lectures/Week-3-Thu.ipynb

{
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     "slide_type": "slide"
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   "source": [
    "# Maximum Likelihood and Applications in Astronomy, part 2\n",
    "\n",
    "A. Connolly, M. Juric & Z. Ivezic 2018"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example 1: Fitting a Line using a Maximum Likelihood Estimator\n",
    "\n",
    "\n",
    "Assume the scatter in our measurements (the residuals) is generated by a gaussian process. I.e.:\n",
    "\n",
    ">$ y_i = a x_i + b + r_i $\n",
    "\n",
    "where $r_i$ is drawn from $N(0, \\sigma)$. Here, $\\sigma$ is the error the measurement induces.\n",
    "\n",
    "Let us compute the likelihood. First, we ask ourselves what is the probability $p(y_i|x_i, M(a, b), \\sigma)$ that a particular point $y_i$ would be measured. It is just the normal distribution:\n",
    "\n",
    ">$ p(y_i|x_i, M(a, b), \\sigma) = N(y_i - M(x)|\\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left( - \\frac{(y_i - M(x_i))^2}{2 \\sigma^2} \\right) $.\n",
    "\n",
    "Given our previous definitions we can write down the $\\ln L$\n",
    "\n",
    ">$ \\ln L(a, b) = constant - \\frac{1}{2 \\sigma^2} \\sum_{i=1}^N (y_i - M(x_i))^2 $\n",
    "\n",
    "This is the expression that we now minimize with respect to $a$ and $b$ to find ML estimators for those parameters. This is equivalent to minimizing the sum of the squares or a _least-squares method_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This astroML function adjusts matplotlib settings for a uniform feel in the \n",
    "textbook. Note that with `usetex=True`, fonts are rendered with $\\LaTeX$. This \n",
    "may result in an error if $\\LaTeX$ is not installed on your system.  In that \n",
    "case, you can set usetex to `False`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MLE with outliers\n",
    "\n",
    "Let's apply the MLE to data with uncertainties where these uncertainties include outliers. To keep this general we will use the scipy optimization function _optimize.fmin_ where we pass a function to be optimized and an initial guess (rather than a standard regression package)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 18.680770\n",
      "         Iterations: 53\n",
      "         Function evaluations: 106\n",
      "Optimization terminated successfully.\n",
      "         Current function value: 289.963723\n",
      "         Iterations: 70\n",
      "         Function evaluations: 133\n"
     ]
    },
    {
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meOQRWLkS1HgoScLtduPxeFi6dCkbNmygpqaGoqIiampqAuNDQiWY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\nIiIiMaOkIiIiMfP/Afgz3JQwNDLyAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c8fd0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from scipy import optimize\n",
    "from astroML.datasets import fetch_hogg2010test\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Get data: this includes outliers\n",
    "data = fetch_hogg2010test()\n",
    "x = data['x']\n",
    "y = data['y']\n",
    "dy = data['sigma_y']\n",
    "\n",
    "# Define the standard squared-loss function\n",
    "def squared_loss(m, b, x, y, dy):\n",
    "    y_fit = m * x + b\n",
    "    return np.sum(((y - y_fit) / dy) ** 2, -1)\n",
    "\n",
    "# define a lambda function that defines the sum of squared errors\n",
    "\n",
    "# initially exclude the outliers\n",
    "f_squared = lambda beta: squared_loss(beta[0], beta[1], x=x[4:], y=y[4:], dy=dy[4:])\n",
    "\n",
    "#include the outliers\n",
    "f_squared_outlier = lambda beta_outlier: squared_loss(beta_outlier[0], beta_outlier[1], x=x, y=y, dy=dy)\n",
    "\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# compute the maximum likelihood \n",
    "beta0 = (1, 30)\n",
    "beta_squared = optimize.fmin(f_squared, beta0)\n",
    "\n",
    "beta_squared_outlier = optimize.fmin(f_squared_outlier, beta0)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Plot the results\n",
    "fig = plt.figure(figsize=(6, 6))\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "x_fit = np.linspace(0, 350, 10)\n",
    "ax.plot(x_fit, beta_squared[0] * x_fit + beta_squared[1], '--k',\n",
    "        label=\"squared loss:\\n $y=%.2fx + %.1f$\" % tuple(beta_squared))\n",
    "\n",
    "ax.plot(x_fit, beta_squared_outlier[0] * x_fit + beta_squared_outlier[1], '--k', color='red',\n",
    "        label=\"squared loss with outliers:\\n $y=%.2fx + %.1f$\" % tuple(beta_squared_outlier))\n",
    "ax.legend(loc=4, prop=dict(size=14))\n",
    "\n",
    "ax.errorbar(x[4:], y[4:], dy[4:], fmt='.k', lw=1, ecolor='gray')\n",
    "ax.errorbar(x[:4], y[:4], dy[:4], fmt='.k', lw=1, ecolor='red')\n",
    "\n",
    "\n",
    "ax.set_xlim(0, 350)\n",
    "ax.set_ylim(100, 700)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How do we deal with outliers: modifying the likelihood\n",
    "\n",
    "The $L_2$ ($\\sum_{i=1}^N (y_i - M(x_i))^2$) norm is sensitive to outliers (i.e. it squares the residuals). A number of approaches exist for correcting for outliers. These include \"sigma-clipping\", using interquartile ranges, taking the median of solutions of subsets of the data, and least trimmed squares (which searchs for the subset of points that minimizes $\\sum_i^K (y_i - \\theta_ix_i)^2$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also change the **loss function** or **likelihood** to reduce the weight of outliers. This is known as the _Huber loss function_\n",
    "\n",
    "> $\n",
    "\\sum_{i=1}^N e(y_i|y),\n",
    "$\n",
    "\n",
    "where\n",
    "\n",
    ">$\n",
    "e(t) = \\left\\{\n",
    "\\begin{array}{ll}\n",
    "\\frac{1}{2} t^2 & \\mbox{if} \\; |t| \\leq c, \\\\\n",
    "c|t| - \\frac{1}{2} c^2 & \\mbox{if} \\; |t| \\geq c,\n",
    "\\end{array}\n",
    "\\right )\n",
    "$\n",
    "\n",
    "this is continuous and differentiable and transitions to an $L_1$ norm ($\\sum_{i=1}^N |y_i - M(x_i)|$) for large excursions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![title](https://upload.wikimedia.org/wikipedia/commons/c/cc/Huber_loss.svg)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1: Create a likelihood function that reproduces the Huber Loss function and apply it to the data with outliers\n",
    "\n",
    "Modify the _squared_\\__loss_ to transition from an L2-norm to and L1-norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 18.680770\n",
      "         Iterations: 48\n",
      "         Function evaluations: 92\n",
      "Optimization terminated successfully.\n",
      "         Current function value: 289.963723\n",
      "         Iterations: 62\n",
      "         Function evaluations: 117\n",
      "Optimization terminated successfully.\n",
      "         Current function value: 43.439758\n",
      "         Iterations: 59\n",
      "         Function evaluations: 115\n"
     ]
    },
    {
     "data": {
      "image/png": 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FCNd5NKkYhhFuGMZywzD2GIaxzTCMfnZlXQzDmGUYxnjDMKYahnGl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\nAQiQIiclnEZ9AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c9277b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from scipy import optimize\n",
    "from astroML.datasets import fetch_hogg2010test\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Get data: this includes outliers\n",
    "data = fetch_hogg2010test()\n",
    "x = data['x']\n",
    "y = data['y']\n",
    "dy = data['sigma_y']\n",
    "\n",
    "\n",
    "# Define the standard squared-loss function\n",
    "def squared_loss(m, b, x, y, dy):\n",
    "    y_fit = m * x + b\n",
    "    return np.sum(((y - y_fit) / dy) ** 2, -1)\n",
    "\n",
    "\n",
    "# Define the log-likelihood via the Huber loss function\n",
    "def huber_loss(m, b, x, y, dy, c=2):\n",
    "    y_fit = m * x + b\n",
    "    t = abs((y - y_fit) / dy)\n",
