Lecture 4 Forecasting

Transcription

1 King Saud University College of Computer & Information Sciences IS 466 Decision Support Systems Lecture 4 Forecasting Dr. Mourad YKHLEF The slides content is derived and adopted from many references Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 2

2 Definitions Forecasting is the process of predicting the future. Forecasting is an integrated part of almost all business enterprises. Examples: Manufacturing firms forecast demand for their product, to schedule manpower and raw material allocation. Service organizations forecast customer arrival patterns to maintain adequate customer service. Firms consider economic forecasts of indicators (housing starts, changes in gross national profit) before deciding on capital investments. IS Forecasting - Dr. Mourad Ykhlef 3 Definitions Good forecasts can lead to Reduced inventory costs. Lower overall personnel costs. Increased customer satisfaction. The forecasting process can be based on: Educated guess. Expert opinions. Past history of data values, known as a time series. IS Forecasting - Dr. Mourad Ykhlef 4

3 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 5 Type of Forecasts by Time Horizon Short-range forecast Up to 1 year; gnerally less than 3 months Job scheduling, worker assignements Medium-range forecast 3 months to 3 years Sales and production planning, budgeting Long-range forecast 3+ years New product planning, facility location or expansion IS Forecasting - Dr. Mourad Ykhlef 6

4 Forecasting approaches Qualtitative forecasts Used when situation is vague & litle data exist New products New technology Involve intuition, experience e.g., forecasting sales on Internet Quantitive forecasts Used when situation is stable and historical data exist Existing products Current technology Use a variety of mathematical models that rely on historical data and/or causal variables e.g., forecasting sales of color televisions IS Forecasting - Dr. Mourad Ykhlef 7 Overview of Qualitative methods Consumer Market Survey Ask the customer Sales force composite Estimates from individual sales person are reviewed for checking realistic, then aggregated Jury of executive opinion Pool opinions of high-level executives, sometines augment by statistical models Delphi method Panel of experts, queried iteratively IS Forecasting - Dr. Mourad Ykhlef 8

5 Overview of Quantitive methods Time-series models (Weighted) Moving average Exponential Smoothing Exponential Smoothing with Trend Adjustment Seasonal and Cyclical Associative models Liner regression Multiple regression Logistic regression IS Forecasting - Dr. Mourad Ykhlef 9 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 10

6 What is a Time Series? Set of evenly spaced numerical data Obtained by observing response variable at regular time periods Forecasting technique That uses a series of past data points to make a forecast Example Year Sales IS Forecasting - Dr. Mourad Ykhlef 11 Time series components Trend Cyclical Seasonal Random IS Forecasting - Dr. Mourad Ykhlef 12

7 Trend component Time series may be relatively stationary or it may exhibit trend over time Trend is the gradual upward or downward movement of data over time Trend indicates that the time series is increasing or decreasing Trend is typically modeled as a linear, quadratic or exponential function Response Mo., Qtr., Yr T/Maker Co. IS Forecasting - Dr. Mourad Ykhlef 13 Seasonal component When a repetitive pattern is observed over some time horizon, the series is said to have seasonal behavior. Response Summer Seasonality is a data pattern that repeats itself after a period of days, weeks, months or quarters. Occurs within 1 year Period of Pattern Season Length Mo., Qtr. Number of Seasons in Pattern Week Day 7 Month Week 4 4 ½ Month Day Year Quarter 4 Year Month 12 Year Week 52 IS Forecasting - Dr. Mourad Ykhlef 14

8 Cyclical component Cycles are patterns in the data that occur every several years. Usually tied into the business cycle and are of the major importance in short-term business analysis and planning. Cycles are upturn or downturn not tied to seasonal variation. Usually result from changes in economic conditions. Usually 2-10 years duration Response Cycle Mo., Qtr., Yr. IS Forecasting - Dr. Mourad Ykhlef 15 Random component Erratic, unsystematic fluctuations Due to random variations or unforeseen events Union strike Tornado Short duration and non repeating IS Forecasting - Dr. Mourad Ykhlef 16

