Multivariate polynomial regression is a generalization of multivariate linear regression. The interest of polynomial regression is to be able to introduce non-linearity in the predictive analysis. Polynomial regression allows finding an adequate prediction function for data that does not seem to have a linear relationship. Polynomial regression describes the variation of an explained random variable, called here z, as a function of a set of explanatory random variables, called here V. We seek, by regression, to link the variables of V by a polynomial of degree n.
When the set V = {x} contains only one variable x, the formula of a polynomial regression model of degree n is of the form z ~ SUM[i=0..n](bi.x^i) = b0.x^0 + b1.x^1 + b2.x^2 + … + bn.x^n. When the set V = {x, y} contains two variables x and y the formula of a polynomial regression model of degree n becomes z ~ SUM[i=0..n](SUM[j=n-i..0](bij.x^i.y^j))
The example introduced below shows how to apply a polynomial regression of degree 4 in order to find a correlation between the value of the oil barrel and the concentration at some depth of an imaginary mineral called : Mendacium.
OIL SEARCHING
Let’s consider that some scientists working for a powerful oil company have the intuition that a correlation exists between the quality of inner ground oil and the concentration at some depth of a mineral called Mendacium. They decide to build a dataset containing the price of the barrel after refinement of the oil found at some depth in the ground with the concentration of Mendacium found at this depth. The dataset built is an array containing 1000 rows whose columns are Mendacium (concentration in g/m^3), Depth (in meters) and Price (in dollars/barrel). The whole dataset is given at the bottom of the page.
The oil company asks a data scientist to find this correlation (if it exists), and after failing to fit the data to one multilinear and the two first bivariate polynomial regression models (of degrees 2 and 3), he builds the polynomial regression model at degree 4.
The dependent variable is P (for Price) and the two independent variables are M (for Mendacium) and D (for Depth). Since the targeted polynomial degree is 4, the formula of the model is P ~ b00.M^0.D^0 + b10.M^1.D^0 + b20.M^2.D^0 + b30.M^3.D^0 + b40.M^4.D^0 + b01.M^0.D^1 + b11.M^1.D^1 + b21.M^2.D^1 + b31.M^3.D^1 + b02.M^0.D^2 + b12.M^1.D^2 + b22.M^2.D^2 + b03.M^0.D^3 + b13.M^1.D^3 + b04.M^0.D^4
In both scripts, in Python and R, we start by preparing the dataset so that it contains one column for each polynomial term (monom) of the targeted model formula above. Building a polynomial regression model is then equivalent to build a multilinear model for this « polynomially structured » dataset. Thus, we apply the backward elimination method, in order to find the optimal multilinear regression model. And finally, we show the observations and the predictions in a 3D graph, that helps to get a visual impression of the correctness of the model.
Here is the script in Python.
And here, the script in R.
