Labs/lab06.ipynb

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   "source": [
    "# Initialize Otter\n",
    "import otter\n",
    "grader = otter.Notebook(\"lab06.ipynb\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "grade": false,
     "grade_id": "intro",
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     "schema_version": 2,
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   "source": [
    "# Lab 6: Modeling, Summary Statistics and Loss Functions\n",
    "\n",
    "This lab will utilize the \"flipped classroom\" model, where you will be introduced and exposed to some concepts in the lab, and then will learn more about the theory in lecture.\n",
    "\n",
    "In this lab, you will:\n",
    "\n",
    "1. Define loss functions and find the arguments that minimize them.\n",
    "2. Explore different statistics and techniques for optimization.\n",
    "\n",
    "**This assignment should be completed and submitted by Tuesday, October 5th, 2021 at 11:59 PM PDT**.\n",
    "\n",
    "\n",
    "### Collaboration Policy\n",
    "\n",
    "Data science is a collaborative activity. While you may talk to others about the labs, we ask that you **write your solutions individually**. If you do discuss the assignments with others, please **include their names** in the following cell:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*List collaborators here*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "q2",
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   "source": [
    "## Constant Model and Loss Functions\n",
    "\n",
    "### Constant Model\n",
    "\n",
    "In the modeling context, $y$ represents our \"true observations\", which are typically what we are trying to model. $\\hat{y}$ represents our prediction (for any model). In this lab, we will use the constant model, where our prediction for any input is a constant:\n",
    "\n",
    "$$\\Large\n",
    "\\hat{y} = \\theta\n",
    "$$\n",
    "\n",
    "$\\theta$ is what we call a **parameter**. Our goal is to find the value of our parameters that **best fit our data**. We represent the optimal parameter(s) with $\\hat{\\theta}$.\n",
    "\n",
    "We call the constant model a **summary statistic**, as we are determining one number that best \"summarizes\" a set of values.\n",
    "\n",
    "\n",
    "### Loss function\n",
    "\n",
    "Loss functions are what we use to determine the optimal parameter(s) for our model.\n",
    "\n",
    "A loss function is a measure of how well a model is able to predict the expected outcome. In other words, it measures the deviations of the predicted values from the observed values. In this lab we will implement the squared loss and absolute loss functions.  \n",
    "\n",
    "In the formulations below $y$ represents the observed values and $\\hat{y}$ stands for our prediction.\n",
    "\n",
    "1. **Squared Loss** (also known as the $L_2$ loss, pronounced \"ell-two\"):\n",
    "\n",
    "$$\\Large L(y, \\hat{y}) = (y - \\hat{y})^2$$\n",
    "\n",
    "2. **Absolute Loss** (also known as the $L_1$ loss, pronounced \"ell-one\"):\n",
    "\n",
    "$$\\Large L\\left(y, \\hat{y} \\right) = \\left| y - \\hat{y} \\right|$$\n",
    "\n",
    "Since we are using the constant model $\\hat{y} = \\theta$ for the remainder of the lab, we will instead refer to these loss functions as being $(y - \\theta)^2$ and $|y - \\theta|$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "_cell_guid": "33c63379-d85b-4638-8183-d008fdb96de7",
    "_uuid": "7ad7f9f24df7dba8ac92d234890835f6b9970834",
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   "source": [
    "# Run this cell\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "np.random.seed(42)\n",
    "plt.style.use('fivethirtyeight')\n",
    "sns.set()\n",
    "sns.set_context(\"talk\")\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1: Implement the Squared Loss Function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
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     "grade_id": "q2a",
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   },
   "source": [
    "\n",
    "### Question 1a\n",
    "\n",
    "$$\\Large\n",
    "L\\left(y,  \\theta \\right) = \\left( y - \\theta \\right)^2\n",
    "$$\n",
    "\n",
    "Based on the comments below, implement the squared loss function. Your answer should not use any loops.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q1a\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def squared_loss(y_obs, theta):\n",
    "    \"\"\"\n",
    "    Calculate the squared loss of the observed data and a summary statistic.\n",
    "    \n",
    "    Parameters\n",
    "    ------------\n",
    "    y_obs: an observed value\n",
    "    theta : some constant representing a summary statistic\n",
    "    \n",
    "    Returns\n",
    "    ------------\n",
    "    The squared loss between the observation and the summary statistic.\n",
    "    \"\"\"\n",
    "    return (y_obs - theta) ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q1a</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q1a results: All test cases passed!"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q1a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Q1b: Plotting the Squared Loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "Let us now consider the case where `y_obs` equals 10. For arbitrary values of `theta`, plot the squared loss using the function above\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q1b\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_obs = 10\n",