9 Components of a Time Series Time series value Linear trend and seasonality Linear trend Future Stationary In Stationary, the mean value of the time series is assumed to be constant Time IS Forecasting - Dr. Mourad Ykhlef 17 Steps in the Time Series Forecasting The goal of a time series forecast is to identify factors that can be predicted. This is a systematic approach involving the following steps. Step 1: Data collection and Hypothesization. Collect historical data. Graph the data vs. time. Hypothesize a form for the time series model. Verify this hypothesis statistically. Step 2: Select a forecasting technique. Determination of input parameter values Performance evaluation on past data of each technique Step 3: Prepare a forecast using the selected techniques IS Forecasting - Dr. Mourad Ykhlef 18

10 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 19 Stationary Forecasting Models In a stationary model the mean value of the time series is assumed to be constant. No trend, seasonal, or cyclical components The general form of such a model is Where: y t = β 0 + ε t y t = the value of the time series at time period t. The values of e t are assumed to be independent The values of e t are assumed to have a mean of 0. β 0 = the unchanged mean value of the time series. ε t = a random error term at time period t If a time series does not have a trend, seasonlity or cyclical components it will be stationary. IS Forecasting - Dr. Mourad Ykhlef 20

11 Checking the Stationary assumption (Homework) Checking for trend Use Linear Regression if e t is normally distributed. Use a nonparametric test if e t is not normally distributed. Checking for seasonality component Autocorrelation measures the relationship between the values of the time series in different periods. Lag k autocorrelation measures the correlation between time series values which are k periods apart. Autocorrelation between successive periods indicates a possible trend. Lag 7 autocorrelation indicates one week seasonality (daily data). Lag 12 autocorrelation indicates 12-month seasonality (monthly data). Checking for Cyclical Components If a time series does not have a trend, seasonlity or cyclical components it will be stationary. IS Forecasting - Dr. Mourad Ykhlef 21 Methods for a stationary time series The Last Period Method The Moving Average Method The Weighted Moving Average Method The Exponential Smoothing Method IS Forecasting - Dr. Mourad Ykhlef 22

12 The Last Period Method The forecast for the next period is the last observed value. F = t+ 1 y t e.g., if May sales were 50 then June sales will be 50 Sometimes cost effective & efficient At least it provides a starting point which more sophisticated models that follow can be compared IS Forecasting - Dr. Mourad Ykhlef 23 The (Weighted) Moving Average Method The forecast is the average of the last n observations of the time series. = y + y y n t t 1 t n+ 1 Ft + 1 More recent values of the time series get larger weights than past values when performing the forecast. F t + 1= w 1 y t + w 2 y t-1 +w 3 y t w n y t-n+1 w 1 w 2 w n Σw i = 1 IS Forecasting - Dr. Mourad Ykhlef 24

13 Example (1/7) Galaxy Industries is interested in forecasting weekly demand for its YoHo brand yo-yos. The yo-yo is a mature product. This year demand pattern is expected to repeat next year. To forecast next year demand, the past 52 weeks demand records were collected. IS Forecasting - Dr. Mourad Ykhlef 25 Example (2 /7) Three forecasting methods were suggested: Last period technique - suggested by Ahmed. Four-period moving average - suggested by Karim. Four-period weighted moving average - suggested by Omar. Management wants to determine: If a stationary model can be used. What forecast will be obtained using each method? IS Forecasting - Dr. Mourad Ykhlef 26

14 Example (3 /7) Collection of demand records Week Week Demand Week Week Demand Week Week Demand Demand Week Week Demand Demand IS Forecasting - Dr. Mourad Ykhlef 27 Example (4 /7) Construct the time series plot Neither seasonality nor cyclical effects can be observed Demand Series1 Weeks IS Forecasting - Dr. Mourad Ykhlef 28

15 Example (5 /7) (Home work) Is the trend present? Run linear regression to test β 1 in the model y t =β 0 +β 1 t+ε t Excel results C oeff. S tand. E rr t-s tat P -value Lower 95%U pper 95% Intercept E W ee ks This large P-value indicates that there is little evidence that trend exists Conclusion: A stationary model is appropriate. IS Forecasting - Dr. Mourad Ykhlef 29 Example (6 /7) Forecast for Week 53 Last period technique (Ahmed s Forecast) $y 53 = y 52 = 484 boxes. Four-period moving average (Karim s forecast) $y 53 = (y 52 + y 51 + y 50 + y 49 ) /4 = ( ) / 4 = 401 boxes. Four period weighted moving average (Omar s forecast) $y 53 =0.4y y y y 49 = 0.4(484) + 0.3(482) + 0.2(393) + 0.1(245) = boxes. IS Forecasting - Dr. Mourad Ykhlef 30

