Principal Component Coefficients Explained
The principal components of a random vector $(X_1, X_2, \cdots, X_p)$ are linear combinations of the variables. The first principal component is the linear combination $\sum_{i=1}^p l_i X_i$ that captures the maximum possible variance.
Relationship with Dispersion Matrix
For a random vector with mean $\mu$ and dispersion matrix $\mathbf{\Sigma}$, the coefficients $(l_1, l_2, \cdots, l_p)$ of the first principal component correspond to the components of the eigenvector associated with the largest eigenvalue of the dispersion matrix $\mathbf{\Sigma}$.
Specifically, the direction of maximum variance in the data is given by this eigenvector.
Identifying the Coefficient Vector
Therefore, the coefficient vector $(l_1, l_2, \cdots, l_p)$ is the eigenvector corresponding to the largest eigenvalue of the dispersion matrix $\mathbf{\Sigma}$.
Important Questions from Principle Component Analysis
Let the random vector $\begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix}$ have the positive definite dispersion matrix $\begin{pmatrix} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{pmatrix}$ Then, which of the following statements are true?
- Let $X$ be a $4 \times 1$ random vector with Multivariate normal distribution with mean $\mu$ and dispersion matrix $\Sigma$. Suppose, the eigenvalues of $\Sigma$ are $\lambda_1 = 6, \lambda_2 = 3, \lambda_3 = 2, \lambda_4 = 1$. Let $Y_1, Y_2, Y_3, Y_4$ be the four principal components. Which of the following statements are correct?
