A textbook of Computer Based Numerical and Statiscal Techniques part 46 pptx

x be the average quarterly or monthly sales in a year and I be the Seasonal index of that quarter or month, then Sale for the Qaurter or Month =1% of = 1 100 The following methods are co

Using (2), a = y

N

∑ = 290

5 = 58 and b =

xy x

∑

∑ 2 = 34

10 = 3.4

From (1), the required equation of the best fitted straight-line is Y = 58 + 3.4x.

Year x Trend Values (y = 58 + 3.4x)

Example 8 Fit a straight-line trend equation by the method of least square and estimate the trend

value.

Sol Here N = Number of years = 8, which is even

Let the straight line trend equation by the method of least squares with the origin at the mid

point of 1964 and 1965, and unit of x as 1/2 year be

Then a and b are given by

N

∑ and b = xy

x

∑

Calculations for fitting the straight-line trend

2

2

Using (2), 734 91.75

8

y a N

2

210 1.25 168

xy b

x

∑

∴From (1), the required equation of the straight-line trend is

91.25 1.25

(y = 91.75 + 1.25x)

Note: If the number of years is even, there is no middle year and in this case the midpoint, which is take as the origin, lies midway between the two middle years In example 8, the

midpoint (i.e., the origin) lies midway between July 1, 1964 and July 1, 1965, which is January

1, 1965 (or December 31, 1964) To avoid fractions, the units of x are taken as 1/2 year (or 6 months)

10.4.2 Analysis of Seasonal Variation

Seasonal variations are short term fluctuations in recorded values due to different circumstances, which affect results at different times of the year, on different days of the week, at different times of day or whatever

Seasonal variations are measured through their indices called the seasonal indices The measurement of seasonal variations requires determining the seasonal component st which indicates how a times series from quarter to quarter, month to month, or week to week, etc through out a year A series of numbers showing relative values of a variable during the quarters or months or weeks etc of the year is called seasonal index for the variable

If Rs x be the average quarterly (or monthly) sales in a year and I be the Seasonal index

of that quarter (or month), then

Sale for the Qaurter or Month =1% of = 1

100

The following methods are commonly used for measuring seasonal variations

(a) Method of Averages (Quarterly, Monthly or Weekly)

(b) Moving Average Method

(c) Ratio to Trend Method

(d) Link Relative Method

(=) Method of Averages: This method is used when trend and cyclical fluctuations, if any, have little effect on the time series

If quarterly data are given, first find quarterly totals for each quarter and the averages for the four-quarter of the years To find these averages, we divide the quarterly totals by the number of the years for which the data are given Then we find grand average of the

4 quarterly averages

Grand average = 1+ 2+ 3+ 4

4

G

If we use multiplicative identity, then the seasonal indices are the 4 quarterly averages expressed as percentages of the grand average G

i.e., x1 100, x2 100, x3 100, x4 100

Similarly, if we use additive model, then seasonal variations for the 4 quarters are

1 , 2 , 3 , 4

x −G x −G x −G x −G

When monthly or weekly data are given, we find monthly (or weekly) averages for the

12 months or (52 weeks)

Example 9 Calculate Seasonal indices for each quarter from the following percentages of wholesale

price indices to their moving averages.

Year Quarter

Sol Calculation for Seasonal Indices

Year Quarter

Total 51.0 53.0 52.0 54.0

Averages 12.75 13.25 13.0 13.5

Grand Average (G) 12.75 13.25 13.0 13.5

4

= 52.5 13.125 4

Using Multiplicative model, Seasonal index ( ) 100

( )

i Average x Grand Average G

∴ Seasonal indices for the first, second, third, and fourth quarters are respectively

12.75 100 97.14

13.125× =

13.25 100 100.95

13.125× =

13.0 100 99.95

13.125× =

13.5 100 102.86

13.125× =

Example 10 Compute the Seasonal Index for the following data:

