Straight line Method of Least Square
The technique of this method, has also been explained in detail in the chapter, “ Regression Analysis”. To reiterate in short, under this method, a straight line called the lien of the best fit, is obtained by the following simple linear equation:
Yc = a + b X
Where, Yc = Computed trend values of Y i.e. that of the value variable
a = intercept of y variable i.e. the computed trend figure of Y variable when X =0
b = Slope of the trend line, or the amount of change in the Y variable with reference to a change of one unit in the X variable.
And X = the time variable, or its deviation from the time variable.
In the above equation, a and b are the numerical constants because, for any given straight line, their values do not at all changes. When the values of these two constants are obtained, the line of the best fit is completely determined. In order to obtain the values of these two constant a and b, the following two normal equations are to be solved simultaneously:
∑Y = Na + b∑X
∑XY = a∑X + b∑X²
It may be noted, here, that the first normal equation stated above is obtained by multiplying each set of relationship disclosed by the equation Yc = a + bX, by the respective coefficient of a, and getting them summed up. Similarly, the second normal equation has been derived by multiplying each set of the relationship by the respective coefficient of b, and getting them summed up.
By taking deviations of the time variable from its mid value or midpoint (in case of even number of items), if the value of (total of deviations of time variable) could be made zero, the values of a and b can be computed directly as under:
a = ∑Y/ N and b = ∑XY/ ∑X²
In view of the above advantage, an attempt should always be made to find the value of a and b directly in the above manner without resorting to the lengthy procedure of solving the simultaneous equations.
Further, if the series consists of even number of items, and therefore, deviations from the midpoint of the time variable come out with decimal points, in order to avoid such decimal numbers, the value of each deviation X is to be adjusted by the following formula:
X = t-midpoint ot t/ (1/2 of the interval in t)
Where, X = deviations of the time variable from its midpoint, and
t = time variable.
Furthermore, if the time variable appears to be advancing by equal intervals, viz : 2001,2002,2003 etc. the successive values of X may be taken as 1,2,3,4, etc. in place of the actual values for the sake of convenience in the computational works. This process of converting the original values into the successive small values is called ‘change of the origin’. If this process is adopted, it will not affect the trend values in any way. Hence, the analysis of time series is independent of the change of the origin.
