15 yaş üstü istihdam ( )

Transcription

1 Forecasting

2 1-2 Forecasting yaş üstü istihdam ( ) What can we say about this data? - Can you guess the employement level for July 2013?

3 1-3 Production planning hierarchy in simple form

4 1-4 Characteristics of forecasting Forecast are almost always wrong! The target is to reduce the forecast error. A good forecast always come with an information on forecast error. (e.g. variance of the forecast error or other measure of forecast accuracy) Aggregate forecasts have less relative error; E.g. Forecast for the demand of a product family has less relative error (% error) than the forecast for a product. Or forecast for a monthly demand is more stable than forecast for a daily demand The further into the future forecast is, the more error it will have. Forecast should not ignore a known information. Quantitative and qualitative method should be combined in forecasting

5 Forecasting Qualitative methods (specially for new products where there is no past sale data, or when past does not represent future) -Customer surveys -Expectations of sales personel -Opinions of managers/experts -Delphi method Quantitative methods (We do have data and past represents future) -Time series methods -Future demand is a function of past demand -Regression (causal) methods -Future demand is a function of another variable The two approaches above are not alternatives to each other, they should be used together.

6 1-6 Demand Forecast Applications Time Horizon Application Short Term Medium Term Long Term (0 3 months) (3 months 2 years) (more than 2 years) Forecasted Items Individual products or services Total sales Groups or families of products or services Total sales Decision Area Inventory management Final assembly scheduling Workforce scheduling Master production scheduling Staff planning Production planning Master production scheduling Purchasing Distribution Facility location Capacity planning Process management Forecasting Technique Time series Causal Judgment Causal Judgment Causal Judgment 2007 Pearson Education

7 1-7 Forecasting process 1. Determine the purpose of the forecast and the type of data needed 6. Cheking the validity of the forecasting model using forecasting error measures as it is used 2. Collection of the past data 5. Validate the forecasting model using past data 3. Plotting the data and determinin g the genera l structure of the data 4. Choosing the proper forecasting model for the data 7. Valid forecasting model? No 8b. Choose a new forecasting model or adjust the parameters of the current model Yes 8a Find the forecasts for the planning horizon 9. Adjust the forecasts using the available qualitative info. 10. Analysing the forecasts and the forecasting erros.

8 Some data structures 1-8

9 1-9 Data structures Demand = Base level + trend + seasonal effect + cyclical effect + random component Stationary data ; Demand = Base level + random component X t = a + e t Data with trend; Demand = Base level + trend + random component X t = a + b.t + e t

10 1-10 Data structures Data with trend and seasonality; Demand = Base level + trend + seasonal effect +random component X t = (a + b.t).c t + e t multiplicative model X t = (a + b.t)+c t + e t additive model Seasonal and stationary data; Demand = Base level +seasonal effect +random component X t = a.c t + e t e t ; i.i.d. Random component, Normally distributed, with mean=0 Forecasting model = Estimating the main trend= Formulating red parts

11 Qualitative forecasting methods Expectations/forecast of the Sales personnel Collect and aggregate expectations from sales personnel from each region or for each product. If the performance of the personnel is based on fulfilling sales quotas, they may give biased expectations. Customer surveys Surveys and exampling plans must be carefully designed.

12 Qualitative forecasting methods Opinions of Managers/Experts Used for especially for new products with no previous demand data. Delphi Method Prevents group dynamics to bias the forecasts constructed by a group of people. Collect the opinions of experts (forecasts) individually. Aggregate/summarize the result and send these results back to experts. Ask them to reconsider their forecast. Keep this cycle until a concensus is reached.

13 2. Quantitative forecasting methods; (1) Regression models 1-13 The variable to be forecasted (dependent variable) is formulated as a function of some other (independent) variables. Let Y be the variable to be forecasted and (X 1, X 2,..., X n ), be the variables to be used in forecasting Y The regression model (causal model) is ; Y = f (X 1, X 2,..., X n ). Typically a linear function is used; Y = a 0 + a 1 X a n X n. Parameters (a s) are found using data analysis.

14 Quantitative forecasting methods; (1) Regression models Example; Forecasting the income for a real estate agent We know that the amount of houses sold depends directly to mortgage interest rates. A simple model for forecasting is Y t : income in year t X t : Mortgage interest rate in year t Y t = a 0 + a 1 X t Parameters a0 and a1 are found using past data. In determining a0 and a1, we minimize the sum of the squared errors.

