Understanding Mardia’s Coefficient of Skewness
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Mardia’s coefficient of skewness is a measure used to assess the symmetry of multivariate distributions. In this article, we will delve into how to calculate and store the Mardia’s coefficients in a vector when dividing data into multiple parts.
Background on Multivariate Skewness
Skewness is a statistical concept that describes the asymmetry of a distribution. In univariate distributions, skewness can be calculated using the formula: $S = \frac{E(X^3) - (E(X))^3}{\sigma^3}$ where $X$ is the random variable, $\mu$ is its mean, and $\sigma$ is its standard deviation.
However, when dealing with multivariate data, things become more complicated. The concept of skewness extends to higher dimensions, making it essential for understanding the underlying structure of a dataset.
Mardia’s Coefficient
In 1970, Andrew Mardia proposed a method for detecting multivariate skewness and kurtosis using the multivariate normal distribution as a reference. This approach involves calculating the Mahalanobis distance (MD) between the given data distribution and the multivariate normal distribution.
The Mahalanobis distance can be thought of as a measure of how many standard deviations away from the mean a given observation is, taking into account the correlations between variables.
The Mardia Test
Mardia’s test calculates two values:
- Skewness ($\lambda_s$): measures the asymmetry or skewness in the distribution.
- Kurtosis ($\lambda_k$): measures the “tailedness” or kurtosis of the distribution.
These values are used to determine whether a dataset is likely to come from a multivariate normal distribution, which serves as a baseline for comparison.
Calculating Mardia’s Coefficient
To calculate Mardia’s coefficient, we can use the following formula:
$$ \begin{aligned} \text{Mardia’s Coefficient} &= \sqrt{(2 - 3)\lambda_s + (1 - 4)\lambda_k}\ &= \sqrt{-\lambda_s - 3\lambda_k}. \end{aligned} $$
This value is then used to assess the multivariate skewness of a dataset.
R Implementation
In this example, we are using R as our programming language to calculate Mardia’s coefficient. We first import the necessary libraries, including Matrix and psych.
library(Matrix)
library(psych)
We then set up some variables for better understanding:
set.seed(10)
N0 <- 1 # number of samples
n0 <- 5 # sample size per iteration
p0 <- 2 # number of variables
q0 <- 4 # number of groups or iterations
# calculate total number of observations (n)
n <- n0 * q0
Next, we create a matrix m2 with the correct dimensions to hold our mean vector and covariance structure.
m2 <- matrix(c(0, 0), p0, 1) # dimension of m2 is (p0 x 1)
dim(m2)
[1] 2 1 # verify that m2 has shape (2x1)
Similarly, we create a covariance structure s2 with the correct dimensions.
s2 <- matrix(c(1, 0, 0, 1), p0, p0)
dim(s2)
[1] 2 2 # verify that s2 has shape (2x2)
We then create an array Dat to hold our multivariate normal distributions.
Dat <- array(data = NA, dim = c(20, 2, q0)) # verify the shape of Dat is (n x p x q)
mardia_lst_vec <- vector('list', length = q0) # list of Mardia values for each group
for (i in 1:q0) {
# calculate multivariate normal distribution
Dat[,, i] <- mvrnorm(n, m2, s2)
# calculate Mardia's coefficient for the current iteration
mardia_lst_vec[[i]] <- mardia(Dat[,, i], plot = FALSE)
}
Finally, we extract and unlist the Mardia values for each group.
the_bp1 <- unlist(unname(c(mardia_lst_vec[[1]][3], mardia_lst_vec[[2]][3],
mardia_lst_vec[[3]][3], mardia_lst_vec[[4]][3])))
# verify the shape of the extracted Mardia values is a vector
dim(the_bp1)
[1] 1 4 # show that we have one row and four columns
print(the_bp1)
Conclusion
In this article, we explored how to calculate and store the Mardia’s coefficient in a vector when dividing data into multiple parts. We used R as our programming language for calculations.
By understanding multivariate skewness and kurtosis, you can gain deeper insights into your dataset and its underlying structure.
Last modified on 2024-01-27