16 Example (7 /7) Forecast for Weeks 54 and 55 Since the time series is stationary, the forecasts for weeks 54 and 55 remain as the forecast for week 53. These forecasts will be revised pending observation of the actual demand in week 53. IS Forecasting - Dr. Mourad Ykhlef 31 Drawbacks of (Weighted) Moving Average Methods Weighted Moving Average can put greater weight on the more recent observations, it uses only last n periods data values and ignores the history of the time series prior to that time. Increasing n makes forecast less sensitive to changes Do not forecast trend well. Require much historical data. Solution: Exponential Smoothing Method IS Forecasting - Dr. Mourad Ykhlef 32

17 Exponential Smoothing Method All the previous values of historical data affect the forecast. For each period create a smoothed value L t of the time series, that represents all the information known by t. The smoothed value L t is the weighted average of The actual value for the current period (with weight of α). The forecast value for the current period (with weight of 1-α). The smoothed value L t becomes the forecast for period t+1. IS Forecasting - Dr. Mourad Ykhlef 33 Exponential Smoothing Method New forecast = last period s forecast +α(last period s actual demand last period s forecast) Define: L t = smoothed value for time t F t+1 = the forecast value for time t+1 y t = the value of the time series at time t α = smoothing constant (weight) between 0 and 1 F t+ 1 = L t = αy An initial forecast is needed to start the process. t + (1 α ) F t IS Forecasting - Dr. Mourad Ykhlef 34

18 Exponential Smoothing Method Generating an initial forecast Approach 1: F = L = y Continue from t=3 with the recursive formula. Approach 2: (when large number of historical values exist) Average the initial n values of the time series. Use this average as the forecast for period n + 1 Begin using exponential smoothing from that time period onward and so on. n+ 2 = αy n+ 1 + ( 1 α ) Fn+ 1 = αy n+ 1 + (1 α F ) y n F n+1 = y n IS Forecasting - Dr. Mourad Ykhlef 35 Exponential Smoothing Method Future Forecasts Since this technique deals with stationary time series, the forecasts for future periods does not change. Assume N is the number of periods for which data are available. Then F N+1 = αy N + (1 α)f N, F N+k = F N+1, for k = 2, 3, IS Forecasting - Dr. Mourad Ykhlef 36

19 Example 1 (1/3) Period Series Forecast #N/A IS Forecasting - Dr. Mourad Ykhlef 37 Example 1 (2/3) An exponential smoothing forecast is suggested, with α = 0.1. An Initial Forecast is created at t=2 by F 2 = y 1 = 415. The recursive formula is used from period 3 onward: F 3 =.1y 2 +.9F 2 =.1(236) +.9(415) = F 4 =.1y 3 +.9F 3 =.1(348) +.9(397.10) = and so on, until period 53 is reached (N+1 = 52+1 = 53). F 53 =.1y F 52 =.1(484) +.9(382.32) = F 54 = F 55 = ( = F 53 ) IS Forecasting - Dr. Mourad Ykhlef 38

20 Example 1 (3/3) Notice the amount of smoothing Included in the smoothed series IS Forecasting - Dr. Mourad Ykhlef 39 Exponential Smoothing Method Relationship with simple moving average Simple moving average of length k ( k)/k = (k+1)/2 Exponential smoothing (1) α +(2) α (1- α)+(3)α(1- α) 2 +(4)α(1- α) 3. = 1/α The two techniques will generate forecasts having the same average age of information if k = 2 α α Exponential smoothing with α=.10 is equivalent, in some sense, to moving average based on 19 periods Exponential smoothing with α=1 is equivalent, in some sense, to moving average based on 1 period IS Forecasting - Dr. Mourad Ykhlef 40

21 Example 2 (1/6) If Drugs uses exponential smoothing to forecast sales, which value for the smoothing constant α,.1 or.8, gives better forecasts? Week Sales Week Sales IS Forecasting - Dr. Mourad Ykhlef 41 Example 2 (2/6) Exponential Smoothing: To evaluate the two smoothing constants, determine how the forecasted values would compare with the actual historical values in each case. Let: Y t = actual sales in week t F t = forecasted sales in week t F 2 = Y 1 = 110 For other weeks, F t+1 =.1Y t +.9F t IS Forecasting - Dr. Mourad Ykhlef 42