15 Quantitative forecasting methods; (2) Time series methods Time series; is the past values of the variable to be forecasted. Time series are the data indexed by time, usually with equal intervals (day, week, quarter etc.) Assumption is that the past data can represent the future. The purpose is to model the main trend in the data and use the main trend in forecasting.

16 1-16 Quantitative forecasting methods we will cover A. Stationary data Moving Average, Simple exponential smoothing B. Data with trend Regression, Double exponential smoothing (Holt s method) C. Stationary data with seasonality D. Data with trend and seasonality Thriple exponential smooting (Winter s method)

17 1-17 Notation Let D 1, D 2,... D n,... be the past values of the series to be predicted (demand). If we are making a forecast in period t, assume we have observed D t, D t-1 etc. Let F t, t + k = forecast made in period t for the demand in period t +k where k = 1, 2, 3, Then F t -1, t is the forecast made in t-1 for t and F t, t+1 is the forecast made in t for t+1. (one step ahead) Use shorthand notation F t = F t - 1, t.

18 1-18 Evaluation of Forecasts The forecast error in period t, e t, is the difference between the forecast for demand in period t and the actual value of demand in t. For a multiple step ahead forecast: e t = F t - k, t - D t. For one step ahead forecast: e t = F t - D t. Let e 1, e 2, e n be a consecutive series of forecast errors.

19 1-19 Biases in Forecasts A bias occurs when the average value of a forecast error tends to be positive or negative. Erros should be randomly distributed when plotted. Mathematically an unbiased forecast is one in which E (e i ) = 0. See Figure 2.3

20 1-20 Forecast Errors Over Time Figure 2.3

21 1-21 Evaluating forecast error e 1,e 2,e 3,e n be the forecast errors for the last n forecasts. The three error measures are; 1. Mean absolute deviation 1 MAD n 2. Mean Squared Error 1 MSE n i1 3. Mean absolute percent error 1 MAPE n n n i1 n i1 2 e i e i ei D i 100

22 1-22 Example 2.1 Artel, SRAM producer, has plants in a few locations. Managers of two plants are asked to forecast process yield rate for the next week as a percentage Based on the forecasts given on the table on the next slide, which manager has better forecasts?

23 1-23 Example 2.1 Hafta F1 D1 e1 e1 2 e1 e1/ D1 F2 D2 e2 e2 2 e2 e2/ D MAD1 = 17/6=2.83 MAD2 = 18/6=3.00 MSE1=79/6=13.17 MSE2=70/6=11.67 MAPE1= MAPE2= Which manager has better forecasts?

24 1-24 A. Forecasting for Stationary Series A stationary time series has the form: D t = m + e t where m is a constant and e t is a random variable with mean 0 and var s 2. Two common methods for forecasting stationary series are moving averages and exponential smoothing.

25 1-25 Moving Averages In words: the arithmetic average of the n most recent observations. For a onestep-ahead forecast: F t = (1/N) (D t D t D t - N ) Such a forecast is called N-Step Moving Average Forecast

26 1-26 Example 2.2 Engine breakdown data for the last 8 quarters Quarter Engine Fixes Engine Fixes

27 1-27 Example 2.2 Moving Average Çeyrek (t) Motor bozulma sayısı Forecast MA(3) Error F4= F5= F6= Tahmin Forecast MA(6) Error F7= F8= F3,15=F3,4=F4= 208

28 1-28 Moving Average In MA, one step ahead and multi-step ahead forecasts are the same (stationary time series) The only parameter to be decided is the number of periods in the moving average. The average age of the data used in forecasting (N(N+1)/2)/N= (N+1)/2

29 1-29 Summary of Moving Averages Advantages of Moving Average Method Easily understood Easily computed Provides stable forecasts Disadvantages of Moving Average Method Requires saving all past N data points Lags behind a trend Ignores complex relationships in data and gives equal weights to all past data.

