Question

Consider the following data matrix D: 8 -20 01 10-19 10 -20 20 (a) Compute the sample mean μ and sample covariance matrix Σ of D. b) Compute the eigenvalues of Σ (c) What is the dimensionality of the subspace that contains most of the variance of the data? d) Compute the first principal component of D. (e) Compute the coordinate of each data point projected on the first principal component.

Answer 1

The student's question involves steps related to principal component analysis. To accurately address the question, a correct data matrix is required. The answer involves calculations such as sample mean, covariance, eigenvalues, and projections onto principal components.

Step-by-step explanation:

The student has presented a problem involving a data matrix and is asking for a series of calculations related to descriptive statistics and principal component analysis (PCA). The steps to solve such a problem generally include: calculating the anddetermining the of the covariance matrix; understanding the dimensionality of the variance; computing the of the data; and finally, projecting the original data points onto the principal component. Unfortunately, the data provided is incompletely formatted and cannot be computed as it stands. However, these processes are integral parts of PCA, which is used for dimensionality reduction in data analysis, making it possible to extract the most informative features from a dataset. To appropriately assist with the original query, corrected and properly formatted data is necessary.

Answer 2

The precise computations for the mean, covariance matrix, eigenvalues, and principal components cannot be performed without the correct data. However, the process involves calculating the average for each variable, determining the covariance among variables, computing eigenvalues of the covariance matrix, identifying the principal components, and projecting data points onto these components.

Step-by-step explanation:

To address the student's question regarding the computation of the sample mean μ, sample covariance matrix Σ, eigenvalues of Σ, and principal component analysis, we first note that the data matrix D appears to be incomplete or incorrectly formatted. However, we can describe the general process for computation:

• To compute the sample mean of a data matrix, we calculate the average of each variable across all samples.
• The sample covariance matrix Σ is calculated by determining the covariance between each pair of variables in the dataset.
• The eigenvalues of the covariance matrix Σ are computed to understand the variance explained by each principal component.
• The dimensionality of the subspace that contains most of the variance is determined by the number of eigenvalues that significantly contribute to the total variance.
• The first principal component is the eigenvector associated with the largest eigenvalue of Σ.
• Data points are projected onto the first principal component to determine their coordinates in the reduced subspace.

Without accurate data, an exact numerical answer cannot be provided, but the student should now understand the steps involved in performing these calculations.