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5.5 Forecasting Models—Trend and Random Variations

If there is trend present in a time series, the forecasting model must account for this and cannot simply average past values. Two very common techniques will be presented here. The first is exponential smoothing with trend, and the second is called trend projection or simply a trend line.

Exponential Smoothing with Trend

An extension of the exponential smoothing model that will explicitly adjust for trend is called the exponential smoothing with trend model. The idea is to develop an exponential smoothing forecast and then adjust this for trend. Two smoothing constants, $α$ $α$ and $β,$ $β,$ are used in this model, and both of these values must be between 0 and 1. The level of the forecast is adjusted by multiplying the first smoothing constant, $α,$ $α,$ by the most recent forecast error and adding it to the previous forecast. The trend is adjusted by multiplying the second smoothing constant, $β,$ $β,$ by the most recent error or excess amount in the trend. A higher value gives more weight to recent observations and thus responds more quickly to changes in the patterns.

As with simple exponential smoothing, the first time a forecast is developed, a previous forecast $(F_{t})$ $(F_{t})$ must be given or estimated. If none is available, often the initial forecast is assumed to be perfect. In addition, a previous trend $(T_{t})$ $(T_{t})$ must be given or estimated. This is often estimated by using other past data, if available; by using subjective means; or by calculating the increase (or decrease) observed during the first few time periods of the data available. Without such an estimate available, the trend is sometimes assumed to be 0 initially, although this may lead to poor forecasts if the trend is large and $β$ $β$ is small. Once these initial conditions have been set, the exponential smoothing forecast including trend $(F I T_{t})$ $(F I T_{t})$ is developed using three steps:

Step1. Compute the smoothed forecast $(F_{t + 1})$ $(F_{t + 1})$ for time period $t + 1$ $t + 1$ using the equation

$\begin{array}{l} Smoothed forecast & = & Previous forecast including trend + α (Last error) \\ F_{t + 1} & = & F I T_{t} + α (Y_{t} - F I T_{t}) \end{array}$ $\begin{array}{l} Smoothed forecast & = & Previous forecast including trend + α (Last error) \\ F_{t + 1} & = & F I T_{t} + α (Y_{t} - F I T_{t}) \end{array}$ (5-10)
Step 2. Update the trend $(T_{t + 1})$ $(T_{t + 1})$ using the equation

$\begin{array}{l} Smoothed trend & = & Previous trend + β (Error or excess in trend) \\ T_{t + 1} & = & T_{t} + β (F_{t + 1} - F I T_{t}) \end{array}$ $\begin{array}{l} Smoothed trend & = & Previous trend + β (Error or excess in trend) \\ T_{t + 1} & = & T_{t} + β (F_{t + 1} - F I T_{t}) \end{array}$ (5-11)
Step 3. Calculate the trend-adjusted exponential smoothing forecast $(F I T_{t + 1})$ $(F I T_{t + 1})$ using the equation

$\begin{array}{l} Forecast including trend (F I T_{t + 1}) & = & Smoothed forecast (F_{t + 1}) + Smoothed trend (T_{t +1}) \\ F I T_{t + 1} & = & F_{t + 1} + T_{t + 1} \end{array}$ $\begin{array}{l} Forecast including trend (F I T_{t + 1}) & = & Smoothed forecast (F_{t + 1}) + Smoothed trend (T_{t +1}) \\ F I T_{t + 1} & = & F_{t + 1} + T_{t + 1} \end{array}$ (5-12)

where

\begin{array}{l} T_{t} & = & smoothed trend for time period t \\ F_{t} & = & smoothed forecast for time period t \\ F I T_{t} & = & forecast including trend for time period t \\ α & = & smoothing constant for forecasts \\ β & = & smoothing constant for trend \end{array}

$\begin{array}{l} T_{t} & = & smoothed trend for time period t \\ F_{t} & = & smoothed forecast for time period t \\ F I T_{t} & = & forecast including trend for time period t \\ α & = & smoothing constant for forecasts \\ β & = & smoothing constant for trend \end{array}$

