关于高斯分布N((1,0)^T,I)的含义及《统计学习基础》数据生成的疑问

Hey there! Let's unpack your two questions clearly, step by step.

This is notation for a multivariate Gaussian (normal) distribution in 2 dimensions. Let's break down each part:

• $N(\mu, \Sigma)$ is the standard symbol for a multivariate Gaussian, where:
  • $\mu = (1,0)^T$ is the mean vector: the superscript $T$ tells us this is a column vector. In plain terms, this means the "center" of the distribution sits at the coordinate (1, 0) on a 2D plane (x=1, y=0).
  • $\Sigma = I$ is the covariance matrix: $I$ stands for the identity matrix, which looks like $\begin{pmatrix}1 & 0 \ 0 & 1\end{pmatrix}$. This means:
    • The two variables (x and y dimensions) are statistically independent—they don't vary together.
    • Each dimension has a variance of 1, so the distribution spreads out equally in both the x and y directions, forming a symmetric "bell curve" shape in 2D.

First, let's restate the confusing passage for clarity:

我们从二元高斯分布N((0,1)^{T,I)中生成10个均值mₖ，并将其标记为蓝色类别。类似地，再从N((0,1)}T,I)中生成10个均值，标记为橙色类别。随后为每个类别生成100个观测样本：对于每个观测样本，以1/10的概率随机选取一个mₖ，然后……

This is describing a way to create clustered, two-class data. Here's a step-by-step breakdown:

Step 1: Create "cluster centers" for each class #

• Blue class: Generate 10 distinct points (called $m_k$, where k ranges from 1 to 10). Each of these points is sampled from the 2D Gaussian distribution $N((0,1)^T, I)$. This means all 10 blue centers are randomly scattered around the coordinate (0, 1), with no correlation between their x and y values, and equal spread in both directions.
• Orange class: Do exactly the same thing—generate another 10 points from the same $N((0,1)^T, I)$ distribution, but label these as the orange class centers.

Step 2: Generate the actual training samples #

• For the blue class's 100 samples:
  1. Pick one of the 10 blue $m_k$ centers at random—each center has a 1/10 chance of being chosen (equal probability).
  2. Then, sample a single point from a Gaussian distribution centered at the chosen $m_k$ (the passage cuts off here, but in most examples from the book, this would use the identity covariance matrix $I$ again, meaning the sample is scattered around the center with equal spread in x and y).
• Repeat this process 100 times for the blue class, and another 100 times for the orange class using its own 10 centers.

Why this setup? #

This creates data where each class has 10 tight "sub-clusters" (each with ~10 samples, since 100 total / 10 centers). Both classes are centered around (0,1) overall, but their internal sub-clusters overlap in interesting ways—perfect for testing classification or clustering algorithms to see how well they can pick out the underlying structure.

内容的提问来源于stack exchange，提问作者Androidy