Time series analysis involves detecting patterns in variables measured over time to aid in forecasting future values. A time series can consist of four components: long-term trend, cyclical effect, seasonal effect, and random variation. Smoothing techniques like moving averages and exponential smoothing are used to remove random variation and identify other components. Seasonal indexes express the degree to which different seasons vary from the average time series value and can be computed using a multiplicative model.
3. Introduction
Any variable that is measured over time in
sequential order is called a time series.
We analyze time series to detect patterns.
The patterns help in forecasting future
values of the time series.
Predicted value
The time series exhibit a
downward trend pattern.
3
4. 20.1 Components of a Time
Series
A time series can consist of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
A trend is a long term relatively
smooth pattern or direction, that
persists usually for more than one year.
4
5. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T)
Cyclical variation (C)
Seasonal variation (S)
Random variation (R)
6/90 6/93 6/96 6/99 6/02
A cycle is a wavelike pattern describing
Cycles are seldom regular, and
a long term behavior (for more than one
often appear in combination with
year).
other components.
5
6. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
The seasonal component of the time series 6/97 12/97 6/98 12/98 6/99
exhibits a short term (less than one year)
calendar repetitive behavior.
6
7. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
We try to remove random variation
Random variation comprises the irregular thereby, identify the other components.
unpredictable changes in the time series.
It tends to hide the other (more predictable)
components.
7
8. 20.2 Smoothing
Techniques
To produce a better forecast we need to
determine which components are present
in a time series.
To identify the components present in the
time series, we need first to remove the
random variation.
This can be done by smoothing
techniques.
8
9. Moving Averages
A k-period moving average for time
period t is the arithmetic average of the
time series values around period t.
– For example: A 3-period moving average at period t
is calculated by (yt-1 + yt + yt+1)/3
9
10. Moving Average: Example
Example 20.1
To forecast future gasoline sales, the last four
years quarterly sales were recorded.
Calculate the three-quarter and five-quarter
moving average. Show the relevant graphs.
Xm20-01 Period Year/Quarter Gas Sales
1 1 1 39
2 2 37
3 3 61
4 4 58
5 2 1 18
6 2 56 10
12. Moving Average: Smoothing
effect
Moving Average
3-period moving average
100
Value
50
0 Comment: We can identify
1 2 3 4 5 6 7 8 9 10 11 12a 13 14component .
trend 15 16
Data Point
Notice how the averaging process removes some of
the random variation.
12
13. Moving Average: Smoothing
effect
3-period moving average
100
80
60
40
The 5-period 20
0
moving average
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
removes more
variation than the 5-period moving average
3-period moving 100
80
average. 60
40
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 13
14. Centered Moving Average
With an even number of observations
included in the moving average, the average
is placed between the two periods in the
middle.
To place the moving average in an actual
time period, we need to center it.
Two consecutive moving averages are
centered by taking their average, and placing
it in the middle between them. (This is used to
graph the points.)
14
15. Centered Moving Average:
Example
Calculate the 4-period moving average and
center it, for the data given below:
Period Time series Moving Avg. Centerd
Mov.Avg.
(2.5)1 15 19.0
(3.5)
2 27 20.25
21.5
(4.5)3 20
17.5
19.50
4 14
5 25
15
6 11
16. Exponential Smoothing
The exponential smoothing – When smoothing the
method provides smoothed time series at time t,
values for all the time > the exponential
periods observed. smoothing method
considers all the data
available at t
– The moving average
method does not the(yt, yt-1,…)
moving average method
considers only the
provide smoothed
observations included in the
values (moving calculation of the average
average values) for value, and “forgets” the
the first and last set of rest.
periods. 16
17. Exponentially Smoothed Time
Series
S = wy + (1-w)St-1
Stt = wytt + (1-w)St-1
St = exponentially smoothed time series at
time t.
yt = time series at time t.
St-1 = exponentially smoothed time series at
time t-1.
w = smoothing constant, where 0 ≤ w ≤ 1.
17
18. The Exponentially Smoothed
Process
Example 20.2 (Xm20-01)
Calculate the gasoline sale smoothed time series
using exponential smoothing with w = .2, and w = .7.
