lectures/Week-3-Thu-class.ipynb

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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": 2,
   "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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Pf/7D2LFj6devH6+99hrdusnfogB8+KE+3PX11/rlV1/Vy9xLUgEksQgRnTq4L2rkyJHE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\ni9e2UMQmhBCi44yeZbZomuZ+6LbZbQOlrzYhhBB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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f39a01ad1d0>"
      ]
     },
     "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": 14,
   "execution_count": 8,
   "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 0x7fde7387ad90>"
      ]
     },
     "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",
    "def huber_loss(m, b, x, y, dy, c):\n",
    "    y_fit = m * x + b\n",
    "    \n",
    "    t = np.abs((y - y_fit)/dy)\n",
    "    end = np.zeros(np.size(t))\n",
    "    end[t<c] = 0.5*t[t<c]**2\n",
    "    end[t>=c] = c*np.abs(t[t>=c]) - 0.5*c**2   \n",
    "   \n",
    "    return np.sum(end)\n",
    "\n",
    "# Define the log-likelihood via the Huber loss function\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": 6,
   "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": 16,
   "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": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7b29sMFcJQIZemZUiswHUQmbHNdqTHnpOtmbsduDuIzMbSjppsj0A+m+dAnzZ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\nUDGEdAAAAKBi/j8e2kxtgEVE9wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fde73bf1290>"
      ]
     },
     "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": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fde73127810>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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S5Opso6reTfLmLOmuqutJzh9ljEf42c8Otp/HQsIOACQj6XHvq+07C7vXFyrl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\n/wbXRB2/0EYvAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fde73c9bb50>"
      ]
     },
     "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_ylabel('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": [
    "### For a single pixel \n",
    "$ p(I(x_i, y_i)|A, PSF(x_i-x_0, y_i-y_0), \\sigma) = N(I(x_i, y_i) - A*PSF(x_0, y_0)|\\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} A\\exp \\left( - \\frac{(I(x_i, y_i) - A*PSF(x_0, y_0))^2}{2 \\sigma^2} \\right) $.\n",
    "\n",
    "### For a whole image then p1*p2*p3....\n",
    "\n",
    "$ P = \\Gamma p(I(x, y)|A, PSF(x_i-x_0, y_i-y_0), \\sigma) = N(I(x_i, y_i) - A*PSF(x_0, y_0)|\\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} A\\exp \\left( - \\frac{(I(x_i, y_i) - A*PSF(x_0, y_0))^2}{2 \\sigma^2} \\right) $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">$ 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 $"
   ]
  },
  {
   "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. The Likelihood **per pixel** is\n",
    "\n",
    "> $L = \\frac{1}{\\sqrt{2 \\pi \\sigma}} A \\exp{\\left[- \\frac{\\left(I(x_i,y_i) - A\\  \\mathrm{PSF}(x_0,y_0)\\right)^2}{2 \\sigma^2} \\right]} $\n",
    "\n",
    "so the net likelihood is\n",
    "\n",
    "> $\\prod_{x,y}{L}$\n",
    "\n",
    "or\n",
    "\n",
    "> $\\sum_{x,y}{\\ln{L}}$"
   ]
  },
  {
   "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",
    "> $\\ln{L} = constant - \\sum_{i=1}^N \\frac{\\left(I(x_i,y_i) - A\\  \\mathrm{PSF}(x_0,y_0)\\right)^2}{2 \\sigma^2}$\n",
    "\n",
    "The best way to think of this expression is as:\n",
    "\n",
    "> \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",
    "> \n",
    "\n",
    "If we set \n",
    "\n",
    "> \n",
    "\n",
    "then the maximum likelihood solution for A becomes\n",
    "\n",
    "> \n",
    "\n",
    "and the $ \\ln L(A) $ is "
   ]
  },
  {
   "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",
