3 Linear Methods for Regression exercises.ipynb

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    "# Ex. 3.9 (using the QR decomposition for fast forward-stepwise selection)\n",
    "\n",
    "Forward stepwise regression. Suppose we have the $QR$ decomposition\n",
    "for the $N\\times q$ matrix $X_1$ in a multiple regression problem with response $y$, and we have an additional $p−q$ predictors in the matrix $X_2$. Denote the current residual by $r$. We wish to establish which one of these additional variables will reduce the residual-sum-of squares the most when included with those in $X_1$. Describe an efficient procedure for doing this.\n",
    "\n",
    "Since $$\\underset{N\\times q}{X_1}=\\underset{N\\times N}{Q}\\underset{N\\times q}{R}$$, If $N\\ge q$, $$\\underset{N\\times q}{X_1}=\\underset{N\\times N}{Q}\\underset{N\\times q}{R}=\\begin{bmatrix}\n",
    "\\underset{N\\times q}{Q_1} & \\underset{N\\times (N-q)}{Q_2}\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "\\underset{q\\times N}{R_1} \\\\\n",
    "\\underset{(N-q)\\times N}{\\mathbf 0}\n",
    "\\end{bmatrix}=\\begin{bmatrix}\n",
    "\\underset{N\\times q}{Q_1} & \\underset{N\\times (N-q)}{Q_2}\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "\\underset{q\\times q}{R_2} & \\underset{q\\times (N-q)}{\\mathbf 0}\\\\\n",
    "\\underset{(N-q)\\times q}{\\mathbf 0} & \\underset{(N-q)\\times (N-q)}{\\mathbf 0}\\\\\n",
    "\\end{bmatrix}$$ where $Q_1$ and $Q_2$ both have orthogonal columns. Each column of $X_1$ is $$x_j=Qr_j$$. $$\\hat{y}=X_1(X_1^TX_1)^{-1}X_1^Ty=QR(R^TQ^TQR)^{-1}R^TQ^Ty\\\\\n",
    "=Q_1R_2R_2^{-1}R_2^{-T}R_2^TQ_1^Ty=Q_1Q_1^Ty$$ \n",
    "Let the $i$-th column is $Q$ is $q_i,\\quad (1\\le i\\le q)$, then $(q_1,\\cdots,q_q)$ is an orthonormal basis of $Q$ or $Q_1$, then the best estimator is $$\\underset{i}{\\text{argmax }}(q_i^Ty)q_i$$\n",
    "\n"
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    "# Ex. 3.10 (using the z-scores for fast backwards stepwise regression) \n",
    "\n",
    "Backward stepwise regression. Suppose we have the multiple regression fit of $y$ on $X$, along with the standard errors and Z-scores as in Table 3.2. We wish to establish which variable, when dropped, will increase the residual sum-of-squares the least. How would you do this?\n",
    "\n",
    "The F statistic,\n",
    "$$\n",
    "F_{p_1−p_0,N−p_1−1}=\\frac{\\left(\\text{RSS}_0-\\text{RSS}_1\\right)/\\left(p_1-p_0\\right)}{\\text{RSS}_1/\\left(N-p_1-1\\right)}\n",
    "$$\n",
    "where $\\text{RSS}_1$ is the residual sum-of-squares for the least squares fit of the bigger model with $p_1+1$ parameters, and $\\text{RSS}_0$ the same for the nested smaller model with $p_0+1$ parameters, having $p_1−p_0$ parameters constrained to be zero. The $F$ statistic measures the change in residual sum-of-squares per additional parameter in the bigger model, and it is normalized by an estimate of $\\sigma^2$. Under the Gaussian assumptions, and the null hypothesis that the smaller model is correct, the $F$ statistic will have a $F_{p_1−p_0,N−p_1−1}$ distribution.\n",
    "\n",
    "The $Z-score$\n",
    "$$z_j=\\frac{\\hat{\\beta}_j}{\\hat{\\sigma}\\sqrt{v_{jj}}}$$ where $v_{jj}$ is the $j$th diagonal element of $(\\mathbf X^T\\mathbf X)^{−1}$. Under the null hypothesis\n",
    "that $\\beta_j = 0$, $z_j$ is distributed as $t_{N−p−1}$ (a $t$ distribution with $N − p − 1$ degrees of freedom), and hence a large (absolute) value of $z_j$ will lead to rejection of this null hypothesis.\n",
    "\n",
    "Since once $\\beta$ is estimated we can compute $\\sigma^2$:\n",
    "$$\\hat{\\sigma}^2=\\frac{1}{N-p-1}\\sum_{i=1}^{N}(y_i-\\hat{y}_i)^2=\\frac{1}{N-p-1}\\sum_{i=1}^{N}(y_i-x_i^T\\beta)^2=\\frac{\\text{RSS}_1}{N-p-1}$$\n",
    "In addition, by just deleting one variable from our regression the difference in degrees of freedom between the two models is one i.e. $p_1 − p_0 = 1$. Thus the $F$-statistic when we delete the $j$th term from the base model simplifies to\n",
    "$$F_{1,N−p_1−1}=\\frac{\\left(\\text{RSS}_j-\\text{RSS}_1\\right)}{\\text{RSS}_1/\\left(N-p_1-1\\right)}=\\frac{\\left(\\text{RSS}_j-\\text{RSS}_1\\right)}{\\hat{\\sigma}^2}$$\n",
    "\n",
    "Since $$\\text{RSS}_j-\\text{RSS}_1=\\left(\\mathbf{y'}-\\mathbf{X'}\\left(\\mathbf{X'}^T\\mathbf{X'}\\right)^{-1}\\mathbf{X'}^T\\mathbf{y'}\\right)^T\\left(\\mathbf{y'}-\\mathbf{X'}\\left(\\mathbf{X'}^T\\mathbf{X'}\\right)^{-1}\\mathbf{X'}^T\\mathbf{y'}\\right)-\\left(\\mathbf{y}-\\mathbf{X}\\left(\\mathbf{X}^T\\mathbf{X}\\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\right)^T\\left(\\mathbf{y}-\\mathbf{X}\\left(\\mathbf{X}^T\\mathbf{X}\\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\right)$$ Where $\\mathbf{X'}$ is $\\mathbf{X}$ with $j$th column deleted and $\\mathbf{y'}$ is $\\mathbf{y}$ with $j$th element deleted.\n",
    "\n",
    "Since that $X^TX(X^TX)^{−1} = I_{p+1}$. Let $u_j$ be the $jth$ column of $X(X^TX)^{−1}$. Using the $v_{ij}$ notation\n",
    "to denote the elements of the matrix $(X^TX)^{−1}$ established above, we have $$u_j=\\sum_{r=1}^{p+1}x_rv_{rj}$$ and $$\\lVert u_j\\rVert^2=u_j^Tu_j=\\sum_{r=1}^{p+1}v_{rj}x_r^Tu_j=\\sum_{r=1}^{p+1}v_{rj}x_r^Tu_j=v_{jj}$$ Then $$\\lVert z_j\\rVert^2=\\frac{\\lVert u_j\\rVert^2}{v_{jj}^2}=\\frac{1}{v_{jj}}$$ Now $$\\frac{z_j}{\\lVert z_j\\rVert}$$ is a unit vector orthogonal to $x_1, \\cdots , x_{j−1}, x_{j+1}, \\cdots , x_{p+1}$. So $$\\text{RSS}_j-\\text{RSS}_1=\\left<y,\\frac{z_j}{\\lVert z_j\\rVert}\\right>^2=\\left(\\frac{<y,z_j>}{<z_j,z_j>}\\right)^2\\lVert z_j\\rVert^2=\\hat{\\beta}_j^2/v_{jj}$$ and $$F_{1,N−p_1−1}=\\frac{\\left(\\text{RSS}_j-\\text{RSS}_1\\right)}{\\text{RSS}_1/\\left(N-p_1-1\\right)}=\\frac{\\left(\\text{RSS}_j-\\text{RSS}_1\\right)}{\\hat{\\sigma}^2}=\\frac{\\hat{\\beta}_j^2}{\\hat{\\sigma}^2v_{jj}}=z_j^2$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.11 (multivariate linear regression with different $\\Sigma_i$) \n",
    "\n",
    "Show that the solution to the multivariate linear regression problem (3.40) is given by (3.39). What happens if the covariance matrices\n",
    "$\\mathbf\\Sigma_i$ are different for each observation?\n",
    "\n",
    "$$\\begin{align}\n",
    "\\text{RSS}(\\mathbf{B}) &= \\sum_{k=1}^K \\sum_{i=1}^N \\left( y_{ik} - f_k(x_i) \\right)^2 \\\\\n",
    "&= \\text{trace}\\left( (\\mathbf{Y}-\\mathbf{XB})^T(\\mathbf{Y}-\\mathbf{XB}) \\right)\n",
    "\\end{align}\\quad(Equation\\;\\;3.38)$$\n",
    "$$\\begin{equation}\n",
    "\\hat{\\mathbf{B}} = \\left(\\mathbf{X}^T\\mathbf{X}\\right)^{-1}\\mathbf{X}^T\\mathbf{Y}.\n",
    "\\end{equation}\\quad(Equation\\;\\;3.39)$$\n",
    "$$\\begin{equation}\n",
    "\\text{RSS}(\\mathbf{B};\\mathbf{\\Sigma}) = \\sum_{i=1}^N (y_i-f(x_i))^T \\mathbf{\\Sigma}^{-1} (y_i-f(x_i))\n",
    "\\end{equation} \\quad(Equation\\;\\;3.40)$$\n",
    "\n",
    "\n",
    "With $N$ training cases we can write the model in matrix notation\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\mathbf{Y}=\\mathbf{XB}+\\mathbf{E},\n",
    "\\end{equation}\n",
    "$$\n",
    "where\n",
    "\n",
    "* $\\mathbf{Y}$ is $N\\times K$ with $ik$ entry $y_{ik}$,\n",
    "* $\\mathbf{X}$ is $N\\times(p+1)$ input matrix,\n",
    "* $\\mathbf{B}$ is $(p+1)\\times K$ parameter matrix,\n",
    "* $\\mathbf{E}$ is $N\\times K$ error matrix.\n",
    "\n",
    "A straightforward generalization of the univariate loss function is\n",
    "$$\n",
    "\\begin{align}\n",
    "\\text{RSS}(\\mathbf{B}) &= \\sum_{k=1}^K \\sum_{i=1}^N \\left( y_{ik} - f_k(x_i) \\right)^2 \\\\\n",
    "&= \\text{trace}\\left( (\\mathbf{Y}-\\mathbf{XB})^T(\\mathbf{Y}-\\mathbf{XB}) \\right)\n",
    "\\end{align}$$\n",
    "\n",
    "The least squares estimates have exactly the same form as before\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat{\\mathbf{B}} = \\left(\\mathbf{X}^T\\mathbf{X}\\right)^{-1}\\mathbf{X}^T\\mathbf{Y}.\n",
    "\\end{equation} \\quad(Equation\\;\\;3.39)$$\n",
    "\n",
    "If the errors $\\epsilon = (\\epsilon_1,\\cdots,\\epsilon_K)$ are correlated with $\\text{Cov}(\\epsilon)=\\mathbf{\\Sigma}$, then the multivariate weighted criterion\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\text{RSS}(\\mathbf{B};\\mathbf{\\Sigma}) = \\sum_{i=1}^N (y_i-f(x_i))^T \\mathbf{\\Sigma}^{-1} (y_i-f(x_i))\n",
    "\\end{equation} \\quad(Equation\\;\\;3.40)$$\n",
    "\n",
    "if the covariance matrices\n",
    "$\\mathbf\\Sigma_i$ are different for each observation $$\n",
    "\\begin{equation}\n",
    "\\text{RSS}(\\mathbf{B}) = \\sum_{i=1}^N (y_i-f(x_i))^T \\mathbf{\\Sigma_i}^{-1} (y_i-f(x_i))\n",
    "\\end{equation}$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.12 (ordinary least squares to implement ridge regression)  \n",
    "\n",
    "Show that the ridge regression estimates can be obtained by\n",
    "ordinary least squares regression on an augmented data set. We augment\n",
    "the centered matrix $X$ with $p$ additional rows $\\sqrt{\\lambda}\\mathbf I$, and augment $y$ with $p$ zeros. By introducing artificial data having response value zero, the fitting procedure is forced to shrink the coefficients toward zero. This is related to the idea of hints due to Abu-Mostafa (1995), where model constraints are implemented by adding artificial data examples that satisfy them.\n",
    "\n",
    "\n",
    "Consider the input centered data matrix $\\mathbf X$ (of size $p\\times p$) and the output data vector $\\mathbf Y$ both appended (to produce the new variables $\\hat{\\mathbf X}$ and $\\hat{\\mathbf Y}$ ) as follows $$\\hat{\\mathbf X}=\\begin{bmatrix}\n",
    "\\underset{p\\times p}{\\mathbf X}\\\\\n",
    "\\sqrt{\\lambda}\\underset{p\\times p}{\\mathbf I}\\\\\n",
    "\\end{bmatrix}$$ and $$\\hat{\\mathbf Y}=\\begin{bmatrix}\n",
    "\\underset{p\\times 1}{\\mathbf Y}\\\\\n",
    "\\underset{p\\times 1}{\\mathbf 0}\\\\\n",
    "\\end{bmatrix}$$ The the classic least squares solution to this new problem is given by  $$\n",
    "\\begin{align}\n",
    "\\hat{\\mathbf{\\beta^{ls}}} &= \\left(\\mathbf{\\hat X}^T\\mathbf{\\hat X}\\right)^{-1}\\mathbf{\\hat X}^T\\mathbf{\\hat Y}\\\\\n",
    "&=\\left(\\begin{bmatrix}\n",
    "\\underset{p\\times p}{\\mathbf X}^T&\\sqrt{\\lambda}\\underset{p\\times p}{\\mathbf I}^T\\\\\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "\\underset{p\\times p}{\\mathbf X}\\\\\n",
    "\\sqrt{\\lambda}\\underset{p\\times p}{\\mathbf I}\\\\\n",
    "\\end{bmatrix}\\right)^{-1}\\begin{bmatrix}\n",
    "\\underset{p\\times p}{\\mathbf X}^T&\\sqrt{\\lambda}\\underset{p\\times p}{\\mathbf I}^T\\\\\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "\\underset{p\\times 1}{\\mathbf Y}\\\\\n",
    "\\underset{p\\times 1}{\\mathbf 0}\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&=\\left(\\mathbf X^T\\mathbf X+\\lambda\\mathbf I\\right)^{-1}\\left(\\mathbf X^T\\mathbf Y\\right)\n",
    "\\end{align}\n",
    "$$ This expression we recognize as the solution to the regularized least squares proving the equivalence.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.13 (principal component regression)  \n",
    "\n",
    "Derive the expression (3.62), and show that $$\\hat{\\beta}^{pcr}(p)=\\hat{β}^{ls}$$\n",
    "\n",
    "\n",
    "The linear combinations $Z_m$ used in principal component regression (PCR) are the principal components. PCR forms the derived input columns\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\mathbf{z}_m = \\mathbf{X} v_m,\n",
    "\\end{equation}\n",
    "$$\n",
    "and then regress $\\mathbf{y}$ on $\\mathbf{z}_1,\\mathbf{z}_2,\\cdots,\\mathbf{z}_M$ for some $M\\le p$. Since the $\\mathbf{z}_m$ are orthogonal, this regression is just a sum of univariate regressions:\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat{\\mathbf{y}}_{(M)}^{\\text{pcr}} = \\bar{y}\\mathbf{1} + \\sum_{m=1}^M \\hat\\theta_m \\mathbf{z}_m = \\bar{y}\\mathbf{1} + \\mathbf{X}\\mathbf{V}_M\\hat{\\mathbf{\\theta}}=\\bar{y}\\mathbf{1} + \\mathbf{X}\\sum_{m=1}^M\\hat\\theta_mv_m,\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "where $\\hat\\theta_m = \\langle\\mathbf{z}_m,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_m,\\mathbf{z}_m\\rangle$. We can see from the last equality that, since the $\\mathbf{z}_m$ are each linear combinations of the original $\\mathbf{x}_j$, we can express the solution in terms of coefficients of the $\\mathbf{x}_j$. $$\\begin{bmatrix}\n",
    "\\mathbf 1 &\\mathbf X \n",
    "\\end{bmatrix}$$ and \n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat\\beta^{\\text{pcr}}(M) = \\begin{bmatrix}\n",
    "\\bar{y}\\\\\n",
    "\\sum_{m=1}^M \\hat\\theta_m v_m\n",
    "\\end{bmatrix}\n",
    "\\end{equation}  \\quad(Equation\\;\\;3.62)\n",
    "$$\n",
    "\n",
    "For multiple linear regression models the estimated regression coefficients in the vector $\\hat{\\beta}$ can be split into two parts a scalar $\\hat\\beta_0$ and a $p \\times 1$ vector $\\hat\\beta^∗$ as $$\\hat{\\beta}=\\begin{bmatrix}\n",
    "\\hat\\beta_0\\\\\n",
    "\\hat\\beta^∗ \n",
    "\\end{bmatrix}$$ In addition the coefficient $\\hat\\beta_0$ can be shown to be related to $\\hat\\beta^∗$ as$$\\hat\\beta_0=\\bar{y}-\\hat\\beta^∗\\bar{x}$$ and since $$\\begin{bmatrix}\n",
    "\\bar{y}\\\\\n",
    "\\sum_{m=1}^M \\hat\\theta_m v_m\n",
    "\\end{bmatrix}=\\begin{bmatrix}\n",
    "\\hat\\beta_0\\\\\n",
    "\\hat\\beta^∗ \n",
    "\\end{bmatrix}$$ then $\\hat\\beta_0=\\bar{y}$ and $\\bar{x}=0$, which means in in principal component regression, the input predictor variables are standardized.\n",
    "\n",
    "\n",
    "To evaluate $\\hat\\beta^{\\text{pcr}}(M)$ when $M = p$, recall that in principal component regression the vectors $v_m$ are from the $SVD$ decomposition of the matrix $X$ given by $X = UDV^T$. This later expression is equivalent to $XV = UD$ from which we get  $$\n",
    "\\begin{align}\n",
    "Xv_1&=d_1u_1\\\\\n",
    "Xv_2&=d_2u_2\\\\\n",
    "\\vdots\\\\\n",
    "Xv_p&=d_pu_p\\\\\n",
    "\\end{align}$$ \n",
    "Since the vectors $z_m$ are given by $z_m = Xv_m$ for $1 \\le m \\le M \\le p$, by the above we have the vectors $z_m$ in terms of the vectors $u_m$ as $z_m = d_mu_m$. Since for this problem we are to show that when $M = p$ the estimate $\\hat\\beta^{\\text{pcr}}(M)$ above becomes the least squares estimate $\\hat\\beta^{\\text{ls}}$ When $\\lambda = 0$ the ridge regression estimate for $\\beta$ is equal to the least\n",
    "squares estimate (when $\\lambda = 0$ we have no $\\lambda\\lVert\\beta\\rVert_2^2$ ridge penalty term) we can use Equation $$\n",
    "\\begin{align}\n",
    "\\hat\\beta^{\\text{ridge}} &= \\left( \\mathbf{X}^T\\mathbf{X} + \\lambda\\mathbf{I} \\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V\\mathbf D^2\\mathbf V^T+\\lambda\\mathbf{V}\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V(\\mathbf D^2+\\lambda\\mathbf{I})\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}^{-T}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "to obtain $$\\hat\\beta^{\\text{ls}}=\\hat\\beta^{\\text{ridge}}(\\lambda=0)=\\mathbf{V}\\mathbf D^{-2}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\=\\mathbf{V}\\mathbf D^{-1}\\mathbf U^T\\mathbf{y}$$\n",