30 Moving Average Lags a Trend 1-30

31 1-31 Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. 1. New Forecast = a (most recent observation) F t+1 = a D t + (1 - a ) F t or equivalently + (1 - a) (last forecast) 2. New Forecast = last forecast - a (last forecast error) F t + 1 = F t - a (F t - D t ) where 0 < a < 1 and generally is small for stability of forecasts ( around.01 to.3)

32 1-32 Exponential Smoothing Method F t+1 = α D t + (1 - α ) F t = α D t + (1 - α ) (α D t-1 + (1 - α ) F t-1 ) = α D t + (1 - α ) α D t-1 + (1 - α) 2 α D t i0 a(1 a) i Dt i Applies exponentially declining weights to past data. Sum the weights is equal to 1 i ( a(1 a). 1) i0

33 1-33 Weights in Exponential Smoothing 0,2500 0,2000 0,1500 0,1000 α (1 - α ) α (1 - α) 2 α alpha= 0,2 alpha=0,05 0,0500 0,0000 t t-1 t-2 t-3 t-4 t-5 t-6 t-7 t-8 t-9 t-10 The larger the alpha is, the more weights the recent data get

34 Demand Impact of Different a 200 Actual demand a = Quarter 2011 Pearson Education, Inc. publishing as Prentice Hall a =.1

35 Demand Impact of Different a Actual demand 200 Chose high values of a when underlying average is likely to 175 change Choose low values of a when underlying 150 average 1 2 is stable Choose a that leads to small error measures Quarter 2011 Pearson Education, Inc. publishing as Prentice Hall a =.1 a =.5

36 1-36 Example 2.3 F1, initial forecast, can be set to the average of past data. We need F1 to start the ES Number of engine Quarter breakdowns Ft=α(Dt-1)+(1-α)Ft-1 ES(0,1) Forecasts et Errors F1= F2=0,1(200)+0,9(200) = F3=0,1(250)+0,9(200) = F4=0,1(175)+0,9(205) = F5=0,1(186)+0,9(202) = F6=0,1(225)+0,9(200) =

37 1-37 MA vs. ES for the engine example Numer of Quarters engines fixed error, MA(3) error, ES(0,1) MA; MAD = ( )/6=57,6 ES; MAD = ( )/6=49,2 MA; MSE = ( )/6=4215,6 ES; MSE = ( )/6=3458,4 -> ES seems to be a better model for this data -> Note that the window length of 3 in the example is too small. Window length is set to larger values generally (e.g.10-50)

38 1-38 Exponential Smoothing Forecast = Last forecast α (Last error) Error is negative -> Adjust the forecast upward. Error is positive -> Adjust the forecast downward. Average age of data is ES = 1/α If we set the average age of data in ES equal to the average age of data in MA 1/α = (N+1)/2 => α = 2/ ( N + 1). One can achieve the similar distribution of forecast error by setting α = 2/ ( N + 1). If you have found good N (or alpha) value, you can also try the corresponding alpha (or N).

39 1-39 Comparison of ES and MA Similarities Both methods are appropriate for stationary series Both methods depend on a single parameter Both methods lag behind a trend Differences ES carries all past history. MA eliminates bad data after N periods into the past, is this a good ides? ES allows us to give more weights to more recent data MA requires all N past data points while ES only requires last forecast and last observation.

40 1-40 B. Forecasting with Trend Data shows an increasing or decreasing tendency Both MA and ES lags the trend and should not be used The methods we will use Regression Double exponential smoothing (Holt s method)

41 1-41 B. Forecasting with Trend: Regression model Fitting a trend line to historical data points to project into the medium to long-range Linear trends can be found using the least squares technique ^ ^ y = a + bx where y = computed value of the variable to be predicted (dependent variable) a = y-axis intercept b = slope of the regression line x = the independent variable 2011 Pearson Education, Inc. publishing as Prentice Hall

42 Values of Dependent Variable 1-42 Least Squares Method Actual observation (y-value) Deviation 7 Deviation 5 Deviation 6 Deviation 3 Deviation 4 Deviation 1 (error) Deviation 2 ^ Trend line, y = a + bx Time period Figure 4.4

43 1-43 Least Squares Method Equations to calculate the regression variables ^ y = a + bx Sxy - nxy b = Sx 2 nx 2 Slope a = y - bx Intercept Each data point has an equal weight in determining a and b. After each observed data, a and b must be updated Pearson Education, Inc. publishing as Prentice Hall

44 Linear regression example Time Electrical Power Year Period (x) Demand x 2 xy x = 28 y = 692 x 2 = 140 xy = 3,063 x = 4 y = xy - nxy 3,063 - (7)(4)(98.86) b = = = x 2 - nx (7)(4 2 ) a = y - bx = (4) = Pearson Education, Inc. publishing as Prentice Hall