    ">$\\psi_{x,y} = \\frac{1}{\\sigma^2} \\sum_{x,y} PSF_{x,y,\\sigma_{PSF}}^2$\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": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A_ML = 10162.0 with chi^2/dof = 1.08\n",
      "Analytically, A_ML =  10278.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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/53bJtJG9CwD2btdQUEp5sJTy48OO11q/0dwc6RuTbxpMhr0LAPZm1EjBN5KcK6V8odnE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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f399634d7f0>"
      ]
     },
     "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\"$\\chi^2/dof$\")\n",
    "\n",
    "phi = ((image - skyBg)*(makePSF(image.shape, 0, 0, sigmaPSF))/skyBg)\n",
    "psi = ((makePSF(image.shape, 0, 0, sigmaPSF))**2/skyBg)\n",
    "print(r'Analytically, A_ML =  %.1f' %(phi.sum()/psi.sum()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## And we can also plot the likelihood instead"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7f3996ebbb70>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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gDgQ+gExR3QNxiGJaHoAKeN/7lr1EdQ/Eg8AHEMYPfrDsJap7IB506QPIBNU9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\naoDABwCgBgh8AABqgMAHAKAG/j8svkXM/g9Y2QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f399711ef60>"
      ]
     },
     "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": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9985.67810152 10000.0\n"
     ]
    }
   ],
   "source": [
    "im = simulatedImage - skyBg\n",
    "psf = makePSF(im.shape, 0, 0, sigmaPSF)\n",
    "\n",
    "phi = im*psf/skyBg\n",
    "psi = psf**2/skyBg\n",
    "        \n",
    "a = phi.sum()/psi.sum()\n",
    "print a, Atrue"
   ]
  },
  {
   "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": 20,
   "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": 21,
   "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": 22,
   "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": 23,
   "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 0x11cbc2be0>"
      ]
     },
     "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 0x11c4746d8>"
      ]
     },
     "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": 24,
   "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/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\nVnZEfGn7QRojsCPp3xyUmqXO2ZfIfxfy3xBcMelg+4GyftU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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c44bcc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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0kg5hGhOmCVGakKqSapbf00u+N1GiHqmToYArLl7m4mcJqSZkCA7KbIVg7z8d\nBTIlS0/uVphlFguY5RuGaQLgsIgcBB4m7zkwtpg3DfyTEZHPichxEUmLIyPfFtEYAH70ox/xL/7F\nv+C6667jgx/8IH/1V3/FDTfcYIHAOuL7UKtCa8ShuSWgcWGN6luaBBe2cLeP4mweQ8Y2oo0WcTmg\n62S0NWQqmWYymmQymmYymmEy6jAZdZmKQ6bjkOkoYjqOaMcxnSSmm8R0ZwOGXoFhXAQDRW1BL3ro\nZQVU0Uzz7sVFQGA7FhqzsCKDfzPwdvIiwkVN7y9U3n0z+UqCid4TxeoCM+See+45HnzwQR555BHu\nuOMOfvjDH9JqtVZ7WGYJejfhQUmoNorNh7aUqV+Y1wg420eQrSPQGEGbTVLfIUrbdNKQmTRkOp5h\nJunSSSI6SUSYZkRZRpxmZAqZKo4IngO+I2SukpHhiBA4DnHmkmQpaZaSOUVzIqQvKaDFNgYnMwNJ\nYtME5swMw2qCIiug5KsIDrHIPgMLBQMPANmc595Y1ujMutfpdHj44Yf54he/yI9//GP27t3L97//\nfbZt27baQzNL5LpQKgvlitDa7FPfGlC9oEJw4Qju9jFk0yiMjqC1FnHgkmiXTpwynbaZTmaYjrtM\nR11mkohOEtNJ0iIQUBLN5/hVwXUU3xESByBDEGInI8kyUs2K1QbZbF3AbJVglkGakSUZWZyRxhlJ\nrMQJpDZNYM6A4K72EFbCI+RTBFdwclOk007vLxQM3A7sE5He371il5Hh+NM0ub/5m79h3759/MEf\n/AFXXnkl9957L+9///utZ8A65rhQqQrNEYfm5oD6BTUqb6kTXDCCu20U2TyKjoygtSaRk9DVLtNp\nyFTcZiqeYTqOmIpjZuI8EOgkad5jIFPSXg5fybMCLpRcEFEcyQiylFj7AwI9ZdVBfzCgSUqWZKSx\nksRKEhfBgEUDZpmGITNAPi2wFzi80KZIcy30L/rnig/rnya4cfnjM+vF5OQk3/zmN9m/fz8/+MEP\n+OhHP8rRo0d561vfutpDM2eB6wrVmsPIqENjc4nahTUqb2nhXjiCs30URkfR+ihZtU6UTNCOp5lO\nppmM20yGbabihKk4YSbO6CQZnaQXCFC0HgYEfAdKxXOOZPgOxNnczMDJ/zuls1CWoWlfZiDJMwMW\nCBhzWreTBwRfLm7mD6vqp073poU6ED42z9Mnlj8+s5a9/vrrfP3rX+fxxx/nqaee4r3vfS8f//jH\n+eVf/mVDedmYAAAgAElEQVSCIFjt4ZmzwPPBD4Ray6W2qURte0B5e51gWwt3yyiMNcgaNdJyQOxm\nRFmXqaTDZNxmImozEXaZDCOm4pSpKKOdpHQSpZsqaZZf9JVix0MgdfMKAAECl9nMQS8b0J9yzLsJ\nZXlWIEnROEWjjDRKSeOMLFULBMwZOw9aDS/G3UCrlxUoeged1kIdCOcr1XkJeMdyR2jWlh/96Ef8\nl//yX3j88cc5duwY1113HR/+8If56le/agWB56FS2aHedIqNhypU3lKldGExNbBplKzVJKtUCF2h\nnXVpR20mwikmohkmwi4TUcxkmDIVp0xHGe0ko5so4WzRYP49npP3EVDyvkF5T4E8GDgZCPSHAr2t\njfNmAloEA1mUkIYpWZyiqUUC5swNyTTBIeBEsarrQRa5CnDBmgFV3d97ICKXAZedyQjN6nv++ef5\nz//5P/P444/z/PPP88EPfpBf+7Vf47rrrqNSqaz28Mw5VCoLjRGHkc29YKCBf2ELd2sRDNSqJOUq\nXUmYSdtMJm3GwxnGozYTYchEmDIZ5YHAdJzSjpUoywjTk3f6guA7+QFOvgOioySZkConMwjFtb2/\nvdApmYEoJYvSIhjI0MyCAXPmzpc+AyJyG3kPgZfmaTd8bbE/wW7yAsJHgNO2gl1ommD/nMfPiMjX\ngC8PeItZg6anpzly5AiHDh3i0KFDHD9+nA996EP81m/9Frt27cL3/dUeojmHRHrdfYVKw6WxIaCx\ntUx1W43SBS2crS3Y2CJrNUgCn9B3aachU2mH8XCS8TDkRJEVmAgzJkNlJs6YjjM6cUacZURpLxDI\nv6/kOqSug0iG57qzWYFUZc7eAlLsEldkBrIMTZJiiiAhC/NgIInyaQJjzKwvAfuK45Tugr1Owar6\nBHkXwkVZaJrgBTgllzeKNR1a89I05dixYxw8eJBDhw5x7NgxfuEXfoFrr72Whx56iEsvvRTXtQUh\nw8LzhUrNoVJzaG4rU7uoQunCJt62UWTjGFmzQVrO9xmYySLacZeJuMN42GE8jDgRxpwI0yIQSJmI\nMtpxfnSTbHYVAeSXdtc5uaGQ5zikvamBYn+B/HXMbk7kiIODg2QKSQxxgoYJ2k1JuilRx4IBc/ac\nR+2IH1XVj893Yu4mg4u10DTBPuDRvsfH+1cWmLXj5Zdf5tChQxw8eJAnn3yS7du3c+2113LPPffw\nS7/0S9RqtdUeolklni/Umw4jGz2a28vFLoQtvK0jyMYx0maNpFLK6wTSiMm4y3jY5kTY4UQvGOim\ns1MEk2EeCHSSjDDJSDUPBqSoDfCK7pMiELhS1AmcugpARIpgQHDEzf+BVoU0QeMYDROybpIHA92M\nOLRgwJwd58s0AfDdBc49spyAYKFpgs+LyCXAreTNhh4FLBhYA1577TWeeuopnnzySQ4ePMj09DR7\n9uzh+uuv5z/+x//IBRdcsNpDNGuE50Ot4bJhs0d1W5nqRXWCC5t5d8GNY2g5IPEdQidjJomZjKeY\niDrF9EDEiW7GeJgyHmZMhhmTYUI3yYOBOD25OsCRvGjQd/LWhoJScnvnT249XJQL5gEBDiJOPlWQ\nZXMyAwlJJyXupMRR3oHQmDN1HhUQbhh0QlUnRORtS/3AhaYJ9pIvUThM/nf4SyLy4HI3KhKRlmUW\nlucf/uEf+LM/+7PZ48c//jHvfe972bVrF48//jgXX3yx7QdgThIolYSg7OQdBrdVKF9YJtjWxN08\nCmMjpPUqWeDRcTJmspipNGIi6nCiG3K8G3Gim3C8mzLeTRkP03wVQZgyHaVERVagtzogA1xHyIqi\nwaD3/JxCQUcoWhQLnuPgioOL4GQgSQZRjHYjtBOTzCTE7ZRORwlDSJNV/PM0Zu0REblaVb894PzO\npX7gQtMEl6vq2+d8+21L/YKionEf+RKH3+17vkXe8vgF8ijncFHwsOxz54sf/ehHp1z8f/rTn/KL\nv/iLXHXVVdx2221ccsklNu9vBnIEyjWH5ohLa2uJ2gVVShfV8baNIBtH0ZEWSbVM7AkdYqaTLhNJ\nJ58emA0EEk500iIrkDAVprSj/EjSjDhR0my2XRDqOqjrINKfDehbQFjsXug5gus4eOLi9YIBVSRJ\ni2AgJutEpO2EsJ3RaUM3hMQyA+YsOF/6DKjqPSLygog8rKr/tv+ciOwAzl5mgHyt4lxL+gIRaTJ4\nC8X9wMHevEbxg92kqs+dwbl16dVXX5298B85coSJiQl+6Zd+iauuuopPfOITXHzxxXbxN4smApWq\nQ2ujx8i2gPoFVUoXNXC2tXA2jZGNNEkCl8gT2knMVNpmIprmRDE1cLwbc7yXGQgTJsKEmTClG6d0\no4w0zcjSjCwrth2mr+GQI339BPrGxMmeA544eI57SmaAJIUwzwxk7Zh0JiZqp3TaeWYgs2DAnAUr\nNU1QXJD3ke8PcBy4WVWfHfDa5d7g3kK+XfHt5NfZZ8g3JbqFPKu/JAsFAzuLpYTfKwZ4E/kPt2i9\n9Y8icrz/+aJnwY1Af6bhMPApEbm/+K4lnSOvbVjzVJWXX375lIt/u93mqquu4qqrruLXf/3X+bmf\n+zkc5/yIYM3KcRxwPSEoC9VRn8bmEvVtVcpb6/hbR9ANTbRXMCgpbRKm04iJqJuvHujGnOjGnOgm\nxfRAwmQ3LbICCVGcERUNgLLejkGSFwQikDr5XX5/d0FgtrjQdfLDc4pgoJcVSBOIYzSM0E5EOhOT\nzMTEnZQoUhKbIjBnyQpmBu4iv8YJeb3dwwxu2LesG9xiuf8O4HeAXk+BceBzZ3U1QVFAuJt8K2OA\nO85iOv5yYHxOs4RjwP3kF/cTyzi3JqkqL7zwwikX/yRJuOqqq9i1axd3330373rXu2zO35wxvyRU\nGy71EZfmtjKVC2v425o4m0dhdIy0XiUJHLqSMpN1mUxCxsM242GYTwuE+dGbGpjopsyECd0wIYpS\nkjglTfIGQFqkAsTJew+n4x22Pv0jtkxHTI1ViN//TpxtjWJqgNk6Ad9x8Rwf3wlwESRJIE7yuYB2\nl3Q6JJ4OiaZi0jC1ZkNm3RGRZv+yPxHZB8y7YdBCN8Ys4ga3qMO7/YwGXFiogPAaYI+q3lE8vuFs\nfGHhcvLUSb/j5L0MlnNuUfs1rwRV5Yc//OEpF3/XdWfv/D/96U/zjne8wy7+5qzzA6HRchnd4tPY\nVqZyQR1/ewt38wiMjZFWAiLfpUPCdBoyGU8zMbuEMOFEN+VEN2W8mwcCk0UgEBbBQJpkJ4OBYg7A\ncR10KuS6x3/AJ5KMsueQ/nSGL//dOP/r//wnuLXmbFbAl7xWwBcPTwJcFSSLIQrRboi2Q7KZiGQq\nIpqOSMIUnbuJujFnYCWmCebpCPgL5LsIzmehG+MVtdA0wS2cOi3whIgcUNXTtjU8QwvdCqzJ24RX\nX32VAwcO8MQTT/Bnf/ZnlMtlrrrqKvbs2cO///f/nh07dtjF35xzQcmhMeKwYatPZXuFygV1vO1N\n2FQEA25G5GZ0s5iZtMtkNM14GDIe5lMDx3sBQXiyViApjjRKyVIlS7I8EOitEwQu/PMfcUeUEHj5\nP7QlR7gjTPmtbz2Pu/fyPDMgvekBbzYz4CQUUwQhhCG0Q7KZIjMwHZGGWGbAnFUr2Wegr3h+B3k9\nwHxp/7N+g1vUIHwJ+Frx1OF5ApQ3WSgYODZPwcOeZY7vTZ/NyemHnjHyXRGPkQciSzk3frovvO++\n+2Z/v2vXLnbt2rWU8Z6i3W5z5MgRDhw4wIEDBzh+/DjXXXcdH/jAB3jggQdsq98FHDlyhCNHjqz2\nMM4bjgOlikOpLLS2BtS2VQguqOFtaSEbRsiaTbJKQOYoHY2ZjsNi5UC3mBYosgFhymSRDZgJE7rd\nhKSbkEQpSZTvD5Cl2SkbBomTdxXcPBVScoq2wuQxQtkVRic6BK5QcoWS61ByPUquT8ktEbhlvDhC\noi60O+h0h2