Baseline Forecasting With Exponential Smoothing Models

Transcription

1 Baseline Forecasting With Exponential Smoothing Models By Hans Levenbach, PhD., Executive Director CPDF Training and Certification Program, URL: Prediction is very difficult, especially if it is about the future NIELS BOHR ( ), Nobel Laureate Physicist This White Paper07 deals with the description, visualization and evaluation of statistical models that: Are widely used in the areas of sales, inventory, logistics, and production planning as well as in quality control, process control, financial planning and marketing planning Are especially suitable for large-scale, automated forecasting applications, because they require little forecaster intervention or parameter adjustments, thereby releasing the time of the demand forecaster to concentrate on the few problem cases Are based on the extrapolation of past patterns with forecasting equations that are simple to update and maintain in a database Can be described in terms of a modern modeling framework that provides uncertainty with prediction intervals and procedures for model selection Capture level (a starting point for the forecasts), trend (a factor for growth or decline) and seasonal factors (for adjustment of seasonal variation) in data patterns White Papers leading up to this White Paper can be freely downloaded by visiting Why Exponential Smoothing? In White Paper03, we introduce forecasting with simple and weighted moving averages as an exploratory smoothing and projection method for the short-term forecasting of level data. With exponential smoothing models, on the other hand, we can create short-term forecasts with a wider variety of data having trends and seasonal patterns; prediction limits (ranges of uncertainty) and prescribed forecast profiles. Exponential Smoothing Models (ESM) provide a viable framework for forecasting disaggregate demand patterns. For short-term planning and control systems, these techniques are extremely valuable and have a more than adequate track record in forecast accuracy. Each technique employs weights that give more emphasis to data from the recent past than to the older data. As a result of weighting of the past data, a fitted model appears smoother (less volatile) than the original data. As part of the State Space forecasting methodology, exponential smoothing models provide a unified approach to weighting past historical data for smoothing and extrapolation purposes. This exponentially declining weighting scheme contrasts with the equal-weighting scheme that underlies the simple moving average method. Exponential smoothing was invented during World War II by Robert G. Brown ( ), who was involved in the design of a tracking system for fire-control information on the location of enemy submarines. Later on, the principles of exponential smoothing were applied to business data, especially in the analysis of the demand for service parts in inventory systems by Brown in a book Advanced Service Parts Inventory Control (1982).

2 2 Exponential smoothing is a method of forecasting that extrapolates historical patterns such as trends and seasonal cycles into the future. There are many types of exponential smoothing models, each appropriate for a particular forecast pattern or forecast profile. As a forecasting tool, exponential smoothing is very widely accepted and a proven tool for a wide variety of short-term forecasting applications. Most inventory planning and production control systems rely on exponential smoothing to some degree. We will see that the process for assigning smoothing weights is simple in concept and versatile for dealing with diverse types of data. Other advantages of exponential smoothing are that the methodology takes account of trend and seasonal patterns in time series; embodies a weighting scheme that gives more weight to the recent past than to the distant past; is easily automated, making it especially useful for largescale forecasting applications; and can be described in a modeling framework needed for deriving useful statistical prediction limits and confidence intervals. When selecting a model for forecasting, focus on forecast profiles, rather than fit statistics and model coefficients. For demand forecasting, the disadvantages are that exponential smoothing models are univariate (i.e., no explanatory drivers) and are not models designed for use with causal variables, and cannot handle economic business cycles. Hence, when forecasting economic variables, we do not expect such techniques to perform well on business data that exhibit cyclical turning points. In applications for which the models are designed, such limitations are not of critical concern. Exponential smoothing provides an essential simplicity and ease of understanding for the practitioner, and has been found to have a reliable track record for accuracy in many business applications. Smoothing Weights To see how exponential smoothing works, we need first to understand the concept of exponentially decaying weights. Consider a time series of production rates (number of completed assemblies per week) for a 4-week period in the following table: Week Production Three periods ago (T - 3) 266 Two periods ago (T - 2) 411 Previous (T - 1) 376 Current t = T 425 In order to predict next period s (T + 1) production rate without having knowledge of or information about future demand, we assume that the following week will have to be an average week for production. A reasonable projection for the following week can be based on taking an average of the production rates during past weeks. However, what kind of average should we propose? Equally Weighted Average. The simplest option is to select an equally weighted average, which is obtained by given equal weight to each of the weeks of available data: ( ) / 4 = 370 This equally weighted average is simply the arithmetic mean of the data. The forecast of next week's production rate is 370 assemblies. Implicitly, we are assuming that events of 2 and 3 weeks prior (e.g., the more distant past) are as relevant to what may happen next week as were events of the most current and prior week. In Exhibit 1, a weight is denoted by w i, where the subscript i represents the number of weeks into the past. For an equally weighted average, the weight given to each of the terms is 1/n, where n is the number of time periods. With n = 4, each weight in column 3 is equal to 1/4. If we consider only the latest week, we have another option, shown in column 4 of Exhibit 1, which is the Naïve_1 forecast; it places all weight on the most recent data value. Thus, the forecast for next

