1. In practice,the method of least squares is widely used
In practice,the method of least squares is widely used. This is the mathematical method with the help of which a Trend Line is fitted to the data in such a way that the two conditions are satisfied ie.
1 𝑌−𝑌𝑐 =0
2 (𝑌−𝑌𝑐) 2 is minimum
Method Of Least Square

2. How to detremine constants ‘a’and ‘b’
Sums up equation,
we get 𝑌=𝑁𝑎+𝑏 𝑋
Now multiply eqn.by X we get 𝑋𝑌=𝑎 𝑋+𝑏 𝑋 2
N denotes the number of years. The equation is the summation of equation whereas equation is the summation of X multiplied to equation

3. Substitute the value of b in any one of the equation and we get 26
Substitute the value of b in any one of the equation and we get 26.9=6a+15(1.7) a = thus 𝑦 𝑐 = X By substituting the diff values of X in this trend line, the computed values of Y (i.e. 𝑦 𝑐 ) can be calculated
X=0
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟎 =𝟎.𝟐𝟑
X=1
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟏 =𝟏.𝟗𝟑
X=2
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟐 =𝟑.𝟔𝟑
X=3
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟑 =𝟓.𝟑𝟑
X=4
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟒 =𝟕.𝟎𝟑
X=5
𝒀 𝒄 =𝟎.𝟐𝟑+ 𝟏.𝟕 𝟓 =𝟖.𝟕𝟑

4. Variable X can be measured from any point of time in origin such as first year. The calculation,ofcourse,becomes simple when the mid point in time is taken as the origin because in that case the negative values in the half of the series balance out the positive values in the second half so that 𝑋=0 , the equation (ii) and (iii) can be written as
𝑌=𝑁𝑎
a = 𝑌 𝑁 = Y
And 𝑋𝑌=𝑏 𝑋 2
Or b = 𝑋𝑌 𝑋 2
The constant ‘a’ gives the arithmetic mean of Y and the ‘b’ shows the rate of change

5. Direct method to fit the Straight Line Trend
Example: fit a straight line to the following data taking X as the independent variable Solution:the straight line to be fitted to the data is Y = a+bX As there are two unknown a and b so we should have two normal equation i.e.
Direct method to fit the Straight Line Trend X:
1991
1992
1993
1994
1995
1996 Y: 1
1.8
3.3
4.5
6.3 10

6. 𝑌=𝑁𝑎+𝑏 𝑋 𝑋𝑌=𝑎 𝑋+𝑏 𝑋 2 X Y XY 𝑿 𝟐 𝒀 𝒄 1 0.23 1.8 1.93 2 3.3 6.6 4 3.631
0.23
1.8
1.93 2
3.3
6.6 4
3.63 3
4.5
13.5 9
5.33
6.3
25.2 16
7.03 5 10 50 25
8.73
𝑿=𝟏𝟓
𝒀=𝟐𝟔.𝟗
𝑿𝒀=𝟗𝟕.𝟏
𝑿 𝟐

7. Solving equation (i) and (ii) 26. 9 = 6a+15b 97
Solving equation (i) and (ii) 26.9 = 6a+15b 97.1 = 15a+55b Multiply eqn(i) by (5) and eqn. (ii) by (2) and subtract = 30a + 75b = 30a + 110b 35b = 59 b = = 1.7

8. By substituting the diff values of X in this trend line, the computed values of Y are calculated above. The straight line fitted to above data is represented with the help of a graph

9. Short Cut Method of Least Squares (odd number of items)
There is a short cut method which simplifies the above above procedure of trend computation. But it it can be applied if the number of items is odd. The middle year is taken as the year of origin and X value equal to zero is given to it. A minus sign is given to the X values prior to the year and a plus sign to those after the origin year. This technique is explained by the following example :

10. Eliminate the trend. What components of the series are thus left over?
Year
2001
2002
2003
2004
2005
2006
2007
production 77 88 94 85 91 98 90
Fit a straight line by least squares method and tabulate the trend value
Eliminate the trend. What components of the series are thus left over?
What is the monthly increase in the production of sugar
Solution: by taking middle year in the case of odd number of years the sum of the X values will be zero. This helps us in modifying the following normal equations. The set of two normal equations is Y = Na+ b 𝑋 𝑋𝑌=𝑎 𝑋+𝑏 𝑋 2

11. Year X Y XY
𝑿 𝟐
𝒀 𝒄
2001 -3 77
-231 9 83
2002 -2 88
-176 4 85
2003 -1 94
-94 1 87
2004 89
2005 91
2006 2 98
196 93
2007 3 90
270 95
N=7
𝑋=0
𝑌=623
𝑋𝑌=56
𝑋 2 =28

12. 𝑌=𝑁𝑎 ∴𝑏 𝑋=0
𝑋𝑌=𝑏 𝑋 2 ∴𝑎 𝑋=0
Substituting the values calculated in the table we get,
623= 7a or a = =89
56= 28b or b = = 2
Now the straight line trend cane be written as
𝑌 𝑐 =89+2𝑋

13. The computed trend values will be
X = -3
𝒀 𝒄 =𝟖𝟗+𝟐 −𝟑 = 83
X = -2
𝒀 𝒄 =𝟖𝟗+𝟐 −𝟐 = 85
X = -1
𝒀 𝒄 =𝟖𝟗+𝟐 −𝟏 = 87
X = 0
𝒀 𝒄 =𝟖𝟗+𝟐 𝟎 = 89
X = 1
𝒀 𝒄 =𝟖𝟗+𝟐 𝟏 = 91
X = 2
𝒀 𝒄 =𝟖𝟗+𝟐 𝟐 = 93
X = 3
𝒀 𝒄 =𝟖𝟗+𝟐 𝟑 = 95

14. (iii) The monthly increase in the production of sugar is 2 12 =0.167
(ii) Components of time series are : (i) secular trend (ii) seasonal variations (iii) cyclical variations and (iv) irregular or random variations. When secular trend is eliminated from the time series data, we are left with seasonal, cyclical and random fluctuations
(iii) The monthly increase in the production of sugar is =0.167

16. Thank You