    "Since $$\\hat\\beta^{\\text{pcr}}(p)=\\sum_{m=1}^M\\hat\\theta_mv_m=\\mathbf{V}\\begin{bmatrix}\n",
    "\\hat\\theta_1\\\\\n",
    "\\hat\\theta_2\\\\\n",
    "\\vdots\\\\\n",
    "\\hat\\theta_p\n",
    "\\end{bmatrix}=\\mathbf{V}\\begin{bmatrix}\n",
    "\\langle\\mathbf{z}_1,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_1,\\mathbf{z}_1\\rangle\\\\\n",
    "\\langle\\mathbf{z}_2,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_2,\\mathbf{z}_2\\rangle\\\\\n",
    "\\vdots\\\\\n",
    "\\langle\\mathbf{z}_p,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_p,\\mathbf{z}_p\\rangle\n",
    "\\end{bmatrix}$$ and since $z_m = d_mu_m$ we have $$\\langle\\mathbf{z}_m,\\mathbf{z}_m\\rangle=d_m^2\\langle\\mathbf{u}_m,\\mathbf{u}_m\\rangle=d_m^2$$ since the vectors $u_m$ are assumed to be orthonormal. Also $\\langle\\mathbf{z}_m,\\mathbf{y}\\rangle = d_m \\langle\\mathbf{u}_m,\\mathbf{y}\\rangle$, so the estimate for $$\\hat\\beta^{\\text{pcr}}(p)=\\mathbf{V}\\begin{bmatrix}\n",
    "\\langle\\mathbf{z}_1,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_1,\\mathbf{z}_1\\rangle\\\\\n",
    "\\langle\\mathbf{z}_2,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_2,\\mathbf{z}_2\\rangle\\\\\n",
    "\\vdots\\\\\n",
    "\\langle\\mathbf{z}_p,\\mathbf{y}\\rangle \\big/ \\langle\\mathbf{z}_p,\\mathbf{z}_p\\rangle\n",
    "\\end{bmatrix}=\\mathbf{V}\\begin{bmatrix}\n",
    "\\langle\\mathbf{u}_1,\\mathbf{y}\\rangle \\big/ d_1\\\\\n",
    "\\langle\\mathbf{u}_1,\\mathbf{y}\\rangle \\big/ d_1\\\\\n",
    "\\vdots\\\\\n",
    "\\langle\\mathbf{u}_1,\\mathbf{y}\\rangle \\big/ d_1\\\\\n",
    "\\end{bmatrix}=\\mathbf{V}\\mathbf D^{-1}\\mathbf U^T\\mathbf{y}\\\\\n",
    "=\\hat\\beta^{\\text{ls}}=\\hat\\beta^{\\text{ridge}}(\\lambda=0)$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.14 (when the inputs are orthogonal PLS stops after m = 1 step) \n",
    "\n",
    "Show that in the orthogonal case, $\\text{PLS}$ stops after $m = 1$ steps, because subsequent $\\hat{\\varphi}_{mj}$ in step 2 in Algorithm 3.3 are zero.\n",
    "\n",
    "Algorithm 3.3 Partial Least Squares:  \n",
    "\n",
    "1.  Standardize each $x_j$ to have mean zero and variance one. Set $\\hat y^{(0)} =\\bar{y}\\mathbf 1$, and $x^{(0)}_j = x_j , j = 1, \\cdots, p$.  \n",
    "\n",
    "2. For $m=1,2,\\cdots,p$  \n",
    "\n",
    "(a) $$z_m=\\sum_{j=1}^{p}\\hat\\varphi_{mj}x_j^{(m-1)}$$ where $$\\hat\\varphi_{mj}=\\langle x_j^{(m-1)}, y\\rangle$$  \n",
    "\n",
    "(b) $$\\hat\\theta_m=\\langle z_m, y\\rangle\\big/\\langle z_m, z_m\\rangle$$  \n",
    "\n",
    "(c) $$\\hat{y}^{(m)}=\\hat{y}^{(m-1)}+\\hat\\theta_mz_m$$  \n",
    "\n",
    "(d) Orthogonalize each $x_j^{(m-1)}$ with respect to $$z_m:x_j^{(m)}=x_j^{(m-1)}-\\left[\\langle z_m, x_j^{(m-1)}\\rangle\\big/\\langle z_m, z_m\\rangle\\right]z_m, j=1,2,\\cdots,p$$  \n",
    "\n",
    "3. Output the sequence of fitted vectors $\\{\\hat y^{(m)}\\}_1^p$. Since the $\\{z_{\\ell}\\}_1^m$ are linear in the original $x_j$, so is $\\hat y^{(m)} = X\\hat{\\beta}^{\\text{pls}}(m)$. These linear coefficients can be recovered from the sequence of $\\text{PLS}$ transformations.\n",
    "\n",
    "\n",
    "In Algorithm 3.3 on partial least squares (PLS) if on any given step $m$ we compute that $$\\hat\\varphi_{mj} =\\langle x_j^{(m-1)}, y\\rangle= 0$$ for all $j$ then all $$z_m=\\sum_{j=1}^{p}\\hat\\varphi_{mj}x_j^{(m-1)}=0$$ it follows that the algorithm must stop. \n",
    "If the $x_j$’s are orthogonal, we have  \n",
    "\n",
    "(a) $$z_1=\\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}$$ where $$\\hat\\varphi_{1j}=\\langle x_j^{(0)}, y\\rangle$$  \n",
    "\n",
    "\n",
    "(b) $$\\hat\\theta_1=\\langle z_1, y\\rangle\\big/\\langle z_1, z_1\\rangle$$  Since $$\\begin{align}\n",
    "\\langle z_1, z_1\\rangle&=\\left\\langle \\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}, \\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}\\right\\rangle\\\\\n",
    "&=\\sum_{j=1}^{p}\\sum_{j'=1}^{p}\\hat\\varphi_{1j}\\hat\\varphi_{1j'}\\left\\langle x_j^{(0)},x_{j'}^{(0)}\\right\\rangle\\\\\n",
    "&=\\sum_{j=1}^{p}\\hat\\varphi_{1j}^2\\\\\n",
    "\\end{align}$$ since the vectors $x_j^{(0)}$ are orthogonal. $$\\langle z_1, y\\rangle=\\left\\langle \\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}, y\\right\\rangle=\\sum_{j=1}^{p}\\hat\\varphi_{1j}\\left\\langle x_j^{(0)}, y\\right\\rangle=\\sum_{j=1}^{p}\\hat\\varphi_{1j}^2$$ and thus we have $$\\hat\\theta_1=\\langle z_1, y\\rangle\\big/\\langle z_1, z_1\\rangle=1$$   \n",
    "\n",
    "\n",
    "(c) $$\\hat{y}^{(1)}=\\hat{y}^{(0)}+\\hat\\theta_1z_1=\\hat{y}^{(0)}+z_1=\\hat{y}^{(0)}+\\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}$$  \n",
    "\n",
    "\n",
    "(d) Orthogonalize each $x_j^{(1)}, j=1,2,\\cdots,p$ with respect to $$z_1:x_j^{(1)}=x_j^{(0)}-\\left[\\langle z_1, x_j^{(0)}\\rangle\\big/\\langle z_1, z_1\\rangle\\right]z_1$$ Since the vectors $x^{(0)}_j$ are orthogonal $$\\langle z_1, x_k^{(0)}\\rangle=\\sum_{j=1}^{p}\\hat\\varphi_{1j}\\langle x_j^{(0)}, x_k^{(0)}\\rangle=\\hat\\varphi_{1k}$$ then $$\\begin{align}\n",
    "x_j^{(1)}&=x_j^{(0)}-\\left[\\langle z_1, x_j^{(0)}\\rangle\\big/\\langle z_1, z_1\\rangle\\right]z_1\\\\\n",
    "&=x_j^{(0)}-\\left[\\hat\\varphi_{1j}\\big/\\sum_{j=1}^{p}\\hat\\varphi_{1j}^2\\right]\\sum_{j=1}^{p}\\hat\\varphi_{1j}x_j^{(0)}\\\\\n",
    "\\end{align}$$ Having finished the first loop of Algorithm 3.3 we let $m = 2$ and compute $\\hat\\varphi_{2j}$ to get $$\\hat\\varphi_{2j}=\\langle x_j^{(1)}, y\\rangle=\\langle x_j^{(0)}, y\\rangle-\\left[\\hat\\varphi_{1j}\\big/\\sum_{j=1}^{p}\\hat\\varphi_{1j}^2\\right]\\sum_{j=1}^{p}\\hat\\varphi_{1j}^2\\\\\n",
    "=\\hat\\varphi_{1j}-\\hat\\varphi_{1j}=0$$\n",
    "\n",
    "\n",
    "# Ex. 3.15 (PLS seeks directions that have high variance and high correlation)  \n",
    "\n",
    "Verify expression (3.64), and hence show that the partial least squares directions are a compromise between the ordinary regression coefficient and the principal component directions.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "PLS begins by computing the weights\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat\\varphi_{1j} = \\langle \\mathbf{x}_j,\\mathbf{y} \\rangle, \\text{ for each } j,\n",
    "\\end{equation}\n",
    "$$\n",
    "which are in fact the univariate regression coefficients, since $\\mathbf{x}_j$ are standardized (only for the first step $m=1$).\n",
    "\n",
    "From this we construct derived input\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\mathbf{z}_1 = \\sum_j \\hat\\varphi_{1j}\\mathbf{x}_j,\n",
    "\\end{equation}\n",
    "$$\n",
    "which is the first PLS direction. Hence in the construction of each $\\mathbf{z}_m$, the inputs are weighted by the strength of their univariate effect on $\\mathbf{y}$.\n",
    "\n",
    "The outcome $\\mathbf{y}$ is regressed on $\\mathbf{z}_1$ giving coefficient $\\hat\\theta_1$, and then we orthogonalize $\\mathbf{x}_1,\\cdots,\\mathbf{x}_p$ w.r.t. $\\mathbf{z}_1$.\n",
    "\n",
    "We continue this process, until $M\\le p$ directions have been obtained. In this manner, PLS produces a sequence of derived, orthogonal inputs or directions $\\mathbf{z}_1,\\cdots,\\mathbf{z}_M$.\n",
    "\n",
    "* As with PCR, if $M=p$, then $\\hat\\beta^{\\text{pls}} = \\hat\\beta^{\\text{ls}}$.\n",
    "* Using $M<p$ directions produces a reduced regression.\n",
    "\n",
    "### Relation to the optimization problem\n",
    "\n",
    "> PLS seeks direction that have high variance *and* have high correlation with the response, in contrast to PCR with keys only on high variance (Stone and Brooks, 1990; Frank and Friedman, 1993).\n",
    "Since it uses the response $\\mathbf{y}$ to construct its directions, its solution path is a nonlinear function of $\\mathbf{y}$.\n",
    "\n",
    "In particular, the $m$th principal component direction $v_m$ solves:\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\max_\\alpha \\text{Var}(\\mathbf{X}\\alpha)\\\\\n",
    "\\text{subject to } \\|\\alpha\\| = 1, \\alpha^T\\mathbf{S} v_l = 0 \\text{ for } l = 1,\\cdots, m-1,\n",
    "\\end{equation}\n",
    "$$\n",
    "where $\\mathbf{S}$ is the sample covariance matrix of the $\\mathbf{x}_j$. The condition $\\alpha^T\\mathbf{S} v_l= 0$ ensures that $\\mathbf{z}_m = \\mathbf{X}\\alpha$ is uncorrelated with all the previous linear combinations $\\mathbf{z}_l = \\mathbf{X} v_l$.\n",
    "\n",
    "The $m$th PLS direction $\\hat\\varphi_m$ solves:\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\max_\\alpha \\text{Corr}^2(\\mathbf{y},\\mathbf{S}\\alpha)\\text{Var}(\\mathbf{X}\\alpha)\\\\\n",
    "\\text{subject to } \\|\\alpha\\| = 1, \\alpha^T\\mathbf{S}\\hat\\varphi_l = 0, \\text{ for } l=1,\\cdots, m-1.\n",
    "\\end{equation}  \\quad (3.64)\n",
    "$$\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "appropriate-opportunity",
   "metadata": {},
   "source": [
    "# Ex. 3.16 (explicit expressions for $\\hat\\beta_j$ when the features are orthonormal)  \n",
    "\n",
    "Derive the entries in Table 3.4, the explicit forms for estimators in the orthogonal case.\n",
    "\n",
    "\n",
    "TABLE 3.4. Estimators of $\\beta_j$ in the case of orthonormal columns of $\\mathbf X$. $M$ and $\\lambda$\n",
    "are constants chosen by the corresponding techniques; sign denotes the sign of its argument ($\\pm 1$), and $x_+$ denotes “positive part” of $x$. Below the table, estimators are shown by broken red lines. The $45^\\circ$ line in gray shows the unrestricted estimate for reference.\n",
    "\n",
    "$$\\begin{array}{l}\n",
    "\\text{Estimator} & \\text{Formula}\\\\\n",
    "\\hline\n",
    "\\text{Best subset (size M)} &\\hat{\\beta}_j\\cdot I(\\lvert\\hat{\\beta}_j\\rvert\\ge\\lvert\\hat{\\beta}_{(M)}\\rvert)\\\\\n",
    "\\text{Ridge} & \\hat{\\beta}_j\\big/(1+\\lambda)\\\\\n",
    "\\text{Lasso} & \\text{sign}(\\hat{\\beta}_j)(\\lvert\\hat{\\beta}_j\\rvert-\\lambda)_+\\\\\n",
    "\\hline\n",
    "\\end{array}$$\n",
    "When the predictors are orthonormal $X^TX = I$ and the ordinary least squared estimate of $\\beta$ is given by $$\\hat\\beta=X^TY$$ \n",
    "\n",
    "In best-subset selection we will take the top $M$ predictors that result in the smallest residual\n",
    "sum of squares. Since the columns of $X$ are orthonormal we can construct a basis for $\\mathbb R^N$ by using the first $p$ columns of $X$ and then extending these with $N −p$ linearly independent\n",
    "additional orthonormal vectors. The Gram-Schmidt procedure guarantees that we can do this. Thus in this extended basis we can write $y$ as $$y=\\sum_{j=1}^{p}\\hat\\beta_jx_j+\\sum_{j=p+1}^{N}\\gamma_j\\tilde{x}_j$$\n",
    "where $\\hat\\beta_j$ equal the components of $\\hat\\beta$ in $$\\hat\\beta=X^TY$$ $\\tilde{x}_j$ are the extended basis vectors required to span $\\mathbb R^N$, and $\\gamma_j$ are the coefficients of $y$ with respect to these extended basis vectors. Then if we seek to approximate $y$ with a subset of size $M$ as in best subset selection,  $\\hat y$ can be written as\n",
    "$$\\hat y=\\sum_{j=1}^{p}I_j\\hat\\beta_jx_j$$ with $I_j=1$ if we keep the predictor $x_j$ and zero otherwise. Now since all the vectors $x_j$ and $\\tilde{x}_j$ are orthonormal we have \n",
    "$$\\begin{align}\n",
    "\\lVert y-\\hat y\\rVert_2^2&=\\lVert y-X\\hat\\beta\\rVert_2^2\\\\\n",
    "&=\\left\\lVert \\sum_{j=1}^{p}\\hat\\beta_jx_j+\\sum_{j=p+1}^{N}\\gamma_j\\tilde{x}_j-X\\hat\\beta\\right\\rVert_2^2\\\\\n",
    "&=\\left\\lVert \\sum_{j=1}^{p}\\hat\\beta_j(\\mathbf 1-I_j)x_j+\\sum_{j=p+1}^{N}\\gamma_j\\tilde{x}_j\\right\\rVert_2^2\\\\\n",
    "&=\\sum_{j=1}^{p}\\hat\\beta_j^2(\\mathbf 1-I_j)^2\\left\\lVert x_j\\right\\rVert_2^2+\\sum_{j=p+1}^{N}\\gamma_j^2\\left\\lVert\\tilde{x}_j\\right\\rVert_2^2\\\\\n",
    "&=\\sum_{j=1}^{p}\\hat\\beta_j^2(\\mathbf 1-I_j)^2+\\sum_{j=p+1}^{N}\\gamma_j^2\\\\\n",
    "\\end{align}$$\n",
    "\n",
    "Thus to minimize $\\lVert y-\\hat y\\rVert_2^2$ we would pick the $M$ values of $I_j$ to be equal to one that have the largest $\\beta_j^2$ values. This is equivalent to sorting the values $\\lvert\\hat\\beta_j\\rvert$ and picking the indices of\n",
    "the largest $M$ of these to have $I_j = 1$. All other indices $j$ would be taken to have $I_j = 0$. Using an indicator function this is equivalent to the expression $$\\hat\\beta_j^{\\text{best subset}}=\\hat\\beta_j\\cdot I\\left( \\lvert\\hat\\beta_j\\rvert \\ge \\lvert\\hat\\beta_{(M)}\\rvert \\right)$$\n",
    "\n",
    "\n",
    "* Best subset (size $M$) drops all variables with coefficients smaller than the $M$th largest; this is a form of \"hard-thresholding\". \n",
    "\n",
    "$$\\begin{equation}\n",
    "\\hat\\beta_j\\cdot I\\left( \\lvert\\hat\\beta_j\\rvert \\ge \\lvert\\hat\\beta_{(M)}\\rvert \\right)\n",
    "\\end{equation}$$\n",
    "\n",
    "For ridge regression, since $X$ has orthonormal columns we have $$\\begin{align}\n",
    "\\hat\\beta^{\\text{ridge}}&= \\left( \\mathbf{X}^T\\mathbf{X} + \\lambda I \\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\\\\n",
    "&=\\left( I + \\lambda I \\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\\\\n",
    "&=\\frac{\\mathbf{X}^T\\mathbf{y}}{1+\\lambda}\\\\\n",
    "&=\\frac{\\hat\\beta^{\\text{ls}}}{1+\\lambda}\n",
    "\\end{align}$$\n",
    "* Ridge does a proportional shrinkage. \n",
    "\n",
    "$$\\begin{equation}\n",
    "\\frac{\\hat\\beta_j}{1+\\lambda}\n",
    "\\end{equation}$$\n",
    "\n",
    "The lasso estimate is defined by\n",
    "$$\\begin{equation}\n",
    "\\hat\\beta^{\\text{lasso}} = \\arg\\min_\\beta \\left\\lbrace \\frac{1}{2}\\sum_{i=1}^N \\left( y_i-\\beta_0-\\sum_{j=1}^p x_{ij}\\beta_j \\right)^2 + \\lambda\\sum_{j=1}^p |\\beta_j| \\right\\rbrace,\n",
    "\\end{equation}$$\n",
    "For the lasso regression procedure we pick the values of $\\beta_j$ to minimize \n",
    "$$\\text{RSS}(\\beta)=\\frac{1}{2}(y-X\\beta)^T(y-X\\beta)+\\lambda\\sum_{j=1}^{p}\\lvert \\beta_j\\rvert$$ Expanding $\\hat y$ as \n",
    "$$\\hat y = \\sum_{j=1}^{p} \\beta_jx_j$$ and with $$y=\\sum_{j=1}^{p}\\hat\\beta_jx_j+\\sum_{j=p+1}^{N}\\gamma_j\\tilde{x}_j$$ we have that $\\text{RSS}(\\beta)$ in this case becomes $$\\begin{align}\n",
    "\\text{RSS}(\\beta)&=\\frac{1}{2}\\left\\lVert \\sum_{j=1}^{p}(\\hat\\beta_j-\\beta_j)x_j+\\sum_{j=p+1}^{N}\\gamma_j\\tilde{x}_j\\right\\rVert_2^2+\\lambda\\sum_{j=1}^{p}\\lvert \\beta_j\\rvert\\\\\n",
    "&=\\frac{1}{2}\\sum_{j=1}^{p}(\\hat\\beta_j-\\beta_j)^2+\\frac{1}{2}\\sum_{j=p+1}^{N}\\gamma_j^2+\\lambda\\sum_{j=1}^{p}\\lvert \\beta_j\\rvert\\\\\n",
    "&=\\frac{1}{2}\\sum_{j=1}^{p}\\left[(\\hat\\beta_j-\\beta_j)^2+2\\lambda\\lvert \\beta_j\\rvert\\right]+\\sum_{j=p+1}^{N}\\gamma_j^2\n",
