PowerPoint Presentation
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Outline
• What is a time series data? • What kinds of variation are in a
time series data?
– Long term trend, circular variation, seasonal variation,
irregular variation
• How to analyze a time series data? – Seasonal effects : seasonal
indices – Long term trend : fit a linear/nonlinear model for
trend
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• A time series is a collection of time-dependent data. • There are
4 components to a time series :
– The long-term trend; – The cyclical variation; – The seasonal
variation; – The irregular variation.
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Secular trend : the smooth long-term direction
• Example. P652 • The number of employees at Home Depot, Inc. • The
number has increased rapidly over the last 10 years.
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Secular trend : the smooth long-term direction • Example. P652 The
number of emergency medical service (EMS) calls in
Horry County, South Carolina, since 1989. • The number increased
from 1989 to 1995. • From 1995 to 2000 the number of calls stayed
about the same • In 2000 it began another increase.
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Secular trend : the smooth long-term direction • Example. P652 The
number of manufactured homes
shipped in USA showed a steady increase from 1990 to 1996, then
remained about the same until 1999, when the number began to
decline.
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Cyclical variation : the rise and fall over periods longer than one
year
• Example. P653 The number of batteries sold by National Battery
Sales, Inc. from 1984 through 2003.
• Cyclical variation : recovery, prosperity, recession,
depression.
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Seasonal variation : patterns of change within a year • Example.
Chart 19-2 The quarterly sales of Hercher Sporting Goods,
Inc. • Most of sales are in the 1st and 2nd quarters of the year. •
Some business are during the holidays, the 4th quarter. • The late
summer, the 3rd quarter, is slow season.
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Data Analysis : a multiplicative model
• A multiplicative model : The time series data = T×C×S×I =
(trend)×(cycle)×(season)×(residual)
• Data analysis : S Find the seasonal indexes
1. Roughly estimate T×C – by the moving-average method; 2. Estimate
(S×I) = (Data)/(T×C); 3. Estimate the seasonal index S -- by simple
average;
D De-seasonalize data = (Data)/S; T Fit long-term trend – The least
squares method
• Linear trends • Nonlinear trends
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1. Determine T×C • Goal :
– Estimate T×C = long-term variations, – Remove short-term
variations, (S×I)
• Strategy : Moving-average method – Calculate 4-quarter moving
averages : remove (S)/
– Center the moving averages : average of successive no.’s, remove
(I)
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Moving average method
• The moving average method : P655 – “Moving” the mean values
through the time series. – If odd period:
Find the moving average and positioned in mid of the period. – If
even period:
1. Find the moving average and positioned in mid of the period. 2.
Center the moving averages and position on a particular date.
• Goals : – Smooth out the short-term fluctuations. – Identify the
long-term trend. – If the duration is constant, and if the
amplitudes() are equal,
the short-term fluctuations can be removed entirely.
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What is a moving average method?
• Example. P655 Data = T×C×I, no seasonal effect • The cycle period
= 7 years, the amplitude of each cycle is 4. • See Table 19-3 &
Chart 19-5 P700 • 7-year moving average :
– Show the increasing long-term trend – The cyclical pattern
vanished.
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• Example. P657 Table 19-4 & Chart 19-6 • 3-year moving
average; 5-year moving average.
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Example. P669
Table 19-6 shows the quarterly sales for Toys International for
1996 through 2001. Determine a quarterly seasonal index using the
ratio-to-moving average method.
The seasonal nature of the sales : the 4th quarter are the largest
and the 2nd quarters are the smallest.
The long-term trend is moderately increased.
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Example. P706
Step 1.1 Find the 4-quarter moving average and positioned between
the quarters. See column (3) in Table 19-7
Step 1.2 Center the moving averages and position on a particular
quarter. See column (4) in Table 19-7
→ (4) = (T×C) = long term effect
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S step :
Step 2 Calculate the specific seasonal for each quarter (S×I) –
(S×I)=(Data)/(the centered moving average)=(TCSI)/(TC)
• Example. P706, (1)/(4) =(5) = (S×I)
Step 3. S=? – Organize the specific seasonal indices – Take simple
average to remove (I) over same season
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Example. –Organize the specific seasonals in Table 19-8 –The mean
is used for the seasonal index for each quarter.
S1 = 0.767, S2 = 0.576, S3 = 1.144, S4 = 1.522
-8
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Step 4 Adjust the indexes to have sum 4. • Method :
– Calculate the correction factor = 4/(S1+S2+S3+S4) – The adjusted
index = index × correction factor
• Example P672 – Since S1+S2+S3+S = 0.767+0.576+1.144+1.522 = 4.009
> 4.000 – The seasonal indices should be adjusted. – Correction
factor = 4/(S1+S2+S3+S4)= 4/4.009=0.997755 – The adjusted quarterly
indices are,
• S1 = 0.767×0.997755=0.765=76.5% • S2 = 0.576×0.997755=0.575=57.5%
• S3 = 1.144×0.997755=1.141=114.1% • S4 =
1.522×0.997755=1.519=151.9%
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• De-seasonalized data = seasonally adjusted data – The seasonal
effect is removed. – The long-term fluctuations, the trend and
cycle effects, are studied.