3 3 week's production rate is 425, the same as the current week s production. This forecast makes sense if only the current week's events are relevant in projecting the following week. Whatever happened before this week is ignored. Exhibit 1 Various Weighting Schemes Exponentially Decaying Weights. Most business forecasters find a middle ground more appealing than either of the two extremes, equally weighted or Naïve_1. In between lie weighting schemes in which the weights decay as we move from the current period to the distant past. w 1 > w 2 > w 3 > w 4 >.... The largest weight, w 1, is given to the most recent data value. This means that to forecast next week's production rate, this week's figure is most important; last week's is less important, and so forth. Many other patterns are possible with decaying weight schemes. As illustrated by column 5 of Exhibit 1, the weight starts at 40% for the most recent week and decline steadily to 10% for week T - 3. Our forecast for week t = T + 1 is the weighted average with decaying weights: 425 x x x x 0.1 = 392 This weighted average gives a production rate forecast that is more than that of the equally weighted average and less than that of the Naïve_1. An exponentially weighted average refers to a weighted average of the data in which the weights decay exponentially. The most useful example of decaying weights is that of exponentially decaying weights, in which each weight is a constant fraction of its predecessor. A fraction of 0.50 implies a decay rate of 50%, as shown in column 6 of Exhibit 1. In forecasting next period s value, the current period s value is weighted 0.5, the prior week half of that at 0.25, and so forth with each new weight 50% of the one before. (These weights must be adjusted to sum to unity as in column 7.) From Exhibit 1, we can see that the adjusted weights are obtained by dividing the exponential decay weights by Exhibit 2 illustrates the weighted average of all past data, with recent data receiving more weight than older data. The most recent data is at the bottom of the spreadsheet. The weight on each data value is shown in Exhibit 3. The weights decline exponentially with time, a feature that gives exponential smoothing its name.

4 4 Exhibit 2 Computation of Weights on Historical Data Exhibit 3 Exponentially Decaying Weights Simple Exponential Smoothing Method All exponential smoothing techniques incorporate an exponential-decay weighting system, hence the term exponential. Smoothing refers to the averaging that takes place when we calculate a weighted average of the past data. To determine a one-period-ahead forecast of historical data, the projection formula is given by Y t (1) = αy t + (1 - α) Y t-1 (1) where Y t (1) is the smoothed value at time t, based on weighting the most recent observation Y t with a weight α (α is a smoothing parameter) and the current period s forecast (or previous smoothed value) with a weight (1 - α). By rearranging the right-hand side, we can rewrite the equation as Y t (1) = Y t-1 (1) + α[y t - Y t-1 (1)] which can be interpreted as the current period s forecast Y t-1 (1) adjusted by a proportion α of the current period s forecast error [Y t - Y t-1 (1)]. The simple exponential smoothing method produces forecasts that are a level line for any period in the future, but it is not appropriate for projecting trending data or patterns that are more complex. We can now show that the one-step-ahead forecast Y t (1) is a weighted moving average of all past observations with the weights decreasing exponentially. If we substitute for Y t-1 (1) in the first smoothing equation, we find that: Y t (1) = α Y t + (1 - α) [α Y t -1 + (1 - α) Y t - 2 (1)] = α Y t + α (1 - α) Y t -1 + (1 - α) 2 Y t - 2 (1)

5 5 If we next substitute for Y t-2 (1), then for Y t-3 (1), and so, we obtain the result Y t (1) = α Y t + α (1 - α) Y t -1 + α (1 - α) 2 Y t α (1 - α) 3 Y t 3 + α (1 - α) 4 Y t α (1 - α) t- 1 Y 1 + (1 - α) t Y 0 (1) The one-step-ahead forecast Y T (1) represents a weighted average of all past observations. For three selected values of the parameter α, the weights that are assigned to the past observations are shown in the following table: Weight Assigned to: α = 0.1 α = 0.3 α = 0.5 α = 0.9 Y T Y T Y T Y T Y T (0.5) (0.5) In Exhibit 4, we calculate a forecast of the production data, assuming that α = 0.5. (The production data are repeated in Exhibit 4, in the Actual column.) To use the formula, we need a starting value for the smoothing operation - a value that represents the smoothed average at the earliest week of our time series, here t = T - 3. The simplest choice for the starting value is the earliest data point. In our example, the starting value for the exponentially weighted average is the production rate for week t = T - 3, which was given as 266. The final result, Y T (1) = 391 (rounded) for week t = T, is called the current level. It is a weighted average of 4 weeks of data, where the weights decline at a rate of 50% per week. Exhibit 4 Updating an Exponentially Weighted Average We defined a one-period-ahead forecast made at time t = T to be Y T (1). Likewise, the m-periodahead forecast is given by Y T (m) = Y T (1), for m = 2, 3,.... For a time series with a relatively constant level, this is a good forecasting technique. This is called simple smoothing, but it is generally known as the simple exponential smoothing method. The forecast profile of the simple exponential smoothing method is a level horizontal line. Simple exponential smoothing works much like an automatic pilot or a thermostat. At each time period, the forecasts are adjusted according to the sign of the forecast error (actual data minus forecast.) If the current forecast error is positive, the next forecast is increased; if the error is negative, the forecast is reduced. To get the smoothing process started (Exhibit 5), we set the first forecast (cell E8) equal to the first data observation (cell D8). We can also use the average of the first few data observations. Thereafter, the forecasts are updated as follows: In column F, each error is equal to actual data minus forecast. In column E, each forecast is equal to the previous forecast plus a fraction of the previous error. This fraction is called the smoothing weight (cell I2). But how can the software program select the smoothing weight? The smoothing weight is usually chosen to minimize the mean square error (MSE), a conventional statistical measure of fit. This smoothing