Period Gas
Sales
Set S1 = y1 1 39 S1 = 39
S2 = wy2 + (1-w)S1 2 37 S2 = (.2)(37) + (1-.2)(39) = 38.6
S3 = wy3 + (1-w)S2 3 61 S3 = (.2)(61) + (1-.2)(38.6) = 43.1
4 58
5 18
6 56
. .
. .
18
19. The Exponentially Smoothed
Process
Example 20.2-continued
Exponential Smoothing, w=.2 Small ‘w’ provides
100 a lot of smoothing
Value
50 Neither value of ‘w’
0 reveals the seasonality.
1
3
5
7
9
11
13
15
Exponential Smoothing , w=.7
100
Value
50
0
1 3 5 7 9 11 13 15
19
20. The Exponentially Smoothed
Process
We will let Minitab do our exponential
smoothing
Stat>Time Series > Single Exp Smoothing
This will give us the smoothed graph
20
21. 20.3 Trend and Seasonal Effects
Trend Analysis
The trend component of a time series
can be linear or non-linear.
It is easy to isolate the trend component
using linear regression.
For linear trend use the model y = β0 + β1t +
ε.
For non-linear trend with one (major) change
in slope use the quadratic model y = β0 + β1t
+ β2t2 + ε 21
22. Seasonal Analysis
Seasonal variation may occur within a
year or within a shorter period (month,
week)
To measure the seasonal effects we
construct seasonal indexes.
Seasonal indexes express the degree to
which the seasons differ from the average
time series value across all seasons.
22
23. Computing Seasonal Indexes
Remove the effects of the seasonal and
random variations by regression analysis
yt = b 0 + b 1 t
ˆ
• For each time period compute the ratio
yt/yt > This is based on the Multiplicative Model.
which removes most of the trend variation
>
• For each season calculate the average of yt/yt
which provides the measure of seasonality.
• Adjust the average above so that the sum of averages
of all seasons is 1 (if necessary)
23
24. The additive and multiplicative models
There are two mostly used general models to
describe the composition of a time series
(Assuming no cyclical effects).
The multiplicative model y t = Tt × S t × R t
The additive model yt = Tt + St + Rt
In the seasonal index analysis performed here we use
the multiplicative model because it is mathematically
easier to handle.
24
25. Seasonal indexes analysis
The multiplicative model
y t Tt × S t × R t
= = St × R t
ˆ
yt Tt
The regression line represents trend.
yt
• We see that the ratio represents the seasonality and
ˆ
yt
the random effects.
• Averaging these ratios for each season type removes
the random effects and leaves the seasonality.
25
26. Seasonal indexes analysis
The multiplicative model
y t Tt × S t × R t
= = St × R t
ˆ
yt Tt
Rate/Predicted rate
1.5
Note how no trend is observed, but
1 seasonality and randomness
0.5
still exist.
0
1 3 5 7 9 11 13 15 17 19
yt
• Averaging the ratios for each season type removes
ˆ
yt
the random effects and leaves the seasonality. 26
27. Computing Seasonal Indexes
Example 20.3 (Xm20-03)
Calculate the quarterly seasonal indexes
for hotel occupancy rate in order to
measure seasonal variation.
Data:
Year Quarter Rate Year Quarter Rate Year Quarter Rate
1996 1 0.561 1998 1 0.594 2000 1 0.665
2 0.702 2 0.738 2 0.835
3 0.8 3 0.729 3 0.873
4 0.568 4 0.6 4 0.67
1997 1 0.575 1999 1 0.622
2 0.738 2 0.708
3 0.868 3 0.806
4 0.605 4 0.632
27
28. Rationale for seasonal indexes
Computing Seasonal Indexes
Perform regression analysis for the model
y = β0 + β1t + ε where t represents the time,
and y represents the occupancy rate.
Time (t) Rate ˆ
y = .639368 + .005246t
1 0.561
2 0.702
3 0.800
4 0.568
Rate
5 0.575
6 0.738
7 0.868
8 0.605 0 5 10 15 20 25
. . The regression line represents trend.
t
. .
28
29. yt / yt
>
The Ratios
t yt ˆ
yt Ratio
1 .561 .645 .561/.645=.870
2 .702 .650 .702/.650=1.08
3 ………………………………………………….
