CLASS B.Sc.III
PAPER
APPLIED STATISTICS
Time SeriesTime Series
““The Art of Forecasting”The Art of Forecasting”
Learning ObjectivesLearning ObjectivesLearning ObjectivesLearning Objectives• Describe what forecasting is
• Explain time series & its components
• Smooth a data series– Moving average– Exponential smoothing
• Forecast using trend models Simple Linear Regression Auto-regressive
What Is Forecasting?What Is Forecasting?What Is Forecasting?What Is Forecasting?• Process of predicting a
future event• Underlying basis of
all business decisions– Production
– Inventory
– Personnel
– Facilities
• Used when situation is vague & little data exist– New products– New technology
• Involve intuition, experience
• e.g., forecasting sales on Internet
Qualitative MethodsQualitative Methods
Forecasting ApproachesForecasting ApproachesForecasting ApproachesForecasting Approaches
Quantitative MethodsQuantitative Methods
• Used when situation is ‘stable’ & historical data exist– Existing products– Current technology
• Involve mathematical techniques
• e.g., forecasting sales of color televisions
Quantitative MethodsQuantitative Methods
Forecasting ApproachesForecasting ApproachesForecasting ApproachesForecasting Approaches
• Used when situation is vague & little data exist– New products– New technology
• Involve intuition, experience
• e.g., forecasting sales on Internet
Qualitative MethodsQualitative Methods
Quantitative ForecastingQuantitative Forecasting Quantitative ForecastingQuantitative Forecasting
• Select several forecasting methods
• ‘Forecast’ the past
• Evaluate forecasts
• Select best method
• Forecast the future
• Monitor continuously forecast accuracy
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Time SeriesModels
CausalModels
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Time SeriesModels
CausalModels
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Time SeriesModels
ExponentialSmoothing
TrendModels
MovingAverage
CausalModels
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Time SeriesModels
RegressionExponentialSmoothing
TrendModels
MovingAverage
CausalModels
Quantitative Forecasting Methods
Quantitative Forecasting Methods
QuantitativeForecasting
Time SeriesModels
RegressionExponentialSmoothing
TrendModels
MovingAverage
What is a Time Series?What is a Time Series?What is a Time Series?What is a Time Series?• Set of evenly spaced numerical data
– Obtained by observing response variable at regular time periods
• Forecast based only on past values– Assumes that factors influencing past, present, &
future will continue
• Example– Year: 1995 1996 1997 1998 1999– Sales: 78.7 63.5 89.7 93.2 92.1
Time Series vs. Time Series vs. Cross Sectional DataCross Sectional Data
Time series data is a sequence of observations
– collected from a collected from a process process
– with with equally spacedequally spaced periods of time periods of time.
Time Series vs. Time Series vs. Cross Sectional DataCross Sectional Data
Contrary to restrictions placed on cross-sectional data, the major purpose of forecasting with time series is to extrapolate beyond the range of the explanatory variables.
Time Series vs. Time Series vs. Cross Sectional DataCross Sectional Data
Time series is dynamic, it does change over time.
Time Series vs. Time Series vs. Cross Sectional DataCross Sectional Data
When working with time series data, it is paramount that the data is plotted so the researcher can view the data.
Time Series ComponentsTime Series Components
Time Series ComponentsTime Series Components
TrendTrend
Time Series ComponentsTime Series Components
TrendTrend CyclicalCyclical
Time Series ComponentsTime Series Components
TrendTrend
SeasonalSeasonal
CyclicalCyclical
Time Series ComponentsTime Series Components
TrendTrend
SeasonalSeasonal
CyclicalCyclical
IrregularIrregular
Trend ComponentTrend Component• Persistent, overall upward or downward
pattern
• Due to population, technology etc.
• Several years duration
Mo., Qtr., Yr.Mo., Qtr., Yr.
ResponseResponse
© 1984-1994 T/Maker Co.