45 Linear regression example Time Electrical Power Year Period (x) Demand x 2 xy The trend 3 line is ^ 2007 y 5 = x x = 28 y = 692 x 2 = 140 xy = 3,063 x = 4 y = xy - nxy 3,063 - (7)(4)(98.86) b = = = x 2 - nx (7)(4 2 ) a = y - bx = (4) = Pearson Education, Inc. publishing as Prentice Hall

46 Power demand 1-46 Linear regression example Trend line, ^ 160 y = x Year 2011 Pearson Education, Inc. publishing as Prentice Hall

47 1-47 Linear regression another example Nodel Construction company renovates old homes in West Bloomfield, Michigan. Companies annual dollar volume of renovations seems to be dependent on the volume of Area payroll. Data for the last six years is available Company wants a mathematical model to predict its renovation volume.

48 Sales 1-48 Linear regression another example Sales Area Payroll ($ millions), y ($ billions), x Area payroll

49 1-49 Linear regression another example Sales, y Payroll, x x 2 xy y = 15.0 x = 18 x 2 = 80 xy = 51.5 x = x/6 = 18/6 = 3 y = y/6 = 15/6 = 2.5 xy - nxy b = = =.25 x 2 - nx (6)(3)(2.5) 80 - (6)(3 2 ) a = y - bx = (.25)(3) = 1.75

50 1-50 Correlation r = nsxy - SxSy [nsx 2 - (Sx) 2 ][nsy 2 - (Sy) 2 ] Shows how strong the linear relationship is between the variables Correlation does not necessarily imply direct causality Coefficient of correlation, r, measures degree of association Values range from -1 to +1

51 Correlation Coefficient y y y (a) Perfect positive correlation: r = +1 x (b) Positive correlation: 0 < r < 1 y x (c) No correlation: r = 0 x x (d) Perfect negative correlation: r = -1

52 1-52 Coefficient of Determination Coefficient of Determination=square of corelation, r 2, measures the percent of change in y predicted by the change in x Values range from 0 to 1 Easy to interpret For the Nodel Construction example: r =.901 r 2 =.81

53 1-53 B. Forecasting with trend: Double exponential smoothing (Holt s method) Instead of giving equal weights to all the data in fiding the regression line, lets give more weights to more recent data. Slope and intercept are updated with every data observed using the idea of exponential smoothing. Two parameters for smoothing slope and intercept ;(α,β) Two equations for updating slope and intercept Allows the ability to give more weight to recent data.

54 1-54 B. Forecasting with trend: Double exponential smoothing (Holt s method) 1, 0 slopein period t in period t intercept ) (1 ) ( ) )( ( a a a a t t t t t t t t t t G S G S S G G S D S Last observation forecast

55 Forecasting with trend: Double exponential smoothing (Holt s method) 1-55 Forecasting model at t Dt St Ft Gt-1 St-1 Forecasting model at t-1 t-1 t

56 1-56 Initial values in Holt s method For So, use Do (Last period s observed value) For Go two choises; Go = D0 D-1 (difference of the last two data) Go = (D0 D-n)/ n (Average difference using all past data) Alternatively; Fit a linear regression line to past data and take the intercept and the slope from the linear regression line. Note; So is the last point of the regression line fitted

57 1-57 Holt s method Forecasting for period in period t F t, t S t t G Compared to regression, it allows us to give more weights to recent data in a flexible way. Updating intercept and slope in holts method is much easier than the regression model. Last intercept and slope are the only values needed. t Example 2.5 (following slides)

58 Example 2.5 Double exponential smoothing (Holt s method) 1-58 Quarter Engine Breakdowns t

59 1-59 Example 2.5 Double exponential smoothing (Holt s method) Assume that we are at the end of quarter 4 (t=0) We will use α=0,1 and β=0,1 The initial intercept, The initial slope at t=0, (S 0, G 0 ) Set S 0 =D 0 S 0 =186 Set G 0 =observed slope in the last n data points; G 0 =( D 0 - D -n )/n G 0 =( D 0 - D -3 )/3= ( )/3=-4.67

60 Example 2.5 Double exponential smoothing (Holt s method) 1-60 The forecasting model at t=0 and two forecasts; F 0,0+τ = S 0 + τ G 0 F 0,0+1 = F 1 = S 0 + (1) G 0 = 186+(-4.67) = F 0,0+5 = S 0 + (5) G 0 = 186+5(-4.67) =