=.639368+.005245(1)
No trend is observed, but
yt seasonality and randomness
Rate/Predicted rate
ˆt
y still exist.
1.5
1
0.5
0
13
17
1
3
5
7
9
11
15
19
29
30. The Average Ratios by Seasons
Rate/Predicted rate
0.870
1.080
• To remove most of the random variation
1.221 but leave the seasonal effects,average
0.860 ˆ
the terms yt / yt for each season.
0.864
1.100
1.284 Rate/Predicted rate
0.888
0.865 1.5
1.067
1.046 1
0.854
0.5
0.879
0.993 0
1.122
1 3 5 7 9 11 13 15 17 19
0.874 Average ratio for quarter 1: (.870 + .864 + .865 + .879 + .913)/5 = .878
0.913
1.138
Average ratio for quarter 2: (1.080+1.100+1.067+.993+1.138)/5 = 1.076
1.181 Average ratio for quarter 3: (1.221+1.284+1.046+1.122+1.181)/5 = 1.171
0.900 30
Average ratio for quarter 4: (.860 +.888 + .854 + .874 + .900)/ 5 = .875
31. Adjusting the Average
Ratios
In this example the sum of all the averaged ratios
must be 4, such that the average ratio per season is
equal to 1.
If the sum of all the ratios is not 4, we need to adjust
them proportionately.
Suppose the(Seasonal averaged ratio) (number of seasons)
sum of ratios is equal to 4.1. Then each
Seasonal index be multipliedSum of averaged ratios
ratio will = by 4/4.1.
In our problem the sum of all the averaged ratios is equal to 4:
.878 + 1.076 + 1.171 + .875 = 4.0.
No normalization is needed. These ratios become the
seasonal indexes.
31
32. Interpreting the Seasonal
Indexes
The seasonal indexes tell us what is the ratio
between the time series value at a certain
season, and the overall seasonal average.
17.1% above the
In our problem: annual average
117.1%
7.6% above the 12.5% below the
annual average annual average
Annual average
occupancy (100%) 107.6%
87.8% 87.5%
12.2% below the
annual average
Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarter 1 Quarter 2 Quarter 3 Quarter 4
32
33. The Smoothed Time Series
The trend component and the seasonality
component are recomposed using the
y = ˆ ˆ t
ˆ
multiplicative model. T × S = (.639 + .0052t ) S
ˆ t t t
In period #1 ( quarter 1): ˆ ˆ ˆ
y1 = T1 × S1 = (.639 + .0052(1))(.878) = .566
ˆ ˆ ˆ
y 2 = T2 × S 2 = (.639 + .0052(2))(1.076) = .699
In period #2 ( quarter 2):
0.9
Actual series Smoothed series
0.8
0.7
0.6
0.5 The linear trend (regression) line
1 3 5 7 9 11 13 15 17 19
33
34. Deseasonalized Time Series
Seasonally adjusted time series = Actual time series
Seasonal index
By removing the seasonality, we can
identify changes in the other components of
the time series, that might have occurred
over time.
34
35. Deseasonalized Time Series
In period #1 ( quarter 1): y 1 / SI1 = .561/ .870 = .639
In period #2 ( quarter 2): y 2 / SI2 = .708 1.076 = .652
In period #5 ( quarter 1): y 5 / SI1 = .575 .870 = .661
There was a gradual increase in occupancy rate
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
35
36. 20.4 Introduction to
Forecasting
There are many forecasting models available
o o o
*
*
? *
o o
*
o * o o
*
* Model 1 * Model 2
Which model
performs better?
36
37. 20.4 Introduction to
Forecasting
A forecasting method can be selected
by evaluating its forecast accuracy
using the actual time series.
The two most commonly used measures of forecast accuracy
are: n
Mean Absolute Deviation ∑ y t − Ft
MAD = t =1
Sum of Squares for Forecast Error n
n
SSE = ∑ ( y t − Ft )2
t =1 37
38. Measures of Forecast
Accuracy
Choose SSE if it is important to avoid
(even a few) large errors. Otherwise,
use MAD.
A useful procedure for model selection.
Use some of the observations to develop several
competing forecasting models.
Run the models on the rest of the observations.
