monte_carlo_with_bootstrap_sampling.ipynb

{
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   "cell_type": "markdown",
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   "source": [
    "# An introduction to numerical uncertainty propogation with Monte Carlo methods and bootstrap sampling\n",
    "\n",
    "1. [Monte Carlo methods](#Monte-Carlo-methods)\n",
    "  - [A basic Monte Carlo example using normally distributed data](#A-basic-Monte-Carlo-example-using-normally-distributed-data)\n",
    "2. [Bootstrap sampling](#Bootstrap-sampling)\n",
    "  - [Bootstrap sampling implementation](#Bootstrap-sampling-implementation)\n",
    "  - [Visualizing bootstrap sampling](#Visualizing-bootstrap-sampling)\n",
    "3. [Weighted bootstrap sampling](#Weighted-bootstrap-sampling)\n",
    "4. [Monte Carlo with bootstrap sampling](#Monte-Carlo-with-bootstrap-sampling)\n",
    "\n",
    "\n",
    "## Monte Carlo methods\n",
    "[Monte Carlo methods](https://en.wikipedia.org/wiki/Monte_Carlo_method) are statistical techniques that use repeated random sampling to obtain numerical results.  A common application is for uncertainty propagation parametric models (e.g. a normal distribution defined by a mean and standard deviation) of different parameter values.\n",
    "\n",
    "For example, suppose we want to add or multiplying two parameters (x1, x2) that have are estimated to be normally distributed with mean of 4 and 5, and standard deviation of 1 and 2 respectively.  A convenient notation to describe these two normally distributed parameters is:\n",
    "- x1: N~(4,1)\n",
    "- x2: N~(5,2)\n",
    "\n",
    "We can use monte carlo methods to estimate the product of x1 and x2 by sampling from their distributions:\n",
    "y = N~(4,1) * N(5,2)\n",
    "\n",
    "A more simplified approach to account for uncertainty might be to generate high, medium and low estimates based on the extreme and mean values that are known for each parameter.  A potential benefit of using monte carlo methods to estimate uncertainty compared to 'medium', 'high' and 'low' estimates is the narrow uncertainty bounds (e.g. based on 2 standard devations) provided by monte carlo estimates.  Tighter uncertainty bounds result from reduced sampling of unlikely events (i.e. the tails of the probability distribution) when monte carlo methods are used. Based on probability theory ([regression to the mean](https://en.wikipedia.org/wiki/Regression_toward_the_mean)), the probability that the result is the combination of extreme values is low compared to other outcomes.  This 'benefit' of tighter uncertainty bounds presumes that uncertainty bounds are fairly well understood which may not the case due to a variety of unknowns.  \n",
    "\n",
    "### A basic Monte Carlo example using normally distributed data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 1.18.1\n",
    "import seaborn as sns  # 0.10\n",
    "from matplotlib import pyplot as plt  # 3.1.3\n",
    "%matplotlib inline\n",
    "\n",
    "from sampling import bootstrap_sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1964ff33648>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_samples = 10000\n",
    "x1_mean = 4\n",
    "x1_SD = 1\n",
    "x2_mean = 5\n",
    "x2_SD = 2\n",
    "x1 = np.random.normal(x1_mean, x1_SD, n_samples)\n",
    "x2 = np.random.normal(x2_mean, x2_SD, n_samples)\n",
    "sns.distplot(x1)\n",
    "sns.distplot(x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Deterministic result = 20\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x1965022a748>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = x1 * x2\n",
    "sns.distplot(y)\n",
    "print(f\"Deterministic result = {x1_mean * x2_mean}\")\n",
    "plt.axvline(x1_mean * x2_mean, color='black', linestyle='--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It might seem odd that some negative values are returned from the multiplication of 4 * 5. This is because a normal distribution is used to represent the parameter distribution. The normal distribution is defined along the range $-\\infty$ to $+\\infty$.  If negative values are not possible, other distributions that do not have negative values can be used like the lognormal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Deterministic result = 20\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x19650363dc8>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = np.random.lognormal(np.log(x1_mean), np.log(x1_SD), n_samples)\n",
    "x2 = np.random.lognormal(np.log(x2_mean), np.log(x2_SD), n_samples)\n",
    "y = x1 * x2\n",
    "sns.distplot(y)\n",
    "print(f\"Deterministic result = {x1_mean * x2_mean}\")\n",
    "plt.axvline(x1_mean * x2_mean, color='black', linestyle='--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrap sampling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes the underlying distribution of a parameter value may be unknown.  Instead we might have a handfull of parameter estimates from different sources that we may want to reconcile with our uncertainty assessment.  One approach might be to take the mean of these values and make an assumption about the parameter distribution.  With a large sample size, sample statistics (mean, standard deviation) can be estimated from using each of the individual estimates.  With smaller sample sizes, bootstrap sampling with replacement is a technique that can be used to estimate the underlying distribution from a set of point estimates.  \n",
    "\n",
    "For example, say we have five estimates of x1 instead of an estimate for x1 of N~(4,1):\n",
    "- x1_1 = 3\n",
    "- x1_2 = 4.5\n",
    "- x1_3 = 7\n",
    "- x1_4 = 3.5\n",
    "- x1_5 = 5\n",
