MathAlgebraLinear algebraMatrix decomposition

Least squares

Comprehension

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In the least squares problem with data matrix $X$ and target $\mathbf{y}$ , select the true statements about the optimal parameter vector $\hat{\mathbf{\beta}}$ .

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It maximizes the error

e(\hat{\beta}) = \lVert \mathbf{y} - X\hat{\beta} \rVert

It always exists

It minimizes the error

e(\hat{\beta}) = \lVert \mathbf{y} - X\hat{\beta} \rVert

It's defined by

\hat{\mathbf{\beta}} = XX^{t} \mathbf{y}

It's given by

\hat{\mathbf{\beta}} = X^{\dag} \mathbf{y}

It only exists when the columns of

X

are linearly independent

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