    "    flag = t > c\n",
    "    return np.sum((~flag) * (0.5 * t ** 2) - (flag) * c * (0.5 * c - t), -1)\n",
    "\n",
    "f_squared = lambda beta: squared_loss(beta[0], beta[1], x=x[4:], y=y[4:], dy=dy[4:])\n",
    "\n",
    "f_squared_outlier = lambda beta: squared_loss(beta[0], beta[1], x=x, y=y, dy=dy)\n",
    "\n",
    "f_huber = lambda beta: huber_loss(beta[0], beta[1], x=x, y=y, dy=dy, c=1)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# compute the maximum likelihood using the huber loss\n",
    "beta0 = (2, 30)\n",
    "beta_squared = optimize.fmin(f_squared, beta0)\n",
    "beta_squared_outlier = optimize.fmin(f_squared_outlier, beta0)\n",
    "\n",
    "beta_huber = optimize.fmin(f_huber, beta0)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Plot the results\n",
    "fig = plt.figure(figsize=(6, 6))\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "x_fit = np.linspace(0, 350, 10)\n",
    "ax.plot(x_fit, beta_squared[0] * x_fit + beta_squared[1], '--k',\n",
    "        label=\"squared loss:\\n $y=%.2fx + %.1f$\" % tuple(beta_squared))\n",
    "ax.plot(x_fit, beta_squared_outlier[0] * x_fit + beta_squared_outlier[1], '--k', color='red',\n",
    "        label=\"squared loss with outliers:\\n $y=%.2fx + %.1f$\" % tuple(beta_squared_outlier))\n",
    "ax.plot(x_fit, beta_huber[0] * x_fit + beta_huber[1], '-k', color='blue',\n",
    "        label=\"huber loss:\\n $y=%.2fx + %.1f$\" % tuple(beta_huber))\n",
    "ax.legend(loc=4, prop=dict(size=14))\n",
    "\n",
    "ax.errorbar(x[4:], y[4:], dy[4:], fmt='.k', lw=1, ecolor='gray')\n",
    "ax.errorbar(x[:4], y[:4], dy[:4], fmt='.k', lw=1, ecolor='red')\n",
    "\n",
    "\n",
    "ax.set_xlim(0, 350)\n",
    "ax.set_ylim(100, 700)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example 2: Optimal Photometry\n",
    "\n",
    "We now turn to examples of applications of ML in astronomy. We'll look at two cases: simple photometry and simultaneous fitting of galaxy size and total flux."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Some useful definitions and functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "from astroML.plotting.mcmc import convert_to_stdev"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plotCutout(img, title = None, clabel='$counts$', xlabel='x (pixels)', ylabel='y (pixels)', loc=None, fig=None):\n",
    "    \"\"\"Make a nice looking plot of a small image\"\"\"\n",
    "    if loc is None:\n",
    "        fig, ax = plt.subplots(figsize=(8,8))\n",
    "        fig.subplots_adjust(right=0.8, bottom=0.34)\n",
    "    else:\n",
    "        ax = fig.add_subplot(loc)\n",
    "\n",
    "    # coordinate of the edge (assuming a square image)\n",
    "    xe = img.shape[0] / 2.\n",
    "\n",
    "    if title is None:\n",
    "        title = \"min = %.0f, max=%.0f\" % (img.min(), img.max())\n",
    "    \n",
    "    ax.set_title(title, fontsize=14)\n",
    "    plt.imshow(img, origin='lower', interpolation='nearest',\n",
    "           extent=(-xe, xe, -xe, xe),\n",
    "           cmap=plt.cm.binary, aspect=1)\n",
    "    plt.clim(-20, 100)\n",
    "    plt.colorbar().set_label(clabel)\n",
    "\n",
    "    ax.set_xlabel(xlabel, fontsize=12)\n",
    "    ax.set_ylabel(ylabel, fontsize=12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## A star and a point spread function\n",
    "\n",
    "The **Point Spread Function** (the \"PSF\") is comprised of various contributions: from minute misalignments in the optics, to jitter in the tracking, to diffraction from the spider, to diffusion of the electrons as they travel through the silicon in the CCD (\"charge diffusion\"), to the homogenization of the PSF due to the refraction and diffraction in the atmosphere.  \n",
    "\n",
    "The dominant effect is the one ***from the atmosphere***.\n",
    "\n",
    "![PSF Formation](figures/psf-formation.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In practice, we don't construct the PSF from *ab initio* considerations (though there are plans in the LSST data management do so (**why?**)). Instead, we measure it by looking at the observed shape of bright, isolated, point sources (i.e., stars). Those observations are noisy, so we fit analytical profiles to estimate the true PSF.\n",
    "\n",
    "Which profiles to fit? A good description of the point spread function is the Moffat profile:\n",
    "\n",
    ">$ I = I_0 \\left[ 1 + \\left( \\frac{r}{\\alpha} \\right)^2 \\right]^{-\\beta} $\n",
    "\n",
    "where $r$ is the radius from the center of the object, $\\alpha$ sets the overall size of the PSF spot, and $\\beta$ controls the shape. This profile can be mathematically tedious to work with, so frequently a sum of Gaussians is used.\n",
    "\n",
    "For simplicity, we will use just a single Gaussian here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1q1+9//gf//Efy+z6hrL52B+peJpzm7ILhAAAAyhbkE9SbnOnTgDJq+hq/ane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\nVNMb+gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ca66198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "Atrue = 10000.0    # the source count normalization\n",
    "xdim = 15         # width of the image\n",
    "ydim = 15         # height of the image\n",
    "muXtrue = 0.0     # source x centroid \n",
    "muYtrue = 0.0     # source y centroid \n",
    "sigmaPSF = 1.5    # in pixels, corresponds to seeing = 1.5*0.2*2.355 = 0.7 arcsec for LSST (0.2\" pix)\n",
    "skyBg = 25        # sky background\n",
    "\n",
    "\n",
    "def makePSF(shape, x0, y0, sigmaPSF):\n",
    "    # Add a (Gaussian) PSF of width sigmaPSF, centered on (x0, y0)\n",
    "    xx = np.arange(shape[0]) - shape[0]/2.+0.5\n",
    "    yy = np.arange(shape[1]) - shape[1]/2.+0.5\n",
    "    \n",
    "    r = np.sqrt((xx[:, None]-x0)**2 + (yy[None, :]-y0)**2)\n",
    "    psf = np.exp(-r**2./2./sigmaPSF) / (2*math.pi*sigmaPSF)\n",
    "    return psf\n",
    "\n",
    "fig = plt.figure(figsize=(12,18))\n",
    "\n",
    "# uniform flux\n",
    "image = np.zeros((xdim, ydim))\n",
    "image[7,7] = Atrue\n",
    "plotCutout(image, loc=321, fig=fig)\n",
    "\n",
    "# random Poisson process\n",
    "image = np.zeros((xdim, ydim))\n",
    "image[7,7] = np.random.poisson(Atrue)\n",
    "plotCutout(image, loc=322, fig=fig)\n",
    "\n",
    "# PSF \n",
    "image = np.zeros((xdim, ydim))\n",
    "image = Atrue * makePSF(image.shape, 0, 0, sigmaPSF)\n",
    "plotCutout(image, loc=323, fig=fig)\n",
    "\n",
    "# PSF with background\n",
    "image += skyBg\n",
    "plotCutout(image, loc=324, fig=fig)\n",
    "\n",
    "# PSF with background and noise\n",
    "image = np.random.poisson(image)\n",
    "plotCutout(image, loc=325, fig=fig)\n",
    "\n",
    "simulatedImage = image\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Photometry (measuring the flux)\n",
    "\n",
    "We now have an image that we observe, and some ***understanding of the processes that created it***. The latter is crucial as we will eventually construct a ***model*** of this process for the ML estimate.\n",
    "\n",
    "Now we wish to estimate the flux. How should we do that? We will assume we have an estimate of the sky background"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Aperture Photometry\n",
    "\n",
    "Before coming up with a more advanced technique, we'll try something simple to estimate the flux."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11d814e48>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kXUp2InBfS/Jia+3P3evVJD9srX1WVReTfNRae29s31PbGdugBoGbqSRwA8DsGDJw303y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\nHgncAADQI4EbAAB69P8BaRS/Tuv3dNoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ca893c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Aperture photometry\n",
    "\n",
    "def apPhot(image, r = 3, x0=0, y0=0):\n",
    "    '''Integrate image given a circular aperture radius r'''\n",
