Lab.06-Curve-Fitting.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Data Analysis and Curve Fitting\n",
    "## Lab 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Name:\n",
    "### Student ID:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Instructions:\n",
    "Skim through the entire lab before beginning.  Read the text carefully and complete the steps as indicated.  Submit your completed lab through the D2L dropbox.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Climate Change"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data in the file `global_tanomaly.dat` represents the mean annual global atmospheric temperature anomaly from 1850 to 2009. You may need to first download this file from D2L. The first column is time (year) and the second is the temperature anomaly (degree Celsius). Import this data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a scatter plot of the data (use small dots as marker symbols and do *not* connect the data points)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a linear fit to these data using the `np.polyfit` command. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the slope of the line of best fit (include units)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now make a second-order polynomial fit to the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the results of each fit onto a single figure and on top of the data. Use different colors to represent each fit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Solitons and Nonlinear Curve Fitting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A soliton is a localized pulselike wave of permanent form that exists in nonlinear systems. Solitons exists as water waves, as light pulses in fiber optics and have also been proposed as a model for elementary particles in Quantum Physics. The shape of a soliton propagating as a water wave is described by the following function:\n",
    "\n",
    "\\begin{align}\n",
    "\\eta(x) &= a\\ \\mathrm{sech}^2\\left(\\frac{x}{\\lambda}\\right) \\quad (1) \\\\\n",
    "\\end{align}\n",
    "\n",
    "where $\\eta$ is the surface displacement, $a$ is the soliton amplitude and $\\lambda$ the soliton lengthscale. \n",
    "Examine the shape of a soliton of amplitude $a$=2 m and lengthscale $\\lambda = 10$ m by executing the following commands"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 2\n",
    "λ = 10\n",
    "\n",
    "def sechf(x):\n",
    "    return a * (1 / np.cosh(x / λ))**2\n",
    "\n",
    "x = np.linspace(-50, 50, 2000)\n",
    "plt.plot(x, sechf(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The file `soliton.dat` provides measurements of the displacement of a water soliton as a function of distance. These are real measurements collected from a student for the senior laboratory course PHYS 3920 (now PHYS4900). The measurements were collected from image analysis.  If you are not on the lab computers, download this file from  D2L and import the data. The first column gives the distance in cm and the second column the vertical displacement of the water also in cm. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that the data have been imported, plot it as a scatter plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fit a nonlinear function of the form given by (1) to these measurements using the `optimize.curve_fit` command (available from `scipy`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What are the best estimates for the parameters $a$ and $\\lambda$ for the curve that best fits the data?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot your fitting function and the data together in the same plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}