统计计算| Statistical Computing代写 MATH6173代考

这是一份southampton南安普敦大学 MATH6173作业代写的成功案

统计计算| Statistical Computing代写 MATH6173代考
问题 1.

We estimate $\beta$ by minimizing the penalized least squares criterion
$$
H(\beta)=(\mathbf{y}-\mathbf{H} \beta)^{T}(\mathbf{y}-\mathbf{H} \beta)+\lambda|\beta|^{2} .
$$
The solution is
$$
\hat{\mathbf{y}}=\mathbf{H} \hat{\beta}
$$


证明 .

with $\hat{\beta}$ determined by
$$
-\mathbf{H}^{T}(\mathbf{y}-\mathbf{H} \hat{\beta})+\lambda \hat{\beta}=0
$$
From this it appears that we need to evaluate the $M \times M$ matrix of inner products in the transformed space. However, we can premultiply by $\mathbf{H}$ to give
$$
\mathbf{H} \hat{\beta}=\left(\mathbf{H H}^{T}+\lambda \mathbf{I}\right)^{-1} \mathbf{H H}^{T} \mathbf{y}
$$

.

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MATH6173 COURSE NOTES :


$$
P(G, X)=\sum_{r=1}^{R} \pi_{r} P_{r}(G, X)
$$
a mixture of joint densities. Furthermore we assume
$$
P_{r}(G, X)=P_{r}(G) \phi\left(X ; \mu_{r}, \Sigma\right) .
$$




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