pca_of_exam_grades.ipynb

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    "Sascha Spors,\n",
    "Professorship Signal Theory and Digital Signal Processing,\n",
    "Institute of Communications Engineering (INT),\n",
    "Faculty of Computer Science and Electrical Engineering (IEF),\n",
    "University of Rostock,\n",
    "Germany\n",
    "\n",
    "# Data Driven Audio Signal Processing - A Tutorial with Computational Examples\n",
    "\n",
    "Master Course #24512\n",
    "\n",
    "- lecture: https://github.com/spatialaudio/data-driven-audio-signal-processing-lecture\n",
    "- tutorial: https://github.com/spatialaudio/data-driven-audio-signal-processing-exercise\n",
    "\n",
    "Feel free to contact lecturer frank.schultz@uni-rostock.de\n",
    "\n",
    "# PCA on Achieved Points of Written Examination "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy\n",
    "from scipy.linalg import svd, diagsvd\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "np.set_printoptions(precision=3, sign=' ', suppress=True)\n",
    "\n",
    "print(np.__version__)  # tested with 1.26.4\n",
    "print(scipy.__version__)  # tested with 1.13.1\n",
    "print(mpl.__version__)  # tested with 3.9.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.loadtxt(open(\"exam_points_meanfree_unitvar.csv\", \"rb\"), delimiter=\";\", skiprows=0)\n",
    "N, F = X.shape\n",
    "print(N, F)  # 34 students, 5 tasks for exam on signals & systems, a typical course in electrical engineering bachelor studies\n",
    "# columns correspond to theses tasks\n",
    "task_label = ['Task 1: Convolution', 'Task 2: Fourier', 'Task 3: Sampling', 'Task 4: Laplace Domain', 'Task 5: z-Domain']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# data in exam_points_meanfree_unitvar.csv is already mean-free and columns have var=1\n",
    "# so the numbers in X do not represent points or percentage,\n",
    "# but rather encode the performance of the students per task in a normalised way\n",
    "# X is however sorted: first row belongs to best grade, last row to worst grade\n",
    "np.mean(X, axis=0), np.std(X, axis=0, ddof=1), np.var(X, axis=0, ddof=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# for completeness of PCA algorithm ->\n",
    "# make X zscore (altough it is already)\n",
    "mu = np.mean(X, axis=0)\n",
    "X = X - mu  # de-mean\n",
    "sigma = np.sqrt(np.sum(X**2, axis=0) / (N-1))\n",
    "X = X / sigma  # normalise to std=1\n",
    "np.mean(X, axis=0), np.std(X, axis=0, ddof=1), np.var(X, axis=0, ddof=1)  # check"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X  # print mean=0 / var=1 data matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# get SVD / CovMatrix stuff\n",
    "[U, s, Vh] = svd(X)\n",
    "V = Vh.T  # we don't use Vh later on!\n",
    "S = diagsvd(s, N, F)  # sing vals matrix\n",
    "D, _ = np.linalg.eig(X.T @ X / (N-1))  # eig vals\n",
    "D = -np.sort(-D)  # sort them, then ==\n",
    "d = s**2 / (N-1)\n",
    "print(np.allclose(d, D))  # so we go for d later on\n",
    "\n",
    "# switch polarities for nicer interpretation ot the\n",
    "# exam data\n",
    "V[:,0] *= -1\n",
    "U[:,0] *= -1\n",
    "\n",
    "V[:,2] *= -1\n",
    "U[:,2] *= -1\n",
    "\n",
    "V[:,3] *= -1\n",
    "U[:,3] *= -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# PCA\n",
    "US = U @ S\n",
    "PC_Features = US @ diagsvd(1 / np.sqrt(d), F, F)  # normalised such that columns have var 1, aka (normalised) PC scores\n",
    "print(np.var(PC_Features, axis=0, ddof=1))\n",
    "#PC_Loadings = (diagsvd(np.sqrt(d), F, F) @ V.T).T  # ==\n",
    "PC_Loadings = V @ diagsvd(np.sqrt(d), F, F)  # aka PC coeff, not unit-length anymore, but normalised such that it shows correlation between PC_Features and X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.allclose(X, PC_Features @ PC_Loadings.T)  # check correct matrix factorisation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# project an x column vector to a pc feature column -> do this for all options -> get all weights for linear comb of pc features\n",
    "# correlation uses unit-length vectors\n",
    "PC_Loadings_manual = np.zeros((F, F))\n",
    "for row in range(F):\n",
    "    tmp_x = X[:, row] / np.linalg.norm(X[:, row])\n",
    "    for column in range(F):\n",
