exercise-multivariate-linear-regression.ipynb

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    "# ML-Fundamentals - Lineare Regression - Exercise: Multivariate Linear Regression"
   ]
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    "## Table of Contents\n",
    "* [Introduction](#Introduction)\n",
    "* [Requirements](#Requirements) \n",
    "  * [Knowledge](#Knowledge) \n",
    "  * [Modules](#Python-Modules)\n",
    "* [Exercises - Multivariate Linear Regression](#Exercises---Multivariate-Linear-Regression)\n",
    "  * [Create Features](#Create-Features)\n",
    "  * [Linear Hypothesis](#Linear-Hypothesis)\n",
    "  * [Generate Target Values](#Generate-Target-Values)\n",
    "  * [Plot The Data](#Plot-The-Data)\n",
    "  * [Cost Function](#Cost-Function)\n",
    "  * [Gradient Descent](#Gradient-Descent)\n",
    "  * [Training and Evaluation](#Training-and-Evaluation)\n",
    "  * [Feature Scaling](#Feature-Scaling)\n",
    "* [Summary and Outlook](#Summary-and-Outlook)\n",
    "* [Literature](#Literature) \n",
    "* [Licenses](#Licenses)"
   ]
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    "## Introduction\n",
    "\n",
    "In this exercise you will implement the _multivariate linear regression_, a model with two or more predictors and one response variable (opposed to one predictor using univariate linear regression). The whole exercise consists of the following steps:\n",
    "\n",
    "1. Generate values for two predictors/features $(x_1, x_2)$\n",
    "2. Implement a linear function as hypothesis (model) \n",
    "3. Generate values for the response (Y / target values)\n",
    "4. Plot the $((x_1, x_2), y)$ values in a 3D plot.\n",
    "5. Write a function to quantify your model (cost function)\n",
    "6. Implement the gradient descent algorithm to train your model (optimizer) \n",
    "7. Visualize your training process and results\n",
    "8. Apply feature scaling (pen & paper)"
   ]
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    "## Requirements\n",
    "### Knowledge\n",
    "\n",
    "You should have a basic knowledge of:\n",
    "- Univariate linear regression\n",
    "- Multivariate linear regression\n",
    "- Squared error\n",
    "- Gradient descent\n",
    "- numpy\n",
    "- matplotlib\n",
    "\n",
    "Suitable sources for acquiring this knowledge are:\n",
    "- [Multivariate Linear Regression Notebook](http://christianherta.de/lehre/dataScience/machineLearning/basics/multivariate_linear_regression.php) by Christian Herta and his [lecture slides](http://christianherta.de/lehre/dataScience/machineLearning/multivariateLinearRegression.pdf) (German)\n",
    "- Chapter 2 of the open classroom [Machine Learning](http://openclassroom.stanford.edu/MainFolder/CoursePage.php?course=MachineLearning) by Andrew Ng\n",
    "- Chapter 5.1 of [Deep Learning](http://www.deeplearningbook.org/contents/ml.html) by Ian Goodfellow \n",
    "- Some parts of chapter 1 and 3 of [Pattern Recognition and Machine Learning](https://www.microsoft.com/en-us/research/people/cmbishop/#!prml-book) by Christopher M. Bishop\n",
    "- [numpy quickstart](https://docs.scipy.org/doc/numpy-1.15.1/user/quickstart.html)\n",
    "- [Matplotlib tutorials](https://matplotlib.org/tutorials/index.html)"
   ]
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    "### Python Modules\n",
    "\n",
    "By [deep.TEACHING](https://www.deep-teaching.org/) convention, all python modules needed to run the notebook are loaded centrally at the beginning. \n"
   ]
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    "# External Modules\n",
    "\n",
    "\n",
    "import numpy as np\n",
    "from mpl_toolkits.mplot3d.axes3d import Axes3D\n",
    "from matplotlib import cm\n",
    "np.random.seed(42)\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "np.set_printoptions(suppress=True)\n",
    "\n",
    "%matplotlib inline"
   ]
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    "## Exercise - Multivariate Linear Regression\n",
    "\n",
    "We will only use two features in this notebook, so we are still able to plot them together with the target in a 3D plot. But your implementation should also be capable of handling more (except the plots). "
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    "### Create Features\n",
    "\n",
    "First we will create some features. The features should be in a 2D numpy array, the rows separating the different feature vectors (training examples), the columns containing the features. Each feature should be **uniformly** distributed in a specifiable range.\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Implement the function to generate a feature matrix (numpy array)."
   ]
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    "def create_feature_matrix(sample_size, n_features, x_min, x_max):\n",
    "    '''creates random feature vectors based on a linear function in a given interval\n",
    "    \n",
    "    Args:\n",
    "        sample_size: number feature vectors\n",
    "        n_features: number of features for each vector\n",
    "        x_min: lower bound value ranges\n",
    "        x_max: upper bound value ranges\n",
    "    \n",
    "    Returns:\n",
    "        x: 2D array containing feature vectors with shape (sample_size, n_features)\n",
    "    '''\n",
    "    x = np.zeros(shape=(n_features,sample_size))\n",
    "    \n",
    "    for i in range(n_features):\n",
    "        x[i] = np.random.uniform(x_min[i],x_max[i],sample_size)\n",
    "        \n",
    "    return np.round(x,2).T\n",
    "    \n",
    "    \n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
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    "sample_size = 50\n",
    "n_features = 2\n",
    "x_min = [1.5, -0.5]\n",
    "x_max = [11., 5.0]\n",
    "\n",
    "X = create_feature_matrix(sample_size, n_features, x_min, x_max)"
   ]
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    "assert len(X[:,0]) == sample_size\n",
    "assert len(X[0,:]) == n_features\n",
    "for i in range(n_features):\n",
    "    assert np.max(X[:,i]) <= x_max[i]\n",
    "    assert np.min(X[:,i]) >= x_min[i]"
   ]
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    "### Linear Hypothesis\n",
    "\n",
    "\n",
    "A short recap, a hypothesis $h_\\theta({\\bf x})$ is a certain function that we believe is similar to a target function that we like to model. A hypothesis $h_\\theta({\\bf x})$ is a function of ${\\bf x}$ with fixed parameters $\\Theta$. \n",
    "\n",
    "Here we have $n$ features ${\\bf x} = \\{x_1, \\ldots, x_n \\}$ and $n+1$ parameters $\\Theta = \\{\\theta_0, \\theta_1 \\ldots, \\theta_n \\}$:\n",
    "\n",
    "$$\n",
    "h_\\theta({\\bf x}) = \\theta_{0} + \\theta_{1} x_1 + \\ldots \\theta_n x_n \n",
    "$$\n",
    "\n",
    "adding an extra element to $\\vec x$ for convenience, this could also be rewritten as:\n",
    "\n",
    "$$\n",
    "h_\\theta({\\bf x}) = \\theta_{0} x_0 + \\theta_{1} x_1 + \\ldots \\theta_n x_n \n",
    "$$\n",
    "\n",
    "with $x_0 = 1$ for all feature vectors (training examples)."
   ]
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    "Or treating ${\\bf x}$ and $\\Theta$ as vectors:\n",
    "\n",
    "$$\n",
    "h(\\vec x) = \\vec x'^T \\vec \\theta\n",
    "$$\n",
    "\n",
    "with:\n",
    "\n",
    "$$\n",
    "\\vec x = \\begin{pmatrix} \n",
    "x_1 & x_2 & \\ldots & x_n \\\\\n",
    "\\end{pmatrix}^T\n",
    "\\text{   and   }\n",
    "\\vec x' = \\begin{pmatrix} \n",
    "1 & x_1 & x_2 & \\ldots & x_n \\\\\n",
    "\\end{pmatrix}^T\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\vec \\theta = \\begin{pmatrix} \n",
    "\\theta_0 & \\theta_1 & \\ldots & \\theta_n \\\\\n",
    "\\end{pmatrix}^T\n",
    "$$\n",
    "\n",
    "Or for the whole data set at once: The rows in $X$ separate the different feature vectors, the columns contain the features. \n",
    "\n",
    "$$\n",
    "\\vec h_\\Theta(X) = X' \\cdot \\vec \\theta\n",
    "$$\n",
    "\n",
    "the vector $\\vec h(X) = \\left( h(\\vec x^{(1)}),h(\\vec x^{(2)}), \\dots, h(\\vec x^{(m)}) )\\right)^T$ contains all predictions for the data batch $X$.\n",
    "\n",
    "with:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "X &= \\begin{pmatrix} \n",
    "x_1^{(1)} & \\ldots & x_n^{(1)} \\\\\n",
    "x_1^{(2)} & \\ldots & x_n^{(2)} \\\\\n",
    "\\vdots &\\vdots &\\vdots \\\\\n",
    "x_1^{(m)} & \\ldots & x_n^{(m)} \\\\\n",
    "\\end{pmatrix}\n",
    "&=\n",
    "\\begin{pmatrix} \n",
    "\\vec x^{(1)T} \\\\\n",
    "\\vec x^{(2)T}  \\\\\n",
    "\\vdots  \\\\\n",
    "\\vec x^{(m)T}  \\\\\n",
    "\\end{pmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "respectively\n",
    "$$\n",
    "\\begin{align}\n",
    "X' = \\begin{pmatrix} \n",
    "1 & x_1^{(1)} & \\ldots & x_n^{(1)} \\\\\n",
    "1 & x_1^{(2)} & \\ldots & x_n^{(2)} \\\\\n",
    "\\vdots &\\vdots &\\vdots &\\vdots \\\\\n",
    "1 & x_1^{(m)} & \\ldots & x_n^{(m)} \\\\\n",
    "\\end{pmatrix}\n",
    "&=\n",
    "\\begin{pmatrix} \n",
    "\\vec x'^{(1)T} \\\\\n",
    "\\vec x'^{(2)T}  \\\\\n",
    "\\vdots  \\\\\n",
    "\\vec x'^{(m)T}  \\\\\n",
    "\\end{pmatrix}\n",
    "\\end{align}\n",
    "$$"
   ]
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    "**Task:**\n",
    "\n",
    "Implement hypothesis $\\vec h_\\Theta(X)$ in the method `linear_hypothesis` and return it as a function. Implement it the computationally efficient (**pythonic**) way by not using any loops and handling all data at once (use $X$ respectively $X'$).\n",
    "\n",
    "**Hint:**\n",
    "\n",
    "Of course you are free to implement as many helper functions as you like, e.g. for transforming $X$ to $X'$, though you do not have to. Up to you."
   ]
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    "def linear_hypothesis(thetas):\n",
    "    ''' Combines given list argument in a linear equation and returns it as a function\n",
    "    \n",
    "    Args:\n",
    "        thetas: list of coefficients\n",
    "        \n",
    "    Returns:\n",
    "        lambda that models a linear function based on thetas and x\n",
    "    '''\n",
    "    \n",
    "    #return lambda x: np.sum(thetas * x_strich(x),axis=1)    \n",
    "    return lambda x:   x_strich(x).dot(thetas)\n",
    "\n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "def x_strich(x):\n",
    "    xt = np.zeros((x.shape[0],x.shape[1]+1))\n",
    "    xt[:,:1] = 1\n",
    "    xt[:,1:] = x\n",
    "    return xt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
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   "outputs": [
    {
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     "text": [
      "[2.561 3.334 3.191 2.864 1.533 2.067 0.507 2.22  1.467 2.133 0.932 2.539\n 3.2   1.242 1.061 1.492 1.062 2.571 1.193 2.433 2.687 0.843 0.815 2.293\n 2.283 2.945 1.902 1.351 1.967 0.53  2.829 1.603 0.92  2.157 2.597 2.323\n 2.031 1.489 3.014 1.866 0.68  2.366 1.57  2.905 2.014 2.324 1.706 1.943\n 1.33  0.779]\n"
     ]
    }
   ],
   "source": [
    "assert len(linear_hypothesis([.1,.2,.3])(X)) == sample_size\n",
    "\n",
    "print(linear_hypothesis([.1,.2,.3])(X))\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate Target Values\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Use your implemented `linear_hypothesis` inside the next function to generate some target values $Y$. Additionally add some Gaussian noise."
   ]
  },
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   "cell_type": "code",
   "execution_count": 87,
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    "def generate_targets(X, theta, sigma):\n",
    "    ''' Combines given arguments in a linear equation with X, \n",
    "    adds some Gaussian noise and returns the result\n",
    "    \n",
    "    Args:\n",
    "        X: 2D numpy feature matrix\n",
    "        theta: list of coefficients\n",
    "        sigma: standard deviation of the gaussian noise\n",
    "        \n",
    "    Returns:\n",
    "        target values for X\n",
    "    '''\n",
    "    \n",
    "    xt = linear_hypothesis(theta)(X)\n",
    "        \n",
    "    return xt + np.random.normal(0, scale=sigma, size=X.shape[0])\n",
    "    \n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta = np.array([2., 3., -4.])\n",
    "sigma = 3.\n",
    "y = generate_targets(X, theta, sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert len(y) == sample_size"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot The Data\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Plot the data $\\mathcal D = \\{((x^{(1)}_1,x^{(1)}_2)^T,y^{(1)}), \\ldots, ((x^{(n)}_1,x^{(n)}_2)^T,y^{(n)})\\}$ in a 3D scatter plot. The plot should look like the following:\n",
    "\n",
    "<img src=\"https://gitlab.com/deep.TEACHING/educational-materials/raw/dev/media/klaus/exercise-multivariate-linear-regression-scatter.png\" width=\"512\" alt=\"internet connection needed\">\n",
    "\n",
    "**Sidenote:**\n",
    "\n",
    "The command `%matplotlib notebook` (instead of `%matplotlib inline`) creates an interactive (e.g. rotatable) plot."
