Averaging and smoothing models

Notes on forecasting with moving averages (pdf)
Moving average and exponential smoothing models
Slides on inflation and seasonal adjustment and Winters seasonal exponential smoothing
Spreadsheet implementation of seasonal adjustment and exponential smoothing
Equations for the smoothing models (SAS web site)

 

Moving average and exponential smoothing models


Simple moving average  model
Brown’s simple exponential smoothing model
Brown’s linear exponential smoothing model
Holt’s linear exponential smoothing model


As a first step in moving beyond mean models, random walk models, and linear trend models, nonseasonal patterns and trends can be extrapolated using a moving-average or smoothing model. The basic assumption behind averaging and smoothing models is that the time series is locally stationary with a slowly varying mean. Hence, we take a moving (local) average to estimate the current value of the mean and then use that as the forecast for the near future. This can be considered as a compromise between the mean model and the random-walk-without-drift-model. The same strategy can be used to estimate and extrapolate a local trend.  A moving average is often called a "smoothed" version of the original series because short-term averaging has the effect of smoothing out the bumps in the original series. By adjusting the degree of smoothing (the width of the moving average), we can hope to strike some kind of optimal balance between the performance of the mean and random walk models. The simplest kind of averaging model is the....


Simple (equally-weighted) Moving Average:

The forecast for the value of Y at time t+1 that is made at time t equals the simple average of the most recent m observations:

 (Here and elsewhere I will use the symbol “Y-hat” to stand for a forecast of the time series Y made at the earliest possible prior date by a given model.) This average is centered at period t-(m+1)/2, which implies that the estimate of the local mean will tend to lag behind the true value of the local mean by about (m+1)/2 periods. Thus, we say the average age of the data in the simple moving average is (m+1)/2 relative to the period for which the forecast is computed: this is the amount of time by which forecasts will tend to lag behind turning points in the data. For example, if you are averaging the last 5 values, the forecasts will be about 3 periods late in responding to turning points. Note that if m=1, the simple moving average (SMA) model is equivalent to the random walk model (without growth). If m is very large (comparable to the length of the estimation period), the SMA model is equivalent to the mean model. As with any parameter of a forecasting model, it is customary to adjust the value of k in order to obtain the best "fit" to the data, i.e., the smallest forecast errors on average.


Here is an example of a series which appears to exhibit random fluctuations around a slowly-varying mean. First, let's try to fit it with a random walk model, which is equivalent to a simple moving average of 1 term:

The random walk model responds very quickly to changes in the series, but in so doing it picks much of the "noise" in the data (the random fluctuations) as well as the "signal" (the local mean). If we instead try a simple moving average of 5 terms, we get a smoother-looking set of forecasts:

The 5-term simple moving average yields significantly smaller errors than the random walk model in this case. The average age of the data in this forecast is 3 (=(5+1)/2), so that it tends to lag behind turning points by about three periods. (For example, a downturn seems to have occurred at period 21, but the forecasts do not turn around until several periods later.)


Notice that the long-term forecasts from the SMA model are a horizontal straight line, just as in the random walk model. Thus, the SMA model assumes that there is no trend in the data. However, whereas the forecasts from the random walk model are simply equal to the last observed value, the forecasts from the SMA model are equal to a weighted average of recent values.

The confidence limits computed by Statgraphics for the long-term forecasts of the simple moving average do not get wider as the forecasting horizon increases. This is obviously not correct! Unfortunately, there is no underlying statistical theory that tells us how the confidence intervals ought to widen for this model.  However, it is not too hard to calculate  empirical estimates of the confidence limits for the longer-horizon forecasts. For example, you could set up a spreadsheet in which the SMA model would be used to forecast 2 steps ahead, 3 steps ahead, etc., within the historical data sample. You could then compute the sample standard deviations of the errors at each forecast horizon, and then construct confidence intervals for longer-term forecasts by adding and subtracting multiples of the appropriate standard deviation.


If we try a 9-term simple moving average, we get even smoother forecasts and more of a lagging effect:

The average age is now 5 periods (=(9+1)/2). If we take a 19-term moving average, the average age increases to 10:

Notice that, indeed, the forecasts are now lagging behind turning points by about 10 periods.

Which amount of smoothing is best for this series?  Here is a table that compares their error statistics, also including a 3-term average:

Model C, the 5-term moving average, yields the lowest value of RMSE by a small margin over the 3-term and 9-term averages, and their other stats are nearly identical.  So, among models with very similar error statistics, we can choose whether we would prefer a little more responsiveness or a little more smoothness in the forecasts.  (Return to top of page.)