    "    xx = np.arange(image.shape[0]) - image.shape[0]/2.+0.5\n",
    "    yy = np.arange(image.shape[1]) - image.shape[1]/2.+0.5\n",
    "    \n",
    "    rpix = np.sqrt((xx[:, None]-x0)**2 + (yy[None, :]-y0)**2)\n",
    "    apPhotImage = image[:]\n",
    "    apPhotImage[rpix > r] = 0\n",
    "\n",
    "    return apPhotImage.sum(), apPhotImage\n",
    "\n",
    "fig = plt.figure(figsize=(12,6))\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "aperture_radius = np.arange(1,10)\n",
    "for r in aperture_radius:\n",
    "    image = simulatedImage - skyBg\n",
    "    count, apPhotImage = apPhot(image, r=r)\n",
    "    #print ('A_ap(r=%.0f) = %.1f' % (r, count))\n",
    "    plt.plot(r, count, 'ro')\n",
    "plt.axvline(x=2.5, color='blue', ls='dashed')\n",
    "ax.set_xlim(0,10)\n",
    "ax.set_title('Growth curve', fontsize=14)\n",
    "ax.set_xlabel('Radius(pixels)', fontsize=12)\n",
    "ax.set_xlabel('Flux (counts)', fontsize=12)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## (Uncalibrated/Uncorrected) Aperture Photometry would introduce bias\n",
    "\n",
    "This is a biased estimator. The result will asymptote to a value lower than the true value (there's flux we miss in the outskirts of the object).\n",
    "\n",
    "In real-life usage, we would calibrate this out by performing *aperture correction*, or consistently using the same aperture for both calibration stars and stars of interest (under the assumption that the PSF is constant!)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## PSF Photometry: A Maximum Likelihood estimation of the flux\n",
    "\n",
    "So our **model** for the image is that it consists of a Gaussian with a flux normalization fo $A$ , plus the background (which we've removed), plus the Gaussian noise (caused by photon quantization).\n",
    "\n",
    "So, <u>once the background is removed</u>, for ***each pixel*** I can write:\n",
    "\n",
    ">$ I(x,y) = A * PSF(x - x_0, y - y_0) + r_i $\n",
    "\n",
    "where $r_i$ is drawn from a Gaussian with\n",
    "\n",
    ">$N(0, \\sigma_{x,y} = \\sqrt{I_{x,y} + B})$\n",
    "\n",
    "and $B$ is the sky background. It's critical not to forget the background when computing the sigma -- for faint sources, this is where most of the measurement uncertainty comes from!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2: Write down the likelihood and the log of the likelihood for fitting the model to the star\n",
    "\n",
    "You can assume that the sky background has been subtracted and that the centroid of the star is known"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood estimation of the flux\n",
    "\n",
    "Note this is practically identical to our case with fitting a line, except that instead of having one index $i$, we have two indices for the data points, $(x, y)$. And ***each pixel*** is an independent data point that constrains our model.\n",
    "\n",
    "So we can readily write out the log-likelihood as:\n",
    "\n",
    ">$ \\ln L(A) = constant - \\sum_{x,y} \\frac{(I_{x,y} - A \\cdot PSF_{x,y,\\sigma_{PSF}})^2}{2 \\sigma_{x,y}^2} $.\n",
    "\n",
    "The best way to think of this expression is as:\n",
    "\n",
    ">$ \\ln L(A) = constant - \\sum_{data} \\frac{(data - model)^2}{2 \\cdot variance} $.\n",
    "\n",
    "where $data$, $model$, and $variance$ are implicitly summed over whichever indices exist in the problem (e.g., $x,y$ coordinates in images, or the $i$ index in fitting a line).\n",
    "\n",
    "As we've seen last week, maximizing the likelihood of a product of normal distribution reduces to minimizing the $\\chi^2$, which is what we'll do next."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3: Solve for the maximum likelihood solution for the flux\n",
    "\n",
    "Think about expanding out the terms in the likelihood and keeping the important terms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The log Likelihood can be write as, \n",
    ">$ \\ln L(A) = C - \\sum_{x,y} \\frac{I_{x,y} A \\cdot PSF(x - x_0, y - y_0)}{\\sigma_{x,y}^2} - \\sum_{x,y} \\frac{A^2 \\cdot PSF(x - x_0, y - y_0)^2}{2 \\sigma_{x,y}^2}$.\n",
    "\n",
    "If we set \n",
    ">$\\phi_{x,y} = \\sum_{x,y} \\frac{I_{x,y} \\cdot PSF_{x,y}}{\\sigma_{x,y}^2}$\n",
    "and \n",
    ">$\\psi_{x,y} = \\sum_{x,y} \\frac{PSF_{x,y}^2}{\\sigma_{x,y}^2}$\n",
    "\n",
    "then the maximum likelihood solution for A becomes\n",
    "\n",
    ">$A_{ML} = \\frac{\\phi}{\\psi}$\n",
    "\n",
    "and the $ \\ln L(A) $ is $\\frac{\\phi^2}{2\\psi}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## If the noise is constant\n",
    "\n",
    ">$\\phi_{x,y} = \\frac{1}{\\sigma^2} \\sum_{x,y} I_{x,y}\\cdot PSF_{x,y,\\sigma_{PSF}}$\n",
    "                             \n",
    ">$\\psi_{x,y} = \\frac{1}{\\sigma^2} PSF_{x,y,\\sigma_{PSF}}^2$\n",
    "\n",
    "\n",
    "Looking at the form of $\\phi_{x,y}$ and considering it for each pixel **what does it represent?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">$\\phi_{x,y} \\equiv \\frac{1}{\\sigma^2} I_{x,y} \\otimes PSF(x, y)$\n",
    "\n",
    "this means that peaks in the likelihood are given by peaks in the image convolved with the PSF and that the height of these peaks (normalized by the size of the PSF squared gives the brightness of the source)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Estimating the true flux\n",
    "\n",
    "To estimate the true flux, we need to maximize the likelihood. That is equivalent to minimizing the $\\chi^2$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A_ML = 10023.0 with chi^2/dof = 1.07\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11d8b8940>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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bQ6FX7PRplsICmLvM9ZprpLvvJmgMAmEDpVOrhZ0aR0fDzyyFBcAy12wRNlBK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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ca53630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def psfPhot(image, Amin, Amax):\n",
    "    \"\"\" Perform PSF photometry with a PSF=N(x=0,y=0,\\sigma=sigmaNoise)\"\"\"\n",
    "    # Estimate the noise in the image\n",
    "    sigmaNoise = np.sqrt(image)\n",
    "\n",
    "    # Subtract the background (we assume we've estimated it somehow)\n",
    "    image = image - skyBg\n",
    "\n",
    "    testA = np.arange(Amin, Amax)\n",
    "    chi2dof = np.empty(testA.shape)\n",
    "    best = (np.nan, np.inf)\n",
    "    psf = makePSF(image.shape, 0, 0, sigmaPSF)\n",
    "    for i, A in enumerate(testA):\n",
    "        model = A*psf\n",
    "        chi2 = np.sum(((image-model)/sigmaNoise)**2)\n",
    "        chi2dof[i] = chi2 / (image.size - 1)\n",
    "        if chi2dof[i] < best[1]:\n",
    "            best = (A, chi2dof[i])\n",
    "    return best, testA, chi2dof\n",
    "\n",
    "image = simulatedImage\n",
    "\n",
    "((bestA, bestChi2dof), testA, chi2dof) = psfPhot(image, Atrue - 200, Atrue + 200)\n",
    "print (\"A_ML = %.1f with chi^2/dof = %.2f\" % (bestA, bestChi2dof))\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,8))\n",
    "ax.plot(testA, chi2dof, '.')\n",
    "ax.set_xlabel(r\"$A$\")\n",
