02_Analyze-data_project.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computational Mechanics Project #02 - Create specifications for a my projectile robot\n",
    "\n",
    "On the first day of class, we threw $2\"\\times~2\"$ dampened paper (spitballs) at a target on the whiteboard. Now, we are going to analyze the accuracy of the class with some cool Python tools and design a robot that has the same accuracy and precision as the class, but we will have the robot move farther away from the target and use a simpler projectile i.e. a tennis ball so we don't need to worry about knuckle-ball physics. \n",
    "\n",
    "The goal of this project is to determine the precision of necessary components for a robot that can reproduce the class throwing distibution. We have generated pseudo random numbers using `numpy.random`, but the class target practice is an example of truly random distributions. If we repeated the exercise, there is a vanishingly small probability that we would hit the same points on the target, and there are no deterministic models that could take into account all of the factors that affected each hit on the board. \n",
    "\n",
    "<img src=\"../images/robot_design.png\" style=\"height: 250px;\"/>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we ask ourselves some questions:\n",
    "\n",
    "1. How do we quantify the class accuracy and precision?\n",
    "\n",
    "2. If we design a robot, what design components can we control?\n",
    "\n",
    "3. How can we relate the controlled components to the class accuracy, and specify the component precision?\n",
    "\n",
    "The first question, we have some experience from our work in [02_Seeing_Stats](../notebooks/02_Seeing_Stats.ipynb). We can define the mean, standard deviation, measure the first, second, and third quartiles,  etc. \n",
    "\n",
    "The second question is a physical question. We cannot control the placement of the robot or the target those are chosen for us. We cannot control temperature, mechanical vibrations, etc. We *can* control the desired initial velocity. The initial velocity will have some speed and direction, and both will be subject to random noise. Once the speed and direction are set, the location on the target is determined by kinematic equations for an object in freefall, as such\n",
    "\n",
    "$x_{impact} = \\frac{v_x}{v_y}d + x(0)~~~~~~~~~~~~~~~~~~~~(1.a)$\n",
    "\n",
    "$z_{impact} = d\\left(\\frac{v_z(0)}{v_y}-\\frac{g}{2v_y^2}d\\right)+ z(0)~~~~~(1.b)$.\n",
    "\n",
    "Where the location of impact is at a $y$-distance of $d$ at a point on the target with coordinates $(x_{impact},~z_{impact})$, and the initial velocity is $\\bar{v}=v_x\\hat{i}+v_y\\hat{j}+v_z(0)\\hat{k}$, the object is released at an initial location $\\bar{r}(0)=x(0)\\hat{i}+0\\hat{j}+z(0)\\hat{k}$, and the only acceleration is due to gravity, $\\bar{a}=-g\\hat{k}$. Equation (1) becomes much easier to  evaluate if we assume that $v_x=0$, resulting in an evalution of the accuracy of the height of the impact, $z_{impact}$, as such\n",
    "\n",
    "$x_{impact} = x(0)~~~~~~~~~~~~~~~~~~~~(2.a)$\n",
    "\n",
    "$z_{impact} = \\frac{d}{\\cos{\\theta}}\\left(\\sin{\\theta}-\\frac{g}{2v_0^2\\cos{\\theta}}d\\right)+ z(0)~~~~~(2.b)$.\n",
    "\n",
    "Where $\\theta$ is the angle of the initial velocity and $v_0$ is the initial speed. Equation (2) restricts the analysis to height accuracy. You can incorporate the 2D impact analysis if you finish the 1D analysis. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The third question, is how we can relate equation (2) to the measured points of impact? For this, we can use Monte Carlo methods *(There are other methods, but Monte Carlo is one of the most straight-forward)*. Our Monte Carlo approach is as such, if we have a desired initial speed, $v_0$, and desired angle, $\\theta$, we can propagate the uncertainty of our actual speeds and angles into the $z_{impact}$ locations. Then, we can choose distributions in speed and angles that match the distributions in $z_{impact}$ locations. Here are the steps:\n",
