Python_FittingData.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fitting Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from astropy.table import QTable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Curve Fitting - Polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "D1 = QTable.read('./MyData/fit_data1.csv', format='ascii.csv')\n",
    "D1[0:2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(D1['x'],D1['y'],marker=\"o\",linestyle=\"None\",markersize=5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First-order fitting:  $f(x) = ax + b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Fit1 = np.polyfit(D1['x'],D1['y'],1)\n",
    "\n",
    "Fit1       # The coefficients of the fit (a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Yfit = np.polyval(Fit1,D1['x'])   # The polynomial of Fit1 applied to the points D1['x']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(D1['x'], D1['y'], marker=\"o\", linestyle=\"None\", markersize=5)\n",
    "\n",
    "ax.plot(D1['x'], Yfit, linewidth=4, color='c', linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### You can use `np.poly1d` to explore the fitted polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "poly01 = np.poly1d(Fit1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly01(5)                 # value of f(x) at x = 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly01.roots              # value of x at f(x) = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(poly01 - 20).roots       # value of x at f(x) = 20"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Second-order fitting:  $f(x) = ax^2 + bx + c$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "D2 = QTable.read('./MyData/fit_data2.csv', format='ascii.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(D2['x'],D2['y'],marker=\"o\",linestyle=\"None\",markersize=5);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Fit2 = np.polyfit(D2['x'],D2['y'],2)\n",
    "\n",
    "Fit2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Yfit = np.polyval(Fit2,D2['x'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(D2['x'], D2['y'], marker=\"o\", linestyle=\"None\", markersize=5)\n",
    "\n",
    "ax.plot(D2['x'], Yfit, linewidth=3, color='y', linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explore the fitted polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "poly02 = np.poly1d(Fit2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly02(5)                 # value of f(x) at x = 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly02.roots              # value of x at f(x) = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(poly02 - 20).roots       # value of x at f(x) = 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(poly02 - 80).roots       # value of x at f(x) = 80, no real root"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Be careful! Very high-order fits may be garbage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Fit3 = np.polyfit(D1['x'],D1['y'],10)\n",
    "\n",
    "xx = np.linspace(0,10,200)\n",
    "\n",
    "Yfit = np.polyval(Fit3,xx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.set_ylim(-20,120)\n",
    "\n",
    "ax.plot(D1['x'], D1['y'], marker=\"o\", linestyle=\"None\", markersize=8)\n",
    "ax.plot(xx, Yfit, linewidth=3, color='m', linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fitting a specific function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "D3 = QTable.read('./MyData/fit_data3.csv', format='ascii.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(D3['x'],D3['y'],marker=\"o\",linestyle=\"None\",markersize=5);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.optimize import curve_fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\Large f(x) = a \\sin(bx) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def ringo(x,a,b):\n",
    "    return a*np.sin(b*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Aguess = 75\n",
    "Bguess = 1.0/5.0\n",
    "\n",
    "fitpars, error = curve_fit(ringo,D3['x'],D3['y'],p0=[Aguess,Bguess])\n",
    "\n",
    "# Function to fit = ringo\n",
    "# X points to fit = D3['x']\n",
    "# Y points to fit = D3['y']\n",
    "# Initial guess at values for a,b = [Aguess,Bguess]\n",
    "\n",
    "print(fitpars)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Z = np.linspace(0,100,1000)\n",
    "\n",
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(Z, ringo(Z, *fitpars), 'r-')\n",
    "ax.plot(Z, ringo(Z,Aguess,Bguess), 'g--')\n",
    "\n",
    "ax.plot(D3['x'],D3['y'],marker=\"o\",linestyle=\"None\",markersize=5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bad initial guesses can lead to very bad fits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Aguess = 35\n",
    "Bguess = 1.0\n",
    "\n",
    "fitpars, error = curve_fit(ringo,D3['x'],D3['y'],p0=[Aguess,Bguess])\n",
    "\n",
    "print(fitpars)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "fig.set_size_inches(6,4)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "ax.set_xlabel(\"This is X\")\n",
    "ax.set_ylabel(\"This is Y\")\n",
    "\n",
    "ax.plot(Z, ringo(Z, *fitpars), 'r-')\n",
    "ax.plot(Z, ringo(Z,Aguess,Bguess), 'g--')\n",
    "\n",
    "ax.plot(D3['x'],D3['y'],marker=\"o\",linestyle=\"None\",markersize=5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Pandas - Plotting and data exploration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "strange_data = pd.read_csv('./MyData/anscombe.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "strange_data.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "strange_data.plot.scatter(x='x1', y='y1', color = 'r', s = 50);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "strange_data.plot.scatter(x='x1', y='y1', color = 'r', s = 50)\n",
    "strange_data.plot.scatter(x='x2', y='y2', color = 'b', s = 50)\n",
    "strange_data.plot.scatter(x='x3', y='y3', color = 'g', s = 50)\n",
    "strange_data.plot.scatter(x='x4', y='y4', color = 'k', s = 50);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ax = strange_data.plot.scatter(x='x1', y='y1', color = 'r', s = 50, label = \"set 1\")\n",
    "\n",
    "strange_data.plot.scatter(x='x2', y='y2', color = 'b', s = 50, label = \"set 2\", ax = ax)\n",
    "strange_data.plot.scatter(x='x3', y='y3', color = 'g', s = 50, label = \"set 3\", ax = ax)\n",
    "strange_data.plot.scatter(x='x4', y='y4', color = 'k', s = 50, label = \"set 4\", ax = ax);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pandas is very good at time series data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "stock_data = pd.read_csv('./MyData/goog.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stock_data[0:1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stock_data.plot(x='Date', y='Close', color = 'r');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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