61 1-61 Example 2.5 Double exponential smoothing (Holt s method) Updating forecasting model at t=1; Assume that we have observed the data for t=1 (quarter 5) D 1 =225. What should be intercept and slope at t= 1 (S 1, G 1 ) S 1 = α(observation for intercept at t=1) + (1- α )(Forecast for intercept at t=1) S 1 = αd 1 + (1- α )F 1 = 0.1(225)+0.9(181.33) = G 1 = β (Observation for slope at t=1) + (1- β )(Forecast for slope at t=1) G 1 = β (S 1 S 0 )+ (1- β )G 0 = 0.1( ) + 0.9(- 4.67) = -4.23

62 Example 2.5 Double exponential smoothing (Holt s method) 1-62 The forecasting model at t=1 and a forecasts; F 1,1+τ = S 1 + τ G 1 F 1,1+1 = F 2 = S 1 + (1) G 1 = (-4.23) = F 1,10 = S 1 + (9) G 1 = (-4.23)

63 1-63 Example 2.5 Double exponential smoothing (Holt s method) Updating forecasting model at t=2; Assume that we have also observed the data for t=2 (quarter 5) D 2 =285. What should be intercept and slope at t=2 (S 2, G 2 ) S 2 = α(observation for intercept at t=2) + (1- α)(forecast for intercept at t=2) S 2 = αd 2 + (1- α )F 2 = 0.1(285)+0.9(181.47) = G 2 = β (Observation for slope at t=2) + (1- β )(Forecast for slope at t=2) G 2 = β (S 2 S 1 )+ (1- β )G 1 = 0.1( ) + 0.9(- 4.23) = -3.19

64 1-64 Example 2.5 Double exponential smoothing (Holt s method) Forecasts at t=2; F 3 = S 2 + (1)G 2 = (-3.19) = F 2,5 = S 2 + (5-2)G 2 = (-3.19) =

65 C. Forecasting for seasonal and stationary data N=5

66 1-66 C. Forecasting for seasonal and stationary data Seasonality is determined by the seasonal factor ci for each peiod in the season N; season length ci = 1.25 means 25% more than the average. ci = 0.5 means 50% less than the average. n i1 c i N Forecast for period i = Ci x Average of the data Forecast is the same for all period i s in the future (stationary series)

67 1-67 C. Forecasting for seasonal and stationary data 1. Find the average of all data points. 2. Find the ratio of each data point to the average. 3. Take the average of the ratios corresponding to the same period in a season. This average is the seasonal factor, Ci for the corresponding period. You will have N seasonal factors. Make sure that sum of the N seasonal factors is equal to N Forecast for period i = Ci x Average of the data

68 1-68 Example 2.6. Number of Cars passed through a toll-bridge (x1000) Monday Tuesday Wednesday Thursday Friday N=5 4 seasons

69 1-69 Example Average of all data: Find the ratio of each data to the average 3. Find the avergae ratio for each day Pazartesi Salı Çarşamba Perşembe Cuma Forecast for period i in a season = CixAvr. e.g. Forecast for Mondays= 0,98 (16425) = 16106,3 5 i1 c i 5

70 1-70 D. Forecasting with trend and seasonality Winter s method (Triple exponential smooting)

71 1-71 Forecasting with trend and seasonality yaş üstü istihdam ( )

72 1-72 Triple exponential smooting (Winter s Method) The assumed data model: D t ( S ( t i ) G ) c i i t e Random value forperiod t t Intercept in period i Slope in period i Seasonal factor for period t Since there are N periods in a season we have We have both seasonality and trend in the assumed model N i1 c i N

73 1-73 Triple exponential smooting (Winter s Method) Previous model is called as multiplicative model; seasonal factor is in multiplicative form Alternative is the additive model; D t ( S ( t i ) G ) i i c t e t We will use the multiplicative model

74 Triple exponential smooting (Winter s Method) 1-74 First we initialize the three parameters and then update them using the exponential smoothing idea. We have one equation for updating each of the three parameters; 1. Intercept (for the data without the seasonal effect) (St) 2. Slope(for the data without the seasonal effect) (Gt) 3. Seasonal factor ( ct) Remember the exponential smoothing in updating; a (last observation) + (1-a) (last forecast)

