{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "116bcc5c",
   "metadata": {},
   "source": [
    "# Summarizing complex landscapes\n",
    "\n",
    "In the previous section, we illustrate of how to infer a complete\n",
    "combinatorial landscape from empirical data using either `VCregression` or `SeqDEFT`. \n",
    "In this section, we illustrate how to obtain low-dimensional summary statistics of the\n",
    "inferred landscapes that characterize the complexity and patterns of genetic interactions\n",
    "across the different sequence positions. We do so by computing the variance explained by\n",
    "genetic interactions of different orders, involving each possible site and pairs of sites.\n",
    "\n",
    "\n",
    "## GB1 protein landscape\n",
    "\n",
    "In this section, we first show how we can use `gpmap-tools` to understand the complexity and patterns of genetic interactions across multiple sites in a previously inferred genotype-phenotype map (GB1) using `VCregression` and [previously published high throughput experimental data](https://doi.org/10.7554/eLife.16965)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "5e5d14fe",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import required libraries\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from gpmap.datasets import DataSet\n",
    "from gpmap.summary import GPmapSummarizer"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb22b7af",
   "metadata": {},
   "source": [
    "We use the inferred complete combinatorial landscape as provided in `DataSet`s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8e0581bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "        vertical-align: middle;\n",
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       "\n",
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       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>y</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>seq</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>AAAA</th>\n",
       "      <td>0.296301</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAC</th>\n",
       "      <td>-2.713474</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAD</th>\n",
       "      <td>-2.912992</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAE</th>\n",
       "      <td>-4.548719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAF</th>\n",
       "      <td>-3.276738</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>YYYS</th>\n",
       "      <td>-4.662925</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>YYYT</th>\n",
       "      <td>-3.223102</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>YYYV</th>\n",
       "      <td>-3.001718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>YYYW</th>\n",
       "      <td>-4.723318</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>YYYY</th>\n",
       "      <td>-4.876429</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>160000 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             y\n",
       "seq           \n",
       "AAAA  0.296301\n",
       "AAAC -2.713474\n",
       "AAAD -2.912992\n",
       "AAAE -4.548719\n",
       "AAAF -3.276738\n",
       "...        ...\n",
       "YYYS -4.662925\n",
       "YYYT -3.223102\n",
       "YYYV -3.001718\n",
       "YYYW -4.723318\n",
       "YYYY -4.876429\n",
       "\n",
       "[160000 rows x 1 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gb1 = DataSet('gb1')\n",
    "gb1.landscape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dc71915",
   "metadata": {},
   "source": [
    "Firts, we define a `GPmapSummarizer` object with the configuration of sequence space and the associated functional values for every possible sequence `f`. \n",
    "\n",
    "> **Note**: Function values must be provided for sequences in alphabetical order."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "abd76f3b",
   "metadata": {},
   "outputs": [],
   "source": [
    "gb1s = GPmapSummarizer(n_alleles=20, seq_length=4, f=gb1.landscape['y'].values)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebe2678b",
   "metadata": {},
   "source": [
    "### How to compute the variance components of interactions of order `k`\n",
    "\n",
    "Now, we only need to call the method `calc_V_k_variance_components` to obtain a `pd.DataFrame` with the total and percentage variance explained by interactions of every possible order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8519786b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>k</th>\n",
       "      <th>variance</th>\n",
       "      <th>variance_perc</th>\n",
       "      <th>variance_perc_cum</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>180067.068327</td>\n",
       "      <td>52.811580</td>\n",
       "      <td>52.811580</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>128538.741722</td>\n",
       "      <td>37.698921</td>\n",
       "      <td>90.510501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>31449.759748</td>\n",
       "      <td>9.223849</td>\n",
       "      <td>99.734350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>905.762731</td>\n",
       "      <td>0.265650</td>\n",
       "      <td>100.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   k       variance  variance_perc  variance_perc_cum\n",
       "0  1  180067.068327      52.811580          52.811580\n",
       "1  2  128538.741722      37.698921          90.510501\n",
       "2  3   31449.759748       9.223849          99.734350\n",
       "3  4     905.762731       0.265650         100.000000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v_k_vcs = gb1s.calc_V_k_variance_components()\n",
    "v_k_vcs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61d81198",
   "metadata": {},
   "source": [
    "which can be easily represented graphically as follows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "3d30ef73",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 300x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 1, figsize=(3, 3))\n",
    "\n",
    "# Plot percentage variance explained\n",
    "axes.bar(x=v_k_vcs[\"k\"], height=v_k_vcs[\"variance_perc\"], color=\"black\")\n",
    "axes.set(\n",
    "    xlabel=\"Interaction order $k$\",\n",
    "    ylabel=\"% variance explained\",\n",
    "    xticks=np.arange(1, 10),\n",
    "    ylim=(0, 100),\n",
    ")\n",
    "\n",
    "# Plot cumulative percentage variance explained\n",
    "axes = axes.twinx()\n",
    "axes.scatter(v_k_vcs[\"k\"], v_k_vcs[\"variance_perc_cum\"], color=\"grey\", s=25)\n",
    "axes.plot(v_k_vcs[\"k\"], v_k_vcs[\"variance_perc_cum\"], color=\"grey\", lw=2)\n",
    "axes.tick_params(axis=\"y\", colors=\"grey\")\n",