   ]
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   "cell_type": "code",
   "execution_count": 90,
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    "%matplotlib notebook\n",
    "\n",
    "def plot_data_scatter(features, targets):\n",
    "    \"\"\" Plots the features and the targets in a 3D scatter plot\n",
    "    \n",
    "    Args:\n",
    "        features: 2D numpy-array features\n",
    "        targets: ltargets\n",
    "    \"\"\"\n",
    "    \n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(111, projection='3d')\n",
    "    \n",
    "    ax.scatter(features[:,0],features[:,1],targets)\n",
    "    \n",
    "    ax.set_xlabel('X1 Label')\n",
    "    ax.set_ylabel('X2 Label')\n",
    "    ax.set_zlabel('Y Label')\n",
    "\n",
    "    return fig\n",
    "    \n",
    "    #raise NotImplementedError(\"You should implement this!\")"
   ]
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position: absolute; left: 0; top: 0; z-index: 1;'\n    );\n\n    // Apply a ponyfill if ResizeObserver is not implemented by browser.\n    if (this.ResizeObserver === undefined) {\n        if (window.ResizeObserver !== undefined) {\n            this.ResizeObserver = window.ResizeObserver;\n        } else {\n            var obs = _JSXTOOLS_RESIZE_OBSERVER({});\n            this.ResizeObserver = obs.ResizeObserver;\n        }\n    }\n\n    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {\n        var nentries = entries.length;\n        for (var i = 0; i < nentries; i++) {\n            var entry = entries[i];\n            var width, height;\n            if (entry.contentBoxSize) {\n                if (entry.contentBoxSize instanceof Array) {\n                    // Chrome 84 implements new version of spec.\n                    width = entry.contentBoxSize[0].inlineSize;\n                    height = entry.contentBoxSize[0].blockSize;\n                } else {\n                    // Firefox implements old version of spec.\n                    width = entry.contentBoxSize.inlineSize;\n                    height = entry.contentBoxSize.blockSize;\n                }\n            } else {\n                // Chrome <84 implements even older version of spec.\n                width = entry.contentRect.width;\n                height = entry.contentRect.height;\n            }\n\n            // Keep the size of the canvas and rubber band canvas in sync with\n            // the canvas container.\n            if (entry.devicePixelContentBoxSize) {\n                // Chrome 84 implements new version of spec.\n                canvas.setAttribute(\n                    'width',\n                    entry.devicePixelContentBoxSize[0].inlineSize\n                );\n                canvas.setAttribute(\n                    'height',\n                    entry.devicePixelContentBoxSize[0].blockSize\n                );\n            } else {\n                canvas.setAttribute('width', width * fig.ratio);\n                canvas.setAttribute('height', height * fig.ratio);\n            }\n            canvas.setAttribute(\n                'style',\n                'width: ' + width + 'px; height: ' + height + 'px;'\n            );\n\n            rubberband_canvas.setAttribute('width', width);\n            rubberband_canvas.setAttribute('height', height);\n\n            // And update the size in Python. We ignore the initial 0/0 size\n            // that occurs as the element is placed into the DOM, which should\n            // otherwise not happen due to the minimum size styling.\n            if (width != 0 && height != 0) {\n                fig.request_resize(width, height);\n            }\n        }\n    });\n    this.resizeObserverInstance.observe(canvas_div);\n\n    function on_mouse_event_closure(name) {\n        return function (event) {\n            return fig.mouse_event(event, name);\n        };\n    }\n\n    rubberband_canvas.addEventListener(\n        'mousedown',\n        on_mouse_event_closure('button_press')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseup',\n        on_mouse_event_closure('button_release')\n    );\n    // Throttle sequential mouse events to 1 every 20ms.\n    rubberband_canvas.addEventListener(\n        'mousemove',\n        on_mouse_event_closure('motion_notify')\n    );\n\n    rubberband_canvas.addEventListener(\n        'mouseenter',\n        on_mouse_event_closure('figure_enter')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseleave',\n        on_mouse_event_closure('figure_leave')\n    );\n\n    canvas_div.addEventListener('wheel', function (event) {\n        if (event.deltaY < 0) {\n            event.step = 1;\n        } else {\n            event.step = -1;\n        }\n        on_mouse_event_closure('scroll')(event);\n    });\n\n    canvas_div.appendChild(canvas);\n    canvas_div.appendChild(rubberband_canvas);\n\n    this.rubberband_context = rubberband_canvas.getContext('2d');\n    this.rubberband_context.strokeStyle = '#000000';\n\n    this._resize_canvas = function (width, height, forward) {\n        if (forward) {\n            canvas_div.style.width = width + 'px';\n            canvas_div.style.height = height + 'px';\n        }\n    };\n\n    // Disable right mouse context menu.\n    this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n        event.preventDefault();\n        return false;\n    });\n\n    function set_focus() {\n        canvas.focus();\n        canvas_div.focus();\n    }\n\n    window.setTimeout(set_focus, 100);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'mpl-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'mpl-button-group';\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'mpl-button-group';\n            continue;\n        }\n\n        var button = (fig.buttons[name] = document.createElement('button'));\n        button.classList = 'mpl-widget';\n        button.setAttribute('role', 'button');\n        button.setAttribute('aria-disabled', 'false');\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n\n        var icon_img = document.createElement('img');\n        icon_img.src = '_images/' + image + '.png';\n        icon_img.srcset = '_images/' + image + '_large.png 2x';\n        icon_img.alt = tooltip;\n        button.appendChild(icon_img);\n\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    var fmt_picker = document.createElement('select');\n    fmt_picker.classList = 'mpl-widget';\n    toolbar.appendChild(fmt_picker);\n    this.format_dropdown = fmt_picker;\n\n    for (var ind in mpl.extensions) {\n        var fmt = mpl.extensions[ind];\n        var option = document.createElement('option');\n        option.selected = fmt === mpl.default_extension;\n        option.innerHTML = fmt;\n        fmt_picker.appendChild(option);\n    }\n\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n};\n\nmpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n    // which will in turn request a refresh of the image.\n    this.send_message('resize', { width: x_pixels, height: y_pixels });\n};\n\nmpl.figure.prototype.send_message = function (type, properties) {\n    properties['type'] = type;\n    properties['figure_id'] = this.id;\n    this.ws.send(JSON.stringify(properties));\n};\n\nmpl.figure.prototype.send_draw_message = function () {\n    if (!this.waiting) {\n        this.waiting = true;\n        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    var format_dropdown = fig.format_dropdown;\n    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n    fig.ondownload(fig, format);\n};\n\nmpl.figure.prototype.handle_resize = function (fig, msg) {\n    var size = msg['size'];\n    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n        fig._resize_canvas(size[0], size[1], msg['forward']);\n        fig.send_message('refresh', {});\n    }\n};\n\nmpl.figure.prototype.handle_rubberband = function (fig, msg) {\n    var x0 = msg['x0'] / fig.ratio;\n    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n    var x1 = msg['x1'] / fig.ratio;\n    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n    x0 = Math.floor(x0) + 0.5;\n    y0 = Math.floor(y0) + 0.5;\n    x1 = Math.floor(x1) + 0.5;\n    y1 = Math.floor(y1) + 0.5;\n    var min_x = Math.min(x0, x1);\n    var min_y = Math.min(y0, y1);\n    var width = Math.abs(x1 - x0);\n    var height = Math.abs(y1 - y0);\n\n    fig.rubberband_context.clearRect(\n        0,\n        0,\n        fig.canvas.width / fig.ratio,\n        fig.canvas.height / fig.ratio\n    );\n\n    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n};\n\nmpl.figure.prototype.handle_figure_label = function (fig, msg) {\n    // Updates the figure title.\n    fig.header.textContent = msg['label'];\n};\n\nmpl.figure.prototype.handle_cursor = function (fig, msg) {\n    var cursor = msg['cursor'];\n    switch (cursor) {\n        case 0:\n            cursor = 'pointer';\n            break;\n        case 1:\n            cursor = 'default';\n            break;\n        case 2:\n            cursor = 'crosshair';\n            break;\n        case 3:\n            cursor = 'move';\n            break;\n    }\n    fig.rubberband_canvas.style.cursor = cursor;\n};\n\nmpl.figure.prototype.handle_message = function (fig, msg) {\n    fig.message.textContent = msg['message'];\n};\n\nmpl.figure.prototype.handle_draw = function (fig, _msg) {\n    // Request the server to send over a new figure.\n    fig.send_draw_message();\n};\n\nmpl.figure.prototype.handle_image_mode = function (fig, msg) {\n    fig.image_mode = msg['mode'];\n};\n\nmpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n    for (var key in msg) {\n        if (!(key in fig.buttons)) {\n            continue;\n        }\n        fig.buttons[key].disabled = !msg[key];\n        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n    }\n};\n\nmpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n    if (msg['mode'] === 'PAN') {\n        fig.buttons['Pan'].classList.add('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    } else if (msg['mode'] === 'ZOOM') {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.add('active');\n    } else {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    }\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Called whenever the canvas gets updated.\n    this.send_message('ack', {});\n};\n\n// A function to construct a web socket function for onmessage handling.\n// Called in the figure constructor.\nmpl.figure.prototype._make_on_message_function = function (fig) {\n    return function socket_on_message(evt) {\n        if (evt.data instanceof Blob) {\n            /* FIXME: We get \"Resource interpreted as Image but\n             * transferred with MIME type text/plain:\" errors on\n             * Chrome.  But how to set the MIME type?  It doesn't seem\n             * to be part of the websocket stream */\n            evt.data.type = 'image/png';\n\n            /* Free the memory for the previous frames */\n            if (fig.imageObj.src) {\n                (window.URL || window.webkitURL).revokeObjectURL(\n                    fig.imageObj.src\n                );\n            }\n\n            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n                evt.data\n            );\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        } else if (\n            typeof evt.data === 'string' &&\n            evt.data.slice(0, 21) === 'data:image/png;base64'\n        ) {\n            fig.imageObj.src = evt.data;\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        }\n\n        var msg = JSON.parse(evt.data);\n        var msg_type = msg['type'];\n\n        // Call the  \"handle_{type}\" callback, which takes\n        // the figure and JSON message as its only arguments.\n        try {\n            var callback = fig['handle_' + msg_type];\n        } catch (e) {\n            console.log(\n                \"No handler for the '\" + msg_type + \"' message type: \",\n                msg\n            );\n            return;\n        }\n\n        if (callback) {\n            try {\n                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n                callback(fig, msg);\n            } catch (e) {\n                console.log(\n                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n                    e,\n                    e.stack,\n                    msg\n                );\n            }\n        }\n    };\n};\n\n// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\nmpl.findpos = function (e) {\n    //this section is from http://www.quirksmode.org/js/events_properties.html\n    var targ;\n    if (!e) {\n        e = window.event;\n    }\n    if (e.target) {\n        targ = e.target;\n    } else if (e.srcElement) {\n        targ = e.srcElement;\n    }\n    if (targ.nodeType === 3) {\n        // defeat Safari bug\n        targ = targ.parentNode;\n    }\n\n    // pageX,Y are the mouse positions relative to the document\n    var boundingRect = targ.getBoundingClientRect();\n    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n\n    return { x: x, y: y };\n};\n\n/*\n * return a copy of an object with only non-object keys\n * we need this to avoid circular references\n * http://stackoverflow.com/a/24161582/3208463\n */\nfunction simpleKeys(original) {\n    return Object.keys(original).reduce(function (obj, key) {\n        if (typeof original[key] !== 'object') {\n            obj[key] = original[key];\n        }\n        return obj;\n    }, {});\n}\n\nmpl.figure.prototype.mouse_event = function (event, name) {\n    var canvas_pos = mpl.findpos(event);\n\n    if (name === 'button_press') {\n        this.canvas.focus();\n        this.canvas_div.focus();\n    }\n\n    var x = canvas_pos.x * this.ratio;\n    var y = canvas_pos.y * this.ratio;\n\n    this.send_message(name, {\n        x: x,\n        y: y,\n        button: event.button,\n        step: event.step,\n        guiEvent: simpleKeys(event),\n    });\n\n    /* This prevents the web browser from automatically changing to\n     * the text insertion cursor when the button is pressed.  We want\n     * to control all of the cursor setting manually through the\n     * 'cursor' event from matplotlib */\n    event.preventDefault();\n    return false;\n};\n\nmpl.figure.prototype._key_event_extra = function (_event, _name) {\n    // Handle any extra behaviour associated with a key event\n};\n\nmpl.figure.prototype.key_event = function (event, name) {\n    // Prevent repeat events\n    if (name === 'key_press') {\n        if (event.which === this._key) {\n            return;\n        } else {\n            this._key = event.which;\n        }\n    }\n    if (name === 'key_release') {\n        this._key = null;\n    }\n\n    var value = '';\n    if (event.ctrlKey && event.which !== 17) {\n        value += 'ctrl+';\n    }\n    if (event.altKey && event.which !== 18) {\n        value += 'alt+';\n    }\n    if (event.shiftKey && event.which !