Here follows the dataset in csv format :
Mendacium,Depth,Price
3.359,1722.533,74.04869105
1.348,2062.571,83.32190745
0.418,2013.507,82.74896396
8.193,1420.607,84.76049381
5.21,951.957,54.58867143
11.189,1574.612,115.3110199
11.657,1372.887,117.1226166
11.895,1251.325,118.1482877
4.879,153.332,57.39187583
3.419,2142.666,83.43013034
6.41,2167.11,97.21816461
6.543,523.084,55.97652351
8.146,2899.092,148.9883104
7.043,1228.714,64.07193605
12.976,228.467,121.589231
10.126,2828.65,152.3597179
2.57,2415.989,111.2053664
5.465,1146.022,71.46200487
10.004,629.207,68.13009814
11.612,2008.463,139.1919038
5.901,291.673,57.40289822
3.681,1958.027,91.22771014
11.091,617.128,80.70150564
8.769,332.9,68.94609119
0.458,1284.385,56.19711916
3.963,521.528,57.65697416
11.752,800.857,94.40546491
12.006,587.324,95.07704595
0.407,370.716,54.85469647
10.609,2289.441,131.6769917
5.821,2437.354,103.4629116
3.197,737.532,57.24212307
11.075,1465.252,96.96059825
6.549,2041.722,84.49466709
3.145,239.938,41.24029038
10.263,614.906,70.05121755
12.507,2652.358,184.0506068
1.86,694.608,42.4766079
5.318,225.73,43.06831796
9.18,2995.655,162.8020367
12.645,2675.286,173.6182134
2.93,2679.509,112.2343512
2.034,2688.027,117.732033
2.549,1378.941,68.11107034
11.398,599.604,92.87324038
7.843,2759.128,139.6123932
8.103,451.831,64.36109441
7.597,798.878,56.0110079
9.867,596.613,74.30641992
3.426,987.717,58.79587309
9.504,1305.213,81.06425009
11.002,2296.367,145.6526972
6.905,298.131,50.23309733
8.39,226.594,69.49867258
12.712,1195.733,122.2505466
4.095,694.645,52.74679322
6.396,2956.143,144.4365121
12.124,1610.064,134.9761855
11.39,288.249,92.33372029
2.466,915.511,52.15375218
9.474,2973.132,151.3622473
3.842,608.705,44.77190597
6.898,166.722,49.09822678
4.26,770.882,46.46693483
2.036,507.169,49.17476416
9.095,692.762,77.37813552
1.468,621.19,46.03041664
4.575,2025.084,92.49672446
8.066,802.113,68.29276007
0.216,1196.052,68.44721179
5.451,1513.991,63.76757587
12.281,1920.929,138.3032677
4.906,2178.795,101.4953382
12.342,72.818,102.5029498
8.886,621.246,69.59159075
8.993,1799.265,92.70308859
8.415,2763.592,141.2193903
4.88,1967.909,80.38668819
5.735,271.646,40.82297531
4.364,1666.37,70.94561791
8.388,2939.418,141.0581617
3.516,902.405,63.92819538
6.796,2684.597,122.7592298
3.483,1570.833,70.3792156
2.595,1578.32,79.78576795
10.286,1811.11,114.4423489
12.663,1225.123,122.3704271
10.837,787.413,79.93125059
1.812,1609.634,75.12856196
2.057,2796.395,130.2731715
1.776,2957.738,131.3569043
10.292,2559.485,138.3066493
6.104,2358.75,96.47063207
1.456,1313.317,63.85687179
12,86.197,108.7702209
10.629,126.216,78.71998252
6.194,2345.663,112.6326047
7.038,2875.937,135.0521069
10.154,2999.077,162.732866
0.824,2777.089,122.4317407
3.427,1957.902,79.43091779
7.54,2046.05,101.9028642
0.311,1924.688,85.19140801
3.145,1416.945,65.51235726
3.656,1404.021,71.48307686
6.267,2686.905,121.141089
9.25,155.845,74.94676025
11.228,1090.658,93.28236211
6.882,1507.391,70.46484215
11.584,1519.722,107.2112491
7.64,953.393,63.58440072
4.164,363.34,44.91151827
3.761,2619.531,122.8388511
11.574,304.296,86.70665158
11.009,93.088,78.52754834
2.418,1862.565,77.5270987
8.836,2234.104,104.0831725
7.023,1064.51,60.35189779
0.366,2.947,47.26013166
1.884,1246.767,58.97145801
0.052,1648.462,77.7852209
5.125,2713.175,122.6707564
