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METODE PENELITIAN
Penelitian Pemodelan Komputer 2
Pertemuan 10
Time Series Model
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. Cross Sectional Data
Time series data is a
sequence of observations
• collected from a process
• with equally spaced periods of
time.
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 is
dynamic, it
does change
over time.
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.
Time Series Components
Trend
Seasonal
Cyclical
Irregular
Trend Component
◼ Persistent, overall upward or downward pattern
◼ Due to population, technology etc.
◼ Several years duration
Mo., Qtr., Yr.
Response
© 1984-1994 T/Maker Co.
Trend Component
◼ Overall Upward or Downward Movement
◼ Data Taken Over a Period of Years
Sales
Time
Cyclical Component
◼ Repeating up & down movements
◼ Due to interactions of factors influencing economy
◼ Usually 2-10 years duration
Mo., Qtr., Yr.
Response
Cycle
Cyclical Component
◼ Upward or Downward Swings
◼ May Vary in Length
◼ Usually Lasts 2 - 10 Years
Sales
Time
Seasonal Component
◼ Regular pattern of up & down fluctuations
◼ Due to weather, customs etc.
◼ Occurs within one year
Mo., Qtr.
Response
Summer
© 1984-1994 T/Maker Co.
Seasonal Component
◼ Upward or Downward Swings
◼ Regular Patterns
◼ Observed Within One Year
Sales
Time (Monthly or Quarterly)
Irregular 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 Component
◼ Erratic, Nonsystematic, Random, ‘Residual’
Fluctuations
◼ Due to Random Variations of
• Nature
• Accidents
◼ Short Duration and Non-repeating
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
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)
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
Moving Average Method
◼ Series of arithmetic means
◼ Used only for smoothing
• Provides overall impression of data over time
Moving Average Graph
0
2
4
6
8
93 94 95 96 97 98
Year
Sales
Actual
Moving Average
[An Example]
You work for Firestone Tire.
You want to smooth random
fluctuations using a 3-period
moving average.
1995 20,000
1996 24,000
1997 22,000
1998 26,000
1999 25,000
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
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
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
Exponential 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
You’re organizing a Kwanza meeting.
You want to forecast attendance for 1998
using exponential smoothing
( = .20). Past attendance (00) is:
1995 4
1996 6
1997 5
1998 3
1999 7
Exponential Smoothing
[An Example]
© 1995 Corel Corp.
Exponential Smoothing
Time Yi
Smoothed 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-1
^
Exponential Smoothing [Graph]
0
2
4
6
8
93 96 97 98 99
Year
Attendance
Actual
Weight
W is... Prior Period
2 Periods
Ago
3 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)
Yi+1 = W·Yi + W·(1-W)·Yi-1 + W·(1-W)2·Yi-2 +...
^
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
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 1998 1999
• Coded year: 0 1 2 3 4
• Sales Y: 78.7 63.5 89.7 93.2 92.1
Linear Time-Series Model
Y
Time, X1

Y b b X
i i
= +
0 1 1
b1 > 0
b1 < 0
Linear Time-Series Model
[An Example]
You’re a marketing analyst for Hasbro Toys.
Using coded years, you find Yi = .6 + .7Xi.
1995 1
1996 1
1997 2
1998 2
1999 4
Forecast 2000 sales.
^
Linear Time-Series
[Example]
Year Coded Year Sales (Units)
1995 0 1
1996 1 1
1997 2 2
1998 3 2
1999 4 4
2000 5 ?
2000 forecast sales: Yi = .6 + .7·(5) = 4.1
The equation would be different if ‘Year’ used.
^
The Linear Trend Model
i
i
i X
.
.
X
b
b
Ŷ 743
143
2
1
0 +
=
+
=
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
Coefficients
In te rce p t 2.14285714
X V a ria b le 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
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
Quadratic Time-Series Forecasting
Model
◼ Used for forecasting trend
◼ Relationship between response variable
Y & time X is a quadratic function
◼ Coded years used

Y b b X b X
i i i
= + +
0 1 1 11
2
1
Y
Year, X1
Y
Year, X1
Y
Year, X1
Quadratic Time-Series Model Relationships
Y
Year, X1
b11 > 0
b11 > 0
b11 < 0
b11 < 0
Quadratic Trend Model
2
2
1
0 i
i
i X
b
X
b
b
Ŷ +
+
=
2
214
33
0
857
2 i
i
i X
.
X
.
.
Ŷ +
−
=
Excel Output
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
Coefficients
Inte rce pt 2.85714286
X V a ria ble 1 -0.3285714
X V a ria ble 2 0.21428571
Time Series Forecasting
Linear
Time
Series
Trend?
Smoothing
Methods
Trend
Models
Yes
No
Exponential
Smoothing
Quadratic Exponential
Auto-
Regressive
Moving
Average
Exponential Time-Series Forecasting
Model
◼ Used for forecasting trend
◼ Relationship is an exponential function
◼ Series increases (decreases) at increasing (decreasing)
rate
Y
Year, X1
Exponential Time-Series Model
Relationships
b1 > 1
0 < b1 < 1
Exponential Weight [Example Graph]
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed
C o e f f ic ie n t s
In t e rc e p t 0 . 3 3 5 8 3 7 9 5
X V a ria b le 10 . 0 8 0 6 8 5 4 4
Exponential Trend Model
i
X
i b
b
Ŷ 1
0
= or 1
1
0 b
log
X
b
log
Ŷ
log i +
=
Excel Output of Values in logs
i
X
i )
.
)(
.
(
Ŷ 2
1
17
2
=
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
antilog(.33583795) = 2.17
antilog(.08068544) = 1.2
Evaluating Forecasts
Forecasting Guidelines
◼ No pattern or direction in forecast error
• ei = (Actual Yi - Forecast Yi)
• Seen in plots of errors over time
◼ Smallest forecast error
• Measured by mean absolute deviation
◼ Simplest model
• Called principle of parsimony
Pattern of Forecast Error
Trend Not Fully
Accounted for
Time (Years)
Error
0
Desired Pattern
Time (Years)
Error
0
Residual Analysis
Random errors
Trend not accounted for
Cyclical effects not accounted for
Seasonal effects not accounted for
T T
T T
e e
e e
0 0
0 0
Questions?
ANOVA

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