== 16) {\n        value += 'shift+';\n    }\n\n    value += 'k';\n    value += event.which.toString();\n\n    this._key_event_extra(event, name);\n\n    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n    return false;\n};\n\nmpl.figure.prototype.toolbar_button_onclick = function (name) {\n    if (name === 'download') {\n        this.handle_save(this, null);\n    } else {\n        this.send_message('toolbar_button', { name: name });\n    }\n};\n\nmpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n    this.message.textContent = tooltip;\n};\n\n///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n// prettier-ignore\nvar _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\nmpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n\nmpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n\nmpl.default_extension = \"png\";/* global mpl */\n\nvar comm_websocket_adapter = function (comm) {\n    // Create a \"websocket\"-like object which calls the given IPython comm\n    // object with the appropriate methods. Currently this is a non binary\n    // socket, so there is still some room for performance tuning.\n    var ws = {};\n\n    ws.close = function () {\n        comm.close();\n    };\n    ws.send = function (m) {\n        //console.log('sending', m);\n        comm.send(m);\n    };\n    // Register the callback with on_msg.\n    comm.on_msg(function (msg) {\n        //console.log('receiving', msg['content']['data'], msg);\n        // Pass the mpl event to the overridden (by mpl) onmessage function.\n        ws.onmessage(msg['content']['data']);\n    });\n    return ws;\n};\n\nmpl.mpl_figure_comm = function (comm, msg) {\n    // This is the function which gets called when the mpl process\n    // starts-up an IPython Comm through the \"matplotlib\" channel.\n\n    var id = msg.content.data.id;\n    // Get hold of the div created by the display call when the Comm\n    // socket was opened in Python.\n    var element = document.getElementById(id);\n    var ws_proxy = comm_websocket_adapter(comm);\n\n    function ondownload(figure, _format) {\n        window.open(figure.canvas.toDataURL());\n    }\n\n    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n\n    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n    // web socket which is closed, not our websocket->open comm proxy.\n    ws_proxy.onopen();\n\n    fig.parent_element = element;\n    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n    if (!fig.cell_info) {\n        console.error('Failed to find cell for figure', id, fig);\n        return;\n    }\n    fig.cell_info[0].output_area.element.on(\n        'cleared',\n        { fig: fig },\n        fig._remove_fig_handler\n    );\n};\n\nmpl.figure.prototype.handle_close = function (fig, msg) {\n    var width = fig.canvas.width / fig.ratio;\n    fig.cell_info[0].output_area.element.off(\n        'cleared',\n        fig._remove_fig_handler\n    );\n    fig.resizeObserverInstance.unobserve(fig.canvas_div);\n\n    // Update the output cell to use the data from the current canvas.\n    fig.push_to_output();\n    var dataURL = fig.canvas.toDataURL();\n    // Re-enable the keyboard manager in IPython - without this line, in FF,\n    // the notebook keyboard shortcuts fail.\n    IPython.keyboard_manager.enable();\n    fig.parent_element.innerHTML =\n        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n    fig.close_ws(fig, msg);\n};\n\nmpl.figure.prototype.close_ws = function (fig, msg) {\n    fig.send_message('closing', msg);\n    // fig.ws.close()\n};\n\nmpl.figure.prototype.push_to_output = function (_remove_interactive) {\n    // Turn the data on the canvas into data in the output cell.\n    var width = this.canvas.width / this.ratio;\n    var dataURL = this.canvas.toDataURL();\n    this.cell_info[1]['text/html'] =\n        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Tell IPython that the notebook contents must change.\n    IPython.notebook.set_dirty(true);\n    this.send_message('ack', {});\n    var fig = this;\n    // Wait a second, then push the new image to the DOM so\n    // that it is saved nicely (might be nice to debounce this).\n    setTimeout(function () {\n        fig.push_to_output();\n    }, 1000);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'btn-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'btn-group';\n    var button;\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'btn-group';\n            continue;\n        }\n\n        button = fig.buttons[name] = document.createElement('button');\n        button.classList = 'btn btn-default';\n        button.href = '#';\n        button.title = name;\n        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    // Add the status bar.\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message pull-right';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n\n    // Add the close button to the window.\n    var buttongrp = document.createElement('div');\n    buttongrp.classList = 'btn-group inline pull-right';\n    button = document.createElement('button');\n    button.classList = 'btn btn-mini btn-primary';\n    button.href = '#';\n    button.title = 'Stop Interaction';\n    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n    button.addEventListener('click', function (_evt) {\n        fig.handle_close(fig, {});\n    });\n    button.addEventListener(\n        'mouseover',\n        on_mouseover_closure('Stop Interaction')\n    );\n    buttongrp.appendChild(button);\n    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n};\n\nmpl.figure.prototype._remove_fig_handler = function (event) {\n    var fig = event.data.fig;\n    if (event.target !== this) {\n        // Ignore bubbled events from children.\n        return;\n    }\n    fig.close_ws(fig, {});\n};\n\nmpl.figure.prototype._root_extra_style = function (el) {\n    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n};\n\nmpl.figure.prototype._canvas_extra_style = function (el) {\n    // this is important to make the div 'focusable\n    el.setAttribute('tabindex', 0);\n    // reach out to IPython and tell the keyboard manager to turn it's self\n    // off when our div gets focus\n\n    // location in version 3\n    if (IPython.notebook.keyboard_manager) {\n        IPython.notebook.keyboard_manager.register_events(el);\n    } else {\n        // location in version 2\n        IPython.keyboard_manager.register_events(el);\n    }\n};\n\nmpl.figure.prototype._key_event_extra = function (event, _name) {\n    var manager = IPython.notebook.keyboard_manager;\n    if (!manager) {\n        manager = IPython.keyboard_manager;\n    }\n\n    // Check for shift+enter\n    if (event.shiftKey && event.which === 13) {\n        this.canvas_div.blur();\n        // select the cell after this one\n        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n        IPython.notebook.select(index + 1);\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    fig.ondownload(fig, null);\n};\n\nmpl.find_output_cell = function (html_output) {\n    // Return the cell and output element which can be found *uniquely* in the notebook.\n    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n    // IPython event is triggered only after the cells have been serialised, which for\n    // our purposes (turning an active figure into a static one), is too late.\n    var cells = IPython.notebook.get_cells();\n    var ncells = cells.length;\n    for (var i = 0; i < ncells; i++) {\n        var cell = cells[i];\n        if (cell.cell_type === 'code') {\n            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n                var data = cell.output_area.outputs[j];\n                if (data.data) {\n                    // IPython >= 3 moved mimebundle to data attribute of output\n                    data = data.data;\n                }\n                if (data['text/html'] === html_output) {\n                    return [cell, data, j];\n                }\n            }\n        }\n    }\n};\n\n// Register the function which deals with the matplotlib target/channel.\n// The kernel may be null if the page has been refreshed.\nif (IPython.notebook.kernel !== null) {\n    IPython.notebook.kernel.comm_manager.register_target(\n        'matplotlib',\n        mpl.mpl_figure_comm\n    );\n}\n"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.HTML object>",
      "text/html": "<div id='819464e8-cbe1-4c33-9a6d-ce5a2716cc96'></div>"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.Javascript object>",
      "application/javascript": "/* Put everything inside the global mpl namespace */\n/* global mpl */\nwindow.mpl = {};\n\nmpl.get_websocket_type = function () {\n    if (typeof WebSocket !== 'undefined') {\n        return WebSocket;\n    } else if (typeof MozWebSocket !== 'undefined') {\n        return MozWebSocket;\n    } else {\n        alert(\n            'Your browser does not have WebSocket support. ' +\n                'Please try Chrome, Safari or Firefox ≥ 6. ' +\n                'Firefox 4 and 5 are also supported but you ' +\n                'have to enable WebSockets in about:config.'\n        );\n    }\n};\n\nmpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n    this.id = figure_id;\n\n    this.ws = websocket;\n\n    this.supports_binary = this.ws.binaryType !== undefined;\n\n    if (!this.supports_binary) {\n        var warnings = document.getElementById('mpl-warnings');\n        if (warnings) {\n            warnings.style.display = 'block';\n            warnings.textContent =\n                'This browser does not support binary websocket messages. ' +\n                'Performance may be slow.';\n        }\n    }\n\n    this.imageObj = new Image();\n\n    this.context = undefined;\n    this.message = undefined;\n    this.canvas = undefined;\n    this.rubberband_canvas = undefined;\n    this.rubberband_context = undefined;\n    this.format_dropdown = undefined;\n\n    this.image_mode = 'full';\n\n    this.root = document.createElement('div');\n    this.root.setAttribute('style', 'display: inline-block');\n    this._root_extra_style(this.root);\n\n    parent_element.appendChild(this.root);\n\n    this._init_header(this);\n    this._init_canvas(this);\n    this._init_toolbar(this);\n\n    var fig = this;\n\n    this.waiting = false;\n\n    this.ws.onopen = function () {\n        fig.send_message('supports_binary', { value: fig.supports_binary });\n        fig.send_message('send_image_mode', {});\n        if (fig.ratio !== 1) {\n            fig.send_message('set_dpi_ratio', { dpi_ratio: fig.ratio });\n        }\n        fig.send_message('refresh', {});\n    };\n\n    this.imageObj.onload = function () {\n        if (fig.image_mode === 'full') {\n            // Full images could contain transparency (where diff images\n            // almost always do), so we need to clear the canvas so that\n            // there is no ghosting.\n            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n        }\n        fig.context.drawImage(fig.imageObj, 0, 0);\n    };\n\n    this.imageObj.onunload = function () {\n        fig.ws.close();\n    };\n\n    this.ws.onmessage = this._make_on_message_function(this);\n\n    this.ondownload = ondownload;\n};\n\nmpl.figure.prototype._init_header = function () {\n    var titlebar = document.createElement('div');\n    titlebar.classList =\n        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n    var titletext = document.createElement('div');\n    titletext.classList = 'ui-dialog-title';\n    titletext.setAttribute(\n        'style',\n        'width: 100%; text-align: center; padding: 3px;'\n    );\n    titlebar.appendChild(titletext);\n    this.root.appendChild(titlebar);\n    this.header = titletext;\n};\n\nmpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n\nmpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n\nmpl.figure.prototype._init_canvas = function () {\n    var fig = this;\n\n    var canvas_div = (this.canvas_div = document.createElement('div'));\n    canvas_div.setAttribute(\n        'style',\n        'border: 1px solid #ddd;' +\n            'box-sizing: content-box;' +\n            'clear: both;' +\n            'min-height: 1px;' +\n            'min-width: 1px;' +\n            'outline: 0;' +\n            'overflow: hidden;' +\n            'position: relative;' +\n            'resize: both;'\n    );\n\n    function on_keyboard_event_closure(name) {\n        return function (event) {\n            return fig.key_event(event, name);\n        };\n    }\n\n    canvas_div.addEventListener(\n        'keydown',\n        on_keyboard_event_closure('key_press')\n    );\n    canvas_div.addEventListener(\n        'keyup',\n        on_keyboard_event_closure('key_release')\n    );\n\n    this._canvas_extra_style(canvas_div);\n    this.root.appendChild(canvas_div);\n\n    var canvas = (this.canvas = document.createElement('canvas'));\n    canvas.classList.add('mpl-canvas');\n    canvas.setAttribute('style', 'box-sizing: content-box;');\n\n    this.context = canvas.getContext('2d');\n\n    var backingStore =\n        this.context.backingStorePixelRatio ||\n        this.context.webkitBackingStorePixelRatio ||\n        this.context.mozBackingStorePixelRatio ||\n        this.context.msBackingStorePixelRatio ||\n        this.context.oBackingStorePixelRatio ||\n        this.context.backingStorePixelRatio ||\n        1;\n\n    this.ratio = (window.devicePixelRatio || 1) / backingStore;\n    if (this.ratio !== 1) {\n        fig.send_message('set_dpi_ratio', { dpi_ratio: this.ratio });\n    }\n\n    var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n        'canvas'\n    ));\n    rubberband_canvas.setAttribute(\n        'style',\n        'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n    );\n\n    // Apply a ponyfill if ResizeObserver is not implemented by browser.\n    if (this.ResizeObserver === undefined) {\n        if (window.ResizeObserver !== undefined) {\n            this.ResizeObserver = window.ResizeObserver;\n        } else {\n            var obs = _JSXTOOLS_RESIZE_OBSERVER({});\n            this.ResizeObserver = obs.ResizeObserver;\n        }\n    }\n\n    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {\n        var nentries = entries.length;\n        for (var i = 0; i < nentries; i++) {\n            var entry = entries[i];\n            var width, height;\n            if (entry.contentBoxSize) {\n                if (entry.contentBoxSize instanceof Array) {\n                    // Chrome 84 implements new version of spec.\n                    width = entry.contentBoxSize[0].inlineSize;\n                    height = entry.contentBoxSize[0].blockSize;\n                } else {\n                    // Firefox implements old version of spec.\n                    width = entry.contentBoxSize.inlineSize;\n                    height = entry.contentBoxSize.blockSize;\n                }\n            } else {\n                // Chrome <84 implements even older version of spec.\n                width = entry.contentRect.width;\n                height = entry.contentRect.height;\n            }\n\n            // Keep the size of the canvas and rubber band canvas in sync with\n            // the canvas container.\n            if (entry.devicePixelContentBoxSize) {\n                // Chrome 84 implements new version of spec.