Solutions to Chapter 9 Learning Aides.ipynb

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   "outputs": [],
   "source": [
    "# Add lib input sys.path\n",
    "import os\n",
    "import sys\n",
    "import time\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import sklearn\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import minimize\n",
    "import math\n",
    "from sklearn.preprocessing import normalize\n",
    "from functools import partial\n",
    "import h5py\n",
    "from scipy.spatial import distance\n",
    "\n",
    "nb_dir = os.path.split(os.getcwd())[0]\n",
    "if nb_dir not in sys.path:\n",
    "    sys.path.append(nb_dir)\n",
    "\n",
    "from matplotlib.colors import ListedColormap\n",
    "import libs.linear_models as lm\n",
    "import libs.data_util as data\n",
    "import libs.nn as nn\n",
    "import libs.plot as myplot\n",
    "from sklearn import decomposition\n",
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 9.1\n",
    "\n",
    "$x_g = (47,35)$, $x_b = (22, 40)$, $x_u = (21, 36)$\n",
    "\n",
    "The distance from $x_u$ to Mr. Bad is closer than the distance to Mr. Good. So the BoL should NOT give hime credit.\n",
    "\n",
    "If the income is measured in dollars, Mr. Unknown is closer to Mr. Good, so the BoL should give him credit."
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Income measured in K\n",
      "--- Distance from unknow to Mr. Good: 26.019223662515376\n",
      "--- Distance from unknow to Mr. Bad: 4.123105625617661\n",
      "--- Income measured in dollars\n",
      "--- Distance from unknow to Mr. Good: 1000.3379428972991\n",
      "--- Distance from unknow to Mr. Bad: 4000.000124999998\n"
     ]
    }
   ],
   "source": [
    "# The distance between $x_u$ and the two points\n",
    "print('--- Income measured in K')\n",
    "xg = np.array([47, 35])\n",
    "xb = np.array([22, 40])\n",
    "xu = np.array([21, 36])\n",
    "\n",
    "d_ug = np.linalg.norm(xu-xg)\n",
    "d_ub = np.linalg.norm(xu-xb)\n",
    "print(f\"--- Distance from unknow to Mr. Good: {d_ug}\")\n",
    "print(f\"--- Distance from unknow to Mr. Bad: {d_ub}\")\n",
    "\n",
    "print('--- Income measured in dollars')\n",
    "# Income measured in dollars\n",
    "xg = np.array([47, 35000])\n",
    "xb = np.array([22, 40000])\n",
    "xu = np.array([21, 36000])\n",
    "\n",
    "d_ug = np.linalg.norm(xu-xg)\n",
    "d_ub = np.linalg.norm(xu-xb)\n",
    "print(f\"--- Distance from unknow to Mr. Good: {d_ug}\")\n",
    "print(f\"--- Distance from unknow to Mr. Bad: {d_ub}\")"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 9.2\n",
    "\n",
    "\\begin{align*}\n",
    "Z &= \\gamma X \\\\\n",
    "&= (I - \\frac{1}{N}1 1^T)X \\\\\n",
    "&= X - \\frac{1}{N} 1 1^T X\\\\\n",
    "&= X - 1\\frac{1}{N} \\begin{bmatrix}\\sum^N_{i=1} x_{i1} & \\sum^N_{i=1} x_{i2} & \\dots & \\sum^N_{i=1} x_{id}\\end{bmatrix}  \\\\\n",
    "&= X - 1\\begin{bmatrix}\\bar{x}_1 & \\bar{x}_2 & \\dots & \\bar{x}_d\\end{bmatrix}\\\\\n",
    "&=  X - 1\\bar{x}^T \\\\\n",
    "\\end{align*}\n",
    "\n",
    "#### Exercise 9.3\n",
    "\n",
    "\\begin{align*}\n",
    "Z &= \\begin{bmatrix}z^T_1 \\\\ \\dots \\\\ z^T_N \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}(Dx_1)^T \\\\ \\dots \\\\ (Dx_N)^T\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}x_1^TD^T \\\\ \\dots \\\\ x_N^TD^T \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}x_1^TD \\\\ \\dots \\\\ x_N^TD \\end{bmatrix}\\\\\n",
    "&= XD \\\\\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "Z^TZ &= (XD)^TXD \\\\\n",
    "&= D^TX^TXD\\\\\n",
    "&= DX^TXD\\\\\n",
    "\\end{align*}\n",
    "\n",
    "#### Exercise 9.4\n",
    "\n",
    "* (a) $\\text{variance}(x_1) = \\text{variance}(\\hat{x}_1) = 1$, $\\text{variance}(x_2) = \\text{variance}(\\sqrt{1-\\epsilon^2}\\hat{x}_1+\\epsilon\\hat{x}_2) = (1-\\epsilon^2)\\text{variance}(\\hat{x}_1)+ \\epsilon^2 \\text{variance}(\\hat{x}_2) = 1$\n",
    "\n",
    "$\\text{covariance}(x_1,x_2) = E[(x_1 - \\bar{x}_1)(x_2 - \\bar{x}_2)] = E[x_1x_2] = E[\\sqrt{1-\\epsilon^2}\\hat{x}^2_1 + \\epsilon\\hat{x}_1\\hat{x}_2] = \\sqrt{1-\\epsilon^2}$\n",
    "\n",
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "f(x) &= w_1x_1 + w_2x_2 \\\\\n",
    "&= w_1\\hat{x}_1 + w_2 (\\sqrt{1-\\epsilon^2}\\hat{x}_1 + \\epsilon \\hat{x}_2) \\\\\n",
    "&= (w_1 + w_2 \\sqrt{1-\\epsilon^2})\\hat{x}_1 + w_2\\epsilon \\hat{x}_2 \\\\\n",
    "&= \\hat{w}_1\\hat{x}_1 + \\hat{w}_2 \\hat{x}_2 \\\\\n",
    "\\end{align*}\n",
    "\n",
    "So if we set $\\hat{w}_1 = w_1 + w_2 \\sqrt{1-\\epsilon^2}, \\hat{w}_2 = w_2\\epsilon$, we see $f$ is linear in $x_1,x_2$.\n",
    "\n",
    "* (c) From problem (b), we have $\\hat{w}_1 = \\hat{w}_2 = 1$, so we have $w_1 = \\frac{\\epsilon - \\sqrt{1-\\epsilon^2}}{\\epsilon}, w_2 = \\frac{1}{\\epsilon}$, so that $C \\ge w^2_1 + w^2_2 = 2\\frac{1-\\epsilon\\sqrt{1-\\epsilon^2}}{\\epsilon^2}$\n",
    "\n",
    "* (d) As $\\epsilon \\to 0$, we have the minimum $C \\to \\infty $. It means that we have to use a huge $C$ to be able to implement the target function, which is impossible here.\n",
    "\n",
    "* (e) If there is significant noise in the data, with correlated inputs, it'll be hard to regularize the learning, and overfitting is likely. So var term can be high while bias can be low.\n",
    "\n",
    "#### Exercise 9.5\n",
    "\n",
    "We compute the distances between $x_1,x_2 $ with $x_{test}$, we have\n",
    "\n",
    "$d^2_1 = |x_1 - x_{test}|^2 = \\sum^d(a_i + 1)^2$ and $d^2_2 = |x_2 - x_{test}|^2 = 4 + \\sum^d(b_i + 1)^2$.\n",
    "\n",
    "Suppose there are $k$ $+1$s in $a_i$ and $l$ $+1$s in $b_i$, then we have\n",
    "\n",
    "$d^2_1 = \\sum^d(a_i + 1)^2 = 4k$ and $d^2_2 = 4 + 4l = 4(l+1)$.\n",
    "\n",
    "To correctly classify $x_{test}$ we want to have $d_1 < d_2$, which indicates that $k < l+1$, i.e. $k\\le l$.\n",
    "\n",
    "So we need compute the probabilities of the number of $+1$ in $a$ is less than or equal to the number of $+1$s in $b$.\n",
    "Both $a$ and $b$ have $d$ elements, by symmetry the probability of $P(k > l) = P(l < k)$, i.e. the probability of $a$ having more $+1$ than $b$ is equal to the probability of $a$ having less $+1$ than $b$. So we have\n",
    "\n",
    "$2P(k>l) + P(k=l) = 1$, thus $P(k\\le l) = P(k < l) + P(k=l) = \\frac{1}{2}(1 + P(k=l))$\n",
    "\n",
    "We only have to solve the probability $P(k=l)$. \n",
    "\n",
    "For a given $k$, the probability of $P\\left[(a \\text{ has } k\\; +1)\\cap (b \\text{ has } k\\; + 1)\\right] = \\frac{d \\choose k}{2^d}\\frac{d \\choose k}{2^d} = \\frac{{d \\choose k}^2}{2^{2d}}$\n",
    "\n",
    "So we have \n",
    "\n",
    "\\begin{align*}\n",
    "P(k=l) &= \\sum^d_{k=0}P\\left[(a \\text{ has } k\\; +1)\\cap (b \\text{ has } k\\; + 1)\\right]\\\\\n",
    "&= \\sum^d_{k=0}\\frac{{d \\choose k}^2}{2^{2d}}\\\\\n",
    "&= \\frac{1}{2^{2d}}\\sum^d_{k=0} {d \\choose k}^2\\\\\n",
    "&= \\frac{1}{2^{2d}}{2d \\choose d}\\\\\n",
    "&= \\frac{1}{2^{2d}}\\frac{(2d)!}{d!d!}\\\\\n",
    "&\\text{apply stirling approximation } n! = \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n\\\\\n",
    "&\\approx \\frac{1}{2^{2d}}\\frac{\\sqrt{2\\pi 2d}\\left(\\frac{2d}{e}\\right)^{2d}}{\\sqrt{2\\pi d}\\left(\\frac{d}{e}\\right)^d \\sqrt{2\\pi d}\\left(\\frac{d}{e}\\right)^d}\\\\\n",
    "&= \\frac{1}{\\sqrt{\\pi d}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $P(k\\le l) = \\frac{1}{2}(1 + P(k=l)) = \\frac{1}{2} + \\frac{1}{2}\\frac{1}{\\sqrt{\\pi d}} = \\frac{1}{2} + O(\\frac{1}{\\sqrt{d}})$\n",
    "\n",
    "This is the probability of classifying $x_{test}$ correctly with two data points. \n",
    "\n",
    "If there's a third data point $x_3$, then to correctly classify the $x_{test}$, we need have both $d_1 < d_2$ and $d_1 < d_3$, so the probability $P = P(k \\le l)P(k \\le l) = \\frac{1}{4} + O(\\frac{1}{\\sqrt{d}})$\n",
    "\n",
    "The probability of correctly classifying the $x_{test}$ drop about half.\n",
    "\n",
    "#### Exercise 9.6\n",
    "\n",
    "* (a) A large offset shouldn't affect the 'natural axes' from PCA. They only change the origin of the natural axes, but not the directions. We don't have to perform centering before PCA.\n",
    "\n",
    "* (b) If one dimension is inflated, this will increase the variance along the dimension and make PCA choose 'natural axes' along the inflated dimension. We should perform input normalized before doing PCA.\n",
    "\n",
    "* (c) If we do input whitening, the 'natural axes' for the inputs can be any orthogonal axes. Rotation to any angle won't reveal that one dimension is more important than another because they are all equalized.\n",
    "\n",
    "We shouldn't perform input whitening before doing PCA, because that will equalize all axes for the inputs. PCA won't be able to find axes that has large variances against other axes.\n",
    "\n",
    "#### Exercise 9.7\n",
    "\n",
    "* (a) \n",
    "\n",
    "\\begin{align*}\n",
    "z &= \\begin{bmatrix}x^Tv_1 \\\\ \\dots \\\\ x^Tv_k \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}v_1^Tx \\\\ \\dots \\\\ v_k^Tx \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}v_1^T \\\\ \\dots \\\\ v_k^T \\end{bmatrix}x \\\\\n",
    "&= V^Tx\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The dimension of $V$ is $d \\times k$, its columns are the basis $v_1, \\dots, v_d$\n",
    "\n",
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "Z &= \\begin{bmatrix} z^T_1\\\\ \\dots \\\\ z^T_N \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix} (V^Tx_1)^T\\\\ \\dots \\\\ (V^Tx_N)^T \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix} x^T_1V \\\\ \\dots \\\\ x^T_NV \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix} x^T_1 \\\\ \\dots \\\\ x^T_N \\end{bmatrix}V\\\\\n",
    "&= XV\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) We consider the $z=V^Tx$ where $z$ has $d$ components, so $V$ is $d\\times d$. $V=\\begin{bmatrix}v_1^T \\\\ \\dots \\\\ v_d^T \\end{bmatrix}$, since $v$s are othrogonal, we have $V^TV = I$, so $V^T = V^{-1}$\n",
    "\n",
    "\\begin{align*}\n",
    "\\sum^d_{i=1} z^2_i &= \\|z\\|^2 \\\\\n",
    "&= z^Tz \\\\\n",
    "&=  (V^Tx)^T(V^Tx) \\\\\n",
    "&= x^TVV^Tx \\\\\n",
    "&= x^TVV^{-1}x \\\\\n",
    "&= x^Tx \\\\\n",
    "&= \\sum^d_{i=1} x^2_i \\\\\n",
    "\\end{align*}\n",
    "\n",
    "If we choose $k\\le d$ for $z$, we thus have $\\|z\\| \\le \\|x\\| $\n",
    "\n",
    "#### Exercise 9.8\n",
    "\n",
    "$U^TX = U^TU\\Gamma V^T = \\Gamma V^T$, $XV = U\\Gamma V^T V = U\\Gamma$, so we have \n",
    "\n",
    "$X^TU = V\\Gamma$, i.e. $X^Tu_i = \\gamma_i v_i$ and $Xv_i = \\gamma_i u_i$\n",
    "\n",
    "#### Exercise 9.9\n",
    "\n",
    "Let $A$ be a $m\\times n$ matrix, and $A=\\begin{bmatrix} a_1 & \\dots & a_n \\end{bmatrix}$, where $a_i$ are column vectors of dimension $m\\times 1$\n",
    "\n",
    "* (a) $\\|A\\|^2_F = \\sum_m \\|\\text{row}_m(A)\\|^2 = \\sum_n \\|\\text{column}_n(A)\\|^2$\n",
    "\n",
    "$AA^T = \\begin{bmatrix} a_1 & \\dots & a_n \\end{bmatrix}\\begin{bmatrix} a^T_1 \\\\ \\dots \\\\ a^T_n \\end{bmatrix} = \\sum^n_{i=1} a_i a^T_i$\n",
    "\n",
    "So $\\text{trace}(AA^T) = \\text{trace}(\\sum^n_{i=1} a_i a^T_i) =\\sum^n_{i=1} \\text{trace}(a_i a^T_i) = \\sum^n_{i=1} \\sum^m_{j=1}a^2_{ij} = \\sum^n_{i=1} \\|a_i\\|^2 = \\|A\\|^2_F$\n",
    "\n",
    "\n",
    "Also $A^TA = \\begin{bmatrix}a^T_1a_1 & a^T_1a_2 & \\dots & a^T_1a_n \\\\ a^T_2a_1 & a^T_2a_2 & \\dots & a^T_2a_n \\\\ \\dots & \\dots & \\dots & \\dots \\\\ a^T_na_1 & a^T_na_2 & \\dots & a^T_na_n \\\\ \\end{bmatrix}$, and $\\text{trace}(A^TA) = \\sum^n_{i=1} a^T_i a_i = \\sum^n_{i=1} \\|a_i\\|^2 = \\|A\\|^2_F$\n",
    "\n",
    "* (b) Apply the result from problem (a) \n",
    "\n",
    "\\begin{align*}\n",
    "\\|UAV^T\\|^2_F &= \\text{trace}((UAV^T)^T(UAV^T)) \\\\\n",
    "&= \\text{trace}(VA^TU^TUAV^T) \\\\\n",
    "&= \\text{trace}(VA^TAV^T) \\\\\n",
    "&= \\text{trace}((AV^T)^T(AV^T)) \\\\\n",
    "&\\text{apply } trace(XX^T) = trace(X^TX), X = AV^T \\\\\n",
    "&= \\text{trace}((AV^T)(AV^T)^T) \\\\\n",
    "&= \\text{trace}((AV^T)(VA^T)) \\\\\n",
    "&= \\text{trace}(AA^T) \\\\\n",
    "&= \\|A\\|^2_F\\\\\n",
    "\\end{align*}\n",
    "\n",
    "#### Exercise 9.10\n",
    "\n",
    "If all the singular values of $X$ are distinct, then the eigenvalues of $\\Sigma$ are all distinct and positive or zero. \n",
    "\n",
    "* (a) Let $X$ has a dimension of $N\\times d$, and with SVD, let $X=U\\Gamma V^T$ where $U$ has a dimension of $N\\times d$ and $V$ has a dimension of $d\\times d$, so we have $U^TU=I_d$ and $V^TV=VV^T=I_d$. Also let $V=\\begin{bmatrix}v_1 & \\dots & v_d\\end{bmatrix}$, where $v_i$ are the $d\\times 1 $ column vector and they are orthonormal (basis). $\\Gamma$ is the diagonal matrix of the singular values of $X$, and by construction it's ordered, i.e. $\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_d \\ge 0 $. \n",
    "\n",
    "Consider any direction $v=\\sum^d_{i=1}q_iv_i$, it should have $\\|v\\|^2 = \\sum^d_{i=1}q^2_i = 1$.\n",
    "\n",
    "\\begin{align*}\n",
    "\\text{var}(z_1,\\dots,z_N) &= v^T\\Sigma v \\\\\n",
    "&= v^T \\frac{1}{N}X^TX v \\\\\n",
    "&= \\frac{1}{N}v^T (U\\Gamma V^T)^T (U\\Gamma V^T) v \\\\\n",
    "&= \\frac{1}{N}v^T V\\Gamma U^TU\\Gamma V^T v \\\\\n",
    "&= \\frac{1}{N}v^T V\\Gamma^2  V^T v \\\\\n",
    "&= \\frac{1}{N}v^T V\\Gamma^2 \\begin{bmatrix}v^T_1 \\\\ \\dots \\\\ v^T_d\\end{bmatrix}  v \\\\\n",
    "&= \\frac{1}{N}v^T V\\begin{bmatrix}\\lambda^2_1 & \\dots & 0 \\\\ 0 & \\lambda^2_2 & \\dots \\\\ \\dots & \\dots & \\dots \\\\ \\dots & 0 & \\lambda^2_d \\end{bmatrix} \\begin{bmatrix}v^T_1v \\\\ \\dots \\\\ v^T_dv\\end{bmatrix} \\\\\n",
    "&= \\frac{1}{N}v^T \\begin{bmatrix}v_1 & \\dots & v_d\\end{bmatrix} \\begin{bmatrix}\\lambda^2_1v^T_1v \\\\ \\lambda^2_2v^T_2v \\\\ \\dots \\\\  \\lambda^2_dv^T_dv \\end{bmatrix}\\\\\n",
    "&= \\frac{1}{N} \\begin{bmatrix}v^Tv_1 & \\dots & v^Tv_d\\end{bmatrix} \\begin{bmatrix}\\lambda^2_1v^T_1v \\\\ \\lambda^2_2v^T_2v \\\\ \\dots \\\\  \\lambda^2_dv^T_dv \\end{bmatrix}\\\\\n",
    "&= \\frac{1}{N} \\sum^d_{i=1}v^Tv_i \\lambda^2_iv^T_iv \\\\\n",
    "&= \\frac{1}{N} \\sum^d_{i=1}\\lambda^2_i v^Tv_i v^T_iv \\\\\n",
    "&= \\frac{1}{N} \\sum^d_{i=1}\\lambda^2_i \\|v^Tv_i\\|^2 \\\\\n",
    "&\\le \\frac{1}{N} \\sum^d_{i=1}\\lambda^2_1 \\|v^Tv_i\\|^2 \\\\\n",
    "&= \\frac{\\lambda^2_1}{N} \\sum^d_{i=1} \\|v^Tv_i\\|^2 \\\\\n",
    "&= \\frac{\\lambda^2_1}{N} \\sum^d_{i=1} q^2_i \\\\\n",
    "&= \\frac{\\lambda^2_1}{N} \\\\\n",
    "&= \\frac{1}{N} v^T_1v_1  \\lambda^2_1 v^T_1v_1\\\\\n",
    "&= \\frac{1}{N} \\sum^d_{i=1} v^T_1v_i  \\lambda^2_1 v^T_1v_i\\\\\n",
    "&= v^T_1\\Sigma v_1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So the variance is highest when the principal direction is $v_1$ , the top right singular vector of $X$.\n",
    "\n",
    "* (b) Follow the proof in problem (a), we see that the top-1 principal direction is $v_1$, next, if we select the next direction $v$ that is orthogonal to $v_1$, it's clear that $\\text{var}(z_1,\\dots,z_N) = v^T\\Sigma v = \\frac{1}{N} \\sum^d_{i=2}\\lambda^2_i \\|v^Tv_i\\|^2$. It's easy to see that the principal direction with the highest variance is $v_2$. Continue this, we see that the top-k principal directions are $v_1, \\dots, v_k$. \n",
    "\n",
    "\n",
    "* (c) If we don't have the data matrix $X$, but knows $\\Sigma$, since $\\Sigma = \\frac{1}{N}X^TX = \\frac{1}{N}V\\Gamma^2  V^T$\n",
    "\n",
    "$\\Sigma$ is symmetric, we can do eigen-decomposition, the principal directions are the top-k eigenvectors of $\\Sigma$."
