diagonals.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Problem\n",
    "Now how many squares? Can we predict how many squares for *this* pattern of any grid with length $\\ell$?\n",
    "\n",
    "![Hard Mode](https://thumbs.dreamstime.com/b/seamlesly-repeatable-diagonal-oblique-slanting-lines-graph-paper-pattern-slope-skew-grid-mesh-draft-drawing-plotting-texture-159489564.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Set up data\n",
    "In this problem, I didn't have the actual mathematical proof for the growth of squares for this pattern. So I don't know what the growth does look like before starting out. What I've done to suppliment this is divide the information regarding the growth of the number of squares into the *solved* simple squares and the *unsolved* diagonal squares.\n",
    "\n",
    "Fortunately, we start to see a repeating pattern in the diagonal squares growth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.DataFrame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "code_folding": [
     3
    ]
   },
   "outputs": [],
   "source": [
    "l = np.array([1,2,3,4,5,6,7,8,9,10,11,12])\n",
    "\n",
    "# Equation 1.1\n",
    "def l_squared_sum(arg):\n",
    "    result = []\n",
    "    try:\n",
    "        for el in arg:\n",
    "            result.append(np.sum( np.arange(1,el+1)**2 ))\n",
    "        return result\n",
    "    except TypeError:\n",
    "         return np.sum( np.arange(1,arg+1)**2 )\n",
    "        \n",
    "l2_sum = l_squared_sum(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = [1,5,16,35,65,108,168,246,345,467,616,793]\n",
    "\n",
    "one_by_one = [0,0,2,4,8,12,18,24,32,40,50,60]\n",
    "two_by_two = [0,0,0,1,2,5,8,13,18,25,32,41]\n",
    "three_by_three = [0,0,0,0,0,0,2,4,8,12,18,24]\n",
    "four_by_four = [0,0,0,0,0,0,0,1,2,5,8,13]\n",
    "five_by_five = [0,0,0,0,0,0,0,0,0,0,2,4]\n",
    "six_by_six = [0,0,0,0,0,0,0,0,0,0,0,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['l'] = l\n",
    "df['l2_sum'] = l2_sum\n",
    "df['1x1'] = one_by_one\n",
    "df['2x2'] = two_by_two\n",
    "df['3x3'] = three_by_three\n",
    "df['4x4'] = four_by_four\n",
    "df['5x5'] = five_by_five\n",
    "df['6x6'] = six_by_six\n",
    "df['n'] = n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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       "    }\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>l</th>\n",
       "      <th>l2_sum</th>\n",
       "      <th>1x1</th>\n",
       "      <th>2x2</th>\n",
       "      <th>3x3</th>\n",
       "      <th>4x4</th>\n",
       "      <th>5x5</th>\n",
       "      <th>6x6</th>\n",
       "      <th>n</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>14</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>16</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>30</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>35</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>55</td>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>65</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "      <td>91</td>\n",
       "      <td>12</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>108</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>7</td>\n",
       "      <td>140</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>168</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8</td>\n",
       "      <td>204</td>\n",
       "      <td>24</td>\n",
       "      <td>13</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>246</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>9</td>\n",
       "      <td>285</td>\n",
       "      <td>32</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>345</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>10</td>\n",
       "      <td>385</td>\n",
       "      <td>40</td>\n",
       "      <td>25</td>\n",
       "      <td>12</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>467</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>11</td>\n",
       "      <td>506</td>\n",
       "      <td>50</td>\n",
       "      <td>32</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>616</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>12</td>\n",
       "      <td>650</td>\n",
       "      <td>60</td>\n",
       "      <td>41</td>\n",
       "      <td>24</td>\n",
       "      <td>13</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>793</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     l  l2_sum  1x1  2x2  3x3  4x4  5x5  6x6    n\n",
       "0    1       1    0    0    0    0    0    0    1\n",
       "1    2       5    0    0    0    0    0    0    5\n",
       "2    3      14    2    0    0    0    0    0   16\n",
       "3    4      30    4    1    0    0    0    0   35\n",
       "4    5      55    8    2    0    0    0    0   65\n",
       "5    6      91   12    5    0    0    0    0  108\n",
       "6    7     140   18    8    2    0    0    0  168\n",
       "7    8     204   24   13    4    1    0    0  246\n",
       "8    9     285   32   18    8    2    0    0  345\n",
       "9   10     385   40   25   12    5    0    0  467\n",
       "10  11     506   50   32   18    8    2    0  616\n",
       "11  12     650   60   41   24   13    4    1  793"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the `l2_sum` column is just the result of Equation 1.1 $( \\sum^{\\ell}_{s=1}s^2 )$ from the simple squares solution of the previous notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualization\n",
