{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Sigmoid Function in Logistic Regression\n",
    "\n",
    "In learning about logistic regression, I was at first confused as to why a sigmoid function was used to map from the inputs to the predicted output. I mean, sure, it's a nice function that cleanly maps from any real number to a range of $-1$ to $1$, but where did it come from? This notebook hopes to explain.\n",
    "\n",
    "## Logistic Regression\n",
    "\n",
    "With classification, we have a sample with some attributes (a.k.a features), and based on those attributes, we want to know whether it belongs to a binary class or not. The probability that the output is 1 given its input could be represented as:\n",
    "\n",
    "$$P(y=1 \\mid x)$$\n",
    "\n",
    "If the data samples have $n$ features, and we think we can represent this probability via some linear combination, we could represent this as:\n",
    "\n",
    "$$P(y=1 \\mid x) = w_o + w_1x_1 + w_2x_2 + ... + w_nx_n$$\n",
    "\n",
    "The regression algorithm could fit these weights to the data it sees, however, it would seem hard to map an arbitrary linear combination of inputs, each would may range from $-\\infty$ to $\\infty$ to a probability value in the range of $0$ to $1$. \n",
    "\n",
    "### The Odds Ratio\n",
    "\n",
    "The odds ratio is a related concept to probability that can help us. It is equal to the probability of success divided by the probability of failure, and may be familiar to you if you ever look at betting lines in sports matchups:\n",
    "\n",
    "$$odds(p) = \\frac{p}{1-p}$$\n",
    "\n",
    "Saying, \"the odds of the output being 1 given an input\" still seems to capture what we're after. However, if we plot the odds function from 0 to 1, there's still a problem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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suSdccEHe1ahs7NrXDbv2SZIkNZ5bboG3vhUGDIC7707d+1Redu2TJEmSauyl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1072dfda0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 281,
       "width": 425
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'retina'\n",
    "\n",
    "def odds(p):\n",
    "    return p / (1 - p)\n",
    "\n",
    "x = np.arange(0, 1, 0.05)\n",
    "odds_x = odds(x)\n",
    "\n",
    "plt.plot(x, odds_x)\n",
    "plt.axvline(0.0, color='k')\n",
