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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Inverse Distance Weighting (IDW) Interpolation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us suppose we have a data that shows the variation of one quantity of interest across space.\n",
"This could be equivalently viewed as { ($\\vec{x_1}, y_1)$,$(\\vec{x_2}, y_2)$,$(\\vec{x_3}, y_3)$, ...}, where the $\\vec{x_i}$'s represent the coordinates of the points where we have data and the $y_i$'s are the actual data at those points. <br><br>\n",
"We would like to perform an interpolation using these data points such that a few things are satisifed.\n",
"1. The interpolation is exact - the value at the known data points is the same as the estimated value, and \n",
"2. We would want far away points from a given source data point to receive less importance than nearby points.\n",
"3. Wikipedia has an excellent article on IDW. I am linking it [here](https://en.wikipedia.org/wiki/Inverse_distance_weighting)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We are using the following approximation for coordinate_type being latlong_small<br>\n",
"$| \\vec{r_2}− \\vec{r_1}| ≈ \\text{R }\\times \\sqrt[]{(Lat_2 - Lat_1)^{2} + (Long_2 - Long_1)^{2}}$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"df = pd.read_csv('../../testdata/30-03-18.csv')\n",
"data = np.array(df[['longitude','latitude','value']])"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def make_grid(X,y,res):\n",
" y_min = y.min()-0.2\n",
" y_max = y.max()+0.2\n",
" x_min = X.min()-0.2\n",
" x_max = X.max()+0.2\n",
" x_arr = np.linspace(x_min,x_max,res)\n",
" y_arr = np.linspace(y_min,y_max,res)\n",
" xx,yy = np.meshgrid(x_arr,y_arr) \n",
" return xx,yy\n",
"\n",
"def idw(dataset, exponent = 2, resolution='standard', coordinate_type='euclidean',verbose='False'):\n",
" \"\"\"\n",
" Here X is the set of spatial locations - Usually assumed to be Lat-Long\n",
" To be extended to higher dimenstions y - estimated value , exponenet - how\n",
" much weight to assign to far off locations to be estimated for each data point, \n",
" extent - interpolate over a grid - what is xmax xmin ymax ymin\n",
" \"\"\"\n",
" if coordinate_type == 'latlong_small':\n",
" \"\"\"\n",
" Assume that the Earth is a Sphere, and use polar coordinates\n",
" $| \\vec{r_2}− \\vec{r_1}| ≈ \\text{R }\\times \\sqrt[]{(Lat_2 - Lat_1)^{2} + (Long_2 - Long_1)^{2}}$\n",
" \"\"\"\n",
" return \"To be done later\"\n",
" if coordinate_type == 'latlong_large':\n",
" \"\"\"\n",
" Code to be written after understanding all the projections.\n",
" \"\"\"\n",
" return \"To be done later\"\n",
" if coordinate_type==\"euclidean\":\n",
" \n",
"# print(dataset)\n",
" X = dataset[:,0]\n",
" y = dataset[:,1]\n",
" if resolution=='high':\n",
" xx,yy = make_grid(X,y,1000)\n",
" \n",
" if resolution=='low':\n",
" xx,yy = make_grid(X,y,10)\n",
" \n",
" if resolution=='standard':\n",
" xx,yy = make_grid(X,y,100)\n",
" \n",
" new = []\n",
" new_arr = dataset\n",
" for points in new_arr:\n",
" mindist = np.inf\n",
" val = 0\n",
" for j in range(len(yy)):\n",
" temp = yy[j][0]\n",
" for i in range(len(xx[0])):\n",
" dist = np.linalg.norm(np.array([xx[0][i],temp]) - points[:2])\n",
" if dist<mindist:\n",
" mindist = dist\n",
" val = (i,j)\n",
" new.append((points,val))\n",
" print(new)\n",
" new_grid = np.zeros((len(xx),len(yy)))\n",
" for i in range(len(new)):\n",
" x = new[i][1][0]\n",
" y = new[i][1][1]\n",
" new_grid[x][y] = new[i][0][2]\n",
" print(new[i])\n",
" x_nz,y_nz = np.nonzero(new_grid)\n",
" list_nz = []\n",
" for i in range(len(x_nz)):\n",
" list_nz.append((x_nz[i],y_nz[i]))\n",
" \n",
" final = np.copy(new_grid)\n",
" \n",
" for i in range(len(xx[0])):\n",
" for j in range(len(yy)):\n",
" normalise = 0\n",
" if (i,j) in list_nz:\n",
" continue\n",
" else:\n",
" \"\"\"\n",
" Could potentially have a divide by zero error here\n",
" Use a try except clause\n",
" \"\"\"\n",
" for elem in range(len(x_nz)):\n",
" source = np.array([x_nz[elem],y_nz[elem]])\n",
" target = np.array([xx[0][i],yy[j][0]])\n",
" dist = (np.abs(xx[0][source[0]] - target[0])**exponent + np.abs(yy[source[1]][0] - target[1])**exponent)**(1/exponent)\n",
" final[i][j]+=new_grid[x_nz[elem],y_nz[elem]]/dist\n",
