gaussian_process_regression.ipynb

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   "source": [
    "# Gaussian Process Light Curve Regression\n",
    "By: Griffin Hosseinzadeh (2019 April 17)"
   ]
  },
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import george\n",
    "from astropy.table import Table\n",
    "\n",
    "%matplotlib inline"
   ]
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   "source": [
    "## Read the Light Curve\n",
    "I have given you light curve data for the Type Ibn supernova PTF11rfh in the file `PTF11rfh.table`, which is readable with the `astropy.table` module. (Hint: `format='ascii.fixed_width'`.) For this example, let's fit the light curve in counts rather than magnitudes. Select only points observed in the $r$ or $R$ filter and have a measurement in counts. Plot it to see what we're working with."
   ]
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   "source": [
    "data = Table.read('PTF11rfh.table', # complete\n",
    "data = # complete\n",
    "data.show_in_notebook()"
   ]
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   "cell_type": "code",
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   "source": [
    "x = # complete\n",
    "y = # complete\n",
    "dy = # complete\n",
    "# complete"
   ]
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   "source": [
    "## Try a Polynomial Fit\n",
    "Hint: use `np.polyfit` and `np.polyval`. Try different degree polynomials to see if you are able to get a good fit."
   ]
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   "source": [
    "x_fit = # complete\n",
    "# complete\n",
    "y_fit_poly = # complete\n",
    "\n",
    "# Plot:\n",
    "# complete\n",
    "# complete\n",
    "# complete\n",
    "# complete"
   ]
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   "source": [
    "Don't spend too much time on this. I could not get it to give a reasonable result."
   ]
  },
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   "source": [
    "## Try a Gaussian Process\n",
    "We are using a package called `george` to make Gaussian process regression easier. It has all of the functions you need build in.\n",
    "\n",
    "First we need to initialize a kernel (`george.kernels`). You should normalize the kernel by the total variance of the data points. Think about what hyperparameters would be appropriate for fitting your data. We can always optimize them later."
   ]
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   "source": [
    "kernel = # complete"
   ]
  },
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   "source": [
    "Initialize a Gaussian process with the kernel you chose. This is all contained within the `george.GP` object, which we will use for the remainder of the calculations."
   ]
  },
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   "cell_type": "code",
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   "source": [
    "gp = george.GP(# complete"
   ]
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   "source": [
    "The first step is to precompute the covariance matrix at the observed times."
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# complete"
   ]
  },
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   "source": [
    "The second and last step is to use the Gaussian process to predict the light curve at the times in `x_fit`, given the observed data points in `y`. This should give you an interpolation **and** and uncertainty on the interpolation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# complete"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the results of your Gaussian process fit and compare to the data and the polynomial fit. Hint: display the uncertainties using `plt.fill_between`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
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   "source": [
    "# complete\n",
    "# complete\n",
    "# complete\n",
    "# complete\n",
    "# complete"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Once you get a reasonable fit, try adjusting the kernel and/or hyperparameters to see what effects they have."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Take-Away Message\n",
    "Gaussian process regression is a very easy and powerful way to interpolate an arbitrary set of data points with a nonparametric function. Furthermore, it gives you uncertainties on the interpolation and handles covariance between points correctly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Other Things to Try\n",
    "- Optimize the Gaussian process using the technique described in the George tutorial: https://george.readthedocs.io/en/latest/tutorials/first/.\n",
    "- Marginalize over the hyperparameter(s) using MCMC.\n",
    "- Try products of different kernels to see if you can get a better fit."
   ]
  }
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