    "plt.ylim(-0.1, 15)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('odds(x)')\n",
    "\n",
    "# y axis ticks and gridline\n",
    "plt.yticks([0.0, 5, 10])\n",
    "ax = plt.gca()\n",
    "ax.yaxis.grid(True)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Log Odds\n",
    "\n",
    "An arbitrary linear combination of the input features may still be less than zero. However, if we take the log of the odds ratio, we now get something that ranges from $-\\infty$ to $\\infty$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wwhTOOy9w3XVTePzxFKC23RYOPRS+/GX46EenAIFvfGMK774LDz0EP/85/OM/\nwsc+Br/85RS6dq3eP7/jjjvueGccbxpr6yhCLt35QggrgR/HGP9pI+dcAnwlxtir3RdshxDC54Cf\nAVfEGM9qY7yp/fmHY4z3beA97M4nSQV7773S0rvmlaUXX0z7mdoyZEjrqtKIEelGtZKk2lSzLc5D\nCC8A60jL+Vrd8aJxz9QsoMv6bcUrLYQwFHgemAfsFpv9AEIIWwJvkDoPbhNjXLGB9zBESVKFvP12\n280dXnut7fO7doXddmu7uYOd8CSp86mZFudtuBq4ELgvhPAtYFqMcV0IoStwMPBdYBhwQU7XK1uM\n8cUQwt3AEcBXgJ80G74Q2AL46YYClCQpfzHCq6+2rirNmQOLFrX9mp49UzBav7nDbrulMUmSOkpe\nlagepO58kxufWge8A2wFdG187jbg5BjjmnZfsJ0aq1EPAdsAtwJzgf1JnQSfAQ7c0P6uxtdbiZKk\nMqxbB/Pmte6CN3cuvP9+26/ZcsvWVaWRI2GXXVLVSZJU32p2OR9ASLP/BPAZYBypQcMS4DHgFzHG\n63O5UE5CCIOBi4CjgIHA68DNwIUxxg21P296rSFKkjZi1Sp47rnWbcOfeSaNtWXrrVtXlUaMgB13\nTDerlSSpLTUdouqJIUqSkhUrUhVpzhyYPbsUmF54IVWd2jJ4cOuq0ogRKURJkpRVLe+JkiR1YsuX\ntw5Ls2dvuBNely6l5g7Nq0t77gn9+lV+/pIk5SnXEBVCGAKcCewNDCAt53scuDbG+HKe15Ik5W/5\n8tISvOZhad68tsNS166w++4pKI0aVQpNu+8OvQq9oYUkSR0nzz1RXwR+BPRoY3g18LUY409zuVjB\nXM4nqdYtW5YqS82D0uzZ8NJLbYelbt1ahqWmx+HDoUdbf+tLklQhNbsnKoTwYeBu4H1SkJoKLAC2\nBw4F/gHoCxwVY7yn3RcsmCFKUq1YurR1WJozZ+NhqaltePOwtNtuhiVJUnWq5RD1B+AAYN8Y4/Nt\njA8jLev7S4zxyHZfsGCGKEnVZunStAyvrbDUlu7dU2WpeVBqCkvdu1d06pIktUstN5bYD/htWwEK\nIMb4Qgjht8CJOV1PkurS8uUpHM2aVVqCN2cOvLyBXafdu6fKUvOw1HRDWsOSJEnlyStE9Qbe3sQ5\nC4EtcrqeJHVqa9bAs8+msNT8eOGFtpfh9eix4bDUzT6skiTlKq9/WucDh23inEmN50mSGjU0pCrS\nrFkwc2akaifpAAAgAElEQVQpLM2dm4LU+rp1S23CR42C0aNLYWnYMMOSJEmVktc/uTcB/xJCuAw4\nN8a4uGkghNAf+A6wP3BxTteTpJoSI7z5Zsuq0syZaTnesmVtv2bo0BSUxoxJj6NHp31MNniQJKlY\neTWW6A88BIwgdeh7CngD2A7YC+gHzAUOiDEuafcFC2ZjCUkbs2RJ62V4M2fCokVtn7/ddi2D0pgx\n6X5LfftWdt6SJNWimu3OBxBCGAD8J3A6aY9UkxXAr4BvxhjfzeViBTNESQJYtSp1xJs5s+VSvFde\nafv8/v1bBqWm5Xhbb13ZeUuS1JnUdIj62xuG0APYA+gPLAHmxhjbWNlfuwxRUn2JERYsgKeeghkz\nSo9z58Lata3P79kz7VNqXl0aPRoGD4ZQsb/eJUmqD50iRGW6eAifAj4VY9xUU4qqYoiSOq9Vq1LL\n8OZh6amnYOHC1ueGAMOHp7DUFJjGjElNHrp2rfzcJUmqR7V8n6hy7Urq2idJFRUjvPFG67A