    "Rather than try to discover the growth rate for the diagonal squares through brute force, let's first visualize the data to help inspire our intuition.\n",
    "\n",
    "First we will see how non-linear the relationship is between $\\ell$ and $n$, and how it compares to the simple \"**straight**\" squares solution (Equation 1.1)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(10,6))\n",
    "ax = fig.add_axes([0,0,1,1])\n",
    "\n",
    "ax.set_xlabel('l')\n",
    "\n",
    "ax.plot(df['l'], df['n'], '^k', label='l vs n', markersize=8)\n",
    "ax.plot(df['l'], df['l2_sum'], label='Equation 1.1', lw=2)\n",
    "ax.plot(df['l'], df['l']**3, label='l^3', lw=2)\n",
    "\n",
    "ax.legend(loc=(0.1,0.8))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we see that the values for $n$ fall just above the values for $n$ for the simple squares solution, plus some growing residual.\n",
    "\n",
    "This residual amount are the growing diagonal squares, so let's turn our attention to those for a moment. We will need to understand how those grow as the grid is expanded, so let's make a graph of the difference between grids of consecutive size $\\ell$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(10,6))\n",
    "ax = fig.add_axes([0,0,1,1])\n",
    "\n",
    "ax.set_xlabel('Δ l')\n",
    "ax.set_ylabel('Δ Square Growth')\n",
    "\n",
    "ax.plot(df['l'][1:], df['1x1'][1:].values - df['1x1'][0:-1].values, label='odds', lw=2)\n",
    "ax.plot(df['l'][1:], df['2x2'][1:].values - df['2x2'][0:-1].values, label='evens', lw=2)\n",
    "\n",
    "ax.legend(loc=(0.8,0.2))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**So we have just stumbled into the actual growth of these two diagonal squares**.  \n",
    "\n",
    "They increase at the same rate; they add two to their growth rate every two iterations. The only difference between these two lines is that the even sized squares (such as $2 \\times 2, 4 \\times 4$, etc.) start later and go up by 1 before growing at a rate of 2 after every other iteration.\n",
    "\n",
    "Since this is now determined, we may wish to write a function that describes these two behaviors. We will do that in the next section, but first let's do the visualization for the model fits of a couple degrees to see which model we might want to use.\n",
    "\n",
    "*Remember, our data has no noise, so this should be fine.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Quick fit and visualization\n",
    "xp = np.linspace(1, 12, 100)\n",
    "\n",
    "z = np.polyfit(df['l'], df['n'], 1)\n",
    "z2 = np.polyfit(df['l'], df['n'], 2)\n",
    "z3 = np.polyfit(df['l'], df['n'], 3)\n",
    "\n",
    "p = np.poly1d(z)\n",
    "p2 = np.poly1d(z2)\n",
    "p3 = np.poly1d(z3)\n",
    "\n",
    "fig = plt.figure(figsize=(10,6))\n",
    "ax = fig.add_axes([0,0,1,1])\n",
    "\n",
    "ax.set_xlabel('l')\n",
    "\n",
    "ax.plot(df['l'], df['n'],'^k', label='l vs n', markersize=8)\n",
    "ax.plot(xp, p2(xp), label='Degree 2 Model', lw=3)\n",
    "ax.plot(xp, p3(xp), label='Degree 3 Model', lw=3)\n",
    "\n",
    "ax.legend(loc=(0.1,0.8))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we will attempt to fit a single model using some linear combination of $\\ell$, $\\ell^2$, and $\\ell^3$, just as before. Except with different coefficients and intercept.\n",
    "\n",
    "\\begin{equation}\n",
    "    n \\approx \\beta_0 + \\beta_1 \\ell + \\beta_2 \\ell^2 + \\beta_3 \\ell^3\n",
    "    \\tag{2.1}\n",
    "\\end{equation}\n",
    "\n",
    "But before we do that, let's first graph what the diffference is in growth between each iterative step of $\\ell$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Writing the diagonal growth function\n",
    "We'll need this to compare and evaluate our trained model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "code_folding": []
   },
   "outputs": [],
   "source": [
    "def diag_growth(l, dim=1):\n",
    "    ''' Enter a side length for the size of the outer grid.\n",
    "        Enter the dimension of the diagonal squares you are interested in. '''\n",
    "    \n",
    "    values = []\n",
    "    accel_growth = False\n",
    "    \n",
    "    if 2*dim > l:\n",
    "        while l > 0:\n",
    "            values.append(0)\n",
    "            l-=1\n",
    "        return np.array(values)\n",
    "    \n",
    "    if dim%2 == 0:\n",
    "        growth_rate = 1\n",
    "        dim = dim*2 - 2\n",
    "    else:\n",
    "        growth_rate = 2\n",
    "        dim = dim*2 - 1\n",
    "        \n",
    "    while dim >= 0:\n",
    "        values.append(0)\n",
    "        dim-=1\n",
    "    \n",
    "    try:\n",
    "        if values[l] == 0:\n",
    "            return np.array(values)\n",
    "    except IndexError:\n",
    "        while len(values) < l:\n",
    "            if accel_growth: \n",
    "                accel_growth = False\n",
    "                values.append(values[-1] + growth_rate)\n",
    "                growth_rate+=2\n",
    "            else:\n",
    "                values.append(values[-1] + growth_rate)\n",
    "                accel_growth = True\n",
    "    \n",
