Statistics lecture 13 (chapter 13)
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Transcript of Statistics lecture 13 (chapter 13)
1
2
• Sequence of observations of the same variable
taken at equally spaced points in time.
• Each observation records both the value of the
variable and the time it was made.
• Tell us where we are and suggest where we are
going.
• Time series data are used to predict future values
for forecasting.
3
• Can reveal the main features of a time series.
• Look for the overall pattern and for deviations from the
pattern.
• Time on x-axis, measured variable on y-axis.
• Plot the points and connect them with straight lines.
Time plots
4
Time plots – Example
SA Fuel Price
0
100
200
300
400
500
600
700
800
J F M A M J J A S O N D J F M A M J J A S
Months 2008 - 2009
Ra
nd
5
• Trend – (T)
• Seasonal variations – (S)
• Cyclical Variations – (C)
• Irregular (random) variations – (I)
Components of a time series
6
• Trend – (T)
– Overall smooth pattern and show long-term upward or
downward movement.
– Trend analysis isolate the long-term movement and is
used to make long-term forecasting.
Components of a time series
7
• Seasonal variation – (S)
– Rises and falls occurring in particular times of the year
and repeated every year.
• Period of times may be years, months, days, hours or
quarters.
– Seasonal effect can be taken into account to evaluate
activity and can be incorporated into forecasts of future
activity.
Components of a time series
8
• Cyclical variation – (C)
– Patterns that repeat over time periods that exceed one
year.
– Time period for the cycle usually differ form each other.
• Business cycles – recession, depression, recovery or boom.
• Changes in governmental monetary and fiscal policy, etc.
Components of a time series
9
• Irregular variation – (I)
– Variation left after the trend, seasonal and cyclical
variations have been removed.
– Have an irregular, saw-tooth pattern.
– Cannot be predicted.
– Unusual events.
• Political events, war, riots, strikes, etc.
– Cannot be analysed statistically or forecasted.
Components of a time series
CONCEPT QUESTIONS
• Questions 1-5 ,p462, textbook
10
11
• Multiplicative time series model:
– Original observed value Yt
– Yt = TSCI
• Decomposing a time series into four components.
• Isolate the influence of each of the four components.
• Statistical methods can isolate trend and seasonal
variations.
• To isolate cyclical and irregular variation is of less value.
Decomposition of a time series
12
• Shows the general direction in which the series is
moving.
– Regression analysis – linear trend line
– Moving average method – smooth curve
Decomposition – Trend analysis
13
• ŷt = a + bx
– ŷt = estimated time series values
– x = time
• List the values of x and yt
– Code x
• 1st time period – x =1
• 2nd time period – x =2 ……….
• nth time period – x = n
– yt = original time series values
Decomposition – Trend analysis – Linear trend
14
• Determine the values of a and b:
Decomposition – Trend analysis – Linear trend
1 12
2 6
1 1
21 12
( 1) ( 1)(2 1)
t tn n
XX XY t tn n
XY
XX
t
x n n x n n n
y y x x
S x x S xy x y
Sb
S
a y bx
15
• ŷt = a + bx
• Substitute each value of xi into the trend equation to
find the trend component.
• Draw the trend line on the same graph as the
original time series.
• The trend line can now be used to estimate future
values of the dependent variable (ŷt).
Decomposition – Trend analysis – Linear trend
16
• The table below lists the quarterly number of foreign visitors at a game ranch in Limpopo for the past 3 years.