\n                canvas.setAttribute(\n                    'width',\n                    entry.devicePixelContentBoxSize[0].inlineSize\n                );\n                canvas.setAttribute(\n                    'height',\n                    entry.devicePixelContentBoxSize[0].blockSize\n                );\n            } else {\n                canvas.setAttribute('width', width * fig.ratio);\n                canvas.setAttribute('height', height * fig.ratio);\n            }\n            canvas.setAttribute(\n                'style',\n                'width: ' + width + 'px; height: ' + height + 'px;'\n            );\n\n            rubberband_canvas.setAttribute('width', width);\n            rubberband_canvas.setAttribute('height', height);\n\n            // And update the size in Python. We ignore the initial 0/0 size\n            // that occurs as the element is placed into the DOM, which should\n            // otherwise not happen due to the minimum size styling.\n            if (width != 0 && height != 0) {\n                fig.request_resize(width, height);\n            }\n        }\n    });\n    this.resizeObserverInstance.observe(canvas_div);\n\n    function on_mouse_event_closure(name) {\n        return function (event) {\n            return fig.mouse_event(event, name);\n        };\n    }\n\n    rubberband_canvas.addEventListener(\n        'mousedown',\n        on_mouse_event_closure('button_press')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseup',\n        on_mouse_event_closure('button_release')\n    );\n    // Throttle sequential mouse events to 1 every 20ms.\n    rubberband_canvas.addEventListener(\n        'mousemove',\n        on_mouse_event_closure('motion_notify')\n    );\n\n    rubberband_canvas.addEventListener(\n        'mouseenter',\n        on_mouse_event_closure('figure_enter')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseleave',\n        on_mouse_event_closure('figure_leave')\n    );\n\n    canvas_div.addEventListener('wheel', function (event) {\n        if (event.deltaY < 0) {\n            event.step = 1;\n        } else {\n            event.step = -1;\n        }\n        on_mouse_event_closure('scroll')(event);\n    });\n\n    canvas_div.appendChild(canvas);\n    canvas_div.appendChild(rubberband_canvas);\n\n    this.rubberband_context = rubberband_canvas.getContext('2d');\n    this.rubberband_context.strokeStyle = '#000000';\n\n    this._resize_canvas = function (width, height, forward) {\n        if (forward) {\n            canvas_div.style.width = width + 'px';\n            canvas_div.style.height = height + 'px';\n        }\n    };\n\n    // Disable right mouse context menu.\n    this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n        event.preventDefault();\n        return false;\n    });\n\n    function set_focus() {\n        canvas.focus();\n        canvas_div.focus();\n    }\n\n    window.setTimeout(set_focus, 100);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'mpl-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'mpl-button-group';\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'mpl-button-group';\n            continue;\n        }\n\n        var button = (fig.buttons[name] = document.createElement('button'));\n        button.classList = 'mpl-widget';\n        button.setAttribute('role', 'button');\n        button.setAttribute('aria-disabled', 'false');\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n\n        var icon_img = document.createElement('img');\n        icon_img.src = '_images/' + image + '.png';\n        icon_img.srcset = '_images/' + image + '_large.png 2x';\n        icon_img.alt = tooltip;\n        button.appendChild(icon_img);\n\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    var fmt_picker = document.createElement('select');\n    fmt_picker.classList = 'mpl-widget';\n    toolbar.appendChild(fmt_picker);\n    this.format_dropdown = fmt_picker;\n\n    for (var ind in mpl.extensions) {\n        var fmt = mpl.extensions[ind];\n        var option = document.createElement('option');\n        option.selected = fmt === mpl.default_extension;\n        option.innerHTML = fmt;\n        fmt_picker.appendChild(option);\n    }\n\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n};\n\nmpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n    // which will in turn request a refresh of the image.\n    this.send_message('resize', { width: x_pixels, height: y_pixels });\n};\n\nmpl.figure.prototype.send_message = function (type, properties) {\n    properties['type'] = type;\n    properties['figure_id'] = this.id;\n    this.ws.send(JSON.stringify(properties));\n};\n\nmpl.figure.prototype.send_draw_message = function () {\n    if (!this.waiting) {\n        this.waiting = true;\n        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    var format_dropdown = fig.format_dropdown;\n    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n    fig.ondownload(fig, format);\n};\n\nmpl.figure.prototype.handle_resize = function (fig, msg) {\n    var size = msg['size'];\n    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n        fig._resize_canvas(size[0], size[1], msg['forward']);\n        fig.send_message('refresh', {});\n    }\n};\n\nmpl.figure.prototype.handle_rubberband = function (fig, msg) {\n    var x0 = msg['x0'] / fig.ratio;\n    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n    var x1 = msg['x1'] / fig.ratio;\n    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n    x0 = Math.floor(x0) + 0.5;\n    y0 = Math.floor(y0) + 0.5;\n    x1 = Math.floor(x1) + 0.5;\n    y1 = Math.floor(y1) + 0.5;\n    var min_x = Math.min(x0, x1);\n    var min_y = Math.min(y0, y1);\n    var width = Math.abs(x1 - x0);\n    var height = Math.abs(y1 - y0);\n\n    fig.rubberband_context.clearRect(\n        0,\n        0,\n        fig.canvas.width / fig.ratio,\n        fig.canvas.height / fig.ratio\n    );\n\n    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n};\n\nmpl.figure.prototype.handle_figure_label = function (fig, msg) {\n    // Updates the figure title.\n    fig.header.textContent = msg['label'];\n};\n\nmpl.figure.prototype.handle_cursor = function (fig, msg) {\n    var cursor = msg['cursor'];\n    switch (cursor) {\n        case 0:\n            cursor = 'pointer';\n            break;\n        case 1:\n            cursor = 'default';\n            break;\n        case 2:\n            cursor = 'crosshair';\n            break;\n        case 3:\n            cursor = 'move';\n            break;\n    }\n    fig.rubberband_canvas.style.cursor = cursor;\n};\n\nmpl.figure.prototype.handle_message = function (fig, msg) {\n    fig.message.textContent = msg['message'];\n};\n\nmpl.figure.prototype.handle_draw = function (fig, _msg) {\n    // Request the server to send over a new figure.\n    fig.send_draw_message();\n};\n\nmpl.figure.prototype.handle_image_mode = function (fig, msg) {\n    fig.image_mode = msg['mode'];\n};\n\nmpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n    for (var key in msg) {\n        if (!(key in fig.buttons)) {\n            continue;\n        }\n        fig.buttons[key].disabled = !msg[key];\n        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n    }\n};\n\nmpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n    if (msg['mode'] === 'PAN') {\n        fig.buttons['Pan'].classList.add('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    } else if (msg['mode'] === 'ZOOM') {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.add('active');\n    } else {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    }\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Called whenever the canvas gets updated.\n    this.send_message('ack', {});\n};\n\n// A function to construct a web socket function for onmessage handling.\n// Called in the figure constructor.\nmpl.figure.prototype._make_on_message_function = function (fig) {\n    return function socket_on_message(evt) {\n        if (evt.data instanceof Blob) {\n            /* FIXME: We get \"Resource interpreted as Image but\n             * transferred with MIME type text/plain:\" errors on\n             * Chrome.  But how to set the MIME type?  It doesn't seem\n             * to be part of the websocket stream */\n            evt.data.type = 'image/png';\n\n            /* Free the memory for the previous frames */\n            if (fig.imageObj.src) {\n                (window.URL || window.webkitURL).revokeObjectURL(\n                    fig.imageObj.src\n                );\n            }\n\n            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n                evt.data\n            );\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        } else if (\n            typeof evt.data === 'string' &&\n            evt.data.slice(0, 21) === 'data:image/png;base64'\n        ) {\n            fig.imageObj.src = evt.data;\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        }\n\n        var msg = JSON.parse(evt.data);\n        var msg_type = msg['type'];\n\n        // Call the  \"handle_{type}\" callback, which takes\n        // the figure and JSON message as its only arguments.\n        try {\n            var callback = fig['handle_' + msg_type];\n        } catch (e) {\n            console.log(\n                \"No handler for the '\" + msg_type + \"' message type: \",\n                msg\n            );\n            return;\n        }\n\n        if (callback) {\n            try {\n                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n                callback(fig, msg);\n            } catch (e) {\n                console.log(\n                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n                    e,\n                    e.stack,\n                    msg\n                );\n            }\n        }\n    };\n};\n\n// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\nmpl.findpos = function (e) {\n    //this section is from http://www.quirksmode.org/js/events_properties.html\n    var targ;\n    if (!e) {\n        e = window.event;\n    }\n    if (e.target) {\n        targ = e.target;\n    } else if (e.srcElement) {\n        targ = e.srcElement;\n    }\n    if (targ.nodeType === 3) {\n        // defeat Safari bug\n        targ = targ.parentNode;\n    }\n\n    // pageX,Y are the mouse positions relative to the document\n    var boundingRect = targ.getBoundingClientRect();\n    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n\n    return { x: x, y: y };\n};\n\n/*\n * return a copy of an object with only non-object keys\n * we need this to avoid circular references\n * http://stackoverflow.com/a/24161582/3208463\n */\nfunction simpleKeys(original) {\n    return Object.keys(original).reduce(function (obj, key) {\n        if (typeof original[key] !== 'object') {\n            obj[key] = original[key];\n        }\n        return obj;\n    }, {});\n}\n\nmpl.figure.prototype.mouse_event = function (event, name) {\n    var canvas_pos = mpl.findpos(event);\n\n    if (name === 'button_press') {\n        this.canvas.focus();\n        this.canvas_div.focus();\n    }\n\n    var x = canvas_pos.x * this.ratio;\n    var y = canvas_pos.y * this.ratio;\n\n    this.send_message(name, {\n        x: x,\n        y: y,\n        button: event.button,\n        step: event.step,\n        guiEvent: simpleKeys(event),\n    });\n\n    /* This prevents the web browser from automatically changing to\n     * the text insertion cursor when the button is pressed.  We want\n     * to control all of the cursor setting manually through the\n     * 'cursor' event from matplotlib */\n    event.preventDefault();\n    return false;\n};\n\nmpl.figure.prototype._key_event_extra = function (_event, _name) {\n    // Handle any extra behaviour associated with a key event\n};\n\nmpl.figure.prototype.key_event = function (event, name) {\n    // Prevent repeat events\n    if (name === 'key_press') {\n        if (event.which === this._key) {\n            return;\n        } else {\n            this._key = event.which;\n        }\n    }\n    if (name === 'key_release') {\n        this._key = null;\n    }\n\n    var value = '';\n    if (event.ctrlKey && event.which !== 17) {\n        value += 'ctrl+';\n    }\n    if (event.altKey && event.which !== 18) {\n        value += 'alt+';\n    }\n    if (event.shiftKey && event.which !== 16) {\n        value += 'shift+';\n    }\n\n    value += 'k';\n    value += event.which.toString();\n\n    this._key_event_extra(event, name);\n\n    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n    return false;\n};\n\nmpl.figure.prototype.toolbar_button_onclick = function (name) {\n    if (name === 'download') {\n        this.handle_save(this, null);\n    } else {\n        this.send_message('toolbar_button', { name: name });\n    }\n};\n\nmpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n    this.message.textContent = tooltip;\n};\n\n///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n// prettier-ignore\nvar _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\nmpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n\nmpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n\nmpl.default_extension = \"png\";/* global mpl */\n\nvar comm_websocket_adapter = function (comm) {\n    // Create a \"websocket\"-like object which calls the given IPython comm\n    // object with the appropriate methods. Currently this is a non binary\n    // socket, so there is still some room for performance tuning.\n    var ws = {};\n\n    ws.close = function () {\n        comm.close();\n    };\n    ws.send = function (m) {\n        //console.log('sending', m);\n        comm.send(m);\n    };\n    // Register the callback with on_msg.\n    comm.on_msg(function (msg) {\n        //console.log('receiving', msg['content']['data'], msg);\n        // Pass the mpl event to the overridden (by mpl) onmessage function.\n        ws.onmessage(msg['content']['data']);\n    });\n    return ws;\n};\n\nmpl.mpl_figure_comm = function (comm, msg) {\n    // This is the function which gets called when the mpl process\n    // starts-up an IPython Comm through the \"matplotlib\" channel.\n\n    var id = msg.content.data.id;\n    // Get hold of the div created by the display call when the Comm\n    // socket was opened in Python.\n    var element = document.getElementById(id);\n    var ws_proxy = comm_websocket_adapter(comm);\n\n    function ondownload(figure, _format) {\n        window.open(figure.canvas.toDataURL());\n    }\n\n    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n\n    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n    // web socket which is closed, not our websocket->open comm proxy.\n    ws_proxy.onopen();\n\n    fig.parent_element = element;\n    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n    if (!fig.cell_info) {\n        console.error('Failed to find cell for figure', id, fig);\n        return;\n    }\n    fig.cell_info[0].output_area.element.on(\n        'cleared',\n        { fig: fig },\n        fig._remove_fig_handler\n    );\n};\n\nmpl.figure.prototype.handle_close = function (fig, msg) {\n    var width = fig.canvas.width / fig.ratio;\n    fig.cell_info[0].output_area.element.off(\n        'cleared',\n        fig._remove_fig_handler\n    );\n    fig.resizeObserverInstance.unobserve(fig.canvas_div);\n\n    // Update the output cell to use the data from the current canvas.\n    fig.push_to_output();\n    var dataURL = fig.canvas.toDataURL();\n    // Re-enable the keyboard manager in IPython - without this line, in FF,\n    // the notebook keyboard shortcuts fail.