    "    return np.array(values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Odd values from data:  [ 0  0  2  4  8 12 18 24 32 40 50 60]\n",
      "Odd values from func:  [ 0  0  2  4  8 12 18 24 32 40 50 60]\n",
      "======================================================================\n",
      "Even values from data:  [ 0  0  0  1  2  5  8 13 18 25 32 41]\n",
      "Even values from func:  [ 0  0  0  1  2  5  8 13 18 25 32 41]\n"
     ]
    }
   ],
   "source": [
    "# Values from our data\n",
    "print(\"Odd values from data: \", df['1x1'].values)\n",
    "\n",
    "# Results from our function\n",
    "print(\"Odd values from func: \",diag_growth(12, dim=1))\n",
    "\n",
    "print(70*'=')\n",
    "\n",
    "# Values from our data\n",
    "print(\"Even values from data: \", df['2x2'].values)\n",
    "\n",
    "# Results from our function\n",
    "print(\"Even values from func: \",diag_growth(12, dim=2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use this function to generate some test data. But let's first finish up our model from Equation 2.1 and then come back to generating test data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Find coefficients to fit the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "lm = LinearRegression()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>l</th>\n",
       "      <th>l2</th>\n",
       "      <th>l3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>27</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   l  l2  l3\n",
       "0  1   1   1\n",
       "1  2   4   8\n",
       "2  3   9  27"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = pd.DataFrame()\n",
    "X['l'] = df['l']\n",
    "X['l2'] = df['l']**2\n",
    "X['l3'] = df['l']**3\n",
    "\n",
    "X.head(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression()"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lm.fit(X,df['n'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept:  -0.22222222222225696\n"
     ]
    }
   ],
   "source": [
    "print('Intercept: ', lm.intercept_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Coeff (𝛽)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>l</th>\n",
       "      <td>0.149110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>l2</th>\n",
       "      <td>0.487235</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>l3</th>\n",
       "      <td>0.417379</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Coeff (𝛽)\n",
       "l    0.149110\n",
       "l2   0.487235\n",
       "l3   0.417379"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf = pd.DataFrame(lm.coef_,X.columns,columns=['Coeff (𝛽)'])\n",
    "cdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have the model\n",
    "\n",
    "\\begin{equation}\n",
    "    n \\approx -0.222222 + 0.149110 \\ell + 0.487235 \\ell^2 + 0.417379 \\ell^3\n",
    "    \\tag{2.2}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generating test data\n",
    "Our model was trained on only 12 data points, $\\ell_{\\text{train}}: 1\\to12$. So let's take a few points just outside that range, as well as a few points far outside the range to see how the model performs for large $\\ell$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    }\n",
       "\n",
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>straights</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1240</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>2870</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>5525</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>42925</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>173880</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>338350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000</th>\n",
       "      <td>333833500</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      straights\n",
       "15         1240\n",
       "20         2870\n",
       "25         5525\n",
       "50        42925\n",
       "80       173880\n",
       "100      338350\n",
       "1000  333833500"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l_test = [15,20,25,50,80,100,1000]\n",
    "l2_test_sum = l_squared_sum(l_test)\n",
    "l2_test_sum = pd.DataFrame(l2_test_sum, index=l_test, columns=['straights'])\n",
    "l2_test_sum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are values for the number of squares without counting the diagonal squares.\n",
    "\n",
    "Let's go ahead and count the diagonals now."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = {}\n",
    "\n",
    "for el in l_test:\n",
    "    results[el] = []\n",
    "    for i in np.arange(1,el):\n",
    "        results[el].append(np.sum(diag_growth(el,dim=i)[-1]))\n",
    "    \n",
    "    results[el] = np.sum(results[el])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>diagonals</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>280</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>665</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>1300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>10412</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>42660</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>83325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000</th>\n",
       "      <td>83333250</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      diagonals\n",
       "15          280\n",