Decomposition – Linear trend – Example
I II III IV I II III IV I II III IV
23 59 64 32 26 45 69 29 15 36 47 38
2006 2007 2008
17
Decomposition – Linear trend – Example
Visitors at a game ranch
0
10
20
30
40
50
60
70
80
I II III IV I II III IV I II III IV
Quarters 2006 - 2008
Nu
mb
er
of
vis
ito
rs
18
• Determine the values of a and b
Decomposition – Linear trend – Example
1 1
2 2
1 12
6 6
1
12
1
12
( 1) 12(12 1) 78
( 1)(2 1) 12(12 1)(2(12) 1) 650
483
3044
(483) 40,25
(78) 6,5
t
t
t
x n n
x n n n
y
xy
y
x
I II III IV I II III IV I II III IV
yt 23 59 64 32 26 45 69 29 15 36 47 38
x 1 2 3 4 5 6 7 8 9 10 11 12
2006 2007 2008
19
21 12 2
12
1 1
12
650 (78) 143
650 (78)(483) 95,5
95,50,668
143
40.25 ( 0,668)(6,5) 44,592
ˆ 44,592 0,668
XX n
XY t tn
XY
XX
t
t
S x x
S xy x y
Sb
S
a y bx
y x
• Determine the values of a and b:
Decomposition – Linear trend – Example
20
• Determine the values of the isolated trend component
Decomposition – Linear trend – Example
ˆIf x = 1 then 44,592 0,668(1) 43,924ty
ˆIf x = 2 then 44,592 0,668(2) 43,256ty
21
• Plot the trend line on the graph
Decomposition – Linear trend – Example
Visitors at a game ranch
0
10
20
30
40
50
60
70
80
I II III IV I II III IV I II III IV
Quarters 2006 - 2008
Nu
mb
er
of
vis
ito
rs
ˆ 44,592 0,668ty x
• Determine the estimated number of visitors for the next
four quarters.
• Determine the x-values for the next four quarters.
22
• Forecast the values for the next four quarters
Decomposition – Linear trend – Example
ˆIf x = 13 then 44,592 0,668(13) 35,908ty
ˆIf x = 14 then 44,592 0,668(14) 35,24ty
IMPORTANT
• Forecasting in this way assumes the same
linear trend holds true for future time
periods
• Remember:-
23
ALSO IMPORTANT
• Without seeing the trend line graphically it is still possible
to determine whether the trend is increasing or
decreasing over time
• Slope of line is given by b. If b is +ve the slope is +ve
and the trend is increasing over time. If b is –ve the
slope is –ve and the trend is decreasing over time
• The strength of the trend influence can be assessed by
looking at b. A string upward/downward trend is shown
by large +ve/-ve values of b. Values of b close to 0
indicate a wek trend
24
EXAMPLE The table below shows the annual expenditure of Exel Ltd
on salaries (in R100,000), for each semester for a 4 year
period.
A. What is the value of the slope of the linear trend line for
this time series?
B. What is the value of the intercept (line crosses Y axis)
of the linear trend line for this time series?
C. Forecast the expenditure on salaries (in rands) for the
second semester in 2006
25
Year Semester1 Semester 2
2002 140.3 160.6
2003 139.6 158.2
2004 141.4 163.8
2005 143.5 167.3
EXAMPLE ANSWER
26
A.
x = 36
x2 = 204
x = 4,5
y = 1 214,7
y = 151,8375
xy = 5 545,8
SXX = 42 SXY = 79,65
b =
SXY
SXX
=
79,65
42
= 1,8964 B.
a =
y – b
x
= 151,8375 – 1,8964(4,5) = 143,3037
C.
ˆ y = 143,3037 + 1,8964(10) = 162,2677
27
• Removes the short term fluctuations in a time series.
– Smoothing a time series
• Remove the effect of seasonal and irregular
variations.
• Reflect the trend and cyclical movements.
