CDF PDF WTF Mastering Distribution Transformations.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "import plotly.figure_factory as ff\n",
    "\n",
    "import plotly.graph_objects as go\n",
    "from plotly.subplots import make_subplots\n",
    "\n",
    "from ipywidgets import VBox, HBox\n",
    "\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(seed=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Let's start by generating some uniform data using our friend Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "unif_1 = np.random.uniform(size = (1000,1))\n",
    "unif_dist  = ff.create_distplot([unif_1.reshape(-1)], group_labels = [' '], show_curve=False, show_rug=False, bin_size=0.05)\n",
    "unif_dist.update_layout(showlegend=False, title_text='Uniform Data', width=800, height=500)\n",
    "unif_dist.show()"
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   "source": [
    "# We want to use it to generate data in the same space as traditional/raw data, say heights of people in the office\n",
    "<img src=\"Images/unif_to_inv_cdf.jpg\"/>\n",
    "\n",
    "## We can use the Inverse Cumulative Distribution Function.\n",
    "## Takes in data in the 0 to 1 space\n",
    "## Output lower bound value of - inf\n",
    "## Output upper bound value of inf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "normal_dist_data = norm.ppf(unif_1, loc=160, scale=7)"
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   "execution_count": 7,
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   "source": [
    "normal_dist  = ff.create_distplot([normal_dist_data.reshape(-1)], group_labels = [' '], show_rug=False, bin_size=0.1)\n",
    "normal_dist.update_layout(showlegend=False, title_text='Uniform to Inverse CDF', width=1000, height=500)\n",
    "normal_dist.show()"
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   "source": [
    "# What if I want to transform raw data into uniform data?\n",
    "<img src=\"Images/data_to_uniform.jpg\"/>\n",
    "\n",
    "## We use the Cumulative Distribution Function, which as the following properties:\n",
    "## Takes in data in the -inf to inf space, SciPy seems to stop at -37 and 8\n",
    "## Output lower bound value of 0\n",
    "## Output upper bound value of 1\n",
    "## For the standard normal you will usually see this symbol $\\Phi$, called Phi\n",
    "## $\\Phi (- x) = 1 -  \\Phi(x)$"
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  },
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   "cell_type": "code",
   "execution_count": 17,
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       "0.9999999999987201"
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   "source": [
    "norm.cdf(7)"
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   "metadata": {},
   "outputs": [],
   "source": [
    "unif_2 = norm.cdf(normal_dist_data, loc=160, scale=7)"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "unif_dist_2  = ff.create_distplot([unif_2.reshape(-1)], group_labels = [' '], show_curve=False, show_rug=False, bin_size=0.05)\n",
    "unif_dist_2.update_layout(showlegend=False, title_text='Raw data to uniform', width=800, height=500)\n",
    "unif_dist_2.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How does PDF fit into  all of this?\n",
    "<img src=\"Images/data_to_pdf.jpg\"/>\n",
    "\n",
    "## Probability Density function is the derivative (or the slope) of the CDF. \n",
    "## Which means we can obtain the CDF by integrating the PDF.  There is no closed form solution ( a nice formula), but this has been calculated for us in all software as it's key to statistics\n",
    "## The output is always bigger than 0\n",
    "## The area under the curve adds up to 1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.linspace(.01, .99, 100)\n",
    "z = np.linspace(-3, 3, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "586f950f7fec49f9b56b3b47a4ab75af",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(HBox(children=(FigureWidget({\n",
       "    'data': [{'name': 'Inverse CDF',\n",
       "              'type': 'scatt…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = go.FigureWidget()\n",
    "f.add_scatter(x = u, y=norm.ppf(u), name = 'Inverse CDF');\n",
    "f.layout.title = 'Inverse Cumulative Distribution Function'\n",
    "f.update_layout(xaxis_title=\"Input\",\n",
    "    yaxis_title=\"Output\")\n",
    "\n",
    "\n",
    "g = go.FigureWidget()\n",
    "g.add_scatter(x = z, y=norm.cdf(z), name = 'CDF');\n",
    "g.layout.title = 'Cumulative Distribution Function'\n",
    "g.update_layout(xaxis_title=\"Input\",\n",
    "    yaxis_title=\"Output\")\n",
    "\n",
    "\n",
    "h = go.FigureWidget()\n",
    "h.add_scatter(x = z, y=norm.pdf(z), name = 'PDF')\n",
    "h.layout.title = 'Probability Density Function'\n",
    "h.update_layout(xaxis_title=\"Input\",\n",
    "    yaxis_title=\"Output\")\n",
    "VBox([HBox([f, g]), h])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Real useful use of the PDF is to find the natural log of the function and use it in the log-likelihood when we are trying to find the parameters of the function which generated this data using an optimiser such as in Maximum Likelihood Estimation or MCMC techniques. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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