Consider the case of Midwestern Manufacturing Company, which has a demand for electrical generators over the period 2007 to 2013 as shown in Table 5.6. To use the trend-adjusted exponential smoothing method, first set initial conditions (previous values for F and T) and choose $α$ $α$ and $β .$ $β .$ Assuming that $F_{1}$ $F_{1}$ is perfect and $T_{1}$ $T_{1}$ is 0 and picking 0.3 and 0.4 for the smoothing constants, we have

F_{1} = 74 T_{1} = 0 α = 0.3 β = 0.4

$F_{1} = 74 T_{1} = 0 α = 0.3 β = 0.4$

This results in

F I T_{1} = F_{1} + T_{1} = 74 + 0 = 74

$F I T_{1} = F_{1} + T_{1} = 74 + 0 = 74$

Following the three steps to get the forecast for 2008 (time period 2), we have

Step 1. Compute $F_{t + 1}$ $F_{t + 1}$ using the equation

$\begin{array}{l} F_{t + 1} & = & F I T_{t} + α (Y_{t} - F I T_{t}) \\ F_{2} & = & F I T_{1} + 0.3 (Y_{1} - F I T_{1}) = 74 + 0.3 (74 - 74) = 74 \end{array}$ $\begin{array}{l} F_{t + 1} & = & F I T_{t} + α (Y_{t} - F I T_{t}) \\ F_{2} & = & F I T_{1} + 0.3 (Y_{1} - F I T_{1}) = 74 + 0.3 (74 - 74) = 74 \end{array}$

Step 2. Update the trend $(T_{t + 1})$ $(T_{t + 1})$ using the equation

\begin{array}{l} T_{t + 1} & = & T_{t} + β (F_{t + 1} - F I T_{t}) \\ T_{2} & = & T_{1} + 0.4 (F_{2} - F I T_{1}) = 0 + 0.4 (74 - 74) = 0 \end{array}

$\begin{array}{l} T_{t + 1} & = & T_{t} + β (F_{t + 1} - F I T_{t}) \\ T_{2} & = & T_{1} + 0.4 (F_{2} - F I T_{1}) = 0 + 0.4 (74 - 74) = 0 \end{array}$

Table 5.6 Midwestern Manufacturing’s Demand

YEAR	ELECTRICAL GENERATORS SOLD
2007	74
2008	79
2009	80
2010	90
2011	105
2012	142
2013	122

Table 5.7 Midwestern Manufacturing Exponential Smoothing with Trend Forecasts

TIME (t)	DEMAND (Y_t)	$F_{t + 1} = F I T_{t} + 0.3 (Y_{t} - F I T_{t})$ $F_{t + 1} = F I T_{t} + 0.3 (Y_{t} - F I T_{t})$	$T_{t + 1} = T_{t + 1} + 0.4 (F_{t + 1} - F I T_{t})$ $T_{t + 1} = T_{t + 1} + 0.4 (F_{t + 1} - F I T_{t})$	$F I T_{t + 1} = F_{t + 1} + T_{t + 1}$ $F I T_{t + 1} = F_{t + 1} + T_{t + 1}$
1	74	74	0	74
2	79	$74 = 74 + 0.3 (74 - 74)$ $74 = 74 + 0.3 (74 - 74)$	$0 = 0 + 0.4 (74 - 74)$ $0 = 0 + 0.4 (74 - 74)$	$74 = 74 + 0$ $74 = 74 + 0$
3	80	$75.5 = 74 + 0.3 (79 - 74)$ $75.5 = 74 + 0.3 (79 - 74)$	$0.6 = 0 + 0.4 (75.5 - 74)$ $0.6 = 0 + 0.4 (75.5 - 74)$	$76.1 = 75.5 + 0.6$ $76.1 = 75.5 + 0.6$
4	90	77.270	1.068	$78.338 = 77.270 + 1.068$ $78.338 = 77.270 + 1.068$
		$= 76.1 + 0.3 (80 - 76.1)$ $= 76.1 + 0.3 (80 - 76.1)$	$= 0.6 + 0.4 (77.27 - 76.1)$ $= 0.6 + 0.4 (77.27 - 76.1)$
5	105	81.837	2.468	$84.305 = 81.837 + 2.468$ $84.305 = 81.837 + 2.468$
		$= 78.338 + 0.3 (90 - 78.338)$ $= 78.338 + 0.3 (90 - 78.338)$	$= 1.068 + 0.4 (81.837 - 78.338)$ $= 1.068 + 0.4 (81.837 - 78.338)$
6	142	90.514	4.952	$95.466 = 90.514 + 4.952$ $95.466 = 90.514 + 4.952$
		$= 84.305 + 0.3 (105 - 84.305)$ $= 84.305 + 0.3 (105 - 84.305)$	$= 2.468 + 0.4 (90.514 - 84.305)$ $= 2.468 + 0.4 (90.514 - 84.305)$
7	122	109.426	10.536	$119.962 = 109.426 + 10.536$ $119.962 = 109.426 + 10.536$
		$= 95.466 + 0.3 (142 - 95.466)$ $= 95.466 + 0.3 (142 - 95.466)$	$= 4.952 + 0.4 (109.426 - 95.466)$ $= 4.952 + 0.4 (109.426 - 95.466)$
8		120.573	10.780	$131.353 = 120.573 + 10.780$ $131.353 = 120.573 + 10.780$
		$= 119.962 + 0.3 (122 - 119.962)$ $= 119.962 + 0.3 (122 - 119.962)$	$= 10.536 + 0.4 (120.573 - 119.962)$ $= 10.536 + 0.4 (120.573 - 119.962)$