   ]
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    "#### Exercise 9.11\n",
    "\n",
    "$x=\\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}$, $z=\\begin{bmatrix}1 \\\\ 1 \\\\ 2\\end{bmatrix}$. The reconstructed test point in $\\mathcal{Z}$ space using top-1 PCA is: \n",
    "\n",
    "$\\hat{z} = z^Tv_1v_1 = 2v_1 = \\begin{bmatrix}0 \\\\ 0 \\\\ 2\\end{bmatrix}$. \n",
    "By the feature transformation $\\Phi$, we don't have a solution for the reconstructed $\\hat{x}$ in $\\mathcal{X}$ space.\n",
    "\n",
    "#### Exercise 9.12\n",
    "\n",
    "* (a) The VC dimension of $\\mathcal{H}_{+},  \\mathcal{H}_{+}$ is 1. The VC dimension of $\\mathcal{H}$ is 2.\n",
    "\n",
    "* (b) I would pick $\\mathcal{H}_{+}$, because if $x$ is the income, then the credit should increase with income, and $w >0$ makes sense here. \n",
    "\n",
    "#### Exercise 9.13\n",
    "\n",
    "If the hybrid strategy stopped at $\\mathcal{H}_{+}$, we can't use the VC bound for $\\mathcal{H}_{+}$ as we implicit assume we have a bigger hypothesis set $\\mathcal{H}$. We should this the VC-bound of $\\mathcal{H}$ instead.\n",
    "\n",
    "#### Exercise 9.14\n",
    "\n",
    "* (a) $sign(x)$ is monotonically increasing function on $x$, to have $h(x)$ is monotonically increasing in $x$, for $x \\ge z$, we want to have \n",
    "\n",
    "$w_0 + w_1 x_1 + w_2 x_2 + w_3 x^2_1 + w_4 x^2_2 + w_5 x_1x_2 \\ge w_0 + w_1z_1 + w_2z_2 + w_3z^2_1 + w_4z^2_2 + w_5z_1z_2$ \n",
    "\n",
    "i.e. $w_1(x_1-z_1) + w_2(x_2 - z_2) + w_3(x^2_1 - z^2_1) + w_4(x^2_2 - z^2_2) + w_5 (x_1x_2 - z_1z_2) \\ge 0$\n",
    "\n",
    "So we want $w_1,w_2,w_3,w_4,w_5 \\ge 0$ for $h(x) to be monotonically increasing in $x$\n",
    "\n",
    "* (b) For $h(x)$ to be invariant under and arbitrary rotation of $x$, it needs to depend only on $\\|x\\| = \\sqrt{x^2_1 + x^2_2}$\n",
    "\n",
    "So we constrain $w_1=w_2 = w_5 = 0$ and $w_2 = w_3$, thus $h(x) = sign(w_0 + w\\|x\\|^2)$\n",
    "\n",
    "* (c) Since the 2D quadratic function is less than 0 in the middle of the bowl, if we want the positive set to be convex, we want to have $w_3 < 0, w_4 < 0$, such that the bowl is now upside down, and the middle part is larger than 0. The set is convex and enclosed by the points where $h(x) = 0$.\n",
    "\n",
    "\n",
    "#### Exercise 9.15\n",
    "\n",
    "Use the data points allow the algorithm to have larger freedom in selecting the hypothesis, thus have a better fit of the data. On the other hand, the virtual examples force the algorithm to select hypothesis that satisfy the constraints, so it doesn't matter much if there are hypotheses that violates the hint. They will have much less probability to be selected as the final hypothesis. \n",
    "\n",
    "If we want to strictly enforce the constraint, we can add more hint examples, so they can help avoid bad hypotheses.\n",
    "\n",
    "#### Exercise 9.16\n",
    "\n",
    "* Rotational invariance: $E_{hint}(h) = \\frac{1}{N}\\sum^N_{n=1}\\left(h(x_n) - h(x'_n)\\right)^2 1\\left[h(x_n) \\ne h(x'_n)\\right]$\n",
    "\n",
    "* Convexity hint: $E_{hint}(h) = \\frac{1}{N}\\sum^N_{n=1}\\left(h(x_n) - h(x'_n)\\right)^2 1\\left[h(\\eta x_n + (1-\\eta)x'_n) > \\eta h(x_n) + (1-\\eta)h(x'_n)\\right]$, where $0 \\le \\eta \\le 1$\n",
    "\n",
    "\n",
    "* Perturbation hint:  $E_{hint}(h) = \\frac{1}{N}\\sum^N_{n=1}\\left(h(x_n) - g(x_n)\\right)^2 1\\left[|h(x_n) - g(x_n) | > \\delta h(x_n) \\right]$, where $g(x)$ is the known function that the target function is close to.\n",
    "\n",
    "#### Exercise 9.17 TODO\n",
    "\n",
    "\n",
    "\n",
    "#### Exercise 9.18"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_Eout_ex_918(Qf, train_xs, train_ys, test_xs, test_ys, lambda_t = 0):\n",
    "    #Fit with models of degree Qf and compute the E_{out}\n",
    "    Z = data.polynomial_transform(Qf, train_xs)\n",
    "    w = np.matmul(np.linalg.inv(np.matmul(Z.transpose(), Z) + lambda_t), \n",
    "                  np.matmul(Z.transpose(), train_ys))\n",
    "    #print(f\"w.shape: {w.shape}\")\n",
    "    Ein = data.calc_Eout(w, train_xs, train_ys)\n",
    "    Eout = data.calc_Eout(w, test_xs, test_ys)\n",
    "    return Ein, Eout\n",
    "\n",
    "def calc_permutation_estimate(d_eff, train_ys, Ein):\n",
    "    #Compute the Permutation Estimate for Linear Regression, equation (9.6)\n",
    "    N = train_ys.shape[0]    \n",
    "    sigma_hat2 = N * np.var(train_ys)/ (N-1)\n",
    "    est = 2 * sigma_hat2 * d_eff/N\n",
    "    est += Ein\n",
    "    return est\n",
    "\n",
    "def calc_fpe_estimate(N, d_eff, Ein):\n",
    "    #Final prediction error, section 9.5.4\n",
    "    p = N/d_eff\n",
    "    est = (p+1)*Ein/(p-1)\n",
    "    return est\n",
    "\n",
    "def calc_vc_estimate(N, d_eff, Ein):\n",
    "    p = N/d_eff\n",
    "    r1 = np.sqrt(p)*Ein\n",
    "    r2 = np.sqrt(p) - np.sqrt(1 + np.log(p)+ np.log(N)/2/d_eff)\n",
    "    est = r1/r2\n",
    "    return est\n",
    "    \n",
    "\n",
    "def compute_leave_one_out_estimate(H, y_hat, y):\n",
    "    N = y_hat.shape[0]\n",
    "    est = 0\n",
    "\n",
    "    for n in np.arange(N):\n",
    "        yhat_n = y_hat[n]\n",
    "        yn = y[n]\n",
    "        Hnn = H[n,n]\n",
    "        r = (yhat_n - yn)/(1-Hnn)\n",
    "        est += r**2\n",
    "    est = est / N\n",
    "    return est\n",
    "    \n",
    "\n",
    "def run_one_experiment_ex_918(N, test_N, max_target_degree, max_hypo_degree, lambda_t):\n",
    "    sigma_square = data.generate_random_numbers01(1, 1)\n",
    "    Qf = np.random.choice(max_target_degree+1, 1)\n",
    "    normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "    train_xs, train_ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "    orders = np.arange(0,max_hypo_degree+1)\n",
    "    Eins, Eouts = [], []\n",
    "    Est_vcs, Est_loocvs, Est_perms, Est_fpes = [], [], [], []\n",
    "    for order in orders:\n",
    "        Ein, Eout = calc_Eout_ex_918(order, train_xs, train_ys, test_xs, test_ys)\n",
    "\n",
    "        Z = data.polynomial_transform(order, train_xs)\n",
    "        H = lm.compute_H(Z, lambda_t) #full H, Need use transformed Z to compute\n",
    "        y_hat = lm.compute_y_hat(H, train_ys)\n",
    "\n",
    "        eff_dof = lm.calc_effective_dof(H) #effective degree of freedom\n",
    "        #print(f\"{eff_dof}\")\n",
    "        if eff_dof <= 1.0e-18:\n",
    "            eff_dof = 1.0e-18    \n",
    "        est_vc = calc_vc_estimate(N, eff_dof, Ein)\n",
    "        est_loocv = compute_leave_one_out_estimate(H, y_hat, train_ys)\n",
    "        est_perm = calc_permutation_estimate(eff_dof, train_ys, Ein)\n",
    "        est_fpe = calc_fpe_estimate(N, eff_dof, Ein)\n",
    "\n",
    "        Eins.append(Ein)\n",
    "        Eouts.append(Eout)\n",
    "        Est_vcs.append(est_vc)\n",
    "        Est_loocvs.append(est_loocv)\n",
    "        Est_perms.append(est_perm)\n",
    "        Est_fpes.append(est_fpe)\n",
    "    return Eouts, Est_vcs, Est_loocvs, Est_perms, Est_fpes, Eins"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Current iteration : 1\n",
      "--- Current iteration : 1001\n",
      "--- Current iteration : 2001\n",
      "--- Current iteration : 3001\n",
      "--- Current iteration : 4001\n",
      "--- Current iteration : 5001\n",
      "--- Current iteration : 6001\n",
      "--- Current iteration : 7001\n",
      "--- Current iteration : 8001\n",
      "--- Current iteration : 9001\n"
     ]
    }
   ],
   "source": [
    "N = 100\n",
    "test_N = 100\n",
    "max_target_degree = 30\n",
    "max_hypo_degree = 20\n",
    "lambda_t = 0\n",
    "\n",
    "max_exps = 10000\n",
    "Eins, Eouts = [], []\n",
    "Est_vcs, Est_loocvs, Est_perms, Est_fpes = [], [], [], []\n",
    "best_outs, best_ins, best_vcs, best_cvs, best_perms, best_fpes = [], [], [], [], [], []\n",
    "reg_ins, reg_vcs, reg_cvs, reg_perms, reg_fpes = [], [], [], [], []\n",
    "for e in np.arange(max_exps):\n",
    "    if e %1000 == 1:\n",
    "        print(f\"--- Current iteration : {e}\")\n",
    "    Eouts1, Est_vcs1, Est_loocvs1, Est_perms1, Est_fpes1, Eins1 = run_one_experiment_ex_918(N, test_N, \n",
    "                                                                             max_target_degree, \n",
    "                                                                             max_hypo_degree, \n",
    "                                                                             lambda_t)\n",
    "    \n",
    "   \n",
    "    best_out = np.argmin(np.array(Eouts1))\n",
    "    optimal = np.min(np.array(Eouts1))\n",
    "    best_in = np.argmin(np.array(Eins1))\n",
    "    best_vc = np.argmin(np.array(Est_vcs1))\n",
    "    best_cv = np.argmin(np.array(Est_loocvs1))\n",
    "    best_perm = np.argmin(np.array(Est_perms1))\n",
    "    best_fpe = np.argmin(np.array(Est_fpes1))\n",
    "    \n",
    "    best_outs.append(best_out)\n",
    "    best_ins.append(best_in)\n",
    "    best_vcs.append(best_vc)\n",
    "    best_cvs.append(best_cv)\n",
    "    best_perms.append(best_perm)\n",
    "    best_fpes.append(best_fpe)\n",
    "    \n",
    "    regret_out = 0\n",
    "    regret_in = (Eouts1[best_in] - optimal)/optimal\n",
    "    regret_vc = (Eouts1[best_vc] - optimal)/optimal\n",
    "    regret_cv = (Eouts1[best_cv] - optimal)/optimal\n",
    "    regret_perm = (Eouts1[best_perm] - optimal)/optimal\n",
    "    regret_fpe = (Eouts1[best_fpe] - optimal)/optimal\n",
    "    \n",
    "    reg_ins.append(regret_in)\n",
    "    reg_vcs.append(regret_vc)\n",
    "    reg_cvs.append(regret_cv)\n",
    "    reg_perms.append(regret_perm)\n",
    "    reg_fpes.append(regret_fpe)\n",
    "    \n",
    "    Eouts.append(Eouts1)\n",
    "    Est_vcs.append(Est_vcs1)\n",
    "    Est_loocvs.append(Est_loocvs1)\n",
    "    Est_perms.append(Est_perms1)\n",
    "    Est_fpes.append(Est_fpes1)\n",
    "    Eins.append(Eins1)\n",
    "\n",
    "    \n",
    "Eouts = np.mean(np.array(Eouts), axis=0)\n",
    "Est_vcs = np.mean(np.array(Est_vcs), axis=0)\n",
    "Est_loocvs = np.mean(np.array(Est_loocvs), axis=0)\n",
    "Est_perms = np.mean(np.array(Est_perms), axis=0)\n",
    "Est_fpes = np.mean(np.array(Est_fpes), axis=0)\n",
    "\n",
    "b_out = np.mean(np.array(best_outs))\n",
    "b_in = np.mean(np.array(best_ins))\n",
    "b_vc = np.mean(np.array(best_vcs))\n",
    "b_cv = np.mean(np.array(best_cvs))\n",
    "b_perm = np.mean(np.array(best_perms))\n",
    "b_fpe = np.mean(np.array(best_fpes))\n",
    "\n",
    "reg_in = np.mean(np.array(reg_ins))\n",
    "reg_vc = np.mean(np.array(reg_vcs))\n",
    "reg_cv = np.mean(np.array(reg_cvs))\n",
    "reg_perm = np.mean(np.array(reg_perms))\n",
    "reg_fpe = np.mean(np.array(reg_fpes))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "orders = np.arange(max_hypo_degree+1)\n",
    "yub = 1.5\n",
    "Est_loocvs_f = np.clip(Est_loocvs, 0, yub)\n",
    "Eouts_f = np.clip(Eouts, 0, yub)\n",
    "xxs = [orders, orders, orders, orders, orders]\n",
    "yys = [Eouts_f, Est_vcs, Est_loocvs_f, Est_perms, Est_fpes]\n",
    "myplot.plt_plot(xxs, yys, 'plot', ['k', 'g', 'r', 'b', 'y'], ['o', 'x', '.', ',', 's'], \n",
    "                ['Out-of-Sample Error', 'VC Penalty', 'LOO-CV', 'Permutation Estimate', \n",
    "                'FPE'], \n",
    "        title = 'Exercise 9.18 (a) E_out/Validation Estimates vs. Model Order', yscale = None, ylb = 0, yub = 1.2,\n",
    "        xlb = None, xub = None, xlabel = 'Model Order', ylabel = 'E_out/Validation Estimates',\n",
    "        legends = ['']*6, legendx = None, legendy = None, marker_sizes=[15]*5) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>name</th>\n",
       "      <th>Regret</th>\n",
       "      <th>Avg. Order</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Eout</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>10.3275</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Ein</td>\n",
       "      <td>28823.186247</td>\n",
       "      <td>19.9878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Ecv</td>\n",
       "      <td>389.119452</td>\n",
       "      <td>9.3558</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Eperm</td>\n",
       "      <td>114.036898</td>\n",
       "      <td>7.2265</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Efpe</td>\n",
       "      <td>889.059910</td>\n",
       "      <td>11.4787</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Evc</td>\n",
       "      <td>441.075303</td>\n",
       "      <td>4.8410</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    name        Regret  Avg. Order\n",
       "0   Eout      0.000000     10.3275\n",
       "1    Ein  28823.186247     19.9878\n",
       "2    Ecv    389.119452      9.3558\n",
       "3  Eperm    114.036898      7.2265\n",
       "4   Efpe    889.059910     11.4787\n",
       "5    Evc    441.075303      4.8410"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#### Exercise 9.18 (d) Order Selection\n",
    "names = ['Eout', 'Ein', 'Ecv', 'Eperm', 'Efpe', 'Evc']\n",
    "regs = [0, reg_in, reg_cv, reg_perm, reg_fpe, reg_vc]\n",
    "orders = [b_out, b_in, b_cv, b_perm, b_fpe, b_vc]\n",
    "df = pd.DataFrame({'name':names, 'Regret': regs, 'Avg. Order': orders})\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#### Exercise 9.18 (b)\n",
    "\n",
    "def run_one_experiment_918_b(N, test_N, max_target_degree, order, lambdas):\n",
    "    sigma_square = data.generate_random_numbers01(1, 1)\n",
    "    Qf = np.random.choice(max_target_degree+1, 1)\n",
    "    normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "    train_xs, train_ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "    #print(f\"sigma_squre = {sigma_square}, Qf={Qf}, order = {order}\")\n",
    "    Eins, Eouts = [], []\n",
    "    Est_vcs, Est_loocvs, Est_perms, Est_fpes = [], [], [], []\n",
    "    for lambda_t in lambdas:\n",
    "        Ein, Eout = calc_Eout_ex_918(order, train_xs, train_ys, test_xs, test_ys, lambda_t)\n",
    "\n",
    "        Z = data.polynomial_transform(order, train_xs)\n",
    "        H = lm.compute_H(Z, lambda_t) #full H, Need use transformed Z to compute\n",
    "        y_hat = lm.compute_y_hat(H, train_ys)\n",
    "        #print(f\"lambda = {lambda_t}, Ein = {Ein}, Eout = {Eout}\")\n",
    "        eff_dof = lm.calc_effective_dof(H) #effective degree of freedom\n",
    "        #print(f\"effective dof: {eff_dof}\")\n",
    "        if eff_dof <= 1.0e-18:\n",
    "            eff_dof = 1.0e-18    \n",
    "        est_vc = calc_vc_estimate(N, eff_dof, Ein)\n",
    "        est_loocv = compute_leave_one_out_estimate(H, y_hat, train_ys)\n",
    "        est_perm = calc_permutation_estimate(eff_dof, train_ys, Ein)\n",
    "        est_fpe = calc_fpe_estimate(N, eff_dof, Ein)\n",
    "        \n",
    "        Eins.append(Ein)\n",
    "        Eouts.append(Eout)\n",
    "        Est_vcs.append(est_vc)\n",
    "        Est_loocvs.append(est_loocv)\n",
    "        Est_perms.append(est_perm)\n",
    "        Est_fpes.append(est_fpe)\n",
    "    return Eouts, Est_vcs, Est_loocvs, Est_perms, Est_fpes, Eins"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Current iteration : 1\n",
      "--- Current iteration : 1001\n",
      "--- Current iteration : 2001\n",
      "--- Current iteration : 3001\n",
      "--- Current iteration : 4001\n",
      "--- Current iteration : 5001\n",
      "--- Current iteration : 6001\n",
      "--- Current iteration : 7001\n",
      "--- Current iteration : 8001\n",
      "--- Current iteration : 9001\n"
     ]
    }
   ],
   "source": [
    "N = 100\n",
    "test_N = 100\n",
    "max_target_degree = 10\n",
    "hypo_degree = 5\n",
    "\n",
    "max_exps = 10000\n",
    "Eins, Eouts = [], []\n",
    "Est_vcs, Est_loocvs, Est_perms, Est_fpes = [], [], [], []\n",
    "best_outs, best_ins, best_vcs, best_cvs, best_perms, best_fpes = [], [], [], [], [], []\n",
    "reg_ins, reg_vcs, reg_cvs, reg_perms, reg_fpes = [], [], [], [], []\n",
    "\n",
    "lambdas = np.hstack([np.arange(0, 0.01, 0.001), np.arange(0.01, 0.1, 0.01), np.arange(0.1, 1, 0.1), \n",
    "                     np.arange(1, 15, 1), np.arange(10, 310, 10)])\n",
    "\n",