75 Triple exponential smooting (Winter s Method) 1-75 Intercept without the seasonal effect (S t ): Last observed data without the seasonal effect Last forecast without the seasonal effect

76 1-76 Winter s Method Slope (G t ): Last observed slope without the seasonal effect Last forecast of the slope without the seasonal effect

77 1-77 Winter s Method Seasonal factor (c t ): Seasonal factor for period t. Last forecast for the same seasonal factor

78 1-78 Winter s method Equations for updating parameters ; 1,, 0 ) (1 ) (1 ) ( ) )( ( ) ( a a a N t t t t t t t t t t t t t c S D c G S S G G S c D S

79 1-79 Winter s method Forecasting model 0 ) ( mod, ) ( 0 ) ( mod, ) (, ) ( mod, N N t t t t N t e t t t t t e if c G S F t e if c G S F N This formulae is simply to find the right seasonal factor for a future period

80 93Q1 93Q2 93Q3 93Q4 94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4 96Q1 96Q2 96Q3 96Q Example Quarter Demand 93Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Demand

81 1-81 Example: Initializing parameters 1. Find the deseasonalized data using centered moving average 2. Using centered moving averages, find the estimates of seasonal factor (normalize seasonal factor if necessary). 3. Deasonalize the data. 4. Fit a linear trend line to the deseasonalized data. Use the intercept and the slope of the line as the initial intercept and slope.

82 1-82 Example: Initializing parameters; Centered moving average 4 period moving average centered at period 3,5 120 Demand Q1 93Q2 93Q3 93Q4 94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4 96Q1 96Q2 96Q3 96Q4 4 period moving average centered at period 2,5

83 Example: Initializing parameters; Centered moving average Avr of 1,2,3,4 Avr of 2,3,4, MA centered at i-0,5 MA centered at i+0,5 Centered 4 period Estimates for Quarter Demand MA seasonal factor (1) (2) (3) (4)= ((2)+(3))/2 (5) = (1)/(4) 93Q Q Q , ,125 1, Q ,5 0, Q ,5 58,75 0, Q ,5 60,75 60,625 1, Q ,75 63,75 62,25 1, Q , ,375 0, Q ,25 67,125 0, Q ,25 67,5 67,375 1, Q ,5 66,25 66,875 1, Q ,25 67,25 66,75 0, Q ,25 71,25 69,25 0, Q ,25 74,5 72,875 1, Q Q

84 1-84 Example; Seasonal factors Quarter Avr. Estimate for seasonal factor Normalized estimates (1) (2)=(1)*4/3, , , , , , , , ,753 Sum 3, Avr. Seasonal factors for Quarter1 (periods 5, 9, 13)

85 1-85 Example;Demand and desasonalized demand Quarter Demand Deseasonalized data (1) (2)=(1)/seasonal factor 93Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q ,01236

86 93Q1 93Q2 93Q3 93Q4 94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4 96Q1 96Q2 96Q3 96Q Example;Demand and desasonalized demand Demand Deseasonalized data 20 0

87 1-87 Example; If we fit a linear line to the deseasonalized data; Intercept a=49,94 and slope b=1,683 The value of the linear line at the last quarter (period 16) is the initial intercept, So So=49,94+1,683(16)=76,86 Initial slope; Go=1,683

88 1-88 Example; So Demand Deseasonalized data linear line Q1 93Q2 93Q3 93Q4 94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4 96Q1 96Q2 96Q3 96Q4

89 1-89 Example; forecasting So=76,86, Go=1,683 Fo,1= (76,86+1,683(1))0,869 = 68,22 Fo,2= (76,86+1,683(2))1,061 = 85,161 Fo,3= (76,86+1,683(3))1,317 = 107,87 Fo,4= (76,86+1,683(4))0,753 = 62,94 Check out the file; winters_method.xlsx

90 1-90 Example; Updating parameters in winter s method

91 1-91 Example; Updating parameters in winter s method

92 1-92 Example; Updating parameters in winter s method We repeat the update procedure after observing each data point. We update level, trend and one of the seasonal factors

93 1-93 Example; Forecasting in Winter s method

94 1-94 Relevant excel functions eğim(x seti;y seti) (slope) kesmenoktası(x seti;y seti) (intercept) tahmin(tahmin için x; X seti; Y seti) (forecast) Regresyon modeli için; Veri>veri çözümleme>regresyon Check out Example excel file linear_trend_tracking_example.xls

95 1-95 Monitoring Forecasts Tracking signal Measures how well the forecast is predicting actual values. Tracking signal t = t i1 ( F i MAD D t i ) Et MAD t Commonly used limits +/-3 or +/-4.