Step 3. Calculate the trend-adjusted exponential smoothing forecast $(F I T_{t + 1})$ $(F I T_{t + 1})$ using the equation

$F I T_{2} = F_{2} + T_{2} = 74 + 0 = 74$ $F I T_{2} = F_{2} + T_{2} = 74 + 0 = 74$

For 2009 (time period 3), we have

Step 1.

$F_{3} = F I T_{2} + 0.3 (Y_{2} - F I T_{2}) = 74 + 0.3 (79 - 74) = 75.5$ $F_{3} = F I T_{2} + 0.3 (Y_{2} - F I T_{2}) = 74 + 0.3 (79 - 74) = 75.5$
Step 2.

$T_{3} = T_{2} + 0.4 (F_{3} - F I T_{2}) = 0 + 0.4 (75.5 - 74) = 0.6$ $T_{3} = T_{2} + 0.4 (F_{3} - F I T_{2}) = 0 + 0.4 (75.5 - 74) = 0.6$
Step 3.

$F I T_{3} = F_{3} + T_{3} = 75.5 + 0.6 = 76.1$ $F I T_{3} = F_{3} + T_{3} = 75.5 + 0.6 = 76.1$

The other results are shown in Table 5.7. The forecast for 2014 would be about 131.35.

To have Excel QM perform the calculations in Excel 2016, from the Excel QM ribbon, select the alphabetical list of techniques and choose Forecasting and then Trend Adjusted Exponential Smoothing. After specifying the number of past observations, enter the data and the values for $α$ $α$ and $β$ $β$ , as shown in Program 5.3.

A screenshot of Excel QM output is titled Forecasting Trend adjusted exponential smoothing. Underneath the word Forecasting an Alpha and Beta table is shown, and a tip box directs you to the forecast for next period output. — Program 5.3 Output from Excel QM in Excel 2016 for Trend-Adjusted Exponential Smoothing Example

Figure 5.3 Full Alternative Text

Trend Projections

Another method for forecasting time series with trend is called trend projection. This technique fits a trend line to a series of historical data points and then projects the line into the future for medium- to long-range forecasts. There are several mathematical trend equations that can be developed (e.g., exponential and quadratic), but in this section we look at linear (straight line) trends only. A trend line is simply a linear regression equation in which the independent variable (X) is the time period. The first time period will be time period 1. The second time period will be time period 2, and so forth. The last time period will be time period n. The form of this is

\hat{Y} = b_{0} + b_{1} X

$\hat{Y} = b_{0} + b_{1} X$

where

\begin{array}{l} \hat{Y} & = & predicted value \\ b_{0} & = & intercept \\ b_{1} & = & slope of the line \\ X & = & time period (i .e ., X = 1, 2, 3, \dots, n) \end{array}

$\begin{array}{l} \hat{Y} & = & predicted value \\ b_{0} & = & intercept \\ b_{1} & = & slope of the line \\ X & = & time period (i .e ., X = 1, 2, 3, \dots, n) \end{array}$

The least squares regression method may be applied to find the coefficients that minimize the sum of the squared errors, thereby also minimizing the mean squared error (MSE). Chapter 4 provides a detailed explanation of least squares regression and the formulas to calculate the coefficients by hand. In this section, we use computer software to perform the calculations.