       "20          665\n",
       "25         1300\n",
       "50        10412\n",
       "80        42660\n",
       "100       83325\n",
       "1000   83333250"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results = pd.DataFrame.from_dict(results, orient='index', columns=['diagonals'])\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we simply add the corresponding values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>diagonals</th>\n",
       "      <th>straights</th>\n",
       "      <th>n</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>280</td>\n",
       "      <td>1240</td>\n",
       "      <td>1520</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>665</td>\n",
       "      <td>2870</td>\n",
       "      <td>3535</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>1300</td>\n",
       "      <td>5525</td>\n",
       "      <td>6825</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>10412</td>\n",
       "      <td>42925</td>\n",
       "      <td>53337</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>42660</td>\n",
       "      <td>173880</td>\n",
       "      <td>216540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>83325</td>\n",
       "      <td>338350</td>\n",
       "      <td>421675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000</th>\n",
       "      <td>83333250</td>\n",
       "      <td>333833500</td>\n",
       "      <td>417166750</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      diagonals  straights          n\n",
       "15          280       1240       1520\n",
       "20          665       2870       3535\n",
       "25         1300       5525       6825\n",
       "50        10412      42925      53337\n",
       "80        42660     173880     216540\n",
       "100       83325     338350     421675\n",
       "1000   83333250  333833500  417166750"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results['straights'] = l2_test_sum['straights']\n",
    "results['n'] = results['straights'] + results['diagonals']\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Make those predictions\n",
    "Okay, now let's make predictions for the test points we just computed *true values* for using the model we built earlier (Equation 2.2).\n",
    "\n",
    "Just first remember that our model requires $\\ell_{\\text{test}}^3$ and $\\ell_{\\text{test}}^2$ as well as $\\ell_{\\text{test}}$, so we will have to write over `l_test` from before to include both."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[15, 225, 3375],\n",
       " [20, 400, 8000],\n",
       " [25, 625, 15625],\n",
       " [50, 2500, 125000],\n",
       " [80, 6400, 512000],\n",
       " [100, 10000, 1000000],\n",
       " [1000, 1000000, 1000000000]]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l_test = [[15,15**2,15**3],\n",
    "          [20,20**2, 20**3],\n",
    "          [25,25**2,25**3],\n",
    "          [50,50**2,50**3],\n",
    "          [80,80**2,80**3],\n",
    "          [100,100**2,100**3],\n",
    "          [1000,1000**2,1000**3]]\n",
    "l_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "predictions = lm.predict(l_test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Predictions:  [1.52029615e+03 3.53668531e+03 6.82957298e+03 5.33976854e+04\n",
      " 2.16828016e+05 4.22265956e+05 4.17866301e+08]\n"
     ]
    }
   ],
   "source": [
    "print('Predictions: ', predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Review performance\n",
    "This gives us a sort of average of the errors, which we will actually break down in the next section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn import metrics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100071.06660004506"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "metrics.mean_absolute_error(results['n'],predictions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "264405.6388453675"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(metrics.mean_squared_error(results['n'],predictions))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Final Thoughts\n",
    "We see that the model doesn't do as well here. For the grid of size $\\ell = 15$, the model predicted 1,520 total squares, which is correct! For $\\ell = 25$, the model predicted 6,830 which is only five squares over the actual answer. But if we're already seeing some inaccuracy, then how does it perform for large $\\ell$?\n",
    "\n",
    "For $\\ell = 1000$ the model predicted 417,866,301 when there are 417,166,750! It overestimated the total squares by almost 700 thousand squares! It's clear that this worsening fit for larger $\\ell$ is due to the lack of data for large values of $\\ell$. In fact, most people might have expected this to happen, since the only data available was data for which I was willing to count by hand, and for $\\ell = 12$ I was already counting almost 150 squares on the diagonal by hand.\n",
    "\n",
    "Of course, now that we have our function, we could use it to give us a better sample of $l \\text{ vs } n$. A better sample of the data, one that wasn't so concentrated around small values of $\\ell$, would certainly improve our model. However, I'll leave that as an exercise for the reader; I'm content with a function that can just give us the answer."
   ]
  }
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