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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 CauchySchwarz 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
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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
 Dr.RajibMaity
 Civil Engineering

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