astrofrog/Models with units.ipynb

## Models with units.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from astropy import units as u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from contextlib import contextmanager\n",
    "\n",
    "@contextmanager\n",
    "def simplified_error():\n",
    "    try:\n",
    "        yield\n",
    "    except Exception as exc:\n",
    "        print('...')\n",
    "        print(exc.__class__.__name__, ':', exc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setting parameters to quantities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Models can now take quantities as parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from astropy.modeling.models import Gaussian1D\n",
    "g1 = Gaussian1D(mean=3 * u.m, stddev=2 * u.cm, amplitude=3 * u.Jy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Accessing the parameter then returns a Parameter object that contains the value and the unit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Parameter('mean', value=3.0, unit=m)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1.mean"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is then possible to access the individual properties of the parameter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'mean'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1.mean.name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1.mean.value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{m}$"
      ],
      "text/plain": [
       "Unit(\"m\")"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1.mean.unit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If a parameter has been initialized as a Quantity, it should always be set to a quantity, but the units don't have to be compatible with the initial ones:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g1.mean = 3 * u.s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Gaussian1D(amplitude=3.0 Jy, mean=3.0 s, stddev=2.0 cm)>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We do this because we don't want to use equivalencies on the parameters (see section below on why this is the case), but in some cases we might want e.g. a temperature parameter to be set to a temperature in e.g. K, C, or F (and these are only convertible via equivalencies).\n",
    "\n",
    "To change the value of a parameter and not the unit, simply set the value property:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g1.mean.value = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Gaussian1D(amplitude=3.0 Jy, mean=2.0 s, stddev=2.0 cm)>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setting a parameter which was originally set to a quantity to a scalar doesn't work because it's ambiguous whether the user means to change just the value and preserve the unit, or get rid of the unit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "...\n",
      "UnitsError : The 'mean' parameter should be given as a Quantity because it was originally initialized as a Quantity\n"
     ]
    }
   ],
   "source": [
    "with simplified_error():\n",
    "    g1.mean = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the other hand, if a parameter previously defined without units is given a Quantity with a unit, we accept this, because it is unambiguous:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g0 = Gaussian1D(mean=3)\n",
    "g0.mean = 3 * u.m"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, once units are attached to a parameter, they can't be removed due to ambiguous meaning."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluating models with quantities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quantities can be passed to model during evaluation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g2 = Gaussian1D(mean=3 * u.m, stddev=5 * u.cm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.13533528 \\; \\mathrm{}$"
      ],
      "text/plain": [
       "<Quantity 0.1353352832366122>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g2(2.9 * u.m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the units of the parameters and the data are no compatible, a UnitsError will be raised:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "...\n",
      "UnitsError : Units of input 'x', s (time), could not be converted to required input units of m (length)\n"
     ]
    }
   ],
   "source": [
    "with simplified_error():\n",
    "    g2(2.9 * u.s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, since the mean and standard deviation have units, the value passed during evaluation also needs units:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "...\n",
      "UnitsError : Units of input 'x', (dimensionless), could not be converted to required input units of m (length)\n"
     ]
    }
   ],
   "source": [
    "with simplified_error():\n",
    "    g2(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In these cases, the error messages are emitted before ``evaluate`` is even called. This is because ``Gaussian1D`` includes the ``input_units`` property that indicates what units the input should be compatible with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'x': Unit(\"m\")}"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g2.input_units"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When implementing models, you don't necessarily have to define input_units, but if you do it will make it easier for the user to know what units they should be using, and also will give clear error messages above (rather than waiting for evaluate to crash).\n",
    "\n",
    "It is actually possible to allow the input value(s) when evaluating the model to be dimensionless, which is done by setting ``input_units_allow_dimensionless``:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g2.input_units_allow_dimensionless = True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But this requires the evaluate method to deal properly with dimesnionless input:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "...\n",
      "UnitsError : Can only apply 'subtract' function to dimensionless quantities when other argument is not a quantity (unless the latter is all zero/infinity/nan)\n"
     ]
    }
   ],
   "source": [
    "with simplified_error():\n",
    "    g2(2.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could consider another option which, if True, means that units are attached automatically to dimensionless input"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equivalencies"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have to be careful with equivalencies - a Gaussian defined in frequency space is not a Gaussian in wavelength space for example, so we need to make sure that we don't set equivalencies on parameters themselves, otherwise the model may be ambiguous.\n",