    "\n",
    "1. Generate random $\\theta_i$ and $v_{0~i}$ variables\n",
    "\n",
    "2. Plug into eqn 2 for random $z_{impact~i}$ locations\n",
    "\n",
    "3. Compare to our measured $z_{impact}$ location statistics\n",
    "\n",
    "4. Repeat 1-3 until the predicted uncertainty matches the desired uncertainty, we can use a number of comparison metrics:\n",
    "    \n",
    "    - standard deviation\n",
    "    \n",
    "    - first, second, and third quartiles\n",
    "    \n",
    "    - visually, with box plots and histograms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Project Deliverables\n",
    "\n",
    "1. Statistical analysis of class accuracy and precision (x- and z-locations) data is in the csv file  [../data/target_data.csv](../data/target_data.csv) _Note: if you want to see how I turned the images into data check out the jupyter notebook [process_target_practice](./process_target_practice.ipynb)\n",
    "\n",
    "2. A Monte Carlo model to generate impact heights based upon uncertainty in $\\theta_0$ and $v_0$. \n",
    "\n",
    "3. The precision required to recreate the class accuracy and precision with a robot. \n",
    "**You must show some validation of your work**\n",
    "\n",
    "4. [BONUS] Repeat 2-3 taking into account the variation in $x_{impact}$ due to misalignment. \n",
    "\n",
    "Given constants and constraints:\n",
    "\n",
    "- $d=$3 m, distance to target\n",
    "\n",
    "- $g=$9.81 m/s$^2$, acceleration due to gravity\n",
    "\n",
    "- $z(0)=$0.3 m,  the initial height is 0.3 m above the bull's eye\n",
    "\n",
    "- 4 m/s$<v_0<$12 m/s, the initial velocity is always higher than 9 mph and less than 27 mph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "#Import rcParams to set font styles\n",
    "from matplotlib import rcParams\n",
    "\n",
    "#Set font style and size \n",
    "rcParams['font.family'] = 'sans'\n",
    "rcParams['font.size'] = 16\n",
    "rcParams['lines.linewidth'] = 3\n",
    "\n",
    "target_data = pd.read_csv(\"../data/target_data.csv\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Part 1: Statistical Analysis**\n",
    "\n",
    "We perform a statistical analysis of x and z position class data by generating mean, standard deviation and quartiles for each distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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rgT0btL01Mzf1aPsEYI+Zd0+S1FSbh5V2Au7uUb4RWDiDtuPvFyJiGbCsfvlARNzYo9pTgbv6rH9WxGAXP2fGOWCO87Hj8TBGmNE4X9e3RnxwekvusKhXYdt/zyF7lDX5TozptM3MlcDKSRccsS4zRxv0YavmOB9bHg/jfDyMEbbecbZ5WOluev+Gv5DeewWdNk7Sdvx9SdIsaTMc1lOdO+i2J/D9Bm13ry+H7W77K+DmsokkaVDaDIeLgX0j4lnjBRExArysfq9f2/nAER1t5wFvAr6cmQ/PoF+THnZ6DHGcjy2Ph3E+HsYIW+k4I7PXof5pLKi60e064EHgr6jOIfwN8BTgBZn5QF1vEXALsCIzV3S0Pw9YArwbuBU4iepszEsz89pWOilJaqS1PYfM/CWwP3AT8GngXKov+f3Hg6EWwLY91v0W4FPA6cClwDOBAw0GSZp9re05SJIeO7baWVkjYpuIOC0ixiLioYi4LiIOa9BuJCJyksebO+oun6DORYMd3Rb9ndY467ZnT9D/s3rU3S8i/r2e9PDOiPhIRDyx/RFN2Nfpbs8dIuJ9dd9/ERH31P9+Q4+6s7Y9Z2MSypl8Ntoy3XFGxGhErIyI/4iITRFxe0ScGxG796g7NsF2K7bxIMxwW070PbN3V72hb8tCZm6VD+ADwMPAKcCrgE8AjwKv7dNuO2DfHo8rgIeAnTrqLqc6d/Kyrrp/MNfHWbc9G/h5j7Eu6qr3AqpzRRcBrwZOpLr8+Py5Pk7gPwF3AmcAB1Gdtzq73m7v6Ko7K9sT2J5qTrHrgTdQTR/zPapzbU9q0P5c4B6q+cpeDXy+3j57t/XZGPY4gQ9TTa3zduCVVHOr3QD8AnhmV90x4LIen+OFc3mMdfukOlze3fft59K27Nn3Ya14hhtsl/oH+f6u8q8A353mB+A+4IKu8vEvk3lb4zjrL8k7GtS7sP4PML+j7Lh67C+ey+MEntT9H62j7e3D2J7AnwGPAHt0lO0ObAbe1aftC+s+vqWjbB5wI3BxW5+NOTDOnXuULaq/EFd0lY8Bq2ZjTG2Osa6bwOl96gx9W/Z6bK2HlZZQzbm0qqt8FfD8Xrumffwp1VVV57TQtza1Pc5CRMwHDgT+JTN/3fHWv1DdY9Jv0