    "\\end{align}$$ We can minimize this expression for each value of $\\beta_j$ for $1 \\le j \\le p$ independently. Thus our vector problem becomes that of solving $p$ scalar minimization problems all of which look like\n",
    "$$\\beta^*=\\text{argmin}_{\\beta}\\left[(\\hat\\beta-\\beta)^2+2\\lambda\\lvert \\beta\\rvert\\right]$$ Then the objective function we want to minimize is $$\n",
    "F(\\beta)=\\begin{cases}\n",
    "(\\hat\\beta-\\beta)^2-2\\lambda\\beta &\\text{ when } \\beta<0\\\\[2ex]\n",
    "(\\hat\\beta-\\beta)^2+2\\lambda\\beta &\\text{ when } \\beta>0\\\\\n",
    "\\end{cases}\n",
    "$$ where $\\hat\\beta$ is the least squares estimate. \n",
    "\n",
    "To find the minimum of this function take the derivative with respect to $\\beta$ and set the result equal to zero and solve for $\\beta$. We find the derivative of $F(\\beta)$ given by $$\n",
    "F'(\\beta)=\\begin{cases}\n",
    "2(\\beta-\\hat\\beta)-2\\lambda &\\text{ when } \\beta<0\\\\[2ex]\n",
    "2(\\beta-\\hat\\beta)+2\\lambda &\\text{ when } \\beta>0\\\\\n",
    "\\end{cases}\n",
    "$$ When we set $F'(\\beta)$ equal to zero we get two possible solutions for $\\beta$ given by \n",
    "$$\\begin{cases}\n",
    "\\beta=+\\lambda+\\hat\\beta &\\text{ when } \\beta<0\\\\[2ex]\n",
    "\\beta=-\\lambda+\\hat\\beta &\\text{ when } \\beta>0\\\\[2ex]\n",
    "\\end{cases}$$\n",
    "\n",
    "* Lasso translates each coefficient by a constant factor $\\lambda$, truncating at zero. This is called \"soft-thresholding\". \n",
    "\n",
    "$$\\begin{equation}\n",
    "\\text{sign}\\left( \\hat\\beta_j \\right)\\left( \\lvert\\hat\\beta_j\\rvert - \\lambda \\right)_+\n",
    "\\end{equation}$$\n",
    "\n",
    "In this expression $\\hat\\beta$ and $\\lambda$ are assumed fixed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "original-disaster",
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext rpy2.ipython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "given-president",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "xx = seq(-15,+15,length=1000) # the range of beta's to sample \n",
    "\n",
    "# the specific parameters:\n",
    "betaHat = -5.0\n",
    "lambda  = 1.0\n",
    "\n",
    "plot( xx, ( xx - betaHat )^2 + 2*lambda*abs(xx) )\n",
    "\n",
    "\n",
    "\n",
    "betaHat = -1.0\n",
    "lambda  = 5.0\n",
    "\n",
    "\n",
    "plot( xx, ( xx - betaHat )^2 + 2*lambda*abs(xx) )\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "betaHat = 5.0\n",
    "lambda  = 1.0\n",
    "\n",
    "\n",
    "plot( xx, ( xx - betaHat )^2 + 2*lambda*abs(xx) )\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "betaHat = 1.0\n",
    "lambda  = 5.0\n",
    "\n",
    "\n",
    "plot( xx, ( xx - betaHat )^2 + 2*lambda*abs(xx) )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "unexpected-springfield",
   "metadata": {},
   "source": [
    "# Ex. 3.17 (linear methods on the spam data set)  \n",
    "\n",
    "Repeat the analysis of Table 3.3 on the spam data discussed in Chapter 1.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "floppy-webmaster",
   "metadata": {},
   "outputs": [],
   "source": [
    "%%R\n",
    "load_spam_data <- function(trainingScale=TRUE,responseScale=TRUE){ \n",
    "\n",
    "  X      = read.table(\"../../data/Spam/spam.data.txt\")\n",
    "  tt     = read.table(\"../../data/Spam/spam.traintest.txt\")\n",
    "\n",
    "  # separate into training/testing sets\n",
    "\n",
    "  XTraining = subset( X, tt==0 )\n",
    "  p = dim(XTraining)[2]-1\n",
    "  \n",
    "  XTesting  = subset( X, tt==1 ) \n",
    "\n",
    "  #\n",
    "  # Sometime data is processed and stored in a certain order.  When doing cross validation\n",
    "  # on such data sets we don't want to bias our results if we grab the first or the last samples.\n",
    "  # Thus we randomize the order of the rows in the Training data frame to make sure that each\n",
    "  # cross validation training/testing set is as random as possible.\n",
    "  # \n",
    "  if( FALSE ){\n",
    "    nSamples = dim(XTraining)[1] \n",
    "    inds = sample( 1:nSamples, nSamples )\n",
    "    XTraining = XTraining[inds,]\n",
    "  }\n",
    "\n",
    "  #\n",
    "  # In reality we have to estimate everything based on the training data only\n",
    "  # Thus here we estimate the predictor statistics using the training set\n",
    "  # and then scale the testing set by the same statistics\n",
    "  # \n",
    "  if( trainingScale ){\n",
    "    X = XTraining \n",
    "    if( responseScale ){\n",
    "      meanV58 = mean(X$V58) \n",
    "      v58 = X$V58 - meanV58 \n",
    "    }else{\n",
    "      v58 = X$V58 \n",
    "    }\n",
    "    X$V58 = NULL\n",
    "    X = scale(X, TRUE, TRUE)\n",
    "    means = attr(X,\"scaled:center\")\n",
    "    stds = attr(X,\"scaled:scale\")\n",
    "    Xf = data.frame(X)\n",
    "    Xf$V58 = v58\n",
    "    XTraining = Xf\n",
    "\n",
    "    # scale the testing predictors by the same amounts:\n",
    "    # \n",
    "    DCVTest  = XTesting\n",
    "    if( responseScale ){\n",
    "      v58Test = DCVTest$V58 - meanV58\n",
    "    }else{\n",
    "      v58Test = DCVTest$V58 # in physical units (not mean adjusted)\n",
    "    }\n",
    "    DCVTest$V58 = NULL \n",
    "    DCVTest  = t( apply( DCVTest, 1, '-', means ) ) \n",
    "    DCVTest  = t( apply( DCVTest, 1, '/', stds ) ) \n",
    "    DCVTestb = cbind( DCVTest, v58Test ) # append back on the response\n",
    "    DCVTestf = data.frame( DCVTestb ) # a data frame containing all scaled variables of interest\n",
    "    names(DCVTestf)[p+1] = \"V58\" # fix the name of the response\n",
    "    XTesting = DCVTestf\n",
    "  }\n",
    "\n",
    "  # Many algorithms wont do well if the data is presented all of one class and\n",
    "  # then all of another class thus we permute our data frames :\n",
    "  #\n",
    "  XTraining = XTraining[sample(nrow(XTraining)),]\n",
    "  XTesting  = XTesting[sample(nrow(XTesting)),]\n",
    "\n",
    "  # Read in the list of s(pam)words (and delete garbage characters):\n",
    "  # \n",
    "  spam_words = read.table(\"../../data/Spam/spambase.names\",skip=33,sep=\":\",comment.char=\"|\",stringsAsFactors=F)\n",
    "  spam_words = spam_words[[1]]\n",
    "  for( si in 1:length(spam_words) ){\n",
    "    spam_words[si] = sub( \"word_freq_\", \"\", spam_words[si] )\n",
    "    spam_words[si] = sub( \"char_freq_\", \"\", spam_words[si] )\n",
    "  }\n",
    "\n",
    "  return( list( XTraining, XTesting, spam_words ) ) \n",
    "}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "imposed-distinction",
   "metadata": {},
   "source": [
    "### OLS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "green-magic",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 0.1212304\n",
      "            [,1]\n",
      "[1,] 0.007212358\n"
     ]
    }
   ],
   "source": [
    "%%R\n",
    "res = load_spam_data(trainingScale=TRUE,responseScale=FALSE)\n",
    "XTraining = res[[1]]\n",
    "XTesting = res[[2]] \n",
    "\n",
    "nrow = dim( XTraining )[1]\n",
    "p = dim( XTraining )[2] - 1 # the last column is the response \n",
    "\n",
    "D = XTraining[,1:p] # get the predictor data\n",
    "\n",
    "# Append a column of ones:\n",
    "# \n",
    "Dp = cbind( matrix(1,nrow,1), as.matrix( D ) )\n",
    "\n",
    "response = XTraining[,p+1]\n",
    "\n",
    "library(MASS)\n",
    "\n",
    "betaHat = ginv( t(Dp) %*% Dp ) %*% t(Dp) %*% as.matrix(response)\n",
    "\n",
    "# this is basically the first column in Table 3.2:\n",
    "#\n",
    "\n",
    "\n",
    "# make predictions based on these estimated beta coefficients:\n",
    "#\n",
    "yhat = Dp %*% betaHat \n",
    "\n",
    "# estimate the variance:\n",
    "# \n",
    "sigmaHat = sum( ( response - yhat )^2 ) / ( nrow - p - 1 )\n",
    "\n",
    "# calculate the covariance of betaHat:\n",
    "#\n",
    "covarBetaHat = sigmaHat * ginv( t(Dp) %*% Dp ) \n",
    "\n",
    "# calulate the standard deviations of betahat:\n",
    "#\n",
    "stdBetaHat = sqrt(diag(covarBetaHat))\n",
    "\n",
    "# this is basically the second column in Table 3.2:\n",
    "#\n",
    "\n",
    "\n",
    "# compute the z-scores:\n",
    "#\n",
    "z = betaHat / stdBetaHat \n",
    "\n",
    "# this is basically the third column in Table 3.2:\n",
    "#\n",
    "\n",
    "\n",
    "\n",
    "cbind(data.frame(`beta estimates` = betaHat), data.frame(`beta standard errors` = as.matrix(stdBetaHat)), data.frame(`beta z-scores` = z))\n",
    "\n",
    "# display the results we get : \n",
    "F = data.frame( Term=c(\"Intercept\",names(XTraining)[1:p]), Coefficients=betaHat, Std_Error=stdBetaHat, Z_Score=z )\n",
    "library(xtable)\n",
    "xtable( F, caption=\"Table of OLS coefficients for the SPAM data set\", digits=2 ) \n",
    "\n",
    "# Run this full linear model on the Testing data so that we can fill in the two\n",
    "# lower spots in the \"LS\" column in Table 3.2\n",
    "#\n",
    "pdt = cbind( matrix(1,dim(XTesting)[1],1), as.matrix( XTesting[,1:p] ) ) %*% betaHat \n",
    "responseTest = XTesting[,p+1]\n",
    "NTest = length(responseTest)\n",
    "mErr = mean( (responseTest - pdt)^2 )\n",
    "print( mErr ) \n",
    "sErr = sqrt( var( (responseTest - pdt)^2 )/NTest ) \n",
    "print( sErr )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "educational-stake",
   "metadata": {},
   "source": [
    "### Runs ridge regression on the SPAM data set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "drawn-desktop",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "R[write to console]: \n",
      "Attaching package: ‘gplots’\n",
      "\n",
      "\n",
      "R[write to console]: The following object is masked from ‘package:stats’:\n",
      "\n",
      "    lowess\n",
      "\n",
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 0.3974\n",
      "           [,1]\n",
      " [1,] -0.012063\n",
      " [2,] -0.018186\n",
      " [3,]  0.024423\n",
      " [4,]  0.017785\n",
      " [5,]  0.048324\n",
      " [6,]  0.035094\n",
      " [7,]  0.082626\n",
      " [8,]  0.038886\n",
      " [9,]  0.012246\n",
      "[10,]  0.010097\n",
      "[11,]  0.008338\n",
      "[12,] -0.022603\n",
      "[13,]  0.000821\n",
      "[14,]  0.005473\n",
      "[15,]  0.008723\n",
      "[16,]  0.080193\n",
      "[17,]  0.025889\n",
      "[18,]  0.025022\n",
      "[19,]  0.033607\n",
      "[20,]  0.020922\n",
      "[21,]  0.063838\n",
      "[22,]  0.046940\n",
      "[23,]  0.058275\n",
      "[24,]  0.041145\n",
      "[25,] -0.041553\n",
      "[26,] -0.013468\n",
      "[27,] -0.036863\n",
      "[28,] -0.013823\n",
      "[29,] -0.002609\n",
      "[30,] -0.022522\n",
      "[31,] -0.004602\n",
      "[32,]  0.033556\n",
      "[33,] -0.017971\n",
      "[34,] -0.023728\n",
      "[35,] -0.009010\n",
      "[36,]  0.016002\n",
      "[37,] -0.009944\n",
      "[38,] -0.012490\n",
      "[39,] -0.017749\n",
      "[40,]  0.023000\n",
      "[41,] -0.001821\n",
      "[42,] -0.025790\n",
      "[43,] -0.012405\n",
      "[44,] -0.020960\n",
      "[45,] -0.039588\n",
      "[46,] -0.034961\n",
      "[47,] -0.009396\n",
      "[48,] -0.018026\n",
      "[49,] -0.030376\n",
      "[50,] -0.027790\n",
      "[51,] -0.004496\n",
      "[52,]  0.041522\n",
      "[53,]  0.061850\n",
      "[54,]  0.012561\n",
      "[55,]  0.005090\n",
      "[56,]  0.021765\n",
      "[57,]  0.038060\n",
      "[1] 0.1212304\n",
      "            [,1]\n",
      "[1,] 0.007212358\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "cv_Ridge <- function( lambda, D, numberOfCV ){\n",
    "  #\n",
    "  # Does Cross Validation of the Ridge Regression Linear Model specified by the formula \"form\"\n",
    "  #\n",
    "  # Input:\n",
    "  #   lambda = multiplier of the penatly term \n",
    "  #   D = training data frame where the last column is the response variable\n",
    "  #   numberOfCV = number of cross validations to do\n",
    "  #\n",
    "  # Output:\n",
    "  #   MSPE = vector of length numberOfCV with components mean square prediction errors\n",
    "  #          extracted from each of the numberOfCV test data sets\n",
    "  \n",
    "  library(MASS) # needed for ginv\n",
    "\n",
    "  nSamples = dim( D )[1]\n",
    "  p        = dim( D )[2] - 1 # the last column is the response \n",
    "  \n",
    "  # needed for the cross vaildation (CV) loops:\n",
    "  # \n",
    "  nCVTest    = round( nSamples*(1/numberOfCV) ) # each CV run will have this many test points\n",
    "  nCVTrain   = nSamples-nCVTest # each CV run will have this many training points \n",
    "\n",
    "  for (cvi in 1:numberOfCV) {\n",
    "\n",
    "    testInds = (cvi-1)*nCVTest + 1:nCVTest # this may attempt to access sample indexes > nSamples \n",
    "    testInds = intersect( testInds, 1:nSamples ) # so we restrict this list here\n",
    "    DCVTest  = D[testInds,1:(p+1)] # get the predictor + response testing data\n",
    "\n",
    "    # select this cross validation's section of data from all the training data:\n",
    "    # \n",
    "    trainInds = setdiff( 1:nSamples, testInds ) \n",
    "    DCV       = D[trainInds,1:(p+1)] # get the predictor + response training data\n",
    "\n",
    "    # Get the response and delete it from the data frame\n",
    "    # \n",
    "    responseName        = names(DCV)[p+1] # get the actual string name from the data frame \n",
    "    response            = DCV[,p+1] \n",
    "    DCV[[responseName]] = NULL\n",
    "\n",
    "    # For ridge-regression we begin by standardizing the predictors and demeaning the response\n",
    "    #\n",
    "    # Note that one can get the centering value (the mean) with the command attr(DCV,'scaled:center')\n",
    "    #                   and the scaling value (the sample standard deviation) with the command attr(DCV,'scaled:scale')\n",
    "    # \n",
    "    DCV          = scale( DCV )\n",
    "    responseMean = mean( response )\n",
    "    response     = response - responseMean \n",
    "    DCVb         = cbind( DCV, response ) # append back on the response \n",
    "    DCVf         = data.frame( DCVb ) # a data frame containing all scaled variables of interest \n",
    "\n",
    "    # extract the centering and scaling information:\n",
    "    #\n",
    "    means = attr(DCV,\"scaled:center\")\n",
    "    stds  = attr(DCV,\"scaled:scale\")\n",
    "\n",
    "    # apply the computed scaling based on the training data to the testing data:\n",
    "    #\n",
    "    responseTest            = DCVTest[,p+1] - responseMean   # mean adjust\n",
    "    DCVTest[[responseName]] = NULL \n",
    "    DCVTest                 = t( apply( DCVTest, 1, '-', means ) )                                             \n",
    "    DCVTest                 = t( apply( DCVTest, 1, '/', stds ) )                                              \n",
