datta meghe institute of management studies quantitative techniques unit no.:04 unit name: time...
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Datta Meghe Institute of
Management Studies
Quantitative Techniques
Unit No.:04
Unit Name: Time Series Analysis and Forecasting
1
2
Datta Meghe Institute of
Management Studies
Time Series Analysis and Forecasting - Components of Time Series, Trend - Moving averages, semi-averages and least-squares, seasonal variation, cyclic variation and irregular variation, Index numbers, calculation of seasonal indices, Additive and multiplicative models, Forecasting, Non linear trend – second degree parabolic trends
SYLLABUS
Datta Meghe Institute of
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Learning ObjectivesLearning Objectives
• Describe what forecasting is• Explain time series & its components• Smooth a data series• Moving average• Exponential smoothing• Forecast using trend models• Measuring forecast error• The multiplicative time series model• Naïve extrapolation• The mean forecast model• Weighted moving average models
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 Methods
Forecasting ApproachesForecasting Approaches
Quantitative Methods
• Used when situation is ‘stable’ & historical data exist– Existing products– Current technology
• Involve mathematical techniques
• e.g., forecasting sales of color televisions
Forecasting ApproachesForecasting Approaches
• Used when situation is vague & little data exist– New products– New technology
• Involve intuition, experience
• e.g., forecasting sales on Internet
Qualitative Methods Quantitative Methods
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Quantitative Forecasting Quantitative Forecasting
• Select several forecasting methods• ‘Forecast’ the past• Evaluate forecasts • Select best method• Forecast the future• Monitor continuously forecast accuracy
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Quantitative Forecasting MethodsQuantitative Forecasting Methods
Datta Meghe Institute of
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Quantitative Forecasting MethodsQuantitative Forecasting Methods
Quantitative
Forecasting
Datta Meghe Institute of
Management Studies
Quantitative Forecasting MethodsQuantitative Forecasting Methods
Quantitative
Forecasting
Time Series
Models
Datta Meghe Institute of
Management Studies
Causal
Models
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative
Forecasting
Time Series
Models
Datta Meghe Institute of
Management Studies
Causal
Models
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative
Forecasting
Time Series
Models
Exponential
Smoothing
Trend
Models
Moving
Average
Datta Meghe Institute of
Management Studies
Causal
Models
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative
Forecasting
Time Series
Models
RegressionExponential
Smoothing
Trend
Models
Moving
Average
Datta Meghe Institute of
Management Studies
Causal
Models
Quantitative Forecasting Methods
Quantitative Forecasting Methods
Quantitative
Forecasting
Time Series
Models
RegressionExponential
Smoothing
Trend
Models
Moving
Average
CasualModels
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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
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Time Series vs. Cross Sectional Data
Time Series vs. Cross Sectional Data
Time series data is a sequence of observations
– collected from a process – with equally spaced periods of time.
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Time Series vs. Cross Sectional Data
Time Series vs. Cross 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. Cross Sectional Data
Time Series vs. Cross Sectional Data
Time series is dynamic, it does change over time.
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Time Series vs. Cross Sectional Data
Time Series vs. Cross Sectional Data
When working with time series data, it is paramount that the data is plotted so the researcher can view the data.
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Time Series ComponentsTime Series Components
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Time Series ComponentsTime Series Components
Trend
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Time Series ComponentsTime Series Components
Trend Cyclical
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Time Series ComponentsTime Series Components
Trend
Seasonal
Cyclical
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Time Series ComponentsTime Series Components
Trend
Seasonal
Cyclical
Irregular
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Trend ComponentTrend Component
• Persistent, overall upward or downward pattern• Due to population, technology etc.• Several years duration
Mo., Qtr., Yr.
Response
© 1984-1994 T/Maker Co.
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Trend ComponentTrend Component
• Overall Upward or Downward Movement• Data Taken Over a Period of Years
Sales
Time
Upward trend
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Cyclical ComponentCyclical Component
• Repeating up & down movements• Due to interactions of factors influencing economy• Usually 2-10 years duration
Mo., Qtr., Yr.
Response Cycle
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Cyclical ComponentCyclical Component
• Upward or Downward Swings• May Vary in Length• Usually Lasts 2 - 10 Years
Sales
Time
Cycle
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• Regular pattern of up & down fluctuations• Due to weather, customs etc.• Occurs within one year
Seasonal ComponentSeasonal Component
Mo., Qtr.
Response
Summer
© 1984-1994 T/Maker Co.
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• Upward or Downward Swings• Regular Patterns• Observed Within One Year
Seasonal ComponentSeasonal Component
Sales
Time (Monthly or Quarterly)
Winter
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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.