22 Example 2 (3/6) Exponential Smoothing (α =.1, 1 - α =.9) F 2 = 110 F 3 =.1Y 2 +.9F 2 =.1(115) +.9(110) = F 4 =.1Y 3 +.9F 3 =.1(125) +.9(110.5) = F 5 =.1Y 4 +.9F 4 =.1(120) +.9(111.95) = F 6 =.1Y 5 +.9F 5 =.1(125) +.9(112.76) = F 7 =.1Y 6 +.9F 6 =.1(120) +.9(113.98) = F 8 =.1Y 7 +.9F 7 =.1(130) +.9(114.58) = F 9 =.1Y 8 +.9F 8 =.1(115) +.9(116.12) = F 10 =.1Y 9 +.9F 9 =.1(110) +.9(116.01) = IS Forecasting - Dr. Mourad Ykhlef 43 Example 2 (4/6) Exponential Smoothing (α =.8, 1 - α =.2) F 2 = 110 F 3 =.8(115) +.2(110) = 114 F 4 =.8(125) +.2(114) = F 5 =.8(120) +.2(122.80) = F 6 =.8(125) +.2(120.56) = F 7 =.8(120) +.2(124.11) = F 8 =.8(130) +.2(120.82) = F 9 =.8(115) +.2(128.16) = F 10 =.8(110) +.2(117.63) = IS Forecasting - Dr. Mourad Ykhlef 44

23 Example 2 (5/6) Mean Squared Error: In order to determine which smoothing constant gives the better performance, calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10 by: [(Y 2 -F 2 ) 2 + (Y 3 -F 3 ) 2 + (Y 4 -F 4 ) (Y 10 -F 10 ) 2 ]/9 Select the forecast with the smallest error value IS Forecasting - Dr. Mourad Ykhlef 45 Mean Squared Error Example 2 (6/6) α =.1 α =.8 Week Y t F t (Y t - F t ) 2 F t (Y t - F t ) Sum Sum MSE Sum/ Sum/ IS Forecasting - Dr. Mourad Ykhlef 46

24 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods (for reading) Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 47 Performance of Forecasting Methods Calculate the value of the evaluation measure using the forecast error equation Error = Actual Forecast t = y F t t Select the forecast with the smallest error value Types of forecast error equations: Mean Squared Error MSE Mean Absolute Deviation MAD Mean Absolute Percent Error MAPE Largest Absolute Deviation LAD IS Forecasting - Dr. Mourad Ykhlef 48

25 Forecast Error Equations MSE = Σ( t ) 2 n MAPE = Σ n t n y t MAD = Σ t n LAD = max t IS Forecasting - Dr. Mourad Ykhlef 49 Example (1/5) Time Time series: Period Moving average: Error for the 3-Period MA: Period Weighted MA(.5,.3,.2) Error for the 3-Period WMA IS Forecasting - Dr. Mourad Ykhlef 50

26 Example (2/5) MSE for the moving average technique: Σ( MSE = t ) 2 = n (-20) 2 +(11.67) 2 +(23.4) 2 3 = MSE for the weighted moving average technique: Σ( MSE = t ) 2 = n (-18) 2 + (16) 2 + (29.5) 2 3 = IS Forecasting - Dr. Mourad Ykhlef 51 Example (3/5) MAD for the moving average technique: Σ t n MAD = = = MAD for the weighted moving average technique: Σ t n MAD = = = IS Forecasting - Dr. Mourad Ykhlef 52

27 Example (4/5) MAPE for the moving average technique: Σ t n MAPE= = -20 / / /115 3 =.188 MAPE for the weighted moving average technique: Σ t n MAPE= = -18 / / /115 3 =.211 IS Forecasting - Dr. Mourad Ykhlef 53 Example (5/5) LAD for the moving average technique: LAD= max t = max { -18, 16, 29.5 } = 29.5 LAD for the weighted moving average technique: LAD= max t = -20, 11.67, 23.4 = 23.4 IS Forecasting - Dr. Mourad Ykhlef 54

28 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 55 Linear Trend Time Series Two methods Expentional Smoothing with Trend Adjustment Trend Projections (Linear Regression) IS Forecasting - Dr. Mourad Ykhlef 56

