统计计算| Statistical Computing代写 MATH6173代考

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
$$\hat{\mathbf{y}}=\mathbf{H} \hat{\beta}$$

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
$$P_{r}(G, X)=P_{r}(G) \phi\left(X ; \mu_{r}, \Sigma\right) .$$