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Random or Irregular ComponentRandom or Irregular Component
• Erratic, Nonsystematic, Random, ‘Residual’
Fluctuations
• Due to Random Variations of
– Nature
– Accidents
• Short Duration and Non-repeating
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Time Series ForecastingTime Series Forecasting
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Time Series ForecastingTime Series Forecasting
TimeSeries
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Time Series ForecastingTime Series Forecasting
TimeSeries
Trend?
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Time Series ForecastingTime Series Forecasting
TimeSeries
Trend?Smoothing
Methods
NoNo
Datta Meghe Institute of
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Time Series ForecastingTime Series Forecasting
TimeSeries
Trend?Smoothing
MethodsTrend
Models
YesYesNoNo
Datta Meghe Institute of
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Time Series ForecastingTime Series Forecasting
TimeSeries
Trend?Smoothing
MethodsTrend
Models
YesYesNoNo
ExponentialSmoothing
MovingAverage
Datta Meghe Institute of
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
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Time Series Analysis
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Plotting 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)
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Moving Average MethodMoving Average Method
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
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Moving Average MethodMoving Average Method
• Series of arithmetic means • Used only for smoothing
– Provides overall impression of data over time
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Moving Average MethodMoving Average Method
• Series of arithmetic means • Used only for smoothing
– Provides overall impression of data over time
Used for elementary forecasting
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Moving Average GraphMoving Average Graph
0
2
4
6
8
93 94 95 96 97 98
Year
Sales Actual
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Moving Average [An Example]
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
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Moving Average [Solution]
Moving Average [Solution]
Year Sales MA(3) in 1,0001995 20,000 NA1996 24,000 (20+24+22)/3 = 221997 22,000 (24+22+26)/3 = 241998 26,000 (22+26+25)/3 = 241999 25,000 NA
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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
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Exponential Smoothing MethodExponential Smoothing Method
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
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Exponential Smoothing MethodExponential Smoothing Method
• 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
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Exponential Smoothing [An Example]
Exponential Smoothing [An Example]
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
© 1995 Corel Corp.
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Exponential 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
Ei = W·Yi + (1 - W)·Ei-1Ei = W·Yi + (1 - W)·Ei-1
^
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Exponential Smoothing [Graph]Exponential Smoothing [Graph]
0
2
4
6
8
93 96 97 98 99Year
Attendance
Actual
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Forecast Effect of Smoothing Coefficient (W)
Forecast Effect of Smoothing Coefficient (W)
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%
Yi+1 = W·Yi + W·(1-W)·Yi-1 + W·(1-W)2·Yi-2 +...^
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Linear Time-Series Forecasting Model
Linear Time-Series Forecasting Model
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Datta Meghe Institute of
Management Studies
Linear Time-Series Forecasting Model
Linear 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 19981999
– Coded year: 0 1 2 34
– Sales Y: 78.7 63.5 89.7 93.292.1
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Linear Time-Series ModelLinear Time-Series Model
Y
Time, X 1
Y b b Xi i 0 1 1
b1 > 0
b1 < 0
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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.
^
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Linear Time-Series [Example]Linear Time-Series [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.
^
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The 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
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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
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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
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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
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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
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Quadratic Time-Series Forecasting Model
Quadratic Time-Series Forecasting Model
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Datta Meghe Institute of
Management Studies
Quadratic Time-Series Forecasting Model
Quadratic Time-Series Forecasting Model
• Used for forecasting trend• Relationship between response variable Y & time X
is a quadratic function• Coded years used
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Quadratic Time-Series Forecasting Model
Quadratic Time-Series Forecasting 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
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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
b11 > 0b11 > 0
b11 < 0b11 < 0
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Quadratic Trend Model
2210 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
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Exponential Time-Series ModelExponential Time-Series Model
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?SmoothingMethods
TrendModels
YesNo
ExponentialSmoothing
Quadratic Exponential Auto-Regressive
MovingAverage
Datta Meghe Institute of
Management Studies
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
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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
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Exponential Time-Series Model Relationships
Exponential Time-Series Model Relationships
Y
Year, X 1
b1 > 1
0 < b1 < 1
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Exponential Weight [Example Graph]
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed
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CoefficientsIntercept 0.33583795X Variable 10.08068544
Exponential Trend Model
iXi 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
81
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• Described what forecasting is
• Explained time series & its components
• Smoothed a data series– Moving average– Exponential smoothing
• Forecasted using trend models
SUMMARY
82
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LONG & SHORT QUESTION
83
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• Statistics : Theory and Practice
- R.S.N. Pillai & Bhagwati
Fundamentals of Statistics
- S.C. Gupta
BOOKS REFERRED
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