96 1-96 Monitoring Forecasts; Tracking signal Errors +3(4) Signal exceeding limit Upper control limit Tracking signal 0 MADs Acceptable range 3 (4) Lower control limit Time 2011 Pearson Education, Inc. publishing as Prentice Hall

97 1-97 Monitoring Forecasts Satistical control diagrams Error are expected to fall within +/-3xStandard deviations of the error. (with more than 99% probability), since we assume the errors follow a normal distribution. An estimate for the standard deviation of error (σ) ; sˆ MSE Another estimate for the standard deviation of error; Upper control limit (UCL) = 3 sˆ Lower control limit (LCL) = -3 sˆ sˆ 1,25MAD

98 1-98 Monitoring Forecasts; Satistical control diagrams Errors +3σ Upper control limit 0 3σ Lower control limit Time

99 1-99 Example; Tracking signal Period D t F t e t e t MAD t error Cumul. Tracking signal 1 37,00 38,00 1,00 1,00 1,00 1,00 1, ,00 37,00-3,00 3,00 2,00-2,00-1, ,00 37,90-3,10 3,10 2,37-5,10-2, ,00 38,83 1,83 1,83 2,23-3,27-1, ,00 38,28-6,72 6,72 3,13-9,99-3, ,00 40,29-9,71 9,71 4,23-19,70-4, ,00 43,20 0,20 0,20 3,65-19,50-5, ,00 43,14-3,86 3,86 3,68-23,36-6, ,00 44,30-11,70 11,70 4,57-35,06-7, ,00 47,81-4,19 4,19 4,53-39,25-8, ,00 49,06-5,94 5,94 4,66-45,19-9, ,00 50,84-3,16 3,16 4,53-48,35-10,66

100 1-100 Example; Tracking signal 3,00 2,00 1,00 0,00-1,00-2,00-3,00-4,00-5,00-6,00-7,00-8,00-9,00-10,00-11,00-12,00 Tracking signal After the 5th error we go beyond the limits, and therefore the forecasting model should be updated. Check out the excel file

101 1-101 Example; Statistical control diagram Estimation1 of σ = SQRT(MSE) Estimation2 of σ =1,25MAD UCL= UCL= Period D F e e e^2 MSE MAD 3xσ -3xσ 1,00 37,00 38,00 1,00 1,00 1,00 1,00 1,00 1,00 1,25 3,75-3,75 2,00 40,00 37,00-3,00 3,00 9,00 5,00 2,00 2,24 2,50 7,50-7,50 3,00 41,00 37,90-3,10 3,10 9,61 6,54 2,37 2,56 2,96 8,88-8,88 4,00 37,00 38,83 1,83 1,83 3,35 5,74 2,23 2,40 2,79 8,37-8,37 5,00 45,00 38,28-6,72 6,72 45,16 13,62 3,13 3,69 3,91 11,74-11,74 6,00 50,00 40,29-9,71 9,71 94,28 27,07 4,23 5,20 5,28 15,85-15,85 7,00 43,00 43,20 0,20 0,20 0,04 23,21 3,65 4,82 4,56 13,69-13,69 8,00 47,00 43,14-3,86 3,86 14,90 22,17 3,68 4,71 4,60 13,79-13,79 9,00 56,00 44,30-11,70 11,70 136,89 34,91 4,57 5,91 5,71 17,13-17,13 10,00 52,00 47,81-4,19 4,19 17,56 33,18 4,53 5,76 5,66 16,99-16,99 11,00 55,00 49,06-5,94 5,94 35,28 33,37 4,66 5,78 5,82 17,47-17,47 12,00 54,00 50,84-3,16 3,16 9,99 31,42 4,53 5,61 5,67 17,00-17,00

102 1-102 Example; Statistical control diagram 20,00 Statistical control chart 15,00 10,00 5,00 0,00-5,00-10,00-15,00-20, UCL LCL Error Check out the excel file

103 1-103 Simple vs. complicated methods More soficticated methods does not necessarily yield better forecasts