    "    DCVTestb                = cbind( DCVTest, responseTest ) # append back on the response                     \n",
    "    DCVTestf                = data.frame( DCVTestb )         # a data frame containing all scaled variables of interest\n",
    "    names(DCVTestf)[p+1]    = responseName                   # fix the name of the response\n",
    "\n",
    "    # fit this linear model and compute the expected prediction error (EPE) using these features (this just estimates the mean in this case):\n",
    "    #\n",
    "    DM      = as.matrix(DCV)\n",
    "    M       = ginv( t(DM) %*% DM + lambda*diag(p) ) %*% t(DM)\n",
    "    betaHat = M %*% as.matrix(DCVf[,p+1])\n",
    "\n",
    "    DTest   = as.matrix(DCVTest)\n",
    "    pdt     = DTest %*% betaHat # get demeaned predictions on the test set\n",
    "    #pdt     = pdt + responseMean # add back in the mean if we want the response in physical units (not mean adjusted)\n",
    "\n",
    "    if( cvi==1 ){\n",
    "      predmat = pdt\n",
    "      y       = responseTest\n",
    "    }else{\n",
    "      predmat = c( predmat, pdt )\n",
    "      y       = c( y, responseTest )\n",
    "    }\n",
    "\n",
    "  } # endfor cvi loop\n",
    "  \n",
    "  # this code is modeled after the code in \"glmnet/cvelnet.R\"\n",
    "  #\n",
    "  N = length(y)\n",
    "  cvraw = ( y - predmat )^2\n",
    "  cvm   = mean(cvraw)\n",
    "  cvsd  = sqrt( var(cvraw)/N )\n",
    "  l = list( cvraw=cvraw, cvm=cvm, cvsd=cvsd, name=\"Mean Squared Error\" )\n",
    "\n",
    "  return(l) \n",
    "}\n",
    "one_standard_error_rule <- function(complexityParam,cvResults){\n",
    "  #\n",
    "  # Input:\n",
    "  #   complexityParam = the sampled complexity parameter (duplicate cross vaidation results expected)\n",
    "  #   cvResults = matrix where\n",
    "  #               each column is a different value of the complexity parameter (with complexity increasing from left to right) ...\n",
    "  #               each row    is the square prediction error (SPE) of a different cross validation sample i.e. ( y_i - \\hat{y}_i )^2 \n",
    "  #\n",
    "  # Output:\n",
    "  #   optCP = the optimal complexity parameter\n",
    " \n",
    "  N = dim(cvResults)[1]\n",
    "\n",
    "  # compute the complexity based mean and standard deviations:\n",
    "  means = apply(cvResults,2,mean)\n",
    "  stds  = sqrt( apply(cvResults,2,var)/N ) \n",
    "\n",
    "  # find the smallest espe:\n",
    "  minIndex = which.min( means )\n",
    "  \n",
    "  # compute the confidence interval around this point:\n",
    "  #cip = 1 - 2*pnorm(-1)\n",
    "  #ciw   <- qt(cip/2, n) * stdev / sqrt(n)\n",
    "  ciw = stds[minIndex] \n",
    "\n",
    "  # add a width of one std to the min point:\n",
    "  maxUncertInMin = means[minIndex] + 0.5*ciw\n",
    "\n",
    "  # find the mean that is nearest to this value and that is a SIMPLER model than the minimum mean model: \n",
    "  complexityIndex = which.min( abs( means[1:minIndex] - maxUncertInMin ) ) \n",
    "  complexityValue = complexityParam[complexityIndex]\n",
    "  \n",
    "  # package everything to send out:\n",
    "  res = list(complexityValue,complexityIndex,maxUncertInMin, means,stds)\n",
    "\n",
    "  return(res)\n",
    "}\n",
    "df_ridge <- function(lambda,X){ \n",
    "  #\n",
    "  # R code to compute the degress of freedom for ridge regression.\n",
    "  # This is formula 3.50 in the book.\n",
    "  #\n",
    "  # Input:\n",
    "  #   lambda: the penalty term coefficent \n",
    "  #   X the data matrix \n",
    "  #\n",
    "  # Output:\n",
    "  #\n",
    "  # dof: the degrees of freedom.\n",
    " \n",
    "\n",
    "  library(MASS) # needed for ginv\n",
    "\n",
    "  XTX = t(X) %*% X\n",
    "  pprime = dim(XTX)[1] \n",
    "\n",
    "  Hlambda = X %*% ginv( XTX + lambda * diag(pprime) ) %*% t(X)\n",
    "    \n",
    "  dof = sum( diag( Hlambda ) )\n",
    "\n",
    "  return(dof)\n",
    "  \n",
    "}\n",
    "\n",
    "opt_lambda_ridge <- function(X,multiple=1){ \n",
    "  #\n",
    "  # Returns the \"optimal\" set of lambda for nice looking ridge regression plots. \n",
    "  #\n",
    "  # Input:\n",
    "  #\n",
    "  #   X = matrix or data frame of features (do not prepend a constant of ones)\n",
    "  #\n",
    "  # Output:\n",
    "  #   lambdas = vector of optimal lambda (choosen to make the degrees_of_freedom(lambda)\n",
    "  #             equal to equal to the integers 1:p\n",
    "  #\n",
    "  #   Notes: these values of lambda are selected to solve (equation 3.50 for various values of dof)\n",
    "  #\n",
    "  #   dof = \\sum_{i=1}^p \\frac{ d_j^2 }{ d_j^2 + \\lambda } \n",
    "\n",
    "  s  = svd( X )\n",
    "  dj = s$d       # the diagional elements d_j.  Note that dj[1] is the largest value, while dj[end] is the smallest \n",
    "  p  = dim( X )[2]\n",
    "\n",
    "  # Do degrees_of_freedom=p first (this is the value for lambda when the degrees of freedom is *p*)\n",
    "  lambdas = 0 \n",
    "\n",
    "  # Do all other values for the degrees_of_freedom next:\n",
    "  # \n",
    "  kRange  = seq(p-1,1,by=(-1/multiple) )\n",
    "  for( ki in 1:length(kRange) ){ \n",
    "    # solve for lambda in (via newton iterations):\n",
    "    #   k = \\sum_{i=1}^p \\frac{ d_j^2 }{ d_j^2 + \\lambda }\n",
    "    #\n",
    "    k = kRange[ki]\n",
    "    \n",
    "    # intialGuess at the root    \n",
    "    if( ki==1 ){\n",
    "      xn = 0.0\n",
    "    }else{\n",
    "      xn = xnp1 # use the oldest previously computed root \n",
    "    }\n",
    "    \n",
    "    f    =   sum( dj^2 / ( dj^2 + xn ) ) - k # do the first update by hand\n",
    "    fp   = - sum( dj^2 / ( dj^2 + xn )^2 ) \n",
    "    xnp1 = xn - f/fp \n",
    "    \n",
    "    while( abs(xn - xnp1)/abs(xn) > 10^(-3) ){\n",
    "      xn   = xnp1\n",
    "      f    =   sum( dj^2 / ( dj^2 + xn ) ) - k \n",
    "      fp   = - sum( dj^2 / ( dj^2 + xn )^2 ) \n",
    "      xnp1 = xn - f/fp       \n",
    "    }\n",
    "\n",
    "    lambdas = c(lambdas,xnp1)\n",
    "  }\n",
    "  # flip the order of the lambdas:\n",
    "  lambdas = lambdas[ rev(1:length(lambdas)) ]\n",
    "  return(lambdas)\n",
    "}\n",
    "\n",
    "PD2 <- load_spam_data(trainingScale=FALSE,responseScale=FALSE)\n",
    "\n",
    "XTraining = PD2[[1]]\n",
    "XTesting  = PD2[[2]]\n",
    "\n",
    "p        = dim(XTraining)[2]-1 # the last column is the response \n",
    "nSamples = dim(XTraining)[1]\n",
    "\n",
    "# Do ridge-regression cross validation (large lambda values => small degrees of freedom):\n",
    "#\n",
    "allLambdas      = opt_lambda_ridge( XTraining[,1:p], multiple=2 ) \n",
    "numberOfLambdas = length( allLambdas ) \n",
    "\n",
    "for( li in 1:numberOfLambdas ){\n",
    "  \n",
    "  res = cv_Ridge(allLambdas[li],XTraining,numberOfCV=10)\n",
    "  \n",
    "  # To get a *single* value for the degrees of freedom use the call on the entire training data set\n",
    "  # We could also take the mean/median of this computed from the cross validation samples \n",
    "  # \n",
    "  dof = df_ridge(allLambdas[li],as.matrix(XTraining[,1:p]))\n",
    "    \n",
    "  if( li==1 ){\n",
    "    complexityParam = dof \n",
    "    cvResults       = res$cvraw\n",
    "  }else{\n",
    "    complexityParam = cbind( complexityParam, dof )  # will be sorted from largest value of dof to smallest valued dof \n",
    "    cvResults       = cbind( cvResults, res$cvraw )  # each column is a different value of the complexity parameter ...\n",
    "                                                     # each row    is a different cross validation sample ...\n",
    "  }\n",
    "}\n",
    "\n",
    "# flip the order of the results: \n",
    "inds            = rev( 1:numberOfLambdas )\n",
    "allLambdas      = allLambdas[ inds ] \n",
    "complexityParam = complexityParam[ inds ] \n",
    "cvResults       = cvResults[ , inds ]\n",
    "\n",
    "# group all cvResults by lambda (the complexity parameter) and compute statistics:\n",
    "#\n",
    "OSE           = one_standard_error_rule( complexityParam, cvResults )\n",
    "complexVValue = OSE[[1]]\n",
    "complexIndex  = OSE[[2]] \n",
    "oseHValue     = OSE[[3]]\n",
    "means         = OSE[[4]] # as a function of model complexity  \n",
    "stds          = OSE[[5]] # as a function of model complexity\n",
    "\n",
    "library(gplots) # plotCI, plotmeans found here, xlim=c(0,8), ylim=c(0.3,1.8)\n",
    "\n",
    "\n",
    "plotCI( x=complexityParam, y=means, uiw=0.5*stds,\n",
    "        col=\"black\", barcol=\"blue\", lwd=1, type=\"l\", \n",
    "        xlab=\"degrees of freedom\", ylab=\"expected squared prediction error (ESPE)\" )\n",
    "abline( h=oseHValue, lty=2 ) \n",
    "abline( v=complexVValue, lty=2 ) \n",
    "\n",
    "# pick the best model, retrain over the entire data set, and predict on the testing data set:\n",
    "#\n",
    "bestLambda = allLambdas[ complexIndex ]\n",
    "\n",
    "DCV  = XTraining\n",
    "#\n",
    "#\n",
    "responseName        = names(DCV)[p+1]  # get the response and delete it from the data frame \n",
    "response            = DCV[,p+1]\n",
    "DCV[[responseName]] = NULL\n",
    "\n",
    "# Standardize the predictors and demean the response\n",
    "#\n",
    "# Note that one can get the centering value (the mean) with the command attr(DCV,'scaled:center')\n",
    "#                   and the scaling value (the sample standard deviation) with the command attr(DCV,'scaled:scale')\n",
    "# \n",
    "DCV          = scale( DCV )\n",
    "responseMean = mean( response )\n",
    "response     = response - responseMean \n",
    "DCVb         = cbind( DCV, response ) # append back on the response \n",
    "DCVf         = data.frame( DCVb ) # a data frame containing all scaled variables of interest \n",
    "# extract the centering and scaling information:\n",
    "#\n",
    "means = attr(DCV,\"scaled:center\")\n",
    "stds  = attr(DCV,\"scaled:scale\")\n",
    "\n",
    "# apply the computed scaling based on the training data to the testing data:\n",
    "#\n",
    "DCVTest                 = XTesting\n",
    "responseTest            = DCVTest[,p+1] - responseMean # in demeaned units (not mean adjusted)\n",
    "DCVTest[[responseName]] = NULL\n",
    "DCVTest                 = t( apply( DCVTest, 1, '-', means ) ) \n",
    "DCVTest                 = t( apply( DCVTest, 1, '/', stds ) ) \n",
    "DCVTestb                = cbind( DCVTest, responseTest  ) # append back on the response\n",
    "DCVTestf                = data.frame( DCVTestb ) # a data frame containing all scaled variables of interest\n",
    "names(DCVTestf)[p+1]    = responseName # fix the name of the response\n",
    "\n",
    "# fit this linear model and compute the expected prediction error (EPE) using these features (this just estimates the mean in this case):\n",
    "#\n",
    "DM      = as.matrix(DCV)\n",
    "M       = ginv( t(DM) %*% DM + bestLambda*diag(p) ) %*% t(DM)\n",
    "betaHat = M %*% as.matrix(DCVf[,p+1])\n",
    "\n",
    "print( responseMean, digits=4 )\n",
    "print( betaHat, digits=3 ) \n",
    "\n",
    "DTest = as.matrix(DCVTest)\n",
    "pdt = DTest %*% betaHat # get predictions on the test set\n",
    "#pdt  = pdt + responseMean # add back in the mean.  Now in physical units (not mean adjusted)\n",
    "mErr = mean( (responseTest - pdt)^2 )\n",
    "print( mErr )\n",
    "NTest = length(responseTest)\n",
    "sErr = sqrt( var( (responseTest - pdt)^2 )/NTest ) \n",
    "print( sErr )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "forbidden-samba",
   "metadata": {},
   "source": [
    "### The lasso"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "integral-underwear",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "R[write to console]: Loading required package: Matrix\n",
      "\n",
      "R[write to console]: Loaded glmnet 4.1-1\n",
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "58 x 1 sparse Matrix of class \"dgCMatrix\"\n",
      "                      s0\n",
      "(Intercept)     4.69e-16\n",
      "V1          .           \n",
      "V2          .           \n",
      "V3              1.60e-02\n",
      "V4              2.10e-03\n",
      "V5              3.88e-02\n",
      "V6              3.08e-02\n",
      "V7              8.27e-02\n",
      "V8              3.07e-02\n",
      "V9              1.11e-02\n",
      "V10         .           \n",
      "V11             1.98e-03\n",
      "V12         .           \n",
      "V13         .           \n",
      "V14         .           \n",
      "V15             4.26e-03\n",
      "V16             7.94e-02\n",
      "V17             2.32e-02\n",
      "V18             1.76e-02\n",
      "V19             2.99e-02\n",
      "V20             1.66e-02\n",
      "V21             7.37e-02\n",
      "V22             2.50e-02\n",
      "V23             6.02e-02\n",
      "V24             3.22e-02\n",
      "V25            -3.55e-02\n",
      "V26            -1.38e-02\n",
      "V27            -1.69e-02\n",
      "V28            -2.85e-03\n",
      "V29         .           \n",
      "V30         .           \n",
      "V31         .           \n",
      "V32         .           \n",
      "V33            -4.85e-03\n",
      "V34         .           \n",
      "V35         .           \n",
      "V36         .           \n",
      "V37            -8.83e-03\n",
      "V38         .           \n",
      "V39            -1.03e-02\n",
      "V40         .           \n",
      "V41         .           \n",
      "V42            -1.54e-02\n",
      "V43            -2.42e-03\n",
      "V44            -3.66e-03\n",
      "V45            -2.29e-02\n",
      "V46            -1.87e-02\n",
      "V47         .           \n",
      "V48            -9.88e-04\n",
      "V49            -2.55e-03\n",
      "V50         .           \n",
      "V51         .           \n",
      "V52             3.22e-02\n",
      "V53             5.71e-02\n",
      "V54         .           \n",
      "V55         .           \n",
      "V56             1.19e-03\n",
      "V57             4.10e-02\n",
      "[1] 0.126\n",
      "        1\n",
      "1 0.00708\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "PD = load_spam_data(trainingScale=TRUE,responseScale=TRUE) # read in unscaled data \n",
    "XTraining = PD[[1]]\n",
    "XTesting = PD[[2]]\n",
    "\n",
    "p        = dim(XTraining)[2]-1 # the last column is the response \n",
    "nSamples = dim(XTraining)[1] \n",
    "\n",
    "library(glmnet)\n",
    "\n",
    "# alpha = 1 => lasso\n",
    "fit = glmnet( as.matrix( XTraining[,1:p] ), XTraining[,p+1], family=\"gaussian\", alpha=1 )\n",
    "\n",
    "# alpha = 0 => ridge\n",
    "#fit = glmnet( as.matrix( XTraining[,1:p] ), XTraining[,p+1], family=\"gaussian\", alpha=0 )\n",
    "\n",
    "#postscript(\"../../WriteUp/Graphics/Chapter3/spam_lasso_beta_paths.eps\", onefile=FALSE, horizontal=FALSE)\n",
