5_1_PCA.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 5-1: PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-3, -2, -1,  0,  1,  2, -2, -1,  0,  1,  2, -2, -1,  0,  1,  2,  3],\n",
       "       [-2, -1,  0,  1,  2,  3, -2, -1,  0,  1,  2, -3, -2, -1,  0,  1,  2]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = np.array( [ (-3,-2), (-2,-1), (-1,0), (0,1), (1,2), (2,3), (-2,-2), (-1,-1), (0,0), (1,1), (2,2), (-2,-3), (-1,-2), (0,-1), (1,0), (2,1), (3,2)]).T\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*(a)* Compute the covariance matrix M."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3.   ,  2.625],\n",
       "       [ 2.625,  3.   ]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cov_m = np.cov(x, rowvar=True)\n",
    "cov_m"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*(b)* Compute the eigenvalues and eigenvectors of M ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.375  5.625]\n",
      "[[-0.70710678  0.70710678]\n",
      " [ 0.70710678  0.70710678]]\n"
     ]
    }
   ],
   "source": [
    "eigenvalues, normalized_eigenvectors = np.linalg.eigh(cov_m)\n",
    "print(eigenvalues)\n",
    "print(normalized_eigenvectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*(c)* Find the smallest eigenvalue and find the related eigenvector as well. The resulted eigenvector builds the\n",
    "basis for the new subspace.\n",
    "Note: Shouldn't we pick the biggest eigenvalue(s) for PCA?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "min_eigenvalue_index = np.argmin(eigenvalues)\n",
    "min_eigenvalue_index"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*(d)* Transform vectors of X in this new subspace. $y = W^T \\times x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.70710678,  0.70710678])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W = normalized_eigenvectors[min_eigenvalue_index]\n",
    "W"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.70710678,  0.70710678,  0.70710678,  0.70710678,  0.70710678,\n",
       "        0.70710678,  0.        ,  0.        ,  0.        ,  0.        ,\n",
       "        0.        , -0.70710678, -0.70710678, -0.70710678, -0.70710678,\n",
       "       -0.70710678, -0.70710678])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W.T.dot(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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