多変量ガウス分布
A 多変量ガウス分布, also known as a 多変量正規分布, is a generalization of the one-dimensional 正規分布 to 高次元. It describes the behavior of a vector of correlated random variables. This distribution is characterized by a mean vector and a 共分散行列.
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 ガウス分布 次のように表現できます：
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 これは共分散行列であり、|Σ|は共分散行列の行列式である。
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 クラスタリングアルゴリズムにおいて重要です.
多変量ガウス分布の重要な性質の一つは、任意の 線形結合 of its variables will also follow a Gaussian distribution. This property makes it a powerful tool in both theoretical studies and practical applications.