wyJJqM6UxlzMxAFNlOhebskgH/PR058hxHjpzdGnRVfUJE9pAXB+4TkaMrVOi+E5Be\nG4AiKDntFP9CwQAi8klOVij+DjDftsbLcZQ3Rz6Xk8+VHFvmuQX1BwNLpap8//vf58CBA/zpn/4p\nTz/9NJdddhn/9J/+U/7wD/+QSy65xAr+FmluIPabv/mbqzeY84DjkrcaHnVpbSlR3V4l2N7A3dJC\nNoyizQZJOSB2lHYaM5V0mIjajEch492Y8TDJj27KRG8JYZgShglxkRFI4zTPCGR54WCvYNARB0V5\nrVEifKNDxTnZfTDOlJmxCpsdIXCEkuvmwYDnE7gBgVvB0wSJY7TTIZvukk1GxLPBQLGKwKYJzArY\ntesSdu26ZPbxb372oVPOF313Lmf+7LQUzz+gqq/0nyge31q097+CN2cHBt0Yn/YGd5DiJv6WvseL\nqvVbcKOiIqK4pRjcA3M3L1ouVX1WRA6LyA19TYx2k9covLqcc2djXP3eeOMNDh06xIEDBzh48CDl\ncpn3ve99/Kt/9a94/PHHaTabZ/srjVkyxxGqNYfRjS7NrQG17VVKFzSQLS2cjaMkzRpJ4BK6SjuJ\nmIrbTIYzTMzWCSSzjYUmwjwz0OkmRGGeGciSlLSXFSgKB8URxBFU8umCH73nAh788RSfSDOqIkSZ\n8nsVn/L/9q4iK9DLDLizmYGSW8HRGZwoQjttdKZLNhkSTya0p9IiM2DBgDnLllmEUlz7zuT691Jx\nzLXQjfGKOiUYEJFL+tMYg3Y9EpG3ku9VsOA8RDF3cQv5nMmtIvKMqj5ZnL4ZeEBEdpL3L7hbVV89\nw3NnJIoi/tt/+2985Stf4ciRI/ziL/4i73vf+7j33nt5+9vffvoPMGaFuG6xCVHdpTrmU99Sorq1\nRrC5gbuphY7UyGoV4pKfrx5IQ6aTkKk4ZCLK6wTyuoA8CJjq9RIIU8IoIQlPZgSyJJtdStgrHOyv\ngXGbFY78ysW88N2/Z2s7ItxYo/LP3sXPbK5T8oSy51D2PEpeQMnx8XFwM0WiGDohOt0hnewSj3eJ\npiKidpZ3HbTeAuZsW4GK1OK6d8WcRnknete+4iZ7p6ruX+jGeJHfdcnZmnqYmxl4uegy+PCgC72I\n3Ej+g71yug8vlj3MG1EVnz/vrkvLPbccqspzzz3HV77yFf7oj/6Id73rXXzkIx/hoYcesrt/s2YF\nZaHWcKiPeTS2lilvr+FvaeBsbKEjLdJaldT3CEVpZyFTWZfJqMtEGM8GABO9NsNRwnSY0g0T4qhv\naqDICPSyAm9KkAqzWQJ/pMLEP3s3WdVnc9VnpBZQdh0qXn6UPY+yGxCIh5tlSNSGTgdm2uhkh3ii\nS3e8SzQd5zUKWbH7odUMmLNpZZanjJHXCCh5vcBxVe1fJngt+Q1y77p4Jje4x0Tk8rMREJwSDBQb\nHDwC/NsienmJvLJxjHxpnxYDXbfd/np+8pOf8NWvfpXf//3fZ2Jigl/91V/lO9/5Dm9725K7OBqz\n4oKSUG+5jG72qW8tU95Ww9/awN3YgpEWWbVCHHh0JaGdRUxFM0xGHSaiaHalwESYMhHldQIzUUIY\npXlGoCgYTJM5DYYgDwDy/0FEZg/HcfBdh8B1KHkO5SIj0AsGKq5P2SsROD5elkHahW4HpjtkUx2S\niQ7hiS7hdJzvUphZIGDWJ1V9GRiYSlbVe+Y8PpMbXIHZrQLuBl4Ebl9OtvxNNQPF3fw9vVQHeWXi\nS8BLxQ+5rr3yyiv8+q//Ot/+9rf55V/+Zf7Df/gPXHXVVVYAaNYFKcr1S2WHxohLa7NPdUuZ0rYa\n7uYmsqGJtlqkvkvsOXTTfIpgKp5hMgqZjOLZQGC82HxoKkpohylxmK8emFs0OLuCoBcI9AcGjiCu\n4LqCX9QHlF2nyAoIFc+l4rmUPZ+yW6IkHm6SInEbbbfzVQQTHaLxLp3xkHAqIQnVagXMuXF+Nq64\nnbyXwRPk1+tnROSypQYEA6+Aqjqhqk8U8xpPnA+BwGOPPcZ73vMerrzySv7n//yffOUrX+Hqq6+2\nQMCsC64LlaowMurQ2BRQ3VIh2FrH29RANrTQZp20Uib2PLqkzCQdpuM2U0UQMBnl0wKTUcZUlO9A\n2I5S4vBkANALAq5755U0S7VT/u1sVeq8/+f/MeLkmQDHzQ/Xcwh8h4rvUvVdqr7kh+dS8TyqXomK\nW6LslvHxceMU2jPoVJt0vEN0IqQznjA5obTb1n7YnENZtrhj/RgHXlTVK1T1DlW9jnwa4neW+kFD\ncxX8xCc+wV133cU3v/lN7rnnHhqNxmoPyZglOSUY2OhT2VKhtKWWBwNjTWjWySoVEt8jlIyZtMNU\n1GYq6jIZJUyGCRNRHgxMRhnTUUonSvPugr0agaJg8P976S/57Ac+RqtcQxWalTq/df3tfOeV789m\nA3qBgOu5+J5L2XOo+Q5V36HmSfGrT9ULqHhlyk6Fknq4UQIz02STM6TjbeLjIe3ZYADieLX/pI1Z\nN3obFM1S1WfI+/IsydAEA2+88QbPPPMM73nPe1Z7KMYsi+MKlarDyJhDc5NPdXOeGXA3NXDG5mYG\nMtq9zEDcywwkTEa9qYH8mA0G+jMDacaJ6Sl+4+v7+OwHb+dnxrbxW9ffzr/7r19iojud1wk4guMK\njncyM1D2Xaq+Q9VzisyAQ8XzqPRnBvRkZiCbapOe6BCd6NI+Ec8GA5YZMOeMZos71o9DzL9D4Y6l\nftCCTYfOJ3/8x39sLYHNuuR5+VFrONRGPaqbAsqbqvibajgbGkirgdbrpKUSkQvdLKKdhkzHIVNR\nlK8WiLLZYybO6Eb5DoRxnBV7DhQ1AmmxCVGmTHSm+b8O/yEvfPYR3nHfzUyGM/nUgOcUQYCbTw8E\nDlXfpeY71HyXuu9S811qfkDNL1P1KgS4eGmKE3bRmQ7ZxAzJeJvweJfuiYhwJiWO1JYTmnNrfV3o\nT6vYUPBXROR7wB+TFxTezjL6FCyYGRCRS0XkkyJyyUKvWw8sEDDrVRBArS40Rx1qG0uUt/QFAiN1\naNTRco3E8wk1YSaeYSbuMB1HTMcx03HKdJwHAe04oxOnRHHRTGhOEJAfeY1gq1Ln31z7z3nHfTfz\nb/b8c0ZrzZOBgO/iBQ6lok6g5jvUAof6bEDgU/MDql6FilclUMGJOtCZRien0RNt4uMd2sdD2uMx\ncdd2KDQr4PzLDABcA7wCfB74t8Bj/bsmLtbppgnGgWeBa0XkeyLyNRG5rWhTbIxZAUEA9To0Rlxq\nGwPKW2r4m+u4G+vIaAPqdbRcJ/E8Qk1oJ9NMxx2m4pCpKGYqTpmONQ8IopR2nBLGxZbE/b0EelkB\nVVrlvEbg331jH6+e+F985ltf5jc/sJfRehPHz4MB33cpBy61wJ0NBOq+Qz1wqfleHgz4eTDgKzhR\nB21PoVMzZCdmiI536LzRpT0eEXfSNboNmTFrW1Hsf7OqOqo6Nnfp4mItGAyo6svFSoLPA/cUjROW\nXJiwFjzzzDOnf5Exa4hIXjRYKjvUmy6NMY/KxjLB5irexjrOaAOaDbJqhTQIiB2hqzHtuM1M3J3N\nCsxEKdNRnhVoJxndOCPu6yw4mxnQ4gD+yc6L+fQ39jERziCOMBXN8NnD/y//eMfFRVbAxQ9cKr0A\nwHeLQMCh7nt5ZsArUXVLlKWEnykSdmB6imximvR4m+h4l854RGcyIe5m1lfAnHvrfDVB0f13qe9Z\nVGZ/KQWEI0XLxMc4g00UVssHPvAB9u7dy2uvvbbaQzHmtFwXKmVoNKA2mgcBpc01vI01nA11nFYD\nqTfz6QFX6GZt2skM7aTDTBLRjhPacUo7yQOATpLRTTLCJCNOM9L+bEARBECvkRD86Q/+nKloBscV\n3GJqYDrt8MQrRwl8l0rgUi+ORuDSDFyaJZdm4NEIAup+laobECB4SYLT7sLENNnxaeLXZ+j+pE33\nRERnKqXThTixJkNmBazzaQJVfUVE7lzsBV5E7lzsZy86GCiCgIaIHCDfSGFd+Zu/+Rvq9To///M/\nzxe+8AViW79k1jDXhXIZGnWojRTBwKYq/sYazlgdadWReqMvGOjQSdq04w7tOGQmSfJAINY8IxCn\ndJOMKMlIkowkzcj66gTmayTUWz7YqxFwfAc3cAmCPBioBd7JgKCUBwWNwKXhB9SDMhW3REmLYKAT\nIpMzZCemSF6fIXytTfdESGc6o9vNVxBYMGDOuXUeDEBeNAi8TUQeLmr6rhGRS/qOvSLyoIh8EfjS\nYjsGLzoYEJH7gQ8DE8y/lGFNGxkZ4Qtf+AJPPfUUBw4c4B/9o3/EN77xDRJbx2TWoNlgoAH1EY9q\nkRk4GQw0oNaEcj2fHkg7fZmBkHYSz04LtJOMTpzRjVOiXmYgydsMZ1kGGXMyA4LjyCkrB3p1Aq7f\nHwy41Et5INAMnDw7EHg0glKeGXCCPBiIE5xOFyZnSHuZgdfadI5HdKbyYMBic2MWT1UfU9VbyLsO\nXg7cURy3Am8UDYg+XnQUXpSlLC38Wq8wQURuAB4/zevXpHe/+91861vf4pvf/Caf/exn+ehHP8qH\nPvQhbrrpJq6++mo8b2hWW5o1yHVPNheqNF1qYy6lDWX8sQruaBVp1qBeJyuXyDwhJiHUmE4a0kki\n2klCp5gW6CRKN1HCRAnTjDhV0iwjy/qmBXr7DDgn2wz3Nh8SR2YzAn7gEZRcSqW+AKCUTw+0Aodm\n4NEMAhp+mapboiIBQSa43Qg6EXpimvT1aaKftpl5PWTyeEp7OiOO1TICZuWs8bv+pVLVZ8mL/IF8\nBSBwuYg0VHVqKZ+1lJqBK4qVBE3y7MC6JSJcf/31fO973+Ppp5/m7W9/O/feey/btm1j7969HDx4\n0KYRzIoTyfsJlMtQqTtUR3yqGyuUxip4YxWcVhVp1JBqjSzwSSQjzEK6aUg3CemkMZ0kPRkIpBlh\ncUSpkmSaBwJzdiDstRcWJ28i1D814Ab54ZdcyiWPWtmjXvJoljxaZZeRkstIyaEZ+EVGoELNLVOW\nEkEiuJ0uOj5BenyK5LVpuj/pMP16xPjxlPaMksQWCZiVo5ou6liviuDgFZZR17eUmoH95CsJHmEZ\n3Y3Wqh07dnDnnXfy3e9+l+9973v87M/+LJ/+9KfZtm0bt912GwcOHLDAwKwY34NyCao1h2oryGsF\nNlRwx6o4I3kwQLVOFgTEThEMZBGdNKKT5MFAO1G6SUa3LzMQpUWdQBEIKEVA0Nt9sJgWkKKrYP/U\ngBd4+CWPSsmjVvJolDxapTwQaJWc4vBoBiXqQeVkZiChmB6YIDs+SfzaDN3XOsy8HjN+PKM9k1m3\nQbOy1vlqgp6FCghV9TD59MGSLKVm4CBwC3lno5GlftF68Na3vpVPfvKTPP300xw7doyf+7mf4777\n7mPbtm3ceuut7Nu3j+eff342xWrM2dJbRuiXoFITKk2P8mhAaUMFf7SCO1JBmlWkVoFSlczziMmD\ngTAN6SYR3SShm6Yng4AiIxBnmk8R5I0FT/73O6dYUFzB9fJpAS8olg+WXIKSS6XkUi25NEoezbJL\nq+SdPAK/mCKoUHMrlPEIUsXtRshkOw8EflrUCbwe0p6ImZlRonBd/LtrzFr0gIgstMHOS0v9wKVM\nE9xd9Bk4zDqfJliMn/mZn+Ff/+t/zXe+8x2ee+453v/+9/M//sf/4Oqrr+Ytb3kLv/qrv8pXvvIV\n/u7v/m61h2rWOccBz82bC5WrLpWWT2U0wB8p4Y6WkVYFqVehWkFLZTLfI3Eg1pgoDQnTiDBLiNJk\n9uKfZEqaQaZKVvQOmF0w0JcJmD36pgW8wMUreQRFNqBe7mUDPEaKqYFW2SkyAwGtUoWGX6XmValI\nmSDOkOlJdHyC7PUp0v81Tee1NlM/DZk6kRB2MmswZFbHebCaoHAtMC4iz4vI50Tk6jnnl9wPaCnV\ncp8SESXfGOFrS/2i9ezCCy/kIx/5CB/5yEdQVZ5//nm+/e1v861vfYu77rqLZrPJNddcw9VXX83V\nV1/N1q1bV3vIZh0RAdcDP4ByzaXa9KmMBARjJZzRMk6rAo0qVKtoqYR6PmmWEGtCmIZEaUyYxoRZ\nSpRmRCnEGUVA0F+g9+aMwGzxIMVzIri+g+c5eVOhkks18GiWPVplj5GSN1snMFJyGQkCWkEeDNS9\nGlXKeNEkMj2Fnhgne32S5CfTdH7SYfK1iKkTMVFnXfxja85H6+NCvxh3k+9W2Nuy+O7i+nyY/Bq9\nB/jUUj5wKcHAPvLtEvcADwBL7n18PhAR3vnOd/LOd76T22+/nSzL+Ku/+iuefPJJvva1r/GJT3yC\n7du3c80117Br1y7e+973smXLltUetlnDTmYGhHLVodryKY+W8EdKOKOl2cyAVitoqUTm+aRJh1gT\noizPDORZgZQoy4iyvuyAMpsZgDzw6M8MqHOymFDcvJDQ8x383hLCkpsXDJZdWsWRZwccWoFDqxTQ\nLFVoeFVqbo1K5kN8Iu80OD5OWgQD3dc6TL4WMn0iIYpW7Y/aDLvzJBgoeg1AURsgIi3yoOAW8v0J\nWkv9zKUEA7cDNwGHlrMJwvnKcRwuvvhiLr74Yn7t136NNE159tlnefLJJ/m93/s9brvt/2/vzoPk\nuOv7/z/f3XOtdqXVYXNjW5Il+SDYOlyEwy4bSTimMAWWj5gkFQLYweSkAAO/QHBS38J2wbegKiTg\nIwcBUsGyCVBcX59yDBXAa1smNthClmTwocPa1d67c/T790f3rFarHWl3tTs9s/N6UFOw073Tn/Hg\nnVe/P9cHOPnkk3nLW94y9li1apU2TpIxmQwUFhjti0IKnVmyi3NkOvMECwvxGIG2AuTbIJOjYkY5\nKlGMShSjMqVKhVIUJV/+UI6q4wI4IgAEBkFghGFA5GARuB2u1QcGYRAQhvF2xPlcvO/AolyYDBbM\nsCQfsiQfsjifZXE+R2e+wMJsOx1hB4UoJDMyAqNDeHcfvr+P4r4BhvcNMbRvhMFDJYoj8YBBjRMQ\nmWYwAbcAACAASURBVF3JegJ3Jo/qukDTMmkYMLNF7t434ekbgaXEmxb9qbtfPN2LtYIwDNmwYQMb\nNmzg+uuvH6sc/PjHP+a+++7j7/7u7xgeHj4iHKxdu5ZsNpt20yUFZpDJxhWBahjILc4TdhYIFhZg\nQQErtEG+Dc/kqBiUoiKlSpFSVKIYVShHEeUkDFQHCVa/bw0IzAjNCAMjDI3IDQtCosAJkjaEZmTD\ngExoFLIhC3LxozMfsigfsKQQjj2qQWBRrp2OTAftYQf5YoVgeBjvH8K7DxHt76O0d5ChvUP07osH\nDRZH4i2KNf5WUjNPKgPHYmanzWSzolqVgUNmthW4phoKkvmLMIMpC61sfOXguuvigspvfvMbfvKT\nn/DjH/+Yr371q+zatYvzzjtvLBy88Y1vZOHCYw0UlfkgLtmPCwOdSWWgM0+4KK4MUK0M5Ap4Jkdk\nZUo+SikqUaqUKUVJZaACZXcq1QGDfnisQBwISMJAgBMHhkrkhGZxN0Vg5MOAXBjEWxInSwzH0wbD\nsarA0kLIknyOxbk2OnPtLEzCQCYawEaGobebqLuXaH8/xX0DDO4dom/fCEP9TnE0UhiQdM3DspSZ\n7SSe7l8dL3CvmX3A3W+fzuvUCgO3uvsHJ1xwI/DwJBUDmaZTTjmFU045hauvvhqAQ4cO8T//8z88\n9NBD/J//83949NFHWbNmDRdccAHnn38+559/PieffHLKrZbZVJ1KmMlAvs0odIS0LcqSW5QlsyhH\nsDCPLchjhTzk8pDN4WGGipcpR2XKUYmyV6hEEZXIk