    "ax.set_ylabel(r\"$\\chi^2/dof$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## And we can also plot the likelihood instead"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x11d64e4e0>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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yAQDgBb3UH0W+Vn/3bpL5AACQAp7xLy76RL6FBbryAQBQF2Tgb1Wy\nR1c+AAACXOpvFfTHxnyDnrNnCfpAR+jVDwQrqKz+KJJ27JAuXFh+jJI9oJjI6gdWonNfl6jTBwCg\nvWAC/+KidPDgciLf2BhBHwCAZkEk91GnDwBAZ0o/4+emOwAAdK7Ugb9V0KdOH0gAvfqBYJUyqz+K\n/P79wYPLvfclkvmAxNCrH8gFvfpbaNeRj6APAEB7pQr8dOQDAKA/pdnjpyMfAAD9K8WMf7Ukvjff\nJOADANCNws/4Vwv67OcDKaJXPxCswmf1Dw87LS4uP0bQB8JAVj+wUhZZ/YUP/NLy+Aj6QDgI/MBK\n3KSnAUEfAID+lSLwc8MdAACSUfis/ulpn9xHC14AAPpX+Bn/xARBH8gcvfqBYBU+ua/I4wOCRa9+\nIBck9wEAgEQR+AEAqBACPwAAFULgBwCgQgj8AFaiVz8QLLL6AeSCrH5gJbL6AQBAogj8AABUCIEf\nAIAKIfADAFAhBH4AK9GrHwgWWf0AVqJXP5ALsvoBAECiCPwAAFQIgR8AgAoh8AMAUCEEfgAr0asf\nCBZZ/QByQVY/sBJZ/QAAIFEEfgAAKoTADwBAhRD4AQCoEAI/gJXo1Q8Ei6x+ACvRqx/IBVn9AAAg\nUQR+AAAqhMAPAECFEPgBAKgQAj+AlejVDwSLrH4AuSCrH1iJrH4AAJAoAj8AABVC4AcAoEII/AAA\nVEjmgd/M1pvZMTN7zMyeMbOJrMcAYA306geClXlWv5mdlPSqc+6H8fcXJH3ZOfdui58lqx/IA736\ngVwEl9VvZjskPSjpZMPD05KezHIcRTAzM5P3EFLF8ZXbTN4DSFnI5y/kY5PCP74sZL3Uf4ekj51z\nVxsee0fSvRmPI3eh/+Xl+MptJu8BpCzk8xfysUnhH18W8gj8l5seuyxpQ8bjAACgksjqBwCgQjJN\n7jOzA5KOOOc+0+6xhudI/QEAVErayX2Dab54C2e0cln/DvkEvxXSPngAAKom06V+59wvJE2b2QMN\nD09IejzLcQAAUFV51PGvk/SspPclfV7Sa865H2c6iIIys1Hn3ELe40BvOH8AyiCTpX4zG5V0UD7Y\n3yHpeDz7l5mNmtkxSRckbZL0jnPu+fi59fIXCRckfUbStHPu9bWey9JaxybpW2pxbPHzL8r3NZAk\nF/9+YY4tHstax7fac4U/d/FYejq++PnCnj8zW++cu5Lle2YpzeMrwgVcr8dXlvOe5vHlff5KcQ6c\nc6l/yZfsbYz/PCrpE0nr4u8vSPrLpp+9Lf7zSUkPNzx3oZPnsvzq49jWSzoq6S8l3SZppOHnCnFs\nHRxfu+cKf+76PL5Cnj/5rbMLkh5reny9pGOSHpP0jKSJfp/r5PmiH1/8/Ivxuf1E0rXGc1WG4+vz\n90p9fEU6fz3+3RyVz337JP7v7Vmcu9ROcMPgvizpN02PnZH0TPznTyTd0/Dce5JGJO2IT+K6hueO\nSXpB0u1q+BBufC7t40ni2BrGe0TSaNPvr3rcWR7bWscnP9Nd7blVz09Rzl0/x1fU8ydpXfxhcLnF\nh09PF2Ltnuvk+RIc36oXcCU6vp5+L5DjK8T56+XY4p8/2TD2M2r47Ejz3GWR3LdTK5v2XJT/gJSk\nH0l60cxuj0v7pp1zl+LfW63L305JH63yXJZ6PTZJGpN0QNL7ZnYmXlaWitXdsN3x3dnmuXbnpyjn\nTur9+KQCnj/n3FXnlxhvGHe7Vtlmdrv8BVBXz631uokcUJOkjy/+83ckXZF01Tn3bsO/z1IcXz+/\nV/bjiz2rApy/Ho7t24oT251z/+38vWomJJmknWn/28si8L8v/yHZ6HL9MefcQ/IfjO9Iutc59/X4\nZ3aodZe/T6s4HQB7PTY55/7W+d4F98pf+b0aP1WUY5PaH98F+eTMVs+tdgxFOndS78dXlvNX1+5i\npJ+LtHYX51nq9fgkv9Ta6gJurdfNUq/jWOv3ynD+1rLaBXi/r5uU1cYw4Zz7ceOFSnzh4ORn/ncq\nxX97WQT+k5JkZs/E/10vP7iL8fcb5D+An5X05frPraEojX26PbYjzS/gnPup/EkcM7N7Mhp3p9od\n34uS3GrH3kZRzp2UwPEV/PzV9XIh1slF2moX51lf5PR8DG0u4Nq9blGOb61xlP38rTmOEpy/jsdg\nZvfKrwpf1drnpq9zl3rgj69i7pC0w8xekPRI/FT9BE1LOuace1LSQ5IeN7OH5a9ebm56uZslfdTm\nuY+TP4LV9XFsrV7ndfmTVohjaxhXq+N7rd1zKsG5k/o6vlavU7jz16F2F2JFukjrVUfHUJILOKwi\nkPN3QNJXsnijTBr4xHsvf+ec2yf/ATkq6bl4FrXDOfff8c+9JOkl+Q/cdl3+3mnzXKa6PLYfxeNs\n5WZJZ9Vld8O0rXZ8bZ47rpKcO6mn43tulZcq5PmL9XIh1slFWlEuchK50Gy6gGv3ukU5vrXGUfbz\n19U4Cnr+OhqDmX1TPnH4aoe/19ex5XGTnpOSHnHORfGJ+sjMbmt4foOkM65Nl792z6U9+DV0cmzv\nxL0L6vXfihM53nbOXSrwsUnLx3e1zXNRSc+d1NnxXS3h+ev1Qmyti7SiXOQkeaFZv4Bb63Wz1Os4\n1vq9sh9fK0U7f2uOIf4seS1O8Ov09/o7tk5S/5P40nK94lebHr9NvszrMfnyqGcanlsnX6rxWPzf\nBzp5Luuvbo9NvqTtgnx532PNYy/SsbU7vjWOvRTnrpfjK/r5U+ta4lNN5+A9Lfco6Pa5kU5et+DH\nNxKf2webzuvRTl+3KMfXw++V6vyt9ntFPH89/N28V36JfzT+2iF/07pUz10mLXvjUjYn6aRrPZsq\nrZCPTWp/fCEce0jHF28vPSTfN+CspG85v/fZtlV2r8918nyRjy9eqXlR/hwfl3SxeewlOb6efq/s\nx1ek89fLsZnZhHzOUHMQftw596/xax5Zbez9HFvmvfoBAEB+8tjjBwAAOSHwAwBQIQR+AAAqhMAP\nAECFEPgBAKgQAj8AABVC4AfQkpm9uvZPASgbAj+AFeI7hU3ETUIABITAD6CV+i2kd+Y6CgCJI/AD\nuEF805Cp+NuxPMcCIHkEfgDNnnDOfVfSgla/jTSAkiLwA7guvjFRfbZ/Ucz4geAM5j0AAIVy0DlX\n39e/KGkiz8EASB6BH4Akycwel+TM7Kgkk0/sY8YPBIbb8gKo30982jl3Z8NjE5JelfR559ylvMYG\nIFns8QOQpO9IOt702IL8zH9H9sMBkBYCP1Bxcfnew5I2NDy2XtKDkpykJ81sJJfBAUgcS/0AAFQI\nM34AACqEwA8AQIUQ+AEAqBACPwAAFULgBwCgQgj8AABUCIEfAIAKIfADAFAhBH4AACrk/wMxfDyF\nuvJgcwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ca53860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lnL = np.exp(-0.5*(chi2dof*(image.size-1)))\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,8))\n",
    "ax.plot(testA, lnL, '.')\n",
    "ax.set_xlabel(r\"$A$\")\n",
    "ax.set_ylabel(r\"$const + L(A)$\")\n",
    "\n",
    "ax.axvline(x=bestA)\n",
    "ax.axvline(x=Atrue, color='red', linestyle='--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4: Calculate the maximum likelihood solution for the flux\n",
    "\n",
    "Using the $\\phi$ and $\\psi$ formalism above estimate the PSF fulx using our MLE. For simplicity you can assume a constant noise and that you know the position of the star.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The MLE estimator of the source flux is 10058.2\n"
     ]
    }
   ],
   "source": [
    "image = simulatedImage - skyBg\n",
    "\n",
    "# create PSF centered on the star\n",
    "psf = makePSF(image.shape, 0, 0, sigmaPSF)\n",
    "\n",
    "#assume the sky variance is constant\n",
    "phi = psf*image/skyBg\n",
    "psi = psf*psf/skyBg\n",
    "\n",
    "A = phi.sum()/psi.sum()\n",
    "print ('The MLE estimator of the source flux is %g'%A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Models with more parameters\n",
    "\n",
    "The iterative example above is a bit contrived as we can, and did, solve the MLE analytically\n",
    "\n",
    "It has the benefit of being general, however. We can apply the same technique to models with more parameters (within the limits of computing power available to us!). The other parameters could be the position of the object ($x$, $y$), or some measure of the shape of the object.\n",
    "\n",
    "As the number of parameters increases, the likelihood curve from the previous slide becomes the ***likelihood (hyper)surface***. The more dimensions there are, the more difficult (computationally expensive) it becomes to find its maximum using brute force solutions; even for 2D cases, we're likely to resort to ***minimization algorithms*** (e.g., Levenberg–Marquardt or others)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example 3: Measuring the size and brightness of a spherical, gaussian, galaxy\n",
    "\n",
    "Let's explore a two-dimensional case.\n",
    "\n",
    "Imagine that, instead of a star, we're observing a galaxy. Even in the ideal (space) case, the Galaxy will not be a point source. We'll imagine we're dealing with a \"Gaussian spherical galaxy\" -- i.e., one whose intensity  falls of as a 2D gaussian would. As with the star, we'll asume the position of the galaxy is known.\n",
    "\n",
    "Our model has two parameters: the measure of the extendedness of the galaxy -- $\\sigma$ -- and its total flux, $C$:\n",
    "\n",
    ">$ I(x, y) = \\frac{C}{2 \\pi \\sigma^2} \\exp \\left( -\\frac{(x-x_0)^2 + (y-y_0)^2}{2\\sigma^2} \\right) $\n",
    "\n",
    "Our code will now have to explore the likelihood surface in $(C, \\sigma)$ space, and find the point of its maximum.\n",
    "\n",
    "Note #2: When generating the simulated image, $I(x, y)$ needs to be ***convolved*** with the PSF. Fortunately (actually, on purpose!) we've chosen the galaxy profile to be gaussian, and the convolution of two gaussians with variances $\\sigma_1^2$ and $\\sigma_2^2$ is also a gaussian with the variance $\\sigma_1^2 + \\sigma_2^2$ (i.e., it's wider). Our function `gauss2D` utilizes this fact to generate the convolved image of the galaxy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Helper functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def FourPanelPlot(nonoise, psf, image, diffimage):\n",
    "    \"\"\"\n",
    "        Make a 4-panel plot of the noiseless image, the psf, the image with noise, and the difference.\n",
    "    \"\"\"\n",
    "    ## nonoise -- noiseless image, N x N array\n",
    "    ## psf -- the image of the PSF, N x N array\n",
    "    ## image -- image with the added noise, N x N array\n",
    "    ## diffimage -- difference image (image - psf)\n",
    "\n",
    "    fig = plt.figure(figsize=(8, 8))\n",
    "    fig.subplots_adjust(left=0.13, bottom=0.15, right=0.95, top=0.90, wspace=0.18, hspace=0.46)\n",
    "\n",
    "    images = [\n",
    "        (221, nonoise, \"noiseless image\",  ['',          r' ',          r'y (pixels)']),\n",
    "        (223, psf,     \"psf image\",        ['',          r'x (pixels)', r'y (pixels)']),\n",
    "        (222, image,   \"image with noise\", [r'$counts$', r' ',          r'']),\n",
    "        (224, diffimage, \"image - psf\",    [r'$counts$', r'x (pixels)', r''])\n",
    "    ]\n",
    "\n",
    "    for (loc, img, title, (clabel, xlabel, ylabel)) in images:\n",
    "        plotCutout(img, title, clabel, xlabel, ylabel, loc, fig)\n",
    "\n",
    "    #plt.savefig('4panels.png')\n",
    "\n",
    "def plotLikelihood(chiPixSig, chiPixCmod, lnL, sigtrue, sigmaML, CmodML):\n",
