Time-Series ForecastingTime-Series Forecasting
Learning Objectives
1.Describe What Forecasting Is
2. Forecasting Methods
3.Explain Time Series & Components
4.Smooth a Data Series
5.Forecast Using Smoothing Methods, & Trend
6.Use MAD to Measure Forecast Error
What Is Forecasting?
1.Process of Predicting a Future Event
2.Underlying Basis of All Business PlanningProduction
Inventory
Personnel
Facilities
Sales will be $200 Million!
Forecasting Methods
1.Qualitative MethodsExpert Opinion
Delphi Method
Surveys
2.Quantitative MethodsTime Series
CausalRegression
Quantitative Forecasting Steps
1.Select Several Forecasting Methods
2.‘Forecast’ the Past
3.Evaluate Forecasts
4.Select Best Method
5.Forecast the Future
6.Monitor Continuously Forecast Accuracy
What’s a Time Series?
1. Set of Numerical Data
2. Obtained by Observing Response Variable at Regular Time Periods
3. Assumes that Factors Influencing Past & Present Will Continue
4. ExampleYear: 1990 1991 1992 19931994
Sales: 78.7 63.5 89.7 93.292.1
Time Series Components
Trend
Seasonal
Cyclical
Irregular
Trend Component
1. Persistent, Overall Upward or Downward Pattern
2. Due to Population, Technology, etc.
3. 15 to 20 Years Duration
Mo., Qtr., Yr.
Response
Linear Increasing Trend Linear Decreasing Trend
Nonlinear Trend No Trend
Examples of Some Time Series Trend Patterns
Toaster Sales in Hundreds, By Quarter, 1990-1998
TIME QUARTER1 QUARTER2 QUARTER3 QUARTER4 ------------------------------------------- 1990 187 243 209 291 1991 198 263 270 297 1992 274 363 294 336 1993 232 273 241 289 1994 206 295 239 317 1995 237 366 300 429 1996 282 424 383 478 1997 375 429 393 560 1998 373 423 387 433
Long-Term Trend in Toaster Sales
YEAR
199119901989198819871986198519841983
Mea
n S
ALE
S
600
500
400
300
200
100
Cyclical Component
1. Repeating Up & Down Movements
2. Due to Interactions of Factors Influencing Economy
3. Usually 2-15 Years Duration
Mo., Qtr., Yr.
ResponseCycle Prosperity
Recession Depression Recovery
Cycles in Toaster Sales
YEAR
199119901989198819871986198519841983
SA
LES
600
500
400
300
200
100
Seasonal Component
1. Regular Pattern of Up & Down Fluctuations
2. Due to Weather, Customs,etc.
3. Occurs Within 1 Year
Mo., Qtr.
Response
Summer
The Seasonal Pattern of Toaster Sales
YEAR
1983
1985
1987
1989
19914321
SA
LES
500
400
300
200
100
QUARTER
Irregular Component
1. Erratic, Unsystematic, ‘Residual’ Fluctuations
2. Due to Random Variation or Unforeseen Events
Union Strike Tornado
3. Short Duration & Nonrepeating
Irregular Fluctuations in Toaster Sales
Quarters
Multiplicative Time-Series Model
1. Any Observed Value in a Time Series Is the Product of Time Series Components
2. If Annual Data Y = T x C x I
3. If Quarterly or Monthly Data Yi = T x S x C x I
Time Series Forecasting
Linear
Time SeriesForecasting
Trend?Smoothing
Methods
TrendModels
YesNo
ExponentialSmoothing
Quadratic ExponentialHolt-
WintersAuto-
Regressive
MovingAverage
Moving Average Method
1. Series of Arithmetic Means
2. Used Only for SmoothingProvides Overall Impression of Data Over
Time
3. Equation
L = Averaging Period (Odd # Years)MA (L)
Y
Li
i t
(L-1)/2
T=(1-L)/2
TimeResponse
Yi
Moving Total(L=3)
MovingAvg (L=3)
1991 4 NA NA
1992 6 4 + 6 + 5 = 15 15/3 = 5.0
1993 5 6 + 5 + 3 = 14 14/3 = 4.7
1994 3 5 + 3 + 7 = 15 15/3 = 5.0
1995 7 3 + 7 + 6 = 16 16/3 = 5.3
1996 6 NA NA
Moving Average Calculation
Moving Average Graph
Year
Sales
0
2
4
6
8
91 92 93 94 95 96
Moving Average with Even Number of Periods
Average of Four-YearElectricity Four-Year Four-Year Centered
Year Purchases Moving Total Moving Totals Moving Average1974 6851975 688
1976 7542913
2975.0 743.75
1977 7863037
3114.0 778.50
1978 8093191
3221.5 805.38
1979 8423252
3272.0 818.00
divide by 4
Time Series Forecasting
Linear
Time SeriesForecasting
Trend?Smoothing
Methods
TrendModels
YesNo
ExponentialSmoothing
Quadratic ExponentialHolt-
WintersAuto-
Regressive
MovingAverage
Exponential Smoothing Method
1. Form of Weighted Moving Average Weights Decline Exponentially Most Recent Data Weighted Most
2. Used for Smoothing & Forecasting Assumes No Trend
3. Requires Smoothing Coefficient (W) Subjectively Chosen Ranges from 0 to 1
Exponential Smoothing Equations
1. Smoothing Equations
Ei = W·Yi + (1 - W)·Ei-1
2. Forecasting Equation
Yi+1 = EiEi = Smoothed
Value
Yi = Actual Value
W = Smoothing Coefficient
Time YiSmoothed Value, Ei
(W = .2)Forecast
Yi+1
1991 4 4.0 NA
1992 6 (.2)(6) + (1-.2)(4.0) = 4.4 4.0
1993 5 (.2)(5) + (1-.2)(4.4) = 4.5 4.4
1994 3 (.2)(3) + (1-.2)(4.5) = 4.2 4.5
1995 7 (.2)(7) + (1-.2)(4.2) = 4.8 4.2
1996 NA NA 4.8
Exponential Smoothing Calculation
^
Ei = W·Yi + (1 - W)·Ei-1
Exponential Smoothing Graph
Year
Sales
0
2
4
6
8
91 92 93 94 95
Exponential Smoothing Thinking Challenge
You’re an economist for GM. You want to get a feel for the long-term trend in car sales. You want to smooth cyclical & random fluctuations using exponential smoothing with W = .25. Yearly sales (million units) are 2, 4, 1, 3.
To obtain starting values: 1.E1 = Y1 = 2
2. E2 = W·Y2 + (1 - W)·E1 = (.25)(4) + (1.00 - .25)(2) = 2.5
3. E3 = W·Y3 + (1 - W)·E2
= (.25)(1) + (1.00 - .25)(2.5) = 2.125
4. E4 = W·Y4 + (1 - W)·E3
= (.25)(3)+(1.00 - .25)(2.125)=2.34
Exponential Smoothing Solution*
Selecting Smoothing Coefficient (W)
1. Subjectively Chosen Computer Search Routines Available
2. To Smooth Cyclical & Irregular, Small W Reveals Long-Term Pattern
3. To Forecast, Large W Forecast Will Reflect Prior Period Data Most
4. Recent Data Weighted Most for All W