1kDh79ozWwsAGQCIxsY8eShOABExOeHlhwP\n4zBQyAS0Z4OxfQbiGQMhSwoZFheyLM5n6cy1sSi7gPagQJsHZEsRweAIfqif6GAvpf19lPYNMLx/\nmKHuIoO9FUZHoFxSEJCUzc/KwL3EAwo3EY8b+CCwBJiVMNA18Ql3v8/MPjDdC8jxLV68mEsuuYRL\nLrkEgNHRUbq6unjooYe4/fbbed/73scrX/nKsXBwwQUXcMopp6TcajkRZpDPQaEN2jsCCgszZBfm\nyHTkCTpy2ILcuCCQxcMMURAQRU4lKlPxChWPiDwiipcQGqsAxN0CjIWAXBhQcSeMnDBIViEkHlMw\n/pxCJn50ZAMWJnsNdOarMwhyLMnnWZyLxwm0hwto8yyZ0SJWHMW7e4j2H6L8Qh8jLwww9EI8lXCo\nr8LoqDYiEplDHwc2JpsJ3jXTF6kVBmotKjQvFxtqNPl8nje/+c28+c1v5hOf+ASVSoVf/OIXPPTQ\nQ3z729/mIx/5CIVC4YhwsGbNGg1KbCJBEH/Pd7RD+8KAtoVZcgtzZBbmCNpzBAvyUMhBPq4KEGbw\nahjwOAxEHlGpdglQ3WsAwmpFIDSykVFxI/KATOBkxk81NMaCQD402jIBbdnxYaC6lkAcBhbnF9CZ\nb2dh2EZHuIBC2ciM9sPAQBIGeqm82MfIiwP0VcNAf4XRIuoekMbQ4JUBM7uGOKd3TXW3QeAO4FEz\nWwFsdfdnZ3LtWmHAzOwyd5+4GdHKmVxETkwYhqxdu5a1a9fyl3/5l7g7O3bs4KGHHuK///u/ufHG\nGxkcHBwLBps3b+bMM89UOGhgQVIZaG+HBR0B+bHKQBwGrC0HhRyWyxFlsngY4kl5P64KJJUB4i6C\n6tSBwIzAnIzFd/zlMA4C8TiBw10J1fEKudAohAH5MGBBNn50ZAMW5YxFuWAsCHTmCnTm21ica2eB\nFVhAG7moFG9C1NsTh4F9hyi/2Mfwi4P07x2iv7vC0CCaSiiNo05hwMyWE0/H3wB0A1eMG3c32fnr\niL/Ur3X3+6d5ud3E3QJXEK9MeC9xKDjxbgJ3/5yZdZnZVcB/Eu+KtAlYMc1GyhwwM9asWcOaNWv4\nwAc+AMBvf/tbHnroIR544AG+8IUvUCqV2Lx5M29729vYtGkTL3vZy1JutUBcEQiTqkCuLSC3ICDT\nniFckCFYkIFCBnJZyGUgk8XDLB7Y2Jd+8vUPOMbh2QJBYGQCyATVu32oeDxOILSITBDgySyDw90J\nh8NAIZOEgUwyXiAbsjCXoTPfliwq1EZ7UKDgIdliiaBUgoFhogM9+P4+Rl/oZ/jFIYb2DjPQU2J4\n0CkmFQGRhlG/AYTXA1uI/3W7k/iLftVkJyZBoIu41P/ADK718epWxcl6A1cRjx2YlTEDuPuGZEbB\nXcT3HbuSC0gDeu1rX8t73vMe3vOe9+Du7Ny5k3vuuYetW7fyoQ99iNNOO423ve1tbN68mbe85S20\ntbWl3eSWFASQzUI+b+QKAdkFGbILsoQLslhbFitksXwmPimTjbsHLEgiQIR7HASqj7GBglYdDEgS\nBDyeQmgQWkA2iEsH8dgCIwzi38mHRiFjtCWVgfZsSEcuw8JsjoXZHIvy7fEUwrDAAjLxegKjIwSD\nw3hPP37gENELvYy+0M/g84P0vTgSh4GhOAxECgPSYpKp+deN+/kW4FhT/e4gvpOfSRCorjEw/n/f\nmjym5ZiLDrn7FRCXPNx993RfXNJhZqxatYpVq1bxoQ99iHK5zM9//nPuuecebrjhBn7xi1/wxje+\nkc2bN7N582Ze//rXEwTT2aZCZipMwkBcGQjJtWfILIgrA9aWiYNALpuEgQyEWQiCOAp4JQkEhzvf\nx68jEFcFoBxAFMZdRGHFyAZO2TmikpCxuC2FahjIGO3ZkPZsyMJslo5sjoXJYMFFuUUsCHLkKk6+\n4thoCevvx7u7ifYfovJiHAYGXhikd+8Ig/0RQ0Ou2QPSeOrQTTDJjLvzgGsmO9fM1hJX3J8xszuA\ndcC9E2fzTfida5Lz7knO7ZtwfLJ1go5rSisQKgg0t0wmw5ve9Cbe9KY38ZnPfIbe3l62bdvG3Xff\nzZVXXsng4CBbtmzhiiuu4M1vfrOCwRwKwqQyUEgqA4UMmbYMQSGDFZIwkM3E6xOHIVg8VqBaC7Cx\nL3UjsGBsMaF4VkBAJYziFQaTWQXZwJMBhEkVgcPTCTOBUagOHMwEtGcztGczdGQLdGTb6Ei6BtoI\nyZcjMqMlwtEi3tNPdOAQlb29jL7Yx+gLgwzuG2GwuxgPGByOpxHOwynd0uxqhIFt//MM2376zKxe\nKpmOfwvxGgA7gckGBG4g/lf7Wnd/Ninz7zazHnevtbdAN/FmgR8E7jSzR0jWGEjGGyw1sxXTGIAY\nt7cVtuM1M2+F9zlTTz31FFu3bmXr1q289NJLRwSDMAzn9NpmhrtPeaSjmfk3WD2XTZpTHe2waBEs\nWhKy6OUFFr28QOGV7WRf0U7m5R3YSYuwZYuwxZ3QsRg6OqlkMpQzGUatwlC5j+FyP4OlkfhRHmW4\nXGK4VGKkUmG0EjFaiZcnLkbxwkIR8R16kHQlhBYHh2xgFDIhhUxIWyZDeyZHezZHe3YBHZl4F8J8\n5OQiyI4UCQeHCAeGqOztofJiD6MvHGLg2X4Gnu2nf/8ofS8VGThUplSKBw3qXzmZS3/Ajmn/7Yj2\nfO74JwLBaR+b1msf45qnEe/lczmwfuIXtJl9DPiEuy8b99xNwBZ3rzXGoHN810ASOjYTj+tbS7yB\nEe5+3nTaOp29CWSeOuOMM/j0pz/Npz/9aZ5++mm2bt3KX/zFX3DgwAEuu+wyrrjiCt7ylrfMeTBo\nBWGyVXE+D7lCQKaQIcjHFQHLh1g2TKoCQTzAIAjAAjBLKgIBRpBUBQIyFpC1gHIYjysw87gbIIJs\nVB1EGE88DJP1B7JBNQyEFDIZ2jIZ2jI5FmQKtGcLtGfaWZBdyIJwAZnRQcLyEDY8hPX24j29RPt6\nKb/Qy+hzfQw8N0TP88MMHiozNBCvKRBFCgLSoGZYrkpK8+uZfPNtS56/2d33jD+Q/HxVskrgBo6u\nDuzi6Cn7z3CMwfrjg0Dy832MWxk4CQfTXgZAYUCOsGbNGj71qU/xqU99ih07drB161b+6q/+in37\n9rFlyxauu+46zj777LSb2ZSMOAxkc0auYGTzAZl8SJgPCXJJEMgG44JA0stvRvyfgKD63xYQBiHZ\nIKQSRlQ8wnBC8/gL3+O9Csb/5cqMLUQUkAsySRjI0ZbJ0pbJsyBsoy1TYEHYzoJwAQUrQGUYRoow\nMEB0qJfoQDelff0UX+hj+PlBBl4coXff6NiAwXI5pX+4IlMRzSyluvttwG0ncOVdyWOiR2HSfv5H\nZ3qhJBxMmzqHpabVq1fzN3/zN2zfvp0HH3yQk046iU2bNnHxxRfzox/9CHW9TE116eFsFjI5I8wG\nhNkQyyZzAUOL/00MII4MJJMF4kUBzMEsILCQMMiSCbJkgyy5IEsuzJIPs8kXei6+w8/maM/kD88I\nyObpzObpzLUlj3jxoM78Qjpzi+jMLWZRtpP2oI02z5ArlQgG++HQS/jBg/j+bsovHqL0XC8jv+1n\n4LeDdD83QvfeEgO9FYqj2oBImkQUTe1xAsysM7k7H/sZ6KmuH2BmG5NKQ3U83p3A+PEBm4nHGtSV\nwoBMyerVq7nhhhvYs2cPV199NR//+Mc5++yzueWWWxgaGkq7eQ1tLAzkIJs1MlkjyAZYJsAy8YR/\nC5JVgKq8ug9xFM8AIK4EhJY5HAaSIFAIs8kdfo4FmTztmTwd2Rwd2TwLs3kW5vIsyhVYlCvEYSC/\ngEW5djpzHXTmx4WBsC1ZR6BMONgPvXEYiPZ1U3mxh+LzvYz8po/+3w7S/