1 Deriving a Pearson’s Correlation Value or Matrix in R

Derive correlation value:

# Enter the data by hand

data_x = c(19.6, 18.0, 18.6, 20.4, 19.4, 19.5)
data_y = c(18.4, 19.2, 15.4, 20.3, 20.6, 19.8)

# Value
cor(data_x, data_y)

[1] 0.4765177

The Pearson’s correlation between data_x and data_y is 0.477.

Derive correlation matrix:

Using the trees data from the "datasets" package.

head(trees, 5)

  Girth Height Volume
1   8.3     70   10.3
2   8.6     65   10.3
3   8.8     63   10.2
4  10.5     72   16.4
5  10.7     81   18.8

# Matrix
cor(trees)

           Girth    Height    Volume
Girth  1.0000000 0.5192801 0.9671194
Height 0.5192801 1.0000000 0.5982497
Volume 0.9671194 0.5982497 1.0000000

The diagonals are 1 because they are the correlation between the variables themselves, while the non-diagonals are the correlation between the different pairs of variables.

The correlation between:

• "Girth" and "Height" is 0.5193.
• "Girth" and "Volume" is 0.9671.
• "Height" and "Volume" is 0.5982.

2 Deriving a Covariance Value or Matrix in R

Derive covariance value:

# Enter the data by hand

data_x = c(18.2, 20.0, 18.5, 16.7, 19.1, 22.0)
data_y = c(18.4, 19.2, 15.4, 20.6, 20.6, 19.8)

# Value
cov(data_x, data_y)

[1] 0.272

The covariance between data_x and data_y is 0.272.

Derive covariance matrix:

Using the trees data from the "datasets" package.

head(trees, 5)

  Girth Height Volume
1   8.3     70   10.3
2   8.6     65   10.3
3   8.8     63   10.2
4  10.5     72   16.4
5  10.7     81   18.8

# Matrix
cov(trees)

           Girth   Height    Volume
Girth   9.847914 10.38333  49.88812
Height 10.383333 40.60000  62.66000
Volume 49.888118 62.66000 270.20280

The diagonals are the variance of the variables, while the non-diagonals are the covariance between the different pairs of variables.

The variance of "Girth" is 9.848, "Height" is 40.6, and "Volume" is 270.203.

The covariance between:

• "Girth" and "Height" is 10.3833.
• "Girth" and "Volume" is 49.8881.
• "Height" and "Volume" is 62.66.

3 Formulas for Sample Covariance and Sample Pearson’s Correlation Coefficient in R

Sample Covariance Formula:

\[cov(x,y) = \frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{n-1}.\]

Sample Pearson’s Correlation Formula:

\[r_{x,y} = \frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}.\] \(\sum\) is the summation sign,

the \(x_i's\), and the \(y_i's\) are the sample data values,

\(\bar x\) and \(\bar y\) are the sample means of the \(x_i's\) and the \(y_i's\) respectively, and

\(n \in \{3, 4, 5 ...\}\) is the number of pairs in the sample.

4 Multiple Correlation Plot in R

Using the trees data from the "datasets" package above.

# Plot
plot(trees)

Multiple Scatter Plot for Correlation Analysis in R

Consistent with the correlation coefficients above, we can see that "Girth" and "Volume" are the most correlated pair, while "Girth" and "Height" are the least correlated.

# With some customization
plot(trees,
     main = "Multiple Scatter Plot for Correlation Analysis in R",
     cex = 1.5, cex.main = 1.25, cex.axis = 1.4,
     font = 2,
     col = "limegreen", col.axis = "red")

Multiple Scatter Plot for Correlation Analysis in R