    "axes.spines[\"right\"].set_color(\"grey\")\n",
    "axes.set(ylim=(0, 105))\n",
    "axes.set_ylabel(\"% cumulative variance\", color=\"grey\")\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cff27023",
   "metadata": {},
   "source": [
    "However, this only provides information about the complexity of the landscape, but not about which sites are involved in more or less complex genetic interactions and whether interactions are widespread or happening between specific sets of sites. \n",
    "\n",
    "### How to compute the variance components of interactions involving a subsets of sites $U$\n",
    "\n",
    "This can be done by calling the method `calc_V_U_variance_components` to obtain a `pd.DataFrame` with the total and percentage variance explained involving every possible combination of sites\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4338153a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>U</th>\n",
       "      <th>k</th>\n",
       "      <th>variance</th>\n",
       "      <th>variance_perc</th>\n",
       "      <th>variance_perc_cum</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>{0}</td>\n",
       "      <td>1</td>\n",
       "      <td>28303.313131</td>\n",
       "      <td>8.301033</td>\n",
       "      <td>8.301033</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>{1}</td>\n",
       "      <td>1</td>\n",
       "      <td>16149.051370</td>\n",
       "      <td>4.736329</td>\n",
       "      <td>13.037362</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>{2}</td>\n",
       "      <td>1</td>\n",
       "      <td>77458.399775</td>\n",
       "      <td>22.717649</td>\n",
       "      <td>35.755012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>{3}</td>\n",
       "      <td>1</td>\n",
       "      <td>58156.304052</td>\n",
       "      <td>17.056569</td>\n",
       "      <td>52.811580</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>{0, 1}</td>\n",
       "      <td>2</td>\n",
       "      <td>5138.287598</td>\n",
       "      <td>1.507000</td>\n",
       "      <td>54.318581</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>{0, 2}</td>\n",
       "      <td>2</td>\n",
       "      <td>25776.797274</td>\n",
       "      <td>7.560035</td>\n",
       "      <td>61.878616</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>{0, 3}</td>\n",
       "      <td>2</td>\n",
       "      <td>17682.891637</td>\n",
       "      <td>5.186187</td>\n",
       "      <td>67.064803</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>{1, 2}</td>\n",
       "      <td>2</td>\n",
       "      <td>8761.142103</td>\n",
       "      <td>2.569541</td>\n",
       "      <td>69.634344</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>{1, 3}</td>\n",
       "      <td>2</td>\n",
       "      <td>5422.459340</td>\n",
       "      <td>1.590344</td>\n",
       "      <td>71.224688</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>{2, 3}</td>\n",
       "      <td>2</td>\n",
       "      <td>65757.163770</td>\n",
       "      <td>19.285813</td>\n",
       "      <td>90.510501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>{0, 1, 2}</td>\n",
       "      <td>3</td>\n",
       "      <td>3723.022582</td>\n",
       "      <td>1.091919</td>\n",
       "      <td>91.602420</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>{0, 1, 3}</td>\n",
       "      <td>3</td>\n",
       "      <td>2675.360421</td>\n",
       "      <td>0.784652</td>\n",
       "      <td>92.387072</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>{0, 2, 3}</td>\n",
       "      <td>3</td>\n",
       "      <td>18786.824440</td>\n",
       "      <td>5.509957</td>\n",
       "      <td>97.897030</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>{1, 2, 3}</td>\n",
       "      <td>3</td>\n",
       "      <td>6264.552305</td>\n",
       "      <td>1.837320</td>\n",
       "      <td>99.734350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>{0, 1, 2, 3}</td>\n",
       "      <td>4</td>\n",
       "      <td>905.762731</td>\n",
       "      <td>0.265650</td>\n",
       "      <td>100.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               U  k      variance  variance_perc  variance_perc_cum\n",
       "0            {0}  1  28303.313131       8.301033           8.301033\n",
       "1            {1}  1  16149.051370       4.736329          13.037362\n",
       "2            {2}  1  77458.399775      22.717649          35.755012\n",
       "3            {3}  1  58156.304052      17.056569          52.811580\n",
       "4         {0, 1}  2   5138.287598       1.507000          54.318581\n",
       "5         {0, 2}  2  25776.797274       7.560035          61.878616\n",
       "6         {0, 3}  2  17682.891637       5.186187          67.064803\n",
       "7         {1, 2}  2   8761.142103       2.569541          69.634344\n",
       "8         {1, 3}  2   5422.459340       1.590344          71.224688\n",
       "9         {2, 3}  2  65757.163770      19.285813          90.510501\n",
       "10     {0, 1, 2}  3   3723.022582       1.091919          91.602420\n",
       "11     {0, 1, 3}  3   2675.360421       0.784652          92.387072\n",
       "12     {0, 2, 3}  3  18786.824440       5.509957          97.897030\n",
       "13     {1, 2, 3}  3   6264.552305       1.837320          99.734350\n",
       "14  {0, 1, 2, 3}  4    905.762731       0.265650         100.000000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v_U_vcs = gb1s.calc_V_U_variance_components()\n",
    "v_U_vcs\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68963729",
   "metadata": {},
   "source": [
    "Whereas the number of possible combinations is relatively small in this case, it scales exponentially with sequence length, making it in practical in other cases to interpret the values in this `DataFrame` directly. Still, we can compute some useful aggregated statistics that we can represent easily as heatmaps.\n",
    "\n",
    "### How to compute the variance explained by interactions of order `k` involving each site\n",
    "\n",
    "`calc_sites_variance_perc` enables aggregation of these variances by whether they involve each possible site and the total number of sites involved in those genetic interactions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "8c90e3d4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "        vertical-align: middle;\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8.301033</td>\n",
       "      <td>14.253222</td>\n",
       "      <td>7.386529</td>\n",
       "      <td>0.26565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.736329</td>\n",
       "      <td>5.666886</td>\n",
       "      <td>3.713892</td>\n",