Calculate the accuracy of each model (by comparing to
actual data at a later time period).
Select the model with the best (lowest) accuracy measure.
38
39. Selecting a Forecasting
Model
Annual data from 1970 to 1996 were used to
develop three forecasting models.
Use MAD and SSE to determine which model
performed best for 1997, 1998, 1999, and
2000.
Forecast value
Year Actual y Model 1 Model 2 Model 3
1997 129 136 118 130
1998 142 148 141 146
1999 156 150 158 170
2000 183 175 163 180
39
40. Solution
For model
Actual y Forecast for y
1 in 1991 in 1991
129 −136 + 142 −148 + 156 −150 + 183 −175
MAD = = 6.75
4
SSE = (129 − 136) + (142 − 148) + (156 − 160) + (183 − 175) 2 = 185
2 2 2
Summary of results
Model 1 Model 2 Model 3
MAD 6.75 8.5 5.5
SSE 185 526 222
40
41. Selecting a Forecasting
Model
In this case, Model 2 is inferior to both Models 1
and 3
Using MAD Model 3 is best
Using SSE Model 1 is most accurate
Choice – prefer model that consistently
produces moderately accurate forecasts (Model
3) or one whose forecasts come quite close
usually but misses badly in a small number of
time periods (Model 1)
41
42. 20.5 Forecasting Models
The choice of a forecasting technique
depends on the components identified in
the time series.
The techniques discussed next are:
Exponential smoothing
Seasonal indexes
Autoregressive models (a brief discussion)
42
43. Forecasting with Exponential
Smoothing
The exponential smoothing model can
be used to produce forecasts when the
time series…
exhibits gradual(not a sharp) trend
no cyclical effects
no seasonal effects
Forecast for period t+k is computed by
Ft+k = St
t is the current period; St = ωyt + (1-ω)St-1 43
44. Forecasting with Exponential
Smoothing - Example
The quarterly sales (in $ millions) of a
department store chain were recorded for
the years 1997 – 2000
44
46. Forecasting with Exponential
Smoothing - Example
Si ngl e Ex ponent i al Smoot hi ng Pl ot f or Sal es
55
Y16 = 45
Variable
Actual
50 Fits
Smoothing Constant
45 Alpha 0.4
40
Accuracy Measures
MAPE 22.4486
S16 =
MAD 7.2311
41.8
Sales
MSD 69.2590
35
30
25
20
2 4 6 8 10 12 14 16
I ndex
46
47. Forecasting with Exponential
Smoothing - Example
Forecast for period t+k is computed by
(t = 16) Ft+k = St
t is the current period; St = ωyt + (1-ω)St-1
F17= S16 = 41.8
F18= S16 = 41.8
F19= S16 = 41.8
F20= S16 = 41.8
47
48. Computing Seasonal Indexes
Example 20.3 (Xm20-03)
Calculate the quarterly seasonal indexes
for hotel occupancy rate in order to
measure seasonal variation.
Data:
ˆ ˆ ˆ
yt = Tt × S t = (.639 + .0052t ) S t
ˆ
Year Quarter Rate Year Quarter Rate Year Quarter Rate
1996 1 0.561 1998 1 0.594 2000 1 0.665
2 0.702 2 0.738 2 0.835
3 0.8 3 0.729 3 0.873
4 0.568 4 0.6 4 0.67
1997 1 0.575 1999 1 0.622
2 0.738 2 0.708
3 0.868 3 0.806
4 0.605 4 0.632
48
49. Forecasting with Seasonal
Indexes; Example
Example 20.5
Use the seasonal indexes calculated in
Example 21.3 along with the trend line, to
forecast each quarter occupancy rate in 2001.
The solution procedure
Use simple linear regression to find the trend line.
Use the trend line to calculate the trend values.
To calculate Ft multiply the trend value for period t by the
seasonal index of period t.