Brown's Simple Exponential Smoothing (exponentially weighted moving average)

The simple moving average model described above has the undesirable property that it treats the last k observations equally and completely ignores all preceding observations. Intuitively, past data should be discounted in a more gradual fashion--for example, the most recent observation should get a little more weight than 2nd most recent, and the 2nd most recent should get a little more weight than the 3rd most recent, and so on. The simple exponential smoothing (SES) model accomplishes this.

Let α denote a "smoothing constant" (a number between 0 and 1).  One way to write the model is to define a series L that represents the current level (i.e., local mean value) of the series as estimated from data up to the present.   The value of L at time t is computed recursively from its own previous value like this:

 

Lt  =  αYt  +  (1-α)  Lt-1

 

Thus, the current smoothed value is an interpolation between the previous smoothed value and the current observation, where α controls the closeness of the interpolated value to the most recent observation. The forecast for the next period is simply the current smoothed value:

 

 

Equivalently, we can express the next forecast directly in terms of previous forecasts and previous observations, in any of the following equivalent versions.  In the first version, the forecast is an interpolation between previous forecast and previous observation:

 

 

In the second version, the next forecast is obtained by adjusting the previous forecast in the direction of the previous error by a fractional amount α:

 

 

where

 

is the error made at time t.  In the third version,  the forecast is an exponentially weighted (i.e. discounted) moving average with discount factor 1-α:

 

 

The interpolation version of the forecasting formula is the simplest to use if you are implementing the model on a spreadsheet: it fits in a single cell and contains cell references pointing to the previous forecast, the previous observation, and the cell where the value of α is stored.

Note that if α=1, the SES model is equivalent to a random walk model (without growth). If α=0, the SES model is equivalent to the mean model, assuming that the first smoothed value is set equal to the mean.  (Return to top of page.)


The average age of the data in the simple-exponential-smoothing forecast is 1/α relative to the period for which the forecast is computed. (This is not supposed to be obvious, but it can easily be shown by evaluating an infinite series.) Hence, the simple moving average forecast tends to lag behind turning points by about 1/α periods. For example, when α=0.5 the lag is 2 periods; when α=0.2 the lag is 5 periods; when α=0.1 the lag is 10 periods, and so on.

For a given average age (i.e., amount of lag), the simple exponential smoothing (SES) forecast is somewhat superior to the simple moving average (SMA) forecast because it places relatively more weight on the most recent observation--i.e., it is slightly more "responsive" to changes occuring in the recent past.  For example, an SMA model with 9 terms and an SES model with α=0.2 both have an average age of 5 for the data in their forecasts, but the SES model puts more weight on the last 3 values than does the SMA model and at the same time it doesn’t entirely “forget” about values more than 9 periods old, as shown in this chart:

Another important advantage of the SES model over the SMA model is that the SES model uses a smoothing parameter which is continuously variable, so it can easily optimized by using a "solver" algorithm to minimize the mean squared error. The optimal value of α in the SES model for this series turns out to be 0.2961, as shown here:

The average age of the data in this forecast is 1/0.2961 = 3.4 periods, which is similar to that of a 6-term simple moving average.


The long-term forecasts from the SES model are a horizontal straight line, as in the SMA model and the random walk model without growth. However, note that the confidence intervals computed by Statgraphics now diverge in a reasonable-looking fashion, and that they are substantially narrower than the confidence intervals for the random walk model. The SES model assumes that the series is somewhat "more predictable" than does the random walk model.

An SES model is actually a special case of an ARIMA model, so the statistical theory of ARIMA models provides a sound basis for calculating confidence intervals for the SES model. In particular, an SES model is an ARIMA model with one nonseasonal difference, an MA(1) term, and no constant term, otherwise known as an "ARIMA(0,1,1) model without constant". The MA(1) coefficient in the ARIMA model corresponds to the quantity 1-α in the SES model. For example, if you fit an ARIMA(0,1,1) model without constant to the series analyzed here, the estimated MA(1) coefficient turns out to be 0.7029, which is almost exactly one minus 0.2961.

It is possible to add the assumption of a non-zero constant linear trend to an SES model. To do this, just specify an ARIMA model with one nonseasonal difference and an MA(1) term with a constant, i.e., an ARIMA(0,1,1) model with constant. The long-term forecasts will then have a trend which is equal to the average trend observed over the entire estimation period. You cannot do this in conjunction with seasonal adjustment, because the seasonal adjustment options are disabled when the model type is set to ARIMA.  However, you can add a constant long-term exponential trend to a simple exponential smoothing model (with or without seasonal adjustment) by using the inflation adjustment option in the Forecasting procedure.  The appropriate "inflation" (percentage growth) rate per period can be estimated as the slope coefficient in a linear trend model fitted to the data in conjunction with a natural logarithm transformation, or it can be based on other, independent information concerning long-term growth prospects.  (Return to top of page.)


