Exams/Midterm-Sample.ipynb

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     "source": [
      "Instructions\n",
      "----\n",
      "\n",
      "This is a \"closed book\" examination - in particular, you are not to use any resources outside of this notebook (except possibly pen and paper). You may consult help from within the notebook using ? but not any online references. You should turn wireless off or set your laptop in \"Airplane\" mode prior to taking the exam. \n",
      "\n",
      "You have 2 hours to complete the exam."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import os\n",
      "import sys\n",
      "import glob\n",
      "import matplotlib.pyplot as plt\n",
      "import numpy as np\n",
      "import pandas as pd\n",
      "%matplotlib inline\n",
      "plt.style.use('ggplot')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.set_printoptions(formatter={'float': '{: 0.3f}'.format})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rpy2.ipython"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 1 (10 points)**. \n",
      "\n",
      "Euclid's algorithm for finding the greatest common divisor of two numbers is\n",
      "\n",
      "```python\n",
      "gcd(a, 0) = a\n",
      "gcd(a, b) = gcd(b, a modulo b)\n",
      "```\n",
      "\n",
      "1. Write a function to find the greatest common divisor in Python (4 points)\n",
      "2. What is the greatest common divisor of 17384 and 1928? (1 point)\n",
      "3. Write a function to calculate the least common multiple (4 points)\n",
      "4. What is the least common multiple of 17384 and 1928? (1 point)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%file dummy.csv\n",
      "humpty,dumpty,sat,on,a,wall\n",
      "humpty,dumpty,had,a,great,fall\n",
      "all,the,king's,horses\n",
      "and,all,the,king's,men\n",
      "couldn't,put,humpty,dumpty,together,again"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 2 (10 points)**. \n",
      "\n",
      "1. Use Unix commands to count the number of words containing 'in' in the last two lines (2 points)\n",
      "2. Using Python, read the `dummy.csv` file, and print a list of each word together with its frequency (4 points)\n",
      "3. What is the average frequency of words that begin with a vowel (aeiou)? (2 points)\n",
      "4. Create a Pandas DataFrame with 2 columns ('Word', 'Frequency') such that each row consists of a word in the nursery rhyme and its frequency. (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 3 (10 points)**.\n",
      "\n",
      "1. What is the gradient and Hessian of the function $f(x, y) = 3x^2y^2 + 2x^2y + xy^2 + 4y + 7$? (4 points)\n",
      "2. What is the condition number of the Hessian? (4 points)\n",
      "3. Is the Hessaian positive definite? (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 4 (20 points)**. \n",
      "\n",
      "Consider the following matrix $A$ with dimensions (4,6), to be interpreted as 4 rows of the measurements of 6 features.\n",
      "```python\n",
      "np.array([[5, 5, 2, 6, 2, 0],\n",
      "          [8, 6, 7, 8, 9, 7],\n",
      "          [9, 5, 0, 4, 6, 8],\n",
      "          [8, 7, 9, 3, 6, 1]])\n",
      "```\n",
      "\n",
      "1. Add 1 to the first row, 2 to the second row, 3 to the third row and 4 to the fourth row using a vector `v = np.array([1,2,3,4])` and broadcasting. (2 points)\n",
      "2. What are the singular values of this matrix? (3 points)\n",
      "3. What are the eigenvalues of the covariance matrixs? (3 points)\n",
      "4. Find the solution vector $x$ if $Ax = y$ where `y = np.reshape([1,2,3,4], (4,1))` (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 5 (10 points)**.\n",
      "\n",
      "1. Prove that $e^{x^2 + y^2}$ is a convex function using `sympy`. Hint: the `hessian` function and `eigenvals` matrix methods may be useful, as is the built-in help using `?`. (5 points)\n",
      "2. Using `scipy.optimize`, find the values of $x$ and $y$ that minimize $e^{x^2 + y^2}$ in the unconstrained case and in the presence of the constraint that $x + y = 3$. (5 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n"
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     "language": "python",
     "metadata": {},
     "outputs": []
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 6 (10 points)**.\n",
      "\n",
      "Given the following covariance matrix\n",
      "```python\n",
      "A = np.array([[2,1],[1,4]])\n",
      "```\n",
      "\n",
      "1. What is the vector representing the first principal component direction? (3 points)\n",
      "2. Find $A^{100}$ without repeated matrix multiplication or the `matrix_power` function. (4 points)\n",
      "3. What is the explained variance if we keep only the projection onto the first principal component? (3 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 7 (10 points)**.\n",
      "\n",
      "A milkmaid is at point A and needs to get to point B. However, she also needs to fill a pail of water from the river en route from A to B. The equation of the river's path is shown in the figure below. What is the minimum distance she has to travel to do this?\n",
      "\n",
      "Hints:\n",
      "\n",
      "- The milkmaid's path must include some point P(x, y) that is on the river\n",
      "- Use Lagrange multipliers\n",
      "- Set up a consttrained optimization problem and solve with `scipy.optimize`\n",
      "\n",
      "![Milkmaid problem](milkmaid.png)"
     ]
    },
    {
     "cell_type": "code",
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     "input": [
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
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     "metadata": {},
     "outputs": []
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