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Unformatted text preview: s, . Thus, correlation is given as: The value of ranges between -1 and +1. i.e. . We can prove this statement in following way:
The Cauchy-Schwarz inequality states that for all vectors
where and of an inner product space, is the inner product. Therefore according to Cauchy- Schwarz’s Inequality, But the left hand side is . Whereas we have, and Hence we can get,
6.1. ; . So, it verifies that . Significance of Correlation Coefficient When , Y and Y are linearly related as shown in figures below: ρ=+1.0 . Y .
. . ..
.. . ρ=-1.0
. . . . . X . . X Fig.1. Scatter plots representing absolute positive and negative correlation
When , the values of X and Y appear as in the figure shown given below:
y 0 0 x Fig. 2. Scatter plot representing zero correlation
For intermediate values of , the scatterplot of vectors x, y would appear as given in figure
. below. Here scatter decreases with increasing
7 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Fig. 3. Scatter plot representing correlation coefficient values between 0 and1 However, from the figures below, we can see that when the relation between
nonlinear, and is (even when there exist a relationship between the variables) Fig. 4. Scatter plot for nonlinear dependence with
This occurs because correlation coefficient is the measure of linear dependence only.
is a measure of
Moreover, a fact about correlation coefficient should be noted. Although,
the degree of linear rela...
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This note was uploaded on 03/18/2014 for the course CE 5730 taught by Professor Dr.rajibmaity during the Spring '13 term at Indian Institute of Technology, Kharagpur.
- Spring '13
- Civil Engineering