29 Expentional Smootthing with Trend Adjustment F t = exponentially smoothed forecast of the data series in period t T t = exponentially smoothed trend in period t y t = times series value in period t α = smoothing constant for the average β = smoothing constant for the trend IS Forecasting - Dr. Mourad Ykhlef 57 Expentional Smootthing with Trend Adjustment Forecast including trend (FIT t ) = exponentially smoothed forecast (F t ) + exponentially smoothed trend (T t ) F t = α(value last period) + (1-α)(Forecast last period +Trend estimate Last period) F t = α y t-1 + (1 α) (F t-1 + T t-1 ) T t = β(forecast this period - Forecast last period) + (1- β)(trend estimate last period) T t = β(f t - F t-1 ) + (1- β)t t-1 IS Forecasting - Dr. Mourad Ykhlef 58

30 Example (1/4) A large Portland manufacturer uses exponential smoothing to forecast demand for a piece of pollution control equipment. It appears that an increasing trend is present. α =.2 and β =.4 F 1 =11 and T 1 =2 Month Demand ? IS Forecasting - Dr. Mourad Ykhlef 59 Forecast for month 2 Example (2/4) F 2 = α y 1 + (1 α) (F 1 + T 1 )=.2(12)+.8(11+2)=12.8 T 2 = β(f 2 - F 1 ) + (1- β)t 1 =.4( )+0.6(2)=1.92 FIT 2 = F 2 +T 2 = units Forecast for month 3 F 3 = α y 2 + (1 α) (F 2 + T 2 )=.2(17)+.8( )=15.18 T 3 = β(f 3 F 2 ) + (1- β)t 2 =.4( )+0.6(1.92)=2.10 FIT 3 = F 3 +T 3 = units IS Forecasting - Dr. Mourad Ykhlef 60

31 Example (3/4) Actual Demand Smoothed Smoothed Forecast Trend Forecast including trend IS Forecasting - Dr. Mourad Ykhlef 61 Example (4/4) Actual Demand D e m a n d Forecast including trend Smoothed forecast Smoothed trend Month IS Forecasting - Dr. Mourad Ykhlef 62

32 Trend Projections This method files a trend line to a series of historical points and then projects the line into the future for medium-to-long-range forecasts (other methods are quadratic or exponential) Values of Dependent Variable Deviation Actual observation Deviation Deviation Deviation Y ˆ = a + bx Time Deviation Deviation Deviation Point on regression line IS Forecasting - Dr. Mourad Ykhlef 63 Trend Projections Least Squares Equations (from statisticians) Equation: Ŷ = a + bx i i x Average of the values of x n number of data points Slope: b = n i = 1 n x i = 1 i x y 2 i i nx y nx 2 Y-Intercept: a = y bx IS Forecasting - Dr. Mourad Ykhlef 64

33 Example (1/4) Year Demand The demand for electrical power at N.Y.Edison over the years is given at the left. Find the overall trend. IS Forecasting - Dr. Mourad Ykhlef 65 Example (2/4) Year Time Period Power Demand x 2 xy Σx=28 Σy=692 Σx 2 =140 Σxy=3,063 IS Forecasting - Dr. Mourad Ykhlef 66

34 Example (3/4) x = Σx n = 28 7 = 4 y = Σy n = = b = Σxy - nxy 2 2 Σx nx = 3,063 (7)(4)(98. 86) (7)(4) = = a = y - bx = (4) = Demand in 2013 = (8) = megawatts Demand in 2014 = (9) = megawatts IS Forecasting - Dr. Mourad Ykhlef 67 Example (4/4) Electric Power Demand Year IS Forecasting - Dr. Mourad Ykhlef 68

35 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 69 General Time Series Models Components of a Time Series T t = Trend of the time series at time t S t = Seasonal C t = Cyclical I t = Irregular Any observed value in a time series is the product (or sum) of time series components Multiplicative model Y t = T t S t C t I t Additive model Y t = T t + S t + C t + I t IS Forecasting - Dr. Mourad Ykhlef 70

