Trend Analysis:
In order to determine the trend relationship, it is necessary to plot several points for the demand levels experienced in the past. A smooth curve should then be drawn through the points by freehand. The form of curve (linear, exponential, or other) may then be specified by inspection or statistical method that best fits the historical data. Trend projection can then be made for future periods.
The free-hand method can be used to fit a line to the points by visual inspection so that the deviation of actual points above and below the trend line is nearly balanced. Though this method may provide quick trend projection, this is not a reliable approach because of its subjective character.
A refined free-hand method involves computing the average actual demand Y for all periods and having a straight line pass through Y marked at the centre of the time series X . It ensures that the sum of all the deviations shall be zero since the positive ones cancel out the negative ones, i.e. ∑ (Y - Y¯ ) = 0 . However, one may still be quite arbitrary. A better method is the method of semi-averages which involves chronological division of historical data into two parts. For each part, a mean is computed and marked at the centre of each sub-period. A line drawn through the two semi-averages must pass through the mean of all the data, again making the sum of all the deviations equal to zero.
For all lines passing through the point (Y¯ , X¯ ) the algebraic sum of deviations equals zero. In order to identify which one fits the data best, it is essential to examine their algebraic sum of squared deviations. The line with the smallest sum of squared deviations fits the data better than any other line and is called the line of least squares. This is unique and represents a reasonably objective approach for fitting a trend relationship to a set of actual points.
If Y = a + bX is the equation of the line of best fit, the intercept a and slope b can be determined from the following equations
Y = N a + b ∑ X
XY = a ∑ X + b ∑ X2
Where N is the number of data periods.