This document discusses techniques for analyzing and forecasting time series data, including decomposition of time series into trend, seasonal, cyclical, and irregular components. It covers plotting time series data, calculating moving averages to smooth the data, and exponential smoothing models. Exponential smoothing assigns weights to observations where more recent observations receive greater weight, allowing forecasting of future values based on past trends. Seasonal indexes can also be calculated by dividing observed values by smoothed values to quantify seasonal effects.

1. Time-Series Decomposition
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Ch. 16-1

2. TIME-SERIES DATA
1. Numerical data ordered over time
2. The time intervals can be annually, quarterly, daily, hourly, etc.
3. The sequence of the observations is important
Example:
Year: 2005 2006 2007 2008 2009
Sales: 75.3 74.2 78.5 79.7 80.2
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Ch. 16-2
16.3

3. TIME-SERIES PLOT
the vertical axis measures
the variable of interest
the horizontal axis
corresponds to the time
periods
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Ch. 16-3
A time-series plot is a two-dimensional
plot of time series data
0.00
5.00
10.00
15.00
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
U.S. Inflation Rate

4. TIME-SERIES COMPONENTS
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Ch. 16-4
Time Series
Cyclical
Component
Irregular
Component
Trend
Component
Seasonality
Component

5. TREND COMPONENT
Long-run increase or decrease over time (overall upward or
downward movement)
Data taken over a long period of time
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Ch. 16-5
Upward trend
Sales
Time

6. TREND COMPONENT
Trend can be upward or downward
Trend can be linear or non-linear
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Ch. 16-6
Downward linear trend
Sales
Time
Upward nonlinear trend
Sales
Time
(continued)

7. SEASONAL COMPONENT
Short-term regular wave-like patterns
Observed within 1 year
Often monthly or quarterly
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Ch. 16-7
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Year n
Year n+1

8. CYCLICAL COMPONENT
Long-term wave-like patterns
Regularly occur but may vary in length
Often measured peak to peak or trough to trough
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Ch. 16-8
Sales
1 Cycle
Year

9. IRREGULAR COMPONENT
Unpredictable, random, residual fluctuations
Due to random variations of
­ Nature
­ Accidents or unusual events
Noise in the time series
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Ch. 16-9

10. TIME-SERIES COMPONENT ANALYSIS
Used primarily for forecasting
Observed value in time series is the sum or product of
components
Additive Model
Multiplicative model (linear in log form)
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Ch. 16-
10
where Tt = Trend value at period t
St = Seasonality value for period t
Ct = Cyclical value at time t
It = Irregular (random) value for period t
ttttt ICSTX ´++=
ttttt ICSTX =

11. MOVING AVERAGES:
SMOOTHING THE TIME SERIES
Calculate moving averages to get an overall impression of the pattern of
movement over time
This smooths out the irregular component
Moving Average: averages of a designated
number of consecutive
time series values
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Ch. 16-
11
16.4

12. (2M+1)-POINT MOVING AVERAGE
A series of arithmetic means over time
Result depends upon choice of m (the number of data values in each
average)
Examples:
­ For a 5 year moving average, m = 2
­ For a 7 year moving average, m = 3
­ Etc.
Replace each xt with
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Ch. 16-
12
å-=
+ -++=
+
=
m
mj
jt
*
t m)n,2,m1,m(tX
12m
1
X !

13. MOVING AVERAGES
Example: Five-year moving average
­ First average:
­ Second average:
­ etc.
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Ch. 16-
13
5
xxxxx
x 54321*
5
++++
=
5
xxxxx
x 65432*
6
++++
=

14. EXAMPLE: ANNUAL DATA
…
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Ch. 16-
14
Year Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
Annual Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sales
…

15. EXAMPLE: QUARTER DATA
…
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Ch. 16-
15
Quarter Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
0
20
40
60
1 2 3 4 5 6 7 8 9 10 11
Sales
Year
AnnualSales
…

16. CALCULATING MOVING AVERAGES
Each moving average is for a consecutive
block of (2m+1) years
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Ch. 16-
16
Year Sales
1 23
2 40
3 25
4 27
5 32
6 48
7 33
8 37
9 37
10 50
11 40
Average
Year
5-Year
Moving
Average
3 29.4
4 34.4
5 33.0
6 35.4
7 37.4
8 41.0
9 39.4
… …
5
3227254023
29.4
++++
=
etc…
§ Let m = 2

17. ANNUAL VS. MOVING AVERAGE
The 5-year moving
average smoothes
the data and shows
the underlying trend
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Ch. 16-
17
Annual vs. 5-Year Moving Average
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sales
Annual 5-Year Moving Average

18. CENTERED MOVING AVERAGES
Let the time series have period s, where s is even
number
­ i.e., s = 4 for quarterly data and s = 12 for monthly data
To obtain a centered s-point moving average series Xt
*:
­ Form the s-point moving averages
­ Form the centered s-point moving averages
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Ch. 16-
18
(continued)
å+-=
++ -++==
s/2
1(s/2)j
jt
*
.5t )
2
s
n,2,
2
s
1,
2
s
,
2
s
(txx !
)
2
s
n,2,
2
s
1,
2
s
(t
2
xx
x
*
.5t
*
.5t*
t -++=
+
= +-
!