6 6 weight is called optimal, because it is our best estimate based on a prescribed criterion (MSE). Forecasts, errors, and squared errors are shown in columns E, F, and G. The one-step-ahead forecast (=16.6 in cell E20) extends one period into the future. The travel expense data, smoothed values, and the one-periodahead forecast are shown graphically in Exhibit 6. Exhibit 5 Forecasting with Simple Exponential Smoothing Company Travel Expenses Exhibit 6 Simple Exponential Smoothing: Company Travel Expenses and One-period Ahead Forecasts Forecast Profiles for Exponential Smoothing Methods A family of exponential smoothing methods can be classified by the type of trend and/or seasonal pattern generated as the forecast profile. The most appropriate technique to use for any forecasting should match the profile expected or desired in an application. Exhibit 7 shows the extended Pegels classification for 12 forecasting profiles for exponential smoothing developed by Everette S. Gardner in a seminal paper Exponential smoothing: The state of the art, J. of Forecasting A Pegels classification of exponential smoothing methods gives rise to 12 forecast profiles for trend and seasonal patterns. After an exploratory analysis of the time series data, we may be able to determine which of the dozen techniques seems most suitable. In Exhibit 7, there are four types of trends to choose from (Nonseasonal column), and two types of seasonality (Additive and Multiplicative).

7 7 Exhibit 7 Pegels Classification of Exponential Smoothing Techniques Extended to Include the Damped Trend Technique For a downwardly trending time series, multiplicative seasonality appears as steadily diminishing swings about a trend. For level data, the constant-level multiplicative and additive seasonality techniques give the same forecast profile. Each profile can be directly associated with a specific exponential smoothing procedure (Exhibit 8), as described the in the next section (some of which are referred to by a common name attributed to their authors). We now explain how each procedure works to generate forecasts; that is, we describe how each procedure produces the appropriate forecasting profile. Model Name Trend profile Seasonal profile State Space Classification Simple (single) None None (N, N) Holt Additive (Linear) None (A, N) Holt-Winters Additive (Linear) Additive or Multiplicative (A, A) or (A, M) Exhibit 8 Most Commonly Implemented Exponential Smoothing Methods Smoothing Levels and Constant Change An exponential smoothing method comprises one or more of the following components: the current level, the current trend, and the current seasonal index. The current level serves as the starting point for the forecast. It is calculated to represent an exponentially weighted average of the time series at the end of the fit period. We can regard the current level as the value the time series would now have if there were nothing at all unusual going on at present. An alternative is to use the last observation as the starting point, but doing so might set the forecasts off at the wrong level if the most recent data is abnormal. The current trend represents the amount by which we expect the time series to grow or decline per time period into the future. It is often calculated as an exponentially weighted average of past

8 8 period-to-period changes in the level of the series. In this way, recent growth or decline in the time series is given more weight than changes farther back in time. The current seasonal index is interpreted the same way as a conventional seasonal index as the amount or degree by which the season s value tends to exceed or fall short of the norm. In the classical decomposition of a time series, a (multiplicative) seasonal index measured the norm as a moving average (see White Paper 03). The determination of the current seasonal index, a key part of Holt-Winters method differs from the conventional indexes in two respects: 1. Difference in weighting years. In a ratio-to-moving-average (RMA) method, the data used over the years to determine a monthly or quarterly index are weighted equally. In contrast, the Holt-Winters method takes an exponentially weighted average of the ratios to level, thus giving more weight to recent years than to those of the past. 2. Representing the norm. The Holt-Winters method uses the current level of the series instead of a moving average of four quarters or 12 months. The formulae to describe the current level, trend, and seasonal indexes can be found in Hyndman, Koehler, Ord and Snyder s book Forecasting with Exponential Smoothing, (2008). It describes in detail the state space framework for forecasting time series with exponential smoothing. The interpretation of the parameters in the smoothing algorithms is a complex matter and we will not deal with the estimation issues. Optimal or near-optimal parameter settings are derived using prescribed criteria for optimal selection found in modern forecasting software tools (e.g., PEERForecaster Addin for Excel, free download and use from In addition, combinations of multiple parameter values severely limit an intuitive feel for their impact on the forecasts. Moreover, for inventoryreplenishment planning purposes, a business planner needs to rely on automatic forecasting features of modern software because of the very large volume of data involved. We now illustrate the use of the equations for (1) calculating the current level, trend, and seasonal indexes and (2) combining these values into the forecasting formula. Consider Exhibit 9 for weekly shipments of a canned beverage for 66 weeks. The series is highly variable but not trending. The average level is approximately units per week with a standard deviation of Some of the high peaks may be attributed to promotions, but we will choose to forecast it using simple exponential smoothing. This model is appropriate for data lacking trend and seasonality. Exhibit 9 Time Plot Weekly Shipments of a Canned Beverage (66 weeks) No Trend or Seasonality (Source: Exhibit 10)