36 Additive and multiplicative models Additive model The magnitude of seasonal component is constant over time. For example the electric power consumption. Multiplicative model The magnitude of seasonal component grows in proportion to the trend of series. For example the cost of electric power consumption. IS Forecasting - Dr. Mourad Ykhlef 71 Forecasting with Trend and Seasonal Components Steps of Multiplicative Time Series Model: 1. Calculate the centered moving averages (CMAs). 2. Center the CMAs on integer-valued periods. 3. Determine the seasonal and irregular factors (S t I t ). 4. Determine the average seasonal factors. 5. Scale the seasonal factors (S t ). 6. Determine the deseasonalized data (Y t /S t ). 7. Determine a trend line of the deseasonalized data. 8. Determine the deseasonalized predictions. 9. Take into account the seasonality. IS Forecasting - Dr. Mourad Ykhlef 72

37 Example (1/11) Business at Cloths Shop can be viewed as falling into three distinct seasons: (1) season 1 (November-December); (2) season 2 (late May - mid-june); and (3) all other times. Average weekly sales (SR) during each of the three seasons during the past four years are shown on the next slide. Determine a forecast for the average weekly sales in year 5 for each of the three seasons. IS Forecasting - Dr. Mourad Ykhlef 73 Past Sales (SR) Example (2/11) Year Season IS Forecasting - Dr. Mourad Ykhlef 74

38 Example (3/11) Moving Remove trend and cyclical components Scaled Year Season Sales (Y t ) Average S t I t S t Y t /S t / = IS Forecasting - Dr. Mourad Ykhlef 75 Example (4/11) 1. Calculate the centered moving averages There are three distinct seasons in each year. Hence, take a three-season moving average to eliminate seasonal and irregular factors (smoothing). Moving Average keeps trend and cyclical factors For example: 1 st MA = ( )/3 = nd MA = ( )/3 = etc. IS Forecasting - Dr. Mourad Ykhlef 76

39 Example (5/11) 2. Center the CMAs on integer-valued periods The first moving average computed in step 1 ( ) will be centered on season 2 of year 1. Note that the moving averages from step 1 center themselves on integer-valued periods because n=3 is an odd number. Step 1 = (1+3)/2 = 2 IS Forecasting - Dr. Mourad Ykhlef 77 Example (6/11) 3. Determine the seasonal & irregular factors (S t I t ) Isolate (or remove) the trend and cyclical components, for each period t, this is given by: S t I t = Y t / (Moving Average for period t ) Y t = T t S t C t I t S t I t = Y t / T t C t S t I t = Y t / (Moving Average for period t ) Moving Average keeps Trend and Cycles and eliminates Season and Irregular factor IS Forecasting - Dr. Mourad Ykhlef 78

40 Example (7/11) 4. Determine the average seasonal factors Averaging all (S t I t ) values corresponding to that season: Season 1: ( ) /3 = Season 2: ( ) /4 = Season 3: ( ) /3 =.587 IS Forecasting - Dr. Mourad Ykhlef 79 Example (8/11) 5. Scale the seasonal factors (S t ) Average the seasonal factors = ( )/3 = Then, divide each seasonal factor by the average of the seasonal factors. Season 1: 1.180/1.002 = Season 2: 1.238/1.002 = Season 3:.587/1.002 =.586 Total = IS Forecasting - Dr. Mourad Ykhlef 80

41 Example (9/11) 6. Determine the deseasonalized data (Y t /S t ) Divide the data point values, Y t, by S t. 7. Determine a trend line of the deseasonalized data Using the least squares method (trend projection) for t = 1, 2,..., 12, gives: T t = t IS Forecasting - Dr. Mourad Ykhlef 81 Example (10/11) 8. Determine the deseasonalized predictions Substitute t = 13, 14, and 15 into the least squares equation: T13 = (33.96)(13) = 2022 T14 = (33.96)(14) = 2056 T15 = (33.96)(15) = 2090 IS Forecasting - Dr. Mourad Ykhlef 82

42 Example (11/11) 9. Take into account the seasonality. Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5: Season 1: (1.178)(2022) = 2382 Season 2: (1.236)(2056) = 2541 Season 3: (.586)(2090) = 1225 IS Forecasting - Dr. Mourad Ykhlef 83 Outline Definitions Forecasting types Time series Stationary forecasting models Performance of forecasting methods Linear trend time series Trend, Seasonal and Cyclical time series Associative forecasting IS Forecasting - Dr. Mourad Ykhlef 84

43 Associative Forecasting Regression analysis (Trend projection) Multiple regression analysis Logistic regression IS Forecasting - Dr. Mourad Ykhlef 85 Final Thought The best way to predict the future is to create it! IS Forecasting - Dr. Mourad Ykhlef 86