Qaurters

Sol Let Cyclical Fluctuations and Trend are absent in the given data

Averages b gx i 87.75 68.75 63.75 79.25 (Total/4)

Grand Average (G) = 87.75 68.75 63.75 79.25

4

= 299.50

4 = 74.875

Now, using Multiplicative model,

Seasonal index Average x

Grand Average G

i

( ) ( )×100 Hence Seasonal indices for the 1st,2nd, 3rdand 4th quarters are respectively

1st = 87 75

74 875 100

× = 117.20 2nd= 68 75

74 875 100

3rd = 63 75

74 875 100

4th= 79 25

74 875 100

× = 105.84 Hence the sum of Seasonal indices is 400

Similarly, we can obtain Seasonal indices using additive model Using additive model,

Seasonal index = Average NE) – Grand average (G) Therefore 1st, 2nd, 3rd and 4th quarters are respectively

12.875, – 6.125, –11.125, 4.375 Hence the sum of seasonal indices for the four quarters is

12.875 + 4.375 – 11.125 − 6.125 = 0 Ans

(>) Moving Average Method (or ratio to moving averages): This is a improved method over method of averages and is widely used for measuring seasonal variation According

to this method, if monthly data are given, we find 12-month centred moving averages, if quarterly data are given, we find 4 quarter centred moving averages and so on This represent trend and then eliminate the effect of trend by using either additive model or multiplicative model

Case 1: If multiplicative model is used, then express the original data as percentage of the corresponding moving averages expressed as percentage that is ratio to moving averages expressed as percentage These percentages for corresponding months or quarters are then averaged by the method of averages, gives the required seasonal indices This method is known as Ratio to Moving Average Method

Example 11 Calculate Seasonal indices by the ratio to moving average method from the following

data.

Iron Prices (In Rupees Per Kg.)

Calculation of Ratio to moving averages

Year/Quarter Prices 4 Quarter 2 Point 4 Quarter Ratio to

total total (2-pt Moving

÷8)

63.375× =

65.375× =

67.125× =

70.875× =

74.000× =

75.375× =

76.625× =

77.625× =

79.500× =

83.000× =

84.750× =

Calculation for Seasonal Indices

Quarter

Grand Average (G) 122.02 92.17 84.46 100.23 99.72

4

3 Seasonal indicies for 4 quarters are respectively

Q1 122.02 100 122.36

99.72

Q2 92.17 100 92.43

99.72

Q3 84.46 100 84.70

99.72

Q4 100.23 100 100.51 99.72

Hence sum of seasonal indices is 400 Ans

Case 2: If the additive model is used, to eliminate trend subtract the moving averages from the original data and also obtain deviations from trend Now apply the method of averages to these deviations to obtain required seasonal variations

Example 12 Obtain Seasonal Fluctuation from the following time series using moving averages

method.

Quaterly output for 4 years

Sol Calculation of moving averages and deviations from trend

Year/Quarter Output 4-Quarter 2-Point 4-Quarter Deviation from

Moving total Moving total Moving Average Trend

Calculate the Seasonal Fluctuations

Deviation from Trend

Grand Average (G) = 5.37 + (–1.29) + (2.96) + (–0.96) = 0.16÷ 4 = 0.04 Therefore the seasonal functions are (x i−G) i.e.,

5.37 – 0.04 = 5.33 –1.29 – 0.04 = –1.33 2.96 – 0.04 = 2.92 –0.96 – 0.04 = –1.00 respectively (? ) Ratio to Trend Method: In this method trend values are first determined by the method of least squares fitting a mathematical curve and the given data are expressed as percentage of the corresponding trend values Using the multiplicative identity these percentages are then averaged by the method of averages

R S|

T|

U V|

W|

(@) Link Relative Method: According to this method for given data for each quarter or m089onth are expressed as percentage of data for the preceding quarter or month These percentages are known as Link relatives The link relative for the first quarter (or 1st month) of the year cannot be determined An appropriate average (Arithmetic Mean or Median) of the link relatives for each quarter (or month) is then found From these average link relatives, we find the chain relative with respect to 1st quarter (or 1st month)

for which the chain relative is taken as 100 If Q1, Q2, Q3, Q4 denotes 4 quarters respectively and chain relative represents by C.R., or link relative represents by L.R then