    "\n",
    "Instead, we take the approach of converting the input data to the parameter space, and any equivalencies should be applied at evaluation time to the data (not the parameters). The following example shows how to evaluate a Gaussian model in wavelength space by giving it frequencies. In this case, the model is truly a Gaussian in wavelength space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g3 = Gaussian1D(mean=3 * u.micron, stddev=1 * u.micron, amplitude=3 * u.Jy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, passing a frequency will not work:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "...\n",
      "UnitsError : Units of input 'x', THz (frequency), could not be converted to required input units of micron (length)\n"
     ]
    }
   ],
   "source": [
    "with simplified_error():\n",
    "    g3(1e2 * u.THz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But you can pass a dictionary of equivalencies to the equivalencies argument (this needs to be a dictionary since some models can contain multiple inputs)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.8889868 \\; \\mathrm{Jy}$"
      ],
      "text/plain": [
       "<Quantity 2.888986819525229 Jy>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g3(110 * u.THz, equivalencies={'x': u.spectral()})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The key of the dictionary should be the name of the inputs according to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('x',)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g3.inputs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also possible to set default equivalencies for the input parameters using the ``input_units_equivalencies`` property:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g3.input_units_equivalencies = {'x': u.spectral()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.8889868 \\; \\mathrm{Jy}$"
      ],
      "text/plain": [
       "<Quantity 2.888986819525229 Jy>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g3(110 * u.THz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting models with units to data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start off by generating synthetic data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.linspace(1, 5, 30) * u.micron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = np.exp(-0.5 * (x - 2.5 * u.micron)**2 / (200 * u.nm)**2) * u.mJy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11c40ba90>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LTZgUIqJnon1mZtad8synYGZm04STgpmZZZwUzMws46RgZmYZJwUzM8s4KZiZWcZJwczM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\nc4GS8Ou4DwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c3e33c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o')\n",
    "plt.xlabel('Wavelength (microns)')\n",
    "plt.ylabel('Flux density (mJy)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the initial guess for the fitting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g4 = Gaussian1D(mean=3 * u.micron, stddev=1 * u.micron, amplitude=1 * u.Jy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we carry out the fit as we would without any units:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from astropy.modeling import fitting\n",
    "fitter = fitting.LevMarLSQFitter()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g4_fit = fitter(g4, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11c965b00>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HGvjwT6BEJ5azptQcGah3+EK4ayUsuBU+8zwMGBJvcCIZoL9y6Z47/O+dMGI8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\n0kFJQUREOigpiIhIByUFERHp8P8B0rkYIV1GmC4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c3e3828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o')\n",
    "plt.plot(x, g4_fit(x), '-')\n",
    "plt.xlabel('Wavelength (microns)')\n",
    "plt.ylabel('Flux density (mJy)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting with equivalencies"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now consider the case where the data is not equivalent to those of the parameters, but they are convertible via equivalencies:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g5 = Gaussian1D(mean=110 * u.THz, stddev=10 * u.THz, amplitude=1 * u.Jy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the equivalencies can either be passed via a dictionary as shown higher up for the evaluation examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g5_fit = fitter(g5, x, y, equivalencies={'x': u.spectral()})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Gaussian1D(amplitude=1.0047966839436813 mJy, mean=121.05352130257793 THz, stddev=9.54642179190809 THz)>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g5_fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11c48b198>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LPDopPk11UDEEBlXFc/zkHkgicph0ex81A78C7iLomvoOYLWZfSLC2KQYNdcFl47iGlE8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\nVRHKTrpDvQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c98f550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o')\n",
    "plt.plot(x, g5_fit(x, equivalencies={'x': u.spectral()}), '-')\n",
    "plt.xlabel('Wavelength (microns)')\n",
    "plt.ylabel('Flux density (mJy)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, you can once again set ``input_units_equivalencies`` on the model itself:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g5.input_units_equivalencies = {'x': u.spectral()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g5_fit = fitter(g5, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11ca9a710>"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LPDopPk11UDEEBlXFc/zkHkgicph0ex81A78C7iLomvoOYLWZfSLC2KQYNdcFl47iGlE8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\nVRHKTrpDvQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ca93f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o')\n",
    "plt.plot(x, g5_fit(x), '-')\n",
    "plt.xlabel('Wavelength (microns)')\n",
    "plt.ylabel('Flux density (mJy)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  Models with n_models > 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from astropy.modeling.models import Gaussian1D\n",
    "g1 = Gaussian1D(mean=[3, 2, 1] * u.m, stddev=[2, 1, 2] * u.m, amplitude=[1,2,2] * u.Jy, n_models=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$[0.8824969,~2,~1.7649938] \\; \\mathrm{Jy}$"
      ],
      "text/plain": [
       "<Quantity [ 0.8824969 , 2.        , 1.76499381] Jy>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g1(2 * u.m)"
   ]
  }
 ],
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