       "      <td>0.26565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>22.717649</td>\n",
       "      <td>29.415389</td>\n",
       "      <td>8.439197</td>\n",
       "      <td>0.26565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>17.056569</td>\n",
       "      <td>26.062344</td>\n",
       "      <td>8.131930</td>\n",
       "      <td>0.26565</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           1          2         3        4\n",
       "0   8.301033  14.253222  7.386529  0.26565\n",
       "1   4.736329   5.666886  3.713892  0.26565\n",
       "2  22.717649  29.415389  8.439197  0.26565\n",
       "3  17.056569  26.062344  8.131930  0.26565"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sites = gb1s.calc_sites_variance_perc(v_U_vcs).T\n",
    "sites"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "5f0fba20",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 300x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 1, figsize=(3, 3))\n",
    "im = axes.imshow(\n",
    "    sites.T.iloc[::-1, :],\n",
    "    cmap=\"Greys\",\n",
    "    vmin=0,\n",
    ")\n",
    "axes.set(\n",
    "    xticks=[0, 1, 2, 3],\n",
    "    xticklabels=[39, 40, 41, 54],\n",
    "    xlabel=\"Site\",\n",
    "    yticks=[0, 1, 2, 3],\n",
    "    yticklabels=[1, 2, 3, 4][::-1],\n",
    "    ylabel=\"Interaction order $k$\",\n",
    "    aspect='equal'\n",
    ")\n",
    "plt.colorbar(im, shrink=0.6, label='% variance explained')\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f6f038e5",
   "metadata": {},
   "source": [
    "This clearly shows that some sites (41 and 54) have a much larger contribution to the phenotypic variance than others (40), and that they contribute mostly through their additive, pairwise and even threeway interactions.\n",
    "\n",
    "### How to compute the variance explained by interactions between every pair of sites\n",
    "\n",
    "Another interesting way to aggregate the variance explain by every subset of sites $U$ is by computing the variance explain by interactions of any order involving a pair of sites. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "9e2ad948",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>site1</th>\n",
       "      <th>site2</th>\n",
       "      <th>variance</th>\n",
       "      <th>variance_perc</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>12442.433333</td>\n",
       "      <td>7.733298</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>49192.407027</td>\n",
       "      <td>30.574370</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>40050.839230</td>\n",
       "      <td>24.892646</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>19654.479720</td>\n",
       "      <td>12.215774</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>15268.134797</td>\n",
       "      <td>9.489546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>91714.303247</td>\n",
       "      <td>57.002842</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   site1  site2      variance  variance_perc\n",
       "0      0      1  12442.433333       7.733298\n",
       "1      0      2  49192.407027      30.574370\n",
       "2      0      3  40050.839230      24.892646\n",
       "3      1      2  19654.479720      12.215774\n",
       "4      1      3  15268.134797       9.489546\n",
       "5      2      3  91714.303247      57.002842"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pairs = gb1s.calc_site_pairs_variance_perc(v_U_vcs)\n",
    "pairs\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7d1f007",
   "metadata": {},
   "source": [
    "We can arrange these values on a matrix and represent them in a heatmap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "3e1cdd04",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>site2</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>site1</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>7.733298</td>\n",
       "      <td>30.574370</td>\n",
       "      <td>24.892646</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>NaN</td>\n",
       "      <td>12.215774</td>\n",
       "      <td>9.489546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>57.002842</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "site2         1          2          3\n",
       "site1                                \n",
       "0      7.733298  30.574370  24.892646\n",
       "1           NaN  12.215774   9.489546\n",
       "2           NaN        NaN  57.002842"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = pd.pivot_table(\n",
    "    pairs, index=\"site1\", columns=\"site2\", values=\"variance_perc\"\n",
    ")\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "4334860c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 300x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 1, figsize=(3, 3))\n",
    "im = axes.imshow(\n",
    "    m,\n",
    "    cmap=\"Greys\",\n",
    "    vmin=0,\n",
    ")\n",
    "axes.set(\n",
    "    xticks=[0, 1, 2],\n",
    "    xlabel=\"Site 1\",\n",
    "    yticks=[0, 1, 2],\n",
    "    xticklabels=[40, 41, 54],\n",
    "    yticklabels=[39, 40, 41],\n",
    "    ylabel=\"Site 2\",\n",
    "    aspect=\"equal\",\n",
    ")\n",
    "plt.colorbar(im, shrink=0.6, label=\"% variance explained\")\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca23a0e1",
   "metadata": {},
   "source": [
    "This plot clearly shows that sites 41 and 54 interact strongly with each other, but also with position 39.\n",
    "\n",
    "## Shine-Dalgarno fitness landscape\n",
    "\n",
    "In the previous example, the total number of subsets of sites $U$ is still relatively small given the short sequence length. To better illustrate the usefulness of this approach, we will use a  9-nucleotide long complete combinatorial landscape inferred with `VCregression` from [previously published high throughput experimental data](https://genome.cshlp.org/content/30/5/711) in the *dmsC* gene context. The inferred landscape is also available as a `DataSet` for illustration purposes.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "998509f8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>f</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>AAAAAAAAA</th>\n",
       "      <td>0.563857</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAAAAAAC</th>\n",
       "      <td>0.701511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAAAAAAG</th>\n",
       "      <td>0.619188</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAAAAAAU</th>\n",
       "      <td>0.628772</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AAAAAAACA</th>\n",
       "      <td>0.594683</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UUUUUUUGU</th>\n",