\n    IPython.keyboard_manager.enable();\n    fig.parent_element.innerHTML =\n        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n    fig.close_ws(fig, msg);\n};\n\nmpl.figure.prototype.close_ws = function (fig, msg) {\n    fig.send_message('closing', msg);\n    // fig.ws.close()\n};\n\nmpl.figure.prototype.push_to_output = function (_remove_interactive) {\n    // Turn the data on the canvas into data in the output cell.\n    var width = this.canvas.width / this.ratio;\n    var dataURL = this.canvas.toDataURL();\n    this.cell_info[1]['text/html'] =\n        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Tell IPython that the notebook contents must change.\n    IPython.notebook.set_dirty(true);\n    this.send_message('ack', {});\n    var fig = this;\n    // Wait a second, then push the new image to the DOM so\n    // that it is saved nicely (might be nice to debounce this).\n    setTimeout(function () {\n        fig.push_to_output();\n    }, 1000);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'btn-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'btn-group';\n    var button;\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'btn-group';\n            continue;\n        }\n\n        button = fig.buttons[name] = document.createElement('button');\n        button.classList = 'btn btn-default';\n        button.href = '#';\n        button.title = name;\n        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    // Add the status bar.\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message pull-right';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n\n    // Add the close button to the window.\n    var buttongrp = document.createElement('div');\n    buttongrp.classList = 'btn-group inline pull-right';\n    button = document.createElement('button');\n    button.classList = 'btn btn-mini btn-primary';\n    button.href = '#';\n    button.title = 'Stop Interaction';\n    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n    button.addEventListener('click', function (_evt) {\n        fig.handle_close(fig, {});\n    });\n    button.addEventListener(\n        'mouseover',\n        on_mouseover_closure('Stop Interaction')\n    );\n    buttongrp.appendChild(button);\n    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n};\n\nmpl.figure.prototype._remove_fig_handler = function (event) {\n    var fig = event.data.fig;\n    if (event.target !== this) {\n        // Ignore bubbled events from children.\n        return;\n    }\n    fig.close_ws(fig, {});\n};\n\nmpl.figure.prototype._root_extra_style = function (el) {\n    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n};\n\nmpl.figure.prototype._canvas_extra_style = function (el) {\n    // this is important to make the div 'focusable\n    el.setAttribute('tabindex', 0);\n    // reach out to IPython and tell the keyboard manager to turn it's self\n    // off when our div gets focus\n\n    // location in version 3\n    if (IPython.notebook.keyboard_manager) {\n        IPython.notebook.keyboard_manager.register_events(el);\n    } else {\n        // location in version 2\n        IPython.keyboard_manager.register_events(el);\n    }\n};\n\nmpl.figure.prototype._key_event_extra = function (event, _name) {\n    var manager = IPython.notebook.keyboard_manager;\n    if (!manager) {\n        manager = IPython.keyboard_manager;\n    }\n\n    // Check for shift+enter\n    if (event.shiftKey && event.which === 13) {\n        this.canvas_div.blur();\n        // select the cell after this one\n        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n        IPython.notebook.select(index + 1);\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    fig.ondownload(fig, null);\n};\n\nmpl.find_output_cell = function (html_output) {\n    // Return the cell and output element which can be found *uniquely* in the notebook.\n    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n    // IPython event is triggered only after the cells have been serialised, which for\n    // our purposes (turning an active figure into a static one), is too late.\n    var cells = IPython.notebook.get_cells();\n    var ncells = cells.length;\n    for (var i = 0; i < ncells; i++) {\n        var cell = cells[i];\n        if (cell.cell_type === 'code') {\n            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n                var data = cell.output_area.outputs[j];\n                if (data.data) {\n                    // IPython >= 3 moved mimebundle to data attribute of output\n                    data = data.data;\n                }\n                if (data['text/html'] === html_output) {\n                    return [cell, data, j];\n                }\n            }\n        }\n    }\n};\n\n// Register the function which deals with the matplotlib target/channel.\n// The kernel may be null if the page has been refreshed.\nif (IPython.notebook.kernel !== null) {\n    IPython.notebook.kernel.comm_manager.register_target(\n        'matplotlib',\n        mpl.mpl_figure_comm\n    );\n}\n"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.HTML object>",
      "text/html": "<div id='3a774f56-9ddd-4e69-bc5c-78cd01891b03'></div>"
     },
     "metadata": {}
    }
   ],
   "source": [
    "plot_data_scatter(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Cost Function\n",
    "A cost function $J$ depends on the given training data $D$ and hypothesis $h_\\theta(\\vec x)$. In the context of the linear regression, the cost function measures how \"wrong\" a model is regarding its ability to estimate the relationship between $\\vec x$ and $y$ for specific $\\Theta$ values. Later we will treat this as an optimization problem and try to minimize the cost function $J_{\\mathcal D}(\\Theta)$ to find optimal $\\theta$ values for our hypothesis $h_\\theta(\\vec x)$. The cost function we use in this exercise is the [Mean-Squared-Error](https://en.wikipedia.org/wiki/Mean_squared_error) cost function:\n",
    "\n",
    "\\begin{equation}\n",
    "    J_{\\mathcal D}(\\Theta)=\\frac{1}{2m}\\sum_{i=1}^{m}{(h_\\Theta(\\vec x^{(i)})-y^{(i)})^2}\n",
    "\\end{equation}\n",
    "\n",
    "Implement the cost function $J_D(\\Theta)$ in the method `mse_cost_function`. The method should return a function that takes the values of $\\Theta$ as an argument.\n",
    "\n",
    "As a sidenote, the terms \"loss function\" or \"error function\" are often used interchangeably in the field of Machine Learning."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def mse_cost_function(x, y):\n",
    "    ''' Implements MSE cost function as a function J(theta) on given traning data \n",
    "    \n",
    "    Args:\n",
    "        x: vector of x values \n",
    "        y: vector of ground truth values y \n",
    "        \n",
    "    Returns:\n",
    "        lambda J(theta) that models the cost function\n",
    "    '''\n",
    "    \n",
    "    return lambda thetas: 1/(2*len(x)) * sum(((linear_hypothesis(thetas))(x) - y)**2)\n",
    "    \n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Review the cell in which you generate the target values and note the theta values, which were used for it (If you haven't edited the default values, it should be `[2, 3, -4]`)\n",
    "\n",
    "**Optional:**\n",
    "\n",
    "Try a few different values for theta to pass to the cost function - Which thetas result in a low error and which produce a great error?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "3.9215915839169964\n"
     ]
    }
   ],
   "source": [
    "J = mse_cost_function(X, y)\n",
    "print(J(theta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "###  Gradient Descent\n",
    "\n",
    "A short recap, the gradient descent algorithm is a first-order iterative optimization for finding a minimum of a function. From the current position in a (cost) function, the algorithm steps proportional to the negative of the gradient and repeats this until it reaches a local or global minimum and determines. Stepping proportional means that it does not go entirely in the direction of the negative gradient, but scaled by a fixed value $\\alpha$ also called the learning rate. Implementing the following formalized update rule is the core of the optimization process:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\theta_{j}^{new} \\leftarrow \\theta_{j}^{old} - \\alpha * \\frac{\\partial}{\\partial\\theta_{j}} J(\\vec \\theta^{old})\n",
    "\\end{equation}\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Implement the function to update all theta values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def update_theta(x, y, theta, learning_rate):\n",
    "    ''' Updates learnable parameters theta \n",
    "    \n",
    "    The update is done by calculating the partial derivities of \n",
    "    the cost function including the linear hypothesis. The \n",
    "    gradients scaled by a scalar are subtracted from the given \n",
    "    theta values.\n",
    "    \n",
    "    Args:\n",
    "        x: 2D numpy array of x values\n",
    "        y: array of y values corresponding to x\n",
    "        theta: current theta values\n",
    "        learning_rate: value to scale the negative gradient \n",
    "        \n",
    "    Returns:\n",
    "        theta: Updated theta vector\n",
    "    '''\n",
    "    \n",
    "    h = linear_hypothesis(theta)(x)\n",
    "\n",
    "    return theta - learning_rate * (1.0 / x.shape[0]) * x_strich(x).T.dot(h - y)\n",
    "\n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "[ 2.00575435  2.72064029 -4.02464557]\n"
     ]
    }
   ],
   "source": [
    "print(update_theta(X,y,theta,0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Using the `update_theta` method, you can now implement the gradient descent algorithm. Iterate over the update rule to find the values for $\\vec \\theta$ that minimize our cost function $J_D(\\vec \\theta)$. This process is often called training of a machine learning model. \n",
    "\n",
    "**Task:**\n",
    "- Implement the function for the gradient descent.\n",
    "- Create a history of all theta and cost values and return them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def gradient_descent(learning_rate, theta, iterations, x, y):\n",
    "    ''' Minimize theta values of a linear model based on MSE cost function\n",
    "    \n",
    "    Args:\n",
    "        learning_rate: scalar, scales the negative gradient \n",
    "        theta: initial theta values\n",
    "        x: vector, x values from the data set\n",
    "        y: vector, y values from the data set\n",
    "        iterations: scalar, number of theta updates\n",
    "        \n",
    "    Returns:\n",
    "        history_cost: cost after each iteration\n",
    "        history_theta: Updated theta values after each iteration\n",
    "    '''\n",
    "    \n",
    "    thetas = np.zeros((iterations,len(theta)))\n",
    "    cost = np.zeros(iterations)\n",
    "    j = mse_cost_function(x, y)\n",
    "    \n",
    "    tmp = theta\n",
    "\n",
    "    for i in range(iterations):        \n",
    "        thetas[i] = tmp\n",
    "        tmp = update_theta(x, y, thetas[i], learning_rate)\n",
    "        cost[i] = j(thetas[i])\n",
    "        #if verbose & (i%500 == 0 ):\n",
    "       #     print(f\"Start: \\n Theta0 = {tmp0}\\n Theta1 = {tmp1}\\n\")\n",
    "       #     print(f'Nach {i} Iterations: \\n',\n",
    "       #         'Kosten = ',cost[i],'\\n',\n",
    "       #         'Theta0 = ',t0s[i],'\\n',\n",
    "       #         'Theta1 = ',t1s[i],'\\n')\n",
    "            #fig = plot_data_with_hypothesis(x, y, tmp0, tmp1,label = f\"Iteration: {i}\")\n",
    "            #fig.savefig(f\"fig{i}.png\")\n",
    "    return cost, thetas\n",
    "    \n",
    "    raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Training and Evaluation\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Choose an appropriate learning rate, number of iterations and initial theta values and start the training"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Your implementation:\n",
    "\n",
    "\n",
    "alpha = 0.001 # assign an appropriate value\n",
    "nb_iterations = 500 # assign an appropriate value\n",
    "start_values_theta = [5., 8., 2.] # assign appropriate values\n",
    "history_cost,history_theta  = gradient_descent(alpha, start_values_theta, nb_iterations, X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that the training has finished we can visualize our results.\n",
    "\n",
    "**Task:**\n",
    "\n",
    "Plot the costs over the iterations. If you have used `fig = plt.figure()` and `ax = fig.add_subplot(111)` in the last plot, use it again here, else the plot will be added to the last plot instead of a new one.\n",
    "\n",
    "Your plot should look similar to this one:\n",
    "\n",
    "<img src=\"https://gitlab.com/deep.TEACHING/educational-materials/raw/dev/media/klaus/exercise-multivariate-linear-regression-costs.png\" width=\"512\" alt=\"internet connection needed\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def plot_progress(costs):\n",
    "    \"\"\" Plots the costs over the iterations\n",
    "    \n",
    "    Args:\n",
    "        costs: history of costs\n",
    "    \"\"\"\n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(111)\n",
    "    \n",
    "    ax.plot(np.arange(len(history_cost)),history_cost)\n",
    "    \n",
    "    ax.set_xlabel('Iterations')\n",
    "    ax.set_ylabel('Cost')\n",
    "    \n",
    "    \n",