Trend ComponentTrend Component
• Overall Upward or Downward Movement
• Data Taken Over a Period of Years
Sales
Time
Upward trend
Cyclical ComponentCyclical Component• Repeating up & down movements
• Due to interactions of factors influencing economy
• Usually 2-10 years duration
Mo., Qtr., Yr.Mo., Qtr., Yr.
ResponseResponseCycle
Cyclical ComponentCyclical Component• Upward or Downward Swings
• May Vary in Length
• Usually Lasts 2 - 10 YearsSales
Time
Cycle
Seasonal ComponentSeasonal Component
• Regular pattern of up & down fluctuations
• Due to weather, customs etc.
• Occurs within one year
Mo., Qtr.Mo., Qtr.
ResponseResponseSummerSummer
© 1984-1994 T/Maker Co.
Seasonal ComponentSeasonal Component• Upward or Downward Swings
• Regular Patterns
• Observed Within One YearSales
Time (Monthly or Quarterly)
Winter
Irregular ComponentIrregular Component
• Erratic, unsystematic, ‘residual’ fluctuations
• Due to random variation or unforeseen events– Union strike– War
• Short duration & nonrepeating
© 1984-1994 T/Maker Co.
Random or Irregular Random or Irregular ComponentComponent
• Erratic, Nonsystematic, Random,
‘Residual’ Fluctuations
• Due to Random Variations of
– Nature
– Accidents
• Short Duration and Non-repeating
Time Series ForecastingTime Series Forecasting
Time Series ForecastingTime Series ForecastingTime
Series
Time Series ForecastingTime Series ForecastingTime
Series
Trend?
Time Series ForecastingTime Series ForecastingTime
Series
Trend?SmoothingMethods
NoNo
Time Series ForecastingTime Series ForecastingTime
Series
Trend?SmoothingMethods
TrendModels
YesYesNoNo
Time Series ForecastingTime Series ForecastingTime
Series
Trend?SmoothingMethods
TrendModels
YesYesNoNo
ExponentialSmoothing
MovingAverage
Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Time Series AnalysisTime Series Analysis
Plotting Time Series DataPlotting Time Series Data
09/83 07/86 05/89 03/92 01/95
Month/Year
0
2
4
6
8
10
12
Number of Passengers
(X 1000)Intra-Campus Bus Passengers
Data collected by Coop Student (10/6/95)
Moving Average MethodMoving Average MethodMoving Average MethodMoving Average Method
Time Series ForecastingTime Series ForecastingTime Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Moving Average MethodMoving Average Method
• Series of arithmetic means
• Used only for smoothing– Provides overall impression of data over time
Moving Average MethodMoving Average MethodMoving Average MethodMoving Average Method
• Series of arithmetic means
• Used only for smoothing– Provides overall impression of data over time
Used for elementary forecasting Used for elementary forecasting
Moving Average GraphMoving Average GraphMoving Average GraphMoving Average Graph
0
2
4
6
8
93 94 95 96 97 98
0
2
4
6
8
93 94 95 96 97 98
YearYear
SalesSalesActualActual
Moving Average Moving Average [An Example]
Moving Average Moving Average [An Example]
You work for Firestone Tire. You want to smooth random fluctuations using a 3-period moving average.
1995 20,0001996 24,0001997 22,0001998 26,0001999 25,000
Moving Average [Solution]
Moving Average [Solution]
Year Sales MA(3) in 1,000
1995 20,000 NA
1996 24,000 (20+24+22)/3 = 22
1997 22,000 (24+22+26)/3 = 24
1998 26,000 (22+26+25)/3 = 24
1999 25,000 NA
Moving Average Moving Average Year Response Moving
Ave
1994 2 NA
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6 NA 94 95 96 97 98 99
8
6
4
2
0
Sales
Exponential Smoothing Exponential Smoothing MethodMethod
Exponential Smoothing Exponential Smoothing MethodMethod
Time Series ForecastingTime Series ForecastingTime Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Exponential Smoothing Exponential Smoothing MethodMethod
Exponential Smoothing Exponential Smoothing MethodMethod
• Form of weighted moving average– Weights decline exponentially– Most recent data weighted most
• Requires smoothing constant (W)– Ranges from 0 to 1– Subjectively chosen
• Involves little record keeping of past data
You’re organizing a Kwanza meeting. You want to forecast attendance for 1998 using exponential smoothing ( = .20). Past attendance (00) is:
1995 41996 61997 51998 31999 7
Exponential SmoothingExponential Smoothing [An Example]
Exponential SmoothingExponential Smoothing [An Example]
© 1995 Corel Corp.