    "for e in np.arange(max_exps):\n",
    "    if e %1000 == 1:\n",
    "        print(f\"--- Current iteration : {e}\")\n",
    "    Eouts1, Est_vcs1, Est_loocvs1, Est_perms1, Est_fpes1, Eins1 = run_one_experiment_918_b(N, \n",
    "                                                                                    test_N, \n",
    "                                                                                    max_target_degree, \n",
    "                                                                                    hypo_degree, \n",
    "                                                                                    lambdas)\n",
    "\n",
    "    best_out = np.argmin(np.array(Eouts1))\n",
    "    optimal = np.min(np.array(Eouts1))\n",
    "    best_in = np.argmin(np.array(Eins1))\n",
    "    best_vc = np.argmin(np.array(Est_vcs1))\n",
    "    best_cv = np.argmin(np.array(Est_loocvs1))\n",
    "    best_perm = np.argmin(np.array(Est_perms1))\n",
    "    best_fpe = np.argmin(np.array(Est_fpes1))\n",
    "    \n",
    "    best_outs.append(lambdas[best_out])\n",
    "    best_ins.append(lambdas[best_in])\n",
    "    best_vcs.append(lambdas[best_vc])\n",
    "    best_cvs.append(lambdas[best_cv])\n",
    "    best_perms.append(lambdas[best_perm])\n",
    "    best_fpes.append(lambdas[best_fpe])\n",
    "    \n",
    "    regret_out = 0\n",
    "    regret_in = (Eouts1[best_in] - optimal)/optimal\n",
    "    regret_vc = (Eouts1[best_vc] - optimal)/optimal\n",
    "    regret_cv = (Eouts1[best_cv] - optimal)/optimal\n",
    "    regret_perm = (Eouts1[best_perm] - optimal)/optimal\n",
    "    regret_fpe = (Eouts1[best_fpe] - optimal)/optimal\n",
    "    \n",
    "    reg_ins.append(regret_in)\n",
    "    reg_vcs.append(regret_vc)\n",
    "    reg_cvs.append(regret_cv)\n",
    "    reg_perms.append(regret_perm)\n",
    "    reg_fpes.append(regret_fpe)\n",
    "\n",
    "    \n",
    "    Eouts.append(Eouts1)\n",
    "    Est_vcs.append(Est_vcs1)\n",
    "    Est_loocvs.append(Est_loocvs1)\n",
    "    Est_perms.append(Est_perms1)\n",
    "    Est_fpes.append(Est_fpes1)\n",
    "\n",
    "Eouts = np.mean(np.array(Eouts), axis=0)\n",
    "Est_vcs = np.mean(np.array(Est_vcs), axis=0)\n",
    "Est_loocvs = np.mean(np.array(Est_loocvs), axis=0)\n",
    "Est_perms = np.mean(np.array(Est_perms), axis=0)\n",
    "Est_fpes = np.mean(np.array(Est_fpes), axis=0)\n",
    "\n",
    "b_out = np.mean(np.array(best_outs))\n",
    "b_in = np.mean(np.array(best_ins))\n",
    "b_vc = np.mean(np.array(best_vcs))\n",
    "b_cv = np.mean(np.array(best_cvs))\n",
    "b_perm = np.mean(np.array(best_perms))\n",
    "b_fpe = np.mean(np.array(best_fpes))\n",
    "\n",
    "reg_in = np.mean(np.array(reg_ins))\n",
    "reg_vc = np.mean(np.array(reg_vcs))\n",
    "reg_cv = np.mean(np.array(reg_cvs))\n",
    "reg_perm = np.mean(np.array(reg_perms))\n",
    "reg_fpe = np.mean(np.array(reg_fpes))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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yK3u5oUZG+ddxJTQW/za57MhGVYqi//z+TtllB5bRf35/oipFFUi57MpebgSOGTMmv3XINpMnTx4zdOjQrAWVfGH8qvFcSL1AWJkwp9+yA8uYt2Meraq1yne5C6kXnDdoWJkw5w07ovmIdHFzWy4/83aXW7p/Kbd9fhvDo4ZTtVRVktOSuZB6gYTkBO74/A7qlq/L1SWu5vu93zNowSCiI6IpW7Qs8efjib8Qz5nzZ7hr4V1UK1mNEoVL8M2ebxjy5RD61etHcKFgTiSc4Pi54/yT+A/3fHUP5YqVI9AE8vXvX3Pv1/fSrVY3AOLOxBF3Jo5Dpw9xPOE4wxYNo2iholxIvcCCnQt44JsHaBvWlsTkRPb/s599f+8jVVKpUboGD337EFv/2sqLP7/IE9c/QZngMuw6sYtdJ3ax8/hOElMSCS0VysPfPcyGoxsYt2Icw6OGUzSoKL8d+42tf21l619bOXP+DFVLVuWxHx5jbdxaXl75Mvc1vo9CAYXYeHQjG49uZMORDZxMPEnlEpV58qcnWfnHSl5b9RqDGw4mVVL5Je4Xfjn8C2vj1vLXub+4psQ1PP3T08TExjB+1Xhur387SSlJrDq0ipV/rGTVH6uIi4/jquJX8ezSZ/lp30+8vuZ1etftTfz5eJYfXE5MbAzLY63f2FOxlC9WnueXPc/hM4d5YfkLzO07l3Zh7bzeh5544YUXjo4ZM2ZytiJdIlfk3mVNmzYVfRkzdxi/ajxRlaLSVdZlB5ax7sg6nmr1VLblHP795/d33gTu5zmVm9NnDm1C2/DTvp8YtGAQH/X4iBZVWpCcmkxKWgor/ljBiO9G8NpNrxFZMZJf4n7huaXPMfrG0TS4uoFTLiUthc1/bmbi2om0D2vPkgNLGNp4KLXK1SIlLYXUtFSn3J6/9zB722yaVGzC+iPr6Vm7J1VKViFVUklNS3X+HjpziCX7l3Bd+ev4/cTvtKraivLFy5Mmaenk0iSN4wnH2fLnFiqFVOJI/BFql69NSOEQ0iTNkhdL7nTSaf44/QdlgsvwT9I/XF3iaoILBTvlHGknpSRxKukUwYWCSUpJolhQMQJMQDo5h2waHr+pp1xhPN/meca2G5vteMaYDSLS1A8qec9TjcyVRW4bBV8aehFxPtG+0/kdIq6J4Oc/fmbkTyMZ1XoUdcrXITElkcTkRJJSkkhMSWT7se3M/G0mkddEsvHoRjrW7EiFYhU4n3recinW75/xf7Lt+DYqFKvAsXPHqBxSmaDAIC6kXuBC6gXnE3ZSShIpaSl5UMLeMRiCAoMINIEEBgRm+D134RzxF+IpXaQ05YuXJ9AEEmACMsgGmAAOxx8m7kwc1UpV49qy11pydpgjToAJcD6Z16tQj4hrIpzhASaAAC7Krj+yng1HNxBVKYrrq16fXs7FLY9dzspDK2lTrQ031bgJY4wzzHDx+Pt93/Pj/h/pVLMTXWt1vSjjJv/l71/y9e6v6Vm7J33q9nHKONJyyM/ZNof5O+fTr14/BjYY6PR3lXMcz9gyg1m/zWJQw0HcE3kPBuMMd/xu+nMTzy97nt51evPFri94uf3LNKvcLIPc+iPrefLHJ7kt/DbmbJ/Dm53epEWVFhnSXBu3lke+fYSBDQcy67dZvNv5Xa6ven2G9FYfWs2wb4ZxV6O7mLFlBlO6T6F1tdYZ5FYcXMHdX93N3ZF3M3XTVGb0msGNoTemkwH4+eDPDFowiKGNhzJ542Rm95lN29C2GeSMMc5784GmD/DB+g9y1JNRI+MjBdHI5KZRAFi6fyn95/fnnc7vUK9CPZbFLmN0zGiGNxtO1ZJVOXP+DGfOn+F00ml2/72bmNgYKoVUIu5MHFVLVsUYw7kL5ziXfI6E5ATSPH9V2ieCAoIIKRJCkcAiFClUJMNv3Jk4Yk/Fcl2562h0dSMKBxYmKDCIwgGFKRx40a06tIoVf6ygfWh7utTqQlBgEEEBQQQFBlEooJDzeP6O+czbMY8B9Qdwd+TdFAoo5HRBAUHOYZDHf3icOxrcwexts/mg6we0qd6GQBPolA0MCGTlHysZ+MXALG/s7DQAvsr+2+R8lfVXT/lylcuubGbkh5Hx++eX/eGaNGkiBY2l+5dK+fHlZen+pR7PRUQSkxNl78m9MnH1RAl5OURu/vhmKfpSUen2aTfpO7evtJveThq830AqvVFJirxYRBhDpi7whUAp82oZqT6xulz1+lXCGKTGWzWk12e9ZODnA2XoV0Pl0e8elVFLRskrK16Rt9e+Ld1mdRPGIP3n9pfv9nwny2OXyy9xv8jWP7fKnpN75NDpQ3Li3An5Zvc3Un58eXl+6fMZrsPTdeelnC9l7Q+5/Mz7cpfLjuxrK1/zmM9rK18rkHLZlc0MYL3kcXud7wYjJ+5KMjK+Vo60tDSZv32+lHqllHT/tLsUG1dMus3qJl1mdZFGHzSScq+V82gozBgjFcZXkLrv1pUbpt4gPWf3lLsX3i1P/fCUvLbyNadRGPT5IPk17lfZdXyXHI0/KucunJO0tDSnPv5o6C/XBu1KaCz+bXLZlVVyhhqZAmhkPDV85V4rJ2+sfkPe+eUdGfb1MGk9tbWUfa1sBgNy1etXSeP/ayzdP+0uDyx6QMb9PE6mb5ouE1ZNkDKvlpEnf3jysnuqF7kyGjRF+TeSH0ZG52T8TGJyIpN+mcTY5WO5qvhVHDp9COFimZcOLk14hXDCK4RTOLAwn2z9hMGNBjN72+w8GY/2x+oyRVEuT3Ti30fy28hk1eAeiT/CN7u/YdGeRfy0/ycSkhMoFFCIlLQUIq+JZHCjwZZhuSqciiUqpls5okZBURR/oUbGR/LbyLgbgCX7l9B3Xl+6X9ed7ce3s/HoRgCql6pO9+u6U7VkVV5f83qmK2bUKCiK4m/UyPhIfhsZsAxA33l9qV6qOpv/3IwgBJgArq96Pd1qdaPbdd2oV6EeMbExubL0UFEU5VLJDyNzRX5+Oa/w1rv49fCvlCxSkvjz8Wz6cxP1KtTjmRue4ZZrb6F8sfLp0lh3ZF06g9IurB1z+85l3ZF1amQURSnwqJHxwvhV4ykUUChdr+PNNW8yaukoaperzea/NhMUEMTDzR7m022fUjmkcgYDA3gc6moX1k4NjKIo/wr8amSMMVOBbsAxEanvIdwAbwFdgAQgWkQ2+lMnX3AYmFdWvsIzNzxDv3n9uLbstfxy+BeCAoL4/eTvlChcgi9v+5L2NdrTq04vHQJTFEXxgL97MtOBd4GPvYR3BmrZrjnwgf2bb7gbmLHLx3L6/GlOHj4JQPuw9jSu2Jiba9ysQ2CKoihZ4FcjIyI/G2NCMxHpCXxsvyS01hhT2hhTUUSO+lMvb3SZ1YXQ0qHM2zGPZ254hv8u+y/nks85w0sULsFT1z9F+xrtM8TVITBFUZSM5PdHyyoDh1zO42y/DBhjhhpj1htj1h8/fjzXFRm/ajyhpUP5cP2H9KvXj5E/jUxnYMoWLcsLbV/gts9vy/BhJUVRFMUz+W1kjAc/j2uqRWSyiDQVkaYVKlTIVSUcQ2TzdsxjWNNhfLD+A5LTkgFrB+HyxcrzXOvnnENo646sy9X8FUVRCir5bWTigKou51WAI3mtRFSlKKcBmb1tttO/cEBhSgWX4pkbnnGGp6Sl6MuRiqIoPpLfRuYrYLCxaAGczuv5GMf3w+f2ncu4FeM4nXQasHowyWnJ9KvXTw2MoihKDvGrkTHGzAbWALWNMXHGmHuMMcOMMcNskcXAfmAvMAV40J/6eCKqUhT95/dHREhJTXFuXlksqBjDmg5zztGogVEURck+/+ptZRxv9AN0mtnJOQ8TXCiYce3H8crKV+hXrx+xp2JZPHDxJeenKIqSn+THtjL5PVyWb7i+0b/qj1XpJvodBuaZG54htHSoGhhFUZQc8q81Mq6T/WOWj3H6Fw4sTOQ1kcztO1eHyBRFUS6Rf62RcbylPyZmDKmSCsCdDe/k6wFf039+f8DzvmOKoiiK7/wrjcz4VeNZdmAZ7cLaEVwoGIDa5Wrzxc4vAJxbxCiKoiiXxr/SyDhWlN298G6OJxynYomK7D65m8GNBmsvRlEUJRf51xkZx3sxz9zwDNO2TAMg/kI8XWt1de5Zpr0YRVGU3MEnI2OMKWmMCbCP6xtjbjfGFPavav7B0YvZdWIXxYOKA5CalspjLR/TyX5FUZRcxteezDKgqDHmGuB7YAgw2W9a5TJ13q3Dg99Y73k6JvynbZ7GueRzFA4oTOHAws4wNTCKoii5h69GxojIOawPkE0RkU5AE/+plbv8fvJ3Plj/AS0/asmyA8uYtHYSKWkpAPSt15fbwm+j//z+uruyoihKLuOrkQk2xhQBOgFLbL9U/6jkP9YeXkuHjzvw1e6vnH6fbvuU2uVr64oyRVEUP+CrkZkDHAeqA6vsYbMkv2mVyywdvNR5LB6+JPDKylcAXVGmKIqS2/hkZETkBaAa0EJE0oCzQB9/KpabtAtrR3BgsMew0JKh2otRFEXxEz59ftkYY7CMynXA00B5oBJw2H+q5R4PfvMgSameO15Te03VTycriqL4CV+Hy94EOgA97fN4YJJfNPIDH6z/wGuYTvgriqL4D1+NTDtgIJAIICInAc/jT1cY+vKloiiK//DVyCSJy4dn7BczjX9Uyn1eu+k154uXoSVDkdHC1cWuBuDF5S/qhL+iKIqf8NXI/GaMGYg1PRMKfACs8JdSuc1TrZ4iMSWR0JKhHHj0AAB/PvknoSVDOXPhTD5rpyiKUnDxaeIfeAxrXqYi8AvwFfC4v5TyB6n/zfhaj8PgKIqiKP7BVyNjROQ+4D6nhzEl/aOSoiiKUlDwdbgsxkc/RVEURXGSaU/GGFMIKAwEGGOKcnGyvxRQzM+6KYqiKFc4WfVknsN6u78BcM4+PgvsBGb5VzVFURTlSidTIyMiL4hIAPCBiAS4uNIi8mIe6agoiqJcofi6d9lwfyuiKIqiFDx8/TJmQ2PMGmNMgjEm1eH8rZyiKIpyZePrEuYPgFFY78rcAjyEtX+ZoiiKonjF54+WicgSIEBEjorIKKCzH/VSFEVRCgC+GpkU+/dvY0wjY0w5rA+YKYqiKIpXfB0um2MblleAlUAg8F+/aaUoiqIUCHwyMiLypn34nTGmLNbwmc7JKIqiKJnia08GY0xNoKYjjjEGEVnsL8UURVGUKx9fP788HrgL+B1wLF0WQI2MoiiK4hVfezK3AmEikpDdDIwxtwBvYc3jfCQir7qFVwemAhWAv4FBIhKX3XwURVGUyw9fV5f9AVzIbuLGmEDgPazlzvWAAcaYem5iE4CPRaQhMBZrcYGiKIpSAPC1J/M48LUx5kcgyeEpIu9nEa8ZsFdE9gMYYz4DegI7XGTqAY/ax8uAhT7qpCiKolzm+NqTGYn1VcwIIMp2TX2IVxk45HIeZ/u5sgXoYx/fCoTYy6XTYYwZaoxZb4xZf/z4cR/VVhRFUfITX3syTYDrRESymb7x4OeexhPAu8aYaOBn4DAXX/68GElkMjAZoGnTptnVQ1EURckHfDUyu4HiWN+SyQ5xQFWX8yrAEVcBETkC9AYwxpQA+ojI6WzmoyiKolyG+GpkzgAbjDHfk35O5qks4q0DahljwrB6KLcDd7gKGGPKA3+LSBrwDNZKM0VRFKUA4KuR2WW7bCEiKcaY4cD3WEuYp4rIdmPMWGC9iHwFtAVeMcYI1nDZQ9nNR1EKGsnJycTFxZGUlJS1sKK4ERwcTJUqVQgKCspvVTDZn2bJf5o2bSrr16/PbzUUxW8cOHCAkJAQypUrhzGepjYVxTMiwsmTJ4mPjycsLCxdmDFmg4j4smgr18i0J2OM6Sci84wxD3oK92EJs6IoOSApKYnQ0FA1MEq2McZQrlw5LpdVuFkNl9UH5mEtWXbnyusCKcoVhBoYJadcTnUnUyMjIqPtw0dE5IxrmDGmpN+0UhRFUQoEvr6MGeOjn6IoBYS4uDh69uxJrVq1qFmzJo888ggXLmS9u9TLL7+c7bzOnz/PTTfdREREBHPmzEkXlpaWxsMPP0z9+vVp0KABUVFRHDhwINt5ZIfQ0FBOnDjhs3zbtm2pXbs2ERERRERE0LdvXz9qd2WRqZExxhQYUvpYAAAgAElEQVQyxhQDAowxRY0xxWxXESiWNyoqipIVs2bNIjQ0lICAAEJDQ5k1a9YlpSci9O7dm169erFnzx52797N2bNnee6557KMmxMjs2nTJpKTk9m8eTO33XZburA5c+Zw5MgRtm7dym+//caCBQsoXbp0tvPwN7NmzWLz5s1s3ryZ+fPnZwhPSUnJ9NwbvspdrmTVk3kO6wXMBsA5+/gssBO4tFqsKEquMGvWLIYOHcrBgwcREQ4ePMjQoUMvydAsXbqU4OBghgwZAkBgYCATJ05k6tSpJCQkMH36dIYPH+6U79atGzExMYwcOZLExEQiIiIYOHBghnT//vtvevXqRcOGDWnRogVbt27l2LFjDBo0iM2bNxMREcG+ffvSxTl69CgVK1YkIMBqrqpUqUKZMmUAeOCBB2jatCnh4eGMHj3aGSc0NJRnn32Wli1b0rRpUzZu3EinTp2oWbMmH374IQAxMTG0adOGW2+9lXr16jFs2DDS0tIy6Dxz5kyaNWtGREQE999/P6mpqRlkvBEdHc1jjz1Gu3btePrppxkzZgxDhw6lY8eODB48mKSkJIYMGUKDBg2IjIxk2bJlAEyfPp1+/frRvXt3Onbs6HN+lyNZzcm8ALxgjHlXRIZnJqsoin8YMWIEmzdv9hq+du1azp8/n84vISGBe+65hylTpniMExERwaRJk7ymuX37dpo0aZLOr2TJklSrVo29e/d6jffqq6/y7rvvetV39OjRREZGsnDhQpYuXcrgwYPZvHkzH330ERMmTGDRokUZ4vTv358bbriBFStW0KFDBwYNGkRkZCQA48aNo2zZsqSmptKhQwe2bt1Kw4YNAahatSpr1qzh0UcfJTo6mlWrVpGUlER4eDjDhg0D4Ndff2XHjh1Ur16dW265hS+++CLdUNfOnTuZM2cOq1atIigoiAcffJBZs2YxePDgDHoOHDiQokWLAnDzzTfz+uuvA7B7925++uknAgMDGTNmDBs2bGDlypUULVqUN954A4DffvuNXbt20bFjR3bv3g3AmjVr2Lp1K2XLlvVa3lcCvr6M+awxJkBE0owx9bFWnX0hItne/l9RlNzF3cBk5e8LIuJxhZI3f19ZuXIln3/+OQDt27fn5MmTnD6d+S5SVapU4ffff2fp0qUsXbqUDh06MG/ePDp06MDcuXOZPHkyKSkpHD16lB07djiNTI8ePQBo0KABZ8+eJSQkhJCQEIKDgzl16hQAzZo1o0aNGgAMGDCAlStXpjMyS5YsYcOGDURFWQtsExMTueqqqzzqOWvWLJo2zfgKSr9+/QgMDHSe9+jRw2mMVq5cyX/+8x8A6tSpQ/Xq1Z1G5uabb77iDQz4bmSWAW2MMSFYb+9vA24Bov2kl6IoNpn1OMAaGjp48GAG/+rVqxMTE5OjPMPDw53GwMGZM2c4dOgQNWvWZMuWLemGlrztTPDee+85e1OLFy/G08vf7kbrl19+4f777wdg7Nix9OjRgyJFitC5c2c6d+7M1VdfzcKFC6lRowYTJkxg3bp1lClThujo6HR6FClSBICAgADnsePcMc/hnrf7uYhw11138corOf/MVfHixb2eZ/YyvHu8KxVfV5cZETkHdAOmiEgnrJ2ZFUXJZ8aNG0exYunX4RQrVoxx48blOM0OHTqQkJDAxx9/DEBqaiqPP/440dHRFCtWjNDQUDZv3kxaWhqHDh3i119/dcYNCgoiOTkZgIceesg5GV6pUiXatGnjnCuKiYmhfPnylCyZ/m2I5s2bO+P06NGDjRs3cuSIta9uWloaW7dupXr16pw5c4bixYtTqlQp/vrrL7799ttsX+evv/7KgQMHSEtLY86cOdxwww0ZymH+/PkcO3YMsOaUPBn0nOJaHrt37+aPP/6gdu3auZb+5YCvPZlgY0wRoBPwtu3n++yXoih+wzHB/txzz/HHH39QrVo1xo0b53Hi3VeMMSxYsIAHH3yQF198kbS0NLp06eJcOdaqVSvCwsJo0KAB9evXp3Hjxs64Q4cOpWHDhjRu3DjD4oMxY8YwZMgQGjZsSLFixZgxY0aWuhw7doz77rvPOfzXrFkzhg8fTnBwMJGRkYSHh1OjRg1atWqV7ets2bIlI0eO5LfffnMuAnClXr16vPTSS3Ts2JG0tDSCgoJ47733qF69eoa0XOdkypcvz08//ZRl/g8++CDDhg2jQYMGFCpUiOnTp6frdRUEfNq7zBgzGuvrmLuAFsBVwEIRaeFf9Tyje5cpBZ2dO3dSt27d/FajQBMTE+N1sUFBwFMdyo+9y3waLrNXmVUDWthb8p/l4tcsFUVRFMUjWb2M2dvltJptYBCRs0BXfyqmKIriT9q2bVtgezGXE1n1ZEa5HE93CxuWu6ooiqIoBY2sjIzxcuzpXFEURVHSkZWRES/Hns4VRVEUJR1ZLWGu4PLBsgpuHy8r7yedFEVRlAJCVj2Zn7A+WBbldhwFLPGvaoqi5Bdt27bl+++/T+c3adIkHnzQes7cvXs3Xbp04dprr6Vu3br079+fv/76K518bGwsRYsWJSIiItMNKC+FMWPGMGHCBMDaVNLx0qZy+ZCpkRGRIZm5vFJSURTvjF81nmUHlqXzW3ZgGeNXjc9xmgMGDOCzzz5L5/fZZ58xYMAAkpKS6Nq1Kw888AB79+5l586dPPDAAx4/91uzZk02b97M1q1b2bFjBwsXLsyxTlmhRubyxNdtZRRFuUyJqhRF//n9nYZm2YFl9J/fn6hKnr6a7ht9+/Zl0aJFzrfsY2NjOXLkCDfccAOffvopLVu2pHv37k75du3aUb9+fa/pFSpUiOuvv965g/Prr79OVFQUDRs2dG7RHxsbS926dbnvvvsIDw+nY8eOJCYmAjBlyhSioqJo1KgRffr0ISEhIV368+fPZ/369QwcOJCIiAi++eabdG/v//jjj/Tu3Rsl7/F1WxlFUfKJEd+NYPOf3rf6B6gUUolOMztRMaQiR+OPUrdCXV5Y/gIvLH/Bo3zENRFMusX7xpvlypWjWbNmfPfdd/Ts2ZPPPvuM2267DWMM27Zty/AZgKxISEhgyZIljB07lh9++IE9e/bw66+/IiL06NGDn3/+mWrVqrFnzx5mz57NlClT6N+/P59//jmDBg2id+/e3HfffQCMGjWK//3vf87di8Eyiu+++y4TJkygadOmiAiPP/44x48fp0KFCkybNs35bRwlb9GejKIUAMoEl6FiSEX+OP0HFUMqUia4zCWn6Tpk5hgqyy779u0jIiKCVq1a0bVrVzp37swPP/zADz/8QGRkJI0bN2bXrl3s2bMHgLCwMCIiIgBo0qQJsbGxAGzbto3WrVvToEEDZs2axfbt2zPN1xjDnXfeycyZMzl16hRr1qyhc+fO2dZfuXS0J6MolzmZ9TgcOIbInm/zPB+s/4DRN46mXVi7S8q3V69ePPbYY2zcuJHExETnJpjh4eEsX77cpzQcczKuiAjPPPOMczt/B7Gxsek2hwwMDHQOl0VHR7Nw4UIaNWrE9OnTffqEwZAhQ+jevTvBwcH069ePQoW0ucsPfOrJGGNqG2M+McasNsb86nD+Vk5RlKxxGJi5fecytt1Y5vadm26OJqeUKFGCtm3bcvfdd6frxdxxxx2sXr2ab775xun33Xff8dtvv/mUbqdOnZg6dSpnz54F4PDhw86t9L0RHx9PxYoVSU5O9vpZ6ZCQEOLj453nlSpVolKlSrz00ktER0f7pJuS+/hq2j8D5gHT0C3+FeWyYt2RdcztO9fZc2kX1o65feey7si6S+7NDBgwgN69e6dbaVa0aFEWLVrEiBEjGDFiBEFBQTRs2JC33nrLpzQ7duzIzp07admyJWAZs5kzZ6b7eqQ7L774Is2bN6d69eo0aNAgnTFxEB0dzbBhwyhatChr1qyhaNGiDBw4kOPHj1OvXr1sXrmSW/i61f8WEWmUB/r4hG71rxR0dKv/3GH48OFERkZyzz335Lcqec4VtdU/sMYY09CvmiiKouQiTZo0YevWrQwaNCi/VflX4+twWXNgiDHmd8D5EW0RaeYXrRRFUS6RDRs25LcKCr4bmRF+1UJRFEUpkPhkZERkOYAxprh9fs6fSimKoigFA1+XMNcwxqwFTgIn7KXMNfyrmqIoinKl4+vE//8Bk4GiQDFgiu2nKIqiKF7x1chUEJGpcpFpQAVfIhpjbjHG/G6M2WuMGekhvJoxZpkxZpMxZqsxpkt2LkBRFP9QokQJj/6TJ0+mTp061KlTh2bNmrFy5Upn2IULFxgxYgQ1a9akVq1a9OzZk7i4OK95fPvttzRt2pS6detSp04dnnjiCWJiYpzv0DhISUnh6quv5ujRo7lzcUqe4auRSTPG1HacGGOuw4eXMo0xgcB7QGegHjDAGOP+VtQoYK6IRAK3A+/7qJOiKHnMokWL+L//+z9WrlzJrl27+PDDD7njjjv4888/AXj22WeJj49n9+7d7Nmzh169etG7d288vY+3bds2hg8fzsyZM9m5cyfbtm2jRo0atGnThri4OOe+ZQA//fQT9evXp2LFinl1qUou4auReRZYYYz5wRjzPbACeMaHeM2AvSKyX0QuYO0c0NNNRoCS9nEpQD8IoSg5Yc0aeOUV69dPvPbaa7z++uuUL299GLdx48bcddddvPfeeyQkJDBt2jQmTpzofHt/yJAhFClShKVLl2ZIa/z48Tz33HPUqVMHsD4H8OCDDxIQEEC/fv2YM2eOUzanG3Qq+Y+vq8u+M8bUxzIaBlgjIid8iFoZOORyHof1zo0rY4AfjDH/AYoDN3lKyBgzFBgKUK1aNV/UVpSCwYgRsDnzrf45fRq2boW0NAgIgIYNoVQp7/IRETAp64033dm+fXuGbf6bNm3KjBkz2Lt3L9WqVaNkyZIZwrdv306HDh3S+W/bto3HH3/cYz4DBgxg6NChPP3005w/f57FixczceLEbOur5D8+b/UvIsdEZJGIfO2jgQHLIGVIyu18ADBdRKoAXYBPjDEZ9BKRySLSVESaVqjg03SQovx7OH3aMjBg/Z4+nWdZiwjGGOevt/DsEBUVxdmzZ/n999/59ttvadGiBWXKXPrnC5S8J9OejDFmiYh0MMYcJ71xMICIyFVZpB8HVHU5r0LG4bB7gFuwElxjjAkGygOZb8uqKP8WfOlxrFkDHTrAhQtQuDDMmgVuk+e5Qb169diwYQPt27d3+m3cuJF69epx7bXXcvDgQeLj4wkJCUkX3r17d9577z2mTJkCwOLFiwkPD2fDhg00auR5W8Tbb7+dzz77jJ07d+pQ2ZWMiHh1QEX7t7onl1lcO14hYD8QBhQGtgDhbjLfAtH2cV0sI2QyS7dJkyaiKAWZHTt2ZD/S6tUiL79s/eYCxYsXz+D35ZdfStOmTeXEiRMiIrJp0yapWrWqHDlyREREHn30Ubn33nslJSVFRERmzJghTZo0kbS0tAxpbdmyRWrWrCm///67iIikpqbKG2+84Qzfvn27XHvttVKhQgU5e/ZsrlzTvwlPdQhYL1m027ntMu3JiIhjveBtIjLeNcwY8xQwPmOsdPFTjDHDge+BQGCqiGw3xoy1L/Yr4HFgijHmUazeUrRdGIqiZIeWLXO195KQkECVKlWc54899hiPPfYYhw8f5vrrr8cYQ0hICDNnznSu+nrllVd44oknuO666wgICKBOnTosWLDA43BZw4YNmTRpEgMGDCAhIQFjDF27dnWG16tXj2LFitGkSROKFy+ea9el5C2+bvW/UUQaZ+WXV+hW/0pBR7f6Vy6Vy2Wr/6zmZG4GOgKVjDGuvZZMlq0oiqIoikVWq8suAGexhrHOubhdQG//qqYoiqJc6WQ1J7McWG6M+VxEtuWRToqiKEoBwdeXMbcZYzoCEUCwi/9YfymmKIqiXPn4ZGSMMa8CUUA48CXW1jA/+VEvRVEUpQDg6xv/XYFOwF8icj/QBPC8RauiKIqi2PhqZJJEJAUQY0yQiBwm/Zv8iqIUMAIDA4mIiKB+/fr069ePhISEPM1/0qRJPuXpLtelSxdOnTp1yfmPGTOGypUrExER4XTe0j116hTvv39xA/kjR47Qt2/fS9YBYOHChezYsSNX0soPfDUy8caYYsBqYIYx5g0gxX9qKYqS3xQtWpTNmzezbds2ChcuzIcffuhz3NTULL8EkiU5NTKLFy+mdOnSl5w/wKOPPsrmzZudzlu67kamUqVKzJ8/P1d0+LcYmQFY3495AtiBtaS5n7+UUhTl8qJ169bs3bsXgJkzZ9KsWTMiIiK4//77nQalRIkS/Pe//6V58+asWbOG0NBQnn32WVq2bEnTpk3ZuHEjnTp1ombNmk6DFRMTQ7du3Zz5DB8+nOnTp/P2229z5MgR2rVrR7t27QB44IEHaNq0KeHh4YwePRrAo1xoaCgnTlh7+L755pvUr1+f+vXrM8neAy42Npa6dety3333ER4eTseOHUlMTPS5LLZv3+68/oYNG7Jnzx5GjhzJvn37iIiI4MknnyQ2Npb69esDMH36dHr16kX37t0JCwvj3Xff5c033yQyMpIWLVrw999/AzBlyhSioqJo1KgRffr0ISEhgdWrV/PVV1/x5JNPEhERwb59+9i3bx+33HILTZo0oXXr1uzatStnf2pekdf72OSG073LlIKO675TjzwicuONvjnwTe6RR7LWwbF3WXJysvTo0UPef/992bFjh3Tr1k0uXLggIiIPPPCAzJgxQ0REAJkzZ44zfvXq1eX9998XEZERI0ZIgwYN5MyZM3Ls2DGpUKGCiIgsW7ZMunbt6ozz0EMPybRp05zxjx8/7gw7efKkiIikpKTIjTfeKFu2bPEo5zhfv3691K9fX86ePSvx8fFSr1492bhxoxw4cEACAwNl06ZNIiLSr18/+eSTTzJc/+jRo6VSpUrSqFEjadSokbRt21ZERIYPHy4zZ84UEZHz589LQkKCHDhwQMLDw51xXc+nTZsmNWvWdF57yZIl5YMPPnCWy8SJE0VEnPvBiYg899xz8vbbb4uIyF133SXz5s1zhrVv3152794tIiJr166Vdu3aZdBd5ArZu8wYs46MW/O7GqhmuWzzFEW5BG68MffSSkxMJCIiArB6Mvfccw+TJ09mw4YNREVFOWWuusrajD0wMJA+ffqkS6NHjx4ANGjQgLNnzxISEkJISAjBwcHZnjeZO3cukydPJiUlhaNHj7Jjxw4aNmzoVX7lypXceuutzn3PevfuzYoVK+jRowdhYWHOa2vSpEm6r3C68uijj/LEE0+k82vZsiXjxo0jLi6O3r17U6tWrSx1b9eunfPaS5UqRffu3QGrXLZu3QpY39cZNWoUp06d4uzZs3Tq1ClDOmfPnmX16tX063dxIOn8+fNZ5p+fZLWE2VG6XYE6wP/s8yHARn8ppSjKRXLwbbFcwTEn44qIcNddd/HKK69kkA8ODnZ+EdNBkSJFAAgICHAeO85TUlIoVKgQaY7v4ABJSUkedTlw4AATJkxg3bp1lClThujoaK+yrrp6w1WXwMDAbA2X3XHHHTRv3pxvvvmGTp068dFHH1GjRo1M47hfu2u5pKRY09vR0dEsXLiQRo0aMX36dGJiYjKkk5aWRunSpTP8L5czmc7JiMhysd76jwJ6isiXIvIl0AdolxcKKopy+dChQwfmz5/PsWPW557+/vtvDh48mOP0qlevzo4dOzh//jynT59myZIlzrCQkBDi4+MBOHPmDMWLF6dUqVL89ddffPvttx7lXGnTpg0LFy4kISGBc+fOsWDBAlq3bp1jXR3s37+fGjVq8PDDD9OjRw+2bt3qVYfsEB8fT8WKFUlOTmbWrFlOf9e0S5YsSVhYGPPmzQMsQ7ply5ZLytff+DrxXwWXN/2BIlifVlYU5V9EvXr1eOmll+jYsSMNGzbk5ptv5ujRo1lH9ELVqlXp378/DRs2ZODAgURGRjrDhg4dSufOnWnXrh2NGjUiMjKS8PBw7r77blq1auVRzpXGjRsTHR1Ns2bNaN68Offee2+69H1h4sSJ6ZYwx8bGMmfOHOrXr09ERAS7du1i8ODBlCtXjlatWlG/fn2efPLJHJXFiy++SPPmzbn55pupU6eO0//222/n9ddfJzIykn379jFr1iz+97//0ahRI8LDw/nyyy9zlF9e4etW/y8B3YA5tld/YJGIPO9H3byiW/0rBR3d6l+5VK6Irf4diMgoY8wvQFusTy+PEpFv/KmYoiiKcuXjk5EBEJGvga/9qIuiKIpSwMhqCfNrIvK0MWYeHpYyi0h/v2mmKIqiXPFk1ZNZaf8u8rciiqIoSsEjq4+WfW3/zsgbdRRFUZSCRFbDZeMzCxeRp3JXHUVRFKUgkdV7MueycIqiFFAcW/27viMSExNDqVKliIyMpG7durzwwgsATn9X+Z9+0u8aKlkPl72QV4ooipIzVq26huTkvzL4BwVdTatWf+Y4XU/bysTGxtK6dWsWLVrEuXPniIiIcO6i7PBXFFd8XsJsjOkIRODy5r+IjPWHUoqi+I4nA5OZf25RvHhxmjRpwr59+5ybZCqKOz4ZGWPMq1j7l4UDXwI9Ae0LK0oesGfPCM6ezdmGiJs2tfXoX6JEBLVqZb7zpusuzGFhYSxYsCBd+MmTJ1m7di3PP/88x48fZ8WKFU55gM8//5yaNWvmSG+l4OBrT6YrEAlsEJH7jTFjgff8p5aiKPmNp+EygBUrVhAZGUlAQAAjR44kPDycmJgYHS5TPOKrkUkSkRRjjBhjgkTksDGmql81UxQFIMseR0yM8RoWGRmTy9ro3IuSPXw1MvHGmGLAamCGMeYokOI/tRRFUZSCgK9b/Q/AMipPADuwtpjpl2kMRVHyhKCgq7Pl7y8cczION3/+/DzNX7k8yXSrf2PMKGCaiBzOO5WyRrf6Vwo6utW/cqlcLlv9Z9WTuRbYYYz5zhjT3xgTlBdKKYqiKAWDrD6/HA1UwvpY2XDgiDHmbWNMRGbxFEVRFAV8mJMRkXMiMk1E2gAtgDPAl8aYjb5kYIy5xRjzuzFmrzFmpIfwicaYzbbbbYw5le2rUJQCiC9frVUUT1xOdcfnN/5tCgNF7N8LWQkbYwKx3qe5GYgD1hljvhKRHQ4ZEXnURf4/WO/jKMq/muDgYE6ePEm5cuUwxvsSZUVxR0Q4efIkwcHBWQvnAVkaGWNMSazVZXcD1YGZQAdXQ5EJzYC9IrLfTuszrN0CvMUdAIz2IV1FKdBUqVKFuLg4jh8/nt+qKFcgwcHBVKlSJb/VALLe6v9ToDuwDHgV+FpEsvN+TGXgkMt5HNDcS17VgTBgaTbSV5QCSVBQEGFhYfmthqJcMln1ZDYDj4pITnfa89TP9zZYeDswX0RSPSZkzFBgKEC1atVyqI6iKIqSl2Q18R8NfGKMeSiH28jEAa7xqgBHvMjeDsz2lpCITBaRpiLStEKFCjlQRVEURclrslrCXA94FCgFzDfGrDfG/NcY08jH9NcBtYwxYcaYwliG5Ct3IWNMbaAMsCZb2iuKoiiXNb4sYd4uIi+LSHOgB3AceM1elvxWFnFTsN6v+R7YCcwVke3GmLHGmB4uogOAz+RyWnenKIqiXDKZbivjFDKmjojscvNrANQTkTn+Us4buq2MoihK9rkct5Vx8KkHvxn5YWAURVGUK4esljCXB64Cgo0xdbm4WqwUUNzPuimKoihXOFktYR4IjMDav2yxi/9pYLy/lFIURVEKBpkaGRF5C3jLGPOsiLycRzopiqIoBQRf9y5baIyp5+7p49YyiqIoyr8UX43MNy7HwcDVwEGsbWAURVEUxSM+GRkRSWdMjDEdgFv8opGiKIpSYPB1CXM6RGQJ0CqXdVEURVEKGD71ZNzmYwKAKKxlzIqiKIrilZzMyaQAe4G7cl8dRVEUpSCRozkZRVEURfEFnz+/bIzpBNyE9T2YH0XkR79ppSiKohQIfJr4N8Y8BbwBnMJ62/9NY8wT/lRMURRFufLxtSczCGgpIvEAxpi3gVXABH8ppig5QQRSUyEtzfr15Tgt7aLL6tzduYd37Qpffmnp4fDzdjx4MEyb5jnc9fc//4FJky6ee5IRgeeeg7FjM/q7n7/6Kjz5ZPpwT3HefRcefDCjnLubOhWiozP6O/4Ph5s9G267LXMZh/vyS+jRI+s0v/8eOnb0HOZ6HhMDN96Y0d89zpo10KJF5nIAGzZA48bpwzz9bt0K9et7l3Ec79oFtWt7T8f1eN++9P6XO75u9f+biDTIyi+v0K3+c5eUFEhMhKQkOH8+a9evn9U4XrgAycmZ/77zDtx7r5VHSorl7+l32TJo2dJqtB2yKSkZzw8fhgoVLvo7DIar4VDSExAAxlx07ufe/HLDwaXJeApz93M99zUst+L4+pvTMNdfx/GIEdA0h5v158dW/74amalYOzBPwZqTuRcIEJEh/lXPM/9WIyMCZ8/C339b7swZiI+3/By/rseO34UL4YYbLEOSkGD9urrk5NzVMygIChe2foOCoFAhyzmO3X8dx4GBF89djx3nDj/XX2+uUCGr4XScZ3UcEHDReTo3JqO/pziOBtvhXM89HWfm52oA3M89GQZP/oriSn4YGUQkS4e1rf+rwHpgg31c3Je4/nBNmjSRgsLp0yI7doj88IPI1KkiY8daneg77xTp2lWkZUuR2rVFKlQQKVQoq4ELyxUpIlK+vEhoqEiDBiLNm4u0ayfSpYtInz4igwaJDB0q8sgjIiNHirzwgsj48SLvvCMyebLIjBkin30msmCByOLFIkuWiKxcKbJuncjWrSK7dons3y8SFyfy118i//wjcu6cyIULImlp+V2iVwirV4u8/LL1e4lyM2fOlFuvuUaeAbn1mmtk5syZV6TclaDjlVA2mQGslzxur33qyWSFMWaUiLx0yQn5yJXUk0lMhN9+s8Zm77vPGncsHmUAAB1XSURBVLs+fBji4iwXH58xTsmSULas5cqUuXjsel6mDJQuDSVKQEjIxd/ixa1egT+YNWsWnz/xBHX+/JNd11xDnwkTGDhw4CXJ5rmcPfkxe+ZMFj79NHWOHWPvVVfR88UX6d+3b4YJlwWff863L75IrRMn2F++PJ1HjqRHt24ZJlgWL1rEkjfeoMbff3OwbFnaP/wwt3Ts6HFSZt20aTT85BMKYb10tu2OO2gycGCGSZdNs2cTPncugUAqsKNPHyL69k0ns3rVKpZ/9BGPpaQ403urUCHa3HUXLZo3d6b56y+/sPqTT3gwNdWZ3uTAQFoMGEBTx+SCCBs3bODXuXO520VuRmAgUb17E9GokVNuy5YtbFi4kDtd5GYHBhLZrRsN6td3Thps++03tixeTH8XuS8CA2nQqRP16tZNNzGxc+dOtv/4Iz1dZL8ODCS8fXtq16rllNu9eze7YmLobMulAd8GBFC3TRuurVnT+p9F2LtvH7tXrKBjWhoBttySgABqXX89NcLCnOkd2L+ffWvX0tZFbnlAADWbNSO0enVnegcPHmT/unW0dpFbFRBAWJMmVKta1Xkthw4d4uCmTbRwkfslIIDqjRpRpXJlZ3qHDx/m0JYtRIlgsIaI1htD1QYNqFSxolMO4MjRoxzeto3GIghwAehapAj3/u9/Xu9BT1y2w2VZJmLMRhFpnAv6+MTlamROn4bNm2HjRti0yXI7d1pzBWANY1SsCFWqQOXK1q/DOc4rVYLgYB8yW7PGms1s29aazPCAzw3zJ5/w9RNPEH7sGAcqVKDHc8/Rq3Nna1Ll/Hm4cIEfFy3ix9df58XkZGdj9mahQrSLjqZFkybWmJs9EbNl/Xq2LFzIALfGosGNN1IrLMw5wXJw715i162jlX0zCrDZGCrXqcM15co5J1pOHjvGP7Gx1ADnzXgUKFW+PCWKFnXKJSUkcCE+nhCXa7sABAUGEuBo6BW/kgbOsToB0tLSMEAgF/+7ZNsFBQU55TCG8+fPUxgo4iKbCCQZQ3BwMGLLJiYmEixCCRe5eOCcMRQrfvFbivHnzlFChFIucqeA08YQEmLVEjGG+DNnKClCWRe5E8ApYyhZqhSOFvL06dP/3965x0lVXfn+u6pfNDTQjSB00zSgI1Hjo0UQHV94NUYzN6LGV0y8JiYyavSSMRk/GOfjdQxxTMzNTZw4KFGiwQdRNErG4OiAooSHAvISRYHwfksjj27o7qp1/9i7qk83daqroYvqotf386lP7X3Or85Z++yqvWqvfc7elKnSJ6DbCuyMROjZs2fieDW7dtE7FqMfzb+v2yIRykpLUR/H3FlTw7GxGP0Dug3AlkiEXmVlTqeKirBz5076xWJUeW0DcD/wwsCBrFmzJu36yWUn86GqntEO9qRFR3Ey69bBwIHwjW84h7J6ddO+8nI44wx3B8oZZ0B1NVRVubGCMFp1CnV1UFPDrHHjGD5+PPm4BnzFqFGcesklbgBm3z7Yu5eVixax9t13Gel/5AqsAvpWVNCzsDAx0h/dt4+8xsaMXB+l6cdTi2sEirp1IypCVISaPXvooUrfgG4tsCESoUdZGVGcM9teU8PAWIyTIPHPcDGwTIQepaU0qtII1HzxBaepcrbXRYH3gDlAYXExUVUaVak7cIDzcA99xZ3gNOBNf+zg66vAlQHdFOBVb2tQdxVwo9c1ApOAyS108fRJwK8h4azvBJb4/RrQngJMAApwjcotwKIWGgWq/fnium/iYtpBjQJnevvzve4q4P0WGsXNGfWfgeNdDsxNojsbeCugu9jrWnI2MD0NXVu0YbqCggJEBPFOK0xXUlKC+MZ+z549obpevXoBICJ8/vnnobq+ffsmjrdly5ZQXWVlZUK3fv36UN3gwYMTOhFh1apVSbXzRIi14c9Thx2Tae0FLDySMb5sjsnEYqozZ7qxjUjE9Y+PP171mmtc6HzaNNUtW5p/JjSWumeP6ooVqm+/rbNuv13H5+VpPWgUtBF0iYjuqqhQ7dfPDbSkMSATFdG6wkLdJKLbQOMBmCjoctApIjq1tFRf6NFDnywp0YdB3/LnU//+EuiNoNeAfh30UtCRoLeC1oE2gNaCfgN0AGg/0GNAe4B2Ac0DPRt0H2i9fz/bt03dunXTnj176jHHHKOk0FVXV+uwYcN0xIgRKXWjRo3Sq6++Wq+99tqUujvuuEPvuusuHTNmTErdgw8+qOPGjdOHHnoope6xxx7T8ePH6xNPPJFS98ILL+jkyZP1xRdf1ClTpqiIJPRjAzoR0WnTpukbb7yhb775ZkrdrFmzdPbs2Tpnzhzt169fUl15ebkuXrxYlyxZokuXLtXy8vKkuoqKCv3ss8905cqVumrVKq2oqEiq69+/v27cuFE3bdqkmzdv1srKyqS6yspK3blzp9bU1OiuXbt0wIABSXUDBgzQ2tparaur0/379+uBAwe0qqoqqbaqqqrZ72ngwIFJdQMHDjwqdW3VpoIcHpM5qsNl8R7LxInw6KMuJFZWBqNHu+cIqqpCPhiNMvVXv2LRT37CvY2N5OG+GeuBiuJiCuvqDvpI8N//KmCxCPXdurG1vp5t9fXsBHoD/0LTP9Jv4f6x7wX2A5FIhFgsFvov6aqrrqJLly4UFxczceLEUN1TTz1FYWEhhYWF3HnnnWzfvp2zgZHAO15TXl7OjBkzKCgooKCggMLCQoYPH86GDRsO0g5s0bUfNGgQa9eu7TS6oLYlYcdsTTd9eil5eV8cpItGe3LxxbtyRpcLNubCtWmNbPRk0p5WphWOypsla2rcIHucW25xD1ZNmAB5eZP5z/v+icd/7kJb148bx/Unn+w80OLFsGgRunQpV9TWcgVNziOGix8/XlfHRki8NgGVwJ9pauxvAuaq8u0rr6SsrIyysjKeePBBwDmFkTQ1aCLCtm3bKCkpoaioiMGDBzN37Vou5uCG75VXXkmUafr06aG6W265JaFraGhg9OjRzK2tTYQvunbtyiOPPMKJJ57Y7Lo9/PDDSbU/+9nPmumeemoXeXlN+a/692h011GpS6YlRJuuLlnDk2x7R9flgo25cG06IimdjIhUqeq6kH1DVXWhz17a7pZlEVXnTJa3WFx6+nS46CJ4/vnneOqWW/hLfT2FAFu2uCcOPfsKC/mkqIg59fXMxzmWx2lyHqNxDfm4ceM4saKCiooKvvvd7zJj8+akjf2kSZMSx37mmWdYu3Ytc2keq66qqqJ3796JfHs3kIMG/YjXX69tcaVqKSj4Ea4v1XZtR/9xp9LFYo1AFNVYSt2BA5tQdaMyrWn37l2S0KbS7do10+tSRyF27Jialm7Llme9JrVu48bH09KtW/fLtHRr1jzoU4mweyirV9/rNal1n332w7TOvWLFba1qAD755LtpnXf58htJpxzLll2T1nmXLr0ioWuPaFM2SRkuC4bBROR9VT0r2b4jTSbDZXv3uluBg1zaewRn7njf9Vh++lM+uuce7qqpoY/fH8P1LP4dNyC9r1cvTq+u5rTTTmPSpEmJAcORhIdSnnvuOUaPHk1tbVPj3LVrVyZMmNBs8D/dbvM774R3LkeO1HbVnXPORmKxelTricUOMH/+aaHak056HtUGVBtYseL7obqqqrGoNhKLNbBx429Cdb17X4VqI6qN7Nw5LVRXUlKNahTVKLW1y0N1+fnH4BxClGh0d6jOOLKIFOBiAYLqgVBdXl4PwA38NzaGh5EKCo5N6Orrt4TqiorisXDhwIGDw5dxiotPSOjq6j4N1XXt+mWnEmHfvmWhupKSauLlBdi7d2GoNvg7bY2OGC4Ltiwtn7446kJkGzbAgAEuXVZWy8m1J3L3gfWM2uEKG92yhQO33sr1wDKgB+4upvjthPNEWL9+PRUVFYk7Q4YNG5ZW6Ki9/v3HYgdobEzdOK5b9wixWC3R6L6Uug8+OI1Y7ACx2P6Uujlz+qfcH+Tjj29MS7d+/a8QyUck9Ve0rm4lInmt6oqKqrwuL6WTOfbYGxCJIJLHhg2/DtUNGvTThG716rGhuiFDngAiiESACCtWhE+S8eUvv5zQLls2KlR3+ukzcA1khEWLLgzVnXnmAuIN1YIF4Td/nnXWp4njzZt3fKjunHM2JXSzZ/cN1Z133m7iDfh775WE6i68sJGmZkSYOTN8vt4LL6xPpFP94Tn//C/S0p177ta0dOecszYt3YgRn6alO+usZWnphg37sFk+lbaj05qT0ZB0snxOs2YNDB4MBQV1XNxwJY/UvMlJuFtS40SAj4B/xN36eVDvpKqK/v2bN7jpOo+Ghq0ko6FhK59//joNDTtoaNiRsgwzZxahWp9SA7B69T2uPJHUD+QUFx9PJNKFSKQLW7Y8HaobMuRxRIqIRAoRKWT58mtDtcOHf0wkUoBIAXPnDgzVXXhh07/VVD+w4cOXpKU79dTX0tINGfLbRDq1k/mXRDqVk6moGN0sn8rJ9Olzdei+IGVlF6Wl6949vUBD164ntC4CiorK09Ll53dvXQSIJBl0Mo46WnMyXUTkJNzfjWAaIJ1HBnOCHTucg8nP38OtDRfxWxYc1E1T3DMNt+McTNeuXVvtnUBq57Fz51vs37+G/fvDu+EAS5f+z7TKUVn5Q/Lze5CX15OVK+8K1Z133h7y8ooRyUvZ4J5yyp8S6VROpqLiH5vlW45lBenW7cTwnUabKSjom/Q7VlDQN6d0uWBjLlybjkhrTqYr8JdAPpg+Knoy0aibfjwvr4HLGi/j31s4mHghG4E7cD2WV1+N0LNn672TWCx1r2LJkvj9Eqn/0Q0dOpeCgt4UFPRm1qzSUN3xx/88kU7lZPLzw0MYR5KO/uPOhYb03HPDxxNySZfNc3d0XVu1HY2UTkZVBx0hO7LGQw/BjBlwAaN5ldkJBxN3LlFgKvAITbcK9+yZ/AnbhoatbNv2Ert3z2H37rns2RM+WAdQXT2TLl0GUVhYwbvvhk841qPHiDaWKjca0o7+486FhtQwOjrt9ZxMTrJ0Kdx/PxzHC8zgaSI0PQgZwz3geC/NbxW+7bbbgPGhx1y+/DoikS507z6Mysq7WL8+fF230tIL2mxzR2/A26o1DOPoptM6GVW47rqt5BPhPe5q5mCiwFsvQ3Ev+LeDPhnuYACGDv2AkpLTiEQKAVI6mSDt7TwMwzA6Ap3Wybz2GnzySV9u4HbK+TypgzkUevRofgu6OQ/DMDozGXcyInIZ8Bvc6PaTqvpwEs11wAO4dn6xqqb3MMUhEovB9763kS7s4Sl+l3AwM6cDEShux3OZ8zAMozOTUScj7kb4x4Cv4JZK+EBEpqrq8oDmBNzQx7mqWiMix2bSJoAxY95j587zGc03KcYt9jLbPQd3WOTC7YSGYRhHkkz3ZM4CVqrqagARmQyMAoJPUtwKPKaqNQCqui2TBkWjMH58H4pZzqO8CMBfX4aGNobH2jKVg2EYRmflMP+7t0p/3Mz2cTb4bUGGAENE5K8iMteH1w5CREaLyHwRmb99+/ZDMuYvf3GLhkWjJ3IzD1BIjNmH4GAMwzCM9Mi0k0n2OHnLLkA+cAJuhpZvAk+KyEFPHKrqBFUdpqrD+vTp03J3WnwYmA7oN0wBDs3BWFjMMAwjPTLtZDYAAwL5StzSKS01r6lqg6r+DViBczrtzt13A3yPX9CXAtSNw7SBgoK+jBypNphvGIaRJpl2Mh8AJ4jIYBEpBG7APUAf5FXgIgAR6Y0Ln63OhDGvvPIc5zCRMWxrc5isoKCvORfDMIw2ktGBf1VtFJE7gf/C3cI8UVU/EpEHcWtNT/X7LhWR5bjHVP5ZVT/PhD1jxozhVvyyxWk6GHMuhmEYh07KRcs6KoeyaJmI8AVQAsxJsxdjDsYwjKOJjrho2VFB3MHEV7kwB2MYhnFkyPSYTIchPrl9uoP95mAMwzAOn07jZOKk24sxDMMwDp9O42Ra3jcdhoXJDMMw2o9O42QGALPSCJWZgzEMw2g/OoWTefbZZwGIthIqszCZYRhG+9Ip7i771re+5VPfTqmzXoxhGEb70il6MgD9+6d2MNaLMQzDaH86hZN5551k83Q2x3oxhmEY7U+ncDKGYRhGdjAng4XKDMMwMoU5GSxUZhiGkSnMyRiGYRgZw5yMYRiGkTHMyRiGYRgZo1M8jDlyZO6tmWMYhnE0YD0ZwzAMI2OYkzEMwzAyhjkZwzAMI2OYkzEMwzAyhjkZwzAMI2OIau7deSUi24G1h/jx3sCOdjQnm1hZOh5HSznAytJROZyyDFTVPu1pTGvkpJM5HERkvqoOy7Yd7YGVpeNxtJQDrCwdlVwri4XLDMMwjIxhTsYwDMPIGJ3RyUzItgHtiJWl43G0lAOsLB2VnCpLpxuTMQzDMI4cnbEnYxiGYRwhzMkYhmEYGaNTORkRuUxEVojIShEZm2172oKIrBGRpSKySETm+229ROQtEfnMv5dl285kiMhEEdkmIssC25LaLo5HfR0tEZGh2bP8YELK8oCIbPR1s0hEvhbYd68vywoR+Wp2rD4YERkgIm+LyMci8pGIjPHbc65eUpQlF+uli4i8LyKLfVn+1W8fLCLzfL38UUQK/fYin1/p9w/Kpv1JUdVO8QLygFXAcUAhsBg4Odt2tcH+NUDvFtt+AYz16bHAz7NtZ4jtFwBDgWWt2Q58DZgGCHA2MC/b9qdRlgeAHyfRnuy/Z0XAYP/9y8t2Gbxt5cBQn+4OfOrtzbl6SVGWXKwXAUp8ugCY56/3i8ANfvvjwO0+fQfwuE/fAPwx22Vo+epMPZmzgJWqulpV64HJwKgs23S4jAKe8elngCuzaEsoqvousLPF5jDbRwF/UMdcoFREyo+Mpa0TUpYwRgGTVfWAqv4NWIn7HmYdVd2sqgt9eg/wMdCfHKyXFGUJoyPXi6rqXp8t8C8F/gcwxW9vWS/x+poCXCwicoTMTYvO5GT6A+sD+Q2k/iJ2NBR4U0QWiMhov62vqm4G90MDjs2adW0nzPZcrac7fRhpYiBsmRNl8SGWM3D/mnO6XlqUBXKwXkQkT0QWAduAt3A9rV2q2uglQXsTZfH7vwCOObIWp6YzOZlk3j2X7t8+V1WHApcDPxCRC7JtUIbIxXoaDxwPVAObgf/rt3f4sohICfAy8ENV3Z1KmmRbRy9LTtaLqkZVtRqoxPWwTkom8+8duizQuZzMBmBAIF8JbMqSLW1GVTf5923An3Bfvq3xkIV/35Y9C9tMmO05V0+qutU3DDHgdzSFXjp0WUSkANcoP6eqr/jNOVkvycqSq/USR1V3Ae/gxmRKRSTf7wramyiL39+T9MO5R4TO5GQ+AE7wd2kU4gbJpmbZprQQkW4i0j2eBi4FluHsv9nLbgZey46Fh0SY7VOB/+XvZjob+CIevumotBibuApXN+DKcoO/A2gwcALw/pG2Lxk+bv8U8LGq/iqwK+fqJawsOVovfUSk1KeLgUtwY0xvA9d4Wct6idfXNcAM9XcBdBiyfefBkXzh7pD5FBfjvC/b9rTB7uNwd8MsBj6K246LvU4HPvPvvbJta4j9L+DCFQ24f17fC7Md1/1/zNfRUmBYtu1PoyyTvK1LcD/68oD+Pl+WFcDl2bY/YNd5uLDKEmCRf30tF+slRVlysV5OAz70Ni8D7vfbj8M5wpXAS0CR397F51f6/cdluwwtXzatjGEYhpExOlO4zDAMwzjCmJMxDMMwMoY5GcMwDCNjmJMxDMMwMoY5GcMwDCNjmJM5yhE3e/MnflbXj0Xk+xk+39MicuchfK5CRN4+jPOOFJFL2+t4Ied4WkQ2+Bl9V4jIw+15/PZCRKpF5Lp2PJ76p+nb63jfEZEprSvbpjU6JuZkOgfXqOrpwLXAf4hIRbYNCiIi+aq6SVUvOozDjMQ9pAq4GRIO83hhPKxuyo+zgOtF5Iq2fFhE8jJgU0uqgUNyMoGnyg2jXTAn04lQ1WVADYHJAEXkHr9+xUIR+bOI9PPbe4rIy74XNF1E/iAiv/T7mvVWwnovInKxiMwRkQ/FrYVzQ2DfOyLykIhMB14TkUEissPv+5o0rQGySETqRWSUiPQTt27IAnFrbfzC608FbsM9kb5IRMYGj+c1l3k7lvjy/J3fPtJ/5gm/b7GIJJsrquW1/AI3i8SXRCRfRP5LROZ7u34vTet9fEdE3hCRSSKyADhVRH4kIh94e+aISHXAThWR+/z+1f4a/pvXLgvaJiI3i1tDZIGIzBCRL4nIMcCDwCW+XI967YjAtVsgIv/gtw8SkR3i1l6ZBaTs6YrIL71ti/11HNjiOHFbPxGRM0Xkd/66zot/tzzx79dib3t/f5xCXxcrRGQGgdmRReRUEXnPf1eXi8gPW6snowOQ7adB7ZXZF24dmlN8+lzcjAGFPv9tYAIQ8fnbcXM/gZtM8Emf7gX8Dfilzz8N3Bk4RyLfIl2GX6cD6It7Qr7M59/BPYWd7/ODgB1J7L8VmI17srkLzdfamAFc5vMPxO1reTzcTMLb8esH4Z7Sn+fTI3FP75/h8/fFr0ESW4JlqwA24qb9EOAYv12APwC3+fx3gL3A8YHj9AmkLwHmBvIK/MCnr/Wf/Qefvwd41qfPB16n6cnvy4G/Bs45JXDMUtxT5OU+X+7rotRfJwWuT/Ed0sB17x3Y/n3clPkEjhO39Z+BXUC1z/8HMC5gXx3wJZ//P3F7gbuAN339dgXmB/Z1D5S3BFgOnJTt35i9Ur+sa9w5mCIigpuR9ip16+kAXAEMAxa63eTjpgoHuAj3g0dVd4rIq4dw3j7ARBE5AWjEOasvAXP9/ue1afrygxC3YuGPgPNVdb+4edseEZG/xzXm/XChoTdasWMEsFhVl/v873Fhw+4+v0JVP/TpucDXUxxrrLhxrUbgF6r63+JCYD8Wkctxi+OVAbWBz8xS1VWB/Jki8hPc9YgBQ1qc44/+fSFuiZHXfX4BcLVPfx04HZjn6078eZPx97jFuaZJ01IjCvwdsAPYj1sUKx0uF5Ef4Br5lu3H3oCtC4ENqrooYPtXAtpZqrrCp5/ETf8C7nv3jKo2AA0i8ixu2hhwTme8iJyOu24VuGvwcZq2G1nAnEzn4BpVXSYi1wLPisgQVd2Ka5jGqerEJJ8RwqcMb6R5qLVLiG48rrdytaqqiHzaQrs3+cfANySPA19V1e1+8924hnSEdzoTUpy72eFIPf35/kA6SurfxcOq+tsW227ENYTnq+oe70CCjiNRTh9GmwJcoKoLxY2PbQyxJwocCLFNgImqen8KWxOnBZao6kHLQ4hbf2WfqrY6v5QPjf0/YLiq/s07++cDkpa2pntdg/WTasGth4AtwHdUtVFE3iS9+jeyiI3JdCJU9SVcKGKs3zQVuEOa1nEv8o07uFlfb/bby2i+iugqYLjfV47795mMUmCNdzBfwf1zbhUfn38ZuElVP21xvM3ewcRXcYyzGzfNeTLmANUicqLP3wx8qG4VxfagFBea2yMiPXFOJ4wuuMY2vmjWHYd4zj/jxqAqIbHQ1Zl+X8trMRs3A3minkRkuEibV1DsAdQDW0QkghsHO1TO9T1ccOGz+J2A04Gb/DhXMc2vZSmw3juYU3AhQ6ODYz2Zzse9wAIR+bmqThKR3sBM395EcLHzxbjB49+LyEe4cZ2/0hRKm4ALwS3GzWo9j+SMxYWlxuJmlV2Spo3fx4XafhtoB/8JeBR4SUQ+xDXS0wOf+ROucVqEW1p7cnyHqm4XkZuA58XdPbUdNx7VXvwBGOWv1UbgPaA4mVBVd4vI/cAHIrIOmHYoJ1TVd0XkPmCqD9cV4mbjXYC7Lj/29TNTVf+3uLvgHhGRX3vtalKHBZOdc6mIvIQb11sHzAQOdfG8mcC/isiXgc+Bm/z2CbiZiD/CjRvNxIX6AMYBk0Tk27g/Ou8e4rmNI4jNwmwkRdwiUHm+19ADmAXcrar/nWXTDMPIIawnY4RRhhsozsOFeJ43B2MYRluxnoxhGIaRMWzg3zAMw8gY5mQMwzCMjGFOxjAMw8gY5mQMwzCMjGFOxjAMw8gY/x9Anz1Aa+tKmwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "yub = 1.5\n",
    "Eouts_f = np.clip(Eouts, 0, yub)\n",
    "Est_loocvs_f = np.clip(Est_loocvs, 0, yub)\n",
    "xxs = [lambdas, lambdas, lambdas, lambdas, lambdas]\n",
    "yys = [Eouts_f, Est_vcs, Est_loocvs_f, Est_perms, Est_fpes]\n",
    "myplot.plt_plot(xxs, yys, 'plot', ['k', 'g', 'r', 'b', 'y'], ['o', 'x', '.', ',', 's'], \n",
    "                ['Out-of-Sample Error', 'VC Penalty', 'LOO-CV', 'Permutation Estimate', \n",
    "                'FPE'], \n",
    "        title = 'Exercise 9.18 (b) E_out/Validation Estimates vs. Regularization', yscale = None, ylb = 0, yub = yub,\n",
    "        xlb = None, xub = None, xlabel = 'Regularization Parameter lambda', ylabel = 'E_out/Validation Estimates',\n",
    "        legends = ['']*6, legendx = None, legendy = None, marker_sizes=[1]*5) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>name</th>\n",
       "      <th>Regret</th>\n",
       "      <th>Avg. Lambda</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Eout</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>42.309662</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Ein</td>\n",
       "      <td>0.016961</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Ecv</td>\n",
       "      <td>0.018275</td>\n",
       "      <td>69.900170</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Eperm</td>\n",
       "      <td>0.021047</td>\n",
       "      <td>81.627672</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Efpe</td>\n",
       "      <td>0.017221</td>\n",
       "      <td>64.018142</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Evc</td>\n",
       "      <td>0.021604</td>\n",
       "      <td>111.009887</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    name    Regret  Avg. Lambda\n",
       "0   Eout  0.000000    42.309662\n",
       "1    Ein  0.016961     0.000000\n",
       "2    Ecv  0.018275    69.900170\n",
       "3  Eperm  0.021047    81.627672\n",
       "4   Efpe  0.017221    64.018142\n",
       "5    Evc  0.021604   111.009887"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#### Exercise 9.18 (d) Lambda Selection\n",
    "names = ['Eout', 'Ein', 'Ecv', 'Eperm', 'Efpe', 'Evc']\n",
    "regs = [0, reg_in, reg_cv, reg_perm, reg_fpe, reg_vc]\n",
    "avg_lambdas = [b_out, b_in, b_cv, b_perm, b_fpe, b_vc]\n",
    "df = pd.DataFrame({'name':names, 'Regret': regs, 'Avg. Lambda': avg_lambdas})\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 9.18 (b)\n",
    "\n",