    "theta_values = np.linspace(0, 20, 100) # some arbitrary values of theta\n",
    "plt.plot(theta_values, squared_loss(y_obs, theta_values))\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('L2 loss')\n",
    "plt.title(r'L2 Loss for different values of $\\theta$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2: Mean Squared Error for the Tips Data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's apply our knowledge to some real world data. In this section, you will try to find the best statistic $\\theta$ to represent the tips given in an array. The simple procedure you will use in this lab includes constructing the mean squared error (MSE) for the tips data and finding the value that minimizes the MSE. Below you are given an array of tips from a restaurant."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "loaddata",
     "locked": true,
     "schema_version": 2,
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "# Run this cell, do not change anything\n",
    "df = sns.load_dataset(\"tips\")\n",
    "tips = np.array(df['tip']) # array of observed tips"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": false,
     "grade_id": "q2d",
     "locked": true,
     "schema_version": 2,
     "solution": false
    }
   },
   "source": [
    "Now, we can extend the above loss functions to an entire dataset by taking the average. Let the dataset $\\mathcal{D}$ be the set of observations:\n",
    "\n",
    "$$\\Large\\mathcal{D} = \\{y_1, \\ldots, y_n\\}$$\n",
    "\n",
    "where $y_i$ is the $i^{th}$ tip.\n",
    "\n",
    "We can define the average loss over the dataset as:\n",
    "\n",
    "$$\\Large\n",
    "R\\left(\\theta\\right) = \\frac{1}{n} \\sum_{i=1}^n L(y_i, \\theta)\n",
    "$$\n",
    "\n",
    "Define the `mean_squared_error` function which computes the mean squared error given the data and a value for `theta`. Assume that `data` will be a numpy array.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q2a\n",
    "manual: false\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.204529508196728"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def mean_squared_error(theta, data):\n",
    "    return sum([squared_loss(e, theta) for e in data]) / len(data)\n",
    "\n",
    "mean_squared_error(5.3, tips)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q2a</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q2a results: All test cases passed!"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q2a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the cell below  we will plot the mean squared error for different theta values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": [
    "theta_values = np.linspace(0, 6, 100)\n",
    "mse = [mean_squared_error(theta, tips) for theta in theta_values]\n",
    "plt.plot(theta_values, mse)\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('L2 loss')\n",
    "plt.title(r'L2 Loss for different values of $\\theta$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "Find the value of `theta` that minimizes the L2 loss above via observation of the plot above. Round your answer to the nearest integer.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q2b\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find the minimal mse value, then get the theta\n",
    "# or, you can find the value is 3 by just looking the figure above\n",
    "min_observed_mse = round([theta for theta in theta_values if mean_squared_error(theta, tips) == min(mse)][0])\n",
    "min_observed_mse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q2b</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q2b results: All test cases passed!"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q2b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3: Find the Minimizing Value for Our Tips Dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cell below plots some arbitrary 4th degree polynomial function. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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": [
    "x_values = np.linspace(-4, 2.5, 100)\n",
    "\n",
    "def fx(x):\n",
    "    return 0.1 * x**4 + 0.2*x**3 + 0.2 * x **2 + 1 * x + 10\n",
    "\n",
    "plt.plot(x_values, fx(x_values));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By looking at the plot, we see that the x which minimizes the function is slightly larger than -2. What if we want the exact value? We will demonstrate grabbing the minimum value and the optimal `x` in the following cell.\n",
    "\n",
    "The function `minimize` from [`scipy.optimize`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html) will attempt to minimize any function you throw at it.\n",
    "\n",
    "Try running the cell below, and you will see that `minimize` seems to get the answer correct.\n",
    "\n",
    "Note: For today, we'll let minimize work as if by magic. We'll discuss how `minimize` works later in the course."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: 8.728505719866614\n",
       " hess_inv: array([[0.50884886]])\n",
       "      jac: array([1.1920929e-07])\n",
       "  message: 'Optimization terminated successfully.'\n",
       "     nfev: 16\n",
       "      nit: 6\n",
       "     njev: 8\n",
       "   status: 0\n",
       "  success: True\n",
       "        x: array([-1.74682779])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import minimize\n",
