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.

In this, we are going to discuss cov(), cor() and cov2cor() functions in R which use covariance and correlation methods of statistics and probability theory.

Covariance in R Programming Language

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

where,
x represents the x data vector
y represents the y data vector
represents mean of x data vector
represents mean of y data vector
N represents total observations

Covariance Syntax in R

Syntax: 
        cov(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 covariance using different methods
print(cov(x, y))
cat("Cov using Pearson=",cov(x, y, method = "pearson"),"\n")
cat("Cov using Kendali=",cov(x, y, method = "kendall"),"\n")
cat("Cov using Spearman=",cov(x, y, method = "spearman"),"\n")

Output:
[1] 30.66667
Cov using Pearson= 30.66667 
Cov using Kendali= 12 
Cov using Spearman= 1.666667 


Correlation in R Programming Language

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 strong the vectors are related. Mathematically,

where,

x represents the x data vector
y represents the y data vector
x bar represents mean of x data vector
y bar 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:
# Data vectors
x <- c(1, 3, 5, 10)

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

# Print correlation using different methods
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


Conversion of Covariance to Correlation in R

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

Syntax: 
        cov2cor(X)

where, X and y represents the covariance square matrix

Example:
# Data vectors
x <- rnorm(2)
y <- rnorm(2)

# Binding into square matrix
mat <- cbind(x, y)

# Defining X as the covariance matrix
X <- cov(mat)

# Print covariance matrix
print(X)

# Print correlation matrix of data
# vector
print(cor(mat))

# Using function cov2cor()
# To convert covariance matrix to
# correlation matrix
print(cov2cor(X))

Output:
    x           y
x  0.3459283 -0.17811105
y -0.1781110  0.09170555

     x  y
x  1 -1
y -1  1

    x  y
x  1 -1
y -1  1


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