9 9 Exhibit 10 Weekly Shipments of a Canned Beverage No Trend or Seasonality The simple exponential smoothing model is fit to all 66 weeks and forecast for 4 weeks. The optimal estimate (based on the MSE criterion) of the smoothing parameter is 0.62 with MSE = 1,668,955. The multi-period forecasts are a constant level (= 5129). Thus, it represents a typical level. Exhibit 11 displays the most recent 20 weeks of historical shipments, 20 weeks of fitted values, and the four forecasts. Because simple exponential smoothing views the future of the time series as lacking both trend and seasonality, the forecasting equation does not contain these terms, leaving the current level as the sole component. The current level L t is calculated by an equation for an exponentially smoothed average of the past data: Y T (m) = L t. Exhibit 11 Forecast Model for Canned Beverage Shipments: Simple Exponential Smoothing: No Trend, No Seasonality (Source: Exhibit 10)

10 10 Exhibit 12 Annual Car Registrations - Linear trend Consider Exhibit 12 for annual car registrations. Because the data are annual, the time series is necessarily nonseasonal. The global trend appears to be linear, although there are a number of local variations on the trend. Exhibit 13 is the graph of the output from the Holt method linear trend, no seasonality. We see that from the vantage point of Year 19, the current level of car registrations is estimated to be 1034 and the current trend is estimated to be an increase of 31 registrations per year. For nonseasonal data, the seasonal index terms are not present in the forecasting equations. What remains in the Holt method, however, is the forecasting equation for a linear trend: Y T (m) = [L t + m x T t ] The forecast for year 22 is 1126, based on calculating a 3-year-ahead projection from the base year T = 19: Y 19 (3) = [ x 30.78] = 1126 Exhibit 13 Time Plot of the Historical Data and Forecast Model for Car Registrations: Holt Method - Linear Trend, No Seasonality (Source: Exhibit 12)

11 11 Damped, Linear and Exponential Trends In the seasonal methods, the trend component, if one is present, is assumed to be linear. Trends can also be nonlinear. For example, damped (upward) trend assumes that the series will continue to grow but that the growth gradually dampens out. An exponential growth trend assumes that the series will grow by a progressively larger amount. Exponential growth is equivalent to a constant percentage rate of growth. As the base grows over time, the constant percentage increase on the base translates into larger and larger increments in volume, in the manner of compound-interest growth on an investment. Everette S. Gardner introduced the damped trend exponential smoothing model in a seminal paper: Exponential Smoothing. The state of the art, J. of Forecasting (1985) and updated 20 years later in an equally important paper Exponential Smoothing. The state of the art Part II, Int. J. Forecasting (2006). These damped trend exponential smoothing models offer practitioners a well tested and consistently top performing approach to trend-seasonal time series forecasting. In the case of a downward trend, the damped and exponential patterns are similar. During a phase out or decline under adverse market conditions, a forecast profile will be decaying without becoming negative. We may refer to the pattern of shipments of a cosmetic product (Exhibit 14) either as a downwardly damped trend or as exponential decay. Exhibit 14 Time Plot of Weekly Shipments of a Cosmetic Product Damped Trend Exponential Smoothing Model Like a no-trend or linear trend, the exponential trend may be used in conjunction with multiplicative, additive, or no seasonality patterns. We use the nonseasonal case here to illustrate damped and exponential trends. (When seasonality is included, it is called the Holt-Winters procedure, discussed later.) Both damped and exponential trends can be represented in a single forecasting equation, given by Y T (m) = L t + φ i x T t where m is the length of the forecast horizon. The symbol φ is called the trend-modification parameter. Depending on the value of φ, the forecast profile can be an exponential trend, linear trend, damped trend, or constant level. Here are the cases. If φ > 1 trend is exponential. If φ = 1 trend is linear. If φ < 1 trend is damped. If φ = 0 there in no trend. Exhibit 15 shows the historical and forecast values for a damped trend and exponential trend model of the annual car registration series. The growth in the forecasts, the change from the prior year s forecast, has dampened. With φ = 0.83, the forecasted trend is slowing by = 0.17 or 17% per period. The estimates of level and trend weights are 0.6 and 0.2, respectively,

12 12 Exhibit 15 Time Plot for Historical Car Registrations with Two Exponential Smoothing Forecasts Damped Trend and Exponential Trend (Source: Exhibits 16 and 17) We can illustrate how the forecast was obtained for Year 20 (= T + 1); the fitted value for year 19 is , L t = , and T t = Setting m = 1, The Year 21 forecasts are calculated as follows: Y 19 (1) = L t + φ x T t = x (18.75) = Y 19 (2) = L t + (φ 1 + φ 2 ) x T t = ( )x (18.75) = Alternatively, we can obtain the forecast for T + 2 by calculating Y T (2) = Y T (1) + φ 2 x T t Exhibits 16 and 17 show a comparison of the forecasts from the exponential trend and the damped trend for car registrations data. Based on the MSE criterion, the MSE of the damped trend model is approximately 16% smaller than the MSE of the linear trend model. The five-period-ahead forecast (fifthyear projection) for the damped trend model is approximately 5% below the linear trend projection. Whether the difference in the 5-year projections is significant depends on the context in which these forecasts are used. A comparison of the forecasts by two techniques is shown graphically in Exhibit 15. Exhibit 16 Exponential Trend Forecast Model for Car Registrations (dampening factor = 1.0)