       "      <td>0.615910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UUUUUUUUA</th>\n",
       "      <td>0.539058</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UUUUUUUUC</th>\n",
       "      <td>0.548782</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UUUUUUUUG</th>\n",
       "      <td>0.536610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UUUUUUUUU</th>\n",
       "      <td>0.564960</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>262144 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                  f\n",
       "AAAAAAAAA  0.563857\n",
       "AAAAAAAAC  0.701511\n",
       "AAAAAAAAG  0.619188\n",
       "AAAAAAAAU  0.628772\n",
       "AAAAAAACA  0.594683\n",
       "...             ...\n",
       "UUUUUUUGU  0.615910\n",
       "UUUUUUUUA  0.539058\n",
       "UUUUUUUUC  0.548782\n",
       "UUUUUUUUG  0.536610\n",
       "UUUUUUUUU  0.564960\n",
       "\n",
       "[262144 rows x 1 columns]"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sd = DataSet('dmsc')\n",
    "sd.landscape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4043bcbf",
   "metadata": {},
   "source": [
    "Lets start again by defining the `GPmapSummarizer` object and compute the variance explained by every possible subset of sites"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "cc1ce857",
   "metadata": {},
   "outputs": [],
   "source": [
    "sds = GPmapSummarizer(n_alleles=4, seq_length=9, f=sd.landscape[\"f\"].values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "d1fcbd68",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>U</th>\n",
       "      <th>k</th>\n",
       "      <th>variance</th>\n",
       "      <th>variance_perc</th>\n",
       "      <th>variance_perc_cum</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>{0}</td>\n",
       "      <td>1</td>\n",
       "      <td>1574.656812</td>\n",
       "      <td>5.500762</td>\n",
       "      <td>5.500762</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>{1}</td>\n",
       "      <td>1</td>\n",
       "      <td>2687.622462</td>\n",
       "      <td>9.388695</td>\n",
       "      <td>14.889458</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>{2}</td>\n",
       "      <td>1</td>\n",
       "      <td>2312.377111</td>\n",
       "      <td>8.077847</td>\n",
       "      <td>22.967305</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>{3}</td>\n",
       "      <td>1</td>\n",
       "      <td>1806.662119</td>\n",
       "      <td>6.311229</td>\n",
       "      <td>29.278533</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>{4}</td>\n",
       "      <td>1</td>\n",
       "      <td>1051.784760</td>\n",
       "      <td>3.674209</td>\n",
       "      <td>32.952742</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>506</th>\n",
       "      <td>{0, 1, 2, 4, 5, 6, 7, 8}</td>\n",
       "      <td>8</td>\n",
       "      <td>2.485503</td>\n",
       "      <td>0.008683</td>\n",
       "      <td>99.956748</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>507</th>\n",
       "      <td>{0, 1, 3, 4, 5, 6, 7, 8}</td>\n",
       "      <td>8</td>\n",
       "      <td>2.545558</td>\n",
       "      <td>0.008892</td>\n",
       "      <td>99.965641</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>508</th>\n",
       "      <td>{0, 2, 3, 4, 5, 6, 7, 8}</td>\n",
       "      <td>8</td>\n",
       "      <td>2.358090</td>\n",
       "      <td>0.008238</td>\n",
       "      <td>99.973878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>509</th>\n",
       "      <td>{1, 2, 3, 4, 5, 6, 7, 8}</td>\n",
       "      <td>8</td>\n",
       "      <td>2.796804</td>\n",
       "      <td>0.009770</td>\n",
       "      <td>99.983648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>510</th>\n",
       "      <td>{0, 1, 2, 3, 4, 5, 6, 7, 8}</td>\n",
       "      <td>9</td>\n",
       "      <td>4.680847</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>100.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>511 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                               U  k     variance  variance_perc  \\\n",
       "0                            {0}  1  1574.656812       5.500762   \n",
       "1                            {1}  1  2687.622462       9.388695   \n",
       "2                            {2}  1  2312.377111       8.077847   \n",
       "3                            {3}  1  1806.662119       6.311229   \n",
       "4                            {4}  1  1051.784760       3.674209   \n",
       "..                           ... ..          ...            ...   \n",
       "506     {0, 1, 2, 4, 5, 6, 7, 8}  8     2.485503       0.008683   \n",
       "507     {0, 1, 3, 4, 5, 6, 7, 8}  8     2.545558       0.008892   \n",
       "508     {0, 2, 3, 4, 5, 6, 7, 8}  8     2.358090       0.008238   \n",
       "509     {1, 2, 3, 4, 5, 6, 7, 8}  8     2.796804       0.009770   \n",
       "510  {0, 1, 2, 3, 4, 5, 6, 7, 8}  9     4.680847       0.016352   \n",
       "\n",
       "     variance_perc_cum  \n",
       "0             5.500762  \n",
       "1            14.889458  \n",
       "2            22.967305  \n",
       "3            29.278533  \n",
       "4            32.952742  \n",
       "..                 ...  \n",
       "506          99.956748  \n",
       "507          99.965641  \n",
       "508          99.973878  \n",
       "509          99.983648  \n",
       "510         100.000000  \n",
       "\n",
       "[511 rows x 5 columns]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v_U_vcs = sds.calc_V_U_variance_components()\n",
    "v_U_vcs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bda04e69",
   "metadata": {},
   "source": [
    "We can see that the number of components now is much larger than before, so we really need to aggregate these components in a meaningful way to obtain low-dimensional summary statistics reflecting the patterns of genetic interactions in the inferred genotype-phenotype map.\n",
    "\n",
    "### How to compute the variance explained by interactions of order `k` involving each site"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "9eaf5911",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>8</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>5.500762</td>\n",
       "      <td>9.388695</td>\n",
       "      <td>8.077847</td>\n",
       "      <td>6.311229</td>\n",
       "      <td>3.674209</td>\n",
       "      <td>2.277808</td>\n",
       "      <td>1.837005</td>\n",
       "      <td>0.835968</td>\n",
       "      <td>0.399951</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>6.390370</td>\n",
       "      <td>10.088863</td>\n",
       "      <td>10.262287</td>\n",
       "      <td>8.163453</td>\n",