49
50. Forecasting with Seasonal
Indexes; Example
The solution procedure
1. Use simple linear regression to find the trend line.
From Minitab (Using decomposition )
Yt = .639 + .00525t
50
51. Forecasting with Seasonal
Indexes; Example
The solution procedure:
2. Use the trend line to calculate the trend values
Yt = .639 + .00525t
We previously used 20 time periods
Quarter 1: 0.639 + 0.00525(21) = 0.749
Quarter 2: 0.639 + 0.00525(22) = 0.755
Quarter 3: 0.639 + 0.00525(23) = 0.760
Quarter 4: 0.639 + 0.00525(24) = 0.765
51
52. Forecasting with Seasonal
Indexes; Example
The solution procedure
3. To calculate Ft multiply the trend value for period t by
the seasonal index of period t.
Quarter 1: 0.749 X 0.878 = 0.658
Quarter 2: 0.755 X 1.076 = 0.812
Quarter 3: 0.760 X 1.171 = 0.890
Quarter 4: 0.765 X 0.875 = 0.670
We forecast that the quarterly occupancy rates for
the next year will be 65.8%, 81.2%, 89% and
67%.
52
53. Forecasting with Seasonal
Indexes: Example (Summary)
Solution
The trend line was obtained from the
regression analysis. y t = .639 + .00525t
ˆ
– For the year 2001 we have:
y21 = .639 + .00525 (21) = .749
ˆ F21 = .749(.878)
t Trend value Quarter SI Forecast
21 .749 1 .878 .658
22 .755 2 1.076 .812
23 .760 3 1.171 .890
24 .765 4 .875 .670
53
54. Problem 20-47
The revenues (in $millions) of a chain of
ice-cream stores are listed for each
quarter during the previous 5 years.
Use the handout with the plots etc. to
answer the questions.
54
56. 20-47
1. Plot the time series. Is there a
pattern? Does there appear to be
A long term trend
A cyclic effect
A seasonal effect
2. Plot the 3-year moving balance. Does
this change any of your answers from #
1?
56
57. 20-47 - Answers
Seems to be a long term trend – going
upward.
Does not seem to be a cyclic effect.
Because we know the data is in quarter
years, this seems to be a strong seasonal
effect.
The three year moving average plot
strengthens these observations.
57
58. 20-47
Why is exponential smoothing not
recommended as a forecasting tool in this
problem?
Look at the exponential smoothing plot (w
= 0.3) Does this confirm your answer
above?
58
59. 20-47 - Answers
Exponential smoothing is not
recommended as a forecasting tool for
this problem because of the strong
seasonal effect.
The 0.3 gives a large smoothing effect,
and the seasonality is no larger as
evident.
59
60. 20-47
Look at the trend analysis plots. Does the
Linear or the Quadratic trend model seem
to be better? Why? What is the equation
of the trend line?
60
61. 20-47 - Answers
The linear model seems to fit well. The
quadratic model doesn’t show much of a
curve. Checking the MAD for both models
shows 6.8716 for the linear model and
6.8784 for the quadratic model. Since the
linear model has a slightly smaller MAD
we will use it.
The equation of the trend line, then, is
Y = 20.2105 + 0.732331t
61
62. 20.47
Predict the sales for the next 4 quarters
using the trend analysis line.
62
64. 20-47
Using the seasonal indexes and the trend
line, forecast revenues for the next four
quarters.
64
65. 20-47 - Answer
The Fitted trend equation for the
decomposition plot is Y = 21.7511 +
0.567732 t
The next time period is number 21.
Y21 = [21.75 + 0.568 (21)] * 0.63581 =
21.41
Y22 = [21.75 + 0.568 (22)] * 1.05732 =
36.21
Y23 = [21.75 + 0.568 (23)] * 1.36851 = 65
66. 20-47 (continued)
If the actual sales numbers for the next
four quarters were 24, 40, 48, 36.
Calculate the MAD for the predictions
from the trend line and seasonal indexes.
66
68. 20-47
Go back to the linear trend equation
forecasts. If the actual sales numbers for
the next four quarters were 24, 40, 48, 36.
Calculate the MAD.
Which method of forecasting did a better
job in this case?
68
69. 20-47 Answers
MAD = |35.58 - 24| + |36.31 – 40| + |37.05 – 48|
+ |37.78 – 36| / 4
= (11.58 + 3.69 + 10.95 + 1.78 ) / 4
= 28/4 = 7 This is the MAD for the trend
analysis.
The MAD for the seasonal trend analysis was
2.43.
The seasonal trend analysis did the better job of
forecasting.
69