Multivariate Gaussian
A multivariate Gaussian, also known as a multivariate normal distribution, is a generalization of the one-dimensional normal distribution to higher dimensions. It describes the behavior of a vector of correlated random variables. This distribution is characterized by a mean vector and a covariance matrix.
The mean vector indicates the expected values of each variable in the distribution, while the covariance matrix captures the relationships between the variables, detailing how they vary together. Specifically, if we have a vector X consisting of n variables, the multivariate Gaussian distribution can be expressed as:
P(X) = (1 / (2π)^(n/2) |Σ|^(1/2)) * exp(-1/2 * (X - μ)ᵀ Σ⁻¹ (X - μ))
where μ is the mean vector, Σ is the covariance matrix, and |Σ| is the determinant of the covariance matrix.
In practical applications, the multivariate Gaussian is widely used in various fields such as statistics, machine learning, and finance. It is particularly useful for modeling phenomena where several interrelated factors influence outcomes, such as in predictive modeling and clustering algorithms.
One important property of the multivariate Gaussian distribution is that any linear combination of its variables will also follow a Gaussian distribution. This property makes it a powerful tool in both theoretical studies and practical applications.