    "minimize(fx, x0 = 1.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `fun` value is the minimum value of the function. The `x` is the x which minimizes the function. We can index into the object returned by `minimize` to get these values. We have to add the additional `[0]` at the end because the minimizing x is returned as an array, but this is not necessarily the case for other attributes (i.e. `fun`). The reason for this is that `minimize` can also minimize multivariable functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8.728505719868085, -1.7468265839690909)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimization_result_for_fx = minimize(fx, x0 = 0)\n",
    "min_of_fx = minimization_result_for_fx['fun']\n",
    "x_which_minimizes_fx = minimization_result_for_fx['x'][0]\n",
    "min_of_fx, x_which_minimizes_fx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameter `x0` that we passed to the `minimize` function is where the `minimize` function starts looking as it tries to find the minimum. For example, above, `minimize` started its search at x = 1.1 because that's where we told it to start. For the function above, it doesn't really matter what x we start at because the function is nice and has only a single local minimum. More technically, the function is nice because it is [convex](https://en.wikipedia.org/wiki/Convex_function), a property of functions that we will discuss later in the course.\n",
    "\n",
    "`minimize` isn't perfect. For example, if we give it a function with many valleys (also known as local minima) it can get stuck. For example, consider the function below:"
   ]
  },
  {
   "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": [
    "w_values = np.linspace(-2, 10, 100)\n",
    "\n",
    "def fw(w):\n",
    "    return 0.1 * w**4 - 1.5*w**3 + 6 * w **2 - 1 * w + 10\n",
    "\n",
    "plt.plot(w_values, fw(w_values));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we start the minimization at w = 6.5, we'll get stuck in the local minimum at w = 7.03. Note that no matter what your actual variable is called in your function, the `minimize` routine still calls the starting point `x0`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: 22.594302881719713\n",
       " hess_inv: array([[0.12308018]])\n",
       "      jac: array([-3.81469727e-06])\n",
       "  message: 'Optimization terminated successfully.'\n",
       "     nfev: 12\n",
       "      nit: 4\n",
       "     njev: 6\n",
       "   status: 0\n",
       "  success: True\n",
       "        x: array([7.03774624])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(fw, x0 = 6.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "Using the `minimize` function, find the value of `theta` that minimizes the mean squared error for our tips dataset. In other words, you want to find the exact minimum of the plot that you saw in question 2.\n",
    "\n",
    "For autograding purposes, assign `min_scipy` to the value of `theta` that minimizes the MSE according to the `minimize` function.\n",
    "\n",
    "Hint: You can't pass your `mean_squared_error` function to `minimize` because `mean_squared_error` has two variables: `theta` and `data`. `minimize` will get confused because it thinks it needs to minimize by picking the best `theta` and best `data` values. We only want it to play around with `theta`.\n",
    "\n",
    "In other words, you need to pass a function of one variable `theta` to the `minimize` function, which means you'll need to create a new function of only ONE variable `theta`. This is very simple, but also very tricky when you do this for the first time. Make sure to ask for help if you get stuck.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q3a\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9982786585626386"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# min_scipy = minimize(..., x0=0.0)['x'][0] \n",
    "def helper(theta):\n",
    "    return mean_squared_error(theta, tips)\n",
    "min_scipy = minimize(helper, x0=5)['x'][0]\n",
    "min_scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q3a</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q3a results: All test cases passed!"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q3a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "In lecture, we will show that the value of `theta` that minimizes the mean squared error is the average of the data for the constant model. Assign `min_computed` to the mean of the tips dataset, and compare this to the values you observed in questions 2b and 3a.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q3b\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.99827868852459"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "min_computed = np.mean(tips)\n",
    "min_computed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q3b</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q3b results: All test cases passed!"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q3b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reflecting on the lab so far, we've now seen 3 ways to find the summary statistic `theta` that minimizes the mean squared error:\n",
    "1. Create a plot of the MSE for the given data array vs. `theta` and eyeball the minimizing `theta`.\n",
    "2. Create a function that returns the MSE for a specific data array as a function of `theta` and use the scipy `minimize` function to find the exact `theta` which minimizes this function.\n",
    "3. Simply compute the `mean` of the data array.\n",
    "\n",
