15_kernel_density_estimation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# ICS 434: DATA SCIENCE FUNDAMENTALS\n",
    "\n",
    "## Kernel Density Estimation (KDE)\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# !pip install seaborn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy  as sp\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import Image"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Introduction to Kernel Density Estimation -- Histograms\n",
    "\n",
    "* The histogram of random draws (e.g., randint(0, 100)) can be generated by tallying the number of data points within each bin of size 10\n",
    "\n",
    "* The bins are: 0-9, 10-19, 20-29, ...\n",
    "\n",
    "* Rather than code the process using Python standard library functionality, we will use `numpy` to generate the counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([19, 70, 56, 55,  2, 92, 41, 74, 28, 49,  0, 51, 23, 14, 90, 60,  3,\n",
       "       36, 63, 95])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_draws = np.random.randint(0, 100, 1000)\n",
    "random_draws[0:20]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0,  10,  20,  30,  40,  50,  60,  70,  80,  90, 100])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bins = np.arange(0, 101, 10)\n",
    "bins"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array([105,  97,  90,  81, 101, 103, 113, 102,  95, 113], dtype=int64), array([  0,  10,  20,  30,  40,  50,  60,  70,  80,  90, 100]))\n",
      "\n",
      " =+==+==+==+==+==+==+==+==+==+= \n",
      "\n",
      "[105  97  90  81 101 103 113 102  95 113]\n",
      "[  0  10  20  30  40  50  60  70  80  90 100]\n"
     ]
    }
   ],
   "source": [
    "counts_and_bins = np.histogram(random_draws, bins)\n",
    "print(counts_and_bins)\n",
    "print(\"\\n\", \"=+=\" * 10, '\\n')\n",
    "print(counts_and_bins[0])\n",
    "print(counts_and_bins[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 4))\n",
    "sns.barplot(x=counts_and_bins[1][:-1], y=counts_and_bins[0]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Frequencies and Probability Distributions\n",
    "\n",
    "* Recall that this is not a valid probability distribution according to the definition of probability distribution in last class\n",
    "\n",
    "  * Why? \n",
    "  \n",
    "  * How can we turn it into a probability distribution?\n",
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "normalized_counts = counts_and_bins[0] / \n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 4))\n",
    "sns.barplot(x=counts_and_bins[1][:-1], y=normalized_counts);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Nonparametric Density Estimation for Discrete Random Variables\n",
    "\n",
    "* Back to histograms: `np.histogram` allows computing normalized counts (`pmf` style)\n",
    "  * `np.histogram(numbers, bins, density=True)`  \n",
    "\n",
    "* A significant problem with histograms is that the choice of bin boundaries can have a disproportionate effect on the resulting visualization\n",
    "\n",
    "* Probabilities for a bin can change based on the bin start and end location of the bin\n",
    "  * The probability distribution and, therefore, the interpretation of the data can become inconsistent when using two different bin boundaries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Nonparametric Density Estimation for Discrete Random Variables\n",
    "\n",
    "* A significant problem with histograms is that the choice of bin boundaries can have a disproportionate effect on the resulting visualization\n",
    "\n",
    "* Value for a bin can change based on the start and end location of the bin\n",
    "  * The probability distribution and, therefore, the interpretation of the data can become inconsistent when using two different bin boundaries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<img src=\"https://www.dropbox.com/scl/fi/8u0k4gzoq1s9w7z0ugojt/bin_diffs.png?rlkey=dqpk1j74voyi3poehdbcepg0p&dl=1\" width=\"800\"> \n",
    "\n",
    "  _Histograms on same data: bins are of the same size, but histogram on the right has bins shifted to the right._"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[115, 120, 125, 130, 135]\n"
     ]
    }
   ],
   "source": [
    "data = [119, 120, 121, 125, 129, 130, 130.1, 130.5, 130.7, 131]\n",
    "bins_2 = range(115, 136, 5)\n",
    "print(list(bins_2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.02 0.04 0.04 0.1 ]\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "counts = np.histogram(data, bins_2, density=True)\n",
    "print(counts[0])\n",
    "\n",
    "plt.figure(figsize=(10, 4))\n",
    "sns.barplot(x=counts[1][:-1], y=counts[0]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[114, 119, 124, 129, 134]\n",
      "[0.   0.06 0.02 0.12]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bins_3 = range(114, 136, 5)\n",
    "print(list(bins_3))\n",
    "\n",
    "counts = np.histogram(data, bins_3, density=True)\n",
    "print(counts[0])\n",
    "plt.figure(figsize=(10, 4))\n",
    "sns.barplot(x=counts[1][:-1], y=counts[0]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1200x400 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12, 4))\n",
    "\n",