C.R for Q2 = (Average L.R for Q2 × C.R for Q1) ÷ 100

C.R for Q3 = (Average L.R for Q3 × C.R for Q2) ÷ 100

C.R for Q4 = (Average L.R for Q4 × C.R for Q3) ÷ 100 and 2nd C.R for Q1 = (Average L.R for Q1 × C.R for Q4) ÷ 100

Generally, the 2nd C.R for Q1 will be either higher or lower than the first C.R 100 for Q1

depending on the presence of an increase or decrease in trend

If d = 2nd C.R for Q1–100, i.e., the difference between 1st and 2nd C.R for Q1 then we subtract

(1÷4) (2÷4)d, (3÷4) (4÷4)d From the chain relatives for Q2, Q3 and Q4 and the 2nd C.R for Q1 respectively to obtain the adjusted chain relatives These adjusted chain relatives expressed as percentages of their A.M gives the required Seasonal indices

Example 13 Calculate seasonal indices by method of link relatives from the data given in

Example 12.

Sol Calculate of Average Link Relatives

Link Relatives Year/Quarter Quarter-I Quarter-II Quarter-III Quarter-IV

The link relative (L.R.) for the 1st quarter Q1 of the first year 1998 cannot be

determined For the other three quarters of 1998,

L.R for Q2 = (58÷65) × 100,

L.R for Q3 = (56÷58) × 100,

L.R for Q4 = (61÷56) × 100, Similarly, we determine the other link relative

From the average link relatives obtained in the last row of the above table, we now

find chain relatives, taking 100 as the chain relative (C.R.) for Q1

C.R for Q1 = 100 C.R for Q2 = (89.46 × 100) ÷ 100 = 89.46

C.R for Q3 = (96.05 × 89.46) ÷ 100 = 85.93 C.R for Q4 = (105.47 × 85.93) ÷ 100 = 90.63 2nd C.R for Q1 = (110.45 × 90.63) ÷ 100 = 100.10

∴ d = 100.10 – 100 = 0.10;

4d= The adjusted chain relatives are respectively

100, 89.46 – 0.025, 85.93 – 0.05, 90.63 – 0.075,

A.M of the adjusted chain relative = (100 + 89.44 + 85.88 + 90.58) ÷ 4 = 91.47

The required seasonal indices are 100 100, 89.44 100, 85.88 100, 90.56 100

91.47 × 91.47× 91.47× 91.47×

i.e., 109.33, 97.78, 93.89, 99.00. Ans

10.4.3 Analysis of Cyclical Fluctuation or Cyclic Variations

To analyse Cyclical fluctuation, we first find trend (T) and seasonal Variation (S) by any suitable

method i.e., by the method of moving averages or other method and then eliminate them from

the original data by using additive or multiplicative identity Irregular movement is removed

by using moving average of appropriate period depending average duration of irregular movement, leaving only cyclical fluctuation

10.4.4 Analysis of Irregular of Random Movements

Irregular movements are obtained by eliminating trend (T), seasonal variation (S) and cyclical fluctuation (C) from the original data Normally irregular movements are found to be of small magnitude

The time series analysis is of great importance not only to a businessman, scientist or economist, but also to people working in various disciplines in natural, social and physical sciences Some

of its uses are:

1 It enables us to study the past behaviour of the phenomenon under consideration, i.e.,

to determine the type and nature of the variations in the data

2 The segregation and study of the various components is of paramount importance to

a businessman in the planning of future operations and in the formulation of executive and policy decisions

3 It enables us to predict or estimate or forecast the behaviour of the phenomenon in future, which is very essential for business planning

4 It helps us to compare the changes in the values of different phenomenon at different times or places, etc