– TC
Decomposition – Trend analysis – Moving average
28
• How to calculate a k-point moving average if k is odd
Decomposition – Trend analysis – Moving average
Time (X) Price (yt)3-point moving
average
5-point moving
average
2008 - O 564.03
N 519.03 480.37
D 358.03 381.02 406.81
2009 - J 265.98 317.00 368.40
F 326.98 321.65 338.09
M 371.98 355.48 339.38
A 367.48 367.98 362.06
M 364.48 370.45 379.94
J 379.38 386.75 383.54
J 416.38 395.25 395.23
A 389.98 410.76
S 425.93
564.03 519.03 358.03
3
480.37
519.03 358.03 265.98
3
381.02
29
• How to calculate a k-point moving average if k is odd
Decomposition – Trend analysis – Moving average
Time (X) Price (yt)3-point moving
average
5-point moving
average
2008 - O 564.03
N 519.03 480.37
D 358.03 381.02 406.81
2009 - J 265.98 317.00 368.40
F 326.98 321.65 338.09
M 371.98 355.48 339.38
A 367.48 367.98 362.06
M 364.48 370.45 379.94
J 379.38 386.75 383.54
J 416.38 395.25 395.23
A 389.98 410.76
S 425.93
EXAMPLE Calculate a 3 point moving average and a 5 point moving
average for the 2011 salaries of Rinto Ltd. The monthly
salary data (in R10,000’s) is as follows:-
30
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Sal (y) 8 4 6 10 8 10 13 5 11 14 10 11
Example answer
• Example 13.3, p470, textbook
31
32
• How to calculate a k-point
moving average if k is
even.
Decomposition
– Trend analysis
– Moving average
33
• Plot the original time series data and moving average.
Decomposition – Trend analysis – Moving average
Visitors at a game ranch
0
10
20
30
40
50
60
70
80
I II III IV I II III IV I II III IV
Quarters 2006 - 2008
Nu
mb
er
of
vis
ito
rs
yt
centred
moving
average
EXAMPLE Calculate a 4 point moving average average for the 2011
salaries of Rinto Ltd. The monthly salary data (in R10,000’s)
is as follows:-
34
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Sal (y) 8 4 6 10 8 10 13 5 11 14 10 11
Example answer
• Example 13.4, p471, textbook
35
TERM OF MOVING AVERAGE
Given a set of time series data how do we
chose an appropriate moving average
term (k) for the series?
Months 12
Quarters 4
Workdays 5
36
37
• Isolates the seasonal components in a time series.
• Dominates short-term movement.
• Find seasonal index for each period.
– Specific seasonal index
• specific year
• short term
– Typical seasonal index
• number of years
• long term
Decomposition – Seasonal analysis
MOVING AVERAGES
• Dampens short term fluctuations
• Shorter terms still show some variations; longer
terms produce a much smoother curve
• Disadvantages of moving averages:-
• Loss of information on both sides of the series
• Not a specific mathematical equation therefore
cannot be used in isolation to make objective
forecasts
38
SEASONAL ANALYSIS
• Isolates the seasonal component in a time
series
• Most business and economic time series
contain seasonal variations
• To isolate the seasonal component we
must find a seasonal index for each time
period
• Seasonal indices important because they
are used to forecast future values 39
SEASONAL INDICES
• Two types:-
– Specific seasonal index – measures
seasonal change during a specific year
– Typical seasonal index – measures
seasonal changes over a number of years
40
41
• Use moving average to smooth time series:
– Isolates trend and cyclical variations – Yt = TC
–
• Find a typical seasonal index for each period.
• Remember that sum of k mean seasonal indices must = k x 100.Calculate a series of seasonally adjusted values: –
• Construct a trend line for the seasonally adjusted data.
• Construct forecasts of the time series values.