Brown's Linear (i.e., double) Exponential Smoothing


The SMA models and SES models assume that there is no trend of any kind in the data (which is usually OK or at least not-too-bad for 1-step-ahead forecasts when the data is relatively noisy), and they can be modified to incorporate a constant linear trend as shown above.  What about short-term trends?  If a series displays a varying rate of growth or a cyclical pattern that stands out clearly against the noise, and if there is a need to forecast more than 1 period ahead, then estimation of a local trend might also be an issue.  The simple exponential smoothing model can be generalized to obtain a linear exponential smoothing (LES) model that computes local estimates of both level and trend.

 

The simplest time-varying trend model is Brown's linear exponential smoothing model, which uses two different smoothed series that are centered at different points in time.  The forecasting formula is based on an extrapolation of a line through the two centers.   (A more sophisticated version of this model, Holt’s, is discussed below.) 

 

The algebraic form of Brown’s linear exponential smoothing model, like that of the simple exponential smoothing model, can be expressed in a number of different but equivalent forms. The "standard" form of this model is usually expressed as follows: Let S' denote the singly-smoothed series obtained by applying simple exponential smoothing to series Y. That is, the value of S' at period t is given by:

 

(Recall that, under simple exponential smoothing, this would be the forecast for Y at period t+1.) Then let S" denote the doubly-smoothed series obtained by applying simple exponential smoothing (using the same α) to series S':

 

 

 

Finally, the forecast for Yt+k, for any k>1, is given by:

 

 

where:

 

 

...is the estimated level at period t, and

 

 


 ...is the estimated trend at period t.

 

For purposes of model-fitting (i.e., calculating forecasts, residuals, and residual statistics over the estimation period), the model can be started up by setting S'1  = S''1  =  Y1, i.e., set both smoothed series equal to the observed value at t=1.  (Return to top of page.)


A mathematically equivalent form of Brown's linear exponential smoothing model, which emphasizes its non-stationary character and is easier to implement on a spreadsheet, is the following:

 

 

 


or equivalently:

 

 

 


In other words, the predicted difference at period t is equal to the previous observed difference  minus a weighted difference of the two previous forecast errors.

Caution: this form of the model is rather tricky to start up at the beginning of the estimation period. The following convention is recommended:

 

 

 


This yields e1 = 0 (i.e., cheat a bit, and let the first forecast equal the actual first observation), and  e2  =  Y2 – Y1, after which forecasts are generated using the equation above. This yields the same fitted values as the formula based on S' and S'' if the latter were started up using S'1  =  S''1  =  Y1.   This version of the model is used on the next page that illustrates a combination of exponential smoothing with seasonal adjustment.


Holt’s Linear Exponential Smoothing

Brown’s LES model computes local estimates of level and trend by smoothing the recent data, but the fact that it does so with a single smoothing parameter places a constraint on the data patterns that it is able to fit:  the level and trend are not allow to vary at independent rates.  Holt’s LES model addresses this issue by included two smoothing constants, one for the level and one for the trend.   At any time t, as in Brown’s model, the there is an estimate Lt of the local level and an estimate Tt of the local trend.  Here they are computed recursively from the value of Y observed at time t and the previous estimates of the level and trend by two equations that apply exponential smoothing to them separately.

 

If the estimated level and trend at time t-1 are Lt‑1 and Tt-1, respectively, then the forecast for Y that would have been made at time t-1 is equal to Lt-1+Tt-1.  When the actual value is observed, the updated estimate of the level is computed recursively by interpolating between Y and its forecast,  Lt-1+Tt-1,  using weights of α and 1-α:

 

 

 


The change in the estimated level, namely Lt ‑ Lt‑1, can be interpreted as a noisy measurement of the trend at time t.  The updated estimate of the trend is then computed recursively by interpolating between Lt ‑ Lt‑1 and the previous estimate of the trend, Tt-1, using weights of β and 1-β:

 

 

 


Finally, the forecasts for the near future that are made from time t are obtained by extrapolation of the updated level and trend:

 

 


The interpretation of the trend-smoothing constant β is analogous to that of the level-smoothing constant α.   Models with small values of β assume that the trend changes only very slowly over time, while models with larger β assume that it is changing more rapidly.  A model with a large β believes that the distant future is very uncertain, because errors in trend-estimation become quite important when forecasting more than one period ahead.  (Return to top of page.)

 

The smoothing constants α and β can be estimated in the usual way by minimizing the mean squared error of the 1-step-ahead forecasts.  When this done in Statgraphics, the estimates turn out to be α =0.3048 and β =0.008.  The very small value of β means that the model assumes very little change in the trend from one period to the next, so basically this model is trying to estimate a long-term trend.   By analogy with the notion of the average age of the data that is used in estimating the local level of the series, the average age of the data that is used in estimating the local trend is proportional to 1/ β, although not exactly equal to it. In this case that turns out to be 1/0.006 = 125.  This isn’t a very precise number inasmuch as the accuracy of the estimate of β isn’t really 3 decimal places, but it is of the same general order of magnitude as the sample size of 100, so this model is averaging over quite a lot of history in estimating the trend.   The forecast plot below shows that the LES model estimates a slightly larger local trend at the end of the series than the constant trend estimated in the SES+trend model.  Also, the estimated value of α is almost identical to the one obtained by fitting the SES model with or without trend, so this is almost the same model.