    "Somehow I can't get the same graph as in Figure 9.8(b), not sure where I did wrong. :(\n",
    "\n",
    "#### Exercise 9.18 (c)\n",
    "\n",
    "The VC estimate is weaker but sufficient. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gEx0dRffKrBhLlvSN0pcsKbYvoy2XYI+IJ9PvWyXdCLwaGPVJgDxG6v150YwZbLn//lFpuwjZ66re2uEVIs0aXcVz7JKOkDS59DNwDlDZNeosd1N6+q7MdN11BXfGrCBLljT+aB3yOXg6A7hb0v3AT4DVEXFrDu1azkrFTKvWuZjJrJFVHOwR8VhEnJx+vSIirqqgrUq7U7iafwxpMRPAaZd5nRmzRlQzpztO2LWLp3fsqP1gHEBE8PSOHUzYtavorgyqbcG8A+vMmFljqZma+ZmPPcYWYNvEiUV3pSITdu1i5mOPFd2NwXV00NbRwa8uSypVfV1Vs8ZRM8E+bu9e2h95pOhuNJ2WZd1w110sXtnLF6/A4W7WAGpmKsYK1NFB25xknRlfmcms/jnYLZGuMzNn9z3M72r3UgRmdczBbgcpXZmpZ6Uv3mFWrxzs9jxTepJwX7PZxUxm9cjBbv2a0uNiJrN65WC3Q8sUM3kpArP64WC3QZUu3rFqnUfuZvXAwW5DMmNpck1VX5nJrPY52G3IWpb1XZnJ8+5mtcvBbsPT0XFg3t3FTGa1ycFuI+JiJrPa5WC3EcsWM3kJYLPa4WC3ikzpSU6JnLZvq0+HNKsRDnbLhYuZzGqHg93yUVbM5HVmzIrjYLdctS3vpm3O0azZ7JG7WVEc7Ja/zk4XM5kVyMFuo8LFTGbFcbDb6EmLmcZPcDGTWTXlFuySxkr6maTv5tWmNYYZS13MZFZNeY7YLwc25tieNZBsMZNH72ajK5dglzQTuAD4ch7tWWMqFTPN2X2PT4c0G0V5jdg/B3wI2H+oDSQtlLRW0tptO3fmtFurR22tyWX3vAyB2eioONglXQhsjYh1A20XESsiYm5EzJ0+aVKlu7V61t23DIGLmczyl8eIfR7wZkmPAyuBsyTdkEO71uDalncfGL17nRmz/FQc7BHRHREzI+I4YAHww4jw0TEbmu6+y+65mMksHz6P3Qo3Y2myDIGLmczykWuwR8SdEXFhnm1ak+jsPFDM5PPdzSrjEbvVlNJFs3tWOtzNRsrBbjWnZZmLmcwq4WC3mpQtZvK8u9nwONitppUu3uFiJrOhc7BbzcsWM/l8d7PBOditLpSKmVat81IEZoNxsFv96E4u3jFt31YvQ2A2AAe71ZeOjgPXVPVBVbP+Odit/qTFTL6uqln/HOxWt1qW9S1F4GImsz4OdqtvnZ2Mn+BiJrMsB7vVvRlLDy5m8ujdmp2D3RpG6bqqS1b5dEhrbg52ayhTelzMZOZgt4aTLWbyKZHWjBzs1pjS66oCLmaypuNgt4bmYiZrRg52a2wuZrIm5GC3ptCyLFlnxsVM1gwc7NY8OjpczGRNwcFuTcXFTNYMHOzWlErFTD0rfVDVGk/FwS5pgqSfSLpf0kOSPplHx8xG25Se5HRIFzNZo8ljxL4HOCsiTgZmA/MlnZ5Du2ajzsVM1ogqDvZI7Ex/HZd+RaXtmlVNppjJl92zRpDLHLuksZLWA1uB2yPi3n62WShpraS123bufH4jZgXLXjTbrJ7lEuwRsS8iZgMzgVdLmtXPNisiYm5EzJ0+aVIeuzXLnYuZrBHkelZMRGwH7gTm59muWTVli5l8vrvVozzOipkuaWr6858AbwR+Xmm7ZoVKL5o9Z/c9DnerO3mM2F8M3CHpAeCnJHPs382hXbNipevMuJjJ6k0eZ8U8EBGnRMSrImJWRPxtHh0zqxXZYibPu1s9cOWp2RBM6UnOd1+8vdfFTFbzHOxmQ9XdV8zk892tljnYzYYjLWbydVWtljnYzUagbXk34ycko3ezWuNgNxuhGUtdzGS1ycFuVoGWZd20zTmaxdt7vRSB1QwHu1mlMtdVdTGT1QIHu1lOWi6e52ImqwkOdrO8dHQcVMzk0bsVxcFulrMpPX3XVfXpkFYEB7vZKHExkxXFwW42WlzMZAVxsJuNsux1Vc2qwcFuVg3dfcVMvb1Fd8YanYPdrEpKxUxrNre7mMlGlYPdrJpczGRV4GA3K4CLmWw0OdjNipAWM42f4GImy5+D3axAM5b2FTN5hUjLi4PdrAaULrvnYibLg4PdrBa4mMly5GA3qyHZYiaHu41UxcEuqVXSHZI2SnpI0uV5dMysaXUnK0SuWudiJhuZPEbse4ElEXEicDrwHkkn5dCuWdOa0uNiJhu5ioM9In4dEfelP+8ANgItlbZr1vQyxUw+392GI9c5dknHAacA9/Zz30JJayWt3bZzZ567NWtoLcuScO9Z6ZG7DU1uwS5pEvBt4P0R8Wz5/RGxIiLmRsTc6ZMm5bVbs6bQsiwpZprf5WImG9xheTQiaRxJqH89Ir6TR5tmdrAZS7uTH7p6oaudW5dvKrZDVrPyOCtGwHXAxohYWnmXzGwgbcuTgHcxkx1KHlMx84B3AmdJWp9+nZ9Du2Z2CC5msoHkcVbM3RGhiHhVRMxOv27Jo3NmdmjZYiaP3i3Lladm9SyzFIGLmazEwW7WAFzMZFkOdrNGUFbM5CWAm5uD3ayBlK6runi752WamYPdrNF0drqYqck52M0a0Iyl3bQt8HVVm5WD3axRpddVnToVlqzy6ZDNxMFu1uCm9LiYqdk42M2agIuZmouD3axZuJipaTjYzZqMi5kan4PdrNm4mKnhOdjNmlS2mMmnQzYWB7tZM0uLmXpWupipkTjYzZqci5kaj4PdzA4UM5VG71bfHOxmdkDpuqouZqpvDnYzO4iLmeqfg93Mni9TzOTz3euPg93MDqlted/UjNUPB7uZDShbzOSlCOqDg93MBlUqZlqz2fPu9SCXYJf0FUlbJW3Io72atX8/PPAArF6dfN+/v+gemVVPZydtC+Yxbd9WL0NQ4w7LqZ2vAl8A/imn9mrP/v1wzTWwaRPs2QPjx0N7O1x+OYzxBx9rEh0dtAGLV/ZCVy9XLNhER0fRnbJyuSRSRNwFPJNHWzVrw4a+UIfk+6ZNye1mzaSsmMmj99pTtaGmpIWS1kpau23nzmrtNj+bN/eFesmePcntZk1oxtLkfHcvIlZ7qhbsEbEiIuZGxNzpkyZVa7f5aW1Npl+yxo9PbjdrVt3JNVV7Vvqgai3x5PBQzZqVzKmXwr00xz5rVrH9MivYlJ6Di5m8FEHx8jp42vjGjEkOlG7YkEy/tLYmoe4Dp2ZAcr77s1f0smpdO7d2biq6O00tr9MdvwH8GDhB0hZJnXm0W3PGjIFXvQouuCD57lA3O8iUnr5KVRczFSevs2IujYgXR8S4iJgZEf4wZtak2pZ3+7qqBfOQ08zyl7muqk+HrD4Hu5mNmpaL57F4e69H7lXmYDez0ZMpZprf5euqVouD3cxG3YylySmRc3bf42KmKnCwm1nVuJipOhzsZlY1LmaqDge7mVVd9rqqlj8Hu5kVo9vFTKPFwW5mhXEx0+hwsJtZsTLFTD6omg8Hu5nVhJaL5x04qGqVcbCbWW1wMVNuHOxmVlOyxUzzu9pd0DQCDnYzq0lty/uuzuTR+/D4QhtmVrOm9HQzBaCrF7rauXjOJjob82oPufKI3cxqXmn0vmqdlyMYCge7mdUFL0cwdA52M6sr2eUIfGpk/xzsZlZ/upPRO3hJgv442M2sbnlJgv452M2svmWWJPDoPeFgN7OG0LLMo/eSXIJd0nxJD0t6VNJH8mjTzGzYykbvj1+xougeFaLiYJc0FrgWOA84CbhU0kmVtluT9u+HBx6A1auT7/v3F90jM+tHy7Ju2hbMY/H23qYcvedRefpq4NGIeAxA0krgLcB/5NB27di/H665BjZtgj17YPx4aG+Hyy+HMZ7RMqs5HR20dXTw1AeScF83YR7blt5QdK+qIo9EagE2Z37fkt7WWDZs6At1SL5v2pTcbmY1qxkXFcsj2NXPbfG8jaSFktZKWrtt584cdltlmzf3hXrJnj3J7WZW85ppUbE8gn0L0Jr5fSbwZPlGEbEiIuZGxNzpkyblsNsqa21Npl+yxo9PbjezulBalqA0em/UZQnyCPafAi+V1C7pcGABcHMO7daWWbOSOfVSuJfm2GfNKrZfZjZsjb6oWMUHTyNir6T3At8HxgJfiYiHKu5ZrRkzJjlQumFDMv3S2pqEug+cmtWl7JLA8xtsSWBFPG86fNTNPfbYWHvllVXfr5lZv3p7eSI9XHbr8k3F9mUAixZpXUTMHWw7DzfNzBpsUTEHu5lZqlEWFXOwm5llNcCyBA52M7N+lBYVq8dlCRzsZmaHko7ex0+or9G7g93MbBAzltbXomIOdjOzoejoOGj0XsvLEjjYzcyGoR4WFXOwm5mNQHZRsVpblsDBbmY2QqVFxabt21pTi4o52M3MKtS2vJu21tpZVMzBbmaWh+7aGb072M3McpQdvRd1aqSD3cwsbwUvKuZgNzMbJUUtKuZgNzMbTQUsKuZgNzOrgpZl1VuWwMFuZlYtZcsSjNbo3cFuZlZl5YuK5b0sgYPdzKwImdF7z8p8FxVzsJuZFWg0FhVzsJuZ1YA8FxVzsJuZ1Yi8FhWrKNglXSzpIUn7Jc2tpC0zM0tUuqjYYRXufwPwNuBLFbZjZmZZ3d20AXQlZ868oXXTkP+0ohF7RGyMiIcracPMzA6tNHpfs3noRU2VjtiHTNJCYGH66x4tWrShWvseBS8Eflt0Jyrg/hfL/S9WPff/2KFsNGiwS/p/wIv6uevKiPjXofYmIlYAK9I210ZE3c7Ju//Fcv+L5f7XvkGDPSLeWI2OmJlZPny6o5lZg6n0dMe3StoCvBZYLen7Q/zT0V+3cnS5/8Vy/4vl/tc4RUTRfTAzsxx5KsbMrME42M3MGkxhwV6PyxFImi/pYUmPSvpI0f0ZLklfkbRVUt3VEEhqlXSHpI3p6+byovs0HJImSPqJpPvT/n+y6D6NhKSxkn4m6btF92W4JD0u6UFJ6yWtLbo/o6nIEXtpOYKcl5gfHZLGAtcC5wEnAZdKOqnYXg3bV4H5RXdihPYCSyLiROB04D119vzvAc6KiJOB2cB8SacX3KeRuBzYWHQnKnBmRMxu9PPYCwv2OlyO4NXAoxHxWET8F7ASeEvBfRqWiLgLeKbofoxERPw6Iu5Lf95BEi4txfZq6CKxM/11XPpVV2cuSJoJXAB8uei+2MA8xz50LcDmzO9bqKNgaSSSjgNOAe4ttifDk05jrAe2ArdHRF31H/gc8CFgf9EdGaEAbpO0Ll3ipGGN6loxeS1HUCPUz211NeJqBJImAd8G3h8Rzxbdn+GIiH3AbElTgRslzYqIujjeIelCYGtErJN0RtH9GaF5EfGkpKOB2yX9PP0U23BGNdgbbDmCLUBr5veZwJMF9aUpSRpHEupfj4jvFN2fkYqI7ZLuJDneURfBDswD3izpfGACMEXSDRGR34U6R1lEPJl+3yrpRpLp1YYMdk/FDN1PgZdKapd0OLAAuLngPjUNSQKuAzZGxNKi+zNckqanI3Uk/QnwRuDnxfZq6CKiOyJmRsRxJK/9H9ZTqEs6QtLk0s/AOdTPm+qwFXm640iXIyhEROwF3gt8n+TA3bci4qFiezU8kr4B/Bg4QdIWSZ1F92kY5gHvBM5KT1dbn44e68WLgTskPUAySLg9IurulME6NgO4W9L9wE+A1RFxa8F9GjVeUsDMrMF4KsbMrME42M3MGoyD3cyswTjYzcwajIPdzKzBONjNzBqMg93MrMH8f5HIq2FC5SolAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 9.1\n",
    "\n",
    "df = pd.DataFrame({'x1':[0, 0, 5], 'x2':[0, 1, 5], 'y':[1, 1, -1]})\n",
    "xsp1 = df.loc[df['y']==1]['x1'].values\n",
    "ysp1 = df.loc[df['y']==1]['x2'].values\n",
    "xsm1 = df.loc[df['y']==-1]['x1'].values\n",
    "ysm1 = df.loc[df['y']==-1]['x2'].values\n",
    "\n",
    "#plt.tight_layout()\n",
    "X_train = df[['x1', 'x2']].values\n",
    "y_train = df['y'].values\n",
    "cls = nn.NearestNeighbors(X_train, y_train, 1)\n",
    "x1_min, x1_max = -1, 6\n",
    "x2_min, x2_max = -1, 6\n",
    "xx1, xx2 = myplot.get_grid(x1_min, x1_max, x2_min, x2_max, step=0.02)\n",
    "myplot.plot_decision_boundaries(xx1, xx2, 2, cls)\n",
    "\n",
    "myplot.plt_plot([xsp1, xsm1], [ysp1, ysm1], 'scatter', \n",
    "               colors = ['r', 'b'], markers = ['o', '+'], labels = ['+1', '-1'], \n",
    "               title = \"Problem 9.1 (a) 1-Nearest Neighbor\", yscale = None, ylb = -1, yub = 6,\n",
    "               xlb = -1, xub = 6, xlabel = None, ylabel = None,\n",
    "               legends = ['+1', '-1'], legendx = None, legendy = None, marker_sizes=[25, 25])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformed data points: [[ 0.18441744 -1.40213773]\n",
      " [-1.30649561  0.54135868]\n",
      " [ 1.12207817  0.86077905]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 9.1 (b)\n",
    "myw = data.Whitening()\n",
    "myw.fit(X_train)\n",
    "\n",
    "def whitening_transform(X):\n",
    "    Z = myw.transform(X)\n",
    "    return Z\n",
    "    \n",
    "#Z_train = data.input_whitening(X_train)\n",
    "Z_train = whitening_transform(X_train)\n",
    "print(f\"Transformed data points: {Z_train}\")\n",
    "cls = nn.NearestNeighbors(Z_train, y_train, 1, 'classification')\n",
    "x1_min, x1_max = -1, 6\n",
    "x2_min, x2_max = -1, 6\n",
    "xx1, xx2 = myplot.get_grid(x1_min, x1_max, x2_min, x2_max, step=0.02)\n",
    "myplot.plot_decision_boundaries(xx1, xx2, 2, cls, whitening_transform)\n",
    "\n",
    "myplot.plt_plot([xsp1, xsm1], [ysp1, ysm1], 'scatter', \n",
    "               colors = ['r', 'b'], markers = ['o', '+'], labels = ['+1', '-1'], \n",
    "               title = \"Problem 9.1 (b) Whitening + 1-Nearest Neighbor\", yscale = None, ylb = -1, yub = 6,\n",
    "               xlb = -1, xub = 6, xlabel = None, ylabel = None,\n",
    "               legends = ['+1', '-1'], legendx = None, legendy = None, marker_sizes=[25, 25])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformed data points: (3, 1)\n",
      "[[-2.5797332 ]\n",
      " [-1.90465535]\n",
      " [ 4.48438855]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 9.1 (c) Use customized PCA\n",
    "\n",
    "pca_tr = data.PCA(1)\n",
    "pca_tr.fit(X_train)\n",
    "\n",
    "def pca_transformer(X):\n",
    "    Z = pca_tr.transform(X)\n",
    "    return Z\n",
    "\n",
    "Z_train = pca_tr.transform(X_train)\n",
    "print(f\"Transformed data points: {Z_train.shape}\\n{Z_train}\")\n",
    "cls = nn.NearestNeighbors(Z_train, y_train, 1, 'classification')\n",
    "x1_min, x1_max = -1, 6\n",
    "x2_min, x2_max = -1, 6\n",
    "xx1, xx2 = myplot.get_grid(x1_min, x1_max, x2_min, x2_max, step=0.1)\n",
    "myplot.plot_decision_boundaries(xx1, xx2, 2, cls, pca_transformer)\n",
    "\n",
    "myplot.plt_plot([xsp1, xsm1], [ysp1, ysm1], 'scatter', \n",
    "               colors = ['r', 'b'], markers = ['o', '+'], labels = ['+1', '-1'], \n",
    "               title = \"Problem 9.1 (c) Top 1 PCA + 1-Nearest Neighbor (Customized PCA)\", yscale = None, ylb = -1, yub = 6,\n",
    "               xlb = -1, xub = 6, xlabel = None, ylabel = None,\n",
    "               legends = ['+1', '-1'], legendx = None, legendy = None, marker_sizes=[25, 25])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_tr: (3, 1)\n",
      " [[-2.5797332 ]\n",
      " [-1.90465535]\n",
      " [ 4.48438855]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 9.1 (c) Use sklearn.decomposition.PCA\n",
    "\n",
    "mpca = decomposition.PCA(n_components=1)\n",
    "mpca.fit(X_train)\n",
    "\n",
    "def pca_t(X):\n",
    "    Z = mpca.transform(X)\n",
    "    return Z\n",
    "    \n",
    "X_tr = pca_t(X_train)\n",
    "print(f\"X_tr: {X_tr.shape}\\n {X_tr}\")\n",
    "cls = nn.NearestNeighbors(X_tr, y_train, 1, 'classification')\n",
    "myplot.plot_decision_boundaries(xx1, xx2, 2, cls, pca_t)\n",
    "\n",
    "myplot.plt_plot([xsp1, xsm1], [ysp1, ysm1], 'scatter', \n",
    "               colors = ['r', 'b'], markers = ['o', '+'], labels = ['+1', '-1'], \n",
    "               title = \"Problem 9.1 (c) Top 1 PCA + 1-Nearest Neighbor (sklearn.decomposition.PCA)\", yscale = None, ylb = -1, yub = 6,\n",
    "               xlb = -1, xub = 6, xlabel = None, ylabel = None,\n",
    "               legends = ['+1', '-1'], legendx = None, legendy = None, marker_sizes=[25, 25])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.2\n",
    "\n",
    "* (a) $x'_n = x_n + b$, so $\\bar{x}'_n = \\bar{x}_n + b$, $z'_n = x'_n - \\bar{x}'_n = (x_n+b) - (\\bar{x}_n + b) = x_n - \\bar{x}_n = z_n$. Centering is robust to this mutation.\n",
    "\n",
    "$(\\sigma'_i)^2 = \\frac{1}{N}\\sum^N_{n=1} (x'_{ni}-\\bar{x}'_{ni})^2 = \\frac{1}{N}\\sum^N_{n=1} (x_{ni}-\\bar{x}_{ni})^2$. Normalization is also robust to this mutation.\n",
    "\n",
    "$\\Sigma' = \\frac{1}{N}\\sum^N_{n=1}(x'_n - \\bar{x}'_n)(x'_n - \\bar{x}'_n)^T = \\Sigma$. It's clear that this whitening is also robust to the mutation. \n",
    "\n",
    "* (b) Uniform scaling: $x'_n = \\alpha x_n$, where $\\alpha > 0$. \n",
    "\n",
    "$\\bar{x}'_n = \\alpha\\bar{x}_n$, $z'_n = \\alpha z_n$, centering is not robust to this mutation. \n",
    "\n",
    "$(\\sigma'_i)^2 = \\frac{1}{N}\\sum^N_{n=1} (x'_{ni}-\\bar{x}'_{ni})^2 = \\frac{1}{N}\\sum^N_{n=1} \\alpha^2(x_{ni}-\\bar{x}_{ni})^2 = \\alpha^2 (\\sigma_i)^2$\n",
    "\n",
    "so $z'_{ni} = \\frac{x'_{ni}}{\\sigma'_i} = \\frac{\\alpha x_{ni}}{\\alpha \\sigma_i} = z_{ni}$. Normalization is robut to this mutation. \n",
    "\n",
    "$\\Sigma' = \\frac{1}{N}\\sum^N_{n=1}(x'_n - \\bar{x}'_n)(x'_n - \\bar{x}'_n)^T = \\alpha^2 \\Sigma$.\n",
    "\n",
    "$z'_n = (\\Sigma')^{-\\frac{1}{2}} x'_n = \\frac{1}{\\alpha} \\Sigma^{-\\frac{1}{2}} \\alpha x_n =  z_n$. Whitening is robust to this mutation. \n",
    "\n",
    "* (c) $x'_n = Ax_n$\n",
    "\n",
    "$\\bar{x}'_n = A\\bar{x}_n$, $z'_n = x'_n -\\bar{x}'_n = Az_n$, centering is not robust to this mutation. \n",
    "\n",
    "$(\\sigma'_i)^2 = \\frac{1}{N}\\sum^N_{n=1} (x'_{ni}-\\bar{x}'_{ni})^2 = \\frac{1}{N}\\sum^N_{n=1} A^2_{ii}(x_{ni}-\\bar{x}_{ni})^2 = A^2_{ii} (\\sigma_i)^2$, so $z'_{ni} = \\frac{x'_{ni}}{\\sigma'_i} = \\frac{A_{ii}x_{ni}}{A_{ii} \\sigma_i} = z_{ni}$. Normalization is robust to this mutation. \n",
    "\n",
    "$\\Sigma' = \\frac{1}{N}\\sum^N_{n=1}(x'_n - \\bar{x}'_n)(x'_n - \\bar{x}'_n)^T = \\frac{1}{N}\\sum^N_{n=1} Az_nz^T_n A^T$. Whitening is not robust to this mutation.\n",
    "\n",
    "* (d) $x'_n = Ax_n$, similar with problem (c), centering is not robust to this mutation. \n",
    "Also, $A$ is no longer diagonal, the normalization and whitening are not robust to this mutation neither. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.3 \n",
    "\n",
    "\\begin{align*}\n",
    "\\Sigma &= U\\Gamma U^T \\\\\n",
    "\\Sigma &= U \\Gamma^{\\frac{1}{2}} \\Gamma^{\\frac{1}{2}} U^T \\\\\n",
    "&= U\\Gamma^{\\frac{1}{2}}U^TU\\Gamma^{\\frac{1}{2}}U^T\\\\\n",
    "&= (U\\Gamma^{\\frac{1}{2}}U^T)^2\\\\\n",
    "&= (\\Sigma^{\\frac{1}{2}})^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $\\Sigma^{\\frac{1}{2}} = U\\Gamma^{\\frac{1}{2}}U^T$\n",
    "\n",
    "\\begin{align*}\n",
    "I &= UU^T \\\\ \n",
    "&= \\Gamma^{\\frac{1}{2}}U^T U\\Gamma^{-\\frac{1}{2}}\\\\\n",
    "&= U\\Gamma^{\\frac{1}{2}}U^T U\\Gamma^{-\\frac{1}{2}}U^T \\\\\n",
    "&= \\Sigma^{\\frac{1}{2}} \\Sigma^{-\\frac{1}{2}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $\\Sigma^{-\\frac{1}{2}} = U\\Gamma^{-\\frac{1}{2}}U^T$\n",
    "\n",
    "$\\Gamma^{\\frac{1}{2}}$ and $\\Gamma^{-\\frac{1}{2}}$ are both diagonal matrices, the former with the square root of the eigenvalues of $\\Sigma$ and the latter has the $\\frac{1}{\\sqrt{\\lambda_i}}$\n",
    "\n",
    "#### Problem 9.4\n",
    "\n",
    "Multiply both sides of $A=V\\psi$ by $V^T$, also note that $VV^T = V^TV = I$\n",
    "\n",
    "\\begin{align*}\n",
    "A &= V\\psi \\\\\n",
    "\\psi &= V^T A \\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then we have $\\psi^T\\psi = A^TV V^T A = I$ and $\\psi \\psi^T = V^T A A^T V = I$\n",
    "so $\\psi$ is orthogonal matrix. \n",
    "\n",
    "#### Problem 9.5\n",
    "\n",
    "* (a) Let $A' = PA$, where $P$ is the permutation matrix that changes the order of the diagonal entries in $A$ such that the entries in $A'$ are ordered from the largest to the smallest. \n",
    "\n",
    "$A=P^{-1}A = U\\Gamma V^T$ where $U=P^{-1}, V^T=I, \\Gamma = A$, notice $U^TU = (P^{-1})^TU = (P^T)^T P^{-1} = PP^{-1} = I$, since for permutation matrix, we have $P^{-1} = P^T$.\n",
    "\n",
    "* (b) $A$ is pairwise orthogonal rows. we have $AA^T = I$, also for any $A=U\\Gamma V^T$, we have $A = \\sum \\gamma_i u_i v^T_i$ where $u_i, v_i$ are the column vectors of matrices $U, V$ and $\\gamma_i$ are the corresponding singular values. Since $A$ has pairwise orthogonal rows, let $u_i =\\begin{bmatrix}0 \\\\ \\dots \\\\ 1 \\\\ \\dots \\\\ 0\\end{bmatrix}$, the $i$-th base in standard basis. let $A$ be a $m \\times n $ matrix, $A = \\begin{bmatrix}a_1 \\\\ \\dots \\\\ a_m\\end{bmatrix}$, N.B. $a_i$ are row vectors here. So we have $A = u_1a_1 + \\dots + u_ma_m = U\\Gamma V^T$, where $U=I, \\Gamma = I$ and $V = A^T$.\n",
    "\n",
    "* (c) $A$ is pairwise orthogonal columns, similarly with problem (b), we have $A = \\begin{bmatrix}a_1 & \\dots & a_n\\end{bmatrix}$, N.B. $a_i$ are column vectors here. So we have $A = a_1u^T_1 + \\dots + a_nu^T_n = U\\Gamma V^T$, where $U=A, \\Gamma = I$ and $V = I$.\n",
    "\n",
    "* (d) $QA = QU\\Gamma V^T = U' \\Gamma V^T$, where $U' = QU$ and $(U')^TU' = U^TQ^T Q U = U^T U = I$\n",
    "\n",