• Method : – De-seasonalized data = original data / seasonal
index
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D step : De-seasonalized data
• Example. Table on page 672 gives the de-seasonalized quarterly
sales of Toys International in Table 19-7. – (1) = original data =
(TSCI); – (2) = seasonal index = (S); – (3) = de-seasonalized sales
= (TSCI)/(S) = (1)/(2). – See P 675 MINITAB output
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• Establish a trend model with de-seasonalized data.
• Trend model – Linear trend models – Nonlinear trend models
• Trend model -- Linear trends : – Linear equation : Y’ = a + b× t
– Y’ = predicted value, t = coded values of time – Recall the least
square estimations for (a, b) – Then b = increment per unit
time.
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Example. P675 Establish a trend model
• Trend model : Linear trend equation : Y’ = a + b t • The time is
coded as t=1 from the winter quarter of 1996. • Then the LSE : a =
8.1096, b = 0.0899 • The trend equation : Y’ = 8.1096 + 0.0899 t •
Over the 24 quarters the de-seasonalized sales increased at
a rate of 0.0899 per quarter.
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F step : Forecast
• Forecast : – Forecast the trend effect in the future with the
trend model – Adjust the trend values to account for the seasonal
factors.
• Method : – A predicted equation is Y’ = g(t), then at time t0,
calculate
Y’ = g(t0) = trend effect – Seasonally adjusted as Y’ ×S
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F step : Forecast • Example. P677 • If we assume that the past 24
periods are a good indicator of future
sales, the trend equation can be used in forecast. • For the winter
quarter of 2002, t=25, the estimated sales total is
Y’ = 8.1096+0.0899(25)=10.3571 • The forecast is then seasonally
adjusted by Y’×S • See Table 19-12 : forecast for the 4 quarters of
2002.
Y’×SY’ = 8.1096+0.0899t
• Log trend equation : – – Then
– Increase/decrease by equal percents over a period. –
tabY =
×+= =
• Log trend equation :
– Fit a linear model with data (t, y*=log(y)) – Then
tlog(b) log(a) )Ylog( ×+=
Example. P665 Charter 19-7
• Data : sales for the Gulf Shores importers from 1988 to 2002 Year
Imports log(Y)=log(imports) Code(t)
1988 124.2 2.0941 1 1989 175.6 2.2445 2
1990 306.9 2.4870 3
1991 524.2 2.7195 4
1992 714.0 2.8537 5
1993 1052.0 3.0220 6
1994 1638.3 3.2144 7
1995 2463.2 3.3915 8
1996 3358.2 3.5261 9
1997 4181.3 3.6213 10
1998 5388.5 3.7315 11
1999 8027.4 3.9046 12
2000 10587.2 4.0248 13
2001 13537.4 4.1315 14
2002 17515.6 4.2434 15
0.0
5000.0
10000.0
15000.0
20000.0
Y’
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Log(Y’)
0.0000
2.0000
4.0000
6.0000
Y*=log(Y’)
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Example. P666 Charter 19-7
• Step 1. Find the log base 10 of each year’s imports • Step 2.
Find the LSE with data (t, log(y))=(t, y*).
– a*=Log(a)=2.0538, b*=Log(b)=0.1534. – Log(Y’) = 2.0538 +
0.1534t.
• The antilog of 0.1534 is b=100.1534 = 1.4235 • Then
1.4235-1=0.4235 is the geometric mean rate of
increase from 1988 to 2002. • The annually increased rate is 42.35%
during the period.
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1 6.5852 6.5852 1065.228 7.36E-14
13 0.0804 0.0062
2.0538 0.0427 48.0741 4.98E-16 1.9615 2.1461
Code(t) 0.1534 0.0047 32.6378 7.36E-14 0.1432 0.1635
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Example. P667 Charter 19-7
• Forecast the imports in the year 2006, t=15+4=19?
000,809,9210Y
97.4191534.00538.2Y
19415t
97.4
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Exercise
• Seasonal indices : 25, 27 • Seasonal indices, long-term trend,
prediction : 27, 31
Ch. 19 Time Series and Forecasting
Outline
Secular trend : the smooth long-term direction
Secular trend : the smooth long-term direction
Cyclical variation : the rise and fall over periods longer than one
year
Seasonal variation : patterns of change within a year
Irregular variation : residual
S step : The Typical Seasonal Index : ratio-to-moving-average
method
Moving average method
What about 4-year moving average? Even period, centered
again!
Example. P669
Example. P706
S step :
S step :
F step : Forecast
F step : Forecast
How to fit a log trend model?
Example. P665 Charter 19-7
Example. P666 Charter 19-7
Example. P667 Charter 19-7