    "    #raise NotImplementedError(\"You should implement this!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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' +\n                'Firefox 4 and 5 are also supported but you ' +\n                'have to enable WebSockets in about:config.'\n        );\n    }\n};\n\nmpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n    this.id = figure_id;\n\n    this.ws = websocket;\n\n    this.supports_binary = this.ws.binaryType !== undefined;\n\n    if (!this.supports_binary) {\n        var warnings = document.getElementById('mpl-warnings');\n        if (warnings) {\n            warnings.style.display = 'block';\n            warnings.textContent =\n                'This browser does not support binary websocket messages. 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Matplotlib will then trigger a resize in the client,\n    // which will in turn request a refresh of the image.\n    this.send_message('resize', { width: x_pixels, height: y_pixels });\n};\n\nmpl.figure.prototype.send_message = function (type, properties) {\n    properties['type'] = type;\n    properties['figure_id'] = this.id;\n    this.ws.send(JSON.stringify(properties));\n};\n\nmpl.figure.prototype.send_draw_message = function () {\n    if (!this.waiting) {\n        this.waiting = true;\n        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    var format_dropdown = fig.format_dropdown;\n    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n    fig.ondownload(fig, format);\n};\n\nmpl.figure.prototype.handle_resize = function (fig, msg) {\n    var size = msg['size'];\n    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n        fig._resize_canvas(size[0], size[1], msg['forward']);\n        fig.send_message('refresh', {});\n    }\n};\n\nmpl.figure.prototype.handle_rubberband = function (fig, msg) {\n    var x0 = msg['x0'] / fig.ratio;\n    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n    var x1 = msg['x1'] / fig.ratio;\n    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n    x0 = Math.floor(x0) + 0.5;\n    y0 = Math.floor(y0) + 0.5;\n    x1 = Math.floor(x1) + 0.5;\n    y1 = Math.floor(y1) + 0.5;\n    var min_x = Math.min(x0, x1);\n    var min_y = Math.min(y0, y1);\n    var width = Math.abs(x1 - x0);\n    var height = Math.abs(y1 - y0);\n\n    fig.rubberband_context.clearRect(\n        0,\n        0,\n        fig.canvas.width / fig.ratio,\n        fig.canvas.height / fig.ratio\n    );\n\n    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n};\n\nmpl.figure.prototype.handle_figure_label = function (fig, msg) {\n    // Updates the figure title.\n    fig.header.textContent = msg['label'];\n};\n\nmpl.figure.prototype.handle_cursor = function (fig, msg) {\n    var cursor = msg['cursor'];\n    switch (cursor) {\n        case 0:\n            cursor = 'pointer';\n            break;\n        case 1:\n            cursor = 'default';\n            break;\n        case 2:\n            cursor = 'crosshair';\n            break;\n        case 3:\n            cursor = 'move';\n            break;\n    }\n    fig.rubberband_canvas.style.cursor = cursor;\n};\n\nmpl.figure.prototype.handle_message = function (fig, msg) {\n    fig.message.textContent = msg['message'];\n};\n\nmpl.figure.prototype.handle_draw = function (fig, _msg) {\n    // Request the server to send over a new figure.\n    fig.send_draw_message();\n};\n\nmpl.figure.prototype.handle_image_mode = function (fig, msg) {\n    fig.image_mode = msg['mode'];\n};\n\nmpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n    for (var key in msg) {\n        if (!(key in fig.buttons)) {\n            continue;\n        }\n        fig.buttons[key].disabled = !msg[key];\n        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n    }\n};\n\nmpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n    if (msg['mode'] === 'PAN') {\n        fig.buttons['Pan'].classList.add('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    } else if (msg['mode'] === 'ZOOM') {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.add('active');\n    } else {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    }\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Called whenever the canvas gets updated.\n    this.send_message('ack', {});\n};\n\n// A function to construct a web socket function for onmessage handling.\n// Called in the figure constructor.\nmpl.figure.prototype._make_on_message_function = function (fig) {\n    return function socket_on_message(evt) {\n        if (evt.data instanceof Blob) {\n            /* FIXME: We get \"Resource interpreted as Image but\n             * transferred with MIME type text/plain:\" errors on\n             * Chrome.  But how to set the MIME type?  It doesn't seem\n             * to be part of the websocket stream */\n            evt.data.type = 'image/png';\n\n            /* Free the memory for the previous frames */\n            if (fig.imageObj.src) {\n                (window.URL || window.webkitURL).revokeObjectURL(\n                    fig.imageObj.src\n                );\n            }\n\n            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n                evt.data\n            );\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        } else if (\n            typeof evt.data === 'string' &&\n            evt.data.slice(0, 21) === 'data:image/png;base64'\n        ) {\n            fig.imageObj.src = evt.data;\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        }\n\n        var msg = JSON.parse(evt.data);\n        var msg_type = msg['type'];\n\n        // Call the  \"handle_{type}\" callback, which takes\n        // the figure and JSON message as its only arguments.\n        try {\n            var callback = fig['handle_' + msg_type];\n        } catch (e) {\n            console.log(\n                \"No handler for the '\" + msg_type + \"' message type: \",\n                msg\n            );\n            return;\n        }\n\n        if (callback) {\n            try {\n                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n                callback(fig, msg);\n            } catch (e) {\n                console.log(\n                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n                    e,\n                    e.stack,\n                    msg\n                );\n            }\n        }\n    };\n};\n\n// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\nmpl.findpos = function (e) {\n    //this section is from http://www.quirksmode.org/js/events_properties.html\n    var targ;\n    if (!e) {\n        e = window.event;\n    }\n    if (e.target) {\n        targ = e.target;\n    } else if (e.srcElement) {\n        targ = e.srcElement;\n    }\n    if (targ.nodeType === 3) {\n        // defeat Safari bug\n        targ = targ.parentNode;\n    }\n\n    // pageX,Y are the mouse positions relative to the document\n    var boundingRect = targ.getBoundingClientRect();\n    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n\n    return { x: x, y: y };\n};\n\n/*\n * return a copy of an object with only non-object keys\n * we need this to avoid circular references\n * http://stackoverflow.com/a/24161582/3208463\n */\nfunction simpleKeys(original) {\n    return Object.keys(original).reduce(function (obj, key) {\n        if (typeof original[key] !== 'object') {\n            obj[key] = original[key];\n        }\n        return obj;\n    }, {});\n}\n\nmpl.figure.prototype.mouse_event = function (event, name) {\n    var canvas_pos = mpl.findpos(event);\n\n    if (name === 'button_press') {\n        this.canvas.focus();\n        this.canvas_div.focus();\n    }\n\n    var x = canvas_pos.x * this.ratio;\n    var y = canvas_pos.y * this.ratio;\n\n    this.send_message(name, {\n        x: x,\n        y: y,\n        button: event.button,\n        step: event.step,\n        guiEvent: simpleKeys(event),\n    });\n\n    /* This prevents the web browser from automatically changing to\n     * the text insertion cursor when the button is pressed.  We want\n     * to control all of the cursor setting manually through the\n     * 'cursor' event from matplotlib */\n    event.preventDefault();\n    return false;\n};\n\nmpl.figure.prototype._key_event_extra = function (_event, _name) {\n    // Handle any extra behaviour associated with a key event\n};\n\nmpl.figure.prototype.key_event = function (event, name) {\n    // Prevent repeat events\n    if (name === 'key_press') {\n        if (event.which === this._key) {\n            return;\n        } else {\n            this._key = event.which;\n        }\n    }\n    if (name === 'key_release') {\n        this._key = null;\n    }\n\n    var value = '';\n    if (event.ctrlKey && event.which !== 17) {\n        value += 'ctrl+';\n    }\n    if (event.altKey && event.which !== 18) {\n        value += 'alt+';\n    }\n    if (event.shiftKey && event.which !== 16) {\n        value += 'shift+';\n    }\n\n    value += 'k';\n    value += event.which.toString();\n\n    this._key_event_extra(event, name);\n\n    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n    return false;\n};\n\nmpl.figure.prototype.toolbar_button_onclick = function (name) {\n    if (name === 'download') {\n        this.handle_save(this, null);\n    } else {\n        this.send_message('toolbar_button', { name: name });\n    }\n};\n\nmpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n    this.message.textContent = tooltip;\n};\n\n///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n// prettier-ignore\nvar _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\nmpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n\nmpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n\nmpl.default_extension = \"png\";/* global mpl */\n\nvar comm_websocket_adapter = function (comm) {\n    // Create a \"websocket\"-like object which calls the given IPython comm\n    // object with the appropriate methods. Currently this is a non binary\n    // socket, so there is still some room for performance tuning.\n    var ws = {};\n\n    ws.close = function () {\n        comm.close();\n    };\n    ws.send = function (m) {\n        //console.log('sending', m);\n        comm.send(m);\n    };\n    // Register the callback with on_msg.\n    comm.on_msg(function (msg) {\n        //console.log('receiving', msg['content']['data'], msg);\n        // Pass the mpl event to the overridden (by mpl) onmessage function.\n        ws.onmessage(msg['content']['data']);\n    });\n    return ws;\n};\n\nmpl.mpl_figure_comm = function (comm, msg) {\n    // This is the function which gets called when the mpl process\n    // starts-up an IPython Comm through the \"matplotlib\" channel.\n\n    var id = msg.content.data.id;\n    // Get hold of the div created by the display call when the Comm\n    // socket was opened in Python.\n    var element = document.getElementById(id);\n    var ws_proxy = comm_websocket_adapter(comm);\n\n    function ondownload(figure, _format) {\n        window.open(figure.canvas.toDataURL());\n    }\n\n    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n\n    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n    // web socket which is closed, not our websocket->open comm proxy.\n    ws_proxy.onopen();\n\n    fig.parent_element = element;\n    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n    if (!fig.cell_info) {\n        console.error('Failed to find cell for figure', id, fig);\n        return;\n    }\n    fig.cell_info[0].output_area.element.on(\n        'cleared',\n        { fig: fig },\n        fig._remove_fig_handler\n    );\n};\n\nmpl.figure.prototype.handle_close = function (fig, msg) {\n    var width = fig.canvas.width / fig.ratio;\n    fig.cell_info[0].output_area.element.off(\n        'cleared',\n        fig._remove_fig_handler\n    );\n    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'\">';\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Tell IPython that the notebook contents must change.\n    IPython.notebook.set_dirty(true);\n    this.send_message('ack', {});\n    var fig = this;\n    // Wait a second, then push the new image to the DOM so\n    // that it is saved nicely (might be nice to debounce this).\n    setTimeout(function () {\n        fig.push_to_output();\n    }, 1000);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'btn-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'btn-group';\n    var button;\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'btn-group';\n            continue;\n        }\n\n        button = fig.buttons[name] = document.createElement('button');\n        button.classList = 'btn btn-default';\n        button.href = '#';\n        button.title = name;\n        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    // Add the status bar.\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message pull-right';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n\n    // Add the close button to the window.\n    var buttongrp = document.createElement('div');\n    buttongrp.classList = 'btn-group inline pull-right';\n    button = document.createElement('button');\n    button.classList = 'btn btn-mini btn-primary';\n    button.href = '#';\n    button.title = 'Stop Interaction';\n    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n    button.addEventListener('click', function (_evt) {\n        fig.handle_close(fig, {});\n    });\n    button.addEventListener(\n        'mouseover',\n        on_mouseover_closure('Stop Interaction')\n    );\n    buttongrp.appendChild(button);\n    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n};\n\nmpl.figure.prototype._remove_fig_handler = function (event) {\n    var fig = event.data.fig;\n    if (event.target !== this) {\n        // Ignore bubbled events from children.\n        return;\n    }\n    fig.close_ws(fig, {});\n};\n\nmpl.figure.prototype._root_extra_style = function (el) {\n    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n};\n\nmpl.figure.prototype._canvas_extra_style = function (el) {\n    // this is important to make the div 'focusable\n    el.setAttribute('tabindex', 0);\n    // reach out to IPython and tell the keyboard manager to turn it's self\n    // off when our div gets focus\n\n    // location in version 3\n    if (IPython.notebook.keyboard_manager) {\n        IPython.notebook.keyboard_manager.register_events(el);\n    } else {\n        // location in version 2\n        IPython.keyboard_manager.register_events(el);\n    }\n};\n\nmpl.figure.prototype._key_event_extra = function (event, _name) {\n    var manager = IPython.notebook.keyboard_manager;\n    if (!manager) {\n        manager = IPython.keyboard_manager;\n    }\n\n    // Check for shift+enter\n    if (event.shiftKey && event.which === 13) {\n        this.canvas_div.blur();\n        // select the cell after this one\n        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n        IPython.notebook.select(index + 1);\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    fig.ondownload(fig, null);\n};\n\nmpl.find_output_cell = function (html_output) {\n    // Return the cell and output element which can be found *uniquely* in the notebook.\n    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n    // IPython event is triggered only after the cells have been serialised, which for\n    // our purposes (turning an active figure into a static one), is too late.\n    var cells = IPython.notebook.get_cells();\n    var ncells = cells.length;\n    for (var i = 0; i < ncells; i++) {\n        var cell = cells[i];\n        if (cell.cell_type === 'code') {\n            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n                var data = cell.output_area.outputs[j];\n                if (data.data) {\n                    // IPython >= 3 moved mimebundle to data attribute of output\n                    data = data.data;\n                }\n                if (data['text/html'] === html_output) {\n                    return [cell, data, j];\n                }\n            }\n        }\n    }\n};\n\n// Register the function which deals with the matplotlib target/channel.\n// The kernel may be null if the page has been refreshed.\nif (IPython.notebook.kernel !== null) {\n    IPython.notebook.kernel.comm_manager.register_target(\n        'matplotlib',\n        mpl.mpl_figure_comm\n    );\n}\n"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.HTML object>",
      "text/html": "<div id='2e635e47-0e3b-44c4-bcbd-2902f9e7af2b'></div>"
     },
     "metadata": {}
    },
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "costs before the training:\t  1182.677294754217\ncosts after the training:\t  4.312760715065726\n"
     ]
    }
   ],
   "source": [
    "plot_progress(history_cost)\n",
    "print(\"costs before the training:\\t \", history_cost[0])\n",
    "print(\"costs after the training:\\t \", history_cost[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:**\n",
    "\n",
    "Finally plot the decision hyperplane (just a plain plane here though) together with the data in a 3D plot.\n",
    "\n",
    "Your plot should look similar to this one:\n",
    "\n",