    "plot(fit)\n",
    "#dev.off()\n",
    "\n",
    "# do crossvalidation to get the optimal value of lambda \n",
    "cvob = cv.glmnet( as.matrix( XTraining[,1:p] ), XTraining[,p+1], family=\"gaussian\", alpha=1 )\n",
    "\n",
    "#postscript(\"../../WriteUp/Graphics/Chapter3/spam_lasso.eps\", onefile=FALSE, horizontal=FALSE)\n",
    "plot( cvob )\n",
    "#dev.off() \n",
    "\n",
    "# get the optimal value of lambda: \n",
    "lambdaOptimal = cvob$lambda.1se\n",
    "\n",
    "# refit with this opimal value of lambda:\n",
    "fitOpt = glmnet( as.matrix( XTraining[,1:p] ), XTraining[,p+1], family=\"gaussian\", lambda=lambdaOptimal, alpha=1 )\n",
    "print( coef(fitOpt), digit=3 )\n",
    "\n",
    "# predict the testing data using this value of lambda: \n",
    "yPredict = predict( fit, newx=as.matrix(XTesting[,1:p]), s=lambdaOptimal )\n",
    "NTest = dim(XTesting[,1:p])[1]\n",
    "print( mean( ( XTesting[,p+1] - yPredict )^2 ), digit=3 ) \n",
    "print( sqrt( var( ( XTesting[,p+1] - yPredict )^2 )/NTest ), digit=3 ) \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "guided-plant",
   "metadata": {},
   "source": [
    "### principal component regression (PCR)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "raised-world",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "R[write to console]: Loading required package: pls\n",
      "\n",
      "R[write to console]: \n",
      "Attaching package: ‘pls’\n",
      "\n",
      "\n",
      "R[write to console]: The following object is masked from ‘package:stats’:\n",
      "\n",
      "    loadings\n",
      "\n",
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Data: \tX dimension: 3065 57 \n",
      "\tY dimension: 3065 1\n",
      "Fit method: svdpc\n",
      "Number of components considered: 57\n",
      "\n",
      "VALIDATION: RMSEP\n",
      "Cross-validated using 10 random segments.\n",
      "       (Intercept)  1 comps  2 comps  3 comps  4 comps  5 comps  6 comps\n",
      "CV          0.4895   0.4687   0.3519   0.3490   0.3486   0.3470   0.3461\n",
      "adjCV       0.4895   0.4688   0.3511   0.3488   0.3490   0.3469   0.3461\n",
      "       7 comps  8 comps  9 comps  10 comps  11 comps  12 comps  13 comps\n",
      "CV      0.3448   0.3429   0.3412    0.3403    0.3394    0.3383    0.3376\n",
      "adjCV   0.3451   0.3423   0.3406    0.3408    0.3393    0.3391    0.3387\n",
      "       14 comps  15 comps  16 comps  17 comps  18 comps  19 comps  20 comps\n",
      "CV       0.3362    0.3356    0.3339    0.3328    0.3326    0.3318    0.3314\n",
      "adjCV    0.3340    0.3332    0.3326    0.3323    0.3326    0.3300    0.3299\n",
      "       21 comps  22 comps  23 comps  24 comps  25 comps  26 comps  27 comps\n",
      "CV       0.3309    0.3306    0.3306    0.3305    0.3308    0.3302    0.3298\n",
      "adjCV    0.3293    0.3294    0.3298    0.3299    0.3302    0.3305    0.3298\n",
      "       28 comps  29 comps  30 comps  31 comps  32 comps  33 comps  34 comps\n",
      "CV       0.3288    0.3287    0.3286    0.3281    0.3279    0.3283    0.3277\n",
      "adjCV    0.3286    0.3289    0.3289    0.3282    0.3283    0.3283    0.3272\n",
      "       35 comps  36 comps  37 comps  38 comps  39 comps  40 comps  41 comps\n",
      "CV       0.3279    0.3274    0.3278    0.3279    0.3271    0.3281    0.3282\n",
      "adjCV    0.3276    0.3272    0.3278    0.3274    0.3266    0.3278    0.3281\n",
      "       42 comps  43 comps  44 comps  45 comps  46 comps  47 comps  48 comps\n",
      "CV       0.3277    0.3255    0.3267    0.3277    0.3277    0.3287    0.3293\n",
      "adjCV    0.3279    0.3253    0.3260    0.3272    0.3272    0.3281    0.3287\n",
      "       49 comps  50 comps  51 comps  52 comps  53 comps  54 comps  55 comps\n",
      "CV       0.3294    0.3292    0.3292    0.3290    0.3294    0.3295    0.3297\n",
      "adjCV    0.3287    0.3286    0.3287    0.3283    0.3288    0.3289    0.3290\n",
      "       56 comps  57 comps\n",
      "CV       0.3313    0.3314\n",
      "adjCV    0.3305    0.3306\n",
      "\n",
      "TRAINING: % variance explained\n",
      "     1 comps  2 comps  3 comps  4 comps  5 comps  6 comps  7 comps  8 comps\n",
      "X     11.932    17.76    21.39    24.23    27.01    29.67    32.14    34.60\n",
      "V58    8.719    47.40    48.93    48.98    49.56    49.88    50.09    50.72\n",
      "     9 comps  10 comps  11 comps  12 comps  13 comps  14 comps  15 comps\n",
      "X      36.93     39.20     41.41     43.42     45.41     47.37     49.29\n",
      "V58    51.29     51.39     51.90     51.91     52.05     53.28     53.54\n",
      "     16 comps  17 comps  18 comps  19 comps  20 comps  21 comps  22 comps\n",
      "X       51.16     52.98     54.78     56.55     58.32     60.06     61.79\n",
      "V58     53.74     53.78     53.81     54.49     54.51     54.74     54.82\n",
      "     23 comps  24 comps  25 comps  26 comps  27 comps  28 comps  29 comps\n",
      "X       63.48     65.14     66.78     68.38     69.95     71.52     73.05\n",
      "V58     54.83     54.86     54.95     54.96     55.13     55.32     55.32\n",
      "     30 comps  31 comps  32 comps  33 comps  34 comps  35 comps  36 comps\n",
      "X       74.54     75.97     77.40     78.78     80.12     81.44     82.74\n",
      "V58     55.50     55.72     55.79     56.01     56.23     56.24     56.35\n",
      "     37 comps  38 comps  39 comps  40 comps  41 comps  42 comps  43 comps\n",
      "X       84.02     85.26     86.47     87.66     88.83     89.96     91.02\n",
      "V58     56.39     56.56     56.69     56.69     56.77     56.81     57.23\n",
      "     44 comps  45 comps  46 comps  47 comps  48 comps  49 comps  50 comps\n",
      "X       92.07     93.06     94.04     94.96     95.81     96.55     97.27\n",
      "V58     57.45     57.46     57.51     57.57     57.57     57.58     57.61\n",
      "     51 comps  52 comps  53 comps  54 comps  55 comps  56 comps  57 comps\n",
      "X       97.86     98.42     98.97     99.38     99.73    100.00    100.00\n",
      "V58     57.63     57.67     57.68     57.69     57.70     57.71     57.71\n",
      "[1] 0.124\n",
      "[1] 0.00633\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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GZGDMEaCwW/SAl8Okgi1rthQYq0oUvPRVdfLaC8YSAAkc7pOsUXrAS8EfDOnNt2wwVJUo+Pf0j06uT/+vofWBDB7BOXrAO8FzkxbU4nuaGaqqxFn08Sm9Jx8ztDqQwu2XNdcD3gmucZAn84JoQ1Xh+2A7UIdJ6r86esA7wZVyVcFn4gxVBcF24NmCI/WAd4L7jitILnrkHkNVQbAdeAQH6wHvBB9uHumKTTd2EC0hOG/pw0tzyywKTMMjOFAPeHkWXbRz7bcGP4gSBee1X7R9cSYM+x+P4CA94J3gZe7pUkNViYKXLVAni14usywwC4/gMD3gneAE91T+LHqK9iuwLeMMrQ9k8AiO0QPeCN66NW6rylvVr1u0NKLg17Xbviz4m6H1gQwewaF6wBvB8fHO+Pj46k3eMVSVKPhS52c3z+100dD6QAaP4AA94N0ueoREVSXOovPfmPp6vkQSYBCPYKcewKUrtPAIVvSAd4KHal/3bco2VBUE24Gk4Arag3jyKhqqCoLtwCPYoQe8E1z5jDo5A8HlH8ljcL8RefziiDuuW7Q0EGwHkoIPpUXVi7rt8HWLlgaC7UDyGMwLd6zd4cNn0fzQ6C6jfzW2PpBBRvDWY3yrB0NViYJPNh3QdUDa8bILA5OQEZywjCd4MFSVKPjZxM/Of570VNmFgUnI7qIlEAVnadcezugonwx4iU2CB965fPKyIX3kkwEvkRFswtWFa8Jja8SGrTC0PpBBRrAJVxe+GNE2OzPiOUPrAxkkd9G+Xl3YdeqiIS/O6mBofSCDpGBfry68vc64d8YndjW0PpBBUrCvVxeOj35uxYIKDxlaH8ggKdjXqwunJDhdzgRcXeh/ZIdJPl5dOCx62+rtMUOMJQASyAr28erC9jXf+c/fa7UytD6QQVKwr1cXdh+SclNKdqeyCwOTkBTs69WFrwauXfde4BJD6wMZJAX7enXh/BqKw1HjGUPrAxkkBft6deHtYZ0mdQ7rVnZhYBKSgn29ujClsjqpUtvQ+kAG2bNoH68uTL41c0xmUwj2P3KCL9WVuIeZKHhkxKZdX0bmlF0YmITkFtxI4rIiUfChqkFBQXH4UZb/kRS8sPvH3+3evdtQVaLgg2nd07OaHjC0PpBBUnCsB0NVlRgmrRrX6tF3MUzyPzb9ZGdaiCvKFTxRPhnwEpsEt3QmRyc7jX2fDGSQFJw3vW503SeMnWmJgiu4q40suzAwCUnBw7qt3/n3tOGGqhIFO1lYTJh4zRvwE5KC62lX5x+PuW7R0oiCFeYMdorVAj8hKbitJvhYnesWLY0omLFARyDz4YAOvERS8OPdPty+utlzhobComCH9igBbMEW4Ns42NBQWBQc5q42xEBDgRw2DZOc7pvc4iTL/9gkWHvWi7qPlk8GvMQmwQ4W27wKjsEWYJPgYHe1QWUXBiZhk+BId7Xh8smAl9gkOFrRao2QTwa8xCbBLqY4FeyiLcAmwQqLCIzASZYF2CaYaTtp+WTAS2z7oCPIGSTefw34CdsEa0Cw/7FtF6048GWDFdh6DIZg/2OfYAbBVmDXlw0e5JMBL8EWTBxswcSBYOJAMHEgmDgQTBwIJg4EE8cPgjdlN6/XPHvTVXEItgPzBS+KyF64fGFOxKLSCyDYDswXHLfZ/fJ1/JXI6g5uarQrXS0E+x/zBYd4noZ0Kqz0AmzBdmC+4P7tN50uOr25/VUPsoNgOzBf8JnsUDVjWM6Z0gsg2A48HR2oB0wYJl3au2PvpavDEGwHno6uoAcwDqaFp6NdegCCaeHpaMNPAJcBgu3A09HCRUIQTAtPRwtjVgimBb5sII4/hkllAMF24Olo4ekaEEwLDJOI4+lo4SowCKYFc18khGMwWXCSRRxPRws3y4BgWkAwcdwfc4h3BYVgWjD3fX1r6gEIpoXnOs6hegCCaRGkbcBsvB6AYFo4QmIqRyn99QAE08IRFR4a5eiiByCYFgGKw+lgf9YDEEyLQBYU7mAT9AAE00IJio6IUtroAQimhRIxoHcf1kEPQDAtnAHVUyIV4VHcEEyLCs7AkICAp/UABNMiueptt7UO/kQPQDAt2jTMaNas3Ro9AMG0uO+b5c98PVR4AiEE0+KnFuu+nzVYCEAwMQ7NGPVWoTAPwcSBYOJAMHEgmDgQTBwIJg4EEweCiQPBxIFg4kAwcSCYOBBMHAgmDgQTB4KJA8HEgWDiWCZYgWBbwBZMHAgmjlVPAFcUBU8AtwNrBDvdG7Cj7MLAX1gjONQtOFg+GZDFGsEVmSPUwaLlkwFZrBEcFRIeGB4aLp8MyGKN4MSAyODIgFryyYAs1ggew5QAhY2QTwZksUZwA8UV4FJS5JMBWSw6iw6qUrGKC2fRNmCRYOfq+asDINgGrBFcizlcDlZdPhmQxRrBaczpcCqp8smALBYNkxxxteOcGCbZgDWC46qEBIbEV5ZPBmSxRnAj5d4NOUpd+WRAFmsEN3PF1okNbiKfDMhijeBWyZmNM1OayycDslgj+LE6iSmJiRPKLgz8hTWCd0RUTqocsUU+GZDFGsFPNBo+cHijqfLJgCzWCB7QJaNDRlZv+WRAFmsE948+xo/H3C6fDMhijeA7W1auUKlVX/lkQBZrBI8Pn/DFpIiH5ZMBWawRPKHJlB6Tmz0inwzIYpbg/PGlIyXGwQ3iGsc1mGSgXcAkzBKce1VJUfCMNH6cN59ioF3AJHwWPNxDznUFT20bFhPW9qqNHPgfnwUHDHpA4/7rCp4SOXHhlMhxxpsHfMVnwanvu1+EXfS2p9xkZOmF+sauO/ZhXA+5JgJf8Fnw/NXul/zJVyI/rXQzUjjktp8xre/Up1rJNRH4gv+GSSte0N/Pqn+B5zZ8XD4ZkMUcwW2uERMF5zWok16nfq73zQJmYY7gaxUTBfPVidXqrPS6UcA8rBG8O+s8v9Bjl/fNAmZhjuBF14iJgp/+UJ1snOl1q4BpWHOSNXu9Ovm/GfLJgCzWCN6V0TGzY+sd8smALNYI3l8vrUfTevvkkwFZrBE8dy0/y9c9XXZh4C+sEfyE9kjqTdPkkwFZrBH8zd1FvOhPm+WTAVmsEcwXtBrRar58LiCNRYL5hX9dkE8F5LFKMLAJCCaOVYI/e+FT+VRAHmsEFw4c+/aj/QrlkwFZrBG8esa3b3775NvyyYAs1gie2jN7UU5P/KrSBqwRPKqbOuk1XD4ZkMUawRMavHt4bUNcm2QD1ghevHjGXdNfWiifDMhijeBT6V/kbko/IZ8MyGLROPjww10fOiSfC0iDT7KIA8HEgWDiQDBxIJg4EEwcCCYOBBMHgonjP8HrUzsIRMWYQ3ikOXkqhJiTJ8asPBERJiWK6VCC5Ot/gGjaQ7vbmpTnmffNyXP4TnPy8HYm/U7lpeXm5MntYqg4BP8REOwBgv8ACPYAwX8ABHuAYA8Q/EdAsIcOJuWZ+6E5eY4OMicP72SS4FffMCdPXtYflxEwTfAZk/JcyDcpkVkNMitP3kWTEhlrkGmCQfkEgokDwcSBYOJAMHEgmDgQTBwIJo4Jgs8MCI3X70Z7IKSz73kuPnBzSMO1PremZMtsaY5fuofzZUlBSf/0ajUTBOdkHtsUvql4rnuG7F8g5Dkzautvi11St74UW1OyZbY0xy/dw9+Pfe+3rQe8Ws13wZfCvuA8O/vy3JoesyX/gpJ5VBJX+JjlqozWN8dP3dP4Va/X813wXnaa84XNPTPnkvbL/gUl8qgcCfzBxyylM9rQHP90T54yu2qVB717QobvgnewIs6X1/PMjHucy/4FJfKof0TbB3zNUiqjHc3xT/fsZxlHf2k4zav1fBO8iLE24n+t75Nypf6C0nnUPVLPfgUyLfLTFizbnJJ5ZLundJ7D7F3OX0rzaj0TjsGhm9UzAM/BYVFobGx4UILPeXh+nx6XfM5SIqM9zfFT91Raa6Fgnt3x1DeR6undXz/iF44cOTI586jPeQoGZP6emyu1zQhZrry3sTl+6R7+aKvjhxp59xgyM8bB/UPjtAFaZ8/D72T3QWKeA0xjnq+tKX5vY3P80j08LycydkyeV6vhkyziQDBxIJg4EEwcCCYOBBMHgokDwcSBYOJAMHEgmDgQTBwIJg4EEweCiQPBxIFg4kAwcSCYOBBcFmNpPIkRgssCgulR4g5OEHyDEPtCYtgjv7QO636O86P9KifM5XxqjfBGn3P+WJWopC3ceYTzMZPVck81Tiku8P1tkVnZEHxjENv++P7oFrvOps/nRS3G5v6c9AF/82jB0psubEs4xv/9qy64W17R5QIFSbMLNrgg+MYgdiPnPVSBzw7hOyMLOF8wxB1O2Lmz8kbt5nNXBG/gxQW2VFRfukHwjUHsbs4HLOR8UV++1pWcnFyrN1+WVi0h8BP+clrkoGO64O94cYE1jdX1RkHwjYEgeEdskRbZG626rKVu2Px499E88mfO757sKXe5wJZ4ddIPgm8MBMGFzR89W/D9lm1Vc/l7bOOer/IvDhzLWy/hP0VdFny5QEGdd/jeUAi+MRAE86ODYqObbeBjU7pMqL/xH40iKt1xiu9o1LTvXZcFFxfY3bzlHUMgGNwAQDBxIJg4EEwcCCYOBBMHgokDwcSBYOJAMHEgmDgQTBwIJg4EEweCiQPBxIFg4kAwcSCYOP8PHNoAkqeT1CQAAAAASUVORK5CYII=\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "require(pls)\n",
    "set.seed (1000)\n",
    "\n",
    "PD = load_spam_data(trainingScale=TRUE,responseScale=TRUE) # read in unscaled data \n",
    "\n",
    "XTraining = PD[[1]]\n",
    "XTesting = PD[[2]]\n",
    "\n",
    "p        = dim(XTraining)[2]-1 # the last column is the response \n",
    "nSamples = dim(XTraining)[1] \n",
    "\n",
    "pcr_model <- pcr(V58~., data = XTraining, scale = FALSE, validation = \"CV\")\n",
    "summary(pcr_model)\n",
    "# Plot the root mean squared error\n",
    "validationplot(pcr_model)\n",
    "# Plot the cross validation MSE\n",
    "validationplot(pcr_model, val.type=\"MSEP\")\n",
    "# Plot the R2\n",
    "validationplot(pcr_model, val.type = \"R2\")\n",
    "\n",
    "#You can plot the predicted vs measured values using the predplot function as below\n",
    "predplot(pcr_model)\n",
    "\n",
    "#while the regression coefficients can be plotted using the coefplot function\n",
    "coefplot(pcr_model)\n",
    "\n",
    "set.seed(777)\n",
    "yPredict<-predict(pcr_model,as.matrix(XTesting[,1:p]),ncomp=20)\n",
    "\n",
    "NTest = dim(XTesting[,1:p])[1]\n",
    "print( mean( ( XTesting[,p+1] - yPredict )^2 ), digit=3 ) \n",
    "print( sqrt( var( ( XTesting[,p+1] - yPredict )^2 )/NTest ), digit=3 ) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "speaking-defensive",
   "metadata": {},
   "source": [
    "### partial least squares regression (PLS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "passive-polls",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Data: \tX dimension: 3065 57 \n",
      "\tY dimension: 3065 1\n",
      "Fit method: kernelpls\n",
      "Number of components considered: 57\n",
      "\n",
      "VALIDATION: RMSEP\n",
      "Cross-validated using 10 random segments.\n",
      "       (Intercept)  1 comps  2 comps  3 comps  4 comps  5 comps  6 comps\n",
      "CV          0.4895   0.3558   0.3309   0.3267   0.3284   0.3296   0.3303\n",
      "adjCV       0.4895   0.3557   0.3309   0.3263   0.3279   0.3289   0.3296\n",
      "       7 comps  8 comps  9 comps  10 comps  11 comps  12 comps  13 comps\n",
      "CV      0.3307   0.3311   0.3312    0.3312    0.3313    0.3313    0.3313\n",
      "adjCV   0.3299   0.3303   0.3304    0.3304    0.3304    0.3305    0.3305\n",
      "       14 comps  15 comps  16 comps  17 comps  18 comps  19 comps  20 comps\n",
      "CV       0.3313    0.3313    0.3313    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3305    0.3305    0.3305    0.3306    0.3306    0.3306    0.3306\n",
      "       21 comps  22 comps  23 comps  24 comps  25 comps  26 comps  27 comps\n",
      "CV       0.3314    0.3314    0.3314    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306\n",
      "       28 comps  29 comps  30 comps  31 comps  32 comps  33 comps  34 comps\n",
      "CV       0.3314    0.3314    0.3314    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306\n",
      "       35 comps  36 comps  37 comps  38 comps  39 comps  40 comps  41 comps\n",
      "CV       0.3314    0.3314    0.3314    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306\n",
      "       42 comps  43 comps  44 comps  45 comps  46 comps  47 comps  48 comps\n",
      "CV       0.3314    0.3314    0.3314    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306\n",
      "       49 comps  50 comps  51 comps  52 comps  53 comps  54 comps  55 comps\n",
      "CV       0.3314    0.3314    0.3314    0.3314    0.3314    0.3314    0.3314\n",
      "adjCV    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306    0.3306\n",
      "       56 comps  57 comps\n",
      "CV       0.3314    0.3314\n",
      "adjCV    0.3306    0.3306\n",
      "\n",
      "TRAINING: % variance explained\n",
      "     1 comps  2 comps  3 comps  4 comps  5 comps  6 comps  7 comps  8 comps\n",
      "X      8.668    17.38    19.94    22.60    24.80    26.64    28.11    29.74\n",