sQ2THucmfnLLOfkAlgH/H57XZyS2ZiEcuCfjQamPc7M3NCj7DZgA7Bry/2ciZlsy6bmwrYsbK3hsBdV0nbfHLe+fu430V+346kOv1w2wfs/iohHIuK2iPjgLB6Lb2Ocu0TEXRGxOSJuioi/jIhtO95/NrCArkkTM/Mhql3nqf4sp6Pt7QnwCuA/Jnhv0NtzNiahHMTPbKpmMs5CRDyP6rfoG3q8fXB9buLhiFg7W+cbaGeMJ9X93hQRqyPi5T3WMextWWh7bqXZshNwT9b7Xh0mnaivl4jYleoS3L+rf0PrdDNwKvBtqt3D1wB/AbwY+JNp9HuqZjrO7wDXUH3IFgCHUh2bfw7VeYXOZUw08WHjn+UMtLY94TcTMu4LHNP11mxtz9mYhLLVn9k0zWScW4jqptd/pNpz+GTX25cA36K6NP5pwH8DLoyIYzOz+7ftts10jKuA/wf8hOqw2buB1RHxJ5n51Y51DHtbFuZEOETEAcC/Nai6JjMXM82J+iZwLNUeVHFIqccH798i4g7grIg4IDOvmMqKZnucmdl9VdIXI+IB4M8j4oOZ+YOOZbX18xzq9oyIxcDfA5/OzHM732t7e/Yx6Eko2/w/MBNt9eFjwEuBpZm5xZdxZp68xcIjLgTWUv2iM+hwgJn9Hzy24+VVEfEFqj2R04H9OpY1F7blFuZEOAD/DjyvQb3xXe2NwMKIiK60nc5EfccB38nM6xrW/yxwFrAP1RVOUzHMcY77LPDnwCjVSejJfjtZyG93badiKOOMiH2opmJZDZzQsK8z2Z4TmekklL0uk+z+WQziszFVMxnnb0TEGVRT7x+fmV/uVz8zH4mIC4APRsTvZeZPm65rGloZ47jMvD8iLmXLz+dc2JaFOREO9fHViY4P97Ke6pLUZ7PlcbrxY3P9JvoDfvNl8jyqQwtTNeW7B4c1zi7dewq3UB3v3GLSxIhYQPVnXy+Y6gqGMc6IeD5wOdWhtMO6Tq430ebdoDOdhPLQiNi+67xD9ySUg/hsTNVMxglARLyX6lDfOzPz01NY92R7vG2a8Rh76N5TmAvbsjSsy6Rm8uC3l379dVf5FcD3prCcj1FdGbLLFNr8BdWGfdXWMs6utn9HdbngszvKLqKa9mReR9kx9ThfMtfHSXUO5U6qK5R2mOK6W9+eVHtmm4FndZSN1J+1/96n7d51f47vKJtHdZL2kkF+NmZznHXdd9Zjfc8U1zuv3ta3zfUx9ljeDsDtVIdU58y27NnXYa24hY12JtVNa+8CFgMfr7/0Du6q9xXg5h7t51P9daaLJ1nHt+svj9dS3WD1kfpD8aW5Pk6qk19fo7rJ6DXAwcA/120/3tV2b6qbrD5PddPVCVS7shcMenwtjHMXquvgNwJLKW822m62tyfVvRc3U90s9XqqyyGvA34IPLlrG20G3tfV/jyqQxYn1tvjX+ufzYu76jX6mQ1wm017nMCb675+qcc227Oj3pH1z+M4qpvD3gxcRRUqb57jYzwF+L9UN/gtproq8ntUe4Avn0vbsufYh7XiFjbatlSzv95GlbrfBQ7vUe+rwFiP8kPrD9hhk6zjPKrDLpvqdXwf+J+dXzhzdZxUx0kvqts9RPXlfy3VlR7b9Gj/CuDquu7PqI7DFzeXzcFxLq6340SPkWFsT6rzBp+jurny/npbjHTVGan7uLyr/IlUwXVnvT2+ASye7s9swNttWuPkt3ex93p8taPevlTnkH5GFeT3Uv1GvWQrGOPBVPdD3FX3/RdU58T+aC5uy+6HE+9Jkgpb601wkqQBMhwkSQXDQZJUMBwkSQXDQZJUMBwkSQXDQZJUMBwkSYX/D9u2/f+vyoLVAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_position = target_data[' x position (m)']\n",
    "z_position = target_data[' y position (m)']\n",
    "plt.hist(x_position, bins=10);\n",
    "plt.hist(z_position, bins=10);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The mean of x_position 0.01207963411779779\n",
      "The standard deviation of x_position 0.25716367278418045\n",
      "The first quartile for x_position -0.16854422326722185\n",
      "The second quartile for x_position 0.028452517775170515\n",
      "The third quartile for x_position 0.1670894539891312\n"
     ]
    }
   ],
   "source": [
    "mean_x = np.mean(x_position)\n",
    "std_x = np.std(x_position)\n",
    "print('The mean of x_position', mean_x)\n",
    "print('The standard deviation of x_position', std_x)\n",
    "\n",
    "quartiles_x = np.percentile(x_position, q = [25,50,75])\n",
    "print('The first quartile for x_position', quartiles_x[0])\n",
    "print('The second quartile for x_position', quartiles_x[1])\n",
    "print('The third quartile for x_position', quartiles_x[2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The mean of z_position -0.047397370492807414\n",
      "The standard deviation of z_position 0.2548611138551949\n",
      "The first quartile for z_position -0.21785495681409592\n",
      "The second quartile for z_position -0.070851135369002\n",
      "The third quartile for z_position 0.13222390283592378\n"
     ]
    }
   ],
   "source": [
    "mean_z = np.mean(z_position)\n",
    "std_z = np.std(z_position)\n",
    "print('The mean of z_position', mean_z)\n",
    "print('The standard deviation of z_position', std_z)\n",
    "\n",
    "quartiles_z = np.percentile(z_position, q = [25,50,75])\n",
    "print('The first quartile for z_position', quartiles_z[0])\n",
    "print('The second quartile for z_position', quartiles_z[1])\n",