    "At this point, you've hopefully convinced yourself that the `mean` of the data is the summary statistic that minimizes mean squared error."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": false,
     "grade_id": "q2c",
     "locked": true,
     "schema_version": 2,
     "solution": false
    }
   },
   "source": [
    "## Question 4: Implement the Absolute Loss \n",
    "\n",
    "\n",
    "In this section, you will follow the exact same steps as above but for the absolute loss function. Absolute loss is defined as:\n",
    "\n",
    "$$\\Large\n",
    "L\\left(y, \\theta \\right) = \\left| y - \\theta \\right|\n",
    "$$\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q4\n",
    "-->\n",
    "In the cell below define the function `abs_loss` which returns the absolute loss given a value of `theta` and `y_obs`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def abs_loss(theta, y_obs):\n",
    "    return abs(theta - y_obs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q4</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q4 results: All test cases passed!"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q4\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the cell below, we will plot the absolute loss for different values of `theta` given that `y_obs = 10`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "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": [
    "y_obs = 10\n",
    "theta_values = np.linspace(0, 20, 100) # some arbitrary values of theta\n",
    "plt.plot(theta_values, abs_loss(theta_values, y_obs))\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('L1 loss')\n",
    "plt.title('L1 Loss for different values of theta');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Thought Question**: How are outliers penalized differently in absolute loss compared to square loss?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 5: Mean Absolute Error for the Tips Data\n",
    "\n",
    "### Question 5a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "Define the `mean_absolute_error` function which computes the mean absolute error (MAE) given the data and a value for `theta`. Assume that `data` will be a numpy array.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q5a\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4527868852459034"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def mean_absolute_error(theta, data):\n",
    "    return sum([abs_loss(e, theta) for e in data]) / len(data)\n",
    "\n",
    "mean_absolute_error(5.3, tips)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q5a</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q5a results: All test cases passed!"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q5a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will plot the mean absolute error for different theta values on the tips dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "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": [
    "theta_values = np.linspace(0, 6, 100)\n",
    "mae = [mean_absolute_error(theta, tips) for theta in theta_values]\n",
    "plt.plot(theta_values, mae)\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('L1 loss')\n",
    "plt.title(r'L1 Loss for different values of $\\theta$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You should see that the plot looks somewhat similar the plot of the mean squared error. Try to identify any key differences you observe and write them down below. This might be more fun with a partner. Note, your answer will not be graded, so don't worry about writing a detailed answer. If you want to see our answer, see the very end of this lab notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Write your answer here, replacing this text._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "To minimize the function, let's zoom in closer to the minimizing `theta`. We will plot the mean absolute error again using the given `theta_values` below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Ar8xbk+FoRERyT3QLtKLC7oTSMEcUlESzRigU4uxPDQdgvpWzKaZZQkREWrZ34fn0DCoCJdGs8qkJB9C3dzcaIhFena/aqIhIIvaOzE1PfygoiWaVLvlhzjh6GACz3ltHze76DEckIpI70r3QAiiJZp2pRx1Efl6I7bvqeOejDZkOR0QkZ+xpzu2r5txOq2/vbhw1fiAAr8xdo+kuIiKtsGPXbnbUBHNEVRPt3E6fciAAayu2s+iTzRmORkQk+8XuI5quJf8A8tP2TC1wzk0C5gAjzGy/o2qccwXAPcAlQAEwC7jRzJbGnFMI3AVcBBQCs4GbzGxek8e6CLgVGA+UAX8EfmRmdal7ZYkbNqg3Y4YWsnRNFa/MXcPBw/tnMhwRkawXTaJdu4Tp3aNL2p43K2qizjkHPE/rkvqTwGeAm4GrgSHAjCBxRj0FXArcAlwM1AIznXMjY57zEuBpfII9B/g1cBNwZ1tfTyqcFtRG3y+toCzY2kdEROKLJtHiwh5pmyMKGa6JOufygRuAnwC7W3H+8cDZwFlmNj049jqwAvgScI9zbgowFbjUzJ4OznkDqACuBW51zoWAnwOPmdmXg4f/t3OuP3BqCl9i0o4YO4B+vbuxubqGf89bwxWnjsl0SCIiWasiqGyksykXMl8TPR74KXAvvmbZkqlANfBy9ICZlQOv4ZMrwEfAscALMdfVAhEg+ts9AhiOr33uYWbfNbMjE30R7SEvHOaUI4YA8PrCdeyqzWgLs4hIVqvIwPQWyHyf6CJgpJmVOeeuacX544BSM2s6gbIUuBzAzHYC7wA45/KAkcA0/BeGR4PzJwa3u51z04GTga3A74AfmllSW6mEQlDYhpUy8vPzgL2Pcd4Jo/jHWyvZWVPP/NJKzjp2eNKPLanVtKwkO6mcckdby2rzthoAhg7snfLybq51OKM1UTPbaGZlCVxSiE92TVUDfeIcvw9YAnwWuNvMFgbHi4PbZ4C5wJnAb4DvAbclEE+76t2rKydO8rXRF99eSUODpruIiDQViUQo3+ybc0v690zrc2e6JpqoEL5ZNt7xeLXHh/GDh84GpjnnwmY2DYgu7/+kmf0g+PcM51w/4Gbn3D1BjTYhkQhUVSU/CCj67Sn2MU6cMJhX5qxmXfl23np/LRNGFiX9+JI68cpKso/KKXe0pay27dw7R7Rnl3DKy7uoqGC/tdFM94kmqor4Nc7ewX2NmNk8M5tpZjfhE+rNQRNvdXDKi00u+Re+33RsyiJuo6ElBYwf1g+Al+euznA0IiLZpzJmjmg6F5+H3EuiBowMRtfGGh3ch3NurHPuujjnzAd6AP2A6JzSbk3OidZQs6rd9LTJQwH4cPkm1lduz3A0IiLZJbrwfLeuefTqnt4G1lxLoi8BfYHTogecc8XAicArwaEJwB+Ak5pcOxVYB1TiF2jYAVzR5Jxzg/sXpTbstjls9IA9I85e1V6jIiKN7J0jmr59RKOyuk80SJCjgI/NbKuZzXLOzQSecM7dBGwCbge2AL8NLnseX+t8xDl3C7ARuBI4D7jKzCLANufcNPy80k3BNVPx80hvNLMW56