13 13 Exhibit 17 Damped Trend Forecast Model for Car Registrations (dampening factor = 0.83) Some Spreadsheet Examples Annual Sales of a Cosmetic Product. Consider the trend shown in Exhibit 18 for the yearly sales of a cosmetic product. The sales for this cosmetic product, for the period , show a declining trend. The forecasts decline exponentially, modeled by a nonlinear trend exponential smoothing technique. Exhibit 19 presents a comparison of a linear trend and damped exponential trend model for the cosmetic product. Note that minimizing MSE should not be the only criterion for selecting a model; the forecast profile should also be considered. In this case, the linear trend model with the lowest MSE also yields much lower forecasts over the forecast period than the damped trend model. It may require some judgment and experience on the part of the forecaster to determine the most appropriate profile for the data at hand. Exhibit 18 Sales of a Cosmetic Product Damped Exponential Trend

14 14 Exhibit 19 Forecast Models for a Cosmetic Product: Damped and Linear Trend Exponential Smoothing (Exhibit 18) To illustrate how the damped exponential trend forecast was obtained for the year 2005 (= T + 3), we set m = 3 and T = 2002: Y 2005 (3) = L t + (φ 1 + φ 2 + φ 3 ) x T t = ( ) x ( ) = The difference in the forecast profiles for the linear trend and damped trend arises from the value of the trend modification parameter, which is below unity (φ = 0.88) for the damped trend model. This value of φ leads to a decrease over prior year s forecast and is characteristic of an exponential decline. On the other hand, the damped exponential trend model, for which the trend modification parameter is constrained to φ = 1, gives rise to a linear trend forecast profile. A comparison of the two techniques yields a statistical summary (Exhibit 20) and forecast profiles for the cosmetics sales data (Exhibit 21) that indicate the difference in the rounded results. Exponential trends should be applied with caution, and careful consideration should be made of the underlying business environment of the data. Blind acceptance of optimal parameter estimates and best MSE values should be avoided. Exhibit 20 A Four-Period Forecast Comparison for Sales of a Cosmetic: Linear and Damped Trend Exponential Smoothing (Source: Exhibit 18) Exhibit 21 A Time Plot of History and Forecast Profiles of a Cosmetic: Damped Trend Exponential Smoothing (φ = 0.88) Compared to Linear trend (φ = 1.0) (Source: Exhibit 18)

15 15 Exhibit 22 Sale of a Cosmetic Product: Log Transform, Linear Trend (Source: Exhibit 18) Exhibit 23 Forecast Model for Sales of a Cosmetic Product (logarithmic transformation): Linear Trend (Source: Exhibit 22) Now consider Exhibit 22. By taking a transformation of the data, exponential growth patterns can also be modeled by applying a linear trend (Holt model) to the natural logarithm of the time series. The results are shown in Exhibit 23. The forecasts in Exhibit 23 are calculated first in terms of the logarithm of the series and then transformed into the original data by exponentiation. To illustrate how the forecast was obtained for the year 2006 (t = T + 4), we set m = 4 and T = 2002: Forecast for log 10 (T + 4) = x (-0.04) = 2.28 Transform back to original data = = 190 The results can be compared with the corresponding forecasts from the damped trend exponential model (DT) in Exhibit 20. The log-transform approach yields forecasts that are slightly lower than those shown in Exhibit 20; however, both depict a forecast profile with a negative exponential trend. Depending on the context in which these projections are used in practice, the differences could become substantial. This illustrates the limitation of using these types of techniques for extrapolating highly trending annual

16 16 time series for more than a couple of periods. More important, when we calculate prediction limits on the forecasts, these two approaches give different interpretations. Because of the transformation, the log-transformed model will result in asymmetric (right-skewed) prediction limits, whereas the original model will give symmetric limits. Company Sales - Linear Trend. Consider Exhibit 24a. The initial values for level and trend are in the top cells of the columns Level and Trend, respectively. Thereafter, adding a fraction of the error to each smoothes the level and trend. This is the same self-correcting idea used in the simple smoothing technique. The cells for LEVEL WEIGHT and TREND WEIGHT show that we used individual weights to smooth the level and trend. Each forecast is just the sum of the latest estimate of level and trend. For example, the linear trend forecast for 1990 is This is the sum of the level and trend at the end of 1989 ( ). At the end of the data, we can forecast as many years ahead as we like. The forecast for 2 years ahead (1991) is 49.96, the 1990 forecast plus another increment of trend ( ). For most business data, the magnitudes of the level and trend can differ greatly. In this example, the numbers in the level column range from 7 to 25 times larger than the numbers in the trend column. Thus, the level weight has to be larger than the trend weight to give reasonable forecasts. The weights shown minimize the MSE. (See Exhibit 25 for the time plot.) Company Sales - Damped Trend. Exhibit 24b is identical to Exhibit 24a except that a new parameter has been added to modify the trend. Each forecast is the latest estimate of the level plus the trend times a trend modifier (shown in the cell for TREND MODIFIER). For example, the forecast for 1990 is The formula is: Forecast = (Level at the end of 1989) + (Trend modifier) x (Trend at the end of 1989) = (0.80 x 1.03) = To generate forecasts for more than one period into the future, the formula is Forecast = (Last forecast) + (Trend modifier) (Number of periods into the future) x (Final trend estimate) The forecast for 1991 is thus x1.03 = Note that the trend modifier in Exhibit 24 is a fraction. Raising a fraction to a power produces smaller numbers as we move farther into the future. The result is called a damped trend because the amount of trend added to each new forecast declines. By changing the trend modifier, we can produce different kinds of trend. A modifier equal to 1.0 yields a linear trend, exactly the same result as the linear trend worksheet. A modifier greater than 1.0 yields an exponential trend, one in which the amount of growth gets larger each time period. A modifier of 0 produces no trend at all and the results are the same as simple exponential smoothing. The weights shown minimize the MSE, given that the trend modifier has been set. (See Exhibit 7.25 for the time plot.) In general, the damped trend techniques appear better suited for smoothing short-term patterns for operational forecasting in inventory and production planning than for smoothing long-term forecasting.