0dy6s\nW9f6/P79Ya+9YOzYdOy1V1qK16dP5ecuSZKKVXSIkqQOt3Jl2ru0/nK8DVWX9tijFJSaHnfayaV4\nkiQpMURJ6lQWLoQnnkjHk0+W9i61VV0aMKB1WBo1CrbwtuCSJGkjDFGSalKMqQve44+XQtMTT8Cr\nr7Y+t0uXVF1qHpbGjrW6JEmSymOIklT11q2DZ59tGZaeeALeeaf1uX36pJC0zz6w995WlyRJUv4M\nUZKqysqV6V5LzcPSjBmwfHnrcwcOhHHjUmBqOnbbzc54kiSpYxmiJBVm6dIUkpovyZszp+17L+28\nc8uwtM8+3ndJkiQVwxAlqSKWL09d8R59tHQ8/XTa29RcCLDnnq0D08CBxcxbkiRpfYYoSblbtSot\nwWsemGbPbt0hr1u3dIPaffdNQWncuNTwwXsvSZKkalZ0iHoS+GXBc5DUDmvWpIDUPDDNmJGeb65L\nFxgzBsaPLx1jx0KvXsXMW5IkqVwhrr+WRpsUQogA/uxUbxoa4Lnn4C9/gb/+NQWmJ59Mlafmmpbk\nNQ9Me+9thzxJkpS/0LhBOsZYsZ3SuVSiQgi/ADaVKBqA94CngdtjjAvyuLakjrNoETzySDr+8heY\nPh0WL2593vDhLQPTPvvAlltWfr6SJEmVkEslKoTQkPEla4B/jTFe3O6LF8BKlDqj1atT44emwPTI\nI/D8863P2357mDAB9t8fPvjBtI9pwIDKz1eSJAmKqUTlFaKGApcAE4EfAA8CbwLbAgcBXwMeAL4H\n7A38K7AjcGKM8ZZ2T6DCDFGqdTHCyy+XwtIjj6Q24+svy+vdOzV9aApN++9vW3FJklRdajlEfR04\nD9g7xvhqG+M7AU8A340xfj+EsDNpWd/DMcaPtHsCFWaIUq1ZvjztYXrwwVJweuut1uftsUcpME2Y\nkDrnde9e+flKkiRtrloOUU8D98cYv7SRcy4DDokxjmz8/n+Bj8QYa+7uL4YoVbsFC1Jgajoef7z1\nDWy32qplYPrgB+EDHyhmvpIkSeWq2cYSwC5AG9vNW1gCDG32/UuAW8+ldmpoSDetbQpM06bBiy+2\nPKdLl9Qdb+JEOOCAFJyGDXNZniRJUjnyClGLgMOBczdyzuGN5zUZQApWkjJYsaK0NG/aNHj4YXj3\n3Zbn9OmTqksTJ8JBB6XQ1K9fMfOVJEnqbPIKUTcA/xBCuA44N8Y4v2kghDCE1FBiH+DHzV4zDngu\np+tLndayZfDQQ/CnP8H996c24+vfyHbHHVNYmjgxHWPHQreib6UtSZLUSeW1J6o/MJUUlNYBr1Hq\nzjcY6AI8CRwaY1wSQtgBuBG4JsZ4WbsnUGHuiVJHev/9VGX605/S8de/ttzPFEIKSU2BaeJE2Hln\nl+ZJkqT6VLONJQBCCL2AfwI+Rcu9T/OAXwIXxxhX5nKxghmilKf33kvL8u6/P4Wmxx6DdetK4126\npHsxHXJIOg4+2PsySZIkNanpENXiTUPoB/QD3osxvpf7BQpmiFJ7vP9+aWne/ffDE0+k5hBNunaF\n8eNLoWniROjfv6jZSpIkVbdOE6I6O0OUsli9Ot2X6Z570jF9esvled26pfbikyal0HTggbClfSsl\nSZI2Sy23OAcghNAHOAHYm1L3vceBm2OMy/K8llStGhpg1qxSaPrzn1NziCZdu6Y244cdloLTAQek\nbnqSJEmqDbmFqBDC0aS9T1u1MfxOCOEzMcbb87qeVE1efRX++McUmqZOhbfeajk+ciR85CPpOOQQ\n241LkiTVsry6840DHgK6AteTOvUtALYHDgU+CawFJsYYH2v3BQvmcj6tWpU66P3hD3DXXany1NyO\nO5ZC02GHwQ47FDNPSZKkzq5m90SFEG4Ejia1MH+4jfH9gT8Bd8YYT2j3BQtmiKpP8+aVQtPUqS2X\n6PXpAx/+MBx+eApOe+xhy3FJkqRKqOUQ9RbwxxjjGRs551rgyBjjNu2+YMEMUfVh5crUPe+uu1J4\nevbZluNjxsBRR6Vj4kTo2bOQaUqSJNW1Wm4s0R+Yv4lzXmk8T6pab78Nd9wBt90Gd9/dstrUv3+q\nNB11FBx5JAweXNw8JUmSVJy8QtQbwH6bOGffxvOkqhEjzJ2bQtNtt8HDD6fnmuy9Nxx9dApOEyak\nduSSJEmqb3n9SngH8KUQwrnAxTHGdU0DIYSuwNeAw4Gf5nQ9qWxr18K0aXD77Sk4Pf98aaxHDzj0\nUJg8GY45Bnbeubh5SpIkqTrltSdqe+BRUje+l4EHSFWn7YCDgF1J3frGxxhfb/cFC+aeqNqzalVq\nP37DDXDrrfDuu6WxgQNTtWnyZDjiCG90K0mSVEtqtrEEQAhhV1Kl6fA2hv8POCvGOC+XixXMEFUb\nVq5M+5puuCFVnJYsKY3tsUcKTZMnp5vddu1a3DwlSZJUvpoOUX97wxAGA/uQmkgsAZ6IMb6a60UK\nZoiqXitWpE56N9yQluu9/35pbOxYOPlkOPFEGDGiuDlKkiQpP50iRNUDQ1R1WbEC7rwTfvc7+P3v\nW3bU22efUnDafffi5ihJkqSOUTMtzkMIvwDKShAxxs+W8zqpuXXr0g1vr7sObrwRli4tjY0fXwpO\nw4YVN0dJkiR1TmVVokIIDeVeMMbYpdzXVgsrUcWIER57LAWn3/wGFiwojX3wg3DKKXDSSbDLLoVN\nUZIkSRVWM5UoYGius5A2Yv58uOYa+NWv4JlnSs/vthucdlo6hg8vbn6SJEmqL+6JKoOVqI63ciXc\ncgtcdVVqTd70ox40CD7+cTj99FR9ChX77w2SJEmqRrVUiZJyFyM8/ngKTr/+NSxenJ7v0QOOPx4+\n9Sk4/HDo5qdWkiRJBfLXURXuvffg2mvhiitgxozS8+PGwWc/C5/4BGy1VXHzkyRJkpozRKkwTz0F\nl12W9jo1tSUfODAt1fvMZ2CvvYqdnyRJktQWQ5QqauXKdCPc//kfePjh0vOTJsGXvgTHHQc9exY2\nPUmSJGmTDFGqiNdeg5/8BH72M1i0KD3Xr1/a53TWWTByZLHzkyRJkjaXIUod6skn4ZJL0n2d1q5N\nz+2zT6o6ffKT0KdPsfOTJEmSsjJEKXcNDXDXXSk83Xdfeq5Ll3Qz3K99DSZMsDW5JEmSapchSrlZ\ntQp++Uv4/vdh7tz0XN++8PnPwznnwC67FDo9SZIkKReGKLXbihVw5ZXwn/+Z9j4BDB6cgtPnPw8D\nBhQ7P0mSJClPhiiVbflyuPxyuPhiWLAgPTdmDHzzm3DyydC9e7HzkyRJkjqCIUqZLV2a7u/0//4f\nvPVWem6ffeD882Hy5LT/SZIkSeqsDFHabCtXwo9/nJbtNbUpHz8eLrgAjj7aZhGSJEmqD4YobVJD\nA1x3HZx3HrzySnpuwoQUno480vAkSZKk+mKI0kb93//BP/9zut8TwNixqRJleJIkSVK9cveK2vTU\nU3DUUXDEESlADR4MV18Njz+enjdASZIkqV5ZiVILb7wB3/pWut9TjNCvH5x7bmpX3rt30bOTJEmS\nimeIEgBr18JPfpI67L3/fmpP/uUvw7e/DVtvXfTsJEmSpOphiBIPPJAC06xZ6ftjjoHvfx92263Y\neUmSJEnVyD1RdWzxYvj7v4cPfSgFqF13hdtvT4cBSpIkSWqblag6ddNNcPbZaQ9U9+5p39M3v+m+\nJ0mSJGlTDFF1ZuHCtHTvd79L3x94IPzsZzByZLHzkiRJkmqFy/nqyB13wJgxKUD17QuXXpr2Qxmg\nJEmSpM1nJaoOLF0K3/hGqjhB2gN19dVpD5QkSZKkbAxRndyMGXDqqTB3LvToAf/+7/D1r0MXa5CS\nJElSWQxRnVSMqfJ0zjmwciWMGgW/+Q2MHl30zCRJkqTaZj2iE1q6FD75ydS+fOVK+NznYPp0A5Qk\nSZKUBytRncy8eXDccTBzJvTpA5dfDqedVvSsJEmSpM7DENWJTJ0Kp5wCixbB8OFw660wYkTRs5Ik\nSZI6F5fzdRKXXgpHHJEC1Mc+lpbvGaAkSZKk/BmialxDA/zLv8DZZ8O6dXDuuXDbbTBgQNEzkyRJ\nkjonl/PVsNWr4bOfheuug27d4Oc/hzPPLHpWkiRJUudmiKpR770HJ54I99yTGkjceCMceWTRs5Ik\nSZI6P0NUDVqyJAWmRx6BbbeFO+6AffctelaSJElSfTBE1ZjFi1OAmj4dhgxJHfmGDi16VpIkSVL9\nMETVkHffTR34Hn0Udt0V7rsvBSlJkiRJlWOIqhGLF8Phh8Njj6XK0333wc47Fz0rSZIkqf7Y4rwG\nrFoFxx+fAtSwYXD//QYoSZIkqSiGqCrX0JDalt9/P2y/Pdx7L+y0U9GzkiRJkuqXIaqKxQjf+Ab8\n9rfQrx/cdZd7oCRJkqSiGaKq2CWXwA9/CN27w803w157FT0jSZIkSSHGWPQcak4IIQJ05M/urrvg\n6KNTNer66+HjH++wS0mSJEk1K4QAQIwxVOqaVqKq0Lx5cNppKUBddJEBSpIkSaomVqLK0JGVqBUr\n4KCD4PHH4Zhj4NZboYtRV5IkSWqTlShx9tkpQA0dCtdcY4CSJEmSqo2/oleRq65KR69ecNNN8IEP\nFD0jSZIkSetzOV8ZOmI530svwZgxsHQpXH01fOpTub21JEmS1Gm5nK9OxQhf+EIKUCedZICSJEmS\nqpkhqgpceSXccw8MHAiXXlr0bCRJkiRtjMv5ypDncr7582H0aHj/fe8HJUmSJGXlcr46EyN88Ysp\nQB1/PJx6atEzkiRJkrQpVqLKkFcl6rrr4PTTYautYPZs2G67XKYnSZIk1Q0rUXVk5Ur41rfS1//1\nXwYoSZIkqVYYogpy6aVpP9SYMXbjkyRJkmqJy/nK0N7lfO++C8OGpcc77oCPfSzX6UmSJEl1w+V8\ndeI//iMFqEMPhY9+tOjZSJIkScrCSlQZ2lOJmj8fdt8dVq2Cv/4Vxo/PfXqSJElS3bASVQfOPz8F\nqFNPNUBJkiRJtchKVBnKrUQ9/3yqQnXrBk8/nfZFSZIkSSqflahO7tJL0w12TzvNACVJkiTVKitR\nZSinErV0KQweDEuWwGOPwbhxHTY9SZIkqW5YierEfvWrFKAOPNAAJUmSJNUyQ1QFxAg/+Un6+qtf\nLXYukiRJktrH5XxlyLqc77774LDDYLvt4OWXoUePDp2eJEmSVDdcztdJ/fjH6fGsswxQkiRJUq2z\nElWGLJWo+fNh112hS5f09fbbd/j0JEmSpLphJaoTuuwyaGiAk082QEmSJEmdgZWoMmxuJaqhAXba\nCV5/HaZNg4kTKzI9SZIkqW5YiepkHn88BajBg1Nrc0mSJEm1zxDVgW6/PT0eeyyEiuViSZIkSR3J\nENWBmocoSZIkSZ2De6LKsDl7ol55BXbeGfr0gYULoVevik1PkiRJqhvuiepEfv/79HjEEQYoSZIk\nqTMxRHWQ225Ljy7lkyRJkjoXl/OVYVPL+ZYuhYEDYc0aWLAAttmmotOTJEmS6obL+TqJu++G1ath\nwgQDlCRJktTZGKI6QFNXvsmTi52HJEmSpPy5nK8MG1vOt24dbL89vP02zJoFo0ZVfHqSJElS3XA5\nXyfwyCMpQA0dCiNHFj0bSZIkSXkzROXsrrvS47HHQqhYFpYkSZJUKYaonD3xRHo86KBi5yFJkiSp\nYxiicjZzZnocM6bYeUiSJEnqGDaWKMOGGkssXgwf+AD06pXuFdW1ayHTkyRJkuqGjSVq3KxZ6XHU\nKAOUJEmS1FkZonI0Y0Z6HDu22HlIkiRJ6jiGqBy5H0qSJEnq/AxRObISJUmSJHV+NpYoQ1uNJWKE\n/v3h/ffhzTdhm20Km54kSZJUN2wsUcNefjkFqG23NUBJkiRJnZkhKifuh5IkSZLqgyEqJ+6HkiRJ\nkuqDISonTSHKSpQkSZLUuRmictK0nM9KlCRJktS52Z2vDOt351u5Evr2TR36li6F3r0LnZ4kSZJU\nN+zOV6OefhrWrYPddzdASZIkSZ2dISoH7oeSJEmS6ochKgfuh5IkSZLqhyEqB1aiJEmSpPphiMqB\nlShJkiSpftidrwzNu/O9/TZss03qzrdkCXQxlkqSJEkVY3e+GtRUhRozxgAlSZIk1YNuRU+g1g0a\nBGefDbvuWvRMJEmSJFWCy/nKsP7NdiVJkiQVw+V8kiRJklTlDFGSJEmSlIEhSpIkSZIyMERJkiRJ\nUgaGKEmSJEnKwBAlSZIkSRkYoiRJkiQpA0OUJEmSJGVgiJIkSZKkDAxRkiRJkpSBIUqSJEmSMjBE\nSZIkSVIGhihJkiRJysAQJUmSJEkZGKIkSZIkKQNDlCRJkiRlYIiSJEmSpAwMUZIkSZKUgSFKkiRJ\nkjIwREmSJElSBoYoSZIkScrAECVJkiRJGRiiJEmSJCkDQ5QkSZIkZWCIkiRJkqQMDFGSJEmSlIEh\nSpIkSZIyMERJkiRJUgaGKEmSJEnKwBAlSZIkSRkYoiRJkiQpA0OUJEmSJGVgiJIkSZKkDAxRkiRJ\nkpSBIUqSJEmSMjBESZIkSVIGhihJkiRJysAQJUmSJEkZGKIkSZIkKQNDlCRJkiRlYIiSJEmSpAzq\nKkSFEK4OITRs4rin6HlKkiRJql7dip5Ahd0MvLiBsTOBocCdlZuOJEmSpFoTYoxFz6FwIYQBwOtA\nAHaMMb6zifMjgD87SZIkqVghBABijKFS16yr5XwbcQbQC7hpUwFKkiRJUn0zRCVfaHy8otBZSJIk\nSap6dR+iQggHAKOBZ2KMfyp6PpIkSZKqW92HKOCLjY8/K3QWkiRJkmpCzTWWCCG8BOyc4SXXxRjP\n2MB79Sc1lOjCZjSUaPY6G0tIkiRJVcDGEpvneWBuhuO1jbzX6UBvymwoEULY4DFp0iRCCEyZMqXN\n106ZMsVxxx133HHHHXfccccd34zxprG2jiLUXCUqTyGEJ4GxwKQY458zvM5KlCRJklQFrERVUAhh\nf1KAeiZLgJIkSZJU3+o2RFFqKGFbc0mSJEmbrS6X84UQ+lFqKDE4634ol/NJkiRJ1cHlfJVzGrAF\ncHM5DSUkSZIk1a+6rES1l5UoSZIkqTpYiZIkSZKkKmeIkiRJkqQMDFGSJEmSlIEhSpIkSZIyMERJ\nkiRJUgaGKEmSJEnKwBAlSZIkSRkYoiRJkiQpA0OUJEmSJGVgiJIkSZKkDAxRkiRJkpSBIUqSJEmS\nMjBESZIkSVIGhihJkiRJysAQJUmSJEkZGKKkKjFlypSip6BOxM+T8uZnSnny86RaF2KMRc+h5oQQ\nIoA/O+UphOBnSrnx86S8+ZlSnvw8KU8hBABijKFS17QSJUmSJEkZGKIkSZIkKQNDlCRJkiRlYIiS\nJEmSpAwMUZIkSZKUgd35ytDUnU+SJElSdbA7nyRJkiRVKStRkiRJkpSBlShJkiRJysAQJUmSJEkZ\nGKIkSZIkKQNDlCRJkiRlYIiSJEmSpAwMUZIkSZKUgSFKkiRJkjIwRAEhhMEhhKtCCK+HEFaGEOaF\nEL4fQhhQxPuo9rX3sxBC2CqE8PkQws0hhOdDCMtDCItDCA+EED4bQqjYHblVHTri75cQwukhhIbG\n43N5zlfVLc/PUwjhw41/Vy1ofK/XQgh/CCF8tCPmruqUx2cqJKeGEO5r/BwtDyG8EEL4bQhhQkfO\nX9UjhHBSCOHHjb/zvNf4b9S1Zb5Xh/1uXvc32w0hDAMeAgYBtwBzgf2BQ4FngIkxxncq9T6qfXl8\nFkIIZwH/A7wO3AfMB7YDTgD6AzfGGE/uqD+DqktH/P0SQtgJmEn6j2l9gc/HGK/Kc96qTnl+nkII\nFwP/H/AKcBewENgGGAfcE2P8Zu5/AFWdHH+XuhL4LOlzdEvj43BgMtANODPGeF1H/BlUPUIITwJj\ngfeB14A9gV/FGM/M+D4d+7t5jLGuD+CPQAPwlfWev6Tx+csq+T4etX/k8Vlo/D/40W08vy3wcuP7\nnFD0n9WjMkfef78AAbgHeA64uPE9Plv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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1038d0860>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 281,
       "width": 424
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def log_odds(p):\n",
    "    return np.log(p / (1 - p))\n",
    "\n",
    "x = np.arange(0.005, 1, 0.005)\n",
    "log_odds_x = log_odds(x)\n",
    "\n",
    "plt.plot(x, log_odds_x)\n",
    "plt.axvline(0.0, color='k')\n",
    "plt.ylim(-8, 8)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('log_odds(x)')\n",
    "\n",
    "# y axis ticks and gridline\n",
    "plt.yticks([-7, 0, 7])\n",
    "ax = plt.gca()\n",
    "ax.yaxis.grid(True)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having a linear combination of arbitary features map to the `log_odds` function allows for any possible input values for each $x_i$ and still represents conceptually what we are trying to represent: that a linear combination of inputs is related to the liklihood that a sample belongs to a certain class. \n",
    "\n",
    "Note: the log of the odds function is often called \"the logistic\" function.\n",
    "\n",
    "So now we have:\n",
    "\n",
    "$$\\text{log_odds}(P(y=1 \\mid x)) = w_o + w_1x_1 + w_2x_2 + ... + w_nx_n$$\n",
    "\n",
    "If we still want to get plain old $P(y=1 \\mid x)$ we can by taking the inverse of the `log_odds` function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Sigmoid\n",