fwI3XuLcRgoQkVbE4tU\nLQVuMbNfJ0v4b3T3q8Ydr/bvA5AcW2xmNyZjCH4+3QWDZoO6CWRa8vk8733ve/njP/5jHnjgAb74\nxS/y6U9/mmuuuYY/+7M/41WvelXaTWw4ZvEwgHjpACPMBYS5kCDpErAwWQGo2itQLe6PDwMWxFMI\ngwwZzyWrD0ZAFD9fMcpBSMUjKlGUFBaqMwgsmVkQkglDskGGXFAgF+YphAsohO3kLEtQGiYoj2Cl\nERgawPv64GAfniwzXHq+n9Hf9tH/fInu50v091SIIidSVUCaRR0Sa3K3f/oxjh+15sD4dQnSosqA\nzIiZ8da3vpXvfve7/OQnP6G3t5ff+Z3f4aabbqJUKqXdvIYTBMm4wAwEY1/+1QAwrmsgcrzi8dJ9\nlQpWKSe33RWCKCIkJGs5ckGefFAgH8aPQqaNtkwbCzJttGcXxDMCcu10ZNtZmOtgYW4hHdkOFmY6\n6AjbaQ/ytHmGfMXJFotkRoYJBwYJevvgYC++/1BSDTjE8HN9DD43QP8LwxzaV2TwUJnRoQqlolPR\nBkTSTCKf2qMFKQzICVu1ahVf+tKX6Orq4sEHH2T9+vX89Kc/TbtZDcMsnhAQhsnkALOxvQGApAKQ\n/HcUxcv2VSpJCIjDgFVKWBSRISAb5smFBXJhG/lwAYVwAW2ZBSzILGBBpp0F2XY6sh3JYyEd2UVj\nj/bMItrDdtrIkY+MbLFEODyA9fVATzd+sBvf3030Yg+V53oZ/W0vQ7/po/+5eEGhnpcqDPZHlJX3\npBnVYcxAs1I3gcya5cuX84Mf/IBvfvObXHbZZbz73e/ms5/9LJ2dnWk3LXWBHTlTsJoGxooC1Y0D\nIodKFIeBqIxXShBkoJwhsADCDEGQIRibXhiSCSpUogqebGZc7R6ozjoILBlrQECIEUROEI1ilXj7\nYRsdheER6O2H3gH84ACV/QOU9w8w+uIggy8M0H+gSG93hb6eiFLRKZVa8+5JZL5SZUBmlZnx+7//\n+zz55JOUSiXOPvtsHnzwwbSblapqFSAOAuOqAu7VYQFx90AUxUGgXInn6JVKUCpCOX5YpUwYRWTc\nyBCSJUvOcuQtRyHIUQjyFII8bWGetqBAW5CnzfIssDxtlqNAhkKUIV8xssWIzEiRcGAY6+2Ldx18\nqYfK/h5KLx6i+EIvw8/1M/j8AIN7hxnYP8pAT5mBvojREdeeA9KcVBmoSZUBmRNLlizh1ltv5Z57\n7uGKK67ga1/7GhdffHHazUqfO56MC/CKQzmCcoSXIny0ArkyVixBtpjsTRBSXWsAPOk2qGDmhNUB\nhB7XBOLO++rgw0o8HMGdwJNljMtJ2CiWYHgIHx7CB4bw/kGi3kHK3YNUDg4wcmCYoX0jDO0fYbin\nzFBvxFCyxLBIU2vR8QBToTAgc2rz5s18+9vf5l3vehf//M//zKWXXpp2k9KVfF975FCJH16OoFhJ\nHuX4y7q6N0EQHN5hyKM4DIRlLIhnF9jYS477I1ctN0QRFlWwqAJRJQ4Zo2UYGcUHh2BwCO8dJjo0\nROXQEKWXBim+NMTA/lH6DhTpO1CiOFSmOOQUR6CkBYWk2bXoXf9UKAzInHvTm97E97//fd7xjnfw\nj//4j1x++eVpNyk9SRCoVga87IcrA8UKjJaTxYiqYcCS23oHr0CUTcJABgvCeOcj7PCCqNUKQRIA\nqJSgXMTLxXhcwPAoPjSM9w/DwBDePUx0cITKwWGK+4cYPjBE/4ESPQcq9BysjI1j9AmrGYrI/KIw\nIHVx3nnncffdd7Np0ybOOusszjrrrLSbVDeezFaKZws6lVJEVKpQKVaojJaxkRBy8VLEQRDggeEY\nlswwsGRAoWfL8dzEsJgEhUwyGhHG0kB1RkKlHI87KJfwYhEvFmFoFB8cIRoYIeofptI3TKl7hOLB\nUYrdowx3jzLSU2agt8LIUDS286BupmTeUDdBTQoDUjfnnHMOn/nMZ7juuuvYtm0bNn7Fvfms+v1c\ngXLJqZQjKsWISrFCebSCDcfVAMvEQSBKigE4WOR4uYKVy5AtJWEgk0xNCI9ctXBsnnQFSmW8XIbR\nEj5axEdKRAOj+MAIlf5RSr3DlHtHGO4pMtxdYrinxEh/iZH+CiODrg2HZH5Ssq1JYUDq6rrrruOr\nX/0q//Zv/8af/MmfpN2cunAOh4FK2YlKUCnFVYHKSLwSYZAJCMKAyCDAxtYdMI+gXMZLZSxbiisC\nhxcsGLdgUTVxxA8vlqBYxkdK+HCJaKhE1D9K1D9KuW+U4qERRg+NMNBdpr+7zGBvhdFhZ3QkolQ6\nvNeAwoDMKwoDNSkMSF2FYcitt97K7/3e77Fly5a0m1MX1S78clK5LxWd0mhEMBIRZMtYGCR7E0CQ\nzDYIKhFWrkC5Eu9mmCvh1XEEYRIEqrsaVr+1q90J5QgfLeGjZXyoRGUwDgPlviLl/iKlviIjvUVG\ne+O9BQYOVRgciHcdLJYUAERakcKA1N3atWt5wxvewHe+8520m1IX1e/pUgmKRRjJxN39ZCrxvgQW\nzwaI3AnLTliKCIoRwUgZHylj2WR740xSEQiTGQaMrwrEsxK8VCEqVYhGy0QjJSpDZcqDcSAY7a8w\n0l+mOFCmOFiiOFBheChiJJkpUNFNk8xz2mm1NoUBScVVV13Ff/zHf6TdjLrwZPCgexwGwhCC0LEg\nIrRysvYAeMWJShFRKSIcrRAWMvhwOd7dMBtiYXW74+pcQ8bWLahOUYyKFaJihfJwmcpIidJQmdJg\nidJgmaH+iMH+iNGhiNJIRGk0olyKVxOs7jqov5Uyr6mboCaFAUnFO9/5Tj70oQ+l3Yy6qY7KLxaT\nsX8GoUVkLAKPkumGEZlShBcjfCTE82XCXDyewDJJRSAgCQOMbWxEMk2xGgQqoxVKw2VKQ8k6AYNl\nRgfK9A9A/wAMDyeTDbSKoLQahYGaFAYkFQsXLuQtb3kLP/jBD9JuSl1FUVySD4qAxaX5XBmypYhs\nsUJ2BDKFiEwuJMwGhLl4i2MLLRlXYBAcXlfI3fEyRBWnXIwoj1Yf8UyF8khEacQpj8TLDJSq0wVV\nARCRcRQGJDWve93rWi4MVCKwMmPTDUtlyJWcbDEiNwrZvJMZNjK5MmEmIMwYQWhYsq+BA5jhkR9e\nSr0cr19QGnWKo06pGFEpxesZjD3K8TYH1TCg7gBpSUrBNSkMSGpaaeGhqigZSFipJBWCUchmndyo\nk8s52VyFXM7GtiUIMza2yVF10ODYTsdRHAKqsxSKRRgdTdYIqMTVgihibExCVF2lWH8PpVWpm6Am\nhQFJzRlnnJF2E1JRHVAYRXHVf2wrgUo8kK9c9LHlBKqzCC2ZQOCeLEEQHV5WYGzKYglGi3EoqAYG\nERlH/1LUpDAgqVm2bFnaTUhVtVQfVaCcrDNUiSAsHx5kGCSrDdv47Qc4/GU/PliUK2ixIBGZEYUB\nSU1bW1vaTUhddQ0Cj3cmPmJhwWoAqO5DBOM2C6pWCPzwo9p1oDAgUoP6yGpSGJDULFiwIO0mNITq\n3b2IzDF1E9QUHP8UkbmRy+XSboKIiJByZcDMlgMfB3YCpwOPuPttybFO4Obk2DLgXne/73jHpHm0\nzK6FItIYVBmoKe1ugnuALe7+OICZdZvZw+6+HbgNuNvdb0+O7TSzy6dwTERE5GgpjBkws5sAd/dP\n1ji+HNgKrAMeBa5x98fq2EQg/W6CFcR39lUHgUNmtg7YAtwx7ti9wCfNbC1w+WTH5ritIiLSzMZW\n6jrOY5aY2Ubi76tax6tV7vcThwGALjNbNGuNmKK0w8CdwFYzW2tm1xCX+/cAG4BD7t437txHgE3J\nsZ4ax0RERFKX3PGvIP5+qmUTcL27P55UtjcSTx7aUIcmHiHVMODuVxL/g3oE2OTu1yWH1gHdE07v\nBpYA62scWzyHTRURkWZX38rAtckYuJqDo9z9ruQGuPpzL/Gs4a7ZasRUpRoGkhLJM8RlksvN7MYp\n/JomioqIyPRFPrXHCUq+yz47g9/bRFwh7zvuybMs7QGE9wHvd/fHzawLuMPMniGuFFwx4dylQE9y\n7MpJjh061oVuuOGGsf994YUXcuGFF55Qw2Vmtm3bxrZt2wAolUrpNkZEWkuNu/5tT7zIg0/unZVL\nJF3ed7h7/wx+/RqO/u6rC/OUlipLqgI97h6Me+4O4kGEtwJd7h6OO/YV4m6Cm2odc/eralzL03qf\nUtvQ0BDt7e24+5TnGJqZf4PVc9ksEWkCf8COaf/tqNz13imdG275tyNeO/mCX8/klenqSuE3u/se\nM9sJLJ/snPHfW5O072PAPWnNikutMuDuvWbWY2bnjnvzi4H/5+6Pmdm9ZnaZu38rObaReFzBs7WO\n1fs9iIhI8/DKzG4Kk77/26Z47unjf05ucp+pNbUwOWcLKQYBSL+bYCPxdMGHgZOIFx365+TYFcDN\nZrYCWAl83N2fncIxERGRozXA3gTJdMMV4xbY20Tc1f1oMgNhCXClu3+inu1KNQwkKWjS0n4ygOK6\n6R4TERGZ1AwrAydo4kU3E3cj3JYEg7snOef6ejRsvLQrAyIiIvPWxLFs4+/4k2X0017vB1AYEBGR\nFuEN0E3QqBQGRESkNaTTTdAUFAZERKQ1VLRrYS0N0VchIiIi6VFlQEREWoLGDNSmMCAiIq1BYwZq\nUhgQEZHWoMpATRozICIi0uJUGRARkZYw070JWoHCgIiItIYaWxiLwoCIiLQKVQZq0pgBERGRFqfK\ngIiItAStM1CbwoCIiLQGdRPUpG4CERGRFqfKgIiItAZVBmpSGBARkZagMQO1KQyIiEhr0BbGNWnM\ngIiISItTZUBERFqCuglqUxgQEZHWoAGENSkMiIhIa1BloCaNGRAREWlxqgyIiEhL0BbGtSkMiIhI\na0ihm8DMbgLc3T9Z4/hy4OPATuB04BF3v62OTQQUBkREpFXUeZ0BM9sIXA5sPcZp9wBb3P3x5He6\nzexhd99ejzZWacyAiIjILEvu+FcAjxzn1BXAsnE/HwQOzVW7alEYEBGRluCRT+kxS65Nyv12nPPu\nBLaa2Vozuwa41933zFYjpkrdBCIi0hrqNIDQzG4EPjuVc939SjO7m7iCsNXdr5rTxtWgMCAiIi2h\n1l3/Q8/38NDzPbNyjeTu/g5375/i+Z3AM8Rh4Hozu8ndPzErjZkGhQEREWlp5796Cee/esnYzzc+\nvOeI48kX/HpgsjRhyfM3J+X9jwPLzY7oHTAzu97dw0l+/z7g/e7+uJl1EXcZ7HT322f+jqZPYUBE\nRFrCTNcZSPr+pzTdz91PH/+zmd0BPDPZ1MKkKrCuOpPA3e8yszuJg0ddw4AGEIqISEuo8wDCSZnZ\nxqTSgLv3Aj1mdu64UxZz/BkIs06VARERaQlROisQTrzoZmA5hysNG4FPmtnDwEnEiw7VtSoACgMi\nIiJzZuLsgImDA5PFhVKZQTCewoCIiLSEue4CaGYKAyIi0hI8qu9yxM1EYUBERFqCdi2sTbMJRERE\nWpwqAyIi0hI0ZqA2hQEREWkJ6iaoTd0EIiIiLU6VARERaQnqJqhNYUBERFpCpDBQk8KAiIi0BI0Z\nqE1jBkRERFqcKgMiItISNGagNoUBERFpCQoDtSkMiIhIS9CYgdo0ZkBERKTFqTIgIiItQbsW1qYw\nICIiLUHdBLUpDIiISEvQAMLaNGZARESkxakyICIiLUHLEdemMCAiIi1BYwZqa5gwYGbXAA50ufv2\ntNsjIiLzi8YM1JZ6GDCzdcAdwLXufv+45zuBm4GdwDLgXne/73jHREREGo2ZLXf33Wm3o5ZUw0AS\nBLqAje7+wITDtwF3u/vtybk7zezypGpwrGMiIiJHqWc3gZltBbZULw2sn8Lv3AS4u39yLts2mbRn\nE9wBbJ0YBJKQsCU5XnUv8EkzWwtcPtmxOW6riIg0MY98So8TlVSvXwLWAuuAlce7WTWzjcTfbalI\nLQwkX+orgMVmdkdyd/+V5PAG4JC79437lUeATcmxnhrHREREJlWvMEDcjd0L9Ln7dnffc6yTzWw5\n8ffhI7Nx8ZlIs5tgA3Hp5Fp3fzZJUrvN7BCwGOiecH43sIS41DLZscVz3F4REZGpWEH8XXW9mT0K\nXHGc8QLXuvsnzWxzfZp3tDS7CRYT3/0/C+DuvcCtxGWSY0UzDQcVEZFp84pP6XHC13F/m7svI65Y\ndwJ31zrXzG4EPnvCFz1BaVYGdnH03fwzwHLiUsmVE44tBXqOcezQsS52ww03jP3vCy+8kAsvziVb\ncQAAG3dJREFUvHC67ZVZsG3bNrZt2wZAqVRKtzEi0lJqLTr084Ehfj4wNOvXc/f7zWwD0G1mbx0/\nYw7GptTf4e79s37xaTL3dG60kz6SncCSav9/8g/m2uTR5e7huPO/QtxNcFOtY+5+VY1reVrvU2ob\nGhqivb0dd7ep/o6Z+TdYPZfNEpEm8AfsmPbfjid+Z82Uzn3d/z59xGsn303rmbwybcnzN9caG2Bm\ndwNfcfdvTXh+J/EN8FGvN/47rh5Sqwy4+24zu5N4FkB1JsBm4n9gj5nZvWZ22bh/eBuBTcn4gkmP\n1fcdiIhIK3D324intM/UUuDRSV739PE/m9kdwDNpTC1MdZ0Bd7/KzL6c9Jl0Az93939ODl8B3Gxm\nK4CVwMer4wuOc0xEROQoUTT310iq3uvc/a7k57XAw9WqQTKFcEUSMBpG6isQuvt1NZ7vA6Z9TERE\nZDL1CAPEY+FuThYQugXYNeF7bjNx18BkYSC1/uzUw4CIiMh84e6PAacf4/gnjnFs0nFv9aAwICIi\nLUH7FNWmMCAiIi2hTt0ETUlhQEREWoLCQG1pb1QkIiIiKVNlQEREWoIqA7UpDIiISEtQGKhNYUBE\nRFqCwkBtGjMgIiLS4lQZEBGRlqDKQG0KAyIi0hIUBmpTGBARkZagMFCbxgyIiIi0OFUGRESkJbhr\nc4JaFAZERKQlqJugNoUBERFpCQoDtWnMgIiISItTZUBERFqCKgO1KQyIiEhLUBioTWFARERagsJA\nbRozICIi0uJUGRARkZagykBtCgMiItISFAZqUxgQEZGWEGkBwpoUBkREROaImV0DONDl7tvTbk8t\nCgMiItIS6tlNYGbrgDuAa939/mOc1wncDOwElgH3uvt99WnlYQoDIiLSEuoVBpIg0AVsdPcHjnP6\nbcDd7n578rs7zezyelcRNLVQRERkdt0BbD1eEEhCw5bk/Kp7gU/OYdsmpTAgIiItIYqm9jgRZrYW\nWAEsNrM7kjv9r9Q4fT1wyN37xj33CLDpxFoxfeomEBGRllCnboINxAMGr3X3Z5MxAbvNrMfdJ97x\nrwe6JzzXDSyuQzuP0DJhwMzSboLMkj9gR9pNEJHm8+wfsOPUKZ677wSus5j4bv9ZAHfvNbNbgctJ\nofw/VS0RBtxdSWCe0GcpIjPh7qfN9HeT6YHrie/4jzqcPH+zu+8BdnH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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c126a58>"
      ]
     },
     "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": {
     "slide_type": "slide"
    }
   },
   "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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