    "* (e) We have $A=U\\Gamma V^T = \\begin{bmatrix}U_1 & \\dots & 0 \\\\ 0 & U_2 & \\dots \\\\ \\dots & \\dots & \\dots \\\\ \\dots & 0 & U_m \\end{bmatrix}\\begin{bmatrix}\\Gamma_1 & \\dots & 0 \\\\ 0 & \\Gamma_2 & \\dots \\\\ \\dots & \\dots & \\dots \\\\ \\dots & 0 & \\Gamma_m \\end{bmatrix}\\begin{bmatrix}V^T_1 & \\dots & 0 \\\\ 0 & V^T_2 & \\dots \\\\ \\dots & \\dots & \\dots \\\\ \\dots & 0 & V^T_m \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_train:  (7291, 256) y_train:  (7291,) X_test:  (2007, 256) y_test:  (2007,)\n",
      "Frequencies of the digits: \n",
      " {0: 0.16376354409546015, 1: 0.13784117405019888, 2: 0.10026059525442327, 3: 0.09024825126868742, 4: 0.08942531888629818, 5: 0.07625840076807022, 6: 0.09107118365107666, 7: 0.08846523110684405, 8: 0.07433822520916199, 9: 0.08832807570977919}\n"
     ]
    }
   ],
   "source": [
    "#### Problem 9.6\n",
    "zip_data_path = './data/usps.h5'\n",
    "\n",
    "\n",
    "X_train, y_train, X_test, y_test = data.load_zip_data(zip_data_path)\n",
    "\n",
    "print('X_train: ', X_train.shape, 'y_train: ',y_train.shape, 'X_test: ',X_test.shape, 'y_test: ', y_test.shape)\n",
    "unique, counts = np.unique(y_train, return_counts=True)\n",
    "freqs = counts/len(y_train)\n",
    "print('Frequencies of the digits: \\n', dict(zip(unique, freqs)))\n",
    "\n",
    "y_train, y_test = data.set_two_classes(y_train, y_test, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pca = decomposition.PCA(n_components=2)\n",
    "pca.fit(X_train)\n",
    "X_tr = pca.transform(X_train)\n",
    "x1s = [X_tr[y_train==1, 0], X_tr[y_train==-1, 0]]\n",
    "\n",
    "#Since the singular vectors are not unique, I flipped the sign of the 2nd feature to make the graph match the textbook\n",
    "x2s = [-X_tr[y_train==1, 1], -X_tr[y_train==-1, 1]] \n",
    "myplot.plt_plot(x1s, x2s, 'scatter', ['b', 'r'], ['o', 'x'], ['Digit 1', 'Non Digit 1'], \n",
    "        title = 'Digits 1 vs non-1 (Centered, sklearn PCA)', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'z1', ylabel = 'z2',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mypca = data.PCA(2)\n",
    "mypca.fit(X_train)\n",
    "X_tr = mypca.transform(X_train)\n",
    "\n",
    "x1s = [X_tr[y_train==1, 0], X_tr[y_train==-1, 0]]\n",
    "\n",
    "#Since the singular vectors are not unique, I flipped the sign of the 2nd feature to make the graph match the textbook\n",
    "x2s = [-X_tr[y_train==1, 1], -X_tr[y_train==-1, 1]]\n",
    "myplot.plt_plot(x1s, x2s, 'scatter', ['b', 'r'], ['o', 'x'], ['Digit 1', 'Non Digit 1'], \n",
    "        title = 'Digits 1 vs non-1 (Centered)', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'z1', ylabel = 'z2',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mypca = data.PCA(2,centering=False)\n",
    "mypca.fit(X_train)\n",
    "X_tr = mypca.transform(X_train)\n",
    "\n",
    "#Since the singular vectors are not unique, I flipped the sign of the features to make the graph match the textbook\n",
    "x1s = [-X_tr[y_train==1, 0], -X_tr[y_train==-1, 0]]\n",
    "x2s = [-X_tr[y_train==1, 1], -X_tr[y_train==-1, 1]]\n",
    "myplot.plt_plot(x1s, x2s, 'scatter', ['b', 'r'], ['o', 'x'], ['Digit 1', 'Non Digit 1'], \n",
    "        title = 'Digits 1 vs non-1 (Not Centered)', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'z1', ylabel = 'z2',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5.4045758e+00, 1.5981337e-06],\n",
       "       [1.5981337e-06, 2.7097924e+00]], dtype=float32)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.matmul(X_tr.transpose(), X_tr)/X_tr.shape[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transformed data has no correlations between the two features but with different variances.  They are kinds of \"whitened\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.7\n",
    "\n",
    "Let $X=\\begin{bmatrix} x^T_1 \\\\ \\dots \\\\ x^T_N \\end{bmatrix}$ be the $N\\times d$ data matrix. From SVD we have $X=U\\Gamma V^T$, where $U$ is $N \\times d$, $\\Gamma, V$ are $d \\times d $ matrices.\n",
    "\n",
    "Let $V_k$ be the first $k$ columns from $V$, so it has dimension of $d\\times k$. Let $U_k$ be the first $k$ columns from $U$, it has dimension of $N\\times k$ and $U^T_k U_k = I_k$. Also $\\Gamma_k$ is the $k\\times k$ diagonal matrix with the first $k$ singular values on the diagonal. \n",
    "\n",
    "From SVD we have $X=U\\Gamma V^T$, so $XV=U\\Gamma$ and $X\\begin{bmatrix}V_k & V_r \\end{bmatrix} = \\begin{bmatrix}U_k & U_r \\end{bmatrix} \\begin{bmatrix}\\Gamma'_k \\\\ \\Gamma'_r \\end{bmatrix}$. \n",
    "\n",
    "Where $V_r$ has a dimension of $d\\times (d-k)$ and $U_r$ has dimension of $N\\times (d-k)$, $\\Gamma'_k$ is the first $k$ rows in $\\Gamma$ and has a dimension of $k\\times d$ and $\\Gamma'_r$ has a dimension of $(d-k)\\times d$. We thus have\n",
    "\n",
    "$XV_k = U_k\\Gamma_k$\n",
    "\n",
    "So we have $Z=\\begin{bmatrix} z^T_1 \\\\ \\dots \\\\ z^T_N \\end{bmatrix} = \\begin{bmatrix} (\\Gamma^{-1}_kV^T_kx_1)^T \\\\ \\dots \\\\ (\\Gamma^{-1}_kV^T_kx_N)^T \\end{bmatrix} = \\begin{bmatrix} x^T_1V_k\\Gamma^{-1}_k \\\\ \\dots \\\\ x^T_NV_k\\Gamma^{-1}_k \\end{bmatrix} = XV_k\\Gamma^{-1}_k$, so $Z$ has a dimension of $N\\times k$\n",
    "\n",
    "so $Z^TZ = \\Gamma^{-1}_kV^T_kX^T XV_k\\Gamma^{-1}_k = \\Gamma^{-1}_k (U_k\\Gamma_k)^TU_k\\Gamma_k\\Gamma^{-1}_k = \\Gamma^{-1}_k \\Gamma_kU^T_kU_k\\Gamma_k\\Gamma^{-1}_k = I$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_train:  (7291, 256) y_train:  (7291,) X_test:  (2007, 256) y_test:  (2007,)\n",
      "Frequencies of the digits: \n",
      " {0: 0.16376354409546015, 1: 0.13784117405019888, 2: 0.10026059525442327, 3: 0.09024825126868742, 4: 0.08942531888629818, 5: 0.07625840076807022, 6: 0.09107118365107666, 7: 0.08846523110684405, 8: 0.07433822520916199, 9: 0.08832807570977919}\n"
     ]
    }
   ],
   "source": [
    "#### Problem 9.8\n",
    "zip_data_path = './data/usps.h5'\n",
    "\n",
    "\n",
    "X_train, y_train, X_test, y_test = data.load_zip_data(zip_data_path)\n",
    "\n",
    "print('X_train: ', X_train.shape, 'y_train: ',y_train.shape, 'X_test: ',X_test.shape, 'y_test: ', y_test.shape)\n",
    "unique, counts = np.unique(y_train, return_counts=True)\n",
    "freqs = counts/len(y_train)\n",
    "print('Frequencies of the digits: \\n', dict(zip(unique, freqs)))\n",
    "\n",
    "y_train, y_test = data.set_two_classes(y_train, y_test, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(256, 1) (256, 1) (256, 2)\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "XX, mean_X = data.input_centering(X_train)\n",
    "\n",
    "Xp = XX[y_train==1]\n",
    "Xn = XX[y_train==-1]\n",
    "\n",
    "mypca1 = data.PCA(1, centering=False)\n",
    "mypca1.fit(Xp)\n",
    "v1 = mypca1.Vk\n",
    "\n",
    "mypca2 = data.PCA(1, centering=False)\n",
    "mypca2.fit(Xn)\n",
    "v2 = mypca2.Vk\n",
    "Vk = np.hstack([v1, v2])\n",
    "print(v1.shape, v2.shape, Vk.shape)\n",
    "\n",
    "X_tr = np.matmul(XX, Vk)\n",
    "\n",
    "\n",
    "x1s = [X_tr[y_train==1, 0], X_tr[y_train==-1, 0]]\n",
    "#Since the singular vectors are not unique, I flipped the sign of the 2nd feature to make the graph match the textbook\n",
    "x2s = [-X_tr[y_train==1, 1], -X_tr[y_train==-1, 1]]\n",
    "myplot.plt_plot(x1s, x2s, 'scatter', ['b', 'r'], ['o', 'x'], ['Digit 1', 'Non Digit 1'], \n",
    "        title = 'Problem 9.8 (a) Digits 1 vs non-1', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'z1', ylabel = 'z2',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.8 (b)\n",
    "\n",
    "The directions $v_1$ and $v_2$ are not necessarily orthogonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "52.1007414905504"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(np.dot(v1,v2.transpose()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.8 (c)\n",
    "\n",
    "Let $\\hat{V}=U_1\\Gamma_1V^T_1$, since $Z=X\\hat{V}$, we have $Z=XU_1\\Gamma_1V^T_1$, multiply $V_1,\\Gamma^{+}_1, U^T_1$ on the right in order, we have $X = ZV_1\\Gamma^{+}_1U^T_1 = Z\\hat{V}^{+}$\n",
    "\n",
    "\n",
    "#### Problem 9.8 (d)\n",
    "\n",
    "This method uses the labels for the data points, so it's a supervised method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.9\n",
    "\n",
    "* (a) If $\\hat{X}$ is a reconstruction of $X$ whose rows are the data points in $X$ projected onto $k$ basis vectors, then suppose the $k$ basis vectors are in $B$ with dimension of $d\\times k$, then we have $\\hat{X} = XBB^T$, with dimension of $N\\times d$. There are at most $k$ independent columns in $\\hat{X}$, so $\\text{rank}(\\hat{X}) \\le k$\n",
    "\n",
    "\n",
    "* (b) Since $U_{N\\times d},V_{d\\times d}, V^T$ have orthonormal columns, i.e. $U^TU=I, (V^T)^TV^T = VV^T=I$,  we have \n",
    "\n",
    "Let $Z=(X-\\hat{X})^TU = YU$, it's also $z=U^Ty$, where $z$ are the columns in $Z$ and $y^T$ are the rows in $Y$.  Since $U$ has orthonormal columns, we can treat $Z$ as a transformed data matrix of $Y = (X-\\hat{X})^T$. Note $Y$ has a dimension of $d\\times N$, so the rows in $Y$, i.e. $y$,  are in a space of $R^N$.  While the basis vectors in $U$ is also in space $R^N$, but we only select $d$ basis vectors in this space (since $U$ has a dimension of $N\\times d$). From exercise 9.7 (c), we have $\\|z\\| \\le \\|y\\| $, we see that $\\|Z\\|^2 = \\sum \\|z\\|^2 \\le \\sum \\|y\\|^2 = \\|Y\\|^2 = \\|(X-\\hat{X})^T\\|^2 = \\|X-\\hat{X}\\|^2$. That is\n",
    "\n",
    "$\\|X-\\hat{X}\\|^2 \\ge \\|(X-\\hat{X})^TU\\|^2_F  = \\|U^T(X-\\hat{X})\\|^2_F = \\|U^T(X-\\hat{X})V\\|^2_F = \\|U^TXV-U^T\\hat{X}V\\|^2_F = \\|\\Gamma-U^T\\hat{X}V\\|^2_F$\n",
    "\n",
    "Where we have applied the results of Exercise 9.9 on $V$ on the second equation above.\n",
    "\n",
    "* (c) Let $\\hat{\\Gamma} = U^T\\hat{X}V$, we have $\\text{rank}(\\hat{\\Gamma}) = \\text{rank}(U^T\\hat{X}V) \\le \\text{rank}(\\hat{X}V) \\le \\text{rank}(\\hat{X}) \\le k $ from problem (a)\n",
    "\n",
    "* (d) We want to minimize the difference $\\|\\Gamma-U^T\\hat{X}V\\|^2_F$, since the Frobenius norm is defined as the sum of squares of each element in the matrix, we want to have the difference between $\\Gamma$ and $\\hat{\\Gamma}$ to be zero. Since $\\Gamma$ has zeros on all off-diagonal elements, it's optimal to choose zeros on the off-diagonal elements in $\\hat{\\Gamma}$ as well, so their differences on off-diagonal are all zero.\n",
    "\n",
    "* (e) Since $\\hat{\\Gamma}$ has a rank of at most $k$, the non-zero diagonals can be at most $k$.\n",
    "\n",
    "* (f) If we select the first $k$ singular values from $\\Gamma$ as the first $k$ elements in $\\hat{\\Gamma}$, we have the optimal choice here. Otherwise, if there's any singular value after $k$ in $\\hat{\\Gamma}$, we can always swap it with the missing singular value from first $k$ items, and obtain a smaller error. \n",
    "\n",
    "* (g) $\\hat{X} = XVV^T =  X$, so we have $\\hat{\\Gamma} = U^T\\hat{X}V=U^TXV=\\Gamma$, we have exact match with $\\Gamma$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.10\n",
    "\n",
    "* (a) Algorithm 1 uses full data points $X$ to construct new features, while algorithm 2 removes the $n$-th data point first, then use the rest of data to construct new features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "In one exeriment, E1=3.3327010677097086, E2=3.740820665349818, Eout = 4.544789128347455\n"
     ]
    }
   ],
   "source": [
    "def pca_cv_problem910_1(X, y, top_k):\n",
    "    # Algorithm 1 in Problem 9.10\n",
    "    pca1 = data.PCA(top_k)\n",
    "    pca1.fit(X)\n",
    "    Z = pca1.transform(X)\n",
    "    N, _ = X.shape\n",
    "    full_indices = np.arange(N)\n",
    "    lr = lm.LinearRegression('pesudo_inv', reg_type=None, reg_param = 0)\n",
    "    ens = []\n",
    "    for n in np.arange(N):\n",
    "        indices = np.hstack([full_indices[:n], full_indices[n+1:]])\n",
    "        Zn = Z[indices]\n",
    "        yn = y[indices]\n",
    "        lr.fit(Zn, yn)\n",
    "        predn = np.matmul(Z[n].transpose(), lr.w)\n",
    "        en = (predn-y[n])**2\n",
    "        ens.append(en)\n",
    "    E1 = np.mean(ens)\n",
    "    return E1\n",
    "\n",
    "def pca_cv_problem910_2(X, y, top_k):    \n",
    "    # Algorithm 2 in Problem 9.10        \n",
    "    N, _ = X.shape\n",
    "    full_indices = np.arange(N)\n",
    "    lr = lm.LinearRegression('pesudo_inv', reg_type=None, reg_param = 0)\n",
    "    ens = []\n",
    "    for n in np.arange(N):\n",
    "        indices = np.hstack([full_indices[:n], full_indices[n+1:]])\n",
    "        Xn = X[indices]\n",
    "        yn = y[indices]\n",
    "        pca2 = data.PCA(top_k)\n",
    "        pca2.fit(Xn)\n",
    "        Zn = pca2.transform(Xn)\n",
    "\n",
    "        lr.fit(Zn, yn)\n",
    "        zn = np.matmul(X[n].transpose(), pca2.Vk)\n",
    "        predn = np.matmul(zn.transpose(), lr.w)\n",
    "        en = (predn-y[n])**2\n",
    "        ens.append(en)\n",
    "    E2 = np.mean(ens)\n",
    "    return E2\n",
    "\n",
    "def generate_data_910(N, d, var):\n",
    "    mu, std = 0, 1\n",
    "    w = np.random.normal(mu, std, d).reshape((d, 1))\n",
    "    X = np.random.normal(mu, std, N*d).reshape((N, d))\n",
    "    epsilons = np.random.normal(mu, np.sqrt(var), N).reshape((N, 1))\n",
    "    y = np.matmul(X, w) + epsilons\n",
    "    return X, y\n",
    "    \n",
    "def calc_Eout_910(X_train, y_train, X_test, y_test, top_k):\n",
    "    lr = lm.LinearRegression('pesudo_inv', reg_type=None, reg_param = 0)\n",
    "    pca = data.PCA(top_k)\n",
    "    pca.fit(X_train)\n",
    "    Z_train = pca.transform(X_train)\n",
    "    lr.fit(Z_train, y_train)\n",
    "    \n",
    "    Z_test = pca.transform(X_test)\n",
    "    Eout = lr.calc_error(Z_test, y_test)[0]\n",
    "    return Eout\n",
    "\n",
    "def run_one_experiment_910(N, N_test, d, var, top_k):\n",
    "    X, y = generate_data_910(N+N_test, d, var)\n",
    "    X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=N_test, random_state=42)\n",
    "    \n",
    "    E1 = pca_cv_problem910_1(X_train, y_train, top_k)\n",
    "    E2 = pca_cv_problem910_2(X_train, y_train, top_k)\n",
    "    Eout = calc_Eout_910(X_train, y_train, X_test, y_test, top_k)\n",
    "    \n",
    "    return E1, E2, Eout\n",
    "\n",
    "E1, E2, Eout = run_one_experiment_910(N, N_test, d, var, top_k)\n",
    "print(f\"In one exeriment, E1={E1}, E2={E2}, Eout = {Eout}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average Errors: E1=2.1437860825253985, E2 = 2.537054958626525, Eout = 2.6116116514240115\n"
     ]
    }
   ],
   "source": [
    "#### Problem 9.10\n",
    "\n",
    "d = 5\n",
    "N = 40\n",
    "N_test = 100\n",
    "var = 0.5\n",
    "top_k = 3\n",
    "maxits = 1e+6\n",
    "\n",
    "\n",
    "\n",
    "E1s, E2s, Eouts = [], [], []    \n",
    "for it in np.arange(maxits):\n",
    "    E1, E2, Eout = run_one_experiment_910(N, N_test, d, var, top_k)\n",
    "    E1s.append(E1)\n",
    "    E2s.append(E2)\n",
    "    Eouts.append(Eout)\n",
    "\n",
    "avg_E1 = np.mean(E1s)\n",
    "avg_E2 = np.mean(E2s)\n",
    "avg_Eout = np.mean(Eouts)\n",
    "print(f\"Average Errors: E1={avg_E1}, E2 = {avg_E2}, Eout = {avg_Eout}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.10 (d)\n",
    "\n",
    "The average estimates $\\bar{E}_1$ is smaller than the average estimates $\\bar{E}_2$ because the algorithm 1 uses the test point when constructing the features. This is data snooping. \n",
    "\n",
    "The average estimate $\\bar{E}_2$ is closer to the $E_{out}(g)$.\n",
    "\n",
    "#### Problem 9.10 (d)\n",
    "\n",
    "The correct estimates should be $\\bar{E}_2$ for $E_{out}(g)$ since it doesn't have data snooping by using the test data to compute SVD."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.11 TODO\n",
    "\n",
    "* (a) For fixed $d_{eff}$, when the order increases, $\\lambda$ should increase to keep the effective degree of freedom down."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_deff_diff(lambda_t, X, d_eff):\n",
    "    H = lm.compute_H(X, lambda_t)\n",
    "    d_eff1 = lm.calc_effective_dof(H)\n",
    "    diff = (d_eff1 - d_eff)**2\n",
    "    return diff\n",
    "\n",
    "def calc_lambda(d_eff, X_train, x0, bounds):\n",
    "    f = partial(calc_deff_diff, X = X_train, d_eff = d_eff)\n",
    "    #res = minimize_scalar(f, method='brent')\n",