    "\n",
    "Let's find the inverse of the `log_odds` function:\n",
    "\n",
    "Starting with:\n",
    "\n",
    "$y = log(\\frac{x}{1-x})$\n",
    "\n",
    "and swapping $y$ and $x$ and solving for $y$ \n",
    "\n",
    "$x = log(\\frac{y}{1-y})$\n",
    "<br>$e^x = \\frac{y}{1-y}$\n",
    "<br>$y = (1-y)*e^x$\n",
    "<br>$y = e^x - y*e^x$\n",
    "<br>$y + ye^x = e^x$\n",
    "<br>$y*(1 + e^x) = e^x$\n",
    "<br>$y = \\frac{e^x}{1+e^x}$\n",
    "<br>$y = \\frac{1}{\\frac{1}{e^x} + 1}$\n",
    "<br>$y = \\frac{1}{1 + e^{-x}}$\n",
    "\n",
    "Let's use $\\phi$ to represent this function and plot it to get a sense of what it looks like:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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euDwvIy7Pk1Qr/v1vGDgQ5s6FBx+E/fbLe6LsuTxPkorN5XmSpIqZ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\nqYZJkyRJkiTV+C/7IK5FgqO64QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1087d01d0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 281,
       "width": 422
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def inverse_log_odds(z):\n",
    "    return 1.0 / (1.0 + np.exp(-z))\n",
    "\n",
    "z = np.arange(-7, 7, 0.1)\n",
    "phi_z = inverse_log_odds(z)\n",
    "\n",
    "plt.plot(z, phi_z)\n",
    "plt.axvline(0.0, color='k')\n",
    "plt.ylim(-0.1, 1.1)\n",
    "plt.xlabel('z')\n",
    "plt.ylabel('$\\phi (z)$')\n",
    "\n",
    "# y axis ticks and gridline\n",
    "plt.yticks([0.0, 0.5, 1.0])\n",
    "ax = plt.gca()\n",
    "ax.yaxis.grid(True)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse form of the logistic function is looks kind of like an S, which, I've read, is why it's called a Sigmoid function.\n",
    "\n",
    "So going back to our:\n",
    "\n",
    "$$\\text{log_odds}(P(y=1 \\mid x)) = w_o + w_1x_1 + w_2x_2 + ... + w_nx_n$$\n",
    "\n",
    "If we call $w_o + w_1x_1 + w_2x_2 + ... + w_nx_n = w^Tx$ simply $z(x)$:\n",
    "\n",
    "$$ \\text{log_odds}(P(y=1 \\mid x)) = z(x) $$\n",
    "\n",
    "And taking the inverse:\n",
    "\n",
    "$$ P(y=1 \\mid x) = \\phi(z) = \\dfrac{1}{1 + e^{-z}} $$\n",
    "\n",
    "and there you have it: Logistic Regression fits weights so that a linear combination of its inputs maps to the log odds the output being equal to 1."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}