ymUzgc4rTJQ3ni36W8+cEGLj5xJD27p29FDhGRbJaJfUSjsr0meg7wNn5eZ9TFwHP4xRIeBtYAp5rZZgAzqwHOwPd33g08i58ac4GZPRZ9EDP7KX6hhzODcy8Cvmxm/699X1Jyjp84mG5d8qjZXc8bC9dnOhwRkayRqYUWAELaJSRltjQ0RAorK7cl/QAtjU7700vGv+evZUBhd37yn8cSDqe32UL20qjP3KByyh1tKatb/zCbteXbueKU0Uw96qBUh0ZRUQHhcKgK353YSLbXRCXGqcEAo4qqXbxfWpHhaEREMi8Siexpzi1Sc640Z3BRLw4d6Xd00XQXERHYvquOmlq/iF26l/wDJdGcE91rdPGqLazaWN3C2SIiHVt5zC5XA/oqiUoLDhnRn8FFflmrf7y1MrPBiIhkWHShhR7d8umVgVkLSqI5JhwKcd5xwwGYZ+WsKUt+IJOISK7L1O4tUUqiOeiocQP31EafU21URDqx6GpFRX2URKWVwuEQ535qOADzFpextly1URHpnPbURDPQHwpKojnr6PEDGdi/JxHUNyoindfeJf8ys2eskmiOCodDnPepYQDMWVTG2gotTC8inYufI+qbc9UnKgk7+uCBlPTrQQR4XrVREelkqnfupna330o6E0v+gZJoTssLhzkv6Bt99+ONqo2KSKdSsSVz+4hGKYnmuGMOGcjAoDb6jzdXZDocEZG0iTbl9uqeT8807yMapSSa4/LC4T3zRucs0khdEek8KvesmZuZplxQEu0Qjj5470jdZ99cmelwRETSojyD+4hGKYl2AHnhMBcEtdG5i8tYrVWMRKQTyPTIXFAS7TCOGh+zitEb6hsVkY6vMsNL/oGSaIcRDoc4/7gRAMxbUq4dXkSkQ4vdR1TNuZISR44r4YABvQB4VrVREenAtm6vZXednyOqmqikhK+NDgdgwdIKPtmg2qiIdEzRWihodK6k0JRxJQwpVm1URDq2aBIt6NGFHt0yM0cUEkyizrnezrljY/5/gnPuL865J5xzJ6Q+PElUOBTigqBv9L3SClZu2JrhiEREUm/PFmgZrIVCAknUOXcwUAr8X/D/kcDLwPnAOcCrzrlT2iNIScwRrpih0dro66qNikjHk+nNuKMSqYneGdx+J7j9AtAFOAkYCCwAfpCyyCRp4dDekbrvL6tkxXrVRkWkY8nFJHoi8Aszmx78/3xgqZm9bWY7gEeAKakOUJLja6MFAPxlRimRSCTDEYmIpE42TG+BxJJoD6AcwDk3DDgYmB5zfwSoT11o0hbhUIhLTxoFwOJVW5hn5RmOSEQkNRoiESqzYLUiSCyJlgLHBf/+PD5pPgvgnAsBnwGWpjQ6aZOJo4o4bFQRAE/+u5Ta3fqOIyK5r2pbLXX1vnUtl5Lob4FrnHMLgVuBD81shnPuUGAevrn3V+0Qo7TBFaeOIS8conLrLqbPXpXpcERE2qwyS+aIQgJJ1Mx+B/wHsBp4EDg75u4ewBfN7E+pDU/aamD/nkw98kAAXnznk0ZvPhGRXFQeNOX27tmF7l0zN0cUIKFnN7PHgMeaHPsQGJ/KoCS1zv3UcN76cANV22t5akYpX77w0EyHJCKStGwZmQuJL7YQcs6NiPn/WOfcz5xzP3bOjU19eJIKPbrl7xlkNGdxGYtWbspwRCIiyavcs9BCZkfmQgI1UefcUOBfQA1whHNuIPAO0Dc45WvOuRPNbEEygTjnJgFzgBFmtqaZ8wqAe4BLgAJgFnCjmS2NOacQuAu4CCgEZgM3mdm8Jq9ndZyn+MjMOlxV7dhDB/Ha++soXVPFn15ewrTrjiI/T6s+ikjuydWa6N3AgfgBRgBfxCfQy4AR+IQ0LZkgnHMOeJ7WJfUn8SOBbwauBoYAM4LEGfUUcClwC3AxUAvMDFZZijosuD0DODbm53PJvIZsFw6FuOr0sYRCsL5yBy/Niff9QUQk+1Vs8Um0OAuSaCJ9olOB+8zs98H/zwdWm9lfAZxzv8eP2m0151w+cAPwE2B3K84/Hj+g6azoog/OudeBFcCXgHucc1OCWC81s6eDc94AKoBrY2I8DNhoZi8lEnMuO2hgb049YiivzFvDc2+u4OjxAzM+sk1EJBENDREqt/okmg2fX4nURAvxyQrnXAkwmcaLLWwnwYFKwPHAT4F78TXLlkwFqvFr9gJgZuXAa+wdLfwRvkb5Qsx1tfh5rbG/8UnAwgTjzXkXnjCSwl5dqd3dwBOvalqviOSWLdtqqG+IzhHNoT5R4BNgQvDvK4Lbf8TcfyZBkk3AImCkmZU5565pxfnjgFIza7pqQClwOYCZ7cT31eKcywNG4puZw8CjMdccBpQFtdQpQBV+6s5tZtZirTieUAgK21Co+fl5QNseoyWFhXDNOQfzy6feY96ScpZvqOZwV9Juz9dRpaOspO1UTrmjtWW1bvPOPf8eeWA/unXNa9e4wH+2708iNdE/A99wzj2H7x9dBUx3zo0Kjl2AT0KtZmYbzawsgUsKgXirqVcDfeIcvw9YAnwWuNvMFgI453oCo/FJ+Q/4ftHfAd8CHkggnpx0wqQDOGRkfwAe+MdHWslIRHLGxk07AOhb0C0tCbQlra6JmtmPnHP1+IE3bwLfNrM651wf/GpFd5rZL9spzqgQvlk23vGGOMcfBp7GN/VOc86FzWwaUIdvGl5pZsuCc19zztUCdzrn7owd7dtakQhUVe1s+cT9iH4Da8tjtNYVp4zh9pXvsqFyB0++bHt2fZHWSWdZSfJUTrmjtWW1OtgjuX+fbmkr16Kigv3WRhNdbOEu/NSRWO8Bxck2gSaoCt8821Tv4L5GYqa0zHTODQBuDhJkLfBqnMd5Ab/l22F08HWAhwzoxelHHsj02at44e1POPaQQRT3VZOXiGS3bJreAokPBMI5Nx4/MncYfsDOavz0FEttaHEZcJpzLmRmsTXS0dHnDxZ9OB54qMk58/Gjc/s553oDpwN/M7OKmHOiWST2WId1/nHDmf3xRjZX1/Dnl5dw42cOa/kiEZEMii5dmg0jcyHxFYt+AnwA/Bg/peQbwM+Aj5xzP019ePt4CT839bSYmIrxzcmvBIcm4Ps5T2py7VRgHVAJ9AP+l33nhF6O73NNasGIXNO9az6fPXUM4DfvXrBU26WJSHYr3+KbcIuzZLBYIisWfQG4CV/rvAs/sjaMH5xzM/DfzrmPzOyPqQouSJCjgI/NbKuZzXLOzQSecM7dBGwCbge2sHcRiOfxtc5HnHO3ABuBK4HzgKuC2un86ACpYATvh/h+028A3zKzfZqGO6rJrphDRvTnoxWb+PPLSzl4eH+6dcl8Z72ISFP1DQ1srq4Bsqc5N5Ga6NeAGWZ2vpnNDpLaFjN7x8wuAmYG56TSOcDbwBExxy4GngN+jh84tAY41cw2A5hZDX607Yv4UcTP4hP9BcEC+lGfA+4Hvo6fqjMVuMHM7kvxa8hqoWAlo/w8v13aC2+vzHRIIiJxbamu3TNHNFuacxPpE3XAt5u5/2/4NW2TYmYP45NiS8c24/s2r23msSqA/2zh+bYD3w1+OrWB/Xty5tHDeP6tlUyfvYpPHTqYQf17ZjosEZFGKmJG4+ZiTbQaGNTM/YMBjSPPUeccO4yiPt2pq4/w2EtGJBJvJpGISOZER+YWFnSlS352dDslkkT/BXzdObfPEM5gB5avE7Mcn+SWbl3y+NzpfpDRRys3M9c0yEhEsku2TW+BxJpzb8H3Nc51zv2LvVNaxuH7E7cAP0hpdJJWh48p5rBRRby/rJInXl3KhJH9M75rvIhIVLQ5NxvWzI1qdU3UzFYBR+FXADoR+GbwcyLwd+BoM1veHkFK+nzu9LF0yQ+zubqG595cmelwRET2iG6Blk010YTmiZrZSjO7Aj9XcxC+H7TQzC5TAu0Yivv24JxjhwHw8pzVrC3fluGIRES8bGzOTSiJRplZg5mVBQvINwA45y51zv1PasOTTDjr6IMo6deD+oYIf3ppiQYZiUjGNZ4jmoPNua1wKnBjCh9PMqRLfh5Xnj4WAFu9hXc+3pjhiESks9u8tYaGSHQf0RyviUrHN2FkEZPHFgPw1L9L2bGrLsMRiUhnVh405YaA/n2URCUHfPa0MXTtEqZqey3PvKEubxHJnOjI3L69u9ElP3tSV/ZEIlmnf5/ue/YZfXXeGlZtrM5wRCLSWWXb7i1RSqLSrKlHHsjgop5EIvCnl5bs6ZMQEUmnbByZC80stuCcOzHBxzqgjbFIFsrPC3PVVMfPHl9A6doq3vxgPSdMVFGLSHpVbMm+hRag+RWLZgKJVDtCCZ4vOWL8sH4cffBAZn+8kb/MWMbhY4op6NEl02GJSCdSsTXHaqLAj1BSlMDlp4zm/dIKtu3czd9mLefqM1ymQxKRTqKuPvv2EY3abxI1s9vTGIdkub4F3bjwhJE88epSXluwluMOHcSoIYWZDktEOoFN1TVEh2NkWxLVwCJptVMnD+HAkgIiwAPPf0xNbX2mQxKRTiDaHxoKZdccUVASlQTkhcNcf8548sIhNm7eyVMzSzMdkoh0AtGRuf16dyM/L7vSVnZFI1nvoIG9uejEkQDMmL+WD5dXZjgiEeno9kxvybJaKCiJShLOPOogxgz1/aF/eHER23buznBEItKRVUb3Ee2bXdNbQElUkhAOh7j+3IPp1jWPqm21/HH6Yu30IiLtpjxLF1oAJVFJUknfHnz21DEAzLNy/j1/bYYjEpGOKluX/AMlUWmDEyYO5piDBwLwxKtLWbF+a4YjEpGOZnddA1uycB/RKCVRSVooFOLqMx2D+vekviHCb5/5kO271D8qIqmzaeuuPav+FGdhTbS5tXNXkPiKRREzG9W2kCSXdO+az1cuPJQ7H5lLRdUuHnxhEV+7eAKhUCjToYlIBxAdmRsOhejXp1uGo9lXczXRB4AhwDBgE/BJK35WtWewkp2GlhRw5dSxACxYWsFLc1ZnOCIR6Sii+4j2692NvHD2NZ42t+zfXc65ucCzQJmZnZ2+sCTXnDDxAJas3sKbH2zgrzOXMWpIIaO1LKCItFG2boEW1WxaN7N/Ad8AznDO3ZCekCRXXTXVMaS4157+0eodtZkOSURy3J4k2jcHkyiAmf0f8CrwLedcc7u+SCfXrUseX7nwULp1yWNzdQ0PPL9Im3iLSJtEm3OzcWQutHJ0rplNNbNxZlbX3gFJbhtc1IvPn+m3SftgeSV/n7VcCzGISNJyujlXJBnHHDKIkyYdAMALb3/C068pkYpI4nbX1VO1zXcLZWsSTVnzrHPuJGCimf0qyesnAXOAEWa2ppnzCoB7gEuAAmAWcKOZLY05pxC4C7gIKARmAzeZ2bz9PGY+8DZQZWanJRO/NPa508eyo6aOdxeV8eI7n1BX38Dlp4zW1BcRabVoLRRyvDm3lS4HfpHMhc45BzxP65L6k8BngJuBq/HTcGYEiTPqKeBS4BbgYqAWmOmcG7mfx/wuMCWZ2CW+/LwwXzzvYI49ZBAAL81ZzSP/MnbXNWQ4MhHJFdHl/vLCIfr27prhaOLL6EChoAZ4