    "    res = minimize(f, x0, method='trust-constr', jac=None, hess=None,\n",
    "               constraints=[],\n",
    "               options={'verbose': 0}, bounds=bounds)\n",
    "    #print(res)\n",
    "    return res.x[0]\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 100\n",
    "test_N = 100\n",
    "sigma_square = data.generate_random_numbers01(1, 1)\n",
    "Qf = np.random.choice(max_target_degree+1, 1)\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "train_xs, train_ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "from scipy.optimize import Bounds\n",
    "bounds = Bounds(0, np.inf)\n",
    "x0 = np.array([0])\n",
    "calc_lambda(7, train_xs, x0, bounds)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def run_one_experiment_prob_911(N, test_N, max_target_degree, max_hypo_degree, d_eff):\n",
    "    sigma_square = data.generate_random_numbers01(1, 1)\n",
    "    Qf = 10 #np.random.choice(max_target_degree+1, 1)\n",
    "    normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "    train_xs, train_ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "    \n",
    "    bounds = Bounds(0, np.inf)\n",
    "    x0 = np.array([0])\n",
    "    lambda_t = calc_lambda(d_eff, train_xs, x0, bounds)\n",
    "    print(f\"lambda_t = {lambda_t}\")\n",
    "    orders = np.arange(0,max_hypo_degree+1)\n",
    "    Eins, Eouts = [], []\n",
    "    for order in orders:\n",
    "        Ein, Eout = calc_Eout_ex_918(order, train_xs, train_ys, test_xs, test_ys, lambda_t)\n",
    "        Eins.append(Ein)\n",
    "        Eouts.append(Eout)\n",
    "    return Eouts, Eins\n",
    "\n",
    "\n",
    "N = 100\n",
    "test_N = 100\n",
    "max_target_degree = 30\n",
    "max_hypo_degree = 30\n",
    "d_eff = 7\n",
    "\n",
    "max_exps = 10000\n",
    "Eins, Eouts = [], []\n",
    "for e in np.arange(max_exps):\n",
    "    if e %1000 == 1:\n",
    "        print(f\"--- Current iteration : {e}\")\n",
    "    Eouts1, Eins1 = run_one_experiment_prob_911(N, test_N, max_target_degree, max_hypo_degree, d_eff)\n",
    "          \n",
    "    Eouts.append(Eouts1)\n",
    "\n",
    "    \n",
    "Eouts = np.mean(np.array(Eouts), axis=0)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "orders = np.arange(max_hypo_degree+1)\n",
    "yub = 1.5\n",
    "Eouts_f = np.clip(Eouts, 0, yub)\n",
    "xxs = [orders]\n",
    "yys = [Eouts_f]\n",
    "myplot.plt_plot(xxs, yys, 'plot', ['b'], ['o'], \n",
    "                ['Out-of-Sample Error'], \n",
    "        title = 'Problem 9.11 (c) E_out vs. Model Order', yscale = None, ylb = 0, yub = 1.2,\n",
    "        xlb = None, xub = None, xlabel = 'Model Order', ylabel = 'E_out',\n",
    "        legends = ['']*6, legendx = None, legendy = None, marker_sizes=[15]*5) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.12\n",
    "\n",
    "* (a) \n",
    "\n",
    "\\begin{align*}\n",
    "E^{\\pi}_{out}(h) &= \\frac{1}{4}E_{x,\\pi}\\left[(h(x)-f_{\\pi}(x))^2\\right] \\\\\n",
    "&= \\frac{1}{4}E_{x}E_{\\pi|x}\\left[(h(x)-f_{\\pi}(x))^2\\right] \\\\\n",
    "&= \\frac{1}{4}\\sum^N_{n=1}P(x=x_n)E_{\\pi|x=x_n}\\left[(h(x_n)-f_{\\pi}(x_n))^2\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}E_{\\pi|x=x_n}\\left[(h(x_n)-f_{\\pi}(x_n))^2\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}E_{\\pi}\\left[(h(x_n)-f_{\\pi}(x_n))^2\\right] \\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (b) \n",
    "\n",
    "Let $\\bar{y} = \\frac{1}{N}\\sum_n y_n$ and $s^2_y = \\frac{1}{N}\\sum_n(y_n - \\bar{y})^2$\n",
    "\n",
    "\\begin{align*}\n",
    "E^{\\pi}_{out}(h) &=\\frac{1}{4N}\\sum^N_{n=1}E_{\\pi}\\left[(h(x_n)-f_{\\pi}(x_n))^2\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}E_{\\pi}\\left[(h(x_n)-\\bar{y} + \\bar{y} - f_{\\pi}(x_n))^2\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}E_{\\pi}\\left[(h(x_n)-\\bar{y})^2 + 2(h(x_n)-\\bar{y})(f_{\\pi}(x_n)-\\bar{y}) + (f_{\\pi}(x_n)-\\bar{y})^2\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}E_{\\pi}[(h(x_n)-\\bar{y})^2] + 2E_{\\pi}[(h(x_n)-\\bar{y})(f_{\\pi}(x_n)-\\bar{y})] + E_{\\pi}[(f_{\\pi}(x_n)-\\bar{y})^2] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}\\left[(h(x_n)-\\bar{y})^2 + 2E_{\\pi}[(h(x_n)-\\bar{y})(f_{\\pi}(x_n)-\\bar{y})] + E_{\\pi}[(f_{\\pi}(x_n)-\\bar{y})^2]\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}\\left[(h(x_n)-\\bar{y})^2 + 2(h(x_n)-\\bar{y})E_{\\pi}[(f_{\\pi}(x_n)-\\bar{y})] + s^2_y\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}\\left[(h(x_n)-\\bar{y})^2 + s^2_y\\right] \\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}(h(x_n)-\\bar{y})^2 + \\frac{1}{4}s^2_y \\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) $E^{\\pi}_{in}(g_{\\pi}) = \\frac{1}{4N}\\sum^N_{n=1} (g_{\\pi}(x_n) - y_{\\pi_n})^2$\n",
    "\n",
    "* (d) The permutation optimism penalty is thus\n",
    "\n",
    "\\begin{align*}\n",
    "E^{\\pi}_{out}(g_{\\pi}) - E^{\\pi}_{in}(g_{\\pi}) &= \\frac{1}{4N}\\sum^N_{n=1}(g_{\\pi}(x_n)-\\bar{y})^2 + \\frac{1}{4}s^2_y - \\frac{1}{4N}\\sum^N_{n=1} (g_{\\pi}(x_n) - y_{\\pi_n})^2\\\\\n",
    " &= \\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})(2g_{\\pi}(x_n) - y_{\\pi_n}-\\bar{y} ) + \\frac{1}{4}s^2_y\\\\\n",
    " &= \\frac{1}{2N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n) + \\frac{1}{4N}\\sum^N_{n=1}(\\bar{y}^2 - y^2_{\\pi_n}) + \\frac{1}{4}\\frac{1}{N}\\sum^N_{n=1}(y_{\\pi_n} - \\bar{y})^2\\\\\n",
    " &= \\frac{1}{2N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n) + \\frac{1}{4N}\\sum^N_{n=1}2\\bar{y}(\\bar{y} - y_n)\\\\\n",
    " &= \\frac{1}{2N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (e) The correlation between $y_{\\pi_n}$ and $g_{\\pi}(x_n)$ is \n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})(g_{\\pi}(x_n) - \\bar{g_{\\pi}}) &= \\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n) - \\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})\\bar{g_{\\pi}}\\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n) - \\bar{g_{\\pi}}\\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})\\\\\n",
    "&= \\frac{1}{4N}\\sum^N_{n=1}(y_{\\pi_n}-\\bar{y})g_{\\pi}(x_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So it is proportional to the permutation optimism.\n",
    "\n",
    "#### Problem 9.13\n",
    "\n",
    "The target distribution for the random problem using sampling of the targets with replacement, then we have  $P[f_{\\pi}(x_n)=y_m] = \\frac{1}{N}$.\n",
    "\n",
    "Follow the steps in problem 9.12, we shall obtain equation (9.5) as well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.14"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.031000000000000028"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def run_one_experiment_914(N, maxit=10000):\n",
    "    dim= 1\n",
    "    max_v = 10000\n",
    "    lb, ub = -10, 10\n",
    "    use_adaline = False\n",
    "    eta = 1    \n",
    "\n",
    "    xs = data.generate_random_numbers(N, dim, max_v, lb, ub)\n",
    "    rands = data.generate_random_numbers01(N, dim, max_v)\n",
    "    df = pd.DataFrame({'x0': np.ones(N), 'x':xs.flatten(), 'rands':rands.flatten()})\n",
    "    df['y'] = 1\n",
    "    df.loc[df['rands'] < 0.5, 'y'] = -1\n",
    "    df = df[['x0', 'x','y']]\n",
    "\n",
    "    w_pocket, w_ts, what_ts, test_w_ts, test_what_ts, Ein = lm.pocket_algo(df.values, dim, \n",
    "                                                                      maxit,eta, randomized = True, \n",
    "                                                                      print_out = False, \n",
    "                                                                      test_points = None)    \n",
    "    opt = 0.5 - Ein\n",
    "    return opt\n",
    "\n",
    "run_one_experiment_914(1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Working on N = 1\n",
      "--- Working on N = 2\n",
      "--- Working on N = 3\n",
      "--- Working on N = 4\n",
      "--- Working on N = 5\n",
      "--- Working on N = 6\n",
      "--- Working on N = 7\n",
      "--- Working on N = 8\n",
      "--- Working on N = 9\n",
      "--- Working on N = 10\n",
      "--- Working on N = 20\n",
      "--- Working on N = 30\n",
      "--- Working on N = 40\n",
      "--- Working on N = 50\n",
      "--- Working on N = 100\n",
      "--- Working on N = 200\n",
      "--- Working on N = 300\n",
      "--- Working on N = 400\n",
      "--- Working on N = 500\n",
      "--- Working on N = 600\n",
      "--- Working on N = 700\n",
      "--- Working on N = 800\n",
      "--- Working on N = 900\n",
      "--- Working on N = 1000\n"
     ]
    }
   ],
   "source": [
    "Ns = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000]\n",
    "opts = []\n",
    "for N in Ns:\n",
    "    print(f\"--- Working on N = {N}\")\n",
    "    opt = run_one_experiment_914(N)\n",
    "    opts.append(opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "vals = 1/np.sqrt(Ns)\n",
    "myplot.plt_plot([Ns, Ns], [opts, vals], 'plot', ['b', 'r'], ['o', 'x'], ['Calculated Optimism', '1/sqrt(N)'], \n",
    "        title = 'Problem 9.14 (a) Rademacher Optimism vs. N', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'N', ylabel = 'Rademacher Optimism',\n",
    "        legends = ['', ''], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Working on 1\n",
      "---- Average optimism: 0.5\n",
      "-- Working on 2\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "D:\\git\\Learning-From-Data-A-Short-Course\\libs\\linear_models.py:119: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  w = w/w[-1]\n",
      "D:\\git\\Learning-From-Data-A-Short-Course\\libs\\linear_models.py:119: RuntimeWarning: invalid value encountered in true_divide\n",
      "  w = w/w[-1]\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "---- Average optimism: 0.4796\n",
      "-- Working on 3\n",
      "---- Average optimism: 0.4063333333333334\n",
      "-- Working on 4\n",
      "---- Average optimism: 0.3649\n",
      "-- Working on 5\n",
      "---- Average optimism: 0.3335999999999999\n",
      "-- Working on 6\n",
      "---- Average optimism: 0.3057\n",
      "-- Working on 7\n",
      "---- Average optimism: 0.28702857142857147\n",
      "-- Working on 8\n",
      "---- Average optimism: 0.2697375\n",
      "-- Working on 9\n",
      "---- Average optimism: 0.2564222222222222\n",
      "-- Working on 10\n",
      "---- Average optimism: 0.24306\n",
      "-- Working on 20\n",
      "---- Average optimism: 0.17616500000000002\n",
      "-- Working on 30\n",
      "---- Average optimism: 0.14551333333333336\n",
      "-- Working on 40\n",
      "---- Average optimism: 0.12594499999999997\n",
      "-- Working on 50\n",
      "---- Average optimism: 0.112944\n",
      "-- Working on 100\n",
      "---- Average optimism: 0.09060000000000003\n",
      "-- Working on 500\n",
      "---- Average optimism: 0.0412528\n",
      "-- Working on 1000\n",
      "---- Average optimism: 0.0306446\n"
     ]
    }
   ],
   "source": [
    "Ns = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 100, 500, 1000]\n",
    "runs = 10000\n",
    "avg_opts = []\n",
    "for N in Ns:\n",
    "    opts = []\n",
    "    print(f\"-- Working on {N}\")\n",
    "    if N < 100:\n",
    "        maxit = 100\n",
    "    elif N <= 500:\n",
    "        maxit = 1000\n",
    "    else:\n",
    "        maxit = 5000\n",
    "    for r in np.arange(runs):\n",
    "        opt = run_one_experiment_914(N, maxit)\n",
    "        opts.append(opt)\n",
    "    avg_opt = np.mean(opts)\n",
    "    print(f\"---- Average optimism: {avg_opt}\")\n",
    "    avg_opts.append(avg_opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "vals = 1/np.sqrt(Ns)\n",
    "myplot.plt_plot([Ns, Ns], [avg_opts, vals], 'plot', ['b', 'r'], ['o', 'x'], ['Calculated Optimism', '1/sqrt(N)'], \n",
    "        title = 'Problem 9. 14 (b)-(c) Rademacher Optimism vs. N', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'N', ylabel = 'Average Rademacher Optimism',\n",
    "        legends = ['', ''], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.14 (c)\n",
    "\n",
    "The Rademacher penalty seems to follow the line of $\\frac{1}{\\sqrt{N}}$. The VC penalty for perceptron is $\\sqrt{\\frac{d\\ln N}{N}}$ according to equation (3.1). It's a bit larger than the Rademacher penalty. So Rademacher penalty is a tigher bound here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.15\n",
    "\n",
    "For a single hypothesis we have $P[|E_{out}(h) - E_{in}(h)| > \\epsilon] \\le 2e^{-2N\\epsilon^2}$. The Rademacher variables $r_n$ are $+1,-1$, so the growth function $m_{\\mathcal{H}}(N)$ is the exact number of hypotheses in $\\mathcal{H}$ for $N$ data points.  We are selecting $g_r$ from  $m_{\\mathcal{H}}(N)$ hypotheses, so we have \n",
    "\n",
    "$P[|E'_{out}(g_r) - E'_{in}(g_r)| > \\epsilon] \\le 2m_{\\mathcal{H}}(N)e^{-2N\\epsilon^2}$\n",
    "\n",
    "Let $\\delta = 2m_{\\mathcal{H}}(N)e^{-2N\\epsilon^2}$, i.e. $\\epsilon = \\sqrt{\\frac{1}{2N}\\ln\\frac{2m_{\\mathcal{H}}}{\\delta}}$. Thus with probability at least $1-\\delta$, we have\n",
    "\n",
    "$E'_{out}(g_r) - E'_{in}(g_r) \\le \\epsilon = \\sqrt{\\frac{1}{2N}\\ln\\frac{2m_{\\mathcal{H}(N)}}{\\delta}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.16 TODO\n",
    "#### Problem 9.17 TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.18\n",
    "\n",
    "* (a) From $\\hat{y}^{(\\pi)} = Hy^{(\\pi)}$, we have $\\hat{y}^{(\\pi)}_n = \\sum_m H_{nm}y^{(\\pi)}_m = \\sum_m H_{nm}y_{\\pi_m}$. The permutation optimism penalty is thus\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{2}{N}\\sum^N_{n=1}(y_{\\pi_n} - \\bar{y})g_{(\\pi)}(x_n) &= \\frac{2}{N}\\sum^N_{n=1}(y_{\\pi_n} - \\bar{y})\\sum_m H_{nm}y_{\\pi_m} \\\\ \n",
    "&= \\frac{2}{N}\\sum^N_{n=1}\\sum_m(y_{\\pi_n} - \\bar{y}) H_{nm}y_{\\pi_m} \\\\\n",
    "&=\\frac{2}{N}\\sum^N_{n=1}\\sum_m H_{nm}(y_{\\pi_n}y_{\\pi_m}  - \\bar{y}y_{\\pi_m} )\\\\\n",
    "&=\\frac{2}{N}\\sum_{m,n=1}H_{mn}(y_{\\pi_m}y_{\\pi_n}  - \\bar{y}y_{\\pi_n} )\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\pi}[y_{\\pi_n}] &= \\sum^N_{k=1}y_k P[y_{\\pi_n} = y_k] \\\\\n",
    "&= \\sum^N_{k=1}y_k \\frac{1}{N} \\\\\n",
    "&= \\bar{y}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "When $m=n$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\pi}[y_{\\pi_m}y_{\\pi_n}] &= \\sum^N_{n=1}y^2_n P[y_{\\pi_m} = y_n]  \\\\\n",
    "&= \\sum^N_{n=1}\\frac{1}{N}y^2_n   \\\\\n",
    "&= \\bar{y}^2 + s^2_y \\\\\n",
    "\\end{align*}\n",
    "\n",
    "When $m\\ne n$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\pi}[y_{\\pi_m}y_{\\pi_n}] &= \\sum^N_{m=1}\\sum^N_{n\\ne m}y_my_n P[y_{\\pi_m} = y_m, y_{\\pi_n}=y_n]  \\\\\n",
    "&= \\sum^N_{m=1}\\sum^N_{n\\ne m}y_my_n P[y_{\\pi_m} = y_m]P[y_{\\pi_n}=y_n]  \\\\\n",
    "&= \\sum^N_{m=1}\\sum^N_{n\\ne m}y_my_n \\frac{1}{N}\\frac{1}{N-1}\\\\\n",
    "&= \\frac{1}{N(N-1)}\\sum^N_{m=1}y_m \\sum^N_{n\\ne m}y_n \\\\\n",
    "&= \\frac{1}{N(N-1)}\\sum^N_{m=1}y_m (N\\bar{y} - y_m) \\\\\n",
    "&= \\frac{1}{(N-1)}\\bar{y}\\sum^N_{m=1}y_m  - \\frac{1}{N(N-1)}\\sum^N_{m=1}y^2_m \\\\\n",
    "&= \\frac{N}{(N-1)}\\bar{y}^2  - \\frac{1}{N(N-1)}(N(\\bar{y}^2 + s^2_y)) \\\\\n",
    "&= \\bar{y}^2  - \\frac{s^2_y}{N-1} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) Take the expectation of the penalty in (a), we have\n",
    "\n",
    "\\begin{align*}\n",
    "E\\left[\\frac{2}{N}\\sum_{m,n=1}H_{mn}(y_{\\pi_m}y_{\\pi_n}  - \\bar{y}y_{\\pi_n} )\\right] &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}E[y_{\\pi_m}y_{\\pi_n}  - \\bar{y}y_{\\pi_n}]\\\\\n",
    " &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}\\left(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}E[y_{\\pi_n}]\\right)\\\\\n",
    " &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}\\left(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2\\right)\\\\\n",
    " &= \\left(\\frac{2}{N}\\sum_{m=n}H_{mn}(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2)\\right)  + \\left(\\frac{2}{N}\\sum_{m\\ne n}H_{mn}(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2)\\right)\\\\\n",
    " &= \\left(\\frac{2}{N}\\sum^N_{n=1}H_{nn}s^2_y\\right) - \\left(\\frac{2}{N}\\sum_{m\\ne n}H_{mn} \\frac{1}{N-1}s^2_y\\right)\\\\\n",