       "      <td>6.898441</td>\n",
       "      <td>4.512101</td>\n",
       "      <td>3.034212</td>\n",
       "      <td>1.908863</td>\n",
       "      <td>1.153861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>5.799007</td>\n",
       "      <td>8.710946</td>\n",
       "      <td>10.348244</td>\n",
       "      <td>11.095941</td>\n",
       "      <td>10.542178</td>\n",
       "      <td>8.887247</td>\n",
       "      <td>5.887798</td>\n",
       "      <td>3.522035</td>\n",
       "      <td>2.133580</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.999159</td>\n",
       "      <td>4.047676</td>\n",
       "      <td>4.776184</td>\n",
       "      <td>5.714224</td>\n",
       "      <td>5.930684</td>\n",
       "      <td>5.423643</td>\n",
       "      <td>4.125461</td>\n",
       "      <td>2.856694</td>\n",
       "      <td>1.932568</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.208228</td>\n",
       "      <td>1.508030</td>\n",
       "      <td>1.640607</td>\n",
       "      <td>1.763638</td>\n",
       "      <td>1.809694</td>\n",
       "      <td>1.735487</td>\n",
       "      <td>1.502939</td>\n",
       "      <td>1.196867</td>\n",
       "      <td>0.925827</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.441740</td>\n",
       "      <td>0.488117</td>\n",
       "      <td>0.516701</td>\n",
       "      <td>0.522051</td>\n",
       "      <td>0.525214</td>\n",
       "      <td>0.525015</td>\n",
       "      <td>0.496194</td>\n",
       "      <td>0.446754</td>\n",
       "      <td>0.389411</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.187551</td>\n",
       "      <td>0.194456</td>\n",
       "      <td>0.197175</td>\n",
       "      <td>0.198214</td>\n",
       "      <td>0.199869</td>\n",
       "      <td>0.199935</td>\n",
       "      <td>0.199599</td>\n",
       "      <td>0.192142</td>\n",
       "      <td>0.179375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.070391</td>\n",
       "      <td>0.071923</td>\n",
       "      <td>0.071268</td>\n",
       "      <td>0.071478</td>\n",
       "      <td>0.071555</td>\n",
       "      <td>0.071555</td>\n",
       "      <td>0.071691</td>\n",
       "      <td>0.071658</td>\n",
       "      <td>0.069766</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
       "      <td>0.016352</td>\n",
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      "text/plain": [
       "          0          1          2          3          4         5         6  \\\n",
       "1  5.500762   9.388695   8.077847   6.311229   3.674209  2.277808  1.837005   \n",
       "2  6.390370  10.088863  10.262287   8.163453   6.898441  4.512101  3.034212   \n",
       "3  5.799007   8.710946  10.348244  11.095941  10.542178  8.887247  5.887798   \n",
       "4  2.999159   4.047676   4.776184   5.714224   5.930684  5.423643  4.125461   \n",
       "5  1.208228   1.508030   1.640607   1.763638   1.809694  1.735487  1.502939   \n",
       "6  0.441740   0.488117   0.516701   0.522051   0.525214  0.525015  0.496194   \n",
       "7  0.187551   0.194456   0.197175   0.198214   0.199869  0.199935  0.199599   \n",
       "8  0.070391   0.071923   0.071268   0.071478   0.071555  0.071555  0.071691   \n",
       "9  0.016352   0.016352   0.016352   0.016352   0.016352  0.016352  0.016352   \n",
       "\n",
       "          7         8  \n",
       "1  0.835968  0.399951  \n",
       "2  1.908863  1.153861  \n",
       "3  3.522035  2.133580  \n",
       "4  2.856694  1.932568  \n",
       "5  1.196867  0.925827  \n",
       "6  0.446754  0.389411  \n",
       "7  0.192142  0.179375  \n",
       "8  0.071658  0.069766  \n",
       "9  0.016352  0.016352  "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sites = sds.calc_sites_variance_perc(v_U_vcs)\n",
    "sites"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "917f5350",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1fT/8oe/ytPWtWxcL4q1ZswazZs3C559/jtTUVDg4OGD06NGYPXu2JlUVRUVF4aOPPkJ2djaysrJQo0YNPHv2DJaWlrCzs5OUFCR1H8XGxpa4JKuNjQ3S09OlhCQiMihddB9ZWVnhm2++wcOHD/HXX3/h/v37CAkJgVwu10qdJ02ahB49euDFixewsLDAhQsX8PDhQ3h5eWHZsmWSYkpKCvb29iWut3H27FnUr19fUkWIiAzJGPdojo6OxpQpU1CpUiWYmJggJycHjo6OWLp0Kb788ktJMSUlhVGjRmHChAm4ePEiZDIZkpOTsX37dkydOhVBQUGSKkJEZEjGmBRMTU3FLi07OzskJCQAKOy1SUxMlBRT0pjCjBkzUFBQgC5duiA7OxsdOnSAmZkZpk6dinHjxkmqCBGRIRnj0tnNmzfH5cuX0bBhQ3Ts2BGzZ8/Gs2fP8MMPP+D999+XFFNSS0Emk+Grr75CWloabt26hQsXLuDp06dYsGCBpEoQERmaMS6dvWjRItSuXRsAsHDhQlSvXh1BQUF4+vQpNmzYICmmRpPX5HI53N3dNQlBRFQuGGNLoWXLluKf7ezscPToUY1jljkpTJ48ucxBuSAeERkbY0wKulDmpBAVFaXy+tq1a8jPz0fjxo0BFC6MZ2JiAi8vL+3WkIhID4wlKTRv3rzM9bh27Zra8cucFF5fBG/FihWwsrLC1q1bUb16dQDAixcvEBgYiPbt26tdiUePHmH69Ok4cuQIsrOz0aBBA2zZskWlaUREpEvGkhT+bhE8bZA0prB8+XL89ttvYkIAgOrVqyMkJAS+vr6YMmVKmWO9ePEC7dq1Q+fOnXHkyBHY2tri3r17KrGJiHRNFzOadWHOnDk6jS8pKSgUCjx9+rTY+adPn+Lly5dqxVqyZAkcHR2xZcsW8ZyLi4uUahERaaQ8tASkuHLlCmJiYgAA7u7uGnXjS0p7n3zyCQIDA7Fv3z4kJSUhKSkJe/fuxYgRI9CnTx+1Yh08eBAtW7ZEv379YGdnh+bNm2Pjxo1SqkVEJJkxTl5LSkpC+/bt0bp1a0yYMAETJkxAq1at4OPjg6SkJEkxJSWFsLAw+Pv7Y9CgQXB2doazszMGDRoEPz8/fPfdd2rFevDgAdatW4eGDRvi2LFjCAoKwvjx47F169YSr8/JyYFCoVA5iIg0ZYxJYeTIkcjLy0NMTAzS0tKQlpaGmJgYFBQUYOTIkZJiqt19lJeXhx49eiAsLAxff/017t+/DwBwdXUttldzWRQUFKBly5ZYtGgRgMKR9Vu3biEsLKzETShCQ0Mxb948tcshIvo7xjLQ/LrIyEj88ccf4lOgANC4cWOsWbNG0kM/gISWgqmpKW7cuAEAqFKlCpo0aYImTZpISggAULt27WIT4P7xj3+Ia3i8KTg4GBkZGeIhdX0PIqLXGeOMZkdHR+Tl5RU7r1Qq4eDgICmmpDscPHgwNm3aJKnAN7Vr1w6xsbEq5+7evQtnZ+cSrzczMxO3wivrlnhERG9jjEnh66+/xrhx43DlyhXx3JUrVzBhwgTJS2dLevooPz8fmzdvxokTJ+Dl5VWslaDOjOZJkyahbdu2WLRoET799FNcunQJGzZskLxuBxGRFMbYfTRs2DBkZ2ejTZs2qFy58OM8Pz8flStXxvDhw1V2eEtLSytTTElJ4datW2jRogWAwm/1r1P3l9aqVSvs378fwcHBmD9/PlxcXPDNN98gICBAStWIiCQxxqTwzTffaD2mpKTw+uxmbfj444/x8ccfazUmEZE6jDEplPQwjqbKVwcZEZGBGOMjqeHh4SWez8/PR3BwsKSYkpNCeno6li9fjpEjR2LkyJFYsWIFMjIypIYjIjIoYxxoHj9+PPr164cXL16I52JjY9GmTRvs3LlTUkxJd3jlyhW4urpi5cqV4oSJlStXwtXVVdKqfEREhmaMLYWoqCgkJSXB09MTx48fx9q1a9GiRQu4ubnh+vXrkmJKGlOYNGkSevbsiY0bN6qMeI8cORITJ07E6dOnJVWGiMhQjHFMwdXVFefOncPEiRPh5+cHExMTbN26FQMHDpQcU3JLYfr06WJCAIDKlSvjiy++UHlelojIWBhjSwEADh06hJ9++gne3t6oVq0aNm3ahOTkZMnxJCUFa2vrEmccJyYmwsrKSnJliIgMxRiTwujRo9GvXz9Mnz4dZ86cwY0bNyCXy+Hp6Yndu3dLiimp+6h///4YMWIEli1bhrZt2wIAzp07h2nTpmnUbCEiMpRKlSrBxMTkrdeUJ+fOncPFixfRtGlTAIC9vT0OHz6MtWvXYvjw4fj000/VjikpKSxbtgwymQxDhgxBfn4+gMI1kYKCgrB48WIpIYmIDMoYxxSuXr0KMzOzYufHjBmDrl27SoopKSnI5XKsWrUKoaGhKqukWlpaSqqENrx69QpyuVxv5RUlQ30pKCjQa3mGKFMQBL2WB+j/P/nr43Dvaplv+7atbbm5uVqJY4xJwczMDPfv38eWLVtw//59rFq1CnZ2djhy5AicnJwkxdSoLWRpaQlPT094enoaNCEQEWlKV2MKjx49wuDBg1GzZk1YWFjA09NTaw/kREZGwtPTExcvXsS+ffuQmZkJALh+/brkbTvLVwcZEZGB6CIpFO1Bb2pqiiNHjuD27dtYvny51vagnzFjBkJCQnD8+HGVnpIPP/wQFy5ckBRT/21ZIqJyqCwzltUdaNb1HvQ3b97Ejh07ip23s7PDs2fPJMVkS4GICLppKeh6D/pq1arh8ePHxc5HRUWhTp06kmIyKRARQb2k8OY+8Tk5OSXGVHcPenUNGDAA06dPR0pKCmQyGQoKCnDu3DlMnToVQ4YMkRRTcvdRREQEIiIikJqaWuwplc2bN0sNS0RkEOp0Hzk6OqqcnzNnDubOnVvsenX3oFfXokWLMGbMGDg6OkKpVMLd3R1KpRKDBg3CzJkzJcWUlBTmzZuH+fPno2XLlqhdu3a5e0yLiEhd6jySmpiYqLIVcElzBYDS96Dfu3evhrUtJJfLsXHjRsyePRs3b95EZmYmmjdvjoYNG0qOKSkphIWFITw8HJ999pnkgomIyhN1Wgpl3R9e3T3opXJ0dCzWepFK0phCbm6uuLwFEdG7QBcDzZMmTcKFCxewaNEixMXFYceOHdiwYQPGjBmjo7vQnKSkMHLkyBIfg5Ji7ty5xX7pbm5uWolNRFRWukgKRXvQ79y5E++//z4WLFhQ7vegl9R99OrVK2zYsAEnTpxAkyZNYGpqqvL+ihUr1Irn4eGBEydO/F+lDLAUABGRLsZHjW0Pekmfvjdu3ECzZs0AALdu3VJ5T8ovtXLlyrC3t5dSFSIirTDGtY90QVJSOHnypFYrce/ePTg4OMDc3Bze3t4IDQ0tdTGnnJwclWeCFQqFVutCRBWTLmY068OZM2ewfv163L9/Hz///DPq1KmDH374AS4uLvDx8VE7nsHvsE2bNggPD8fRo0exbt06xMfHo3379nj58mWJ14eGhsLGxkY8tDXiTkQVW1FSeNtRnuzduxfdunWDhYUFoqKixC/MGRkZ4twIdUm+w/T0dCxfvhwjR47EyJEjsWLFCmRkZKgdx9/fH/369UOTJk3QrVs3HD58GOnp6aXuGhQcHIyMjAzxSExMlHoLREQiY9x5LSQkBGFhYdi4caPK2G67du1w7do1STEl79Hs6uqKlStXIi0tDWlpaVi5ciVcXV0lV6RItWrV0KhRI8TFxZX4vpmZmfiMcFmfFSYiehtjTAqxsbHo0KFDsfM2NjZIT0+XFFNSUpg0aRJ69uyJP//8E/v27cO+ffsQHx+Pjz/+GBMnTpRUkSKZmZm4f/8+ateurVEcIiJ1GGNSsLe3L/EL9NmzZ1G/fn1JMSW3FKZPn67y6GjlypXxxRdfqL15xNSpUxEZGYk///wTf/zxBz755BOYmJhwr2ci0isTE5MyHeXJqFGjMGHCBFy8eBEymQzJycnYvn07pk6diqCgIEkxJT19ZG1tjYSEhGKTzBITE2FlZaVWrKSkJAwcOBDPnz+Hra0tfHx8cOHCBdja2kqpGhGRJMb4SOqMGTNQUFCALl26IDs7Gx06dICZmRmmTp2KcePGSYopKSn0798fI0aMwLJly8TlLs6dO4dp06ap/Q3/p59+klIFIiKtMsakIJPJ8NVXX2HatGmIi4tDZmYm3N3dUbVqVckxJSWFZcuWQSaTYciQIcjPz4cgCJDL5QgKCsLixYslV4aIyFCMMSlkZGRAqVSiRo0aKquxpqWloXLlypIexJE0piCXy7Fq1Sq8ePEC0dHRuH79uvgEUmlLyBIRlWfGOE9hwIABJfa27N69GwMGDJAUs8wthcmTJ2PBggWoUqUKJk+e/LfXqrv2ERGRoclksrd+6Je3lsLFixdL/Lzt1KkTvvrqK0kxy5wUoqKikJeXJ/65NOXtl0ZEVBbG2H2Uk5OD/Pz8Yufz8vLw119/SYpZ5qTw+npH2l77iIjI0Ixx7aPWrVtjw4YNWLNmjcr5sLAweHl5SYopaaA5ISEBjo6OJWbNhISEUhezIyIqr4yxpRASEoKuXbvi+vXr6NKlCwAgIiICly9fxm+//SYppqS05+LigqdPnxY7//z5c7i4uEiqCBGRIRnjjOZ27drh/PnzcHR0xO7du/Hrr7+iQYMGuHHjBtq3by8ppqSWgiAIJf5yMjMzYW5uLqkiRESGZIzdRwDQrFkzbN++XWvx1EoKRU8dyWQyzJo1C5aWluJ7SqUSFy9eFDff0bekpCSNJmyoq6SWki49e/ZMr+UBwOPHj/VaXm5url7LA6D2DHxNNWjQQK/lAYCDg4Ney6tWrZpeyyttmX11GWtSKCgoQFxcHFJTU1FQUKDyXkmL5b2NWkmh6KkjQRBw8+ZNyOVy8T25XI6mTZti6tSpaleCiMjQjHFM4cKFCxg0aBAePnwIQRBU3pPJZFAqlWrHVCspFD11FBgYiNWrV+v9WxYRka4YY1L4z3/+g5YtW+LQoUOoXbu2VuonaUyhYcOG2LNnD4YPH65yfvPmzXj69CmmT5+uccWIiPTJGJPCvXv38PPPP2u1W1JSB9mGDRuKrZAKAB4eHggLC9O4UkRE+maMS2e3adOm1A3JpJLUUkhJSSlxExxbW1u9D04SEWmDMbYUxo0bhylTpiAlJQWenp4qW3ICQJMmTdSOKSkpODo64ty5c8XmJJw7d07vTzoQEWmDMSaFvn37AoBKV75MJhOnDeh8oLnIqFGjMHHiROTl5eHDDz8EUDiL7osvvsCUKVOkhCQiMihjTArx8fFajykpKUybNg3Pnz/H559/Lj5bbm5ujunTpyM4OFirFSQi0gdjXCXV2dlZ6zElJQWZTIYlS5Zg1qxZiImJgYWFBRo2bMi9FIjIaBnr5DUAuH37NhISEopNAO3Zs6fasSQlhSJVq1ZFq1atNAmhYvHixQgODsaECRPwzTffaC0uEdHbGGP30YMHD/DJJ5/g5s2b4lgC8H/1lDKmoFHau337No4ePYqDBw+qHFJcvnwZ69evlzRaTkSkKV3vvLZ48WLIZDJMnDhRa3WeMGECXFxckJqaCktLS/zvf//D6dOn0bJlS5w6dUpSTEktBW1np8zMTAQEBGDjxo0ICQmRUiUiIo3osqWgqy+958+fx++//4733ntPTFo+Pj4IDQ3F+PHj/3ZDtNJISnvazk5jxoxB9+7d0bVr17dem5OTA4VCoXIQEZVXr3/prV69ulZjK5VKcbmh9957D8nJyQAKB6BjY2MlxZSUFM6fP4/58+eXmp3U8dNPP+HatWsIDQ0t0/WhoaGwsbERD0dHRym3QESkQp39FN78YpqTk1NqXHW+9Krr/fffx/Xr1wEUzm5eunQpzp07h/nz56N+/fqSYkpKCtrKTomJiZgwYQK2b99e5n0YgoODkZGRIR6JiYnq3wAR0RvUGVNwdHRU+XJa2pdadb/0qmvmzJnictnz589HfHw82rdvj8OHD2P16tWSYkoaUyjKTi4uLmJ2ksvl2LBhg1rZ6erVq0hNTUWLFi3Ec0qlEqdPn8a3336LnJycYmuNmJmZ8dFXItI6dcYUEhMTYW1tLZ4v6TOp6Evv8ePHdbb5WLdu3cQ/N2jQAHfu3EFaWhqqV68uefxDUlKYOXMmsrKyABRmp48//hjt27dHzZo1sWvXrjLH6dKlC27evKlyLjAwEG5ubpg+fXq5W3yKiN5d6iQFa2trlaRQEilferWhRo0aGv28pKSgrexkZWWF999/X+VclSpVULNmzWLniYh0SdtPH+nqS2+fPn0QHh4Oa2tr9OnT52+v3bdvn9rx1U4KeXl58PPzQ1hYGBo2bCie1zQ7EREZkraTgq6+9NrY2Ij1sLGxkRynNGonBVNTU9y4cUPrFSkidcIFEZEmjGVG85YtWwAUbos8b9482NrawsLCQmvxJT19NHjwYGzatElrlSAiMjR1HkmV6tSpU1pbwkcQBDRo0ABJSUlaiVdE0phCfn4+Nm/ejBMnTsDLywtVqlRReX/FihVaqRwRkb4YS0uhSKVKldCwYUM8f/5cpStfU5KSwq1bt8QR9bt376q8V55+aURE6jC2z6/Fixdj2rRpWLdundYezpGUFE6ePKmVwomIygtjaykAwJAhQ5CdnY2mTZtCLpcXG1tIS0tTO6bkpbPPnDmD9evX48GDB9izZw/q1KmDH374AS4uLvDx8ZEalojIIIxxPwVdbDEgKSns3bsXn332GQICAnDt2jVx3Y+MjAwsWrQIhw8f1moliYiouKFDh2o9pqS0FxISgrCwMGzcuBGmpqbi+Xbt2uHatWtaqxwRkb7o4+kjXXr16pVWVpCW1FKIjY1Fhw4dip23sbFBenq6pIpo6uTJkzpbX6Qk+m4NGSLZPn/+XK/lFS2yqE/NmzfXa3m9evXSa3lA4Q6J73J5+fn5WoljjGMKWVlZmD59Onbv3l3i/1e97bxmb2+PuLi4YufPnj0reblWIiJDMsaWwhdffIHff/8d69atg5mZGb7//nvMmzcPDg4O2LZtm6SYkloKo0aNwoQJE7B582bIZDIkJyfj/PnzmDp1KmbNmiWpIkREhmSMA82//vortm3bhk6dOiEwMBDt27dHgwYN4OzsjO3btyMgIEDtmJKSwowZM1BQUIAuXbogOzsbHTp0gJmZGaZOnYpx48ZJCUlEZFDG2H2UlpYm9s5YW1uLj6D6+PggKChIUkxJaU8mk+Grr75CWloabt26hQsXLuDp06dYsGCBpEoQERmaMXYf1a9fH/Hx8QAANzc37N69G0BhC6JatWqSYkpKCgkJCRAEAXK5HO7u7mjdurU4uJSQkCCpIkREhmSMSSEwMFDcjnPGjBlYu3YtzM3NMWnSJEybNk1STEndRy4uLnj8+DHs7OxUzj9//hwuLi6SRryJiEg9kyZNEv/ctWtX3LlzB1evXkWDBg3QpEkTSTElJQVBEErMmJmZmXp9LJSISFuMcaA5MTERjo6O4mtnZ2c4OztrFFOtpDB58mQAhc2sWbNmwdLSUnxPqVTi4sWLaNasmUYVIiIyBGMcaK5Xrx58fHwwePBg/Otf/0L16tU1jqlWUoiKigJQ2FK4efMm5HK5+J5cLkfTpk0xdepUjStFRKRvxpgUrly5gh07dmD+/PkYN24c/Pz8MHjwYPTo0QNmZmaSYqqVFIpWRw0MDMTq1au1MgN13bp1WLduHf78808AgIeHB2bPng1/f3+NYxMRlZUxJoXmzZujefPmWLp0KU6dOoUdO3bg3//+NwoKCtCnTx9s3rxZ7ZiSxhS2bNmCiIgIREREIDU1FQUFBSrvq1ORunXrYvHixWjYsCEEQcDWrVvRq1cvREVFwcPDQ0r1iIjUZoxJoYhMJkPnzp3RuXNnBAUFYcSIEdi6daukpCBp1GT+/Pnw9fVFREQEnj17hhcvXqgc6ujRowc++ugjNGzYEI0aNcLChQtRtWpVXLhwQUrViIgkM6bHUV+XlJSEpUuXolmzZuIUgbVr10qKJamlsG7dOoSHh+Ozzz6TVGhplEol9uzZg6ysLHh7e5d4TU5OjrhUNwDJKwESEb3OGFsK69evx44dO3Du3Dm4ubkhICAAv/zyi0ZPIElKCrm5uWjbtq3kQt908+ZNeHt749WrV6hatSr2798Pd3f3Eq8NDQ3FvHnztFY2EZGxCgkJwcCBA7F69Wo0bdpUKzEldR+NHDkSO3bs0EoFAKBx48aIjo7GxYsXERQUhKFDh+L27dslXhscHIyMjAzxSExM1Fo9iKjiMsYZzQkJCVi6dKnWEgIgsaXw6tUrbNiwASdOnECTJk1UNtoBgBUrVqgVTy6Xo0GDBgAALy8vXL58GatWrcL69euLXWtmZib5USsiotIYY/eRLuojKSncuHFDnKR269Ytlfe0UcmCggKVcQMiIl0zxqSgC5KSQtF8BW0IDg6Gv78/nJyc8PLlS+zYsQOnTp3CsWPHtFYGEdHbMCkUkpQUtCk1NRVDhgzB48ePYWNjgyZNmuDYsWP45z//aeiqEVEFwqRQSK2k0KdPnzJdt2/fvjLH3LRpkzpVICKiEjx79gwXL16EUqlEq1atULt2bUlx1EoKNjY2kgohIirvdNFSCA0Nxb59+3Dnzh1YWFigbdu2WLJkCRo3bqxJVYvZu3cvRowYgUaNGiEvLw+xsbFYu3YtAgMD1Y6lVlLYsmWL2gUQERkDXSydHRkZiTFjxqBVq1bIz8/Hl19+CV9fX9y+fRtVqlSRXNfMzExxYzMAmDdvHi5duoRGjRoBAA4dOoRRo0ZJSgrla3FwIqJ3yNGjRzFs2DB4eHigadOmCA8PR0JCAq5evapRXC8vL/zyyy/i68qVKyM1NVV8/eTJE5VVrNVh8IFmIqLyQB8DzRkZGQCAGjVqaBTn2LFjGDNmDMLDw7F27VqsWrUK/fv3h1KpRH5+PipVqoTw8HBJsZkUiIigXlJ4c821skyqLSgowMSJE9GuXTu8//77GtW1Xr16OHToEHbu3ImOHTti/PjxiIuLQ1xcHJRKJdzc3CTvgsnuIyIiqLfMhaOjI2xsbMQjNDT0rfHHjBmDW7du4aefftJanQcOHIjLly/j+vXr6NSpEwoKCtCsWTONtkVmS4GICOq1FBITE2FtbS2ef1srYezYsfjvf/+L06dPo27duppXFsDhw4cRExODpk2b4vvvv0dkZCQCAgLg7++P+fPnw8LCQlJcthSIiKBeS8Ha2lrlKC0pCIKAsWPHYv/+/fj999/h4uKilbpOmTIFgYGBuHz5MkaPHo0FCxagY8eOuHbtGszNzdG8eXMcOXJEUux3pqWQkZGh1/WS/ve//+mtLAAwMTHRa3kA0K9fP72W5+TkpNfyAKBjx456LU9bHwrqeO+99/RaniaPWkpRnmcZjxkzBjt27MAvv/wCKysrpKSkACic8yX1mzwAhIeH47fffoOXlxfS0tLwwQcfYNasWZDL5ViwYAEGDhyI0aNHS9rWmC0FIiLoZunsdevWISMjA506dULt2rXFY9euXRrVtUqVKoiPjwdQ2JX15hiCu7s7zpw5Iyn