Quantitative Techniques to Transport Planning

Page 44 of 88 pages. Chapter: 8: Time Series and Forecasting

Techniques for Extracting the Trend

There are three techniques that can be used to extract a trend from a set of time series values.

a) Semi-Averages. This is the simplest technique, involving the calculation of two (x,y) averages which, when plotted on a chart as two separate points and joined up, form a straight line.
b) Least Squares Regression. This method, is covered in correlation and regression analysis topic.
c) Moving Average. This is the most commonly used method for identifying a trend and involves the calculation of set of averages.

The Method of Semi-Averages

The Semi-Average Method of extracting the trend is demonstrated with the example below;

Suppose the following sales (£00, to nearest £10) were recorded for a firm and it is required to obtain a semi-average trend

-
-
Week 1
-
-
-
-
Week 2
-
-
Day
Mon
Tue
Wed
Thu
Fri
Mon
Tue
Wed
Thu
Fri
Sales (y)
250
320
340
520
410
260
380
410
670
420

Procedure

a) Split the data into lower and upper group
Lower Group is 250 320 340 520 410
Upper Group is 260 380 410 670 420
b) Find the mean value of each group
Mean of Lower group (L) = <insert formula>
= 368
Mean of upper group (U) = <insert formula>
= 428
c) Plot on the graph, each mean against an appropriate time point.
An appropriate time point can always be taken as the median time point of the respective group, i.e. L would be plotted against Wednesday of week 1 and U against Wednesday of week 2.
d) The line joining the two plotted points is the required Trend.
Note that it is important that the two groups in question have an equal number of data values. If the given data, however, contains an odd number of data values, the middle value can be ignored (for the purposes of obtaining the trend line)

Once a trend line has been obtained, the trend values corresponding to each time point can be read off from the graph.

Example1

Using the data given below;

a) Use the method of semi-averages to obtain and plot a trend line
b) Draw up a table showing the original data (Y) values against the trend (T)

UK Outward Passenger Movement by Sea

Year
Quarter
Number of Passengers (millions)
1
1
2.2
2
5.0
3
7.9
4
3.2
2
1
2.9
2
5.2
3
8.2
4
3.8
3
1
3.2
2
5.8
3
9.1
4
4.1

Solution

The mean values both L and U must be plotted against a hypothetical point between the middle two time points in their respective sets. Thus L is plotted between Q₃ and Q₄ of Year 1 whilst U is plotted between Q₁ and Q₂ of Year 3. As shown below.

The two means (L and U) have been plotted joined by a straight line to form the trend line.

b) The trend values have been read from the graph and are tabulated below, together with the original data values.

Year
Quarter
Number of Passengers (millions)
Trend Values
1
1
2.2
3.9
2
5.0
4.1
3
7.9
4.3
4
3.2
4.5
2
1
2.9
4.7
2
5.2
4.9
3
8.2
5.2
4
3.8
5.4
3
1
3.2
5.6
2
5.8
5.8
3
9.1
6.0
4
4.1
6.2

The Method of Least Squares Regression

Procedure

1. Take the physical time points (i.e. quarters) as the independent variable x
2. Take the actual data values as dependent variable y.
3. Translate the Regression line of y on x , y = a + bx

Example2

Use the data in example 1 to calculate least squares regression and extract a trend component for each point given.

Solution

Make y = number of passenger and x = time point

Use the general formula y = a + bx.

Where:

And

Quarter
x
Number of Passengers (millions)
y
xy
x²
1
2.2
2.2
1
2
5.0
10.0
4
3
7.9
23.7
9
4
3.2
12.8
16
5
2.9
14.5
25
6
5.2
31.2
36
7
8.2
57.4
49
8
3.8
30.4
64
9
3.2
28.8
81
10
5.8
58.0
100
11
9.1
100.1
121
12
4.1
49.2
144
78
60.6
418.3
650

Therfore

b = 0.1706

and

a = 3.94

Thus the regression line for the Trend is T = 3.94 + 0.17x, the normal y has been replaced by ‘T’.

The time point values (x = 1,2,3,4,5,6, etc) can now be substituted into the above regression line to give the trend values required.

When x = 1 (i.e. Q₁, year 1) T= 3.94 + 0.17 (1) = 4.11
When x = 2 (i.e. Q₂, year 1) T= 3.94 + 0.17 (2) = 4.28 . . . etc

The Trend values from Least Squares regression are tabulated below:

x
y
Trend (T)
1
2.2
4.11
2
5.0
4.28
3
7.9
4.45
4
3.2
4.62
5
2.9
4.79
6
5.2
4.96
7
8.2
5.13
8
3.8
5.30
9
3.2
5.47
10
5.8
5.64
11
9.1
5.81
12
4.1
5.98

The Method of Moving Average

A moving average is an average of the results of a fixed number of periods

The moving average method is a technique used to find the Trend. This method attempts to remove seasonal variations from actual data by a process of averaging.

Moving Average of an Odd Number Results

Example:

Output at a factory appears to vary with the day of the week. Output over the last three weeks has been as follows:

Week 1
Week 2
Week 3
000 units
000 units
000 units
Monday
80
82
84
Tuesday
104
110
116
Wednesday
94
97
100
Thursday
120
125
130
Friday
62
64
66

Find

(a) The moving average trend values.
(b) Draw the graph of both output and trend on the same graph paper.

Solution

An average of five items, which coincide, with the length of the natural cycle of the series will be used.

(a)

Output (Y)
Five day Moving Total
Trend (T)
Week 1
Monday
80
-
-
Tuesday
104
-
-
Wednesday
94
460
92.00
Thursday
120
462
92.40
Friday
62
468
93.60
Week 2
Monday
82
471
94.20
Tuesday
110
476
95.20
Wednesday
97
478
95.60
Thursday
125
480
96.00
Friday
64
486
97.20
Week 3
Monday
84
489
97.80
Tuesday
116
494
98.80
Wednesday
100
496
99.20
Thursday
130
-
-
Friday
66
-
-

Procedure

The average in the 1st Five days (Mo, Tu, We, Th, Fr) period were

The average output in 2nd Five days ( Tu, We, Th, Fr,Mo) period were;

The average output in 3rd Five days (, We, Th, Fr,Mo,Tu) period were

Similarly, the other averages are calculated and are tabulated above.

(b)

Moving Average over an even number of Periods (Centered Moving Average)

When calculating moving averages with an even period (i.e. 4, 6, or 8), the resulting moving average would seem to have to be placed in between two corresponding time points. However, a trend value is required to coincide with a particular time points. Therefore centering process is deployed in this type of situation. (A moving average of two will be calculated on the first average trend.)

Example:

Calculate a moving average trend line of the following results

Year
Quarter
Volume of Sales in '000 units
2001
1
600
2
840
3
420
4
720
2002
1
640
2
860
3
420
4
740
2003
1
670
2
900
3
430
4
760

Solution

Year
Quarter
Volume of Sales
'000 units
4Qtr
Moving Total
8Qtr
Moving Total
Trend
Centered Moving Average
2001
1
600
-
-
-
2
840
-
-
-
3
420
2580
5200.00
650.00
4
720
2620
5260.00
657.50
2002
1
640
2640
5280.00
660.00
2
860
2640
5300.00
662.50
3
420
2660
5350.00
668.75
4
740
2690
5420.00
677.50
2003
1
670
2730
5470.00
683.75
2
900
2740
5500.00
687.50
3
430
2760
-
-
4
760
-
-
-

Procedure

The 1st 4 Qtr Moving Total =

The 1st 8 Qtr Moving Total =

The 1st Trend Value =

The other results are tabulated in the table above