Gaussienne multivariée
A Gaussian multivarié, also known as a distribution normale multivariée, is a generalization of the one-dimensional distribution normale to dimensions supérieures. It describes the behavior of a vector of correlated random variables. This distribution is characterized by a mean vector and a matrice de covariance.
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 distribution gaussienne peut être exprimée comme :
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 est la matrice de covariance, et |Σ| est le déterminant de la matrice de covariance.
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 algorithmes de clustering.
Une propriété importante de la distribution gaussienne multivariée est que toute combinaison linéaire of its variables will also follow a Gaussian distribution. This property makes it a powerful tool in both theoretical studies and practical applications.