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   "metadata": {},
   "source": [
    "# Chapter 15 - Prediction"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9dcf322-4648-4eaf-a878-f10fc0417247",
   "metadata": {},
   "source": [
    "## Chapter 15.5 - Visual Diagnostics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b22e8371-8a05-42dc-9396-01d50dab6cbf",
   "metadata": {},
   "source": [
    "Previously seen material ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9538feb6-a0fb-4e3f-9a4f-5d45072870b5",
   "metadata": {},
   "outputs": [],
   "source": [
    "from datascience import *\n",
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aee0c8ef-e8da-4f9f-b691-ead4c9da9eb8",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generate somewhat random data for the equation y = 2.5x - 50\n",
    "def generate_table(number_items):\n",
    "    result = Table(make_array(\"x\", \"y\"))\n",
    "    for _ in range(number_items):\n",
    "        x = np.random.random() * 100\n",
    "        delta = 20 * np.random.random() - 10\n",
    "        y = 2.5 * (x + delta) - 50\n",
    "        result = result.with_row(make_array(x,y))\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c07f4a89-a070-48fc-993b-661e14f48979",
   "metadata": {},
   "outputs": [],
   "source": [
    "data = generate_table(100)\n",
    "data.scatter(\"x\")\n",
    "data.show(3)\n",
    "plots.plot([0, 100], [-50, 200], color=\"red\", lw=2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f092b11d-0f1a-4030-a90b-3c0a809caa19",
   "metadata": {},
   "outputs": [],
   "source": [
    "def standard_units(numbers):\n",
    "    \"Convert any array of numbers to standard units.\"\n",
    "    return (numbers - np.mean(numbers))/np.std(numbers)    \n",
    "    \n",
    "def correlation(t, label_x, label_y):\n",
    "    \"Calculate r\"\n",
    "    return np.mean(standard_units(t.column(label_x))*standard_units(t.column(label_y)))\n",
    "\n",
    "def slope(t, label_x, label_y):\n",
    "    \"Calculate m of y = mx + b\"\n",
    "    r = correlation(t, label_x, label_y)\n",
    "    return r*np.std(t.column(label_y))/np.std(t.column(label_x))\n",
    "\n",
    "def intercept(t, label_x, label_y):\n",
    "    \"Calculate b of y = mx + b\"\n",
    "    return np.mean(t.column(label_y)) - slope(t, label_x, label_y)*np.mean(t.column(label_x))\n",
    "\n",
    "def fit(table, x, y):\n",
    "    \"\"\"Return the height of the regression line at each x value.\"\"\"\n",
    "    a = slope(table, x, y)\n",
    "    b = intercept(table, x, y)\n",
    "    return a * table.column(x) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0b3e5bb-3d6b-4978-bf33-98676a8abfcc",
   "metadata": {},
   "source": [
    "New material ..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eebfed71-0d6c-48a4-891b-799108cbb36b",
   "metadata": {},
   "source": [
    "**residual** = observed_value - regression_estimate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "91d83174-389b-4768-8036-0dc0af9d58db",
   "metadata": {},
   "outputs": [],
   "source": [
    "def residual(table, x, y):\n",
    "    return table.column(y) - fit(table, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "abe0f381-1313-4b31-885e-4d31fbd860a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "data = data.with_columns(\n",
    "        \"Fitted Value\", fit(data, \"x\", \"y\"),\n",
    "        \"Residual\", residual(data, \"x\", \"y\")\n",
    "    )\n",
    "data.show(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a10e6992-46b5-4af8-b0d1-a7cb62dd326e",
   "metadata": {},
   "source": [
    "Plotting the (x, Residual) pairs let us make a visual diagnosis of the quality of the\n",
    "linear regression analysis.  **The residual plot of a good regression shows no pattern. \n",
    "The residuals look about the same, above and below the horizontal line at 0, across the range of \n",
    "the predictor variable.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7d8e21d1-1d3d-46ab-87c2-fc8a9f637a12",
   "metadata": {},
   "outputs": [],
   "source": [
    "residual_table = Table().with_columns(\n",
    "    \"x\", data.column(\"x\"),\n",
    "    \"residuals\", residual(data, \"x\", \"y\")\n",
    ")\n",
    "residual_table.scatter(\"x\", \"residuals\", color=\"r\")\n",
    "xlims = make_array(min(data.column(\"x\")), max(data.column(\"x\")))\n",
    "plots.plot(xlims, make_array(0, 0), color=\"darkblue\", lw=4)\n",
    "plots.title(\"Residual Plot\");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc65c219-f5aa-4cf9-85cb-5a574663f20d",
   "metadata": {},
   "source": [
    "**When a residual plot shows a pattern, there may be a non-linear relation between the variables.**  \n",
    "Terminology: *Heteroscedasticity* means an uneven spread of the data."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c35e601a-eb62-4986-8680-534a26ff9286",
   "metadata": {},
   "source": [
    "Is there a pattern in the above plot?  What does this tell us about the likely relation between x and y?"
   ]
  }
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