Exponential SmoothingExponential SmoothingExponential SmoothingExponential Smoothing
Time YiSmoothed Value, Ei
(W = .2)Forecast
Yi + 1
1995 4 4.0 NA
1996 6 (.2)(6) + (1-.2)(4.0) = 4.4 4.0
1997 5 (.2)(5) + (1-.2)(4.4) = 4.5 4.4
1998 3 (.2)(3) + (1-.2)(4.5) = 4.2 4.5
1999 7 (.2)(7) + (1-.2)(4.2) = 4.8 4.2
2000 NA NA 4.8
Time YiSmoothed Value, Ei
(W = .2)Forecast
Yi + 1
1995 4 4.0 NA
1996 6 (.2)(6) + (1-.2)(4.0) = 4.4 4.0
1997 5 (.2)(5) + (1-.2)(4.4) = 4.5 4.4
1998 3 (.2)(3) + (1-.2)(4.5) = 4.2 4.5
1999 7 (.2)(7) + (1-.2)(4.2) = 4.8 4.2
2000 NA NA 4.8
Ei = W·Yi + (1 - W)·Ei-1Ei = W·Yi + (1 - W)·Ei-1
^
Exponential SmoothingExponential Smoothing [Graph]
Exponential SmoothingExponential Smoothing [Graph]
0
2
4
6
8
93 96 97 98 99
0
2
4
6
8
93 96 97 98 99
YearYear
AttendanceAttendanceActualActual
Weight
W is... Prior Period2 Periods
Ago3 Periods
Ago
W W(1-W) W(1-W)2
0.10 10% 9% 8.1%
0.90 90% 9% 0.9%
Weight
W is... Prior Period2 Periods
Ago3 Periods
Ago
W W(1-W) W(1-W)2
0.10 10% 9% 8.1%
0.90 90% 9% 0.9%
Forecast Effect of Smoothing Coefficient (W)
Forecast Effect of Smoothing Coefficient (W)
YYii+1+1 = = W·YW·Yii + + W·W·(1-(1-WW))·Y·Yii-1-1 + + W·W·(1-(1-WW))22·Y·Yii-2 -2 +...+...^
Linear Time-Series Linear Time-Series Forecasting ModelForecasting ModelLinear Time-Series Linear Time-Series Forecasting ModelForecasting Model
Time Series ForecastingTime Series ForecastingTime Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear Time-Series Forecasting ModelLinear Time-Series Forecasting Model
• Used for forecasting trend
• Relationship between response variable Y & time X is a linear function
• Coded X values used often– Year X: 1995 1996 1997 1998 1999– Coded year: 0 1 2 3 4– Sales Y: 78.7 63.5 89.7 93.2 92.1
Linear Time-Series Linear Time-Series ModelModelLinear Time-Series Linear Time-Series ModelModel
Y
Time, X 1
Y
Time, X 1
Y b b Xi i 0 1 1Y b b Xi i 0 1 1
bb11 > 0 > 0
bb11 < 0 < 0
Linear Time-Series Model [An Example]
Linear Time-Series Model [An Example]
You’re a marketing analyst for Hasbro Toys. Using coded years, you find Yi = .6 + .7Xi.
1995 11996 11997 21998 21999 4
Forecast 2000 sales.
^
Linear Time-Series Linear Time-Series [Example][Example]Linear Time-Series Linear Time-Series [Example][Example]
Year Coded Year Sales (Units)1995 0 11996 1 11997 2 21998 3 21999 4 42000 5 ?
2000 forecast sales: Yi = .6 + .7·(5) = 4.1
The equation would be different if ‘Year’ used.