19. CENTERED MOVING AVERAGESUsed when an even number of values is used in the moving average
Average periods of 2.5 or 3.5 don t match the original periods, so we
average two consecutive moving averages to get centered moving
averages
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Ch. 16-
19
Average
Period
4-Quarter
Moving
Average
2.5 28.75
3.5 31.00
4.5 33.00
5.5 35.00
6.5 37.50
7.5 38.75
8.5 39.25
9.5 41.00
Centered
Period
Centered
Moving
Average
3 29.88
4 32.00
5 34.00
6 36.25
7 38.13
8 39.00
9 40.13
etc…

20. CALCULATING THE
RATIO-TO-MOVING AVERAGE
Now estimate the seasonal impact
Divide the actual sales value by the centered moving average for that period
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Ch. 16-
20
*
t
t
x
x
100

21. CALCULATING A SEASONAL INDEX
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Ch. 16-
21
Quarter Sales
Centered
Moving
Average
Ratio-to-
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
83.7
29.88
25
(100)
x
x
100 *
3
3
==

22. CALCULATING SEASONAL INDEXES
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Ch. 16-
22
Quarter Sales
Centered
Moving
Average
Ratio-to-
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
1. Find the mean of
all of the same-
season values
2. Adjust so that
the average over
all seasons is
100
Fall
Fall
Fall
(continued)

23. INTERPRETING SEASONAL INDEXES
Suppose we get these seasonal
indexes:
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Ch. 16-
23
Season
Seasonal
Index
Spring 0.825
Summer 1.310
Fall 0.920
Winter 0.945
S = 4.000 -- four seasons, so must sum to 4
Spring sales average 82.5% of the
annual average sales
Summer sales are 31.0% higher
than the annual average sales
etc…
§ Interpretation:

24. EXPONENTIAL SMOOTHING
A weighted moving average
­ Weights decline exponentially
­ Most recent observation weighted most
Used for smoothing and short term forecasting (often one or two periods
into the future)
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Ch. 16-
24
16.5

25. EXPONENTIAL SMOOTHING MODEL
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Ch. 16-
25
§ Exponential smoothing model
where:
= exponentially smoothed value for period t
= exponentially smoothed value already
computed for period i - 1
xt = observed value in period t
a = weight (smoothing coefficient), 0 < a < 1
11 xx =ˆ
txˆ
1-txˆ
n),1,2,t1;(0 !=<< α

26. EXPONENTIAL SMOOTHING
The weight (smoothing coefficient) is a
­ Subjectively chosen
­ Range from 0 to 1
­ Smaller a gives more smoothing, larger a gives less smoothing
The weight is:
­ Close to 0 for smoothing out unwanted cyclical and irregular components
­ Close to 1 for forecasting
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Ch. 16-
26
(continued)

27. EXPONENTIAL SMOOTHING EXAMPLE
Suppose we use weight a = .2
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Ch. 16-
27
Time
Period
(i)
Sales
(Yi)
Forecast
from prior
period (Ei-1)
Exponentially Smoothed
Value for this period (Ei)
1
2
3
4
5
6
7
8
9
10
etc.
23
40
25
27
32
48
33
37
37
50
etc.
--
23
26.4
26.12
26.296
27.437
31.549
31.840
32.872
33.697
etc.
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
etc.
= x1
since no
prior
information
exists
1xˆ

28. SALES VS. SMOOTHED SALES
Fluctuations have
been smoothed
NOTE: the smoothed
value in this case is
generally a little low,
since the trend is upward
sloping and the weighting
factor is only .2
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28
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10
Time Period
Sales
Sales Smoothed

29. FORECASTING TIME PERIOD (T + 1)
The smoothed value in the current period (t) is used as the forecast
value for next period (t + 1)
At time n, we obtain the forecasts of future values, Xn+h of the series
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29
)1,2,3(hxx nhn !==+
ˆˆ

30. YOUR TURN
1. Take your tourism data
www.tourism.go.th
2. Calculate Seasonal Index using 12 Month Moving Average
3. Submit your calculation and answer (12 monthly seasonal index)
before leaving
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30