Covariance and Correlation in R Programming

Covariance and Correlation are terms used in statistics to measure relationships between two random variables. Both of these terms measure linear dependency between a pair of random variables or bivariate data. They both capture a different component of the relationship, despite the fact that they both provide information about the link between variables.

Covariance in R

In R programming, covariance can be measured using the cov() function. Covariance is a statistical term used to measure the direction of the linear relationship between the data vectors.

Mathematically,

\operatorname{Cov}(x, y)=\frac{\Sigma\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{n-1} 

Where:

• x represents the x data vector 
• y represents the y data vector 
• \bar{x} represents mean of x data vector 
• \bar{y} represents mean of y data vector 
• n: number of observations

Syntax:

cov(x, y, method)

Where:

• x and y represents the data vectors

Example: 

x <- c(1, 3, 5, 10)
y <- c(2, 4, 6, 20)

print(cov(x, y))
print(cov(x, y, method = "pearson"))

Output: 

[1] 30.66667
[1] 30.66667
[1] 12
[1] 1.666667

Correlation in R

cor() function in R programming measures the correlation coefficient value. Correlation is a relationship term in statistics that uses the covariance method to measure how strongly the vectors are related.

Mathematically,

\operatorname{Corr}(x, y)=\frac{\sum\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\sum\left(x_{i}-\bar{x}\right)^{2}.\sum\left(y_{i}-\bar{y}\right)^{2}}} 

Where:

• x represents the x data vector 
• y represents the y data vector 
• \bar{x} represents mean of x data vector 
• \bar{y} represents mean of y data vector 

Syntax:

cor(x, y, method)

Where:

• x and y represents the data vectors
• method defines the type of method to be used to compute covariance. Default is "pearson".

Example: 

x <- c(1, 3, 5, 10)

y <- c(2, 4, 6, 20)

print(cor(x, y))
print(cor(x, y, method = "pearson"))
print(cor(x, y, method = "kendall"))
print(cor(x, y, method = "spearman"))

Output: 

[1] 0.9724702
[1] 0.9724702
[1] 1
[1] 1

Covariance and Correlation For data frame

We can calculate the covariance and correlation for all columns in data frame.

data(iris)
library(dplyr)

data=select(iris,-Species)

cor(data)
cov(data)

Output:

Conversion of Covariance to Correlation in R

cov2cor() function in R programming converts a covariance matrix into a corresponding correlation matrix.

Syntax:

cov2cor(X)

Where:

• X represents the covariance square matrix

Example: 

x <- rnorm(2)
y <- rnorm(2)

mat <- cbind(x, y)
X <- cov(mat)

print(X)
print(cor(mat))
print(cov2cor(X))

Output: 

Correlation describes the intensity and direction of the linear link between two variables, whereas covariance shows how much two variables vary together.

Covariance vs. Correlation