      "V58   47.367    54.82    57.36    57.63    57.69    57.70    57.70    57.70\n",
      "     9 comps  10 comps  11 comps  12 comps  13 comps  14 comps  15 comps\n",
      "X      31.23     32.56     34.18     35.81     37.28     38.87     40.34\n",
      "V58    57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     16 comps  17 comps  18 comps  19 comps  20 comps  21 comps  22 comps\n",
      "X       41.85     43.10     44.22     45.78     47.25     48.53     49.74\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     23 comps  24 comps  25 comps  26 comps  27 comps  28 comps  29 comps\n",
      "X       51.24     52.90     54.18     55.49     57.19     58.58     59.95\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     30 comps  31 comps  32 comps  33 comps  34 comps  35 comps  36 comps\n",
      "X       61.25     62.53     63.74     64.95     66.33     67.65     69.00\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     37 comps  38 comps  39 comps  40 comps  41 comps  42 comps  43 comps\n",
      "X       70.50     72.02     73.58     74.98     76.48     77.90     79.31\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     44 comps  45 comps  46 comps  47 comps  48 comps  49 comps  50 comps\n",
      "X       80.65     82.00     83.50     84.94     86.48     88.05     89.63\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "     51 comps  52 comps  53 comps  54 comps  55 comps  56 comps  57 comps\n",
      "X       91.21     92.79     94.37     95.94     97.52     99.10    100.68\n",
      "V58     57.71     57.71     57.71     57.71     57.71     57.71     57.71\n",
      "[1] 0.121\n",
      "[1] 0.00748\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "require(pls)\n",
    "set.seed (1000)\n",
    "\n",
    "PD = load_spam_data(trainingScale=TRUE,responseScale=TRUE) # read in unscaled data \n",
    "\n",
    "XTraining = PD[[1]]\n",
    "XTesting = PD[[2]]\n",
    "\n",
    "p        = dim(XTraining)[2]-1 # the last column is the response \n",
    "nSamples = dim(XTraining)[1] \n",
    "\n",
    "plsr_model <- plsr(V58~., data = XTraining, scale = FALSE, validation = \"CV\")\n",
    "summary(plsr_model)\n",
    "# Plot the root mean squared error\n",
    "validationplot(plsr_model)\n",
    "# Plot the cross validation MSE\n",
    "validationplot(plsr_model, val.type=\"MSEP\")\n",
    "# Plot the R2\n",
    "validationplot(plsr_model, val.type = \"R2\")\n",
    "\n",
    "#You can plot the predicted vs measured values using the predplot function as below\n",
    "predplot(plsr_model)\n",
    "\n",
    "#while the regression coefficients can be plotted using the coefplot function\n",
    "coefplot(plsr_model)\n",
    "\n",
    "set.seed(777)\n",
    "yPredict<-predict(plsr_model,as.matrix(XTesting[,1:p]),ncomp=4)\n",
    "\n",
    "NTest = dim(XTesting[,1:p])[1]\n",
    "print( mean( ( XTesting[,p+1] - yPredict )^2 ), digit=3 ) \n",
    "print( sqrt( var( ( XTesting[,p+1] - yPredict )^2 )/NTest ), digit=3 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "black-tuner",
   "metadata": {},
   "source": [
    "# Ex. 3.18 (conjugate gradient methods) \n",
    "\n",
    "Read about conjugate gradient algorithms (Murray et al., 1981, for example), and establish a connection between these algorithms and partial least squares.\n",
    "\n",
    "\n",
    "The conjugate gradient method is an algorithm that can be adapted to find the minimum of nonlinear functions. It also provides a way to solve certain linear algebraic equations\n",
    "of the type $Ax = b$ by formulating them as a minimization problem. Since the least squares solution $\\hat\\beta$ is given by the solution of a quadratic minimization problem given by\n",
    "$$\\hat\\beta=\\text{argmin}_{\\beta}\\text{RSS}(\\beta)=\\text{argmin}_{\\beta}(y-X\\beta)^T(y-X\\beta)$$ which has the explicit solution given by the normal equations or \n",
    "$$(X^TX)\\beta=X^Ty$$\n",
    "One obvious way to use the conjugate gradient algorithm to derive refined estimates of $\\hat\\beta$ the least squared solution is to use this algorithm to solve the normal Equations $$(X^TX)\\beta=X^Ty$$ Thus after finishing each iteration of the conjugate gradient algorithm we have a new approximate value of $\\hat\\beta(m)$ for $m = 1, 2, \\cdots, p$ and when $m = p$ this estimate $\\hat\\beta(p)$ corresponds to the least squares solution. The value of $m$ is a model selection parameter and can be selected by\n",
    "cross validation.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.19 (increasing norm with decreasing $\\lambda$) \n",
    "\n",
    "Show that $$\\lVert \\hat{\\beta}^{\\text{ridge}}\\rVert$$ increases as its tuning parameter $\\lambda\\to 0$. Does the same property hold for the lasso and partial least squares estimates? For the latter, consider the “tuning parameter” to be the successive steps in the algorithm.\n",
    "Since $$\n",
    "\\begin{align}\n",
    "\\hat\\beta^{\\text{ridge}} &= \\left( \\mathbf{X}^T\\mathbf{X} + \\lambda\\mathbf{I} \\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V\\mathbf D^2\\mathbf V^T+\\lambda\\mathbf{V}\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V(\\mathbf D^2+\\lambda\\mathbf{I})\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}^{-T}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "\\end{align}\n",
    "$$ then $$\\begin{align}\n",
    "\\lVert \\hat{\\beta}^{\\text{ridge}}\\rVert_2^2&=\\left(\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\right)^T\\left(\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\right)\\\\\n",
    "&=y^T\\mathbf U\\mathbf D(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf{V}^T\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\left(\\mathbf U^T\\mathbf y\\right)^T\\left(\\mathbf D(\\mathbf D^2+\\lambda\\mathbf{I})^{-2}\\mathbf D\\right)\\left(\\mathbf U^T\\mathbf y\\right)\n",
    "\\end{align}$$\n",
    "The matrix in the middle of the above is a diagonal matrix with elements $$\\frac{d_j^2}{(d_j^2+\\lambda)^2}$$ Thus $$\\lVert \\hat{\\beta}^{\\text{ridge}}\\rVert_2^2=\\sum_{i=1}^{p}\\frac{d_j^2\\left(\\mathbf U^T\\mathbf y\\right)_j^2}{(d_j^2+\\lambda)^2}$$ where $\\left(\\mathbf U^T\\mathbf y\\right)_j$ is the $j$th component of the vector $\\mathbf U^T\\mathbf y$. As $\\lambda\\to 0$, the fraction $$\\frac{d_j^2}{(d_j^2+\\lambda)^2}$$ increases, and because $\\lVert \\hat{\\beta}^{\\text{ridge}}\\rVert_2^2$ is made up of sum of such terms, it too must increase.  \n",
    "\n",
    "\n",
    "To determine if the same properties hold For the lasso, note that in both ridge-regression and the lasso can be represented as the minimization problem\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat\\beta = \\arg\\min_\\beta \\left\\lbrace \\frac{1}{2}\\sum_{i=1}^N \\left( y_i-\\beta_0-\\sum_{j=1}^p x_{ij}\\beta_j \\right)^2 + \\lambda\\sum_{j=1}^p |\\beta_j|^q \\right\\rbrace,\n",
    "\\end{equation}$$\n",
    "When $q = 1$ we have the lasso and when $q = 2$ we have ridge-regression. This form of the problem is equivalent to the Lagrangian from.  \n",
    "Because as $\\lambda\\to 0$ the value of $\\sum\\lvert\\beta_j\\rvert^q$ needs to increase to have the product $\\lambda\\sum\\lvert\\beta_j\\rvert^q$ stay constant and have the same error minimum value. Thus there is an\n",
    "inverse relationship between $\\lambda$ and $t$, where $\\sum\\lvert\\beta_j\\rvert^q\\le t$ in the two problem formulations, in that as $\\lambda$ decreases $t$ increases and vice versa. Thus the same behavior of $t$ and\n",
    "$\\sum\\lvert\\beta_j\\rvert^q$ with decreasing $\\lambda$ will\n",
    "show itself in the lasso.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.20   \n",
    "\n",
    "Consider the canonical-correlation problem (3.67). Show that the leading pair of canonical variates $u_1$ and $v_1$ solve the problem max\n",
    "$$\\underset{v^T(\\mathbf X^T\\mathbf X)v=1}{\\underset{u^T(\\mathbf Y^T\\mathbf Y)u=1}{\\text{max}}}u^T(\\mathbf Y^T\\mathbf X)v \\quad(3.86)$$\n",
    "a generalized SVD problem. Show that the solution is given by $$u_1 = (\\mathbf Y^T\\mathbf Y)^{-\\frac{1}{2}}u_1^*$$ and $$v_1 = (\\mathbf X^T\\mathbf X)^{-\\frac{1}{2}}v_1^*$$ where $u_1^∗$ and $v_1^*$ are the leading left and right singular vectors in $$(\\mathbf Y^T\\mathbf Y)^{-\\frac{1}{2}}(\\mathbf Y^T\\mathbf X)(\\mathbf X^T\\mathbf X)^{-\\frac{1}{2}}= \\mathbf {U^*}\\mathbf {D^*}\\mathbf {V^*}^T \\quad (3.87)$$\n",
    "Show that the entire sequence $$u_m, v_m, m = 1, \\cdots ,\\text{min}(K, p)$$ is also given by (3.87).\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.21  \n",
    "\n",
    "Show that the solution to the reduced-rank regression problem (3.68), with $\\mathbf\\Sigma$ estimated by $\\mathbf Y^T\\mathbf Y/N$, is given by (3.69). Hint: Transform $\\mathbf Y$ to $\\mathbf Y^* = \\mathbf Y\\mathbf\\Sigma^{-\\frac{1}{2}}$, and solved in terms of the canonical vectors $u_m^*$. Show that $\\mathbf U_m = \\mathbf\\Sigma^{-\\frac{1}{2}}\\mathbf U_m^*$, and a generalized inverse is $$\\mathbf U_m^-={\\mathbf U_m^*}^T\\mathbf\\Sigma^{-\\frac{1}{2}}$$.\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.22   \n",
    "\n",
    "Show that the solution in Exercise 3.21 does not change if $\\mathbf\\Sigma$ is estimated by the more natural quantity \n",
    "$$(\\mathbf Y −\\mathbf X\\hat{\\mathbf B})^T (\\mathbf Y −\\mathbf X\\hat{\\mathbf B})/(N − pK)$$.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.23 ($(X^TX)^{−1}X^Tr$ keeps the correlations tied and decreasing)\n",
    "\n",
    "Consider a regression problem with all variables and response having mean zero and standard deviation one. Suppose also that each variable has identical absolute correlation with the response:\n",
    "$$\\frac{1}{N} \\left\\lvert\\langle \\mathbf x_j , \\mathbf y\\rangle\\right\\rvert = \\lambda, j = 1, \\cdots, p$$\n",
    "Let $\\hat{\\beta}$ be the least-squares coefficient of $y$ on $\\mathbf X$, and let $\\mathbf u(\\alpha) = \\alpha \\mathbf X\\hat{\\beta}$ for $\\alpha \\in [0, 1]$ be the vector that moves a fraction $\\alpha$ toward the least squares fit $\\mathbf u$. Let RSS be the residual sum-of-squares from the full least squares fit.\n",
    "\n",
    "(a) Show that $$\\frac{1}{N}\\left\\lvert\\langle \\mathbf x_j , \\mathbf y − \\mathbf u(\\alpha)\\rangle\\right\\rvert = (1 − \\alpha)\\lambda, \\quad j = 1, \\cdots , p$$ and hence the correlations of each $\\mathbf x_j$ with the residuals remain equal in magnitude as we progress toward $\\mathbf u$.  \n",
    "\n",
    "Since $$\\frac{1}{N}\\left\\lvert\\langle \\mathbf x_j , \\mathbf y − \\mathbf u(\\alpha)\\rangle\\right\\rvert$$ is the absolute value of the $j$th component of $$\\begin{align}\n",
    "\\frac{1}{N}\\mathbf X^T\\left(\\mathbf y − \\mathbf u(\\alpha)\\right)&=\\frac{1}{N}\\mathbf X^T\\left(\\mathbf y − \\alpha\\mathbf X\\left(\\mathbf X^T\\mathbf X\\right)^{-1}\\mathbf X^T\\mathbf y\\right)\\\\\n",
    "&=\\frac{1}{N}\\left(\\mathbf X^T\\mathbf y − \\alpha\\mathbf X^T\\mathbf y\\right)\\\\\n",
    "&=\\frac{1}{N}\\left(1 − \\alpha\\right)\\mathbf X^T\\mathbf y\\\\\n",
    "\\end{align}$$ since the absolute value of each element of $\\mathbf X^T\\mathbf y$ is equal to $N\\lambda$ we have from the above that $$\\left\\lvert\\frac{1}{N}\\mathbf X^T\\left(\\mathbf y − \\mathbf u(\\alpha)\\right)\\right\\rvert=\\left(1 − \\alpha\\right)\\lambda$$ or the $j$th column of $\\mathbf X$ is $$\\frac{1}{N}\\left\\lvert\\langle \\mathbf x_j , \\mathbf y − \\mathbf u(\\alpha)\\rangle\\right\\rvert=(1 − \\alpha)\\lambda, \\quad j = 1, \\cdots , p$$ In words, the magnitude of the projections of $x_j$ onto the residual $\\mathbf y − \\mathbf u(\\alpha)= \\mathbf y − \\alpha\\mathbf X \\hat\\beta$ is the same for every value of $j$.  \n",
    "\n",
    "\n",
    "(b) Show that these correlations are all equal to \n",
    "$$ \\lambda(\\alpha) = \\frac{(1 − \\alpha)}{(1 − \\alpha)^2 + \\frac{\\alpha(2−\\alpha)}{N}\\cdot RSS} \\cdot λ$$\n",
    "and hence they decrease monotonically to zero.  \n",
    "\n",
    "Since all variables having mean zero and standard deviation one, then $\\left\\langle \\mathbf x_j ,\\mathbf x_j\\right\\rangle=N\\sigma^2=N$. The correlations (not covariances) would be given by $$\n",
    "\\lambda(\\alpha) = \\frac{\\frac{1}{N}\\left\\langle \\mathbf x_j , \\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{\\left(\\frac{\\left\\langle \\mathbf x_j ,\\mathbf x_j\\right\\rangle}{N}\\right)^{1/2}\\left(\\frac{\\left\\langle \\mathbf y − \\mathbf u(\\alpha),\\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{N}\\right)^{1/2}}=\\frac{(1 − \\alpha)\\lambda}{\\left(\\frac{\\left\\langle \\mathbf y − \\mathbf u(\\alpha),\\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{N}\\right)^{1/2}}$$ Since the normal equations for linear regression $$\\mathbf X^T\\mathbf X\\hat\\beta=\\mathbf X^T\\mathbf y$$ then  $$\\begin{align}\n",
    "\\left\\langle \\mathbf y − \\alpha\\mathbf X\\hat\\beta,\\mathbf y − \\alpha\\mathbf X\\hat\\beta\\right\\rangle&=\\mathbf y^T\\mathbf y-\\alpha\\mathbf y^T\\mathbf X\\hat\\beta-\\alpha\\hat\\beta^T\\mathbf X^T\\mathbf y+\\alpha^2\\hat\\beta^T\\left(\\mathbf X^T\\mathbf X\\hat\\beta\\right)\\\\\n",
    "&=\\mathbf y^T\\mathbf y-\\alpha\\mathbf y^T\\mathbf X\\hat\\beta-\\alpha\\hat\\beta^T\\mathbf X^T\\mathbf y+\\alpha^2\\hat\\beta^T\\left(\\mathbf X^T\\mathbf y\\right)\\\\\n",
    "&=\\mathbf y^T\\mathbf y-2\\alpha\\mathbf y^T\\mathbf X\\hat\\beta+\\alpha^2\\hat\\beta^T\\left(\\mathbf X^T\\mathbf y\\right)\\\\\n",
    "&=\\mathbf y^T\\mathbf y+\\alpha(\\alpha-2)\\mathbf y^T\\mathbf X\\hat\\beta\\\\\n",
    "\\end{align}$$ If $\\alpha=1$, then $$\\left\\langle \\mathbf y − \\mathbf X\\hat\\beta,\\mathbf y − \\mathbf X\\hat\\beta\\right\\rangle=\\text{RSS}=\\mathbf y^T\\mathbf y-\\mathbf y^T\\mathbf X\\hat\\beta$$ so $$\\mathbf y^T\\mathbf X\\hat\\beta=\\mathbf y^T\\mathbf y-\\text{RSS}$$ then we have $$\\begin{align}\n",
    "\\left\\langle \\mathbf y − \\alpha\\mathbf X\\hat\\beta,\\mathbf y − \\alpha\\mathbf X\\hat\\beta\\right\\rangle&=\\mathbf y^T\\mathbf y-2\\alpha\\mathbf y^T\\mathbf X\\hat\\beta+\\alpha^2\\hat\\beta^T\\left(\\mathbf X^T\\mathbf y\\right)\\\\\n",
    "&=\\mathbf y^T\\mathbf y - 2\\alpha\\left(\\mathbf y^T\\mathbf y-\\text{RSS}\\right)+\\alpha^2\\left(\\mathbf y^T\\mathbf y-\\text{RSS}\\right)\\\\\n",
    "&=(1-\\alpha)^2\\mathbf y^T\\mathbf y+\\alpha(2-\\alpha)\\cdot\\text{RSS}\n",
    "\\end{align}$$ As $\\mathbf y$ has a mean zero and a standard deviation of one means that $$\\mathbf y^T\\mathbf y=N$$ so $$\\left\\langle \\mathbf y − \\alpha\\mathbf X\\hat\\beta,\\mathbf y − \\alpha\\mathbf X\\hat\\beta\\right\\rangle=N(1-\\alpha)^2+\\alpha(2-\\alpha)\\cdot\\text{RSS}$$\n",
    "The correlation is $$\\begin{align}\n",
    "\\lambda(\\alpha) &= \\frac{\\frac{1}{N}\\left\\langle \\mathbf x_j , \\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{\\left(\\frac{\\left\\langle \\mathbf x_j ,\\mathbf x_j\\right\\rangle}{N}\\right)^{1/2}\\left(\\frac{\\left\\langle \\mathbf y − \\mathbf u(\\alpha),\\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{N}\\right)^{1/2}}\\\\\n",
