ProblemSets_PartII/Math227C20Sp_P07_PowerLaws.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Math 227C Problem Set 7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How does the size of the mitochondria in a cell depend on the size of the cell itself?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2qQFkK+DHL23c4wAbfTdu/va2kOMS3MH0E+k0XkBPNMSGtzzqJwGNOckGcMyG/g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\nQAMa0IAGNNB1Bv4fe+fo4HChqz4AAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot settings\n",
    "options(repr.plot.width=4, repr.plot.height=3)\n",
    "\n",
    "# Simulate power-law data\n",
    "\n",
    "N <- 30\n",
    "alpha <- 2.3\n",
    "Y0 <- 100\n",
    "sigma <- 500\n",
    "\n",
    "CellSize <- runif(N,3,10) # independent variable\n",
    "eps <- rnorm(N,0,sigma) # residuals\n",
    "\n",
    "MitoSize <- Y0*CellSize^alpha + eps\n",
    "\n",
    "plot(CellSize,MitoSize)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1 \n",
    "\n",
    "__(a)__ Use nonlinear least squares (`nls`) to estimate the parameter $\\alpha$ from the simulated data. Use (`b=1,a=5`) as starting guesses in the `nls` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the __error__ as $e=|\\hat{\\alpha}-\\alpha|$ where $\\hat{\\alpha}$ is the estimate and $\\alpha$ is the true value (which is known because this is simulated data). \n",
    "\n",
    "__(b)__ Write a function to execute the simulation and nls fitting, taking in the true model parameters $\\alpha, y_0, \\sigma$ and the sample size $N$. Plot the error in $\\alpha$ estimation, $e$, as a function of the magnitude of the noise, $\\sigma$, for a range of $\\sigma=(100,2000)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2\n",
    "\n",
    "__(a)__ Transform the simulated data to $lY = \\mbox{log}(y)$ and $lX = \\mbox{log}(x)$. Use linear regression model fit (lm) to estimate the parameter $\\alpha$ from the simulated data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__(b)__ Write a function that executes the simulation, log-transforms the data, and performs linear regression fitting. Plot the error in $\\alpha$ estimation, $e$, as a function of the magnitude of the noise, $\\sigma$, for a range of $\\sigma=(100,2000)$. \n",
    "\n",
    "Note you may need additional code to handle negative measurements.\n",
    "\n",
    "__(c)__ Overplot the error in $\\alpha$ from nls and the error from lm (as functions of $\\sigma$). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 3\n",
    "\n",
    "In the above simulations, the independent variable $X$ was chosen from a uniform distribution.\n",
    "\n",
    "__(a)__ Find a distribution of $X$ data that makes a very big difference between error from nls and error from lm.\n",
    "\n",
    "__(b)__ Find a distribution of $X$ data that makes a very small difference between the error from nls and error from lm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "language": "R",
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