    "bins_2 = range(115, 136, 5)\n",
    "counts = np.histogram(data, bins_2, density=True)\n",
    "plt.subplot(1, 2, 1)\n",
    "sns.barplot(x=counts[1][:-1], y=counts[0])\n",
    "\n",
    "bins_3 = range(114, 136, 5)\n",
    "counts = np.histogram(data, bins_3, density=True)\n",
    "plt.subplot(1, 2, 2)\n",
    "sns.barplot(x=counts[1][:-1], y=counts[0]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Estimating Densities Using Kernel\n",
    "\n",
    "* How can we bypass the use of bin boundaries? \n",
    "  * For instance, each data point could contribute **uniform** height to the histogram\n",
    "\n",
    "* For each position along the $x$-axis - including values with no data - we draw bars the heights of which equal to the sum of overlapping blocks at that position "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "\n",
    "<img src=\"https://www.dropbox.com/scl/fi/33q1dx6cgn6kpdcxqm6kl/top_hat_1.png?rlkey=0344yyi1gsxdqhwvq99bc7vhf&dl=1\" width=\"600\" height=\"300\"> \n",
    "\n",
    "_Each data point contributes a uniform `height` value. The shaded regions representing two or more overlapping blocks_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "<img src=\"https://www.dropbox.com/scl/fi/4v1sxuzs0gy39icyup27d/top_hat_2.png?rlkey=mzrx28hxc95gqut8zel4bi33u&dl=1\" width=\"600\" height=\"300\"> \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<center><img src=\"https://www.dropbox.com/scl/fi/gw8jsaiqqs92s1ce9vbyh/top_hat_example.png?rlkey=bvhwxz6kp3jwzfmblrnvjsh5y&dl=1\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Estimating Densities Using Kernels -- Cont'd\n",
    "\n",
    "* Applying this approach to the problematic histogram above produces a single, consistent visualization\n",
    "\n",
    "* Centering the box around each data point is a deterministic operation\n",
    "  * Not dependent on where we start our bin\n",
    "\n",
    "* We get an implied height for areas without a coverage \n",
    "  * That's good because samples are often unavailable for some areas of the sample space\n",
    "\n",
    "* Dense areas produce taller peaks\n",
    "* We can make inferences about the population based on a finite data sample\n",
    "  \n",
    "* The square block we used here is called a `kernel`, and the approach above is called `Kernel Density Estimation` (KDE)\n",
    "\n",
    "* Problem: leads to a more jagged histogram\n",
    "  * Can very close values lead to such drastic disagreement in density?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Using Different Kernels\n",
    "\n",
    "* We are used to density curves being smooth (as opposed to the jagged plot with the square kernel)\n",
    "\n",
    "* Instead of using a square block, what if we use a smoother curve as the kernel of each of the point\n",
    "\n",
    "  * Which curve should we try?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Using Different Kernels -- Cont'd\n",
    "\n",
    "* We could, for instance, use a Gaussian distribution, centered at the data point and with a scale of 0.1\n",
    "\n",
    "* Rather than having each point contribute uniformly, we can determine the contribution of a point using its `pdf`\n",
    "\n",
    "<center><img src=\"https://www.dropbox.com/scl/fi/fgn1ujp6fekzic67hy9ax/gaussian_kernel.png?rlkey=1lrf9cfk3oe3kroy0ge32cl85&dl=1\" width=\"300\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Using Different Kernels -- Cont'd\n",
    "\n",
    "- The contribution of a point at position $x$ is simply the sum of the \n",
    "`pdf`s with regard to each of the pdfs that overlap it\n",
    "\n",
    "- Contribution of each data point is :\n",
    "\n",
    "$$ \n",
    "\\LARGE\n",
    "\\sum_{ x \\in |X|} \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{\\frac{(x-\\mu_x)^2}{2\\sigma^2}}\n",
    "$$\n",
    "\n",
    "* Or in Python: `norm.pdf(x, `$\\mu, \\sigma$`)`\n",
    "  \n",
    "* The result is a smooth density-like estimate which is derived from the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "\n",
    "<img src=\"https://www.dropbox.com/scl/fi/kvgfsi09t62ob1u8z1ra6/final_kernel.png?rlkey=xnuch4gpyfhac3ua55huhbmm7&dl=1\" width=\"700\"> \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### KDE Kernels\n",
    "\n",
    "* Other kernels are also available.\n",
    "<center><img src=\"https://www.dropbox.com/scl/fi/tvpag0g9ox6d2r8vp259f/kernels.png?rlkey=us7qxt3astlmygcp3r83dzh8c&dl=1\" width=\"900\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Differences Between the Kernels \n",
    "\n",
    "- Each kernel emphasizes the contribution of neighboring points differently\n",
    "\n",
    "- KDE with some kernels is faster to compute than with others (ex. tophat versus Gaussian)\n",
    "  - With a tophat kernel of width $k$, the complexity is at most $O(n)$ \n",
    "  - With  the Gaussian kernel, the complexity is at most $O(n^2)$"
   ]
  },
  {
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    "### Differences Between the Kernels \n",
    "\n",
    "- Each kernel emphasizes the contribution of neighboring points differently\n",
    "\n",
    "- KDE with some kernels is faster to compute than with others (ex. tophat versus Gaussian)\n"
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