A data scientist is developing a multiple linear regression model using ordinary least squares (OLS). The feature matrix X is a 1000x15 matrix (1000 samples, 15 features). During model fitting, the process fails because the matrix (X^T * X) is singular and cannot be inverted. This problem indicates perfect multicollinearity among the features. What does this singularity imply about the rank of the feature matrix X?
The correct answer is that the rank of X is less than 15. The rank of a matrix is the number of linearly independent columns or rows. For a feature matrix X with n columns (features), perfect multicollinearity exists when one or more features can be expressed as a linear combination of others. This means the columns are not all linearly independent, so the rank of the matrix must be less than the total number of columns (rank(X) < n).
In OLS regression, the coefficient vector is calculated as (X^T * X)^-1 * X^T * y. The matrix (X^T * X) is invertible only if the feature matrix X has full column rank (i.e., its rank is equal to the number of columns). If rank(X) < 15, the matrix (X^T * X) is singular (non-invertible), which confirms the diagnosis of perfect multicollinearity and explains why the OLS estimation fails.
A rank of 15 would mean the matrix has full column rank, which is the desired condition for OLS regression, as all features would be linearly independent. The rank of a 1000x15 matrix cannot be greater than the minimum of its dimensions, so it cannot be greater than 15 or equal to 1000.
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