    "    \"\"\"Plot the likelihood densities\"\"\"\n",
    "    fig = plt.figure(figsize=(8, 8))\n",
    "    fig.subplots_adjust(left=0.08, bottom=0.15, right=0.95, top=0.90, wspace=0.29, hspace=0.46)\n",
    "\n",
    "    ## renormalize and truncate for clean plotting\n",
    "    lnL -= lnL.max()\n",
    "    lnL[lnL < -6] = -6\n",
    "\n",
    "    ## lnL image \n",
    "    ax = fig.add_axes([0.35, 0.35, 0.45, 0.6], xticks=[], yticks=[])\n",
    "    ax.set_title('ln(L) image', fontsize=14)\n",
    "    # pretty color map\n",
    "    plt.imshow(lnL, origin='lower', extent=(chiPixSig[0], chiPixSig[-1], chiPixCmod[0], \\\n",
    "                                            chiPixCmod[-1]), cmap=plt.cm.RdYlGn, aspect='auto')\n",
    "\n",
    "    # colorbar\n",
    "    cax = plt.axes([0.82, 0.35, 0.02, 0.6])\n",
    "    cb = plt.colorbar(cax=cax)\n",
    "    cb.set_label(r'$lnL(\\sigma, C_{mod})$', fontsize=14)\n",
    "    plt.clim(np.min(lnL), np.max(lnL))\n",
    "\n",
    "    # contours\n",
    "    # plt.contour(chiPixSig, chiPixCmod, convert_to_stdev(lnL), levels=(0.683, 0.955, 0.997), colors='w')\n",
    "\n",
    "    # mark true values     \n",
    "    ax.plot(sigtrue, 1000.0, 'o', color='red', alpha=0.75)\n",
    "    # mark ML solution: (sigmaML, CmodML)\n",
    "    ax.plot(sigmaML, CmodML, 'x', color='white', alpha=0.99, lw=35)\n",
    "\n",
    "    # marginal projections\n",
    "    p_sigma = np.exp(lnL).sum(0)\n",
    "    p_Cmod = np.exp(lnL).sum(1)\n",
    "  \n",
    "    ax1 = fig.add_axes([0.35, 0.1, 0.45, 0.23], yticks=[])\n",
    "    ax1.plot(chiPixSig, p_sigma, '-k')\n",
    "    ax1.set_xlabel(r'$\\sigma$ (pixel)', fontsize=12)\n",
    "    ax1.set_ylabel(r'$p(\\sigma)$', fontsize=12)\n",
    "    ax1.set_xlim(np.min(chiPixSig), np.max(chiPixSig))\n",
    "\n",
    "    ax2 = fig.add_axes([0.15, 0.35, 0.18, 0.6], xticks=[])\n",
    "    ax2.plot(p_Cmod, chiPixCmod, '-k')\n",
    "    ax2.set_ylabel(r'$C_{mod}$ (counts)', fontsize=12)\n",
    "    ax2.set_xlabel(r'$p(C_{mod})$', fontsize=12)\n",
    "    ax2.set_xlim(ax2.get_xlim()[::-1])  # reverse x axis\n",
    "    ax2.set_ylim(np.min(chiPixCmod), np.max(chiPixCmod))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Functions that simulate the image, and compute the likelihood surfaces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "### computations \n",
    "def gauss2D(muX, muY, sig, A, skyBg, sigPSF, Xpixels, Ypixels):\n",
    "    \"\"\"\n",
    "    Generate an image of size (Xpixels, Ypixels) with\n",
    "    a 2D circular Gaussian of total flux A\n",
    "    with standard deviation sigma^2=sig^2 + sigPSF^2\n",
    "    superimposed on a uniform background B and centered\n",
    "    on (muX, muY).\n",
    "    \"\"\"\n",
    "    r = np.sqrt((Xpixels-muX)**2 + (Ypixels-muY)**2)\n",
    "    # make and set image to the background value\n",
    "    image = np.empty(r.shape)\n",
    "    image.fill(skyBg)\n",
    "\n",
    "    ## now add circular gaussian profile (normalized to A)\n",
    "    # source gauss convolved with single-gauss PSF  \n",
    "    sigConvSquare = sig**2 + sigPSF**2\n",
    "    image += A*np.exp(-r**2/2/sigConvSquare) / (2*math.pi*sigConvSquare)\n",
    "    return image\n",
    "\n",
    "\n",
    "def addnoise(inimage, sigNoise, sourceImage, addsourcenoise=0): \n",
    "    \"\"\"Add gaussian noise to the image, and return the image and variance plane\"\"\"\n",
    "    image = np.copy(inimage)\n",
    "    image += np.random.normal(0, sigNoise, image.shape)\n",
    "    variance = 0*image + sigNoise**2\n",
    "\n",
    "    if (addsourcenoise):\n",
    "        gain = 1.0  # as a reminder...\n",
    "        sourceVariance = sourceImage/gain\n",
    "        image += np.random.normal(0, np.sqrt(sourceVariance), image.shape)\n",
    "        variance += sourceVariance\n",
    "\n",
    "    return image, variance \n",
    "\n",
    "\n",
    "### make chi2(sigma, Cmod) image, find the ML solution, and return the best-fit model image \n",
    "def getlnLimage(xpix, image, muX, muY, skyBg, sigmaPSF, sigmaNoise): \n",
    "    \"\"\"\n",
    "    Compute the log-likelihood image.\n",
    "    \"\"\"\n",
    "\n",
    "    # a little helper \n",
    "    def lnLinit(yPixels, xPixels):\n",
    "        lnL = np.zeros(xPixels.shape, dtype=float) - 0 * np.log(yPixels)\n",
    "        return lnL\n",
    "\n",
    "    ## first make the log likelihood (lnL) image\n",
    "    # define the grid\n",
    "    chiPixelsSigma = np.linspace(0, 2.5, 101)\n",
    "    chiPixelsCmod = np.linspace(500, 1500, 101)\n",
    "    lnL = lnLinit(chiPixelsCmod[:, None], chiPixelsSigma)\n",
    "\n",
    "    # brute force loop over all sigma and Cmod\n",
    "    chi2min = np.inf\n",
    "    sigmaML = -1.0\n",
    "    CmodML = -1.0\n",
    "    for i in range(0, chiPixelsSigma.size):\n",
    "        sigma = chiPixelsSigma[i]\n",
    "        for j in range(0,chiPixelsCmod.size):\n",
    "            Cmod = chiPixelsCmod[j]\n",
    "            # model image: a circular 2D gaussian \n",
    "            model = gauss2D(muX, muY, sigma, Cmod, skyBg, sigmaPSF, xpix[:, np.newaxis], xpix)\n",
    "            thisChi2 = np.sum(((image-model)/sigmaNoise)**2)\n",
    "            lnL[j][i] = thisChi2   # NB this is chi2, not yet lnL (see below)\n",
    "            if (thisChi2 < chi2min): \n",
    "                chi2min = thisChi2\n",
    "                sigmaML = sigma\n",
    "                CmodML = Cmod\n",
    "                \n",
    "    ## go from chi2 to ln(L) \n",
    "    # since L = exp(-chi2/2): lnL = -1/2 * chi2\n",
    "    lnL = -0.5*lnL\n",
    "            \n",
    "    # ML model\n",
    "    bestModel = gauss2D(muX, muY, sigmaML, CmodML, skyBg, sigmaPSF, xpix[:, np.newaxis], xpix)\n",
    "    # chi2 per degree of freedom\n",
    "    chi2minDOF = chi2min / (xpix.size**2 - 2)\n",
    "    return chiPixelsSigma, chiPixelsCmod, lnL, bestModel, sigmaML, CmodML, chi2minDOF"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Driver functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def simulate(muXtrue, muYtrue, sigtrue, Atrue, skyBg, sigmaPSF, sigmaNoise, seed=42):\n",
    "    # set seed\n",
    "    np.random.seed(seed)\n",
    "    \n",
    "    # define the (square) grid\n",
    "    xpix = np.linspace(-7, 7, 15)\n",
    "\n",
    "    ## make psf (sigtrue=0) \n",
    "    psf = gauss2D(muXtrue, muYtrue, 0, Atrue, skyBg, sigmaPSF, xpix[:, np.newaxis], xpix)\n",
    "\n",
    "    ## make noiseless image (convolved with psf given by sigmaPSF, image size given by 1Dpixels) \n",
    "    nonoise = gauss2D(muXtrue, muYtrue, sigtrue, Atrue, skyBg, sigmaPSF, xpix[:, np.newaxis], xpix)\n",
    " \n",
    "    ## now add noise\n",
    "    image, variance = addnoise(nonoise, sigmaNoise, 0) \n",
    "\n",
    "    ## difference object - psf\n",
    "    diffimage = image - psf\n",
    "\n",
    "    return nonoise, psf, image, diffimage\n",
    "\n",
    "def printMLResults(sigmaML, CmodML, chi2minDOF, bestModel, image, psf):\n",
    "    print ('the Maximum Likelihood point:')    \n",
    "    print ('sigmaML = ', sigmaML)\n",
    "    print (' CmodML = ', CmodML)\n",
    "    print ('chi2DOF = ', chi2minDOF)\n",
    "\n",
    "    ## compute Cmod from bestModel and compare to CmodML\n",
    "    modelNorm = bestModel / np.sum(bestModel)\n",
    "    wModel = modelNorm / np.sum(modelNorm**2)\n",
    "    CountModel = np.sum(image*wModel)\n",
    "    print ('Cmod from bestModel profile: ', CountModel)\n",
    "\n",
    "    ## compute Cpsf from the PSF profile \n",
    "    psfNorm = psf / np.sum(psf)\n",
    "    wPSF = psfNorm / np.sum(psfNorm**2)\n",
    "    CountPSF = np.sum(image*wPSF)\n",
    "    print ('Cpsf from the PSF profile: ', CountPSF)\n",
    "\n",
    "def demo(muXtrue, muYtrue, sigtrue, Atrue, skyBg, sigmaPSF, sigmaNoise, seed=42):\n",
    "    # define the (square) grid\n",