17 Exhibit 24 Company Sales: Comparison of Exponential Smoothing of (a) Linear and (b) Damped-Exponential Trends 17

18 18 Exhibit 25 Time Plot of Company Sales: Linear and Damped Trend Smoothing Handling Special Events with Smoothing Methods Special events arise whenever periodic actions of the organization, such as special promotions, scheduled disruptions for maintenance, unusual weather, and holiday effects, cannot be treated as seasonality because they do not fall within the same period (week) each year. A special event adjustment can be combined with an exponential smoothing method to improve the accuracy of a forecast. A special event refers to a sudden change in the level of the time series that is expected to recur. Event Adjustments for Outliers. When dealing with an outlier, it is generally not possible to simply delete it because doing so would leave a gap or missing value event between adjacent time periods. One procedure to remedy missing values is to interpolate by using a method to project the missing value(s), or replace the value by averaging the two adjacent values. For example, a missing value for June can be replaced by the average for May and July. An alternative to interpolating missing values is to use dummy variable events. We first identify the specific time period(s) when the outlier occurs and then create dummy variable events - a time series of 0s and 1s with 1 for each outlier time period and 0 for each normal time period. Once this dummy time series is incorporated into a forecasting technique, the outlying values are effectively removed from the calculations of the coefficients of the forecasting formula. The resulting forecasts are generated as if the unusual time periods never occurred. Dummy variable events can be used with other forecasting techniques as well. For exponential smoothing, event adjustments can be applied both to outliers and special events. First, the dummy variable event is created as an index in the exponential smoothing forecasting formula. For example, in the Winters method, the forecasting formula is an extension of the exponential smoothing technique for multiplicative and additive forms: Additive: Y T (m) = L t + m x T t + Seasonal index + Event index Multiplicative: Y T (m) = [L t + m x T t ] x Seasonal index x Event index A multiplicative event index is interpreted as a multiple of the level of the series. For example, if the current level is 100 and the event index is 0.75, the result indicates that the average effect of the outlier value(s) was to reduce the level of the series by 25%. The same result in the additive model would become event index of -25. Note that the effect of the event index in the procedure is to remove the outlier values from the calculation of the trend component and seasonal indexes. Then set the event index (= 0 for additive, = 1 for multiplicative) to generate forecasts on the assumption that no outliers will occur over the forecasting horizon. Event Adjustments for Promotions and Other Special Events. When data are available on the timing and magnitude of product promotions (price changes), we can effectively use this information in a causal forecasting approach, such as a regression model, in which the promotion variable enters as one or more explanatory (independent) variable.

19 19 Although exponential smoothing does not explicitly incorporate explanatory variables, it can be extended to incorporate dummy variables. In this manner, a dummy (0, 1) coding can be used to distinguish those periods in which a promotion occurred from those free of promotion. The introduction of such a dummy event series effectively models the timing of the promotions, without specifying their magnitude. Exhibit 26 shows a summary from an analysis of the monthly sales of a consumer product during a 24 month period. There were two peaks in the data associated with promotions during December of Year 1 and a promotion in August of Year 2. Three versions of a Holt-Winters (multiplicative) procedure were fit. Version 1 is a direct application of a multiplicative Winters seasonal exponential smoothing method. The Mean Absolute Percentage Error (MAPE) is 16.3% for the 24-month fit period. The strong effect of the twin promotions is viewed as seasonality, giving the months December, January, August, and September the high seasonal indexes of the year. However, promotions will appear as the seasonal indexes only if they are scheduled for the same months every year. This was not the case for this product in Year 3. Version 2 employs an event adjustment model in which the occurrence of promotions is represented by a (0, 1) dummy variable. For the historical period, the dummy variable is set equal to 1 in December of Year 1 and August of Year 2 and set equal to zero otherwise. The Winters procedure results in a substantial reduction in the MAPE from 16.3 to 11.2%, along with a much altered seasonal pattern. Now the high seasonal indexes are attributed to January, February, and September because December of Year 1 and August of Year 2 are identified as promotion months. The event index is reported as 1.57, a result that implies that, on average, sales during a promotions month will be approximately 57% higher than sales in a normal month. Although this is an improvement over Version 1, the seasonal pattern in Version 2 is still problematic. The high indexes for January and September are felt to be, at least in part, a response to the promotions occurring in December of Year 1 and August of Year 2 and to be unrepresentative of the plans for Year 3. Version 3 replaces the (0, 1) coding for promotions by a (0, 1, 2) coding. In general, a (0, 1, 2) coding indicates that there are two types of special events being dealt with: Event 1 and Event 2. In this example, the Event 2 was used to represent the delayed response of sales to an earlier promotion. Hence, the promotion time series was set equal to 1 for December of Year 1 and August of Year 2, 2 during January of Year 2 and September of Year 2, and 0 for all other months during these 24 months. Version 3 very substantially improves the fit to the historical data, with the MAPE falling to 6.9%. The event indexes imply that when a promotion occurs, sales can be expected to rise by 71% in the same month (Event 1) and 75% in the following month (Event 2). The January seasonal index has fallen from Version 2 because part of the January strength is now attributed to the December of Year 1 promotion. Exhibit 26 Forecast Performance with Three Models (n.a., not available) Prediction Limits for Exponential Smoothing Models The State-Space Approach Prediction limits on forecasts have generally not been available for exponential smoothing methods. To derive prediction limits, one needs a model which explicitly incorporates an error distribution assumption in the model specification. To use models, we need a modeling framework in which uncertainty (randomerror component) is made explicitly part of the forecasting technique. Up to this point, we have limited ourselves to the deterministic component (change) of the technique. Now, we will introduce measured uncertainty component (chance) into the forecasting framework.