    "\n",
    "In this example each of these estimates is a point value with no descriptive statistics to indicate uncertainty.  Bootstrap sampling with replacement works by sampling a number of values (e.g. 3) from the set of x1_1 to x1_5 using replacement (each time a value is sampled from the set it is not removed and so can be resampled again on the next draw) and taking the average of the sampled values.  Bootstraping undertakes this sampling procedure repeatedly to build a distribution to represent the entire sample\n",
    "\n",
    "Now let's implement a method for bootstrap sampling and apply it to this set of estimates for x1.\n",
    "\n",
    "### Bootstrap sampling implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bootstrap_sample(x, n_samples, smoothed=True, replace=True, p=None):\n",
    "    \"\"\"\n",
    "    Parameters\n",
    "    --------------\n",
    "    x : 1-D array-like.\n",
    "        Sample to compute bootstrap statistics from.\n",
    "    n_samples : int\n",
    "        Number of bootstrap samples to return.\n",
    "    smoothed : bool\n",
    "        Whether smoothed sampling is used.\n",
    "    replace : bool\n",
    "        Whether sampling with replacement is used.\n",
    "    p : 1-D array-like\n",
    "        Probability for each value of x for weighted sampling.\n",
    "        \n",
    "    Notes\n",
    "    --------------\n",
    "    References:  https://en.wikipedia.org/wiki/Bootstrapping_(statistics)#Smoothed_bootstrap\n",
    "    \"\"\"\n",
    "    size = (n_samples, len(x))\n",
    "    bootstrap_samples = np.random.choice(x, size, replace, p)\n",
    "    bootstrap_result = np.mean(bootstrap_samples, axis=1)\n",
    "    \n",
    "    if smoothed:\n",
    "        noise = gaussian_noise(x, n_samples)\n",
    "        return bootstrap_result + noise\n",
    "\n",
    "    return bootstrap_result\n",
    "\n",
    "def gaussian_noise(x, n_samples)->np.ndarray:\n",
    "    \"\"\"Create guassian noise of len(n_samples).\n",
    "    \n",
    "    Parameters\n",
    "    ------------\n",
    "    x : ndarray\n",
    "    n_samples : int\n",
    "    \"\"\"\n",
    "    sigma = np.std(x)/np.sqrt(len(x))\n",
    "    noise = np.random.normal(0, sigma, n_samples)\n",
    "    return noise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Visualizing bootstrap sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "histogram of raw data\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = [3,4.5,7,3.5,5]\n",
    "p = [1/len(x1)]*len(x1)\n",
    "sns.distplot(x1, bins=20, kde=False, color='white')\n",
    "plt.bar(x1, p, width=0.2, alpha=0.7)\n",
    "print(\"histogram of raw data\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "bootstrapped sample\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1_bootstrap = bootstrap_sample(x1, n_samples)\n",
    "sns.distplot(x1_bootstrap)\n",
    "print(\"bootstrapped sample\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Statistics statistics of x1\n",
      "mean: 4.6\n",
      "std: 1.3928388277184118\n",
      "-----------\n",
      "Bootstrap statistics\n",
      "mean: 4.588024976973153\n",
      "std: 0.8804850415295014\n"
     ]
    }
   ],
   "source": [
    "print(\"Statistics statistics of x1\")\n",
    "print(\"mean: {}\".format(np.mean(x1)))\n",
    "print(\"std: {}\".format(np.std(x1)))\n",
    "print(\"-----------\")\n",
    "print(\"Bootstrap statistics\")\n",
    "print(\"mean: {}\".format(np.mean(x1_bootstrap)))\n",
    "print(\"std: {}\".format(np.std(x1_bootstrap)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Monte Carlo with bootstrap sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x196505ceb48>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = x1_bootstrap * np.random.normal(x2_mean, x2_SD, n_samples)\n",
    "sns.distplot(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Weighted bootstrap sampling\n",
    "\n",
    "The weighted bootsrap method we have defined above also given us the opportunity to apply a weighting factor to each of the individual estimates.  This can be useful if there is some evidence to suggest that certain parameter estimates are more likely or valid than others (e.g. measurement procedures follow standard practices).  We can use this information to influence the likelihood that specific estimates will be sampled."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "histogram of weighted data.\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = [3,4.5,7,3.5,5]\n",
    "p = [0.1,0.1,0.5,0.1,0.20]\n",
    "sns.distplot(x1, bins=20, kde=False, color='white');\n",
    "plt.bar(x1,p, width=0.2, alpha=0.7);\n",
    "print(\"histogram of weighted data.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample statistics from bootstrap sampling with replacement\n",
      "mean: 5.576287659519408\n",
      "std: 0.9056229451199955\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x196503f94c8>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1_bootstrap = bootstrap_sample(x1, n_samples=1000, p=[0.1,0.1,0.5,0.1,0.2])\n",
    "print(\"sample statistics from bootstrap sampling with replacement\")\n",
    "print(\"mean: {}\".format(np.mean(x1_bootstrap)))\n",
    "print(\"std: {}\".format(np.std(x1_bootstrap)))\n",
    "sns.distplot(x1_bootstrap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1964ffc2808>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "samples = 1000\n",
    "y = np.random.normal(4,1, samples)*x1_bootstrap\n",
    "sns.distplot(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:analytics]",
   "language": "python",
   "name": "conda-env-analytics-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}