    "&=\\frac{(1 − \\alpha)\\lambda}{\\left(\\frac{\\left\\langle \\mathbf y − \\mathbf u(\\alpha),\\mathbf y − \\mathbf u(\\alpha)\\right\\rangle}{N}\\right)^{1/2}}\\\\\n",
    "&=\\frac{(1 − \\alpha)\\lambda}{\\left(\\frac{N(1-\\alpha)^2+\\alpha(2-\\alpha)\\cdot\\text{RSS}}{N}\\right)^{1/2}}\\\\\n",
    "\\end{align}$$\n",
    "\n",
    "\n",
    "(c) Use these results to show that the LAR algorithm in Section 3.4.4 keeps the correlations tied and monotonically decreasing, as claimed in (3.55).\n",
    "\n",
    "When $\\alpha=0$ we have $\\lambda(0)  = \\lambda$, when $\\alpha=1$ we have that $\\lambda(1)  = 0$, where all correlations are tied and decrease from $\\lambda$ to zero as $\\alpha$ moves from 0 to 1.\n",
    "\n",
    "\n",
    "# Ex. 3.24 LAR directions.   \n",
    "\n",
    "Using the notation around equation (3.55) on page 74, show that the LAR direction makes an equal angle with each of\n",
    "the predictors in $A_k$.  \n",
    "\n",
    "\n",
    "* $\\mathcal{A}_k$ is the active set of variables at the beginning of the $k$th step,\n",
    "* $\\beta_{\\mathcal{A}_k}$ is the coefficient vector for these variables at this step.\n",
    "There will be $k-1$ nonzero values, and the one just entered will be zero. If $\\mathbf{r}_k = \\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}$ is the current residual, then the direction for this step is\n",
    "\n",
    "$$\\begin{equation}\n",
    "\\delta_k = \\left( \\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k} \\right)^{-1}\\mathbf{X}_{\\mathcal{A}_k}^T \\mathbf{r}_k \\quad (3.55)\\\\\n",
    "=\\left(\\mathbf{V}\\mathbf D^2\\mathbf{V}^T\\right)^{-1}\\mathbf{V}\\mathbf D\\mathbf U^T\\mathbf{r}_k\\\\\n",
    "=\\mathbf{V}^{-T}\\mathbf D^{-2}\\mathbf D\\mathbf U^T\\mathbf{r}_k\n",
    "\\end{equation}$$ and the LAR direction vector $\\mathbf u_k$ is $$\\mathbf u_k=\\mathbf{X}_{\\mathcal{A}_k}\\delta_k$$ then $$\\begin{align}\n",
    "\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf u_k&=\\mathbf{X}_{\\mathcal{A}_k}^T\\left(\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right)\\\\\n",
    "&=\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k}\\left(\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k}\\right)^{-1}\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{r}_k\\\\\n",
    "&=\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{r}_k\n",
    "\\end{align}$$  \n",
    "\n",
    "Since the cosign of the angle of $\\mathbf u_k$ with each predictor $\\mathbf x_j$ in $\\mathcal{A}_k$ is given by \n",
    "$$\\frac{\\mathbf x_j^T\\mathbf u_k}{\\lVert\\mathbf x_j\\rVert\\lVert\\mathbf u_k\\rVert}=\\frac{\\mathbf x_j^T\\mathbf u_k}{\\lVert\\mathbf u_k\\rVert}$$ each element of the vector $\\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf u_k$ corresponds to a cosign of an angle between a predictor $\\mathbf x_j$ and the vector $\\mathbf u_k$. Since the procedure for LAR adds the predictor $\\mathbf x_j'$ when the absolute value of $\\mathbf x_j'^T\\mathbf r$ equals that of $\\mathbf x_j^T\\mathbf r$ for all predictors $\\mathbf x_j$ in $\\mathcal{A}_k$, the direction $\\mathbf u_k$ makes an equal angle with all predictors in $\\mathcal{A}_k$.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.25 LAR look-ahead.   \n",
    "\n",
    "Starting at the beginning of the $k$-th step of the LAR algorithm, derive expressions to identify the next variable to enter the active set at step $k+1$, and the value of $\\alpha$ at which this occurs (using the notation around equation (3.55) on page 74).\n",
    "\n",
    "\n",
    "Algorithm 3.2. Least Angle Regression\n",
    "\n",
    "1. Standardize the predictors to have mean zero and unit norm.  \n",
    "Start with the residual $r = y - \\bar{y}$,and coefficients $\\beta_0 = \\beta_1 = \\cdots = \\beta_p = 0$. Since  with all $\\beta_j = 0$ and standardized predictors the constant coefficient $\\beta_0=\\bar{y}$.\n",
    "2. Set $k = 1$ and begin start the $k$-th step. Since all values of $\\beta_j$ are zero the first residual is $r_1 = y - \\bar{y}$.  Find the predictor $x_j$ that is most correlated with this residual $r_1$. Then as we begin this $k = 1$ step we have the active step given by $A_1 = \\{x_j\\}$ and the active\n",
    "coefficients given by $\\beta_{A_1} = [0]$.\n",
    "3. Move $\\beta_j$ from $0$ towards its least-squares coefficient $$\\delta_1=\\left(X_{A_1}^TX_{A_1}\\right)^{-1}X_{A_1}^Tr_1=\\frac{x_j^Tr_1}{x_j^Tx_j}=x_j^T r_1$$\n",
    "until some other competitor $x_k$ has as much correlation with the current residual as does $x_j$. \n",
    "The path taken by the elements in $\\beta_{A_1}$ can be parameterized by $$\\beta_{A_1}(\\alpha)\\equiv\\beta_{A_1}+\\alpha\\delta_1=0+\\alpha x_j^T r_1=(x_j^T r_1)\\alpha,\\quad(0\\leq\\alpha\\leq1)$$\n",
    "This path of the coefficients $\\beta_{A_1}(\\alpha)$ will produce a path of fitted values given by $$\\hat{f}_1(\\alpha)=X_{A_1}\\beta_{A_1}(\\alpha)=(x_j^T r_1)\\alpha x_j$$ and a residual of $$r(\\alpha)=y-\\bar{y}-\\hat{f}_1(\\alpha)=y-\\bar{y}-(x_j^T r_1)\\alpha x_j=r_1-(x_j^T r_1)\\alpha x_j$$\n",
    "Now at this point $x_j$ itself has a correlation with this residual as $\\alpha$ varies given by $$x_j^T(r_1-(x_j^T r_1)\\alpha x_j)=x_j^Tr_1-(x_j^T r_1)\\alpha=(1-\\alpha)x_j^T r_1$$\n",
    "\n",
    "4. Move $\\beta_j$ and $\\beta_k$ in the direction defined by their joint least squares coefficient of the current residual on $(x_j,x_k)$, until some other competitor $x_l$ has as much correlation with the current residual.\n",
    "5. Continue in this way until all $p$ predictors have been entered. After $\\min(N-1,p)$ steps, we arrive at the full least-squares solution.\n",
    "\n",
    "The termination condition in step 5 requires some explanation. If $p > N-1$, the LAR algorithm reaches a zero residual solution after $N-1$ steps (the $-1$ is because we have centered the data).\n",
    "\n",
    "* $\\mathcal{A}_k$ is the active set of variables at the beginning of the $k$th step,\n",
    "* $\\beta_{\\mathcal{A}_k}$ is the coefficient vector for these variables at this step.\n",
    "There will be $k-1$ nonzero values, and the one just entered will be zero. If $\\mathbf{r}_k = \\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}$ is the current residual, then the direction for this step is\n",
    "\n",
    "$$\\begin{equation}\n",
    "\\delta_k = \\left( \\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k} \\right)^{-1}\\mathbf{X}_{\\mathcal{A}_k}^T \\mathbf{r}_k \\quad (3.55)\\\\\n",
    "=\\left(\\mathbf{V}\\mathbf D^2\\mathbf{V}^T\\right)^{-1}\\mathbf{V}\\mathbf D\\mathbf U^T\\mathbf{r}_k\\\\\n",
    "=\\mathbf{V}^{-T}\\mathbf D^{-2}\\mathbf D\\mathbf U^T\\mathbf{r}_k\n",
    "\\end{equation}$$\n",
    "\n",
    "At step $k$ we have $k$ predictors in the model (the elements in $\\mathcal{A}_k$) with an initial coefficient vector $\\beta_{\\mathcal{A}_k}$. In that coefficient vector, one element will be zero corresponding to the predictor\n",
    "we just added to our model. We will smoothly update the coefficients of our model until we decide to add the next predictor. We update the coefficients according to \n",
    "$$\\beta_{\\mathcal{A}_k}(\\alpha)=\\beta_{\\mathcal{A}_k}+\\alpha\\delta_k$$ with $\\alpha$ sliding between $\\alpha=0$ and $\\alpha=1$, and with $\\delta_k$ is the least-angle direction at step $k$ given by $$\\begin{equation}\n",
    "\\delta_k = \\left( \\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k} \\right)^{-1}\\mathbf{X}_{\\mathcal{A}_k}^T \\mathbf{r}_k\n",
    "\\end{equation}$$ The residuals for each $\\alpha$ are then given by $$\\mathbf{r}_{k+1}(\\alpha) = \\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}(\\alpha)$$ From Exercise 3.23 the magnitude of the correlation between the predictors in $\\mathcal{A}_k$ and the above residual $\\mathbf{r}_{k+1}(\\alpha)$ is constant and decreases as $\\alpha$ moves from zero to one. The signed correlation with a predictor $x_a$ in the active set is \n",
    "$$x_a^T\\mathbf{r}_{k+1}(\\alpha)$$. Notionally we will increase $\\alpha$ from zero to one and “stop” with a value of $\\alpha^*$ where $0 < \\alpha^* < 1$ when the new residual given by $$\\begin{align}\n",
    "\\mathbf{r}_{k+1}(\\alpha) &= \\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}(\\alpha)\\\\\n",
    "&=\\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}(\\beta_{\\mathcal{A}_k}+\\alpha\\delta_k)\\\\\n",
    "&=\\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}-\\alpha\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\\\\n",
    "&=\\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}-\\alpha\\mathbf{X}_{\\mathcal{A}_k}\\left( \\mathbf{X}_{\\mathcal{A}_k}^T\\mathbf{X}_{\\mathcal{A}_k} \\right)^{-1}\\mathbf{X}_{\\mathcal{A}_k}^T \\mathbf{r}_k\\\\\n",
    "\\end{align}$$ has the same absolute correlation with one of the predictors not in the active set.  \n",
    "\n",
    "A different way to describe this is the following. As we change $\\alpha$ from zero to one $\\alpha\\approx 0$ the absolute correlation of $\\mathbf{r}_{k+1}(\\alpha)$ with all of the predictors in $\\mathcal{A}_k$ will be the same.  \n",
    "\n",
    "For all predictors not in $\\mathcal{A}_k$ the absolute correlation will be less than this constant value. If\n",
    "$x_b$ is a predictor not in $\\mathcal{A}_k$ then this means that $$\\left|x_b^T \\mathbf{r}_{k+1}(\\alpha)\\right| < \\left|x_a^T\\mathbf{r}_{k+1}(\\alpha)\\right|, \\quad \\forall b \\notin \\mathcal{A}_k;\\quad a \\in \\mathcal{A}_k$$ We “stop” at the value of $\\alpha = \\alpha^∗$ when one of the predictors not in $\\mathcal{A}_k$ has an absolute correlation equal to the value $|x_a^T\\mathbf{r}_{k+1}(\\alpha^∗)|$. The variable $x_b$ that has this largest absolute correlation will be the next variable added to form the set $\\mathcal{A}_{k+1}$ and the process continued.  \n",
    "\n",
    "What we want to do is determine which variable next enters the active set (to form $\\mathcal{A}_{k+1}$) and what the value of $\\alpha^∗$ is where that happens.  \n",
    "\n",
    "Let $x_a$ be any of the predictors in the active set $\\mathcal{A}_{k}$ and $x_b$ be one of the $p−k$ predictors not in the set $\\mathcal{A}_{k}$. Then from the discussion above and Exercise 3.23 the magnitude of the correlation between $x_a$ and $\\mathbf{r}_{k+1}(\\alpha)$ will be the same for\n",
    "all predictors $x_a$ in the active set $\\mathcal{A}_{k}$. This means that $$\\begin{align}\n",
    "x_a^T\\mathbf{r}_{k+1}(\\alpha)&=x_a^T\\left(\\mathbf{y} - \\mathbf{X}_{\\mathcal{A}_k}\\beta_{\\mathcal{A}_k}-\\alpha\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right)\\\\\n",
    "&=x_a^T\\mathbf{r}_{k}-\\alpha x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\n",
    "\\end{align}$$ will have the same magnitude for all $a \\in \\mathcal{A}_{k}$.  \n",
    "\n",
    "Next consider the correlation of $x_b$ with\n",
    "$\\mathbf{r}_{k+1}(\\alpha)$. In the same was as above we will find $$\\begin{align}\n",
    "x_b^T\\mathbf{r}_{k+1}(\\alpha)\n",
    "&=x_b^T\\mathbf{r}_{k}-\\alpha x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\n",
    "\\end{align}$$ \n",
    "We now consider what would need to be true if the predictor $x_b$ was to be the next predictor added. If it was the next one added then we would have $$\\left|x_a^T\\mathbf{r}_{k}-\\alpha x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|=\\underset{b}{\\max}\\left|x_b^T\\mathbf{r}_{k}-\\alpha x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|$$ Another way to say this is that $x_b$ would be the predictor with the largest absolute correlation among all of the predictors not in $\\mathcal{A}_{k}$. We will now turn this procedure around and ask the question if predictor $x_b$ was to be the next one added what would be the value of $\\alpha$. In this case we would need to have $$\\left|x_a^T\\mathbf{r}_{k}-\\alpha x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|=\\left|x_b^T\\mathbf{r}_{k}-\\alpha x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|$$ Now depending on the sign of $x_a^T\\mathbf{r}_{k}-\\alpha x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k$ and $x_b^T\\mathbf{r}_{k}-\\alpha x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k$ we can consider the different possible values for the absolute values and would find that either $$\\alpha=\\frac{x_b^T\\mathbf{r}_{k}-x_a^T\\mathbf{r}_{k}}{x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k-x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k}$$ or $$\\alpha=\\frac{x_b^T\\mathbf{r}_{k}+x_a^T\\mathbf{r}_{k}}{x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k+x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k}$$   \n",
    "\n",
    "Here $x_a$ is any predictor in $\\mathbf{X}_{\\mathcal{A}_k}$ (they will all have the same correlations with the residuals)\n",
    "and everything on the right-hand-side is known at this step. Only one of these expressions will give a value for $\\alpha$ that is in the range $0 < \\alpha < 1$. Lets call this value $\\alpha_b$ as it depends on the $b$th predictor. The steps of the algorithm are then clear. We compute $\\alpha_b$ for all of\n",
    "the $b$ predictors not in $\\mathcal{A}_k$. Given that value of $\\alpha = \\alpha_b$ we can compute the right-hand-side\n",
    "of Equation $$\\left|x_a^T\\mathbf{r}_{k}-\\alpha x_a^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|=\\underset{b}{\\max}\\left|x_b^T\\mathbf{r}_{k}-\\alpha x_b^T\\mathbf{X}_{\\mathcal{A}_k}\\delta_k\\right|$$ and select the value of $b$ that gives the largest correlation. This will be the next variable added and the corresponding $\\alpha$ at which it occurs.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.26 (forward stepwise regression vs. LAR)\n",
    "\n",
    "\n",
    "Forward stepwise regression enters the variable at each step that most reduces the residual sum-of-squares. LAR adjusts variables that have the most (absolute) correlation with the current residuals. Show that these two entry criteria are not necessarily the same. [Hint: let $\\mathbf x_{j\\cdot A}$ be the $j$th variable, linearly adjusted for all the variables currently in the model. Show that the first criterion amounts to identifying the $j$ for which $\\text{Cor}(\\mathbf x_{j\\cdot A}, \\mathbf r)$ is largest in magnitude.\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.27 Lasso and LAR  \n",
    "\n",
    "\n",
    "Consider the lasso problem in Lagrange multiplier form: with  $$L(\\beta) = \\frac{1}{2}\\sum_{i}(y_i-\\sum_{j}x_{ij}\\beta_j)^2$$ we minimize\n",
    "$$L(\\beta) + \\lambda\\sum_{j}|\\beta_j|=L(\\beta) + \\lambda\\sum_{j}\\left(\\beta_j^++\\beta_j^-\\right) \\quad (3.88)$$ for fixed $\\lambda>0$.\n",
    "\n",