20 20 The Pegels Classification for Trend-Seasonal Models A state-space framework for forecasting models using exponential smoothing appeared in the literature in the 1990 s and is published in a book in 2008 by Rob Hyndman, Anne Koehler, Keith Ord and Ralph Snyder, entitled Forecasting with Exponential Smoothing The State Space Approach. The state-space models that underlie the exponential smoothing techniques come in two forms: a model with additive errors and a model with multiplicative errors. Although the forecast profiles (Exhibit 7) for the two models are identical, there are important differences in the prediction limits produced by the two models. With this distinction in error structure, the state-space framework today effectively describes 30 models in the Pegels classification (Exhibit 27). A Pegels classification of exponential smoothing techniques in a state-space modeling framework gives rise to 30 trend-seasonal forecast profiles with prediction limits for trend and seasonal patterns Exhibit 27 Classification of the Exponential Smoothing Methods for State Space Forecasting (Source: Hyndman, et al. (2008) Forecasting with Exponential Smoothing The State Space Approach.) There are differences in their use as well. The multiplicative error models are not well defined if there are zeros or negative values in the data. Similarly, additive error models should not be used with multiplicative trend or multiplicative seasonality if any value is zero. The estimation and model selection steps are beyond the scope of this White Paper; the reader is referred to Hyndman et al. (2008) for the theoretical details. Fully functional software (PEERForecaster Add-in for Excel) for exponential smoothing using the state-space algorithms is available with a free download from Trend-Seasonal Exponential Smoothing Models The forecasting equations for additive and multiplicative seasonality are: Additive: Y T (m) = L t + m x T t + Seasonal index Multiplicative: Y T (m) = [L t + m x T t ] x Seasonal index

21 21 where Y T (m) denotes a forecast made at time T, the final season in the fit period, for m periods into the future. This technique is known as the Holt-Winters procedure. Unlike the nonseasonal exponential smoothing techniques, seasonality brings in a complexity that makes interpreting the smoothing equations directly less intuitive. Fortunately, these algorithms are available in some software, such as the free Excel Add-in PEERForecaster.xla ( We show here only that the forecast for m periods ahead can be completed using the steps: 1. Start at the current level, L t. 2. Add the product of current trend, T t, and the number of periods m ahead that we are projecting trend. 3. Adjust the resulting sum of level and trend for seasonality through a multiplicative or additive seasonal index. Exhibit 28 is a time plot of a quarterly automobile sales time series. For these data (Exhibit 29), an exploratory ANOVA decomposition shows that about 68% of the total variation was attributed to seasonality. The trend appears to be linear with seasonal peaks occurring in the second quarter of each year. The low seasonal peak in Year 2 was not adjusted because it occurred very early in the dataset and would be have little weight in the exponential smoothing calculations. Exhibit 28 Time Plot with 95% Prediction Limits of a Quarterly Automobile Sales Series Exhibit 29 Quarterly Automobile Sales Series in Quebec City

22 22 The key results are summarized in Exhibit 30. (Recall that the various error measures - MAPE, MAE, MAD, and RMSE - are defined and interpreted in White Paper 03). Each measure for the model is compared with an appropriate naïve technique, such as Naïve_3 for seasonal data. The fit coefficient is 1 [MSE of your model)/(mse of Naïve_3)]. Naïve_3 is computed as follows: 1. The series is deseasonalized so that the seasonal pattern is removed. 2. The preliminary forecast for each period is taken as the seasonally adjusted value from the previous period. 3. The final forecasts are computed by re-seasonalizing the preliminary forecasts. 4. Forecast errors are computed by subtracting final forecasts from actual data. The Naïve_3 technique assumes no trend, but has valuable information about seasonality. In a seasonal series with little trend, we cannot expect to do much better than Naïve_3. The prediction limits are shown in Exhibit 28. These 95% prediction limits indicate that we are 95% sure that the true (in the sense that the model is correct) forecasts will lie within these limits. The prediction limits appear asymmetrical, which suggests that the model errors are multiplicative. The forecasts are more likely to be high than low, which makes sense with the consistent upward trend in the historical data. Exhibit 30 Model-Fitting Summary for Quarterly Automobile Sales; Model (A, A) (Source: Exhibit 29) The smoothing weights (level = 0.02, trend = and seasonal = 0.00) in Exhibit 31 show the relative emphasis given to the data from the recent and more distant past in the calculation of the current level, trend, and seasonal indexes. The values for the current level, trend, and seasonal indexes are called final values. Note that period T is the second quarter (spring) of Year 6. To forecast from this time origin, the current level is 46,841, the current trend is (units per quarter), and each season has its own seasonal index. The summer index (Q3= -2,225.8) indicates that automobile sales during the summer tend to be approximately 2,226 units below the norm Exhibit 31 Forecast Model for Automobile Sales: Additive Winters - Linear Trend, Additive Seasonality Model (A, A) (Source: Exhibit 29)