    " &= \\frac{2}{N}\\left(s^2_y\\text{trace}(H) - \\sum_{m\\ne n}H_{mn} \\frac{1}{N-1}s^2_y\\right)\\\\\n",
    " &= \\frac{2s^2_y}{N}\\left(\\text{trace}(H) -\\frac{1}{N-1} \\sum_{m\\ne n}H_{mn}\\right)\\\\\n",
    " &= \\frac{2s^2_y}{N}\\left(\\text{trace}(H) -\\frac{1}{N-1} (1^TH1 - \\text{trace}(H))\\right)\\\\\n",
    " &= \\frac{2s^2_y}{N-1}\\left(\\frac{N-1}{N}\\text{trace}(H) -\\frac{1}{N} (1^TH1 - \\text{trace}(H))\\right)\\\\\n",
    " &= \\frac{2s^2_y}{N-1}\\left(\\text{trace}(H) -\\frac{1^TH1 }{N}\\right)\\\\\n",
    " &= \\frac{2\\hat{\\sigma}^2_y}{N}\\left(\\text{trace}(H) -\\frac{1^TH1 }{N}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.19\n",
    "Follow the same derivation as in problem 9.18, we have \n",
    "\n",
    "When $m=n$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\pi}[y_{\\pi_m}y_{\\pi_n}] &= \\bar{y}^2 + s^2_y \\\\\n",
    "\\end{align*}\n",
    "\n",
    "When $m\\ne n$, note we can still have $y_{\\pi_m} = y_{\\pi_n}$ even when $m\\ne n$, so we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\pi}[y_{\\pi_m}y_{\\pi_n}] &= \\sum^N_{i=1}\\sum^N_{j=1}y_iy_j P[y_{\\pi_m} = y_i, y_{\\pi_n}=y_j]  \\\\\n",
    "&= \\sum^N_{i=1}\\sum^N_{j=1}y_iy_j P[y_{\\pi_m} = y_i]P[y_{\\pi_n}=y_j]  \\\\\n",
    "&= \\sum^N_{i=1}\\sum^N_{j=1}y_iy_j \\frac{1}{N}\\frac{1}{N}\\\\\n",
    "&= \\frac{1}{N^2}\\sum^N_{i=1}y_i \\sum^N_{j=1}y_j \\\\\n",
    "&= \\bar{y}^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take the expectation of the penalty in (a), we have\n",
    "\n",
    "\\begin{align*}\n",
    "E\\left[\\frac{2}{N}\\sum_{m,n=1}H_{mn}(y_{\\pi_m}y_{\\pi_n}  - \\bar{y}y_{\\pi_n} )\\right] &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}E[y_{\\pi_m}y_{\\pi_n}  - \\bar{y}y_{\\pi_n}]\\\\\n",
    " &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}\\left(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}E[y_{\\pi_n}]\\right)\\\\\n",
    " &= \\frac{2}{N}\\sum_{m,n=1}H_{mn}\\left(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2\\right)\\\\\n",
    " &= \\left(\\frac{2}{N}\\sum_{m=n}H_{mn}(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2)\\right)  + \\left(\\frac{2}{N}\\sum_{m\\ne n}H_{mn}(E[y_{\\pi_m}y_{\\pi_n}]  - \\bar{y}^2)\\right)\\\\\n",
    " &= \\frac{2}{N}\\sum^N_{n=1}H_{nn}s^2_y\\\\\n",
    " &= \\frac{2s^2_y}{N}\\text{trace}(H)\\\\\n",
    " \\end{align*}\n",
    " \n",
    " #### Problem 9.20\n",
    " \n",
    " See the answer in [Problem 5.2](https://github.com/niuers/Learn-From-Data-A-Short-Course/blob/master/Solutions%20to%20Chapter%205%20Exercises%20and%20Problems.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.21\n",
    "\n",
    "* (a) From [Problem 4.26](https://github.com/niuers/Learn-From-Data-A-Short-Course/blob/master/Solutions%20to%20Chapter%204%20Exercises%20and%20Problems.ipynb), we have the error on $x_n$ from data set $\\mathcal{D}_m$ as $e_n = \\left(\\frac{\\hat{y}^{(m)}_n - y_n}{1-H^{(m)}_{nn}}\\right)^2$, so the cross validation error for this data set is \n",
    "\n",
    "$E_{CV}(\\mathcal{D}_m) = \\frac{1}{N-1}\\sum_{n\\ne m} \\left(\\frac{\\hat{y}^{(m)}_n - y_n}{1-H^{(m)}_{nn}}\\right)^2$ \n",
    "\n",
    "as there are only $N-1$ data points.\n",
    "\n",
    "* (b) Let $H, A$ correspond to the full data set $\\mathcal{D}$\n",
    "\n",
    "From problem 4.26, we have $H^{(m)}_{nk} = z^T_n(A^{(m)})^{-1}z_k$ and $A^{(m)} = A - z_mz^T_m$, so \n",
    "\n",
    "\\begin{align*}\n",
    "H^{(m)}_{nk} &= z^T_n(A^{(m)})^{-1}z_k \\\\\n",
    "&= z^T_n\\left(A - z_mz^T_m\\right)^{-1}z_k\\\\\n",
    "&\\text{apply problem 4.26(b)}\\\\\n",
    "&= z^T_n\\left(A^{-1} + \\frac{A^{-1}z_mz^T_mA^{-1}}{1-z^T_mA^{-1}z_m}\\right)z_k\\\\\n",
    "&= z^T_nA^{-1}z_k + \\frac{z^T_nA^{-1}z_mz^T_mA^{-1}z_k}{1-z^T_mA^{-1}z_m}\\\\\n",
    "&= H_{nk} + \\frac{H_{mn}H_{mk}}{1-H_{mm}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) We have $\\hat{y} = Hy$, so \n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y}^{(m)}_n &= \\sum^{N}_{k=1, k\\ne m} H^{(m)}_{nk}y_k \\\\\n",
    "&= \\sum^N_{k=1,k\\ne m} \\left(H_{nk} + \\frac{H_{mn}H_{mk}}{1-H_{mm}}\\right)y_k\\\\\n",
    "&= \\sum^N_{k=1,k\\ne m} \\left(H_{nk}y_k + \\frac{H_{mn}H_{mk}}{1-H_{mm}}y_k\\right)\\\\\n",
    "&= \\sum^N_{k=1} \\left(H_{nk}y_k + \\frac{H_{mn}H_{mk}}{1-H_{mm}}y_k\\right) - (H_{nm} + \\frac{H_{mn}H_{mm}}{1-H_{mm}})y_m\\\\\n",
    "&= \\hat{y}_n + \\sum_k \\frac{H_{mn}H_{mk}}{1-H_{mm}}y_k - \\frac{H_{mn}}{1-H_{mm}}y_m\\\\\n",
    "&= \\hat{y}_n + \\frac{H_{mn}}{1-H_{mm}} \\sum_k H_{mk}y_k- \\frac{H_{mn}}{1-H_{mm}}y_m \\\\\n",
    "&= \\hat{y}_n + \\frac{H_{mn}}{1-H_{mm}} \\hat{y}_m - \\frac{H_{mn}}{1-H_{mm}}y_m\\\\\n",
    "&= \\hat{y}_n + \\frac{\\hat{y}_m - y_m}{1-H_{mm}}H_{mn} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) Take the results of problem (b), (c) into the formula for $E_{CV}(\\mathcal{D}_m)$ we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{CV}(\\mathcal{D}_m) &= \\frac{1}{N-1}\\sum_{n\\ne m} \\left(\\frac{\\hat{y}^{(m)}_n - y_n}{1-H^{(m)}_{nn}}\\right)^2 \\\\\n",
    "&= \\frac{1}{N-1}\\sum_{n\\ne m} \\left(\\frac{\\hat{y}_n + \\frac{\\hat{y}_m - y_m}{1-H_{mm}}H_{mn}  - y_n}{1-(H_{nn} + \\frac{H_{mn}H_{mn}}{1-H_{mm}})}\\right)^2\\\\\n",
    "&= \\frac{1}{N-1}\\sum_{n\\ne m} \\left(\\frac{\\hat{y}_n - y_n + \\frac{\\hat{y}_m - y_m}{1-H_{mm}}H_{mn}  }{1-H_{nn} - \\frac{H^2_{mn}}{1-H_{mm}}}\\right)^2 \\\\\n",
    "&= \\frac{1}{N-1}\\sum_{n=1} \\left(\\frac{\\hat{y}_n - y_n + \\frac{\\hat{y}_m - y_m}{1-H_{mm}}H_{mn}  }{1-H_{nn} - \\frac{H^2_{mn}}{1-H_{mm}}}\\right)^2  - \\frac{1}{N-1}\\left(\\frac{\\hat{y}_m - y_m}{1-2H_{mm}}\\right)^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (e) We also have $E_{CV}(\\mathcal{D}) = \\frac{1}{N}\\sum_{n=1} \\left(\\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right)^2$\n",
    "\n",
    "so the leverage $l_m = E_{CV}(\\mathcal{D}) - E_{CV}(\\mathcal{D}_m)$\n",
    "\n",
    "To compute the leverage, we only need calculate the hat matrix $H$ for the full data set $\\mathcal{D}$ once, and make predictions $\\hat{y}$ once. \n",
    "So this is $N$ times more cost than a normal linear regression."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "#### Problem 9.22\n",
    "\n",
    "def compute_full_Ecv(H, y_hat, y):\n",
    "    N = y_hat.shape[0]\n",
    "    Ecv = 0\n",
    "\n",
    "    for n in np.arange(N):\n",
    "        yhat_n = y_hat[n]\n",
    "        yn = y[n]\n",
    "        Hnn = H[n,n]\n",
    "        r = (yhat_n - yn)/(1-Hnn)\n",
    "        Ecv += r**2\n",
    "    Ecv = Ecv / N\n",
    "    return Ecv\n",
    "\n",
    "def compute_loo_Ecv(H, y_hat, y, m):\n",
    "    #m is the point that was left out\n",
    "    N = y.shape[0]\n",
    "    Ecv = 0\n",
    "    yhat_m = y_hat[m]\n",
    "    ym = y[m]\n",
    "    Hmm = H[m, m]    \n",
    "    r0 = (yhat_m - ym) /(1-Hmm)\n",
    "    #print(ym, yhat_m, Hmm)\n",
    "    for n in np.arange(N):\n",
    "        yhat_n = y_hat[n]\n",
    "        yn = y[n]\n",
    "        Hnn = H[n,n]  \n",
    "        Hmn = H[m,n]\n",
    "        \n",
    "        r1 = yhat_n - yn + r0 * Hmn\n",
    "        r2 = 1 - Hnn - Hmn*Hmn/(1-Hmm)\n",
    "        r = r1/r2\n",
    "        Ecv += r**2\n",
    "        \n",
    "    Ecv = Ecv / (N-1)\n",
    "    r3 = (yhat_m - ym)/(1-2.0*Hmm)\n",
    "    Ecv =  Ecv - r3**2/(N-1)\n",
    "    return Ecv\n",
    "    \n",
    "        \n",
    "def leverage(X, y, lambda_t):\n",
    "    #Compute the leverage of each point m\n",
    "    H = lm.compute_H(X, lambda_t) #full H\n",
    "    \n",
    "    y_hat = lm.compute_y_hat(H, y)\n",
    "    Ecv_full = compute_full_Ecv(H, y_hat, y)\n",
    "    N, _ = X.shape\n",
    "    full_indices = np.arange(N)\n",
    "    lms = []\n",
    "    for m in np.arange(N):\n",
    "        indices = np.hstack([full_indices[:m], full_indices[m+1:]])\n",
    "        Xm = X[indices]\n",
    "        ym = y[indices]\n",
    "        #print(f\"--Current point {m}: {ym}\")\n",
    "        Ecv_m = compute_loo_Ecv(H, y_hat, y, m)\n",
    "        lm = Ecv_full - Ecv_m\n",
    "        lms.append(lm)\n",
    "    return lms\n",
    "\n",
    "X = np.array([[1, 0.51291], [1, 0.46048], [1, 0.3504], [1, 0.095046], [1, 0.43367], [1, 0.70924], [1, 0.11597]])\n",
    "y = np.array([0.36542, 0.22156, 0.15263, 0.10355, 0.10015, 0.26713, 2.3095])\n",
    "lambdas = np.hstack([np.arange(0., 0.1, 0.01), np.arange(0.1, 1, 0.05), np.arange(1, 10, 1)])\n",
    "last_lvs = []\n",
    "for lambda_t in lambdas:\n",
    "    lvs = leverage(X, y, lambda_t)\n",
    "    #print(lvs)\n",
    "    last_lvs.append(lvs[-1])\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "No handles with labels found to put in legend.\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "myplot.plt_plot([lambdas], [last_lvs], 'plot', ['b', 'r'], ['o', 'x'], ['', ''], \n",
    "        title = 'Problem 9.22 Last Point Leverage vs. Lambda', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'Lambda', ylabel = 'Leverage',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.22\n",
    "\n",
    "As $\\lambda$ increases, we see the leverage of the last point decreases, which means that with larger regularization, the algorithm tries less harder to fit the outlier. \n",
    "\n",
    "#### Problem 9.23"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_train:  (500, 256) y_train:  (500,) X_test:  (6791, 256) y_test:  (6791,)\n",
      "Frequencies of the digits: \n",
      " {0: 0.164, 1: 0.138, 2: 0.1, 3: 0.09, 4: 0.09, 5: 0.076, 6: 0.092, 7: 0.088, 8: 0.074, 9: 0.088}\n"
     ]
    }
   ],
   "source": [
    "#### Problem 9.23\n",
    "zip_data_path = './data/usps.h5'\n",
    "data_splits = data.split_zip_data(zip_data_path, splits = 1)\n",
    "\n",
    "X_train, y_train, X_test, y_test = data_splits[0]\n",
    "\n",
    "print('X_train: ', X_train.shape, 'y_train: ',y_train.shape, 'X_test: ',X_test.shape, 'y_test: ', y_test.shape)\n",
    "unique, counts = np.unique(y_train, return_counts=True)\n",
    "freqs = counts/len(y_train)\n",
    "print('Frequencies of the digits: \\n', dict(zip(unique, freqs)))\n",
    "\n",
    "y_train, y_test = data.set_two_classes(y_train, y_test, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "lambda_t = 0\n",
    "X_tr, X_te = data.compute_features(X_train, X_test)\n",
    "lvs = leverage(X_tr, y_train, lambda_t)\n",
    "lvs = np.array(lvs)\n",
    "ids = np.argsort(-lvs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "large_leverage_ids = ids[:10]\n",
    "\n",
    "x1s = [X_tr[y_train==1, 0], X_tr[y_train==-1, 0], X_tr[large_leverage_ids, 0]]\n",
    "x2s = [X_tr[y_train==1, 1], X_tr[y_train==-1, 1], X_tr[large_leverage_ids, 1]]\n",
    "myplot.plt_plot(x1s, x2s, 'scatter', ['b', 'r', 'g'], ['o', 'x', '+'], ['Digit 1', 'Non Digit 1', 'Outliers'], \n",
    "        title = 'Problem 9.23 Digits 1 vs non-1 (with outliers)', yscale = None, ylb = None, yub = None,\n",
    "        xlb = None, xub = None, xlabel = 'Average Intensity', ylabel = 'Symmetry',\n",
    "        legends = ['+', '-'], legendx = None, legendy = None, marker_sizes=[15, 15, 100]) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.23 \n",
    "\n",
    "Most of the points with large leverage are on the boundaries between digit '1' and non digit '1'. Remove them will help achieve better cross validation errors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 9.24\n",
    "\n",
    "* (a) For $\\hat{t}_n$, we use $N-1$ data points to estimate $t$, since $\\hat{t}(\\mathcal{Z}_n)$ is asymptotically unbiased, we have\n",
    "\n",
    "$E_{\\mathcal{Z}}[\\hat{t}(\\mathcal{Z})] = t + \\frac{a_1}{N-1} + \\frac{a_2}{(N-1)^2} + \\dots $\n",
    "\n",
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{Z}}[\\hat{\\tau}_n] &= E_{\\mathcal{Z}}[N\\hat{t}(\\mathcal{Z}) - (N-1)\\hat{t}_n]\\\\\n",
    "&= NE_{\\mathcal{Z}}[\\hat{t}(\\mathcal{Z})] - (N-1)E_{\\mathcal{Z}}[\\hat{t}_n]\\\\\n",
    "&= N(t+\\frac{a_1}{N} + \\frac{a_2}{N^2} + \\dots) - (N-1)(t + \\frac{a_1}{N-1} + \\frac{a_2}{(N-1)^2} + \\dots)\\\\\n",
    "&= t - \\frac{a_2}{N(N-1)} + O(\\frac{1}{N^2})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) It's clear that with $N-1$ data points, we have \n",
    "\n",
    "\\begin{align*}\n",
    "s^2_n &= \\frac{1}{N-1}\\left(\\sum^N_{m=1}x^2_m - x^2_n\\right) - \\frac{1}{(N-1)^2}\\left(\\sum^N_{m=1}x_m - x_n\\right)^2\\\\\n",
    "&= \\frac{1}{N-1}\\sum^N_{m=1}x^2_m - \\frac{1}{N-1}x^2_n- \\frac{1}{(N-1)^2}\\left(\\left(\\sum^N_{m=1}x_m\\right)^2 -2 x_n\\sum^N_{m=1}x_m + x^2_n\\right)\\\\\n",
    "&= \\frac{1}{N-1}\\sum^N_{m=1}x^2_m - \\frac{1}{(N-1)^2}\\left(\\sum^N_{m=1}x_m\\right)^2 - \\frac{N}{(N-1)^2}x^2_n + \\frac{2x_n}{(N-1)^2} \\sum^N_{m=1}x_m\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "s^2_J &= Ns^2 - \\frac{N-1}{N}\\sum_n s^2_n \\\\\n",
    "&= \\sum^N_{n=1}x^2_n - \\frac{1}{N}\\left(\\sum^N_{n=1}x_n\\right)^2 - \\sum_n \\left[\\frac{1}{N}\\sum^N_{m=1}x^2_m - \\frac{1}{N(N-1)}\\left(\\sum^N_{n=1}x_n\\right)^2 - \\frac{1}{N-1}x^2_n + \\frac{2x_n}{N(N-1)} \\sum^N_{n=1}x_n  \\right] \\\\\n",
    "&= \\frac{1}{N(N-1)}\\left(\\sum^N_{n=1}x_n\\right)^2 - \\frac{2}{N(N-1)}\\sum^N_{n=1}x_n \\sum^N_{m=1}x_m + \\frac{1}{N-1}\\sum^N_{n=1}x^2_n  \\\\\n",
    "&= \\frac{1}{N-1}\\sum^N_{n=1}x^2_n - \\frac{1}{N(N-1)}\\left(\\sum^N_{n=1}x_n\\right)^2 \\\\\n",
    "&= \\frac{N}{N-1}\\left(\\frac{1}{N}\\sum^N_{n=1}x^2_n - \\frac{1}{N^2}\\left(\\sum^N_{n=1}x_n\\right)^2\\right) \\\\\n",
    "&= \\frac{N}{N-1}s^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) If $t(\\mathcal{Z})$ has an asymptotic bias, the $E_{\\mathcal{Z}}[\\hat{\\tau}_n]$ will have the same bias. The jackknife estimate will have the same bias as well.\n",
    "\n",
    "* (e) If the leading order term in the bias was $\\frac{1}{N^2}$, since $\\frac{1}{N}$ term doesn't appear in $E_{\\mathcal{Z}}[\\hat{\\tau}_n]$, so it's not affected by the $\\frac{1}{N}$ term, the Jackknife estimate will have the same bias.\n",
    "\n",
    "#### Problem 9.25\n",
    "\n",
    "* (a) The $E_{in}$ is downward biased, i.e. tend to lower the estimate of $\\mathcal{E}_{out}(N)$. When the number of points $N$ is large, it is asymptotically unbiased.\n",
    "\n",
    "* (b) The $E_{in}(g^{(\\mathcal{D})})$ is the estimate with full data set, and $E_{in}(g^{(\\mathcal{D}_n)})$ is the estimate with leave one out sample set.\n",
    "  * (i) The pseudo-values are $\\hat{\\tau}_n = NE_{in}(g^{(\\mathcal{D})}) - (N-1)E_{in}(g^{(\\mathcal{D}_n)})$ by definition.\n",
    "  * (ii) The jackknife estimate is thus $E_J = NE_{in}(g^{(\\mathcal{D})}) - \\frac{N-1}{N}\\sum^N_{n=1}E_{in}(g^{(\\mathcal{D}_n)})$\n",
    "  \n",
    "* (c) Take expectation on   $E_{in}(g^{(\\mathcal{D}_n)})$, since it has $N-1$ data points, we have\n",
    "\n",
    "$E_{\\mathcal{D}}[E_{in}(g^{(\\mathcal{D}_n)})] = \\mathcal{E}_{out}(N-1) + \\frac{a_1}{N-1} + \\frac{a_2}{(N-1)^2} + \\dots$\n",
    "\n",
    "Take expectation on $E_J$ we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D}}[E_J] &= N E_{\\mathcal{D}}\\left[ E_{in}(g^{(\\mathcal{D})})\\right] - \\frac{N-1}{N}\\sum^N_{n=1} E_{\\mathcal{D}}\\left[E_{in} (g^{(\\mathcal{D}_n)}) \\right]\\\\\n",
    "&= N\\left[\\mathcal{E}(N)+\\frac{a_1}{N}+\\frac{a_2}{N^2}\\right] - \\frac{N-1}{N}\\sum^N_{n=1}\\left[\\mathcal{E}(N-1)+\\frac{a_1}{N-1}+\\frac{a_2}{(N-1)^2}\\right]\\\\\n",
    "&= \\mathcal{E}(N) + (N-1)(\\mathcal{E}(N) - \\mathcal{E}(N-1)) + O(\\frac{1}{N^2})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) If the learning curve converges, then we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D}}[E_J] &= \\mathcal{E}(N) + (N-1)(\\mathcal{E}(N) - \\mathcal{E}(N-1)) + O(\\frac{1}{N^2}) \\\\\n",
    "&= \\mathcal{E}(N)  + (N-1)\\left(E + \\frac{b_1}{N}+\\frac{b_2}{N^2} - (E + \\frac{b_1}{N-1}+\\frac{b_2}{(N-1)^2})\\right) + O(\\frac{1}{N^2})\\\\\n",
    "&= \\mathcal{E}(N)  - \\frac{b_1}{N} +  O(\\frac{1}{N^2})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "When the learning curve bias term is smaller than the in-sample bias term, for example, when the sample is not representative of the data distribution. Jackknife estimate will be helpful.\n",
    "\n",
    "#### Problem 9.26 TODO\n",
    "#### Problem 9.27 TODO\n",
    "#### Problem 9.28 TODO"
   ]
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