2O9NSICLShC4eSRUEQZMqlSo0NBRDhgzB+PHjkZ2dja1bt2otNpMCERGMa0G8gIAA+Pn54cGDB2jYsCGqVaumtdhMCkRE/195+dAvi5o1a6JmzZpaj8ukQEQE42op6BIHmomISMSWAhER2FIowqRARAQmhSJMCkREYFIoYvAxhdDQULRq1QpWVlaws7ND7969ERsba+hqERFVSAZPCkU7E124cAHHjx9HXl4efH19kZWVZeiqEVEFoosZzcbI4N1HR48eVXkdHh4OOzs7XL16FR06dDBQrYiIKiaDJ4U3vW1nopycHJWF797c7IKISAqOKRQyePfR68qyM1FoaKjK5haOjo56riURvYvYfVSoXCWFsuxMFBwcjIyMDPFITEzUYw2JiN5t5ab7qKw7E5VlL1QiInWx+6iQwZOCIAgYN24c9u/fj1OnThlkExIiIipk8KSgq52JiIjUwZZCIYOPKehqZyIiInVwoLmQwVsKutqZiIiI1GfwlgIREZUfBm8pEBGVBxxTKMSWAhERidhSICICWwpFmBSIiMCkUITdR0REJGJLgYgIbCkUeWeSgp2dnV5nQNva2uqtLABwc3PTa3kA4OHhodfy/vnPf+q1PABwcHDQa3mGWLfL2tpar+VVrqzfj5Xc3Fy9lveue2eSAhGRJthSKMSkQEQEJoUiHGgmItKhtWvXol69ejA3N0ebNm1w6dIlQ1fpbzEpEBFBNwvi7dq1C5MnT8acOXNw7do1NG3aFN26dUNqaqqO7kJzTApERDqyYsUKjBo1CoGBgXB3d0dYWBgsLS2xefNmQ1etVBxTICIC8PLly7e2BF6+fAkAUCgUKudL2hEyNzcXV69eRXBwsHiuUqVK6Nq1K86fP6+lWmsfkwIRVWhyuRz29vZwdHQs0/VVq1Ytdu2cOXMwd+5clXPPnj2DUqlErVq1VM7XqlULd+7c0ajOusSkQEQVmrm5OeLj48s830EQhGItindp33gmBSKq8MzNzWFubq7VmO+99x5MTEzw5MkTlfNPnjyBvb29VsvSJoMPNJ8+fRo9evSAg4MDZDIZDhw4YOgqERFpTC6Xw8vLCxEREeK5goICREREwNvb24A1+3sGTwpZWVlo2rQp1q5da+iqEBFp1eTJk7Fx40Zs3boVMTExCAoKQlZWFgIDAw1dtVIZvPvI398f/v7+hq4GEZHW9e/fH0+fPsXs2bORkpKCZs2a4ejRo8UGn8sTgycFdeXk5CAnJ0d8/eajYURE5cnYsWMxduxYQ1ejzAzefaSu0NBQ2NjYiEdZHyMjIqK3M7qkEBwcjIyMDPFITEw0dJWIiN4ZRtd9VNLMQSIi0g6jaykQEZHuGLylkJmZibi4OPF1fHw8oqOjUaNGDTg5ORmwZkREFY/Bk8KVK1fQuXNn8fXkyZMBAEOHDkV4eLiBakVEVDEZPCl06tQJgiAYuhpERASOKRAR0WuYFIiISMSkQEREIiYFIiISMSkQEZGISYGIiERMCkREJDL4PAVNFc1xePXqlV7LVSqVei0vLy9Pr+UB+v+dZmVl6bU8AHj58qVeyyvrPsDGrHJl/X6sFP0dcr6TdsgEI/9NJiUlcflsIkJiYiLq1q1r6GoYPaNPCgUFBUhOToaVlRVkMlmZf06hUMDR0RGJiYmwtrbWYQ0rVpm8x3ejTGO6R0EQ8PLlSzg4OKBSJfaIa8rou48qVaqk0bcDa2trvf2jr0hl8h7fjTKN5R5tbGx0VJuKh2mViIhETApERCSqsEnBzMwMc+bM0esubhWhTN7ju1FmRbhHKpnRDzQTEZH2VNiWAhERFcekQEREIiYFIiISMSkQEZGowiSFffv2wdfXFzVr1oRMJkN0dHSxa0aPHg1XV1dYWFjA1tYWvXr1wp07d3RSXlpaGsaNG4fGjRvDwsICTk5OGD9+PDIyMiSVV5YyAWDDhg3o1KkTrK2tIZPJkJ6ertPyXr16hTFjxqBmzZqoWrUq+vbtiydPnkgu801PnjzBsGHD4ODgAEtLS/j5+eHevXtai1+SzMxMjB07FnXr1oWFhQXc3d0RFham0zJlMlmJx9dff62zMmNiYtCzZ0/Y2NigSpUqaNWqFRISEnRW3rBhw4rdn5+fn87Ko5JVmKSQlZUFHx8fLFmypNRrvLy8sGXLFsTExODYsWMQBAG+vr6SFr97W3nJyclITk7GsmXLcOvWLYSHh+Po0aMYMWKE2mWVtUwAyM7Ohp+fH7788kvJ5ahT3qRJk/Drr79iz549iIyMRHJyMvr06aNx2UDh8ga9e/fGgwcP8MsvvyAqKgrOzs7o2rWrThfXmzx5Mo4ePYoff/wRMTExmDhxIsaOHYuDBw/qrMzHjx+rHJs3b4ZMJkPfvn11Ut79+/fh4+MDNzc3nDp1Cjdu3MCsWbNgbm6uk/KK+Pn5qdznzp07dVoelUCoYOLj4wUAQlRU1FuvvX79ugBAiIuL00t5u3fvFuRyuZCXlye5vLKWefLkSQGA8OLFC43K+rvy0tPTBVNTU2HPnj3iuZiYGAGAcP78eY3LjY2NFQAIt27dEs8plUrB1tZW2Lhxo8bxS+Ph4SHMnz9f5VyLFi2Er776SmdlvqlXr17Chx9+qLP4/fv3FwYPHqyz+CUZOnSo0KtXL72WScVVmJaCurKysrBlyxa4uLjobRXWjIwMWFtb633pYV25evUq8vLy0LVrV/Gcm5sbnJyccP78eY3j5+TkAIDKt9dKlSrBzMwMZ8+e1Th+adq2bYuDBw/i0aNHEAQBJ0+exN27d+Hr66uzMl/35MkTHDp0SKNW5d8pKCjAoUOH0KhRI3Tr1g12dnZo06YNDhw4oJPyXnfq1CnY2dmhcePGCAoKwvPnz3VeJqliUnjDd999h6pVq6Jq1ao4cuQIjh8/DrlcrvNynz17hgULFuDf//63zsvSl5SUFMjlclSrVk3lfK1atZCSkqJx/KIEExwcjBcvXiA3NxdLlixBUlISHj9+rHH80qxZswbu7u6oW7cu5HI5/Pz8sHbtWnTo0EFnZb5u69atsLKy0lo33JtSU1ORmZmJxYsXw8/PD7/99hs++eQT9OnTB5GRkTopEyjsOtq2bRsiIiKwZMkSREZGwt/fX+97l1R072RS2L59u/jBXrVqVZw5c6bMPxsQEICoqChERkaiUaNG+PTTT9+62Ywm5QGFSwZ3794d7u7umDt3bpl+RtMy1aXv8spShwsXLmDfvn24e/cuatSoAUtLS5w8eRL+/v5aW0K5pPtes2YNLly4gIMHD+Lq1atYvnw5xowZgxMnTuiszNdt3rwZAQEBWuvff7O82NhYAECvXr0wadIkNGvWDDNmzMDHH3+stQH1ku5xwIAB6NmzJzw9PdG7d2/897//xeXLl3Hq1CmtlEll8270U7yhZ8+eaNOmjfi6Tp06Zf5ZGxsb2NjYoGHDhvjggw9QvXp17N+/HwMHDtRJeS9fvoSfnx+srKywf/9+mJqalunnNClTCinl2dvbIzc3F+np6SqthSdPnsDe3l4rdbCwsEB0dDQyMjKQm5sLW1tbtGnTBi1btlQ7flnL7NKlC/bv34/u3bsDAJo0aYLo6GgsW7ZMpatMm2UWOXPmDGJjY7Fr1y6NyymtPFtbW1SuXBnu7u4q1/3jH//QWrdcWf491a9fH++99x7i4uLQpUsXrZRLb/dOJgUrKytYWVlpHEcQBAiCIPZda7s8hUKBbt26wczMDAcPHlTrm5+27lGX5Xl5ecHU1BQRERHiUzKxsbFISEiAt7e3VutQtJ7+vXv3cOXKFSxYsEDt+GUpU6FQIC8vr1hLxMTEBAUFBTop83WbNm2Cl5cXmjZtqpWySiuvVatWYouhyN27d+Hs7KyzMt+UlJSE58+fo3bt2lopk8rmnUwKJUlLS0NCQgKSk5MBQPwHb29vD3t7ezx48AC7du2Cr68vbG1tkZSUhMWLF8PCwgIfffSR1stTKBTw9fVFdnY2fvzxRygUCigUCgCF39RMTEy0XiZQ2M+fkpKCuLg4AMDNmzdhZWUFJycn1KhRQ6vl2djYYMSIEZg8eTJq1KgBa2trjBs3Dt7e3vjggw/Uvr+S7NmzB7a2tnBycsLNmzcxYcIE9O7dW2eDvtbW1ujYsSOmTZsGCwsLODs7IzIyEtu2bcOKFSt0UmYRhUKBPXv2YPny5TotBwCmTZuG/v37o0OHDujcuTOOHj2KX3/9VWddOZmZmZg3bx769u0Le3t73L9/H1988QUaNGiAbt266aRMKoWBn37Smy1btggAih1z5swRBEEQHj16JPj7+wt2dnaCqampULduXWHQoEHCnTt3dFJe0SOhJR3x8fE6KVMQBGHOnDklXrNlyxadlPfXX38Jn3/+uVC9enXB0tJS+OSTT4THjx9Lur+SrFq1Sqhbt65gamoqODk5CTNnzhRycnK0Fr8kjx8/FoYNGyY4ODgI5ubmQuPGjYXly5cLBQUFOi13/fr1goWFhZCenq7Tcops2rRJaNCggWBubi40bdpUOHDggM7Kys7OFnx9fQVbW1vB1NRUcHZ2FkaNGiWkpKTorEwqGZfOJiIi0Tv59BEREUnDpEBERCImBSIiEjEpEBGRiEmBiIhETApERCRiUiAiIhGTAhklmUyml6WciSoaJgUql54+fYqgoCA4OTnBzMwM9vb26NatG86dOwegcCcyf39/AMCff/5Z6nagRKSeCrP2ERmXvn37Ijc3F1u3bkX9+vXx5MkTREREiJuuSFlllYjejstcULmTnp6O6tWr49SpU+jYsWOJ18hkMuzfvx+9e/eGTCZTea9jx47iwm3ff/89li9fjvj4eNSrVw/jx4/H559/rutbIDJabClQuVO08cqBAwfwwQcfwMzM7G+vv3TpElq3bo0TJ07Aw8ND3Clv+/btmD17Nr799ls0b94cUVFRGDVqFKpUqYKhQ4fq41aIjA7HFKjcqVy5MsLDw7F161ZUq1YN7dq1w5dffokbN26UeL2trS0AoGbNmrC3txeXAJ8zZw6WL1+OPn36wMXFBX369MGkSZOwfv16vd0LkbFhUqByqW/fvkhOTsbBgwfh5+eHU6dOoUWLFggPDy/Tz2dlZeH+/fsYMWKEyraPISEhuH//vm4rT2TEOKZARmPkyJE4fvw4Hj58qDKm8Oeff8LFxQVRUVFo1qwZgP/b8vPHH39U2fYRKNwlzcXFxQB3QFT+cUyBjIa7u3uJcxOKxhCUSqV4rlatWnBwcMCDBw8QEBCgryoSGT0mBSp3nj9/jn79+mH48OFo0qQJrKyscOXKFSxduhS9evUqdr2dnR0sLCxw9OhR1K1bF+bm5rCxscG8efMwfvx42NjYwM/PDzk5Obhy5QpevHiByZMnG+DOiIyAIbd9IyrJq1evhBkzZggtWrQQbGxsBEtLS6Fx48bCzJkzhezsbEEQBAGAsH//fvFnNm7cKDg6OgqVKlUSOnbsKJ7fvn270KxZM0EulwvVq1cXOnToIOzbt0/Pd0RkPDimQEREIj59REREIiYFIiISMSkQEZGISYGIiERMCkREJGJSICIiEZMCERGJmBSIiEjEpEBERCImBSIiEjEpEBGRiEmBiIhE/w+MVjl/fzJiyQAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 400x400 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "orders = np.arange(1, 10)\n",
    "positions = np.arange(-13, -4)\n",
    "ticks = np.arange(9)\n",
    "\n",
    "fig, axes = plt.subplots(1, 1, figsize=(4, 4))\n",
    "im = axes.imshow(\n",
    "    sites.iloc[::-1, :],\n",
    "    cmap=\"Greys\",\n",
    "    vmin=0,\n",
    ")\n",
    "axes.set(\n",
    "    xticks=ticks,\n",
    "    xticklabels=positions,\n",
    "    xlabel=\"Site\",\n",
    "    yticks=ticks,\n",
    "    yticklabels=orders[::-1],\n",
    "    ylabel=\"Interaction order $k$\",\n",
    "    aspect=\"equal\",\n",
    ")\n",
    "plt.colorbar(im, shrink=0.6, label=\"% variance explained\")\n",
    "fig.tight_layout()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7c279bf",
   "metadata": {},
   "source": [
    "This plot shows that whereas some positions have large additive and pairwise contributions, other sites have more context-dependent contributions to phenotypic variation, with the largest contributions arising from interactions of order 3 or higher.\n",
    "\n",