    "print('The third quartile for z_position', quartiles_z[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Part 2-3: Monte Carlo Model**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def z_impact(theta,v_0):\n",
    "    d=3\n",
    "    g=9.81\n",
    "    z_0=0.3\n",
    "    factor = np.divide(d,np.cos(theta))\n",
    "    stuff = np.sin(theta) - np.divide(((g*d)/2),(v_0**2*np.cos(theta)))\n",
    "    return factor*stuff + z_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are generating a Monte Carlo model in order to determine what range of angle and initial velocities we can use for a robot so that it can get as close as possible to the class data. Any random number generator is only pseudo-random so we want to analyze how close we can get to modeling a truly random process such as the class data.\n",
    "\n",
    "1. First, we make a contour plot in order to determine our phase space. The goal is to find a value for the initial velocity and angle such that z_impact will be close to the mean of the class z-values above. These will also be the fixed mean values for the angle and intial velocity. \n",
    "\n",
    "2. Once we have the mean, we generate a normal distribution of the angle and intial velocity and we vary their standard deviations until the standard deviation of z-impact matches that of the class data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f1d06de5588>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "V,A = np.meshgrid(np.linspace(6,6.7,100),np.linspace(0,np.pi/12,100))\n",
    "\n",
    "Z = z_impact(A,V)\n",
    "\n",
    "plt.contourf(V,A,Z)\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.24966442830554178\n",
      "-0.04771200042249688\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": [
    "v_0 = np.random.normal(6.6, 0.695, 150)\n",
    "theta = np.random.normal(0.25, 0.0095,150)\n",
    "z_result = z_impact(theta, v_0)\n",
    "plt.hist(z_result, range = [-0.6,.5], color=\"teal\", label= \"robot\");\n",
    "plt.hist(z_position,color=\"purple\", label =\"original\")\n",
    "plt.legend(loc=\"best\")\n",
    "print(np.std(z_result))\n",
    "print(np.mean(z_result))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We obtain a standard deviation of about 0.24966 m and a mean of -0.047712 m for the the robot compared to a standard deviation of 0.25486 m  mean of -0.04739 m for the class data. Since, we are using a small number of counts for the normal distribution for the intial velocity and angle there is a lot of variation in the data however, we can see visually that the mean and standard deviation of the the two distributions is pretty close.\n",
    "\n",
    "Below, a boxplot is plotted to further indicate how the distributions compare visually. The Monte Carlo generated model is boxplot 1 and the original data is boxplot 2. We can see that the medians and quartiles are approximately the same. There is some outliers in boxplot 1 due to having more counts than the original data had. However, as the number of counts is increased, the Monte Carlo model will approach the original data but will never actually get there due to truncation errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "v_0 = np.random.normal(6.6, 0.695, 10000)\n",
    "theta = np.random.normal(0.25, 0.0095,10000)\n",
    "z_result = z_impact(theta, v_0)\n",
    "plt.figure(2)\n",
    "impacts = [z_result, z_position]\n",
    "plt.boxplot(impacts);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Part 4: BONUS**\n",
    "\n",
    "Now, we want to consider than x_impact may vary so we have to account for the different velocity components. Using our results from the previous part, we can find the $V_{z}$ and $V_{y}$ velocity components using the angle $\\theta$.\n",
    "\n",
    "1. $$V_{z} = V_{0} sin(\\theta)$$\n",
    "\n",
    "2. $$V_{y} = V_{0} cos(\\theta)$$\n",
    "\n",
