often called Exponential Smoothing methods. All Smoothing methods require that certain parameters be defined (their values lie between 0 and 1). I will explain on Exponential Smoothing methods in the next page.

Pulse	With Random Error	No Trend Effect	No Seasonal Effect	Additive Seasonal	Multiplicative Seasonal
Ramp		Additive Trend
Step		Multiplicative Trend
Figure 2.2  Some Common Test Patterns For Testing Forecasting Procedures

 

Single Moving Averages
The term moving average means using a method of computation when each new observation data point becomes available, a new average is then computed by dropping the oldest observation point and including the new one. This moving average will then be the forecast for the next period. To illustrate this, let's say there are N data points and a decision to use t observations for each average (I call it moving average of order t, or simply MA (t). Statistically, it can be shown as below:

	Initialization Set	Test Set
	X1  X2  ......  Xt	Xt + 1   Xn
Time	Moving Average	Forecast
t	= ( X1  X2 ......  Xt ) / t	Ft + 1 = =
t + 1	= ( X2  X3 ......  Xt + 1 ) / t	Ft + 2 = =

Table 2.1 below illustrates using a 3-month, 4-month and a 12-month Moving Average method
to forecast a time series of product sales, based on past 24 months data points.

Month	Period  t	Historical Sales Qty Xt	3-Month Moving Average MA(3)	4-Month Moving Average MA(4)	12-Month Moving Average MA(12)
Jan-06	1	200	-	-	-
Feb-06	2	135	-	-	-
Mar-06	3	195	-	-	-
Apr-06	4	197	176.7	-	-
May-06	5	310	175.7	181.8	-
Jun-06	6	175	234.0	209.3	-
Jul-06	7	155	227.3	219.3	-
Aug-06	8	130	213.3	209.3	-
Sep-06	9	220	153.3	192.5	-
Oct-06	10	277	168.3	170.0	-
Nov-06	11	235	209.2	195.5	-
Dec-06	12	210	244.2	215.5	-
Jan-07	13	161	240.9	235.5	203.3
Feb-07	14	146	202.2	220.8	200.0
Mar-07	15	208	172.5	188.0	200.9
Apr-07	16	256	171.9	181.3	202.0
May-07	17	180	203.5	192.8	206.9
Jun-07	18	234	214.9	197.5	196.1
Jul-07	19	190	223.5	219.5	201.0
Aug-07	20	132	201.5	215.0	203.9
Sep-07	21	124	185.5	184.0	204.1
Oct-07	22	295	148.9	170.0	196.1
Nov-07	23	212	183.9	185.3	197.6
Dec-07	24	185	210.5	190.8	195.7
Jan-08	25	-	230.9	204.0	193.6		Figure 2.3  Actual and Moving Average Values for 12-Months Data
			Test Periods
			Month 4 -24	Month 5 - 24	Month 12 - 24
Mean Mean Error Mean Absolute Error Mean Absolute Percentage Error Std Deviation of Error Mean Squared Error U-Statistic Kurtosis			198.18 3.34 60.35 28.89 71.95 4930.57 1.01 -0.95	198.66 3.09 55.44 30.99 68.12 4408.93 0.97 -0.81	200.63 -7.04 42.43 24.15 53.63 2636.82 0.84 -1.01

Here are the workout of the moving averages as the forecast:

April-06's forecast at MA(3) = (200+135+195) / 3 = 176.7
May-06's forecast at MA(3) = (135+195+197) / 3
May-06's forecast at MA(4) = (200+135+195+197) / 4 = 181.8
Jan-08's forecast at MA(12) = the average for Jan-07 through to Dec-07

Comparing F_{t + 1} (176.7) and F_{t + 2} (175.7) in MA(3), the value of F_{t + 2} requires dropping X₁ (200) and adding X_{t + 1} (197), so another way of know F_{t + 2} is:

      F_{t + 2}  =  F_{t + 1} + (X_{t + 1} - X₁) / t  =  176.7+((197-200)/3)  = 175.7

Some points to observe here as to what number of t periods you would choose for the moving average. MA(1) - a moving average of order 1 - taking the last known data as the next period's forecast is truly impractical. Using MA(3) or MA(4) would effectively smooth out seasonal effects (especially for additive seasonal data pattern) as shown in Figure 2.3 above, but if it is used as a forecast for the next period, it self will not be able to accommodate trend or seasonality pattern. However, you can use MA(4) as a Centered Moving Average rather than as a forecast, to help to examine the deviations, spreads and other components within the time series.

MA(12) or larger, as you can see in Figure 2.3, would smooth seasonal effects out of the time series, and would be useful in decomposing the series into trend, seasonal and other components (explained in the next page). Again, if used on its own, is not very useful as a forecasting tool for data that exhibits trend and seasonality. The larger the order of the moving averages, the greater the smoothing effect, and the lesser attention it pays to the fluctuations in the series.

In the above chart, you can see that a small value of t in MA(t) allows the moving averages to follow the pattern, and these MA forecasts can trail the pattern by lagging behind one or a few periods.

Double Moving Averages
The Double Moving Average method (or 2nd moving averages of the 1st -order moving averages) gives you unequally weighted averages, and intended to handle data series with trend better than the single moving averages. Double-Moving Average can be denoted as MA (M x N) which means an M-period MA of an N-period MA. In order to mitigate against the systemic error that occurs when moving averages are applied to data series with trend, we therefore use a method called linear moving averages. Table 2.2 below shows you a few ways of forecasting the series with trend pattern, using a MA(MxN) Linear Moving Average.

The general Linear Moving Average has the following equations, from which the series in Table 2.2 are computed:

For Equations on the left, Single-Moving Average, MA(N), is denoted as St .The Double-Moving Averages are denoted as Dt and will only compute when all the single-MA  averages are computed. at refers to the adjustment of the single-MA, St by the difference St - Dt , meaning the adjusted smoothed value for period t. The trend component bt is the estimate of the trend from one time period to another. The last equation Ft + m refers to how to obtain forecasts for m periods ahead of t . The forecast for m periods ahead is equal to at + bt (m). Note that 2/(n-1) in bt arises because n period of moving average should be centered at a time period (n+1)/2. For instance, the first moving average at n time period for MA(4) is using n - (n+1)/2 which is the difference of (n-1)/2 or (4-1)/2 =1.5 periods.  For n =4, as in periods _1___2_x_ 3___4_ the cross mark is where 1.5 periods (i.e. correction for lag) should be centered, and MA(4) is computed at period 4.

See below for a few workout examples from Table 2.2:

4-Month MA, MA(4) value at period 4                    = (140+159+136+157) / 4
4-Month Double-MA, MA(4X4) value at period 7      =  (148.0+156.3149.3+159.5) / 4
Double-MA Adjusted Forecast at period 10, F₁₀    = [MA(4) at period 9] + [MA(4) - MA(4X4) at period 9]      = (162.5+(162.5-159.6))    = 165.4
Adjusted smoothed value for period 10, a₁₀          = 2(174.5)-165.9  =183.1
Trend Estimate for period 24, or b₂₄                           = 2(234.8 - 227)/(4-1)   = 5.17
Forecast a+b(m) with m=1 for period 25, F_{t + 25}      = a₂₄  + b₂₄  (1)   = 242.5 + 5.17  = 247.7
Forecast a+b(m) with m=2 for period 26, F_{t + 26}      = a₂₅  + b₂₅  (2)   = 258.2 + 8.13(2)  = 274.4
Forecast a+b(m) with m=3 for period 27, F_{t + 27}      = a₂₆  + b₂₆  (3)   = 258.2 + 8.13(3)  = 282.6

Download worksheet for Table 2.2.

This site was created in February 2007.
contact Tan, William     email: vbautomation@yahoo.com