    "<img src=\"https://gitlab.com/deep.TEACHING/educational-materials/raw/dev/media/klaus/exercise-multivariate-linear-regression-scatter_and_boundary.png\" width=\"512\" alt=\"internet connection needed\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def evaluation_plt(x, y, final_theta):\n",
    "    ''' Plots the data x, y together with the final model\n",
    "    \n",
    "    Args:\n",
    "        cost_hist: vector, history of all cost values from a opitmization\n",
    "        theta_0: scalar, model parameter for boundary\n",
    "        theta_1: scalar, model parameter for boundary\n",
    "        x: vector, x values from the data set\n",
    "        y: vector, y values from the data set\n",
    "    '''\n",
    "    \n",
    "    x_1 = np.linspace(np.min(x[:,0]),np.max(x[:,0]),100)\n",
    "    x_2 = np.linspace(np.min(x[:,1]),np.max(x[:,1]),100)\n",
    "    x1,x2 = np.meshgrid(x_1,x_2)\n",
    "    \n",
    "    #xG = np.column_stack((x_1, x_2))\n",
    "    #yl = linear_hypothesis(final_theta)(xG) ???\n",
    "    \n",
    "    z = final_theta[0] + final_theta[1] * x1 + final_theta[2] * x2 \n",
    "    \n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(111, projection='3d')\n",
    "    ax.set_xlabel('X1 Label')\n",
    "    ax.set_ylabel('X2 Label')\n",
    "    ax.set_zlabel('Y Label')\n",
    "    ax.scatter(x[:,0], x[:,1], y, c=\"r\") \n",
    "    \n",
    "    ax.plot_surface(x1, x2, z, cmap=cm.jet,linewidth=0, antialiased=False)\n",
    "    return fig\n",
    "\n",
    "    raise NotImplementedError(\"You should implement this!\")  \n",
    "    \n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.Javascript object>",
      "application/javascript": "/* Put everything inside the global mpl namespace */\n/* global mpl */\nwindow.mpl = {};\n\nmpl.get_websocket_type = function () {\n    if (typeof WebSocket !== 'undefined') {\n        return WebSocket;\n    } else if (typeof MozWebSocket !== 'undefined') {\n        return MozWebSocket;\n    } else {\n        alert(\n            'Your browser does not have WebSocket support. ' +\n                'Please try Chrome, Safari or Firefox ≥ 6. ' +\n                'Firefox 4 and 5 are also supported but you ' +\n                'have to enable WebSockets in about:config.'\n        );\n    }\n};\n\nmpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n    this.id = figure_id;\n\n    this.ws = websocket;\n\n    this.supports_binary = this.ws.binaryType !== undefined;\n\n    if (!this.supports_binary) {\n        var warnings = document.getElementById('mpl-warnings');\n        if (warnings) {\n            warnings.style.display = 'block';\n            warnings.textContent =\n                'This browser does not support binary websocket messages. ' +\n                'Performance may be slow.';\n        }\n    }\n\n    this.imageObj = new Image();\n\n    this.context = undefined;\n    this.message = undefined;\n    this.canvas = undefined;\n    this.rubberband_canvas = undefined;\n    this.rubberband_context = undefined;\n    this.format_dropdown = undefined;\n\n    this.image_mode = 'full';\n\n    this.root = document.createElement('div');\n    this.root.setAttribute('style', 'display: inline-block');\n    this._root_extra_style(this.root);\n\n    parent_element.appendChild(this.root);\n\n    this._init_header(this);\n    this._init_canvas(this);\n    this._init_toolbar(this);\n\n    var fig = this;\n\n    this.waiting = false;\n\n    this.ws.onopen = function () {\n        fig.send_message('supports_binary', { value: fig.supports_binary });\n        fig.send_message('send_image_mode', {});\n        if (fig.ratio !== 1) {\n            fig.send_message('set_dpi_ratio', { dpi_ratio: fig.ratio });\n        }\n        fig.send_message('refresh', {});\n    };\n\n    this.imageObj.onload = function () {\n        if (fig.image_mode === 'full') {\n            // Full images could contain transparency (where diff images\n            // almost always do), so we need to clear the canvas so that\n            // there is no ghosting.\n            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n        }\n        fig.context.drawImage(fig.imageObj, 0, 0);\n    };\n\n    this.imageObj.onunload = function () {\n        fig.ws.close();\n    };\n\n    this.ws.onmessage = this._make_on_message_function(this);\n\n    this.ondownload = ondownload;\n};\n\nmpl.figure.prototype._init_header = function () {\n    var titlebar = document.createElement('div');\n    titlebar.classList =\n        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n    var titletext = document.createElement('div');\n    titletext.classList = 'ui-dialog-title';\n    titletext.setAttribute(\n        'style',\n        'width: 100%; text-align: center; padding: 3px;'\n    );\n    titlebar.appendChild(titletext);\n    this.root.appendChild(titlebar);\n    this.header = titletext;\n};\n\nmpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n\nmpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n\nmpl.figure.prototype._init_canvas = function () {\n    var fig = this;\n\n    var canvas_div = (this.canvas_div = document.createElement('div'));\n    canvas_div.setAttribute(\n        'style',\n        'border: 1px solid #ddd;' +\n            'box-sizing: content-box;' +\n            'clear: both;' +\n            'min-height: 1px;' +\n            'min-width: 1px;' +\n            'outline: 0;' +\n            'overflow: hidden;' +\n            'position: relative;' +\n            'resize: both;'\n    );\n\n    function on_keyboard_event_closure(name) {\n        return function (event) {\n            return fig.key_event(event, name);\n        };\n    }\n\n    canvas_div.addEventListener(\n        'keydown',\n        on_keyboard_event_closure('key_press')\n    );\n    canvas_div.addEventListener(\n        'keyup',\n        on_keyboard_event_closure('key_release')\n    );\n\n    this._canvas_extra_style(canvas_div);\n    this.root.appendChild(canvas_div);\n\n    var canvas = (this.canvas = document.createElement('canvas'));\n    canvas.classList.add('mpl-canvas');\n    canvas.setAttribute('style', 'box-sizing: content-box;');\n\n    this.context = canvas.getContext('2d');\n\n    var backingStore =\n        this.context.backingStorePixelRatio ||\n        this.context.webkitBackingStorePixelRatio ||\n        this.context.mozBackingStorePixelRatio ||\n        this.context.msBackingStorePixelRatio ||\n        this.context.oBackingStorePixelRatio ||\n        this.context.backingStorePixelRatio ||\n        1;\n\n    this.ratio = (window.devicePixelRatio || 1) / backingStore;\n    if (this.ratio !== 1) {\n        fig.send_message('set_dpi_ratio', { dpi_ratio: this.ratio });\n    }\n\n    var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n        'canvas'\n    ));\n    rubberband_canvas.setAttribute(\n        'style',\n        'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n    );\n\n    // Apply a ponyfill if ResizeObserver is not implemented by browser.\n    if (this.ResizeObserver === undefined) {\n        if (window.ResizeObserver !== undefined) {\n            this.ResizeObserver = window.ResizeObserver;\n        } else {\n            var obs = _JSXTOOLS_RESIZE_OBSERVER({});\n            this.ResizeObserver = obs.ResizeObserver;\n        }\n    }\n\n    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {\n        var nentries = entries.length;\n        for (var i = 0; i < nentries; i++) {\n            var entry = entries[i];\n            var width, height;\n            if (entry.contentBoxSize) {\n                if (entry.contentBoxSize instanceof Array) {\n                    // Chrome 84 implements new version of spec.\n                    width = entry.contentBoxSize[0].inlineSize;\n                    height = entry.contentBoxSize[0].blockSize;\n                } else {\n                    // Firefox implements old version of spec.\n                    width = entry.contentBoxSize.inlineSize;\n                    height = entry.contentBoxSize.blockSize;\n                }\n            } else {\n                // Chrome <84 implements even older version of spec.\n                width = entry.contentRect.width;\n                height = entry.contentRect.height;\n            }\n\n            // Keep the size of the canvas and rubber band canvas in sync with\n            // the canvas container.\n            if (entry.devicePixelContentBoxSize) {\n                // Chrome 84 implements new version of spec.\n                canvas.setAttribute(\n                    'width',\n                    entry.devicePixelContentBoxSize[0].inlineSize\n                );\n                canvas.setAttribute(\n                    'height',\n                    entry.devicePixelContentBoxSize[0].blockSize\n                );\n            } else {\n                canvas.setAttribute('width', width * fig.ratio);\n                canvas.setAttribute('height', height * fig.ratio);\n            }\n            canvas.setAttribute(\n                'style',\n                'width: ' + width + 'px; height: ' + height + 'px;'\n            );\n\n            rubberband_canvas.setAttribute('width', width);\n            rubberband_canvas.setAttribute('height', height);\n\n            // And update the size in Python. We ignore the initial 0/0 size\n            // that occurs as the element is placed into the DOM, which should\n            // otherwise not happen due to the minimum size styling.\n            if (width != 0 && height != 0) {\n                fig.request_resize(width, height);\n            }\n        }\n    });\n    this.resizeObserverInstance.observe(canvas_div);\n\n    function on_mouse_event_closure(name) {\n        return function (event) {\n            return fig.mouse_event(event, name);\n        };\n    }\n\n    rubberband_canvas.addEventListener(\n        'mousedown',\n        on_mouse_event_closure('button_press')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseup',\n        on_mouse_event_closure('button_release')\n    );\n    // Throttle sequential mouse events to 1 every 20ms.\n    rubberband_canvas.addEventListener(\n        'mousemove',\n        on_mouse_event_closure('motion_notify')\n    );\n\n    rubberband_canvas.addEventListener(\n        'mouseenter',\n        on_mouse_event_closure('figure_enter')\n    );\n    rubberband_canvas.addEventListener(\n        'mouseleave',\n        on_mouse_event_closure('figure_leave')\n    );\n\n    canvas_div.addEventListener('wheel', function (event) {\n        if (event.deltaY < 0) {\n            event.step = 1;\n        } else {\n            event.step = -1;\n        }\n        on_mouse_event_closure('scroll')(event);\n    });\n\n    canvas_div.appendChild(canvas);\n    canvas_div.appendChild(rubberband_canvas);\n\n    this.rubberband_context = rubberband_canvas.getContext('2d');\n    this.rubberband_context.strokeStyle = '#000000';\n\n    this._resize_canvas = function (width, height, forward) {\n        if (forward) {\n            canvas_div.style.width = width + 'px';\n            canvas_div.style.height = height + 'px';\n        }\n    };\n\n    // Disable right mouse context menu.\n    this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n        event.preventDefault();\n        return false;\n    });\n\n    function set_focus() {\n        canvas.focus();\n        canvas_div.focus();\n    }\n\n    window.setTimeout(set_focus, 100);\n};\n\nmpl.figure.prototype._init_toolbar = function () {\n    var fig = this;\n\n    var toolbar = document.createElement('div');\n    toolbar.classList = 'mpl-toolbar';\n    this.root.appendChild(toolbar);\n\n    function on_click_closure(name) {\n        return function (_event) {\n            return fig.toolbar_button_onclick(name);\n        };\n    }\n\n    function on_mouseover_closure(tooltip) {\n        return function (event) {\n            if (!event.currentTarget.disabled) {\n                return fig.toolbar_button_onmouseover(tooltip);\n            }\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'mpl-button-group';\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'mpl-button-group';\n            continue;\n        }\n\n        var button = (fig.buttons[name] = document.createElement('button'));\n        button.classList = 'mpl-widget';\n        button.setAttribute('role', 'button');\n        button.setAttribute('aria-disabled', 'false');\n        button.addEventListener('click', on_click_closure(method_name));\n        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n\n        var icon_img = document.createElement('img');\n        icon_img.src = '_images/' + image + '.png';\n        icon_img.srcset = '_images/' + image + '_large.png 2x';\n        icon_img.alt = tooltip;\n        button.appendChild(icon_img);\n\n        buttonGroup.appendChild(button);\n    }\n\n    if (buttonGroup.hasChildNodes()) {\n        toolbar.appendChild(buttonGroup);\n    }\n\n    var fmt_picker = document.createElement('select');\n    fmt_picker.classList = 'mpl-widget';\n    toolbar.appendChild(fmt_picker);\n    this.format_dropdown = fmt_picker;\n\n    for (var ind in mpl.extensions) {\n        var fmt = mpl.extensions[ind];\n        var option = document.createElement('option');\n        option.selected = fmt === mpl.default_extension;\n        option.innerHTML = fmt;\n        fmt_picker.appendChild(option);\n    }\n\n    var status_bar = document.createElement('span');\n    status_bar.classList = 'mpl-message';\n    toolbar.appendChild(status_bar);\n    this.message = status_bar;\n};\n\nmpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n    // which will in turn request a refresh of the image.\n    this.send_message('resize', { width: x_pixels, height: y_pixels });\n};\n\nmpl.figure.prototype.send_message = function (type, properties) {\n    properties['type'] = type;\n    properties['figure_id'] = this.id;\n    this.ws.send(JSON.stringify(properties));\n};\n\nmpl.figure.prototype.send_draw_message = function () {\n    if (!this.waiting) {\n        this.waiting = true;\n        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    var format_dropdown = fig.format_dropdown;\n    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n    fig.ondownload(fig, format);\n};\n\nmpl.figure.prototype.handle_resize = function (fig, msg) {\n    var size = msg['size'];\n    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n        fig._resize_canvas(size[0], size[1], msg['forward']);\n        fig.send_message('refresh', {});\n    }\n};\n\nmpl.figure.prototype.handle_rubberband = function (fig, msg) {\n    var x0 = msg['x0'] / fig.ratio;\n    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n    var x1 = msg['x1'] / fig.ratio;\n    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n    x0 = Math.floor(x0) + 0.5;\n    y0 = Math.floor(y0) + 0.5;\n    x1 = Math.floor(x1) + 0.5;\n    y1 = Math.floor(y1) + 0.5;\n    var min_x = Math.min(x0, x1);\n    var min_y = Math.min(y0, y1);\n    var width = Math.abs(x1 - x0);\n    var height = Math.abs(y1 - y0);\n\n    fig.rubberband_context.clearRect(\n        0,\n        0,\n        fig.canvas.width / fig.ratio,\n        fig.canvas.height / fig.ratio\n    );\n\n    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n};\n\nmpl.figure.prototype.handle_figure_label = function (fig, msg) {\n    // Updates the figure title.