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   "source": [
    "# Practice Problems\n",
    "### Lecture 16\n",
    "Answer each number in a separate cell\n",
    "\n",
    "Rename the notebook with your lastName first initial and the lecture number\n",
    "    \n",
    "    ex. Cych_B_16\n",
    "    \n",
    "Turn this notebook into Canvas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
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   "metadata": {
    "collapsed": true
   },
   "source": [
    "# 1. Linear regression\n",
    "- Read in the dataset **Datasets/SpreadingRates/km_age.txt** as a Pandas DataFrame \n",
    "    - Each data point is the age and distance from the South Atlantic ridge crests to a particular magnetic anomaly. The ages (Ma) were published by Gradstein et al. (2004) \n",
    "- Plot the age against distance; use age as the x-axis. Label your figure\n",
    "\n",
    "- Calculate the spreading rate by using the function **np.polyfit( )** and a linear fit\n",
    "- Evaluate the linear fit using **np.polyval( )**\n",
    "- Plot the best fit line as a red dashed line along with the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Lambda functions\n",
    "- Write a lambda function that converts z to velocity\n",
    "- Write a second lambda function that converts mu to distance\n",
    "- Rewrite the code cell that follows to call your new functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "16.865755533316488\n"
     ]
    }
   ],
   "source": [
    "new_data=pd.read_csv('../../Datasets/redShiftDistance/mu_z.csv',header=1)\n",
    "\n",
    "# function returns age in Ga for Ho\n",
    "age_from_Ho= lambda Ho : 1e-9*3.09e19/(Ho*np.pi*1e7)\n",
    "\n",
    "# convert z to velocity in 10^3 km/s (or 10^6 m/s)\n",
    "c=3e8 # speed of light in m/s\n",
    "new_data['velocity']=1e-6*c* \\\n",
    "    (((new_data.z+1.)**2-1.)/((new_data.z+1.)**2+1.)) # the formula for v from z (and c)\n",
    "# convert mu to distance in 10^3 Mpc (a Gpc):\n",
    "new_data['distance']=10.*(10.**((new_data['mu'])/5.))*1e-9 # convert mu to Gpc\n",
    "# and filter for the closest objects\n",
    "close_data=new_data[new_data.distance<0.7]\n",
    "\n",
    "close_fit= np.polyfit(close_data['distance'],close_data['velocity'],1) # calculate the coefficients\n",
    "close_modelYs=np.polyval(close_fit,close_data['distance']) # get the model values\n",
    "\n",
    "age=age_from_Ho(close_fit[0]) # get a new age \n",
    "print (age)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Curve Fitting\n",
    "\n",
    "- generate a set of data from 0 to 720 degrees at a spacing of 10 degree intervals.\n",
    "- convert it to radians\n",
    "- make a cosine wave from your radians array. \n",
    "- create a normal distribution generator with **stats.norm( )** that has a mean of 0 and a standard deviation of .2.\n",
    "- generate a set of simulated normally distributed \"noise\" that has the same number of data points as your cosine curve. \n",
    "- add your \"noise\" to your cosine curve. \n",
    "- use **np.polyfit** and **np.polyval** to fit a polynomial curve to your noisy cosine data (use degrees 5 to 8).   \n",
    "- Make a plot of the data with the opacity set to .5.\n",
    "- Plot  each fit (using a different colored line) and add a legend to the right of the plot using with the degree as the label. Hint: use the  **bbox\\_to\\_anchor( )** method. \n",
    "- Add the cosine curve to your plot. \n",
    "- Which degree fits  the best without over fitting? "
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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