    "        tmp_pc = PC_Features[:, column] / np.linalg.norm(PC_Features[:, column])\n",
    "        PC_Loadings_manual[row, column] = np.inner(tmp_pc, tmp_x)\n",
    "np.allclose(PC_Loadings_manual, PC_Loadings)  # we get the PC_Loadings matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# explained variance\n",
    "d, np.var(US, axis=0, ddof=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# explained cum variance in %\n",
    "cum_var = np.cumsum(d) / np.sum(d) * 100\n",
    "cum_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Check via Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12,8))\n",
    "\n",
    "plt.subplot(2,1,1)\n",
    "for f in range(F):\n",
    "    plt.plot(X[:, f], 'o-', color='C'+str(f), label='Task '+str(f+1), ms=3)\n",
    "plt.legend(loc='lower left')\n",
    "plt.xticks([0, N-1], labels=['best grade', 'worst grade'])\n",
    "plt.ylabel('normalised points (mean-free, var=1)')\n",
    "plt.grid(True)\n",
    "plt.title(task_label)\n",
    "\n",
    "plt.subplot(2,1,2)\n",
    "for f in range(F):\n",
    "    plt.plot(US[:, f], 'o-', color='C'+str(f), label='PCA v '+str(f+1), lw=(F-f)*2/3, ms=(F-f)*3/2)\n",
    "plt.legend(loc='lower left')\n",
    "plt.xticks([0, N-1], labels=['best grade', 'worst grade'])\n",
    "plt.ylabel('PC features (mean-free, sorted var)')\n",
    "plt.xlabel('student index (sorted grade)')\n",
    "plt.grid(True)\n",
    "plt.title(['cum var in %:', cum_var])\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# correlation between task and pc\n",
    "pc_label = ['PC 1', 'PC 2', 'PC 3', 'PC 4', 'PC 5']\n",
    "cmap = plt.get_cmap('Spectral_r', 8)\n",
    "fig = plt.figure(figsize=(6,4))\n",
    "ax = fig.add_subplot(111)\n",
    "cax = ax.matshow(PC_Loadings, cmap=cmap, vmin=-1, vmax=+1)\n",
    "fig.colorbar(cax)\n",
    "ax.set_xticks(np.arange(len(pc_label)))\n",
    "ax.set_yticks(np.arange(len(task_label)))\n",
    "ax.set_xticklabels(pc_label)\n",
    "ax.set_yticklabels(task_label)\n",
    "ax.set_title('Loading Matrix = PC x contributes to Task y')\n",
    "plt.tight_layout()\n",
    "\n",
    "# a rank 3 approximation of the data,\n",
    "# i.e. using only PC1, PC2 and PC3 in the linear combination to reconstruct X\n",
    "# would only change one grade by a 1/3 grade step\n",
    "# so 85.899 % explained variance would be enough to figure the actual grading\n",
    "\n",
    "# PC1 and PC2 might allow an intuitive interpretation:\n",
    "# students are very well prepared to convolution, Laplace and z-Domain tasks\n",
    "# as theses tasks are always very similar and definitiely will be queried in the exam\n",
    "# so, PC1 indidcates the performance on 'fulfilled' expectations and is\n",
    "# highly correlated with the achieved grade\n",
    "# the Fourier task and the sampling task were chosen out of a wide range of options\n",
    "# here students have rather 'unknown' expectations, which is why we need PC 2 to cover this\n",
    "#\n",
    "# PC 3 to 5 show positive vs. negative correlations, i.e. mostly one good task vs. one bad task performance\n",
    "# some of these results are intuitive: we know that students sometimes have preferences for Laplace vs. z-Domain  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sum(PC_Loadings**2, axis=0)  # that's again the explained variance of the PCs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sum(PC_Loadings**2, axis=1)  # communalities, must sum to 1 in our normalised handling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Copyright\n",
    "\n",
    "- the notebooks are provided as [Open Educational Resources](https://en.wikipedia.org/wiki/Open_educational_resources)\n",
    "- feel free to use the notebooks for your own purposes\n",
    "- the text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/)\n",
    "- the code of the IPython examples is licensed under the [MIT license](https://opensource.org/licenses/MIT)\n",
    "- please attribute the work as follows: *Frank Schultz, Data Driven Audio Signal Processing - A Tutorial Featuring Computational Examples, University of Rostock* ideally with relevant file(s), github URL https://github.com/spatialaudio/data-driven-audio-signal-processing-exercise, commit number and/or version tag, year."
   ]
  }
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