\n    fig.header.textContent = msg['label'];\n};\n\nmpl.figure.prototype.handle_cursor = function (fig, msg) {\n    var cursor = msg['cursor'];\n    switch (cursor) {\n        case 0:\n            cursor = 'pointer';\n            break;\n        case 1:\n            cursor = 'default';\n            break;\n        case 2:\n            cursor = 'crosshair';\n            break;\n        case 3:\n            cursor = 'move';\n            break;\n    }\n    fig.rubberband_canvas.style.cursor = cursor;\n};\n\nmpl.figure.prototype.handle_message = function (fig, msg) {\n    fig.message.textContent = msg['message'];\n};\n\nmpl.figure.prototype.handle_draw = function (fig, _msg) {\n    // Request the server to send over a new figure.\n    fig.send_draw_message();\n};\n\nmpl.figure.prototype.handle_image_mode = function (fig, msg) {\n    fig.image_mode = msg['mode'];\n};\n\nmpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n    for (var key in msg) {\n        if (!(key in fig.buttons)) {\n            continue;\n        }\n        fig.buttons[key].disabled = !msg[key];\n        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n    }\n};\n\nmpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n    if (msg['mode'] === 'PAN') {\n        fig.buttons['Pan'].classList.add('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    } else if (msg['mode'] === 'ZOOM') {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.add('active');\n    } else {\n        fig.buttons['Pan'].classList.remove('active');\n        fig.buttons['Zoom'].classList.remove('active');\n    }\n};\n\nmpl.figure.prototype.updated_canvas_event = function () {\n    // Called whenever the canvas gets updated.\n    this.send_message('ack', {});\n};\n\n// A function to construct a web socket function for onmessage handling.\n// Called in the figure constructor.\nmpl.figure.prototype._make_on_message_function = function (fig) {\n    return function socket_on_message(evt) {\n        if (evt.data instanceof Blob) {\n            /* FIXME: We get \"Resource interpreted as Image but\n             * transferred with MIME type text/plain:\" errors on\n             * Chrome.  But how to set the MIME type?  It doesn't seem\n             * to be part of the websocket stream */\n            evt.data.type = 'image/png';\n\n            /* Free the memory for the previous frames */\n            if (fig.imageObj.src) {\n                (window.URL || window.webkitURL).revokeObjectURL(\n                    fig.imageObj.src\n                );\n            }\n\n            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n                evt.data\n            );\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        } else if (\n            typeof evt.data === 'string' &&\n            evt.data.slice(0, 21) === 'data:image/png;base64'\n        ) {\n            fig.imageObj.src = evt.data;\n            fig.updated_canvas_event();\n            fig.waiting = false;\n            return;\n        }\n\n        var msg = JSON.parse(evt.data);\n        var msg_type = msg['type'];\n\n        // Call the  \"handle_{type}\" callback, which takes\n        // the figure and JSON message as its only arguments.\n        try {\n            var callback = fig['handle_' + msg_type];\n        } catch (e) {\n            console.log(\n                \"No handler for the '\" + msg_type + \"' message type: \",\n                msg\n            );\n            return;\n        }\n\n        if (callback) {\n            try {\n                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n                callback(fig, msg);\n            } catch (e) {\n                console.log(\n                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n                    e,\n                    e.stack,\n                    msg\n                );\n            }\n        }\n    };\n};\n\n// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\nmpl.findpos = function (e) {\n    //this section is from http://www.quirksmode.org/js/events_properties.html\n    var targ;\n    if (!e) {\n        e = window.event;\n    }\n    if (e.target) {\n        targ = e.target;\n    } else if (e.srcElement) {\n        targ = e.srcElement;\n    }\n    if (targ.nodeType === 3) {\n        // defeat Safari bug\n        targ = targ.parentNode;\n    }\n\n    // pageX,Y are the mouse positions relative to the document\n    var boundingRect = targ.getBoundingClientRect();\n    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n\n    return { x: x, y: y };\n};\n\n/*\n * return a copy of an object with only non-object keys\n * we need this to avoid circular references\n * http://stackoverflow.com/a/24161582/3208463\n */\nfunction simpleKeys(original) {\n    return Object.keys(original).reduce(function (obj, key) {\n        if (typeof original[key] !== 'object') {\n            obj[key] = original[key];\n        }\n        return obj;\n    }, {});\n}\n\nmpl.figure.prototype.mouse_event = function (event, name) {\n    var canvas_pos = mpl.findpos(event);\n\n    if (name === 'button_press') {\n        this.canvas.focus();\n        this.canvas_div.focus();\n    }\n\n    var x = canvas_pos.x * this.ratio;\n    var y = canvas_pos.y * this.ratio;\n\n    this.send_message(name, {\n        x: x,\n        y: y,\n        button: event.button,\n        step: event.step,\n        guiEvent: simpleKeys(event),\n    });\n\n    /* This prevents the web browser from automatically changing to\n     * the text insertion cursor when the button is pressed.  We want\n     * to control all of the cursor setting manually through the\n     * 'cursor' event from matplotlib */\n    event.preventDefault();\n    return false;\n};\n\nmpl.figure.prototype._key_event_extra = function (_event, _name) {\n    // Handle any extra behaviour associated with a key event\n};\n\nmpl.figure.prototype.key_event = function (event, name) {\n    // Prevent repeat events\n    if (name === 'key_press') {\n        if (event.which === this._key) {\n            return;\n        } else {\n            this._key = event.which;\n        }\n    }\n    if (name === 'key_release') {\n        this._key = null;\n    }\n\n    var value = '';\n    if (event.ctrlKey && event.which !== 17) {\n        value += 'ctrl+';\n    }\n    if (event.altKey && event.which !== 18) {\n        value += 'alt+';\n    }\n    if (event.shiftKey && event.which !== 16) {\n        value += 'shift+';\n    }\n\n    value += 'k';\n    value += event.which.toString();\n\n    this._key_event_extra(event, name);\n\n    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n    return false;\n};\n\nmpl.figure.prototype.toolbar_button_onclick = function (name) {\n    if (name === 'download') {\n        this.handle_save(this, null);\n    } else {\n        this.send_message('toolbar_button', { name: name });\n    }\n};\n\nmpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n    this.message.textContent = tooltip;\n};\n\n///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n// prettier-ignore\nvar _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\nmpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n\nmpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n\nmpl.default_extension = \"png\";/* global mpl */\n\nvar comm_websocket_adapter = function (comm) {\n    // Create a \"websocket\"-like object which calls the given IPython comm\n    // object with the appropriate methods. Currently this is a non binary\n    // socket, so there is still some room for performance tuning.\n    var ws = {};\n\n    ws.close = function () {\n        comm.close();\n    };\n    ws.send = function (m) {\n        //console.log('sending', m);\n        comm.send(m);\n    };\n    // Register the callback with on_msg.\n    comm.on_msg(function (msg) {\n        //console.log('receiving', msg['content']['data'], msg);\n        // Pass the mpl event to the overridden (by mpl) onmessage function.\n        ws.onmessage(msg['content']['data']);\n    });\n    return ws;\n};\n\nmpl.mpl_figure_comm = function (comm, msg) {\n    // This is the function which gets called when the mpl process\n    // starts-up an IPython Comm through the \"matplotlib\" channel.\n\n    var id = msg.content.data.id;\n    // Get hold of the div created by the display call when the Comm\n    // socket was opened in Python.\n    var element = document.getElementById(id);\n    var ws_proxy = 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}\n        };\n    }\n\n    fig.buttons = {};\n    var buttonGroup = document.createElement('div');\n    buttonGroup.classList = 'btn-group';\n    var button;\n    for (var toolbar_ind in mpl.toolbar_items) {\n        var name = mpl.toolbar_items[toolbar_ind][0];\n        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n        var image = mpl.toolbar_items[toolbar_ind][2];\n        var method_name = mpl.toolbar_items[toolbar_ind][3];\n\n        if (!name) {\n            /* Instead of a spacer, we start a new button group. */\n            if (buttonGroup.hasChildNodes()) {\n                toolbar.appendChild(buttonGroup);\n            }\n            buttonGroup = document.createElement('div');\n            buttonGroup.classList = 'btn-group';\n            continue;\n        }\n\n        button = fig.buttons[name] = document.createElement('button');\n        button.classList = 'btn btn-default';\n        button.href = '#';\n        button.title = name;\n        button.innerHTML = '<i 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version 3\n    if (IPython.notebook.keyboard_manager) {\n        IPython.notebook.keyboard_manager.register_events(el);\n    } else {\n        // location in version 2\n        IPython.keyboard_manager.register_events(el);\n    }\n};\n\nmpl.figure.prototype._key_event_extra = function (event, _name) {\n    var manager = IPython.notebook.keyboard_manager;\n    if (!manager) {\n        manager = IPython.keyboard_manager;\n    }\n\n    // Check for shift+enter\n    if (event.shiftKey && event.which === 13) {\n        this.canvas_div.blur();\n        // select the cell after this one\n        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n        IPython.notebook.select(index + 1);\n    }\n};\n\nmpl.figure.prototype.handle_save = function (fig, _msg) {\n    fig.ondownload(fig, null);\n};\n\nmpl.find_output_cell = function (html_output) {\n    // Return the cell and output element which can be found *uniquely* in the notebook.\n    // Note - this is a bit hacky, but it is 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refreshed.\nif (IPython.notebook.kernel !== null) {\n    IPython.notebook.kernel.comm_manager.register_target(\n        'matplotlib',\n        mpl.mpl_figure_comm\n    );\n}\n"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<IPython.core.display.HTML object>",
      "text/html": "<div id='c8d0d1a0-3162-4106-91c3-fa6445b4e248'></div>"
     },
     "metadata": {}
    },
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "thetas before the training:\t [5. 8. 2.]\nthetas after the training:\t [ 3.89329895  2.39794986 -3.25098443]\n"
     ]
    }
   ],
   "source": [
    "evaluation_plt(X, y, history_theta[-1])\n",
    "\n",
    "\n",
    "print(\"thetas before the training:\\t\", history_theta[0])\n",
    "print(\"thetas after the training:\\t\", history_theta[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Feature Scaling\n",
    "\n",
    "Now suppose the following features $X$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "[[   0.0001 2000.    ]\n [   0.0002 1800.    ]\n [   0.0003 1600.    ]]\n"
     ]
    }
   ],
   "source": [
    "X = np.array([[0.0001, 2000],\n",
    "       [0.0002, 1800],\n",
    "       [0.0003, 1600]], dtype=np.float32)\n",
    "\n",
    "sample_size = len(X[:,0])\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [],
   "source": [
    "def scale(x):\n",
    "    mean = np.mean(x)\n",
    "    scaled = (x - mean) / np.std(x)\n",
    "    return scaled"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "[[   0.0001 2000.    ]\n [   0.0002 1800.    ]\n [   0.0003 1600.    ]]\n[[-0.99187     1.2122852 ]\n [-0.99186987  0.9918697 ]\n [-0.99186975  0.77145416]]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(suppress=True)\n",
    "print(X)\n",
    "print(scale(X))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:**\n",
    "\n",
    "This task can be done via **pen & paper** or by inserting some code below. Either way, you should be able to solve both tasks below on paper only using a calculator.\n",
    "\n",
    "1. Apply feature scaling onto `X` using the *mean* and the *standard deviation*. What values do the scaled features have?\n",
    "    * *Optional:*\n",
    "\n",
    "        You can even execute the cell above and start running your notebook again from top (all **except** executing the cell to generate your features, which would overwrite these new features).\n",
    "\n",
    "        When you start training you should notice that your costs do not decrease, maybe even increase, if you have not adjusted your learning rate (training might also throw an overflow warning)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:**\n",
    "\n",
    "2. After the training with scaled features your new $\\\\vec \\theta'$ values will something like: $\\vec \\theta'=\\left(-7197,  326, -326\\right)^T$ (you can try training with but you do not have to). \n",
    "\n",
    "    Suppose $\\vec \\theta'=\\left(-7197,  326, -326\\right)^T$. What are the corresponding $\\theta_j$ values for the unscaled data?\n",
    "\n",
    "    * (If you did train your model with the scaled features, the resulting $\\theta_j$ should really be $\\vec \\theta'=\\left(-7197,  326, -326\\right)^T$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Summary and Outlook\n",
    "\n",
    "During this exercise, the linear regression was extended to multidimensional feature space and feature scaling was practiced. You should be able to answer the following questions:\n",
    "- How does the implementation of the multivariate regression differ from the univariate one?\n",
    "- Why do we apply feature scaling?\n",
    "- Why does feature scaling help?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "## Licenses\n",
    "\n",
    "### Notebook License (CC-BY-SA 4.0)\n",
    "\n",
    "*The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).*\n",
    "\n",
    "Exercise: Multivariate Linear Regression <br/>\n",
    "by Christian Herta, Klaus Strohmenger<br/>\n",
    "is licensed under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/).<br/>\n",
    "Based on a work at https://gitlab.com/deep.TEACHING.\n",
    "\n",
    "\n",
    "### Code License (MIT)\n",
    "\n",
    "*The following license only applies to code cells of the notebook.*\n",
    "\n",
    "Copyright 2018 Christian Herta, Klaus Strohmenger\n",
    "\n",
    "Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the \"Software\"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:\n",
    "\n",
    "The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.\n",
    "\n",
    "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE."
   ]
  }
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