 

    

 

Now, do these look like reasonable forecasts for a model that is supposed to be estimating a local trend?  If you “eyeball” this plot, it looks as though the local trend has turned downward at the end of the series!  What has happened?  The parameters of this model have been estimated by minimizing the squared error of 1-step-ahead forecasts, not longer-term forecasts, in which case the trend doesn’t make a lot of difference.  If all you are looking at are 1-step-ahead errors, you are not seeing the bigger picture of trends over (say) 10 or 20 periods.  In order to get this model more in tune with our eyeball extrapolation of the data, we can manually adjust the trend-smoothing constant so that it uses a shorter baseline for trend estimation.   For example, if we choose to set β =0.1, then the average age of the data used in estimating the local trend is 10 periods, which means that we are averaging the trend over that last 20 periods or so.  Here’s what the forecast plot looks like if we set β =0.1 while keeping α =0.3.  This looks intuitively reasonable for this series, although it is probably dangerous to extrapolate this trend any more than 10 periods in the future.

 

 

What about the error stats?  Here is a model comparison for the two models shown above as well as three SES models.  The optimal value of α.for the SES model is approximately 0.3, but similar results (with slightly more or less responsiveness, respectively) are obtained with 0.5 and 0.2.

 

Models

(A) Holt's linear exp. smoothing with alpha = 0.3048 and beta = 0.008

(B) Holt's linear exp. smoothing with alpha = 0.3 and beta = 0.1

(C) Simple exponential smoothing with alpha = 0.5

(D) Simple exponential smoothing with alpha = 0.3

(E) Simple exponential smoothing with alpha = 0.2

 

Estimation Period

Model

RMSE

MAE

MAPE

ME

MPE

(A)

98.9302

76.3795

16.418

-6.58179

-7.0742

(B)

100.863

78.3464

16.047

-3.78268

-5.63482

(C)

101.053

76.7164

14.605

1.70418

-4.94903

(D)

98.3782

75.0551

18.9899

3.21634

-4.85287

(E)

99.5981

76.3239

12.528

5.20827

-4.7815

 

Model

RMSE

RUNS

RUNM

AUTO

MEAN

VAR

(A)

98.9302

OK

OK

OK

OK

OK

(B)

100.863

OK

OK

OK

OK

OK

(C)

101.053

OK

OK

OK

OK

OK

(D)

98.3782

OK

OK

OK

OK

OK

(E)

99.5981

OK

*

OK

OK

OK

 

Their stats are nearly identical, so we really can’t make the choice on the basis of 1-step-ahead forecast errors within the data sample.  We have to fall back on other considerations.  If we strongly believe that it makes sense to base the current trend estimate on what has happened over the last 20 periods or so, we can make a case for the LES model with α = 0.3 and β = 0.1.  If we want to be agnostic about whether there is a local trend, then one of the SES models might be easier to explain and would also give more middle-of-the-road forecasts for the next 5 or 10 periods.  (Return to top of page.)


Which type of trend-extrapolation is best: horizontal or linear? Empirical evidence suggests that, if the data have already been adjusted (if necessary) for inflation, then it may be imprudent to extrapolate short-term linear trends very far into the future. Trends evident today may slacken in the future due to varied causes such as product obsolescence, increased competition, and cyclical downturns or upturns in an industry. For this reason, simple exponential smoothing often performs better out-of-sample than might otherwise be expected, despite its "naive" horizontal trend extrapolation.  Damped trend modifications of the linear exponential smoothing model are also often used in practice to introduce a note of conservatism into its trend projections.  These can be implemented as special cases of ARIMA models..


It is possible to calculate confidence intervals around long-term forecasts produced by exponential smoothing models, by considering them as special cases of ARIMA models. (Beware: not all software calculates confidence intervals for these models correctly.)  The width of the confidence intervals depends on (i) the RMS error of the model, (ii) the type of smoothing (simple or linear);  (iii) the value(s) of the smoothing constant(s); and (iv) the number of periods ahead you are forecasting. In general, the intervals spread out faster as α gets larger in the SES model and they spread out much faster when linear rather than simple smoothing is used.  This topic is discussed further in the ARIMA models section of the notes.  (Return to top of page.)

Go on to next topic:  spreadsheet implementation of seasonal adjustment and exponential smoothing

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