    "(a) Setting $\\beta_j = \\beta_j^+-\\beta_j^-$ with $\\beta_j^+, \\beta_j^-\\ge 0$, expression (3.88) becomes $L(\\beta) + \\lambda\\sum_{j}(\\beta_j^++\\beta_j^-)$. Show that the **Lagrange dual function** is $$L(\\beta) + \\lambda\\sum_{j}(\\beta_j^++\\beta_j^-)-\\sum_{j}\\lambda_j^+\\beta_j^+-\\sum_{j}\\lambda_j^-\\beta_j^-\\quad\\quad(3.89)$$ and Two of the **Karush-Kuhn-Tucker (KKT)** conditions are obtained by taking the derivatives of the above objective function with respect to $\\beta_j^+$ and $\\beta_j^-$ and setting the results equal to zero. This for $\\beta_j^+$ gives $$\\nabla L(\\beta)_j + \\lambda − \\lambda_j^+ =0\\\\$$ and for $\\beta_j^-$ gives $$-\\nabla L(\\beta)_j + \\lambda − \\lambda_j^- =0$$  \n",
    "\n",
    "The other two KKT conditions are the **complementary slackness conditions** which in this case take the form\n",
    "$$\\begin{align}\n",
    "\\lambda_j^+\\beta_j^+&=0\\\\\n",
    "\\lambda_j^-\\beta_j^-&=0\\\\\n",
    "\\end{align}$$ along with the non-negativity constraints on the parameters and all the Lagrange multipliers. \n",
    "then the Karush–Kuhn–Tucker optimality conditions are\n",
    "$$\\begin{align}\n",
    "\\nabla L(\\beta)_j + \\lambda − \\lambda_j^+ &=0\\\\\n",
    "-\\nabla L(\\beta)_j + \\lambda − \\lambda_j^- &=0\\\\\n",
    "\\lambda_j^+\\beta_j^+&=0\\\\\n",
    "\\lambda_j^-\\beta_j^-&=0\\\\\n",
    "\\end{align}$$\n",
    "\n",
    "(b) Show that $|\\nabla L(\\beta)_j| \\leq \\lambda ; \\quad\\forall j$, and that the **KKT conditions** imply one of the following three scenarios:\n",
    "$$\\begin{align}\n",
    "\\lambda = 0 &\\Rightarrow \\nabla L(\\beta)_j= 0 \\forall j\\\\\n",
    "\\beta_j^+ > 0, \\lambda > 0 &\\Rightarrow \\lambda_j^+ = 0, \\nabla L(\\beta)_j = −\\lambda < 0, \\beta_j^− = 0\\\\\n",
    "\\beta_j^− > 0, \\lambda > 0 &\\Rightarrow \\lambda_j^−= 0, \\nabla L(\\beta)_j = \\lambda > 0, \\beta_j^+ = 0\\\\\n",
    "\\end{align}$$\n",
    "Hence show that for any “active” predictor having $\\beta_j\\ne 0$, we must have $\\nabla L(\\beta)_j = −\\lambda$ if $\\beta_j > 0$, and $\\nabla L(\\beta)_j = \\lambda$ if $\\beta_j < 0$. Assuming the predictors are standardized, relate $\\lambda$ to the correlation between the $j$th predictor and the current residuals. \n",
    "\n",
    "Summing the first two KKT equations gives $$\\lambda_j^++\\lambda_j^-=2\\lambda\\ge0$$ Taking the differences of the same two equations we get $$\\lambda_j^--\\lambda_j^+=2\\nabla L(\\beta)_j$$  \n",
    "Taking the absolute value of this gives \n",
    "$$|\\nabla L(\\beta)_j|=\\frac{1}{2}|\\lambda_j^--\\lambda_j^+|\\le\\frac{1}{2}(\\lambda_j^-+\\lambda_j^+)=\\lambda$$ \n",
    "\n",
    "Now if $\\lambda=0$ then we see that $|\\nabla L(\\beta)_j| \\le 0$ so $\\nabla L(\\beta)_j=0$. Another way to argue this is to note that when $\\lambda=0$ then the Lasso problem reduces to ordinary least squares which is minimized by enforcing that the partial derivative of $L$ w.r.t any $\\beta_j$ are zero.  \n",
    "\n",
    "If $\\lambda>0$ then assuming that $β_j^+\\ne 0$, then from the complementary slackness conditions\n",
    "$$\\begin{align}\n",
    "\\lambda_j^+\\beta_j^+&=0\\\\\n",
    "\\lambda_j^-\\beta_j^-&=0\\\\\n",
    "\\end{align}$$ we know that $\\lambda_j^+= 0$. Combined with Equation $$\\nabla L(\\beta)_j + \\lambda − \\lambda_j^+ =0\\\\$$ we get, $∇L(β)j = −λ < 0$. Plug this into the Equation $$-\\nabla L(\\beta)_j + \\lambda − \\lambda_j^- =0$$ we get $λ_j^- = 2λ > 0$ so finally using Equation $\\lambda_j^-\\beta_j^-=0$ we get\n",
    "$\\beta_j^-=0$.  \n",
    "\n",
    "\n",
    "If $\\lambda>0$ but now $\\beta_j^-\\ne0$, then from the complementary slackness conditions Equation $\\lambda_j^-\\beta_j^-=0$ we know that $\\lambda_j^- = 0$. Combined with Equation $$-\\nabla L(\\beta)_j + \\lambda − \\lambda_j^- =0$$ we get, $$\\nabla L(\\beta)_j = \\lambda>0$$. Plug this into the Equation $$\\nabla L(\\beta)_j + \\lambda − \\lambda_j^+ =0$$ we get $\\lambda_j^+=2\\lambda>0$ so finally using Equation $\\lambda_j^+\\beta_j^+=0$ we get $\\beta_j^+=0$.\n",
    "Thus in the above we have shown that $$\\nabla L(\\beta)_j = -\\lambda\\cdot\\text{sign}(\\beta_j)$$ \n",
    "\n",
    "\n",
    "Next from the expression given for $L(\\beta)$ we have that $$\\nabla L(\\beta)_j =-x_j^T(y-X\\beta)=-\\lambda\\cdot\\text{sign}(\\beta_j)$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "(c) Suppose that the set of active predictors is unchanged for $\\lambda_0 \\ge \\lambda \\ge \\lambda_1$. Show that there is a vector $\\gamma_0$ such that $$\\hat{\\beta}(\\lambda) = \\hat{\\beta}(\\lambda_0) − (\\lambda − \\lambda_0)\\gamma_0 \\quad (3.90)$$\n",
    "Thus the lasso solution path is linear as $\\lambda$ ranges from $\\lambda_0$ to $\\lambda_1$ (Efron et al., 2004; Rosset and Zhu, 2007).\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.28  \n",
    "\n",
    "Suppose for a given $t$ in (3.51), the fitted lasso coefficient for variable $X_j$ is $\\hat{\\beta}_j = a$. Suppose we augment our set of variables with an identical copy $X_j^* = X_j$. Characterize the effect of this exact collinearity by describing the set of solutions for $\\hat{\\beta}_j$ and $\\hat{\\beta}_j^*$, using the same value of $t$.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat\\beta^{\\text{lasso}} = \\arg\\min_\\beta \\sum_{i=1}^N \\left( y_i-\\beta_0-\\sum_{j=1}^p x_{ij}\\beta_j \\right)^2 \\text{ subject to } \\sum_{j=1}^p |\\beta_j| \\le t,\\quad (3.51)\n",
    "\\end{equation}$$\n",
    "\n",
    "\n",
    "Define the new, augmented observation matrix as two copies of $X$ concatenated as $X^{\\text{new}} =[X,X]$. The new coefficient vector is then as $(\\beta^{\\text{new}})^T = (\\beta_1^T, \\beta_2^T)^T$. With this notation $$X^{\\text{new}}\\beta^{\\text{new}}=[X,X]\\beta^{\\text{new}}=X\\beta_1+X\\beta_2=X(\\beta_1+\\beta_2)$$ The Lagrange multiplier expression (that we minimize) for the Lasso optimization is $$\\begin{align}\n",
    "L&=\\left\\lVert y-X^{\\text{new}}\\beta^{\\text{new}}\\right\\rVert^2+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}\\rvert+\\sum_{j=1}^{p}\\lvert\\beta_{2j}\\rvert\\right)\\\\\n",
    "&=\\left\\lVert y-X(\\beta_1+\\beta_2)\\right\\rVert^2+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}\\rvert+\\sum_{j=1}^{p}\\lvert\\beta_{2j}\\rvert\\right)\\\\\n",
    "&=\\left\\lVert y-X(\\beta_1+\\beta_2)\\right\\rVert^2+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)-\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}\\rvert+\\sum_{j=1}^{p}\\lvert\\beta_{2j}\\rvert\\right)\\\\\n",
    "\\end{align}$$  Note that the first two terms represent the same minimization problem for the original lasso (before we provided duplicate features) and that because $$\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\le\\lvert\\beta_{1j}\\rvert+\\lvert\\beta_{2j}\\rvert$$ the $$-\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}\\rvert+\\sum_{j=1}^{p}\\lvert\\beta_{2j}\\rvert\\right)=\\lambda\\left(\\sum_{j=1}^{p}\\left(\\lvert\\beta_{1j}\\rvert+\\lvert\\beta_{2j}\\rvert-\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)\\right)$$ term is positive or zero. It will be zero if $\\beta_{1j}$ and $\\beta_{2j}$ are either both positive or both negative.  \n",
    "\n",
    "We know that the minimum of the first two terms $$\\left\\lVert y-X(\\beta_1+\\beta_2)\\right\\rVert^2+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)$$ of this objective function happens for the solution $\\beta_{1} +\\beta_{2} = \\hat\\beta = a$ as mentioned in the problem. As the $$-\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)+\\lambda\\left(\\sum_{j=1}^{p}\\lvert\\beta_{1j}\\rvert+\\sum_{j=1}^{p}\\lvert\\beta_{2j}\\rvert\\right)=\\lambda\\left(\\sum_{j=1}^{p}\\left(\\lvert\\beta_{1j}\\rvert+\\lvert\\beta_{2j}\\rvert-\\lvert\\beta_{1j}+\\beta_{2j}\\rvert\\right)\\right)$$ term is guaranteed to be positive (thus increasing the objective value) if $\\beta_{1j}$ and $\\beta_{2j}$ have different signs, and the solution set for $\\beta_{1}$ and $\\beta_{2}$ to this problem must satisfy $$\\beta_{1} +\\beta_{2} = \\hat\\beta = a$$ and the components of $\\beta_{1}$ and $\\beta_{2}$ must have the same sign.\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.29   \n",
    "\n",
    "Suppose we run a ridge regression with parameter $\\lambda$ on a single variable $X$, and get coefficient $a$. We now include an exact copy $X^∗ = X$, and refit our ridge regression. Show that both coefficients are identical, and derive their value. Show in general that if $m$ copies of a variable $X_j$ are included in a ridge regression, their coefficients are all the same.\n",
    "\n",
    "The ridge regression optimization selects $\\beta$ to minimize  $$\\lVert y-\\beta X\\rVert^2+\\lambda\\lVert\\beta\\rVert^2$$ The minimum of this objective function is given by $$\n",
    "\\begin{align}\n",
    "\\hat\\beta^{\\text{ridge}} &= \\left( \\mathbf{X}^T\\mathbf{X} + \\lambda\\mathbf{I} \\right)^{-1}\\mathbf{X}^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V\\mathbf D^2\\mathbf V^T+\\lambda\\mathbf{V}\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=(\\mathbf V(\\mathbf D^2+\\lambda\\mathbf{I})\\mathbf{V}^T)^{-1}\\mathbf V\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}^{-T}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "&=\\mathbf{V}(\\mathbf D^2+\\lambda\\mathbf{I})^{-1}\\mathbf D\\mathbf U^T\\mathbf{y}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "In this problem, we are assuming that $X \\in \\mathbb R^{n\\times1}$. That is, there is only one feature $p = 1$. In this case $X^TX$ is a scalar and the inverse is a scalar and we get\n",
    "$$\\beta=\\frac{X^Ty}{X^TX+\\lambda}$$ When we augment our inputs to include a duplicate predictor a second time the optimization problem we seek to solve is to find the minimum of $$\\lVert y-\\beta_1 X-\\beta_2 X\\rVert^2+\\lambda\\lVert\\beta_1\\rVert^2+\\lambda\\lVert\\beta_2\\rVert^2$$ As $\\beta_1$ and $\\beta_2$ are scalars the above is the same as $$\\sum_{i=1}^{n}\\left(y_i-x_i\\beta_1-x_i\\beta_2\\right)^2+\\lambda\\beta_1^2+\\lambda\\beta_2^2$$  The first order conditions for this objective function are $$2\\sum_{i=1}^{n}\\left(y_i-x_i\\beta_1-x_i\\beta_2\\right)(-x_i)+2\\lambda\\beta_1=0$$\n",
    "$$2\\sum_{i=1}^{n}\\left(y_i-x_i\\beta_1-x_i\\beta_2\\right)(-x_i)+2\\lambda\\beta_2=0$$ or back in matrix/vector form $$X^T\\left(y-X\\beta_1-X\\beta_2\\right)-\\lambda\\beta_1=0$$\n",
    "$$X^T\\left(y-X\\beta_1-X\\beta_2\\right)-\\lambda\\beta_2=0$$ and $$\\left(X^TX+\\lambda\\right)\\beta_1+X^TX\\beta_2=X^Ty$$\n",
    "$$X^TX\\beta_1+\\left(X^TX+\\lambda\\right)\\beta_2=X^Ty$$ Or \n",
    "$$\\begin{bmatrix}\n",
    "\\left(X^TX+\\lambda\\right)&X^TX\\\\\n",
    "X^TX&\\left(X^TX+\\lambda\\right)\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "\\beta_1\\\\\n",
    "\\beta_2\n",
    "\\end{bmatrix}=\\begin{bmatrix}\n",
    "X^Ty\\\\\n",
    "X^Ty\n",
    "\\end{bmatrix}$$\n",
    "This is a system of two equations and two unknowns and has a solution given by (using Cramer’s rule) $$\\beta_1=\\frac{\\begin{vmatrix}\n",
    "X^Ty&X^TX\\\\\n",
    "X^Ty&\\left(X^TX+\\lambda\\right)\n",
    "\\end{vmatrix}}{\\begin{vmatrix}\n",
    "\\left(X^TX+\\lambda\\right)&X^TX\\\\\n",
    "X^TX&\\left(X^TX+\\lambda\\right)\n",
    "\\end{vmatrix}}$$ and $$\\beta_2=\\frac{\\begin{vmatrix}\n",
    "\\left(X^TX+\\lambda\\right)&X^Ty\\\\\n",
    "X^TX&X^Ty\n",
    "\\end{vmatrix}}{\\begin{vmatrix}\n",
    "\\left(X^TX+\\lambda\\right)&X^TX\\\\\n",
    "X^TX&\\left(X^TX+\\lambda\\right)\n",
    "\\end{vmatrix}}$$ then $$\\begin{bmatrix}\n",
    "\\beta_1\\\\\n",
    "\\beta_2\n",
    "\\end{bmatrix}=\\frac{X^Ty}{2X^TX+\\lambda}\\begin{bmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{bmatrix}$$ Notice that $\\beta_1 = \\beta_2$. We could have derived this result if we notice that exchanging $\\beta_1 ↔ \\beta_2$ leaves the system unchanged. This means that we know (before we solve the system) that\n",
    "the solution must satisfy $\\beta_1 = \\beta_2$.  \n",
    "\n",
    "For $m$ copies of the original predictor $X$ the optimization problem we have is to minimize $$\\left(y-X\\sum_{j=1}^{m}\\beta_j\\right)^T\\left(y-X\\sum_{j=1}^{m}\\beta_j\\right)+\\lambda\\sum_{j=1}^{m}\\beta_j^2$$ One of the first order conditions for this optimization is found by taking the derivative of the above with respect to $\\beta_k$ for $1 \\le k \\le m$ and setting the result equal to zero. $$-X^T\\left(y-X\\sum_{j=1}^{m}\\beta_j\\right)-\\left(y-X\\sum_{j=1}^{m}\\beta_j\\right)^TX+2\\lambda\\beta_k=0$$ or $$2X^TX\\sum_{j=1}^{m}\\beta_j-2X^Ty+2\\lambda\\beta_k=0$$ or $$X^TX\\sum_{j=1}^{m}\\beta_j+\\lambda\\beta_k=X^Ty$$ This means that the values of $\\beta_k$ are all equal for $1 ≤ k ≤ m$ and then  $$mX^TX\\beta_k+\\lambda\\beta_k=X^Ty$$ or $$\\beta_k=\\frac{X^Ty}{mX^TX+\\lambda},\\quad (1\\le k\\le m)$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Ex. 3.30 (solving the elastic net optimization problem with the lasso)  \n",
    "\n",
    "Consider the elastic-net optimization problem:\n",
    "$$\\underset{\\beta}{\\text{min}} \\lVert y − X\\beta\\rVert^2 + \\lambda \\left[\\alpha\\lVert \\beta\\rVert_2^2 + (1 − \\alpha)\\lVert \\beta\\rVert_1\\right] \\quad (3.91)$$\n",
    "Show how one can turn this into a lasso problem, using an augmented version of $X$ and $y$.\n",
    "\n",
    "if we augment $X$ with a multiple of the $p \\times p$ identity to get $$\\tilde{X}=\\begin{bmatrix}\n",
    "X\\\\\n",
    "\\gamma I\n",
    "\\end{bmatrix}$$ then $$\\tilde{X}\\beta=\\begin{bmatrix}\n",
    "X\\beta\\\\\n",
    "\\gamma \\beta\n",
    "\\end{bmatrix}$$ If we next augment $y$ with $p$ zero values as $$\\tilde{y}=\\begin{bmatrix}\n",
    "y\\\\\n",
    "0\\\\\n",
    "\\end{bmatrix}$$ Then we have $$\\left\\lVert\\tilde y-\\tilde X\\beta\\right\\rVert_2^2=\\left\\lVert\\begin{bmatrix}y-X\\beta\\\\\n",
    "-\\gamma \\beta\\\\\n",
    "\\end{bmatrix}\\right\\rVert_2^2=\\left\\lVert y-X\\beta\\right\\rVert_2^2+\\gamma^2\\left\\lVert\\beta\\right\\rVert_2^2$$ \n",
    "\n",
    "Now in the this augmented space a lasso problem for $\\beta$ is $$\\begin{align}\n",
    "\\hat\\beta&=\\underset{\\beta}{\\text{argmin}}\\left(\\left\\lVert\\tilde y-\\tilde X\\beta\\right\\rVert_2^2+\\tilde\\lambda\\left\\lVert\\beta\\right\\rVert_1\\right)\\\\\n",
    "&=\\underset{\\beta}{\\text{argmin}}\\left(\\left\\lVert y-X\\beta\\right\\rVert_2^2+\\gamma^2\\left\\lVert\\beta\\right\\rVert_2^2+\\tilde\\lambda\\left\\lVert\\beta\\right\\rVert_1\\right)\\\\\n",
    "&=\\underset{\\beta}{\\text{argmin}}\\left(\\left\\lVert y-X\\beta\\right\\rVert_2^2+\\lambda\\alpha\\left\\lVert\\beta\\right\\rVert_2^2+\\lambda(1-\\alpha)\\left\\lVert\\beta\\right\\rVert_1\\right)\\\\\n",
    "\\end{align}$$ where we set $\\gamma=\\sqrt{\\lambda \\alpha}$ and $\\tilde\\lambda=\\lambda(1-\\alpha)$ \n",
    "\n",
    "\n"
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