This method of trend fitting is resorted to only when the value variable Y shows a geometric progression viz : 1,2,4,8,16,32, and so on, and the time variable (t) shows an arithmetic progression viz : 1,2,3,4,5,6, and the like In such cases, the trend line is to be drawn on a semi logarithmic chart in the form of a straight line, or a non-linear curve to show the increase, or decrease of the value variable Y at a constant rate rather than a constant amount. When the trend takes the form of a non-linear curve on a semi-logarithmic chart, an upward curve indicates the increase at varying rates depending upon the shape of the slopes. The steeper the slope, the higher is the rate of increase.
However, under this method, the trend line is fitted by the following model:
Yc = abX
Using the logarithmic operation, the above equation is modified as under:
Yc = A.L. (log a + X log b)
In the above equation, a and b is the two constants the values of which are determined by solving the following two normal equations and finding the antilogarithms thereof:
∑ log y = N log a + log b ∑ X
∑ X log y = log a ∑ X + log b ∑ X²
If by taking the time deviations X from the mid-point of the time variable, t, ∑ X could be made zero, the logarithm of the two constants a and b can be determined directly as under:
log a = ∑ X log y/ ∑ X²
log b = ∑ X log y/ ∑ X²
After obtaining the values of a and b in the above manner, and substituting their values in the equation Yc = abx, we can fit the trend lien equation under this method, and with such an equation we can very well estimate the trend values of the time series, and predict the value for any past future year as well.