Let us consider the case of Midwestern Manufacturing’s demand for generators that was presented in Table 5.6. A trend line can be used to predict demand (Y) based on the time period using a regression equation. For the first time period, which was 2007, we let $X = 1.$ $X = 1.$ For 2008, we let $X = 2,$ $X = 2,$ and so forth. Using computer software to develop a regression equation as we did in Chapter 4, we get the following equation:

\hat{Y} = 56.71 + 10.54 X

$\hat{Y} = 56.71 + 10.54 X$

To project demand in 2014, we first denote the year 2014 in our new coding system as $X = 8 :$ $X = 8 :$

\begin{array}{l} (Sales in 2014) & = & 56.71 + 10.54 (8) \\ = & 141.03, or 141 generators \end{array}

$\begin{array}{l} (Sales in 2014) & = & 56.71 + 10.54 (8) \\ = & 141.03, or 141 generators \end{array}$

We can estimate demand for 2015 by inserting $X = 9$ $X = 9$ in the same equation:

\begin{array}{l} (Sales in 2015) & = & 56.71 + 10.54 (9) \\ = & 151.57, or 152 generators \end{array}

$\begin{array}{l} (Sales in 2015) & = & 56.71 + 10.54 (9) \\ = & 151.57, or 152 generators \end{array}$

Program 5.4 provides output from Excel QM in Excel 2016. To run this model, from the Excel QM ribbon, select Alphabetical to see the techniques. Then select Forecasting and Regression / Trend Analysis. When the input window opens, enter the number of past observations (7), and the spreadsheet is initialized, allowing you to input the seven Y values and the seven X values, as shown in Program 5.4.

This problem could also be solved using QM for Windows. To do this, select the Forecasting module, and then enter a new problem by selecting New – Time Series Analysis. When the next window opens, enter the number of observations (7) and press OK. Enter the values (Y) for the seven past observations when the input window opens. It is not necessary to enter the values for X (1, 2, 3, . . . , 7) because QM for Windows will automatically use these numbers. Then click Solve to see the results shown in Program 5.5.

Figure 5.4 provides a scatter diagram and a trend line for these data. The projected demand in each of the next three years is also shown on the trend line.

The title of this screenshot from QM for Windows is Forecasting Simple linear regression.

Figure 5.4 Full Alternative Text

A screenshot showing trend line output for Midwestern Manufacturing Company. Tip boxes point out the forecast for future time periods and the location of the trend line formula in the output table. — Program 5.5 Output from QM for Windows for Trend Line Example

Figure 5.5 Full Alternative Text

A line graph and scatter diagram. — Figure 5.4 Generator Demand and Projections for Next Three Years Based on Trend Line

Figure 5.4 Full Alternative Text

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Table of Contents for
5.5 Forecasting Models—Trend and Random VariationsForecasting Models—Trend and Random Variations

5.5 Forecasting Models—Trend and Random Variations

Exponential Smoothing with Trend

Table 5.6 Midwestern Manufacturing’s Demand

Table 5.7 Midwestern Manufacturing Exponential Smoothing with Trend Forecasts

Program 5.3 Output from Excel QM in Excel 2016 for Trend-Adjusted Exponential Smoothing Example

Trend Projections

Program 5.4 Output from Excel QM in Excel 2016 for Trend Line Example

Program 5.5 Output from QM for Windows for Trend Line Example

Figure 5.4 Generator Demand and Projections for Next Three Years Based on Trend Line

Table of Contents for 5.5 Forecasting Models&#8212;Trend and Random VariationsForecasting Models—Trend and Random Variations

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Table of Contents for
5.5 Forecasting Models—Trend and Random VariationsForecasting Models—Trend and Random Variations