    "    xpix = np.linspace(-7, 7, 15)\n",
    "\n",
    "    nonoise, psf, image, diffimage = simulate(muXtrue, muYtrue, sigtrue, Atrue, skyBg, sigmaPSF, sigmaNoise, seed)\n",
    "    chiPixSig, chiPixCmod, lnLimage, bestModel, sigmaML, CmodML, chi2minDOF = getlnLimage(xpix, image, muXtrue, muYtrue, skyBg, sigmaPSF, sigmaNoise)\n",
    "\n",
    "    # visualize results\n",
    "    FourPanelPlot(nonoise, psf, image, diffimage)\n",
    "    printMLResults(sigmaML, CmodML, chi2minDOF, bestModel, image, psf)\n",
    "    plotLikelihood(chiPixSig, chiPixCmod, lnLimage, sigtrue, sigmaML, CmodML)\n",
    "\n",
    "    return xpix, nonoise, psf, image, diffimage\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sanity check: a star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the Maximum Likelihood point:\n",
      "sigmaML =  0.0\n",
      " CmodML =  1010.0\n",
      "chi2DOF =  0.944291679508\n",
      "Cmod from bestModel profile:  6621.8036442\n",
      "Cpsf from the PSF profile:  6629.48567646\n"
     ]
    },
    {
     "data": {
      "image/png": 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bYSol/QA67H3K/RmsZ7IWwPZyRcd2SV2EmbiTaFyKaDrJaIy5KBj/RBHTIf5V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\npKSfkbSSCtm9Jv0ZSddtH5Z0RNLlnAqlRUn3tPUO8NK4sbe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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d814240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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7bjO7ltBNPVtDYvH9PTPT+93avhnHHAQ+7u5PxjbxXuC65oi55bJimYHYP3ImcCgOrTgD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\nAOQcYQAAgJz7/81HTH77DjkKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11cb74dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sigmaPSF = 1.5    # in pixels, corresponds to seeing = 1.5*0.2*2.355 = 0.7 arcsec for LSST (0.2\" pix)\n",
    "muXtrue = 0.0     # source x centroid \n",
    "muYtrue = 0.0     # source y centroid \n",
    "sigmaNoise = 5.0  # gaussian sigma for white noise (counts), e.g. due to sky \n",
    "sigtrue = 0.0     # the intrinsic gaussian source width, sigma (in pixels) \n",
    "Ctrue = 1000.0    # the source count normalization\n",
    "skyBg = 25        # sky background\n",
    "\n",
    "demo(muXtrue, muYtrue, sigtrue, Ctrue, skyBg, sigmaPSF, sigmaNoise);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the Maximum Likelihood point:\n",
      "sigmaML =  0.9\n",
      " CmodML =  970.0\n",
      "chi2DOF =  0.939255229631\n",
      "Cmod from bestModel profile:  6617.82108974\n",
      "Cpsf from the PSF profile:  6440.93837768\n"
     ]
    },
    {
     "data": {
      "image/png": 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eLTCVkn4AbfY+5f4M1jPZAGBXuaJju6SuwEzcSTQuRTSbZDTGXBSMf6KI2RD/\nqlquKzhJOsvnLIAzMAukNEdXerUryh2DWXs/tfgsXrvAmwLgrY4UJM836soU45+oUkjxH5e0YrJb\nVZ+b4r0ryGH1ONsF1M9uXArVbGgxpcX4p9CFHP9J55hMVSkpj2tN+X4VtmHifdcVent7x57H7/Em\nqrdSqYRSqZT6/JAvTEgQ/z/4wQ/Gnq9YsQIrVqzIu0xEib3xxht44403Up8fcvznuo6JiGyDuVd6\nspvUxR7vjE+gFZGnAeydqbXU0dGRUUmJshevLPf19TmdX/Su3Lzj/ytf+UpGJSXK3sqVK7Fy5cqx\n19GKdBJFj/9q5FoxsbNwE8/EBcZ2LD7KzQEpdEVvMTH+idIrevxXw6sqmR1XbgIwYtc0WCMi3P2U\nguS6wFLRMf6JxoUW/1HeLElvbw88gondvlyZkoIUUouJ8U9UKaT4j/OmYqKqx+BJD47rL4RL7dV1\nT5jGRrddqV3yd90TxGUPCdfv6Zre5Wde1ACf7a2iqDTxv2jRosRpXX7XXfdWcdmH5+LFi055u3rg\ngQcSp3W9trj8XFz+bwBgdNRpYVCn/0+X6xYA/PrXv06c9pFHHnHK20VI8R8X7jcn8lyKJamj5zYk\nyL85s8ISUaZCjn9WTIg8lWaMWUTWi8ggzO228fcO2X1obtm9K2a8eBFRfYQc/94M5RBRJdchKBFZ\nBLNSatMk7zUAuASzb4UAGM5qnysiyl7I8c+KCZGnUmzidRUARGRokrc7AQwDuGoXRSQij4Uc/xzK\nIfJUNWPMk2iB6d49IyIDPo8vE1HY8c+KCZGnslzHQFW/pKpLYHbxbYC5NZeIPBVy/LNiQuSpjFtM\nAABVfRlAO4AWEVmXaYGJKDMhxz/nmBB5Kn7x+elPf4qf/exnVeerqiMicgyA20IWRFQzIcc/KyZE\nnop3165atQqrVq0ae/3d7363muybAJysJgMiyk/I8c+hHCJPZdWVa/ed2Rh5vRrAcZ9vFyQKXcjx\nz4oJkadcJ7+JSIOIbAPQDGBrZAy5EUCniJwWkZ0AmlV1R+2+CRG5Cjn+OZSTAZcZ0q77Nrju2zFv\n3rzEaVXj+6VNz6WWPnfuXKe8XWeZh7CPhOsEN1UdAdBtH9HjrwJYnl3J/ODyO+CS9p133nEqh8s+\nT8uWLXPK++2333ZKf/vttydO+4lPfMIp72vXriVOe+PGDae8Xa8X9913X+K0H330kVPeCxcuTJz2\n1KlTTnk+RBYtAAAgAElEQVS7CDn+va2Y2O3OVVWfqXdZiOohhMrXVBj/FLqQ49/Lb263QN9U73IQ\n1VMetwsWAeOfKNz4BzzsMbEr0rUAOFHvshDVU4gtJsY/kRFi/Jf5+M23q2o3zEZDRMEKtMXE+CdC\nsPEPwLMeExHZC+DZepeDyAehtZgY/0TjQov/KG8qJvY2px5VTTz1u7e3d+x5a2srWltb8ygaUSql\nUgmlUin1+bO9VRSVJv4PHz489vyzn/0sPvvZz+ZRNKJUTp06hdOnT6c+P6T4j8u1YmIvNm0AJrsv\nVezxTrvQy24AzbH/DBGRXao66b1kHR0d2RaYKEPxynJfX5/T+UW/MOUd/5s2cX4s+WvFihVYsWLF\n2Osf/vCHTucXPf6rkWvFxI4Vd8+Y0KStuM9aRHoAnOHtghSqonflMv6J0it6/FfDm6EcIqoUcouJ\nKHQhx7/PFRO3ZUmJZpmQW0xg/FPgQo5/bysmqrq13mXIg+svm2ut2WVp5zyXpHcVcutgKiH/TJLE\nv8vv77vvvps4retS7XfddVfitBcvXnTK+5577nFKPzw8nDit6/dctGhRLuUA3JeNv379euK0rtuA\nuGzrsWTJEqe8XYQc/95WTIhCl6bFJCINADoBDAJYAqBfVY9lXDQiylnI8c+KCZGnUraYugEcUdWD\nNo9BEdmkqq9lWjgiylXI8R/uIBaR51Jse74GwEYAPZHD/QB4ZwtRwYQc/6yYEHkqxZLUbQCGVfVq\n5NgJABtqUmAiykzI8c+hHCJPpRhjbgMwFDs2BKAxkwIRUc2EHP+smBB5KuRZ+UShCzn+WTEh8lS8\nxfTKK6/g+PHj051yAsDm2LEmAG73bhJR3YUc/6yYEHkq3mJ69NFH8eijj469/ta3vhU/ZQATu23b\nYCbAEVGBhBz/nPxK5CnXyW+q+iqAfhF5PHJ4PYBdtSozEWUj5PhnjwmRp1IuSb0ZQKeItAB4EMBu\nVT2facGIKHchxz8rJkSeSjP5zd4quCP70hBRLYUc/95WTERkG8xGXgNFW7UuS9yfJlwhb+KVJP5d\n9sq5//77E6e9evXqzIkibt26lTjt0qVLnfK+cOGCU/qmpqbEaS9duuSUd2Nj8rtO77zzTqe8XX/X\nXfbtuXLlilPeLnsfvffee055uwg5/r375iKyRkQGAZxR1YMhV0oobCkWWCo8xj+REWL8l3nVY2KX\n1B0AsF5Vf1zv8hDVU2gtJsY/0bjQ4j/Kq4oJzBr/h3hRIgpyqI3xT2QFGP9jvKmSichqAC0AGkWk\nx+6KeKDe5SKqF9dNvIqM8U9UKaT4j/Opx6QdZrLbdlU9LyINAM6JyBVVLdzuiETVCqzFxPgniggs\n/iv4VDFphNkZ8TwAqOqIiDwPYBOm2La5t7d37HlraytaW1trUU6iREqlEkqlUurzZ3urKMY5/l98\n8cWx5w8//DA++9nP1qKcRIn84he/wC9+8YvU5wcW/xVyrZjYW/7aYFpCE962xztV9U0AZzFxOd0z\nMN27k+ro6MimoEQ5iFeW+/r6nM4veosp7/jfuHFjNgUlysHDDz+Mhx9+eOz1P/3TPzmdX/T4r0au\nFRNV7QbQnTD5SQAQkUV2kZiK40ShKfqFifFPlF7R478a3vQVqeo5AIdR2W37GICu+pSIqL5CmvzG\n+CeqFFL8x/k0xwSqulVE9ovIXgBDAF5R1YP1LhdRPYTWYmL8E40LLf6jvKqYAICqFn6df6IszPZW\n0WRc4v/69euJ8503b17itKOjo4nTAsDFixcTp12yZIlT3i5LzAPA5cuXE6edP3++U94uP+8PPvjA\nKW/XJexd8neNo4ULFyZO67ItgqsQ478s3G9O5LlaLkltb8+dKU1zph9KRFMKOf5ZMSHyVC3GmEVk\nvd2bZtsk7x0SkVv2MQpgxosXEWUj5Pj3biiHiIy8x5hFZBHM3jQTxgtsC+oSgNUwt/YO29t6iagG\nQo5/VkyIPJX3GHP5tlwRGZrk7U4AwwCu2jtmiKiGQo5/DuUQearO2563wHTvnhGRAc4vIaqtkOOf\nPSZEnoq3mP793/8d//Ef/1GTz1bVLwGAiKyDWUvkCICHavLhRBR0/LNiQuSpeKvoi1/8Ir74xS+O\nvX722WdzL4Oqviwi7QCGRGSdqr6c+4cSUdDxH/RQTjUbrNUz77zzZ961zXsqrrPyRWSbiBywi5TF\nH+Xjy1zLoaojAI5h4l42hVbNBmszGRwczC3vPMt96tSp3PJ+4403cssbyPfn8rOf/Sy3vKcScvyz\nYlLAvPPOn3nXNu+puI4xq2q3qj6lqjsmeZSPv5myOE2YZfvW5Pl/yorJRKyYuAk5/oOumBD5rF6T\n30SkWUQ2Rl6vBnCctwsT1U7I8c+KCZGn8l5gSUQaRGQbgGYAW+1EN8B02XaKyGkR2QmgmVtFENVW\nyPEvea71nycRKWbBKWiqmqipIyI6054tc+fOTZzfbMP4pyJi/CdT2LtyZut/CJF1fu7cuQ/MlKYm\nJfEQ459muaDjv7A9JkRERDT7cI4JEREReaOwQzlEtWQniSmAAVV9rd7lIaLaYfzXFntMJmEXqnlC\nRFbVuyy1YmdoHxCRnSKyV0TWZ5h3s91v4Zb9d3VWecc+Z5+I7M04zzVitgU/o6oHeVGa3UKMfYDx\nP02ejP86YMUkoha/hFkHT4ZB3w3TGvimqj4DoCuLi7OY7bM7AXwdwBp7eEDMltuZsRfSTRnnuQZm\nW/BtXIp9dqvVHyDGPwDGP82AFROrFr+EWQdPVkFvv/tGAD2Rw/0AnsmgmBsA7FLV1+3Ffj0AAdCe\nQd4AzMUZZjfME1nlafUAOKSqP844X/JIrf4AMf4Z/5QMKybjcv0lzCl4sgr6NgDDqno1cuyEzb8q\nqvpidMVAu++CwvwhyMp2Ve2G+e6ZsC3PFgCNItIjIoMiciCr/Mkruf8BYvyPvWb804xYMUHNfgkz\nD54Mg74NwFDs2BBy2LRJRDYA6I9dBKvJby+APLbZbIf5WW5X1S0wP6MtWY9hU33V8A8Q4x+Mf0qG\nFRMj11/CHIMn/jmZBn1OtgHYnEVGdqZ8j6peyyK/mEaYVuR5YOyi/zwyHsemusv9DxDjvwLjn2bE\niomR2y9hzsETlzboT8DsHhnVBGC46hJFiMjTAPZmeOHcDTOmPioiozDj5Lvt82qdxcQW4xmY1jXN\nHrn+AWL8j2P8U1Kzdh0Te0Fog2kNTXjbHu+0XaFOv4SOee8G0CyVu0GKiOxS1blV5h09r5qgH8DE\n798GMwEuE2J2qzya5d0Oqro89hk9MHdVZDFp76TNc1HsZ1qzrb8pnTxjP0X+jH8w/skNl6TH2MS0\nQQCLy7+E9gKxXVXXZvxZWQZPOc+NNs/UQS8iPwLQpaov2denAWwotySrLN8GmB0syxe6xQC2qOqe\navOOfU6mP1sReQHA2XJ+Nv8jqnowi/yp/moZ+zZvxj/jn2Ywa3tMXKjqORE5DHN7XPmX+jEAXfUr\nVTI26JsAnLQX2bRBvxlmq+sWAA8C2J3RRWk9gCOY2ALcVW3ek8i0lq2qW0Vkv50jMATgFV6UZpci\nxz7A+I9h/M8S7DGJEJH9MOOqQwBUVb+Zw2dU1MKrzGs9gKOYJOhV9f9Vmz9RKGoR+/ZzGP9EM2DF\nhIiIiLzBu3KIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBigkRERF5gxUTIiIi8gYr\nJkREROQNVkyIiIjIG6yYEBERkTdYMSEiIiJvsGJCRERE3mDFhIiIiLzBisksJiJPi8g2ETktIoti\n7w2IyOP1KhsRTS2k+JzuOkVhElWtdxkoByKyEcAGVd0hIstU9c3Y+6tU9bX6lI6IphNKfM50naIw\nscdk9lpbfjJZsIdw0SMqqoDic9rrFIWJFZMCEJGNtmv3aRHpEZEhEXk68v4+EXncvrdORNYD2ACg\nXUSemGQYZ7WIHBGRnZHX5fwPiMigfb7evh4SkVWxz1tn/90XOd5cPiYit0TkRyKyLvIdttky7gMR\nTaqo8Zn1dYoCpqp8FOAB4BaAx+3z1fb1Kvv8iD2+CMA6+/wAgP3T5HcEwM7I68Hy60j+y+zrfQBe\nsM8bAJyOliuWx+ft8wEAeyP59UTOH/sufPDBx8RHUeMz6+sUH2E+2GNSLGcBQFVfBXASZmz2VQBN\nIjIA4EFVfbna/O2/CmDIvj4DoMV+9oiqPmRbXxsBaKSl0wJA7PP+SL5bbbonAGwBcKiKMhKFqijx\nmfd1imY5VkyKawjjF4B2mAvNgL245EZEGkSkx3ysvojxCx0AHIW5yAGm5fWP9nkjgLOqelBVu1V1\nq6q+lGc5iUKUZ3zaoZ5B+ziesEh1uU5RsbFiUlzNAPrtOHOzqu4B8CUAmzPIW1B5QYvaAmC1qr4p\nIo0wLbeypwA8aFtqh1X1dXv8KIDtIrIaGBvr5oWJKJ26xKettCy3j7Xx96eQ53WKZilWTIrlSTux\nbC+Azap61R4/Yi8kGwDstheYDQA2lCe3Rdn32wA8JiLL7Otm+3oRgG0wF7TtItIA4DEALSKyDLYL\nWER+BGA9TFdtt826E2Ys+Xlbph8BgG25PQ9zgToNYLuqHszyB0M0W8yC+MzkOkXh4jomBSEit2Ba\nQq/PmLgO7AVyj6o+Ezm2DgA4nkxUX7WKT9+vU1QM7DEplqm6b33wzCTHGmFm/xNRfdUyPn2+TlEB\nsGJSAOXZ9TBdpL7e638AZvz6tIi8ICL7AfRHunGJqH5yj8+CXKeoADiUQ+QhEUkSmOdVdVneZSGi\n2go9/gtbMUn4H0fkFVVN1M0tIioyfVK7GFGQ3eaMfyoixn8yt9W7ANX49re/XdX5vb296OjoyKg0\ntcs77/yZdz55f+1rX3NKn+TCFLJXX301cdqbN29OONbd3Y1t27ZNOD5njtsI99y5cyccO3DgAJ56\n6qkJxxcsWOCU9/Xr1ycc6+rqwpNPPjlp+sbGxsR5T/Yz+Zu/+Rv82Z/92aTpJ/ueU/nwww8nHPvW\nt76FP/3TP500/bvvvps4bwD43Oc+N+HYc889h6effnrC8bfeessp76VLl0449s1vfhM7d+6ccPy9\n995LnO+KFSucyhFy/Be6YkI0m810YSKi2Svk+GfFhMhTM7XcR0dHa1QSIqq1kOM/6IpJa2trIfPO\nO3/mXdu8pxJyi6kW1qxZk1ve7e3tueXd1taWW96/8zu/k1vea9cmXSw2nd/93d8tZN5TCTn+Cz35\ntdo5JkS19LWvfc1p8tv8+fOnTXPjxo1ZO/ltJiKi1c4xmUoWc0ymksUck+lUO8dkOtXOMZlOFnNM\nppLFHJOpuM4xYfwnE3SPCZHPQm4xEYUu5PhnxYTIU64td8BswAbgSQBnYPZb6bJbzpeXJe8EMAhg\nCcwCW8cyKzARZSbk+GfFhMhTKVtMJ2D2KjkvIv0AzohIo13hsxvAkfIGbXb7+k2q+lp2pSaiLIQc\n/1ySnshTc+bMmfYRJyKbAKiqnod5cg5md9ln7E6umwD0RE7px+R7qBBRnYUc/6yYEHlKRKZ9TKId\nwFDs2FkAa+x7V2J7o5yA2XaeiDwTcvx7PZQjIttgNoUa8LG7iShPKcaYzwBoiR0bArAaZrw5ftEa\ngtlh1jtJYt/l53PPPfckTut6J4yLO++80ym9690tV68m35PP9a4cF/fee69T+rvvvtsp/cWLFxOn\ndR0ScblD6MyZM055uwg5/r3sMRGRNSIyCOCMqh5kpYRClKLF1GPP22v/bYBpEZ2tXamrw9gnMkKM\n/zLvekxEZA2AAQDrVfXH9S4PUb24tphUdURE2gB0isgLMHEEAEcBjADYHDulCcBwteXMCmOfaFxo\n8R/lXcUEptZ3iBcmCl28VXTz5k189NFH055jexi+bM9fA2AfgC4AyzGx27YNZgKcLxj7RFaA8T/G\nq6EcO3O4BUCjiPTY25kO1LtcRPUQ77qdN28e7r777rFHAj0AtqvqNbuWQb+IPB55fz2AXXmU3RVj\nn6hSSPEf51uPSTvMhLft9j7sBgDnROSKqnp5WxNRXtIssASMLbJ0CMBeVf27yFubYbp5WwA8CGB3\n+dZCDzD2iSICi/8KvlVMGgEMR+7DHhGR52Huv+bFiYKSZoGlyN0s62K3BsK+3pFN6TLH2CeKCCz+\nK/hWMTmLieNgk90CBQDo7e0de97a2lqXHWCJplIqlVAqlVKfn6bFpKrdqT+wvpxiHwD2798/9ry9\nvT333WuJXPz85z/Hf//3f6c+P7D4r+BbxeQkAIjIolht7+RkiTs6OmpSKKI04pXlvr4+p/MD28TL\nKfYBYMeOQjT+KFCPPPIIHnnkkbHXhw4dcjo/sPiv4NXkV7uE7mFUdt0+BjOrmCgorktSFxljn6hS\nSPEf51uPCVR1q4jst4vEDAF4pbzpEFFIQmsxMfaJxoUW/1HeVUwAQFXZR0vBm+2tosm4xP7o6Ggu\nZVi4cKFT+uHh5GtUvf/++055z7RuRZzL78yCBQuc8nZZTt/1j6rr8vgu//fz5s1zyruhoSFx2ttu\ny+9PaIjxX+ZlxYSIwm4xEYUu5PhnxYTIUyG3mIhCF3L8s2JC5KmQW0xEoQs5/lkxIfJUyBcmotCF\nHP+smBB5KuSuXKLQhRz/rJgQeSrkFhNR6EKOf1ZMiDxVixaTiDTbxc2IyCMhx3+4fUVEnotvex5/\nTHHOLREZtf+WH/sj7x+KHB8FkHzRBiKqmZDjnz0mRJ5ybTHZnUW3A+iPHO6E2QIdItIA4BKA1QAE\nZjffN7MoKxFlK+T4Z8WEyFMpxphfiG91LiJtqvqyfdkJYBjAVR+7b4loXMjxz6EcIk+5buI1yUVp\nPYATkUMtALYBOCMiAyLSnOsXIKLUQo5/9ph4TlWd0t+6dSu3vF1q8Cm6IZ3ShyCDn8lmAC+UX6jq\nl2y+62B27T0C4KFqP6ReXH7HhoaGEqd97733nMrxqU99KnFa171VXPZtAdzi/8aNG055j4yMJE57\n7do1p7xd9wRy2bfn6tWrMyeKuP322xOnfeutt5zydhFy/HvdYyIi++xOo0TByWDb8y2q+lL8oO3a\nbQfQYi9S3mHsU+hCjn9ve0xsN9Qm2Ik7RKGJt5iuXr2auCVq4+foVO+r6oiIHAPQWE0Z88DYJwo3\n/gFPKyZ27KsFleNjREGJt4oaGxvR2Dh+Hbl48eJ0p1d0406hCcDJlMXLBWOfyAgx/st8HcrZrqrd\nMLc0EQUpzToGERXduCLSLCIbI69XAzju4e2CjH0iBBv/ADysmNhx5WfrXQ6iekt7YbLduMdjhxsB\ndIrIaRHZCaBZVXfkWHxnjH2icaHFf5RXQzl2gZgeVXWb0k00C6VdklpVjwE4Fjv2KoDlGRQrF4x9\nokohxX+cVxUTALsBNMdqgyIiu1R1bjxxb2/v2PPW1la0trbmX0KihEqlEkqlUurzA7uF2in2AWD/\n/rGVttHe3o61a9fmW0IiB6+//jp++tOfpj4/sPiv4FXFRFUranQi0gPgjKo+M1n6jo6OmpSLKI14\nZbmvr8/p/JC2PXeNfQDYscPbnmgifP7zn8fnP//5sdf/8A//4HR+SPEf51XFhIjGhdxiIgpdyPHv\ne8XEbWlSolkk5BYTGPsUuJDj3+uKiapurXcZiOol5BYTY59CF3L8e10xmY1c9rIAgJs3bzqld9n/\n4uOPP3bK22Wfj/nz5zvlfccddzilD6E1EcJ3rIbLz8dl/5PPfOYzTuW4fv164rQffPCBU94ue8IA\nwK9+9avEaV334Vm8eHHitJcvX3bKe8mSJU7ph4eHE6edO3fSudNTcvldcS23i5DjnxUTIk+F3GIi\nCl3I8c+KCZGnQm4xEYUu5PhnxYTIUyG3mIhCF3L8s2JC5KmQL0xEoQs5/lkxIfJUtV25dpl3BTCg\nqq9lUigiqomQ458VEyJPpW0xicgaAD0wO/W+HDneAKATwCCAJQD67b4aROSZkOOfFRMiT6VpMdmL\n0gCA9ar649jb3QCOqOpBm3ZQRDYVrTVFFIKQ4z/cab9Enku57XkPgEPxi5K9YG2075f1A5hyLxoi\nqp+Q458VEyJPzZkzZ9pHnIisBtACoFFEemyL6IB9ux3AsKpejZxyAsCG3L8IETkLOf45lEPkqRRj\nzO0wk922q+p5O6Z8TkSGATQCGIqlH7LHicgzIcc/KyYZUE2+35jrEvNXrlxxSv/OO+8kTnvt2jWn\nvBcuXJg47dKlS53ydlnuGgDmzZuXOG1Rb7tLMcbcCNMqOg8AqjoiIs8D2ATgaMbFq7uRkZHEae+9\n997EafPcBsJ1CXOX7wi4xWhLS4tT3m+//XbitC7LugPApUuXnNK7XF9cl6R3WU7f5TrkKuT4965i\nIiLNAA4BWAPgJIBtqvpqfUtFVHvxCtWvf/1r/OY3v5nulLOY2AI6A6AZptt2S+y9JgDJNx3JGWOf\naFxo8R/lVcUkcjvT1wEIgIMABkRkcWxsjGjWi7eYli5dWtFS/J//+Z/4KScBQEQWxeLlJMyFKX7R\naoOZAFd3jH2iSiHFf5xvk183ANilqq/bW5jWw1yk2utbLKLac52Vr6rnABxG5Uz7xwAcsD0P/SLy\neOS99QB25fkdHDD2iSICi/8KXvWYqOqLsdcjIqIw92UTBSXN3BhV3Soi+0VkL8zktldU9e/s25sB\ndIpIC4AHAewuj0fXG2OfqFJI8R/nVcUkTkQ2wKxOx65cCk7aJalVdccUx68CmPQ93zD2KXQhx7/X\nFRMA22BqeUTBKerdRBlh7FPQQo5/bysmIvI0gL3TtZh6e3vHnre2tqK1tbUWRSNKpFQqoVQqpT6/\n2k28iipJ7APA3//93489X716NVavXp130YgSO378OI4fP576/FDjH3ComIjIKphV5YYBnFXVN/Mq\nlIhsBHB0pjX8Ozo68ioCUdXileW+vj6n80NsMSWNfQD4kz/5kxqUiCidtWvXYu3atWOvDxw4ME3q\niUKM/7IZKya29fIMgMsAzsFUTFpE5DLM5JlMNwCyY8tNAE7adQ0WA9iiqnuy/Bwi34XWYmLsE40L\nLf6jpq2Y2ErJYVV9bor3N4pIU3Rr5WqIyHoAR2CW1Y3y8pYmojyF1GJi7BNVCin+46asmIjIqqkq\nJGWq+qKINE+yoEsqqnoM/q2tQlQXIbWYGPtElYoa/yLyBICBakZTpqyYTJdptCJiF3UJ2q1btxKn\nddlXA3Db+wYAXn/99dzydt3/xsWdd97plN5lLw7XvTJ8EXKLKQmX3xmXn6Xrvi2/9Vu/lTita8x9\n8MEHTunb25OvR+e6z8vHH3+cOK3rPly33eZ2H4bLHkINDQ1Oebtw2ZvIVYHjfw/M/jwAxioqjar6\nzaQZJKqS2QVb1tnnAwBOiMhOx8ISkQPXbc+JaPYocPx3qeprIrLOzkX9BoDfEZEXkmaQ9NudVNWX\nRWQ/gCFVfQhmwyAiyonrktRENHsUOP7XishpmB2Nu1W1SVW3wGEV56T9Z412IuxWAMvssScBvORQ\nWCJy4HmriIhyVNT4V9Ut9rb/k+WpHnaTzi0App23Wpbom9tJsCcBNKvqVfuhh9MVm4iSqEWLyd6W\nS0SeKXL8q+qLsfmnzwC4kvT8KSsmIrIs+gBwBsBi+/wEgPxmFBFR1RcmEdlnN/OKHjskIrfsYxSM\nYyIvFTX+7WKsFVR1j6p+KWke0w3l7IHperkMs/14XDOAxLNsichNNV25dl2QTQAORY41ALgEYDVM\nTA/nuYIzEaVXtPgXkW0w9YW1AF6Lvee0pMh0FZNDqvrUNIXgxhREOUrbXWu7Z1tgejajOmFWbr7K\n2/yJ/FbA+O8BcAzAahHZBKAfZgJsP4AmEWlPuhjrlFUyu+ARAFPbsbcMP2tfPwEztENEOanidsHt\nqtqNiT2dLTC79p4RkQHOLyHyV9HiX1VHYHYEfw7AUwBGAPwlzNySowC6kuaV9K6cwzC3B4stwEER\n+RGALycvNhG5SNNismPKz072XnmM165J1AWzBPxDVRSRiHJSxPi3PTHlva2inRvrATQmzSfpINZR\nO6wTXbtk7VSJiah6ri0mO8bbo6rTLrtpu1PbYTbjXJdP6YmoGrMp/lX1mKq+mDR94nWA7fBNi4g8\nDnPrT3+K8hFRQvEW04ULF/DLX/5yulN2A2iOnScisktVK9blV9URETkGh1YMEdVOUePf1hHWwnRo\nvGyPrQNwNulk20QVE1V9zq5dshjA/4aplSVaKMWVnTncCWAQwBIA/dH5Lj5SjW+IOjWX/SYA9z0n\nXPbiOH/+vFPeLlpaWpzSu/5cXH7mRRVvFS1btgzLli0be/2Tn/yk4n1VXR59LSI9AM6o6jNTfEQT\nzPpEXnCNfZf9VS5cuJA47eLFixOnBYDLly8nTjt//nynvFeuXOmU3qX732W/KQC4ejX5Pq2ud5S4\n/AwBt7K7Dol8+OGHidMODQ055e2iwPH/DZvvZjuP5SRMR0YLzCKtM0oc2bYbZqwrRkTWJZ1h66gb\nwBFVPWg/Z1BENlWzUyFREWW57LS9QKwpd6fau+qOe3a7MGOfyCpw/HfB3NX7pog0AlgP4DFkscBa\n1GnlSMgAABf1SURBVCSLtGyDmWWbKRFZA2AjzG1HZf0wQ0dEQclgE69ot1IjgE4ROS1mA85mVd2R\nS8FTYOwTVSpw/HfBTPtYparDdhXYp6ZbfiQuaY/JkK2cHIfpaj0HcztQ1tpgFn2J9hmeALAvh88i\n8lq1LSZV3Rp5/iqA5dMkrzfGPlFEgeP/JEzvyICIDAN4dqYJuXFJBwIV5sLxDQCvAthk75POWhuA\n+KDdEDhBjwJUi70yPMLYJ4oocPz3A3geZp7YkwCGReS47alJJGmPyTdgFm0pj/0+bVdxSzSRhYjc\nFXV3USKqXoHjf5ft+ewGABFpwfg8k0SSVkzabFcQgIq7dLJ2AmbluKgmmGV0J+jt7R173traitbW\n1hyKRJROqVRCqVRKfb7nraKsOcU+AOzfv3/seXt7O9au5dJK5I+BgQGcOBFfFT65Asf/SRE5Arsc\nvaqetRv77UqaQdKKyRUxuwpH5bGc9QAmdt22YYo1Uzo6OnIoAlE24pXlvr4+p/ML3GJKwyn2AWDH\nDm/m7hJN0N7ejvb29rHXzz//vNP5BY7/PQC2w8xDVRE5i/GJ7IlaD1NWTERkEGYuyWs206ZYkgZk\nvLuwqr4qIv0i8riqvmQPrwewIcvPISqCAreYnDH2iSoVNf5V9TCAw/ZW4W0wvZ4PIrJE/UymrJjE\nFmt5Mr7QkV37Pg+bYW5raoH5MrtVNb+VwIg8VeAWU1qMfSKr6PGvqsMAnrNrppx1WSg16cqvxyLL\nzF6GWTwll9VY7aQZ9tFS8IraYkqLsU80rqjxLyK3YOaM9cPMMxmAmQ6yd5pVaCskqpiIyAsYH+8V\nAN0i0lO+Syd0Lr9ALstoA8DChQud0i9dutQpfV55u5bb9edS1KB1UfQWU97mzp07cyJrwYIFuaQF\ngBs3bjild/HBBx/kln50dNQp74aGhsRph4ennLM8qaam+EyB6bl8T9ef4T333JM47c2bN53ydlHg\n+P8GzJpnj8HcNtwMswFwExIumOhyV058Hf6nk5eTiFyFUPkioskVNf5VdY99Wl7+vgFmN+PEGwsl\nrZJ1iUi8CTy23K0d5iGiDGWwJDURFVRR419E1tl9ri6LyD8CWKaqx6JLjswkaY/JVgD77PKyZYtF\n5BmYoZ0GAMn7VYloRkVtMRFR9Qoc/08B2A0zdNMG4Mci8n9V9f8lzSBpxaQLZn7JZF0xAnNLEBFl\nKE2ryO4i2oXxrtPN5ZaK7VLtBDAIs1x0f16T2ImoOgWO/xfKuxjDrP76lIjsiy0FMK3p1jFZVd5u\nfLp9cezCa24rxxDRjFK2mHbB7NIrAA7D7Nb7kH2vG8CRyNYSgyJSXquIiDxS4PhvEZFF0Q05VXWP\ny5SP6apk50TkCRFZNFUCuyx9i6qOJP1AIkrGdRMvezHYoarX7EWhC3ZJdxFZA3PB6omcUl6NkYg8\nU+D4fx5mWfqvx463JM1gugXWRkTkEIC/tIupnYXpGmrC+NLRu9naIsqHa1dutIVircX4MGsbgOFY\nmhMA9qUuIBHlpqjxb+sOWwD0iMjzAE7at4aQcLX4aeeY2J6QPZHbfVpgKihnVfVc6pIT0YzSTn6z\nDYkumPUDBgG8BnNhis8RG8LE/WmIyANFjn9VPQlguYhsgqk7XIbDlI+kK7+OwKxzz4lyRDWS9pZA\nu1LzBpiJbl0iMpBpwYgod0WO//Ic1fK+Oa7nuy23SUQ1E28xnTp1CqdPn050rqq+CWCriJyGabGc\ngNmLJqoJdgyaiPxS1PgXkR6Y3pq1kWPLbJkSYcWEyFPxFlNraytaW1vHXv/whz9Mks05mOHXK5jY\nbVveZoKIPFPg+I/eLly2WERaVPXlJBkk3StnXdIMQ+TS5TZ//nynvPPc+6alJfEkaQBu+9+4ltv1\n5+LzyodZcR1jLs8FK69NYF9fKceuiPTH1hJYD2BDhkWuqQ8//DBx2jvvvDNx2o8//tipHLdu3Uqc\ndsmSJU55X7p0ySm9i7feessp/eLFixOnVdWZE0Vcv349t/Su1xaXPZjy3CepwPE/4Q+Lqr4qIjsB\nZFcxAbBFRB6DmcBy2KVLhojSSVH5aoIZU1aY8eUhVd0aeX8zgE4RaQHwIMxddeczKSwRZarA8X9Y\nRI4D2KWqP44cfwxZ3JVTpqpPlZ+LyOP2NuJ+AF1ZVlLsqnWHAKyBucVom8v6+kSziWuLyd4pt3ya\n968C2FFlsXLD+CcaV9T4V9VzIvIkzO3CzTCxvAYOw0aJqmQiskxEVonIAQAHYcat+gG0ichOEVnl\nXvwJn1FeLvfrMF8CAAamW+CNaDYr6iZeaTD+iSoVOf5V9aSqLgfwZZhF3Z5S1S8nPT/pUM5ZAGdg\n7kNujq70aleUOwaz9n41NsB0/bxp810PM2GnHQnHpYhmkwJv4pUG458oYjbEv6r2I8UE26QVk92q\n+twU711BBqvHxWfx2tXjFADXYKAgzYYLU1KMf6JKIcV/XNI5JlNVSsrjWlO+n5ZdIKZ/kmV2x/T2\n9o49j99KRVRvpVIJpVIp9fm+d9fmKUn8d3V1jT1va2tDe3t7LYpGlMjAwAAGBtLXq0OOf5/XMdmG\niQvCVOjo6KhRUYjcxSvLfX19TueH3GJCgvh/8skna1QUInft7e0VleVoRTqJkOM/14qJiGyDWcRl\nspvaxR7vjN/ZIyJPA9g7XWuJaLYreouJ8U+UXtHjvxq5VkxUtRtAt8s5IrIRwFHuWkyhK3qLifFP\nlF7R478aXlXJ7LhyE4AREWkWkTUiwm3ZKUhFvl0wDcY/0bjQ4j/Kmzkm9vbAI5jY7burDsUhqruQ\nWkyMf6JKIcV/nDcVE7u+fyGrgS6/QHfccYdT3i77UwD57gly223Jf11c96dw/bmEELSzvVUUlSb+\nFy1Kvvba6Oho4rQuMeSa/le/+pVT3i77AQFAQ0ND4rSu+7xcuXIlcdq77rrLKe+bN286pf/oo48S\np3XZ4wsA3n333cRpf/u3f9spbxchxX9cuN+cyHMiMu1jhnNn/Atll4smIg+FHP+smBB5Ks0Ys4is\nF5FBmNtt4+8dEpFb9jEKIHnzmohqKuT492Yoh4gqpdj2fBHMSqlNk7zXAOASgNUwt+oOc5dwIn+F\nHP+smBB5ynWMubzuh4gMTfJ2J4BhAFftas1E5LGQ459DOUSeqmaMeRItMN27Z0RkwOfxZSIKO/5Z\nMSHyVJYXJlX9kqougdnFtwHm1lwi8lTI8c+KCZGn8lhgSVVfBtAOoEVE1mVaYCLKTMjxzzkmRJ6K\nt4p++tOf4mc/+1nV+arqiIgcA9BYdWZElIuQ458VEyJPxVtFq1atwqpVq8Zef+c736km+yYAJ6vJ\ngIjyE3L8cyiHyFNZjTHbfWc2Rl6vBnDc59sFiUIXcvyzx6TGXMcG582b55T+9ttvT5xWdbLd6Kfm\nEgyu3zOEJeZdpfgZNgDYAqAZwFYROWnHlBsBdNoN8boAnFXVHVmXt9bmzp2bS74uS5IDbtsv3H//\n/U55v/POO07pXeL/E5/4hFPe165dS5x2ZGTEKW/X3/V77703cVqX5esBt60O3njjDae8XYQc/6yY\nEHkqxcz7EQDd9hE9/iqA5dmVjIjyFnL8e1sxsbU7VdVn6l0WonoIeRMvxj+FLuT49/Kb2y3QN9W7\nHET1lPECS4XB+CcKN/4BD3tM7Ip0LQBO1LssRPUUYouJ8U9khBj/ZT5+8+2q2g2z0RBRsAJtMTH+\niRBs/APwrMdERPYCeDZp+t7e3rHnra2taG1tzaNYRKmUSiWUSqXU58/2i0+ca/z/1V/91djzRx99\nFF/4whfyKBZRKidOnMCJE+k7/kKL/yhvKiYisg1Aj6omvieto6MjxxIRVSdeWe7r63M6P6Su3DTx\n/xd/8Rc5loioOm1tbWhraxt7ffDgQafzQ4r/uFwrJvZi0wZgsgUzxB7vtAu97AbQHKsliojsUtV8\nFiwg8ljRW0yMf6L0ih7/1ci1YmLHirtnTGjSVtxnLSI9AM7wdkEKVdFbTIx/ovSKHv/V8GYoh4gq\nhdxiIgpdyPHvc8XEbb10olkm5BYTGP8UuJDj39uKiapurXcZfOBaa85r/xCqvZBbTEni/9atW4nz\ne/vttxOndd1D5q677kqc9uLFi05533PPPU7ph4eHE6f91Kc+5ZS3y/d0KQcA3Lx50yn9+++/nzit\n6zXRZX8y198VFyHHv7cVE6LQpWkx2Y28OgEMAlgCoF9Vj2VcNCLKWcjxz4oJkadStpi6ARxR1YM2\nj0ER2aSqr2VaOCLKVcjxH+4gFpHn5syZM+0jTkTWANgIoCdyuB8A72whKpiQ458VEyJPpViSug3A\nsKpejRw7AWBDTQpMRJkJOf45lEPkqRRduW0AhmLHhgA0ZlIgIqqZkOOfFRMiT4V8uyBR6EKOf1ZM\niDwVbzG98sorOH78+HSnnACwOXasCYDbvZtEVHchxz8rJkSeireYvvCFL1TsoPutb30rfsoAJnbb\ntsFMgCOiAgk5/sPtKyLynOvkN1V9FUC/iDweObwewK5alZmIshFy/LPHhMhTKceYNwPoFJEWAA8C\n2K2q5zMtGBHlLuT4Z8WEyFNpFliytwruyL40RFRLIce/txUTEdkGs5HXQNFWrSPKQsiz8rOO/wce\neCBx2pGREae8XfbsWbp0qVPe58+7NXabmpoSp/3Nb37jlPfixYsTp50/f75T3q5/hBctWpQ4reu+\nPQsWLEic9tq1a055uwg5/r375iKyRkQGAZxR1YOslFCoUiywVHiMfyIjxPgv86rHxC6pOwBgvar+\nuN7lIaqn0FpMjH+icaHFf5RXFROYNf4P8aJEFOS254x/IivA+B/jTZVMRFYDaAHQKCI9dlfEA/Uu\nF1G9uG7iVWSMf6JKIcV/nE89Ju0wk922q+p5EWkAcE5ErqjqpLsj9vb2jj1vbW1Fa2trbUpKlECp\nVEKpVEp9fmAtJuf4/+u//uux548++igeffTR2pSUKIH/+q//wiuvvJL6/MDiv4JPFZNGmJ0RzwOA\nqo6IyPMANmGKbZs7OjpqWDwiN/HKcl9fn9P5gV2YnOP/z//8z2tYPCI38cry3/7t3zqdH1j8V8i1\nYmJv+WuDaQlNeNse71TVNwGcxcTldM/AdO8SBafo3bWMf6L0ih7/1ci1YqKq3QC6EyY/CQAissgu\nElNxnCg0RW8xMf6J0it6/FfDmyqZqp4DcBiV3baPAeiqT4mI6iukyW+Mf6JKIcV/nE9zTKCqW0Vk\nv4jsBTAE4BVVPVjvchHVQ2gtJsY/0bjQ4j/Kq4oJAKhq4df5J8pCLVtFItKgqtOuxS4izbZnIzcu\n8X/9+vXE+d5xxx2J046OjiZOCwBvv/124rQuS8anSX/p0qXEaV2WXgeA999/P3Fal/8bALjrrruc\n0n/44YeJ0+a53L3LdgSuQoz/stndH0RUYLVYklpE1tsl4LdN8t4hEbllH6MAGjL5UCKaUcjx712P\nCREZebeYRGQRzBLwE5rldh2RS8D/3979LEdxnWEcfl9lj0C+gCCRC4iQslchmQsAgy8giEr2/Ily\nARBVnDUi8g1E2NmDMamCgkUYSc7aQpgLQChkbX9Z9Bk0Gmk0kqa7p7vn91RRRjPD163xvD2n+5w+\nR9PK7qDZTXfPACjBKOefhglQUUX3MbfvfrG9c8jTy5J2JX0s6/ItgD2jnH+6coCKGvKo/Clll3ff\n2G7Znix6gwD2jHL+aZgAFTXMZc8j4nJEfCZpQVnf8pNCNwhgn1HOP105QEV1H3yeP3+uFy9elLoP\nEfHM9qykHduXIuJZqTsAjKhRzj8NE6Ciui/Xzs3NaW5u7tPP9+7d2/f8CaeAP7a0bs33OjhlPICC\njHL+R7orZ5CVX4dZu+j61C63di8nvZQbEasR8YeI+OMhf9qP/3TK3ZlQw6aHH2Tl1342Nop7q4rc\n71arVcvaUraab1FevnxZWO1eRjn/NExqWLvo+tQut3Yvwxr8ZnvS9tWOn6clvW7a7cJFfsFvbm4W\nVrvI/V5fXy+sdp0bJq9evSqsdi+jnP+RbpgAVVb04Dfb4+ny76SkL21fSk+dlbRs+0fbtyRNMiMz\nUK5Rzj9jTICKKvqWwDQF9YEVgCNiU9JvCt04gCONcv4dcdg4meqzXc8dx0iLiGOd6tiOfutwjI2N\nHbte05B/1BH5P57aXjFp6v8QIHk3Njb2636vKWVPKoj8o+FGOv+1vWICAACah8GvAACgMmrblQOU\nKY1eD0mtiPhh2PsDoDzkv1xcMTmE7UXbN2z/dtj7UpZ069iK7Vu279uez7H2ZFoI6pf03+m8andt\n5y+27+dc86LtLUlvIuJrDkrNNorZl8j/ETXJ/xDQMOlQxocw7/DkGPpVZWcDX0XEkqSHeRycbY8r\nW0L795Iupodbts8MWrtrO/OSvsi55kVJLUmLrBHTbGV9AZF/SeQffdAwScr4EOYdnrxCn373q5LW\nOh5+Kmkph91ckHQnIv6TDvbzytZtmM2htqTs4Kxsme68p61ck/QoIv6Vc11USFlfQOSf/ON4aJjs\nKfRDWFB48gr9jKTdiPjY8dh6qj+QiPi2cyrjNKlPKPsiyMvNiFhV9rvnIp15Tkk6a3vN9pbtlbzq\no1IK/wIi/59+Jv/oi4aJSvsQ5h6eHEM/I2mn67EdFbCapO0FSU+7DoKD1Lsv6V7fF57crLL38mZE\nXFf2Hl3Puw8bw1XiFxD5F/nH8dAwyRT6ISwwPN3byTX0BVmUdC2PQmmk/FpE/C+Pel3OKjuLfCd9\nOuj/XTn3Y2PoCv8CIv/7kH/0RcMkU9iHsODwdDtt6NeVLWvdaULS7sB71MH2bUn3czxw3lXWp/6z\n7Z+V9ZPfTX8f1LYOnjG+UXZ2jeYo9AuI/O8h/ziuxs5jkg4IM8rOhg48nR5fTpdCT/QhPGHtu5Im\nvX81SNu+ExG/GrB2578bJPQtHfz9Z5QNgMuFs2W0v8vzboeI2LfQlO01ZXdV5DFobyPVPNP1nm7k\nUBsFKjL7p6hP/kX+cTJMSa9PA9O2JJ1rfwjTAeJmRPwu523lGZ52zaup5qlDb/uxpIcR8c/084+S\nFtpnkgPu34KypbXbB7pzkq5HxJ8Grd21nVzfW9v/kLTdrpfqP4mIr/Ooj+ErM/upNvkn/+ijsVdM\nTiIi3tr+Rtntce0P9eeSHg5vr44nhX5C0kY6yJ429NckLdueknRB0t2cDkrzkp7o4BngnUFrHyLX\nVnZEfGn7QRojsCPp3xyUmqXO2ZfIfxfy3xBcMelg+4GyftU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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d814208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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0kg5hGhOmCVGakKqSapbf00u+N1GiHqmToYArLl7m4mcJqSZkCA7KbIVg7z8d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\nofw/VS0RBtxdSWCe0GcpIjPh7qfN9HeT6YHrie/4jzqcPH+zu+8BdnH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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11cbc02e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sigtrue = 1.0     # the intrinsic gaussian source width, sigma (in pixels) \n",
    "demo(muXtrue, muYtrue, sigtrue, Ctrue, skyBg, sigmaPSF, sigmaNoise);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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    }
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   "source": [
    "## Summary\n",
    "\n",
    "ML estimators are the standard workhorse for estimation of parameter values all across astronomy. Today we've seen how they can be used to perform stellar PSF photometry and shape measurement of objects. We've also learned a bit about image processing in astronomy.\n",
    "\n",
    "Though the details may be more complicated, ***what you've seen today is very similar in spirit to what LSST DM image processing codes do today***.\n",
    "\n",
    "For more on this topic, read \"Photon Rates and SNR Calculations\" by Ivezic, Jones & Lupton (http://ls.st/lse-40; especially Section 4)."
   ]
  }
 ],
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