Introduction
The package provides a suite of functions for computing various
distance metrics between pairs of groups within a list of data frames.
Each data frame represents observations of a species, including multiple
factors. In addition to the Mahalanobis distance, which is a
dissimilarity measure based on the covariance matrix and useful for
statistical matching or data merging, the package includes:
Mahalanobis distance: Also defined as a measure of dissimilarity
between two random vectors and with the same probability density
function and with covariance matrix.
Euclidean Distance: A direct geometric measure between two points
in a multidimensional space, defined as the square root of the sum of
the squares of the differences between corresponding coordinates of the
points.
Manhattan Distance: Also known as taxi cab distance, it computes
the sum of the absolute differences between the coordinates of the
points, representing the path a taxi would take in a grid-like road
system.
Chebyshev Distance: Defined as the maximum difference between the
coordinates of the points, this metric is useful when the largest
distance dominates the overall effect.
These metrics are fundamental in various fields, such as cluster
analysis, classification, and other applications of machine learning and
data mining, where assessing similarity or dissimilarity between data is
crucial. The package is designed to be flexible and easily integrated
into data analysis workflows, providing reliable tools for evaluating
distances in multidimensional contexts.
Applications
Application on iris dataset
library(cmahalanobis)
# Load iris dataset
data(iris)
# Split data into three parts
setosa <- subset(iris, Species == "setosa")
setosa <- setosa[,-5] # Remove the column of specie
versicolor <- subset(iris, Species == "versicolor")
versicolor <- versicolor[,-5] # Remove the column of specie
virginica <- subset(iris, Species == "virginica")
virginica <- virginica[,-5] # Remove the column of specie
# Create a list with the three groups of flowers
groups <- list(setosa, versicolor, virginica)
cmahalanobis(groups, plot = TRUE, p.value = TRUE)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.0000 335.19989 727.42056
#> [2,] 107.1736 0.00000 26.71618
#> [3,] 171.7689 16.88654 0.00000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 2.748687e-71 4.023276e-156
#> [2,] 2.915001e-22 NA 2.268568e-05
#> [3,] 4.363119e-36 2.033555e-03 NA
ceuclide(groups, plot = TRUE, p.value = TRUE, num.permutations = 10)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.000000 3.208281 4.754507
#> [2,] 3.208281 0.000000 1.620489
#> [3,] 4.754507 1.620489 0.000000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.6 0
#> [2,] 0.9 NA 1
#> [3,] 0.0 1.0 NA
cmanhattan(groups, plot = TRUE, p.value = TRUE, num.permutations = 10)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.000 5.466 7.906
#> [2,] 5.466 0.000 2.848
#> [3,] 7.906 2.848 0.000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.0 0.0
#> [2,] 0 NA 0.8
#> [3,] 0 0.9 NA
cchebyshev(groups, plot = TRUE, p.value = TRUE, num.permutations = 50)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.000 2.798 4.090
#> [2,] 2.798 0.000 1.292
#> [3,] 4.090 1.292 0.000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.98 0.42
#> [2,] 0.96 NA 1.00
#> [3,] 0.44 1.00 NA
Application on mtcars dataset
# Split the data into 2 parts for each type of transmission
auto <- subset(mtcars, am == 0)
auto <- auto[,-9]
manual <- subset(mtcars, am == 1)
manual <- manual[,-9]
# Create a list with the two groups of cars
groups <- list(auto, manual)
cmahalanobis(groups, plot = TRUE, p.value = TRUE)
#> $distances
#> [,1] [,2]
#> [1,] 0.0000 156.1163
#> [2,] 735.5919 0.0000
#>
#> $p_values
#> [,1] [,2]
#> [1,] NA 2.050145e-28
#> [2,] 1.429549e-151 NA
ceuclide(groups, plot = TRUE, p.value = TRUE, num.permutations = 10)
#> $distances
#> [,1] [,2]
#> [1,] 0.0000 150.8032
#> [2,] 150.8032 0.0000
#>
#> $p_values
#> [,1] [,2]
#> [1,] NA 1
#> [2,] 0 NA
cmanhattan(groups, plot = TRUE, p.value = TRUE, num.permutations = 10)
#> $distances
#> [,1] [,2]
#> [1,] 0.0000 193.8557
#> [2,] 193.8557 0.0000
#>
#> $p_values
#> [,1] [,2]
#> [1,] NA 1
#> [2,] 0 NA
cchebyshev(groups, plot = TRUE, p.value = TRUE, num.permutations = 50)
#> $distances
#> [,1] [,2]
#> [1,] 0.0000 146.8482
#> [2,] 146.8482 0.0000
#>
#> $p_values
#> [,1] [,2]
#> [1,] NA 1
#> [2,] 0 NA
Application on simulated data
# Load cmahalanobis package
library(cmahalanobis)
# Define the number of observations and variables for each groups
num_observations <- 100
num_variables <- 5
# We generate three groups of simulated data with normal distribution
set.seed(123) # For the reproducibility of results
group1 <- as.data.frame(matrix(rnorm(num_observations * num_variables), nrow = num_observations))
group2 <- as.data.frame(matrix(rnorm(num_observations * num_variables), nrow = num_observations))
group3 <- as.data.frame(matrix(rnorm(num_observations * num_variables), nrow = num_observations))
# Create a list of three groups of data
groups <- list(group1, group2, group3)
# Calculate Mahalanobis distance with cmahalanobis function
cmahalanobis(groups, plot = TRUE, p.value = TRUE)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.000000 5.639257 5.567479
#> [2,] 4.722923 0.000000 5.029954
#> [3,] 5.329901 5.783087 0.000000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.3429174 0.3506032
#> [2,] 0.4506217 NA 0.4122355
#> [3,] 0.3769584 0.3279009 NA
ceuclide(groups, plot = TRUE, p.value = TRUE, num.permutations = 190)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.0000000 0.2282174 0.156693
#> [2,] 0.2282174 0.0000000 0.302220
#> [3,] 0.1566930 0.3022200 0.000000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.32105263 0.65789474
#> [2,] 0.3789474 NA 0.06315789
#> [3,] 0.5736842 0.02105263 NA
cmanhattan(groups, plot = TRUE, p.value = TRUE, num.permutations = 10)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.0000000 0.4442511 0.2777603
#> [2,] 0.4442511 0.0000000 0.6671049
#> [3,] 0.2777603 0.6671049 0.0000000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0 0.3
#> [2,] 0.0 NA 0.0
#> [3,] 0.1 0 NA
cchebyshev(groups, plot = TRUE, p.value = TRUE, num.permutations = 50)
#> $distances
#> [,1] [,2] [,3]
#> [1,] 0.0000000 0.1327059 0.1230044
#> [2,] 0.1327059 0.0000000 0.1622405
#> [3,] 0.1230044 0.1622405 0.0000000
#>
#> $p_values
#> [,1] [,2] [,3]
#> [1,] NA 0.94 0.88
#> [2,] 0.92 NA 0.66
#> [3,] 0.80 0.54 NA