A/AToMWlbpxzxwNnA2eZ2fTg2OvACuBLwD3OuSnAVOBSM3s6OOcNoAK4Fri1yWMeBnwf2JCilyWB6P6j+XkhXl+4ntfeW8fSNVVcd/Z4Rh7QJ9PhiUiWi91HNBvniELm+0SPB34K3IuvWbZkKlANvBw9YGblwGv45ArwEXAs8ELMdbX41ZcaNao757oAfwR+BVhSr0CaFQ6H+PxZ4zj/uOGEQyHWVWznrkfn8tSMUmp312c6PBHJYtk+qAgyXBMFFgEjzazMOXdNK84fB5SaWdNP31J8czJmthN4B8A5lweMBKbhvzA82uS6HwJdg9t/JfkapAXhUIgLTxjJpDEDePCFRawp38702at4b2kF1509ntFDtSiDiOyrIov3EY3KaBI1s40JXlIIxNsqpBqI1z54H/C14N+3mdnC6B3OuSOBbwMnmlmN75Ztm1AICtvQ+Z2fnwe07TGy2WGFPfj5qGL+NrOUp2eUsmHTDn782DzO+dQIPjfV0a1rXqZDbLWOXlYdhcopd8Qrq83ByNyhJb0zWobNjYdsbgH6gxJ8nt4Jnp+MEPEXxQ8B8UasPAw8jW/qneacC5vZNOdcd3wz7n1m9m57BSv76pIf5vLTxnLUwQP59V8XsmL9Vp5/cwVzF2/kK5dM5JARRZkOUUSyRPlmXxMt7pe9X4Kaq4muJLFdXPaX4FKpCt8821Tv4L5GYqa0zHTODQBuds7dCdyJb969I2YVphAQCv5fb2YJv5ZIBKqC5odkRL9pteUxckX/Xl353lVH8M93PuG5N1eyoXIHt/3fO5w6eSiXfnpU1tdKO1NZ5TKVU+5oWlY1u+vZss3vI9qra15Gy7CoqGC/tdHmkugjtH9STJQBpznnQk2S3OjgPpxzY/EDlh5qcs58/OjcfvjpL8OAbXGeYzdwMjAz5dFLI/l5Yc47bgSHjy3mDy8s4pMN1bw6bw3vl1Zw7dnjGT+sX6ZDFJEMqWw0RzQHBxaZ2TVpjKO1XsLP/TyNYISuc64YOBG4OzhnAvAH/LSXGTHXTgXWAZXAeUDTjen+F6gDvopG6qbV0OICfnD1ZKbPXsWzb6ygomoXP3t8AScfPoRLTxpFj26ZHv8mIulWETtHtCD79hGNyupPpyBBjgI+NrOtZjbLOTcTeMI5dxN+n9PbgS3Ab4PLnsfXOh9xzt0CbASuxCfOq4La6QdxnqsaqDOzue36oiSuvHCYc44dzqQxxTz04iKWr9vKjAVrWbiskmvOHschw/tnOkQRSaPKoPm2qE93wuHsXeks0/NEW3IOfjm+I2KOXQw8B/wcP3BoDXCqmW0GMLMa4AzgRXzt9Fn81JgLzOyxtEUuSRkyoBffv2oyl508mi75YSq37uLeJ97j4X8uZscu7X8g0lmUZ/kWaFEhLQyeMlsaGiKFlZXxullbR4MgGtuwaQcPvriI0jV+zFi/3t245qxxTBiZ+RG8KqvcoHLKHU3L6jfPfMjcxWWceNhgrjlrfCZDo6iogHA4VAX0bXpfttdEpRMb1L8n3/3cEXz21DF0zQ+zubqGXzz1Pg++sIgdu1pcJVJEctie5twsn+OrJCpZLRwOcfqRBzLt+qMYe2BfAN74YD0/eGA275VWZDY4EWk3ubDkHyiJSo4Y2K8nN33ucK48fSzduuSxZVstv/rrQn7/j4/ZtlO1UpGOZFdtHdU7/N91sWqiIqkRDoU4dfJQfnT9UXvmkL790QZufWA285eUZzg6EUmV2DmiRaqJiqRWcd8efPuKSVx9hqN71zyqttdy/98+4HfPfkj1jtpMhycibRRtys3PC1FYkJ37iEYpiUpOCoVCnHT4EO64/mgOGeHnkL67qIwfPDCbOYvLMhydiLRFNIkW9elOuLnV37OAkqjktKLC7nzrssO49qxx9OiWR/WO3fz2mQ/59d8/oGq7aqUiuSgXtkCLUhKVnBcKhTjhsAO44/qjmTjKzyGdZ+Xc+sBs3vl4A5oLLZJbcmVkLiiJSgfSv093brx0ItefM56e3fLZtnM3//fcx9z/tw/27AYhItlPSVQkQ0KhEMdNGMwdXziaSaMHALBgaQW3PjCbtz5cr1qpSA6o2BI052b59BZQEpUOql/vbnz9kgnccN7B9Oqez/ZddTzw/CJ++deFbK5WrVQkW+2sqWN7sE62aqIiGRQKhTjmkEHc+cVjmDy2GICFyyr5wQOzeX3hOtVKRbJQruwjGqUkKh1eYa+ufOWiQ/nSBYdQ0KMLO2vqeOjFxfziqffZtHVXyw8gImkT7Q/tkh+mT6/sniMKSqLSSYRCIY4aP5A7v3A0R44rAeDDFZv4wQOzee29taqVimSJ8uj0lsLuhLJ8jigoiUon06dXV7584aF85cJD6dOzC7tq6/njdOPeJ9/bM5hBRDIn2pyb7cv9RSmJSqc0ZVwJd37xGI45eCAAH6/czK0Pvsu/56+hQbVSkYzZO70l+0fmgpKodGIFPbpww/mH8PVLJlDYqys1tfX86aUl/PzxBZSpViqSEdEWoWLVREVyw+Fjirnzi0fzqUMHAbB41RZu+8NsXp67WrVSkTSrUHOuSO7p1b0LXzj3YG68dCL9enejdncDj7+ylHsem8/GTTsyHZ5Ip7B952521ETniKo5VyTnHDZ6AHdcfxQnTBwMwNI1Vdz24Lv8691VNDSoVirSnspjulEG9FVNVCQn9ezehWvPHs+3LjuM/n26sbuugSf/XcqPH5vH+srtmQ5PpMOKtvp07RKmd48uGY6mdZRERfbj0JFF3HH90Zw06QAAlq3dyg8fnMM/3/mE+vqGDEcn0vGUb967Zm4uzBEFJVGRZvXols/VZ47j21dMYkBhd+rqG/jLzGV8/3dvsWpjdabDE+lQyjb7mmguLPcXpSQq0goHD+/Pj64/ilOOGAJA6ZoqvvP/3uAfb62kTrVSkZQo27J3taJcoSQq0krdu+Zz1VTHzZ87nEH9e1JX38DfZy3nrkfmsbpsW6bDE8l5ZZuiNdHcGJkLSqIiCXMH9ePeG0/g3ONGEAI+2VjNjx6ew7NvrFCtVCRJkUgkpk9UNVGRDq1713yuPfdgvnvVEQzs14P6hgjPvrGCO/44V32lIknYvqtu7xzRHJneAkqiIm0yZmhfpl13FGcedRChEKwu28Ydf5zL32ctV61UJAFlMYuaqDlXpBPp2iWPy04Zzfevmszgop7UN0T4x1srmfbwHFas35rp8ERywroKPwe7W9c8enXPz3A0rackKpIio4YUcvu1R3L2McMIhWBt+XbufGQuf525jN119ZkOTyRrNTREeGbWMgBGDu6TM3NEAbIm3TvnJgFzgBFmtqaZ8wqAe4BLgAJgFnCjmS2NOacQuAu4CCgEZgM3mdm8mHO6ArcC/wGUAO8DPzSzl1L7yqQz6ZKfx6UnjWKyK+bBFxextnw7L77zCQuWlnPd2eMZNaQw0yGKZJ03P1zPinW+1ebiE0dmOJrEZEVN1DnngOdpXVJ/EvgMcDNwNTAEmBEkzqingEuBW4CLgVpgpnMutnTuB/4L+DlwPrAUeME596k2vRgRYMTgPtz2+SM591PDCYdCrK/cwd1/msdT/y6ldrdqpSJRu2rr+NtrywE44bADcu6LZkaTqHMu3zn3FXwNtMWeZOfc8cDZwNVm9kcz+xtwGtAX+FJwzhRgKvBVM3s4qFleAuQB1wbnlADXAbeb2f1m9gpwDbAa+HJKX6R0Wl3yw1x84khu/fwUDiwpIBKB6e+u4ocPzWHpmi2ZDk8kK7z4zidUba+la5cwV505LtPhJCzTNdHjgZ8C9+Jrli2ZClQDL0cPmFk58Bo+uQJ8BBwLvBBzXS0QAaLjpjcDRwEPxjxOA1AXc45ISgwb1JtbPz+FC44fQV44xMZNO/jJn+bz+CtLqVGtVDqxiqqdTJ+9GoALThjFgL65Myo3KtN9oouAkWZW5py7phXnjwNKzazpJ08pcDmAme0E3gFwzuUBI4Fp+C8Mjwbn7AbmB+eE8U3C/w2MAr7etpcksq/8vDAXHD+Cw8cM4MEXF7Fq4zZenrua90sruPbscbiD+mU6RJG0WVexnblWxlsfbqCuvoG+BV258NO51RcaldEkamYbE7ykEIg3Z6Aa6BPn+H3A14J/32ZmC+Oc8018vyjA/wGvJBjTHqEQFLZhflN+fh7QtseQ9Ei2rCYU9uBnowbwzKxl/OXVUsq27OSePy/gzGOGcdWZ4+jRLdPfazsW/U1lj1Ubq3n7g/W89cF61sQskxkKwXXnHUJBz25AdpZVc4OFc+0vNoRvlo13PN7M9oeBp/FNvdOcc2Ezm9bknH8Cc4HjgNuArgR9pyLtIT8vzKUnj+Gogwfx67++T+maKqa/8wnzrYyvXDKRCaMGZDpEkTaLRCKs2ljNWx+s550PNzRKnAAD+vbg2EMHcfxhBzB6aN/MBJkCoUgkXk5Kv6A59yHgwP1NcXHO/QXf/Du5yfFfAZeY2ZBmHv9B4Aqgd5zm4Og5twG3A0PNbF2CL2FLQ0OksLIy+YXIo9/Aqqp2tnCmZFqqyqq+oYGX3l3N31/fu+7uSZMO4DMnj1atNAX0N5VekUiE1WXbmGtlzF1czoaYVYgAivp058hxJUweV7zPfNBsLquiogLC4VAVfhBrI7n2V2rAac65kJnFZv/RwX0458biByw91OSc+fgaZj/nXDf8IKW/mNm2JueEgAOARJOoSMLywmHOOmYYk4K+0mVrtzLzvXV8sLySz581jkNHFGU6RJFm+RpnNHGWsXFz4yQ4oLA7U8aVcOS4EoYP6p1TCym0Rq4l0Zfwcz9PIxih65wrBk4E7g7OmQD8AVgBzIi5dio+MVYCR7B3ZO5DTc6pAZa0T/gi8Q0u6sX3rpzMy3NX87dZy6ncWsP/PPk+J0wczOWnjKFnDi2DJh1fJBLhk43VzFlcxrzF5Xv2AY0q6duDyeOKOXJcCcMGdrzEGSur/zKDBDkK+NjMtprZLOfcTOAJ59xNwCZ88+sW4LfBZc/ja5SPOOduATYCVwLnAVeZWcQ5Nz8473+ccz3wSfMc4Kv4AUha8FTSLhwOccZRBzFptK+VLl1TxesL1/Phik18/kzHRPWVSgZFIhFWbvCJc+7iMiqqdjW6f2C/HkwZV8IUV8JBAws6dOKMldVJFJ/YHgJOBmYGxy4G/gc/ojYMvAFcZmabAcysxjl3Bn7Zv7uBAcBC4AIzey44J+KcuwK/7N938M23S4EbzOwP6XlpIvEN7N+Tm688glfnreHp15axubqG+/6ykOMOHcQVp42hV/cumQ5ROolIJMLy9VuZu9j3cVZubZw4B/XvGSTOYg4s6TyJM1bWDCzqADSwqBNJV1mVbd7Bw/9czOJVW/zzFnTl6jMch48pbtfn7Sj0N5W4hkiE5euCxGllbNpa0+j+Awb0YoorZsq4EoYM6JWyxJnNZdXcwCIl0dRREu1E0llWDZEIMxes5S8zlu1Z4WjS6AGccsQQDh7Rn3An/PbfWvqbap2GSITSNVXMtTLmWTmbqxsnziHFvTjSlTA5SJztIZvLqiONzhXpdMKhEKccMZQJI4t4+J+LWfTJZt4rreC90goGFHbnxMMO4OiDB1Kcg0umSeY0NERYumYLc62ceVbGlm21je4fWlzAlGBw0OCi9kmcHYFqoqmjmmgnkqmyikQizFlcxoz5a7HVWxrdN2xQb44M+qdK+vVMa1zZSn9TjTU0RFiyesueGmfV9saJ86CSAiYH76F0J85sLis156aHkmgnkg1ltb5yO6+9t453Pt7I1qYfhgMLgoRawsD+nTehZkM5ZVp9QwNLVgU1ziXl+7xXhg3szZRxxRl/r2RzWSmJpoeSaCeSTWW1p1lucTlzl5RR1aRZ7sCSgj0jKDtbs1w2lVM61Tc0sHjVFuYuLmP+knKqd+xudP+Iwb2ZEvRxlmRJN0A2l5WSaHooiXYi2VpWewaIBCMr9+3n6sUUV8KUcSUc0E4DRLJJtpZTe6irb2Dxqs1B4qxg286mibPPnub+bNxyLJvLSkk0PZREO5FcKKuGSITla7f6yfFWtu+IywG9mOz8wJEhxQUZirJ95UI5tUVdfQOLPtnMnMVlLFhSzvZddY3uH3VAnz0LIBQVZvdWydlcVkqi6aEk2onkWlnFzv2bZ2VUNpn7N7ioJ1NcSZBQUzf3L9NyrZxao66+gY9WbGKulbFgSQU7ahonztFDC/10FFdM/z7ZnThjZXNZKYmmh5JoJ5LLZRVdhWbe4nLm2r7Lt/lVaPxAk1xfhSaXyynW7roGPlq5ibmLy1iwtIKdMYkzhE+c0Rpnv97dMhdoG2RzWSmJpoeSaCfSUcoquh5qtA+1fEvHWg81l8tpd109H67wifO90gp21uzdwTEEjD2wL1PGlXDE2OKcTZyxsrmslETTQ0m0E+mIZRXd0irah1rWZEurXNyZI9fKqXZ3PR8s38Q884lzV21M4gyBCxLn5LHFFBbkfuKMlc1lpSSaHkqinUhHL6vo5srRHTv2t0fkFFfCiMHZm1BzoZxqdtfzwbJK5loZ7y+rpKZJ4hx3UL89Nc7CXl0zGGn7yuayUhJNDyXRTqQzlVUkEmFN+fY9CXXDph2N7i/q043JwaCkEQf0yaq1fLO1nGpq61m4vJK5i8t4f1kFtbsb9twXDoUYP8zXOA8fW0yfnh03ccbK1rICJdF0URLtRDprWUUiEdZWbGfu4jLmLC5jfWXjhNqvd7c9o3xHDsl8Qs2mctpVW8fCZT5xLlxWSW3d3sSZFw4xftjeGmdBj8633V02lVVTSqLpoSTaiaisvLUV25m3uIw5Vsba8u2N7uvXuxuTnR/lO3poYUYSaqbLaWdNHe8vq2Du4nI+WF7J7iaJ8+Dh/ZkyrpjDx3TOxBkr02XVHCXR9FAS7URUVvtaXxmtoZazprzx30FhQVemjC1hyrhixgztSzicnoSaiXLasauO90srmGtlfLB8E3X1jRPnISP6c+S4EiaNGaAN1mNk89+Ukmh6KIl2Iiqr5m3YtMNPm1lcxqqyJgm1V9c9NdSxB7ZvQk1XOe3YtZsFSyuYZ+V8uKKSuvq9n6v5eSEOHVHElHHFTBpdTM/u2oEynmz+m1ISTQ8l0U5EZdV6GzdHE2o5n2ysbnRfn55dOMKVcMjwfoweUpjyaRvtWU5V22p4r7SCBUsr+GjFJuobYhNnmAkj+zNlXAmTRg+gRzclzpZk89+Ukmh6KIl2Iiqr5JRt2en7UBeXsXJD9T73F/ftzqD+vUhV92l+fhiAupi+yFSo3lHLyvXVxH56dskPM3FkEZPHFXPYKCXORGXz35SSaHooiXYiKqu2q9iyk7lWzntLy1m+vrpR32Gu6NEtjwkjizhibDETRxXRvasSZ7Ky+W+quSSqEheRjBjQtwdnHn0QZx59EHX1DXyysZpla6r22b6tLboGtcHaJou0t1VeXgh3UF/GHdSP/LxwSh9bcouSqIhkXH5emFEHFDLqgMKUPm42126kY9BXKBERkSQpiYqIiCRJSVRERCRJSqIiIiJJUhIVERFJkpKoiIhIkrTYQuo0RCKRUFt+ndFVWlQk2U9llRtUTrkjm8sqFIJQKBQhTsVTSTR16vC/4K2ZDkRERFKqD9BAnLUVlERFRESSpD5RERGRJCmJioiIJElJVEREJElKoiIiIklSEhUREUmSkqiIiEiSlERFRESSpCQqIiKSJCVRERGRJCmJioiIJElJVEREJEn7LKYr7cM5FwZuAL4CjAQ2As8CPzSz6jjnPwx8fn+PZ2ah9olUEi2r4Jpi4KfAGUB34C3gm2a2NC1Bd0JJltNEfDmdAOwEXgS+a2br0hJ0J+WcCwE34svqQGAJcI+Z/bmZawqAe4BLgAJgFnBjtv1NqSaaPjcB9wMvABcC9+KT5F/2c/4dwLFNfv4Dv5PA79o51s4uobIKPiD+DpwFfBdfToOAGc65fmmIt7NKtJxGAa/jE+4NwNXAYOAt51zf9g+3U/se8HPgj8C5wMvAY865y5q55kngM8DN+LIagv+bKmznWBOiXVzSIPiQrQQeN7Ovxhy/HHgCONzM3mvhMfKAd4A84Bgzq22/iDuvZMrKOTcWMODzZvZIcGwksAy4xsz+mKbwO40ky+lXwBeAcWa2KjjWHV92j5nZ99MUfqfinOuCbyV4zMy+HnN8JpBnZifEueZ4/Bees8xsenCsGFgB3GFm96Qj9tZQTTQ9egN/Apo2XSwObke14jG+BBwBfEkJtF0lU1bdg9vYJsRNwW1R6kKTGMmUkwM+iCZQADPbBbwLnN0eQQoA9cCngR83OV7L3r+dpqbi/55ejh4ws3LgNbKsrNQnmgZmthX4Rpy7LgxuP2ru+qBvYBrwqJm9m9roJFYyZWVmC51zM4DbnHOL8DWke4FtwDPtE2nnluTf1CrgEOdcvpnVxRwfGfxIOzCzBuAD2NOCUAJcC5wG/Od+LhsHlJpZfZPjpcDl7RRqUpREM8Q5dzS+/+wZM1vcwunXAf2Au9s9MNlHK8vqy8C/gEXB/2uAC81seRpCFFpVTo8A1wN/cM7dCuzCD3Y5BOiStkA7t4uBvwb/fgHfmhBPIbA1zvFqoE87xJU0NedmgHPuOGA6vn3/C6245KvAs2a2pF0Dk320pqycc+Px/dXlwEX4Ebr/AJ52zu3T3yOp15pyMrPX8V9Izwc+ATbgazy/BXakJ9JObz6+affrwHH4RBpPCIg3YCeEH1yZNZRE0ywY+PAKvmnpVDOrbOH8icBY4NE0hCcxEiirbwa3U83sGTN7CbgMWAD8ov0j7dwS+Zsys4eBAfj+0cFmdgm+1rNpf9dI6pjZCjObZWb341sBTnbOHRvn1Cri1zh7B/dlDSXRNHLOfQt4HHgbONHM1rfisnPxfWsvtmds0liCZTUM+NjMNkcPmFkEeAPfVCjtJJFycs6Nd85dZWb1ZrbEzDYGdx2BryFJO3DO9XfO/Ydz7oAmd0V/50PiXGbAyKAPNdbo4L6soSSaJs656/GDTZ4CzjSz1n6bOgaYa2Y17RacNJJEWRlwaJw5occAK1MfoUBS5TQBeDSYfhR9jJOAw9AAsPYUxs8PbTqIaGpw+0Gca14C+uIHHwF7priciG91yBqaJ5oGzrkSfF9NOXAVUNfklFJ8W/8ofI1ma8y1K4AXY+fCSftJpqycc8Px36pLgZ/g+9euBj4LXGFmT6Yn+s4jyXLqhf/ALgd+iB8l+gtgKXB8kxG7kkLOufuBLwK3AXOB4/ELMDxqZl8MEmSjz79gxPtE/KIam4Db8VPGJsS2+mSaRuemx5lAT3yz3+tx7v8PfFk8BJwMzIy5byCQNW+YTiDhsjKzlcHAlnuAh/EDHz4ATjezrPrW3IEkU07bnXNnAr/Er4azHb8wwy1KoO3um/g+6+vx0/XW4L/I/Cy4/xz2/fy7GPgf/EpHYXz3yGXZlEBBNVEREZGkqU9UREQkSUqiIiIiSVISFRERSZKSqIiISJKUREVERJKkJCoiIpIkzRMVkYQ4587AT5SfAuzEz7n8drA3p0inopqoiLSac+6/8bulrMdPoP8HfpehX2YyLpFM0WILItIqzrnT8Gua3mRmP485Ph2/0kxx7JKVIp2BaqIi0iLnXBhf21yAX/Q91kygK3BomsMSyTj1iYpIa5wBHAxcE2zzFqs2uC1Mb0gimackKiKtcTlQD7zunBvQ5L6BwW11ekMSyTwlURFpjZOBPGBZM+csT1MsIllDA4tEpFlBzbMc+DvwmzinPAXUmNngtAYmkgVUExWRlowMbuc03R/VOTcC6Af8Oe1RiWQBjc4VkZYUBLfx+jwvDW6fTFMsIllFSVREWhKd+9kn9qBzrivwZcCAF9IdlEg2UBIVkZZ8DOzAT3OJdRcwHPiGmdWnOyiRbKCBRSLSIufcL4FvAI8BrwFnARcB34ldvUiks9HAIhFpje8AEeBKfPKcB5xlZtMzGpVIhqkmKiIikiT1iYqIiCRJSVRERCRJSqIiIiJJUhIVERFJkpKoiIhIkpRERUREkqQkKiIikiQlURERkSQpiYqIiCRJSVRERCRJ/x/j00NqO01eCwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta_values = np.linspace(2.7, 3.02, 100)\n",
    "mae = [mean_absolute_error(theta, tips) for theta in theta_values]\n",
    "plt.plot(theta_values, mae)\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('L1 Loss')\n",
    "plt.title(r'L1 Loss for different values of $\\theta$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "### Question 5b\n",
    "\n",
    "This time, observe that the function is piecewise linear and has a slope of zero near its minimum. Because of the large flat region at the minimum, there are multiple values of `theta` that minimize the L1 loss.\n",
    "\n",
    "Give a `theta` rounded to the nearest tenth that minimizes L1 loss. By \"rounded to the nearest tenth\" we mean you'd say 7.6 instead of 7.55.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q5b\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "min_observed_mae = 2.9\n",
