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{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import openpathsampling as paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fake data for examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "exact = [1.0, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625]\n",
    "iface1 = [2.0, 1.0, 0.5, 0.25, 0.125, 0.0625, 0.0]\n",
    "iface2 = [1.0, 1.0, 1.0, 0.5, 0.25, 0.125, 0.0625]\n",
    "iface3 = [3.0, 3.0, 3.0, 3.0, 3.0, 1.5, 0.75]\n",
    "\n",
    "index = [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
David W.H. Swenson's avatar
David W.H. Swenson committed
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    "If we plot that data as it comes from the reverse crossing probabilities (as might be calculated by OPS), we get the following:"
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   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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YSWMZc9oY6ifVM+/6eXR2dhZ7eBWrKugG7v4g8CCAmVmATTe7+5ag+yu673wH/vjHWP/C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\nICIiIn36/xQ4d3GnvGyCAAAAAElFTkSuQmCC\n",
      "text/plain": [
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       "<matplotlib.figure.Figure at 0x110e64090>"
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(index, exact, '-ok', lw=2)\n",
    "for (iface, style) in zip([iface1, iface2, iface3], ['-or', '-ob', '-og']):\n",
    "    plt.plot(index, iface, style)"
   ]
  },
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also normalize those probabilities such that they all start at 1 (the normal way we plot crossing probabilities)."
   ]
  },
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fhQH2to0b32XLliL+hpeqYGbUbqntf1crAIUFqTp7TZrEo/3c91PgLOCaBx/k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\nPiNAxHUAAAAASUVORK5CYII=\n",
      "text/plain": [
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       "<matplotlib.figure.Figure at 0x111c96050>"
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(index, exact, '-ok', lw=2)\n",
    "for (iface, style) in zip([iface1, iface2, iface3], ['-or', '-ob', '-og']):\n",
    "    plt.plot(index, [i/iface[0] for i in iface], style)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "input_df = pd.DataFrame(data=np.array([iface1, iface2, iface3]).T,\n",
    "                        index=index,\n",
    "                        columns=[\"I'face 1\", \"I'face 2\", \"I'face 3\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>I'face 1</th>\n",
       "      <th>I'face 2</th>\n",
       "      <th>I'face 3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0.0</th>\n",
       "      <td>2.0000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.1</th>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.2</th>\n",
       "      <td>0.5000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.3</th>\n",
       "      <td>0.2500</td>\n",
       "      <td>0.5000</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.4</th>\n",
       "      <td>0.1250</td>\n",
       "      <td>0.2500</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.5</th>\n",
       "      <td>0.0625</td>\n",
       "      <td>0.1250</td>\n",
       "      <td>1.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.6</th>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0625</td>\n",
       "      <td>0.75</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     I'face 1  I'face 2  I'face 3\n",
       "0.0    2.0000    1.0000      3.00\n",
       "0.1    1.0000    1.0000      3.00\n",
       "0.2    0.5000    1.0000      3.00\n",
       "0.3    0.2500    0.5000      3.00\n",
       "0.4    0.1250    0.2500      3.00\n",
       "0.5    0.0625    0.1250      1.50\n",
       "0.6    0.0000    0.0625      0.75"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "input_df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preparing the system based on cutoffs\n",
    "\n",
    "The data comes from OPS in the form of the `input_df` above. In order to be prepared for the WHAM analysis, we first:\n",
    "\n",
    "1. Remove repeated leading copies of the maximum value.\n",
    "2. Remove contributions that fall below `cutoff * max_value` from each individual histogram.\n",
    "\n",
    "Importantly, this should work regardless of `max_value` for each interface."
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 7,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
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    "wham = paths.numerics.WHAM(cutoff=0.1, interfaces=[0.0, 0.2, 0.4])"
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   ]
  },
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we show the prepared dataframe:"
   ]
  },
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  {
   "cell_type": "code",
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   "execution_count": 8,
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   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>I'face 1</th>\n",
       "      <th>I'face 2</th>\n",
       "      <th>I'face 3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0.0</th>\n",
       "      <td>2.00</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.1</th>\n",
       "      <td>1.00</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.2</th>\n",
       "      <td>0.50</td>\n",
       "      <td>1.000</td>\n",
       "      <td>0.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.3</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.500</td>\n",
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       "      <td>0.00</td>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.4</th>\n",
       "      <td>0.00</td>\n",
       "      <td>0.250</td>\n",
       "      <td>3.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.5</th>\n",
       "      <td>0.00</td>\n",
       "      <td>0.125</td>\n",
       "      <td>1.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.6</th>\n",
       "      <td>0.00</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.75</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     I'face 1  I'face 2  I'face 3\n",
       "0.0      2.00     0.000      0.00\n",
       "0.1      1.00     0.000      0.00\n",
       "0.2      0.50     1.000      0.00\n",
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       "0.3      0.25     0.500      0.00\n",
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       "0.4      0.00     0.250      3.00\n",
       "0.5      0.00     0.125      1.50\n",
       "0.6      0.00     0.000      0.75"
      ]
     },
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     "execution_count": 8,
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     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
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    "prepared = wham.prep_reverse_cumulative(input_df)\n",
    "prepared"
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   ]
  },
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we plot the prepared data, showing the old data in dashed lines."
   ]
  },
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  {
   "cell_type": "code",
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   "execution_count": 9,
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   "metadata": {
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    "collapsed": false
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   },
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   "outputs": [
    {
     "data": {
      "image/png": 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lEo/MCwug9SNERFJE8KyOqi6kr8wMC7B//YgJEzR+QUTERYE1I1RdSF+ZGxYC60do/IKI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\n+CF9gZwQu73g8fDWlCn8au7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GvG2M+cIYs90Y844xZmIy25sKov25FnTcBP//0afbu42p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\n",
      "text/plain": [
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       "<matplotlib.figure.Figure at 0x119659c10>"
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
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   "source": [
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    "plt.plot(index, exact, '-ok', lw=2)\n",
    "for iface, color in zip(prepared.columns, ['r', 'g', 'b']):\n",
    "    # NaN out the 0 data, to prevent it from plotting\n",
    "    values = prepared[iface].apply(lambda x: np.nan if x==0.0 else x)\n",
    "    plt.plot(values, \"-o\"+color)\n",
    "    plt.plot(input_df[iface], \"--\"+color)"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Combining the results from multiple interfaces\n",
    "\n",
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    "Of course, the fundamental goal is to join the data from all the interfaces into one result. This is done (starting from the original input) with the following command:"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 10,
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   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "final_result = wham.wham_bam_histogram(input_df)"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 11,
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   "metadata": {
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    "collapsed": false
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   },
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   "outputs": [
    {
     "data": {
      "text/plain": [
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       "[<matplotlib.lines.Line2D at 0x1198b32d0>]"
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      ]
     },
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     "execution_count": 11,
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     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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NGxkcrMfSlvbmvFXOOauO81Y552xyzr/0Uk4++mjO3bKFN7/WvHOt3r+iBY4R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\nprkwuVNCdQAAAABJRU5ErkJggg==\n",
      "text/plain": [
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       "<matplotlib.figure.Figure at 0x1198b3210>"
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
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   "source": [
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    "plt.plot(index, exact, '-ok', lw=2)\n",
    "plt.plot(final_result, '-or')"
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   ]
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  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0    1.000000\n",
       "0.1    0.500000\n",
       "0.2    0.250000\n",
       "0.3    0.125000\n",
       "0.4    0.062500\n",
       "0.5    0.031250\n",
       "0.6    0.015625\n",
       "Name: WHAM, dtype: float64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "final_result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
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  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}