    "### How to compute the variance explained by pairwise and higher order interactions between every pair of sites\n",
    "\n",
    "In this example, we do an finer aggregation and separate the variance explained by pairwise and higher order interactions between pairs of sites on the upper and lower triangular parts of the heatmap representation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "76874f0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>site2</th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>site1</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>20.235400</td>\n",
       "      <td>19.297067</td>\n",
       "      <td>12.736994</td>\n",
       "      <td>8.528527</td>\n",
       "      <td>7.706634</td>\n",
       "      <td>5.887429</td>\n",
       "      <td>4.593244</td>\n",
       "      <td>3.814681</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8.562998</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>28.937652</td>\n",
       "      <td>22.225871</td>\n",
       "      <td>13.779223</td>\n",
       "      <td>9.638418</td>\n",
       "      <td>7.624355</td>\n",
       "      <td>5.565630</td>\n",
       "      <td>4.245642</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>9.734242</td>\n",
       "      <td>13.710318</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>28.779777</td>\n",
       "      <td>20.837386</td>\n",
       "      <td>13.780875</td>\n",
       "      <td>7.749691</td>\n",
       "      <td>5.786126</td>\n",
       "      <td>4.398405</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.380292</td>\n",
       "      <td>9.409153</td>\n",
       "      <td>6.562947</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>31.867450</td>\n",
       "      <td>23.175051</td>\n",
       "      <td>12.769771</td>\n",
       "      <td>6.189255</td>\n",
       "      <td>5.449330</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.371878</td>\n",
       "      <td>2.865711</td>\n",
       "      <td>4.951972</td>\n",
       "      <td>7.331872</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>31.061032</td>\n",
       "      <td>19.615363</td>\n",
       "      <td>10.645316</td>\n",
       "      <td>6.161446</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.202204</td>\n",
       "      <td>1.320697</td>\n",
       "      <td>2.437803</td>\n",
       "      <td>3.856483</td>\n",
       "      <td>4.213326</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>21.838515</td>\n",
       "      <td>13.053678</td>\n",
       "      <td>7.791366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1.367760</td>\n",
       "      <td>1.333584</td>\n",
       "      <td>0.959993</td>\n",
       "      <td>1.554637</td>\n",
       "      <td>3.281415</td>\n",
       "      <td>1.401816</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>13.716746</td>\n",
       "      <td>7.937204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.635882</td>\n",
       "      <td>0.887281</td>\n",
       "      <td>0.527592</td>\n",
       "      <td>0.344046</td>\n",
       "      <td>1.860773</td>\n",
       "      <td>1.435976</td>\n",
       "      <td>1.094349</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>9.259406</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.129675</td>\n",
       "      <td>0.408224</td>\n",
       "      <td>0.274862</td>\n",
       "      <td>0.711385</td>\n",
       "      <td>0.446727</td>\n",
       "      <td>1.349362</td>\n",
       "      <td>0.584660</td>\n",
       "      <td>0.498108</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "site2         0          1          2          3          4          5  \\\n",
       "site1                                                                    \n",
       "0      0.000000  20.235400  19.297067  12.736994   8.528527   7.706634   \n",
       "1      8.562998   0.000000  28.937652  22.225871  13.779223   9.638418   \n",
       "2      9.734242  13.710318   0.000000  28.779777  20.837386  13.780875   \n",
       "3      1.380292   9.409153   6.562947   0.000000  31.867450  23.175051   \n",
       "4      1.371878   2.865711   4.951972   7.331872   0.000000  31.061032   \n",
       "5      1.202204   1.320697   2.437803   3.856483   4.213326   0.000000   \n",
       "6      1.367760   1.333584   0.959993   1.554637   3.281415   1.401816   \n",
       "7      0.635882   0.887281   0.527592   0.344046   1.860773   1.435976   \n",
       "8      0.129675   0.408224   0.274862   0.711385   0.446727   1.349362   \n",
       "\n",
       "site2          6          7         8  \n",
       "site1                                  \n",
       "0       5.887429   4.593244  3.814681  \n",
       "1       7.624355   5.565630  4.245642  \n",
       "2       7.749691   5.786126  4.398405  \n",
       "3      12.769771   6.189255  5.449330  \n",
       "4      19.615363  10.645316  6.161446  \n",
       "5      21.838515  13.053678  7.791366  \n",
       "6       0.000000  13.716746  7.937204  \n",
       "7       1.094349   0.000000  9.259406  \n",
       "8       0.584660   0.498108  0.000000  "
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pairs_pw = sds.calc_site_pairs_variance_perc(v_U_vcs.loc[v_U_vcs['k'] == 2, :])\n",
    "pairs_ho = sds.calc_site_pairs_variance_perc(v_U_vcs, min_k=3)\n",
    "\n",
    "m_bottom = pairs_pw.pivot(index=\"site1\", columns=\"site2\", values=\"variance_perc\")\n",
    "m_bottom = m_bottom.reindex(ticks).T.reindex(ticks).fillna(0)\n",
    "m_top = pairs_ho.pivot(index='site1', columns='site2', values='variance_perc')\n",
    "m_top = m_top.reindex(ticks).T.reindex(ticks).fillna(0).T\n",
    "m = m_top + m_bottom\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "63840aa5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x400 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 1, figsize=(4, 4))\n",
    "im = axes.imshow(\n",
    "    m,\n",
    "    cmap=\"Greys\",\n",
    "    vmin=0,\n",
    ")\n",
    "axes.set(\n",
    "    xticks=ticks,\n",
    "    xlabel=\"Site 1\",\n",
    "    yticks=ticks,\n",
    "    xticklabels=positions,\n",
    "    yticklabels=positions,\n",
    "    ylabel=\"Site 2\",\n",
    "    aspect=\"equal\",\n",
    ")\n",
    "plt.colorbar(im, shrink=0.6, label=\"% variance explained\")\n",
    "fig.tight_layout()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9fd01c5a",
   "metadata": {},
   "source": [
    "Overall, this summary plots show that positions in the Shine-Dalgarno sequence tend to interact mostly with adjacent positions. Moreover, higher order interactions can involve sites that are further appart from each other in the primary sequence."
   ]
  }
 ],
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