    "min_observed_mae"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q5b</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q5b results: All test cases passed!"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q5b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 6: Find the Minimizing Value Using Absolute Error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "### Question 6a\n",
    "\n",
    "As before, we will use the `minimize` function to find a solution. Note that we will be minimizing the mean absolute error for our tips dataset. Assign `min_abs_scipy` to the value of `theta` that minimizes the MAE according to the `minimize` function for the `tips` data. Note: Depending on the `x0` value you specify, you will get different results! \n",
    "\n",
    "Extra: Try various `x0` values and record the different outputs you get from `minimize`. Use the plot you created above to verify that these are all valid minimizing statistics.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q6a\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9090163677930834"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#min_abs_scipy = minimize(..., x0=0.0)['x'][0]\n",
    "def helper(theta):\n",
    "    return mean_absolute_error(theta, tips)\n",
    "min_abs_scipy = minimize(helper, x0=2.95)['x'][0]\n",
    "min_abs_scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q6a</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q6a results: All test cases passed!"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q6a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just like the MSE, there are three ways to compute the summary statistic `theta` that minimizes the MAE:\n",
    "1. Create a plot of the MAE for the given data array vs. `theta` and eyeball a minimizing `theta`.\n",
    "2. Create a function that returns the MAE for a specific data array as a function of `theta` and use the scipy `minimize` function to find an exact `theta` which minimizes this function.\n",
    "3. Simply compute the ?????? of the data array.\n",
    "\n",
    "Try to figure out what to substitute in for the ?????? above. To this, try out various statistics functions provided by `np`. A list and documentation is available at [https://docs.scipy.org/doc/numpy/reference/routines.statistics.html](https://docs.scipy.org/doc/numpy/reference/routines.statistics.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "### Question 6b\n",
    "\n",
    "Assign `min_abs_computed` to the correct summary statistic using method `#3` outlined in the the previous problem.\n",
    "\n",
    "<!--\n",
    "BEGIN QUESTION\n",
    "name: q6b\n",
    "-->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "min_abs_computed = 2.90\n",
    "min_abs_computed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<p><strong><pre style='display: inline;'>q6b</pre></strong> passed!</p>"
      ],
      "text/plain": [
       "q6b results: All test cases passed!"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check(\"q6b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Extra Notes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Our Observations on Differences Between MAE vs. MSE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Earlier in this lab, we said we'd describe our observations about the differences between the MAE and MSE.\n",
    "\n",
    "There are three key differences that we identified between the plots of the MSE and MAE.\n",
    "\n",
    "1. The minimizing $\\theta$ is different.\n",
    "2. The plot for MAE increases linearly instead of quadratically as we move far away from the minimizing $\\theta$.\n",
    "3. The plot for MAE is piecewise linear instead of smooth. Each change in slope happens at the same $\\theta$ value as a data point in our dataset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Congratulations! You finished the lab!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "---\n",
    "\n",
    "To double-check your work, the cell below will rerun all of the autograder tests."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "q1a results: All test cases passed!\n",
       "\n",
       "q2a results: All test cases passed!\n",
       "\n",
       "q2b results: All test cases passed!\n",
       "\n",
       "q3a results: All test cases passed!\n",
       "\n",
       "q3b results: All test cases passed!\n",
       "\n",
       "q4 results: All test cases passed!\n",
       "\n",
       "q5a results: All test cases passed!\n",
       "\n",
       "q5b results: All test cases passed!\n",
       "\n",
       "q6a results: All test cases passed!\n",
       "\n",
       "q6b results: All test cases passed!"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grader.check_all()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "## Submission\n",
    "\n",
    "Make sure you have run all cells in your notebook in order before running the cell below, so that all images/graphs appear in the output. The cell below will generate a zip file for you to submit. **Please save before exporting!**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "            <p>Your submission has been exported. Click <a href=\"lab06_2022_03_31T14_08_03_321063.zip\" download=\"lab06_2022_03_31T14_08_03_321063.zip\" target=\"_blank\">here</a>\n",
       "            to download the zip file.</p>\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Save your notebook first, then run this cell to export your submission.\n",
    "grader.export(pdf=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " "
   ]
  }
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