^
The Linear Trend ModelThe Linear Trend Model
iii X..XbbY 743143210 Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
0
1
2
3
4
5
6
7
8
1993 1994 1995 1996 1997 1998 1999 2000
Projected to year 2000
CoefficientsIntercept 2.14285714X Variable 1 0.74285714
Excel Output
Time Series Plot
01/93 01/94 01/95 01/96 01/97
Month/Year
16
17
18
19
20
Number of Surgeries
(X 1000)
Surgery Data(Time Sequence Plot)
Source: General Hospital, Metropolis
Time Series Plot [Revised]
01/93 01/94 01/95 01/96 01/97
Month/Year
183
185
187
189
191
193
Number of Surgeries
(X 100)
Revised Surgery Data(Time Sequence Plot)
Source: General Hospital, Metropolis
Seasonality Plot
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
99.7
99.9
100.1
100.3
100.5
Monthly Index
Revised Surgery Data
(Seasonal Decomposition)
Source: General Hospital, Metropolis
Trend Analysis
12/92 10/93 8/94 6/95 9/96 2/97 12/97
Month/Year
18
18.3
18.6
18.9
19.2
19.5
Number of Surgeries
(X 1000)
Revised Surgery Data(Trend Analysis)
Source: General Hospital, Metropolis
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
Time Series ForecastingTime Series ForecastingTime Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
• Used for forecasting trend
• Relationship between response variable Y & time X is a quadratic function
• Coded years used
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
Quadratic Time-Series Quadratic Time-Series Forecasting ModelForecasting Model
• Used for forecasting trend
• Relationship between response variable Y & time X is a quadratic function
• Coded years used
• Quadratic model
Y b b X b Xi i i 0 1 1 11
21
Y b b X b Xi i i 0 1 1 11
21
Y
Year, X 1
Y
Year, X 1
Y
Year, X 1
Quadratic Time-Series Model Relationships
Quadratic Time-Series Model Relationships
Y
Year, X 1
bb1111 > 0 > 0bb1111 > 0 > 0
bb1111 < 0 < 0bb1111 < 0 < 0
Quadratic Trend ModelQuadratic Trend Model2
210 iii XbXbbY
22143308572 iii X.X..Y
Excel Output
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
CoefficientsIntercept 2.85714286X Variable 1 -0.3285714X Variable 2 0.21428571
Exponential Time-Series Exponential Time-Series ModelModel
Exponential Time-Series Exponential Time-Series ModelModel
Time Series ForecastingTime Series ForecastingTime Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Exponential Time-Series Forecasting Model
Exponential Time-Series Forecasting Model
• Used for forecasting trend
• Relationship is an exponential function
• Series increases (decreases) at increasing (decreasing) rate
Exponential Time-Series Forecasting Model
Exponential Time-Series Forecasting Model
• Used for forecasting trend
• Relationship is an exponential function
• Series increases (decreases) at increasing (decreasing) rate
Y
Year, X 1
Y
Year, X 1
Exponential Time-Series Model Relationships
Exponential Time-Series Model Relationships
bb11 > 1 > 1
0 < 0 < bb11 < 1 < 1
Exponential Weight Exponential Weight [Example Graph][Example Graph]
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed
CoefficientsIntercept 0.33583795X Variable 10.08068544
Exponential Trend ModeliX
i bbY 10 or 110 blogXblogYlog i
Excel Output of Values in logs
iXi ).)(.(Y 21172
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
antilog(.33583795) = 2.17antilog(.08068544) = 1.2
ASSIGNMENT
1. WHAT ARE THE FOUR COMPONENTS OF TIME SERIES? EXPLAIN ONE BY ONE.
2.EXPLAIN THE MOVIN AVG METHOD
3.WHAT DO YOU MEAN BY EXPONENTIAL TREND
TEST
1.WHAT ARE CYCLIC COMPONENTS OF TIME SERIES?
2.WHAT DO YOU MEAN BY TIME SERIES ANALYSIS? WHAT ARE ITS COMPONENTS? 3.WHAT DO YOU MEAN BY LINEAR TREND MODEL?