    "3. $$V_{x} = \\sqrt{V_{0}^{2} - V_{y}^{2} - V_{z}^{2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def z_impact2(V_z,V_y):\n",
    "    d=3\n",
    "    g=9.81\n",
    "    z_0=0.3\n",
    "    stuff = np.divide(V_z,V_y) - np.divide((g*d/2),V_y**2)\n",
    "    return d*stuff+z_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We generate a phase space plot for the z_impact values so that we can match the mean once again, but this time our function depends on the $V_{y}$ and $V_{z}$ velocity components. We repeated the same procedure as above (once finding the mean, we generated normal distributions of the random variables and varied the standard deviations until z_impact matched the class data)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f1d06ed1d68>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "V_z,V_y = np.meshgrid(np.linspace(1,2,100),np.linspace(6,7,100))\n",
    "\n",
    "Z = z_impact2(V_z,V_y)\n",
    "\n",
    "plt.contourf(V_z,V_y,Z)\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2476576876738001\n",
      "-0.04239309514507956\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": [
    "V_z = np.random.normal(1.61, 0.4, 150)\n",
    "V_y = np.random.normal(6.3, 0.5,150) \n",
    "z_result = z_impact2(V_z, V_y)\n",
    "print(np.std(z_result))\n",
    "print(np.mean(z_result))\n",
    "plt.hist(z_result, range = [-0.6,.5], color=\"blue\", label= \"robot\");\n",
    "plt.hist(z_position,color=\"green\", label =\"original\");\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We obtain a standard deviation of around 0.2476 m and a mean of -0.0424 m for the Monte Carlo z_impact compared to a standard deviation of 0.25486 m  mean of -0.04739 m for the class data.\n",
    "\n",
    "Then, we make another countour plot to understand the phase space of x_impact which depends on random values of $V_{x}$ and $V_{y}$. From the plot we understand that $V_{x}$ should be small so that we can obtain appropriate values that allow us to match the mean of the random model to that of the class data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f1d06bf1da0>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "V_x,V_y = np.meshgrid(np.linspace(0,0.5,100),np.linspace(6,7,100))\n",
    "\n",
    "X = (V_x/V_y)*3\n",
    "\n",
    "plt.contourf(V_x,V_y,X)\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.014094491606673566\n",
      "0.2592432083999937\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"
    },
    {
     "data": {
      "image/png": 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f75zLX5j2e1vK0yjnheahwb+jaegcR/Og1AM0Dam3Tvu9TXOyi2Y6rgR20Dzc8ZM0D8qclZm392y3L6/sRvs+TT/vZcDBNK32h4ALgRvGXHMJmfl8RJwA/DXNoHhB8x/pRfnKQfGC5vvf+yn3XOCDNBfvfoImhE7OzK+Ou/bKRjwvD9PcTXY68HrgOZr759+dmV+eQPl7LAcbk6Si7IOXpKIMeEkqyoCXpKIMeEkqyoCXpKIMeEkqyoCXpKIMeEkq6v8Br8Yee08yWLoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "V_y = np.random.normal(6.3, 0.5,150) \n",
    "V_x = np.random.normal(0.046,0.54,150)\n",
    "x_impact = (V_x/V_y)*3\n",
    "plt.scatter(x_position,z_position, label = 'actual')\n",
    "plt.scatter(x_impact,z_result, label = 'random')\n",
    "plt.legend(loc = 'best');\n",
    "print(np.mean(x_impact))\n",
    "print(np.std(x_impact))\n",
    "plt.figure(2)\n",
    "\n",
    "plt.hist(x_impact);\n",
    "plt.hist(x_position);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We determine a mean of 0.0141 m and a standard deviation of 0.25924 m for x_impact compared to a mean of 0.01207 m and a standard deviation of 0.25716 m for the x-position class data. We also plotted scatter plot of the actual x vs z data and the random x vs z data and we can see that agreement is good. There are more points for the random data compared to the class data which is why not all the points match!"
   ]
  }
 ],
 "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}