多元正态分布

多变量正态分布亦称为多变量高斯分布。它是单维正态分布向多维的推广。它同矩阵正态分布有紧密的联系。

N维随机矢量 $\ X = [X_1, \dots, X_N]^T$ 如果服从多变量正态分布，必须满足下面的三个等价条件：

任何线性组合 $\ Y = a_1 X_1 + \cdots + a_N X_N$ 服从正态分布。

存在随机矢量 $\ Z = [Z_1, \dots, Z_M]^T$ ( 它的每个元素服从独立标准正态分布），矢量 $\ \mu = [\mu_1, \dots, \mu_N]^T$ 及 $N \times M$ 矩阵 $\ A$ 满足 $\ X = A Z + \mu$ .

存在 $\mu$ 和一个对称正定阵 $\ \Sigma$ 满足 $\ X$ 的特征函数

\phi_X\left(u;\mu,\Sigma\right) = \exp \left( i \mu^T u - \frac{1}{2} u^T \Sigma u \right)

如果 $\ \Sigma$ 是非奇异的，那幺该分布可以由以下的PDF来描述：

f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}|\Sigma|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^T \Sigma^{-1}(x - \mu) \right)

注意这里的 $|\Sigma|$ 表示协方差矩阵的行列式。

单词	Multivariate normal distribution
释义	Multivariate normal distribution 中文百科多元正态分布多变量正态分布亦称为多变量高斯分布。它是单维正态分布向多维的推广。它同矩阵正态分布有紧密的联系。 N维随机矢量 $\ X = [X_1, \dots, X_N]^T$ 如果服从多变量正态分布，必须满足下面的三个等价条件：任何线性组合 $\ Y = a_1 X_1 + \cdots + a_N X_N$ 服从正态分布。存在随机矢量 $\ Z = [Z_1, \dots, Z_M]^T$ ( 它的每个元素服从独立标准正态分布），矢量 $\ \mu = [\mu_1, \dots, \mu_N]^T$ 及 $N \times M$ 矩阵 $\ A$ 满足 $\ X = A Z + \mu$ . 存在 $\mu$ 和一个对称正定阵 $\ \Sigma$ 满足 $\ X$ 的特征函数 $\phi_X\left(u;\mu,\Sigma\right) = \exp \left( i \mu^T u - \frac{1}{2} u^T \Sigma u \right)$ 如果 $\ \Sigma$ 是非奇异的，那幺该分布可以由以下的PDF来描述： $f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}\|\Sigma\|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^T \Sigma^{-1}(x - \mu) \right)$ 注意这里的 $\|\Sigma\|$ 表示协方差矩阵的行列式。英语百科 Multivariate normal distribution 多元正态分布 In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One possible definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
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Multivariate normal distribution

多元正态分布

Multivariate normal distribution 多元正态分布