" normalise+=1/(dist)\n",
" final[i][j]/=normalise\n",
" \n",
" return final\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[(array([ 77.234291, 28.581197, 194. ]), (60, 39)), (array([ 77.245721, 28.739434, 267. ]), (62, 60)), (array([ 77.101961, 28.822931, 273. ]), (42, 72)), (array([ 76.991463, 28.620806, 129. ]), (27, 44)), (array([ 77.0325413, 28.60909 , 176. ]), (33, 42)), (array([ 77.072196, 28.570859, 172. ]), (38, 37)), (array([ 77.1670103, 28.5646102, 168. ]), (51, 36)), (array([ 77.1180053, 28.5627763, 105. ]), (45, 36)), (array([ 77.272404, 28.530782, 203. ]), (66, 32)), (array([ 77.26075 , 28.563827, 192. ]), (64, 36)), (array([77.0996943, 28.610304 , 95. ]), (42, 43)), (array([ 77.2273074, 28.5918245, 148. ]), (59, 40)), (array([ 77.09211 , 28.732219, 203. ]), (41, 59)), (array([ 77.317084, 28.668672, 221. ]), (72, 51)), (array([ 77.1585447, 28.6573814, 141. ]), (50, 49)), (array([ 77.2011573, 28.6802747, 192. ]), (56, 52)), (array([ 77.237372, 28.612561, 203. ]), (61, 43)), (array([ 77.305651, 28.632707, 152. ]), (70, 46)), (array([ 77.1473105, 28.6514781, 185. ]), (49, 48)), (array([ 77.16482 , 28.699254, 290. ]), (51, 55)), (array([ 77.170221, 28.728722, 273. ]), (52, 59)), (array([ 77.2005604, 28.6372688, 173. ]), (56, 46)), (array([ 77.2011573, 28.7256504, 269. ]), (56, 58)), (array([ 77.136777, 28.669119, 160. ]), (47, 51)), (array([77.267246, 28.49968 , 78. ]), (65, 27)), (array([ 77.2494387, 28.6316945, 211. ]), (62, 45)), (array([ 77.2735737, 28.5512005, 252. ]), (66, 34)), (array([ 77.2159377, 28.5504249, 133. ]), (58, 34)), (array([77.1112615, 28.7500499, 77. ]), (44, 62)), (array([77.22445, 28.63576, 96. ]), (59, 46))]\n",
"(array([ 77.234291, 28.581197, 194. ]), (60, 39))\n",
"(array([ 77.245721, 28.739434, 267. ]), (62, 60))\n",
"(array([ 77.101961, 28.822931, 273. ]), (42, 72))\n",
"(array([ 76.991463, 28.620806, 129. ]), (27, 44))\n",
"(array([ 77.0325413, 28.60909 , 176. ]), (33, 42))\n",
"(array([ 77.072196, 28.570859, 172. ]), (38, 37))\n",
"(array([ 77.1670103, 28.5646102, 168. ]), (51, 36))\n",
"(array([ 77.1180053, 28.5627763, 105. ]), (45, 36))\n",
"(array([ 77.272404, 28.530782, 203. ]), (66, 32))\n",
"(array([ 77.26075 , 28.563827, 192. ]), (64, 36))\n",
"(array([77.0996943, 28.610304 , 95. ]), (42, 43))\n",
"(array([ 77.2273074, 28.5918245, 148. ]), (59, 40))\n",
"(array([ 77.09211 , 28.732219, 203. ]), (41, 59))\n",
"(array([ 77.317084, 28.668672, 221. ]), (72, 51))\n",
"(array([ 77.1585447, 28.6573814, 141. ]), (50, 49))\n",
"(array([ 77.2011573, 28.6802747, 192. ]), (56, 52))\n",
"(array([ 77.237372, 28.612561, 203. ]), (61, 43))\n",
"(array([ 77.305651, 28.632707, 152. ]), (70, 46))\n",
"(array([ 77.1473105, 28.6514781, 185. ]), (49, 48))\n",
"(array([ 77.16482 , 28.699254, 290. ]), (51, 55))\n",
"(array([ 77.170221, 28.728722, 273. ]), (52, 59))\n",
"(array([ 77.2005604, 28.6372688, 173. ]), (56, 46))\n",
"(array([ 77.2011573, 28.7256504, 269. ]), (56, 58))\n",
"(array([ 77.136777, 28.669119, 160. ]), (47, 51))\n",
"(array([77.267246, 28.49968 , 78. ]), (65, 27))\n",
"(array([ 77.2494387, 28.6316945, 211. ]), (62, 45))\n",
"(array([ 77.2735737, 28.5512005, 252. ]), (66, 34))\n",
"(array([ 77.2159377, 28.5504249, 133. ]), (58, 34))\n",
"(array([77.1112615, 28.7500499, 77. ]), (44, 62))\n",
"(array([77.22445, 28.63576, 96. ]), (59, 46))\n"
]
},
{
"data": {
"text/plain": [
"(100, 100)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"idw(data).shape\n"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"temp = data[10]"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(array([10, 10, 10]), array([0, 1, 2]))"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.where(data==temp)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [],
"source": [
"result = np.nonzero(data==temp)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.unique(result[0])[0]"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [],
"source": [
"listOfCoordinates= list(zip(result[0], result[1]))"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(10, 0), (10, 1), (10, 2)]"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"listOfCoordinates"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
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