23 23 Starting from spring Q2 of Year 6, the automobile sales forecast for Q3 is (setting m = 1): Y T (m) =[L t + m x T t ] + Seasonal index Y T (1) = [46, ] + ( ) = 45,177 units To forecast three periods ahead to winter Q1 of Year 7, we set m = 3 and use the seasonal index for Q1: Y T (3) = [L t + 3 x T t ] + Seasonal index = [46, x 561.8] + ( ) = 38,924 units The forecast for the winter of Year 7 is lower than that for the previous summer for two reasons. First, the trend is only growing by approximately 562 units per quarter. Second, and more substantially, the seasonal index for winter is 7377 units lower than in summer. In Exhibit 32, the multiplicative seasonal model produces the following forecast for the winter quarter of Year 7 Y T (3) = [L t + 3 x T t ] x Seasonal index = [46, x 957.1] x = 37,624 units Exhibit 32 Forecast Model for Automobile Sales: Multiplicative Winters - Linear Trend, Multiplicative Seasonality Model (A, M) (Source: Exhibit 29) In this example, the two versions of Holt-Winters procedure give very similar values for the current level and trend. The seasonal indexes are in a different form and result in different projections. The additive index for the summer season tells us that summer sales tend to be approximately 15,000 units above the norm; this will be a constant amount for all future years. In contrast, the multiplicative index for the summer season is estimated to be approximately 39% above the norm, which represents an increasing amount as long as the data are trending up. Which model is preferable? On a purely statistical basis, the multiplicative model has the preferred summary results. However, this may not necessarily mean that forecast performance will be better as well.

24 24 Predictive Visualization of Change and Chance Hotel/Motel Demand In the examples so far, we have been concerned with how to calculate forecasts with exponential smoothing. However, fitting summaries may not be a good indication of how to achieve forecast accuracy. The reason is that the same data are used to estimate the parameters and evaluate its suitability as a forecasting tool. Forecast accuracy measurement and evaluation procedures are taken up in White Paper 10. However, in the meantime, we discuss some steps we can take with exponential smoothing techniques that can be used for creating credible models. In Exhibit 34, we provide a model fitting summary and model performance comparison for an additive and multiplicative exponential smoothing model of the monthly time series of hotel/motel (DCTOUR) demand. The historical period consists of 96 monthly values (Exhibit 33). The smoothing parameters come from minimizing MSE over the fit period. On the basis of the error measures for bias and precision, the multiplicative version appears preferable to the additive version. Hence, we will continue the analysis with the multiplicative Holt-Winters model. However, in practice it would be advisable to maintain several models at all times. Exhibit 33 Historical Data of Monthly Hotel/Motel Demand (DCTOUR) (Source: D. C. Frechtling, Practical Tourism Forecasting 1996, Appendix. 1) Fit Statistics (A, A) (A, M) Level (alpha) Trend (beta) Season (gamma) MAPE (Mean Absolute 2.2% 1.9% Percentage Error) AIC ME (Mean ResidualError) MdE (Median Residual Error) MAD (Mean Absolute Deviation) MdAD (Median Absolute Deviation) RMSE (Root Mean Square Error) Exhibit 34 Model-Fitting Summary for Monthly Hotel/Motel Demand (DCTOUR) - Additive Trend, Additive Seasonality (A, A) and Additive Trend, Multiplicative Seasonality (A, M) (Source: Exhibit 33) The preferred way to simulate forecast accuracy is by creating a holdout period and distinguishing a fit period (for estimating parameters) from a forecast period (for determining forecast performance).

25 25 Because our primary interest is in forecasting performance, we start with a holdout period of the last 12 months and create forecasts based on the first 84 monthly values. In Exhibit 35, the summary FIT statistics, Smoothing parameters, and Seasonal suggest we have a credible model for DCTOUR. Exhibit 35 Monthly Hotel/Motel Demand Series - Linear Trend, Multiplicative Seasonality (A, M) Summary Fit Statistics, Smoothing Parameters and Seasonal Factors for 12-Month Hold-Out Sample Exhibit 36 Monthly Hotel/Motel Demand Series - Linear Trend, Multiplicative Seasonality (A, M) Forecast Bias and Precision over 12-Month Horizon The hold-out sample evaluation of forecasts in Exhibit 36 is a static test in the sense that all forecasts are made from a single time point (T = 1993:12) for a fixed horizon (m = 12). The forecast errors are all negative, implying that the model is biased (over-forecasting). In fact, May, November and December 1994 are outside the lower prediction limits; most severely over-forecasted by the model. The one-step-ahead forecast errors play a special role in the analysis of forecast performance because prediction limits are based on them. Exhibit 37 displays the 12 period-ahead forecasts with the upper and lower prediction limits over the holdout period made with the linear trend, multiplicative seasonal model (A, M). These forecasts are compared with the actuals in the holdout sample and forecast errors are calculated. Evidently, the peak month is May (index = 1.187), meaning that May is almost 19% above norm. This is attributable to the attractions of the cherry blossoms during spring season in