Decomposition – Seasonal analysis
– Ratio-to-moving-average method
100 100 - seasonal and irregulart
TSCIY SI
TC
100 100t
TSCIY TCI
S
42
Decomposition
– Seasonal analysis
– Ratio-to-moving-average
method
43
64100 142.62
44.88
36100 109.51
32.88
TSCISI
TC
TSCISI
TC
44
Yt =
TCSI/TC=SIYt = TCI
42.81
53.52
142.62 40.10
73.56 20.05 I II III IV
61.36 48.39 2006 142.62 73.56
105.57 40.82 2007 61.36 105.57 168.81 75.57
168.81 43.23 2008 43.48 109.51 Total
75.57 37.94 Mean 52.42 107.54 155.715 74.565 390.24
43.48 27.92 Typical SI 53.73 110.23 159.61 76.43 400
109.51 32.66
29.45
49.72
Calculate the mean
index for each quarter:
45
Yt =
TCSI/TC=SIYt = TCI
42.81
53.52
142.62 40.10
73.56 20.05 I II III IV
61.36 48.39 2006 142.62 73.56
105.57 40.82 2007 61.36 105.57 168.81 75.57
168.81 43.23 2008 43.48 109.51 Total
75.57 37.94 Mean 52.42 107.54 155.715 74.565 390.24
43.48 27.92 Typical SI 53.73 110.23 159.61 76.43 400
109.51 32.66
29.45
49.72
100 400adjustment factor = 1,025
mean SI 390,24
52,42 1,025 53,73
k
Sum of mean indices = 390,24
Must be adjusted to 400
46
Time
(x)
yt =
TCSI
Yt =
TCSI/TC=SIYt = TCI
2006 - I 1 23 42.81
II 2 59 53.52
III 3 64 142.62 40.10
IV 4 32 73.56 20.05
2007 - I 5 26 61.36 48.39
II 6 45 105.57 40.82
III 7 69 168.81 43.23
IV 8 29 75.57 37.94
2008 - I 9 15 43.48 27.92
II 10 36 109.51 32.66
III 11 47 29.45
IV 12 38 49.72
I II III IV
2006 142.62 73.56
2007 61.36 105.57 168.81 75.57
2008 43.48 109.51 Total
Mean 52.42 107.54 155.715 74.565 390.24
Typical SI 53.73 110.23 159.61 76.43 400
Seasonal adjusted value = TCI
23100 42,81
53,73
47
Time
(x)
yt =
TCSI
Yt =
TCSI/TC=SIYt = TCI
2006 - I 1 23 42.81
II 2 59 53.52
III 3 64 142.62 40.10
IV 4 32 73.56 20.05
2007 - I 5 26 61.36 48.39
II 6 45 105.57 40.82
III 7 69 168.81 43.23
IV 8 29 75.57 37.94
2008 - I 9 15 43.48 27.92
II 10 36 109.51 32.66
III 11 47 29.45
IV 12 38 49.72
I II III IV
2006 142.62 73.56
2007 61.36 105.57 168.81 75.57
2008 43.48 109.51 Total
Mean 52.42 107.54 155.715 74.565 390.24
Typical SI 53.73 110.23 159.61 76.43 400
Seasonal adjusted value = TCI
29100 37,94
76,43
48
• Determine the trend line:
Decomposition – Seasonal analysis
1 1
2 2
1 12
6 6
1 1
12 12
( 1) 12(12 1) 78
( 1)(2 1) 12(12 1)(2(12) 1) 650
466,61
2941,78
(466,61) 38,88 (78) 6,5
t
t
t
x n n
x n n n
y
xy
y x
49
21 12 2
12
1 1
12
650 (78) 143
2941,78 (78)(466,61) 91,185
91,1850,638
143
38,88 ( 0,638)(6,5) 43,027
ˆ 43,027 0,638
XX n
XY t tn
XY
XX
t
t
S x x
S xy x y
Sb
S
a y bx
y x
Decomposition – Seasonal analysis
• Determine the trend line:
Seasonally
adj data
50
• Determine the values of the isolated trend component:
Decomposition – Seasonal analysis
ˆIf x = 9 then 43,027 0,638(9) 37,29
ˆIf x = 14 then 43,027 0,638(14) 34,10
t
t
y
y
51
• Determine the real predicted values:
Decomposition – Seasonal analysis
34,73 53,73 33,46 159,6118,7 53,401
100 100
52
• Represent the real predicted values graphically:
Decomposition – Seasonal analysis
Visitors at a game ranch
0.00
10.00
20.00
30.00
40.00
50.00
60.00
I II III IV I II III IV I II III IV I II III IV
Quarters 2006 - 2009
Nu
mb
er
of
vis
ito
rs
EXAMPLE MONTH PRICE
Jan 355
Feb 326
Mar 371
Apr 375
May 389
Jun 365
Jul 362
Aug 351
Sep 346
Oct 364
Nov 399
Dec 338
53
Example Answer
See answer sheet
54