Trends
The trends were plotted on the arithmetic scales. The Trends may also be plotted on a semi-logarithmic chart in the form of a straight line or a non-linear curve.
The trends are usually computed by using logarithms by;
Exponential trends
The equation of the exponential curve is of the form
Y = abx
Putting the equation in logarithmic form, we get
Log Y = log a + X log b.
The Exponential trends when plotted on the semi-logarithmic graph, the curve provides a starlight line. However, in an arithmetic chart the curve gives a non-linear trend. As to obtain the value of the constants a & b, the two normal equations to be solved are as follows:
∑log Y = N log a + log b ∑X
∑(X. log Y) = log a ∑ X + log b ∑ X2.
A is the Y intercept and b the slope of the curve.
When deviations are taken from middle year, ∑X = 0, the above equation takes the following form:
∑ Log y = N log a ∴ log a = ∑ log Y / N
And ∑ (X.log Y) = log b ∑ X2 ∴ log b = ∑(X log Y) / ∑X2
Illustration: - the sales of a company in millions of dollars for the years 2003 to 2009 are given below:
|
Year
|
2003
|
2004
|
2005
|
2006
|
2007
|
2008
|
2009
|
|
Sales [100000($)]
|
32
|
47
|
65
|
92
|
132
|
190
|
275
|
Find trend values by using the equation Yc = abx and estimate the value for 2010.
Solution: -
Fitting equation of the form Y = abx
|
Year
|
Sales (Y) [100000 ($)]
|
X
|
Log Y
|
X2
|
X. log Y
|
|
2003
|
32
|
-3
|
1.5051
|
9
|
-4.5153
|
|
2004
|
47
|
-2
|
1.6721
|
4
|
-3.3442
|
|
2005
|
65
|
-1
|
1.8129
|
1
|
-1.8129
|
|
2006
|
92
|
0
|
1.9638
|
0
|
0
|
|
2007
|
132
|
+1
|
2.1206
|
1
|
+2.1206
|
|
2008
|
190
|
+2
|
2.2788
|
4
|
+4.5576
|
|
2009
|
275
|
+3
|
2.4393
|
9
|
+7.3179
|
|
N=7
|
|
∑X = 0
|
∑log Y =13.7926
|
∑X2 = 28
|
∑X. log Y=4.3237
|
Log a = ∑ logy / N = 13.7926 / 7 = 1.9704;
Log b = ∑X. log Y/∑X2 =4.3237/28 = 0.154
Hence log Y = 1.9704 + 0.154 X
For 2010 X would be + 4. When X = 4, log Y will be
Log Y = AL 2.5864 = 385.9
Thus the estimated sale for the year 2010 is $ 38.59 million.