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   "cell_type": "markdown",
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   "source": [
    "# Inverse Distance Weighting (IDW)\n",
    "\n",
    "The **Inverse Distance Weighting (IDW)** is a deterministic interpolation method.\n",
    "\n",
    "The assigned values to unknown points are calculated with a *weighted average* of the values available at the known points. \n",
    "The weights are calculated as a *function of the distance* between the known and unknown points. The function used is the inverse of the distance, elevated to a given power, to each known point.\n",
    "\n",
    "The model can be tuned by the following parameters:\n",
    "- `power`: the power of the inverse distance function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mathematical formulation\n",
    "\n",
    "Given a set of sample points:\n",
    "\n",
    "$\\{ \\mathbf{x}_i, u_i | \\text{for } \\mathbf{x}_i \\in \\mathbb{R}^n,  u_i \\in \\mathbb{R}\\}_{i=1}^N$\n",
    "\n",
    "\n",
    "the IDW interpolation function $u(\\mathbf{x}): \\mathbb{R}^n \\to \\mathbb{R}$ is defined as:\n",
    "\n",
    "$u(\\mathbf{x}) = \\begin{cases}\n",
    " \\dfrac{\\sum_{i = 1}^{N}{ w_i(\\mathbf{x}) u_i } }{ \\sum_{i = 1}^{N}{ w_i(\\mathbf{x}) } }, & \\text{if } d(\\mathbf{x},\\mathbf{x}_i) \\neq 0 \\text{ for all } i, \\\\\n",
    " u_i, & \\text{if } d(\\mathbf{x},\\mathbf{x}_i) = 0 \\text{ for some } i,\n",
    "\\end{cases}$\n",
    "\n",
    "where\n",
    "\n",
    "$w_i(\\mathbf{x}) =  \\frac{1}{d(\\mathbf{x},\\mathbf{x}_i)^p}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "import py3dinterpolations as p3i\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 22,
     "metadata": {},
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   "source": [
    "df = pd.read_csv(\n",
    "    \"../../../tests/fixtures/griddata_default_colnames.csv\"\n",
    ")\n",
    "df.tail()"
   ]
  },
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   "cell_type": "code",
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       "                                 V\n",
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       "ID30 62.163 14.336 20.0   7.523950\n",
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       "                   16.0  12.431670\n",
       "                   14.0  12.653931\n",
       "                   12.0  17.956143\n",
       "...                            ...\n",
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       "\n",
       "[283 rows x 1 columns]"
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     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gd = GridData(df)\n",
    "gd.data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "interpolated, model = p3i.interpolate(\n",
    "    gd,\n",
    "    model_name = \"idw\",\n",
    "    model_params = {\n",
    "        \"power\": 2,\n",
    "    },\n",
    "    grid_resolution=5,\n",
    "    preprocess_kwags={\n",
    "        \"normalize_xyz\": True,\n",
    "        \"standardize_v\": True,\n",
    "    },\n",
    "    return_model=True,\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
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          "colorway": [
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           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
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           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = p3i.plot_3d_model(\n",
    "    model,\n",
    "    plot_points=True,\n",
    "    scale_points=10,\n",
    "    volume_kwargs={\n",
    "        \"surface_count\": 10,\n",
    "    }\n",
    ")\n",
    "fig.show()"
   ]
  }
 ],
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