10.409,149.125,72.57185203
4.643,4.371,38.82187454
7.149,703.005,53.34939921
8.485,2453.965,118.8291042
2.333,456.668,48.34051242
11.046,2009.143,116.877523
1.797,28.624,38.73426258
7.974,1490.998,69.58324344
2.548,2761.116,127.8326868
4.76,1269.401,72.01424043
6.582,2982.853,150.6559949
0.295,1520.509,63.32035055
1.154,1537.094,65.497558
12.2,1048.214,106.9352131
5.638,506.286,44.72869433
5.006,2214.527,88.91548282
11.756,2453.618,162.0789303
1.721,824.114,48.3936002
12.329,648.79,107.9798648
6.709,1192.14,60.08353894
6.402,1872.648,94.86638349
10.318,1940.046,119.6029077
12.365,747.979,109.3750761
1.735,1142.347,56.57177031
6.965,214.151,44.3417372
11.49,2626.113,160.444737
11.995,565.229,110.5278936
11.788,1049.04,104.9818123
5.01,323.029,57.97944811
2.369,1251.608,63.08972689
2.326,1727.477,80.4898606
7.937,148.671,58.38534372
4.733,1206.442,55.22428584
12.729,1957.473,151.6912149
12.864,121.592,109.841504
4.944,2660.614,117.227581
0.45,1788.518,74.42060811
10.335,1014.539,79.5770751
6.864,862.283,53.7199708
8.034,2782.933,140.1564428
11.422,2946.428,169.0024023
8.854,496.388,73.65444239
7.121,1691.862,85.61967479
8.678,1824.517,103.0015617
4.617,2367.393,104.1982565
6.078,490.203,54.23472469
11.693,627.74,104.9277992
0.67,2715.857,117.2370211
6.821,1859.013,91.21461597
4.036,1388.72,59.01966793
2.083,2382.117,113.6775081
12.493,1015.593,123.6887993
10.715,2378.936,140.3605964
4.629,1348.232,69.99859873
11.13,2647.5,164.6577721
9.472,1464.947,92.5653145
6.705,2632.467,113.6801452
5.215,2538.974,120.3325489
1.642,274.283,50.33048802
4.931,2837.823,131.7691465
8.308,530.118,66.45849228
8.802,2461.15,113.8470731
4.178,899.691,51.47324205
12,1232.951,110.8568123
0.112,2759.294,126.5900055
2.74,1824.969,82.46108239
6.887,252.536,44.02127423
4.472,577.423,49.65324748
0.972,2545.247,117.0100247
7.103,2911.895,140.7485543
10.768,2756.892,147.6812971
5.155,2720.859,114.2868607
6.358,168.672,56.22868434
9.85,2924.74,156.9131977
11.867,2168.882,151.0917957
6.598,1052.592,57.84437558
2.762,1553.275,80.95085886
8.59,1749.286,85.88895347
10.569,49.984,73.6470264
2.56,775.011,59.25376276
12.511,143.938,117.350731
9.421,731.179,66.63746841
3.283,2215.793,93.3528461
6.579,2644.142,129.7046797
5.435,2619.553,117.5827161
3.406,2374.979,107.1750608
7.658,887.934,61.82122903
3.565,2165.395,98.57007221
10.152,2487.657,144.1228122
11.79,1104.545,106.1946822
4.664,2789.654,124.7902566
0.352,670.194,54.49051547
9.525,1038.713,76.87943325
2.331,1289.094,70.35628032
12.793,1376.413,137.4526116
2.531,1661.759,81.24680698
3.829,598.573,59.54105743
10.843,1295.488,107.4525826
6.212,2956.492,131.1365603
1.627,1892.321,91.21737031
2.966,1472.55,72.95617925
2.759,2411.868,100.8551753
7.738,2399.828,111.4972735
1.914,460.23,49.5524219
12.057,1029.53,115.1146541
11.512,2767.488,173.8490699
10.836,640.432,89.4872276
7.921,1284.396,82.03955991
9.781,267.3,60.76429335
10.665,1702.629,98.68240098
2.439,1751.723,72.94989224
6.292,1186.082,68.50633829
6.416,602.337,49.36261064
4.323,920.946,56.37056042
5.339,2597.407,125.8133472
0.683,1687.342,84.5589055
5.892,1962.978,97.50573742
7.544,2784.292,137.0647872
3.552,561.87,42.17445336
8.296,1504.506,78.000092
9.405,803.523,68.98680437
12.49,1038.182,122.4464524
6.123,2807.673,126.2995596
10.888,1643.06,117.4936814
3.222,609.786,57.28895738
6.153,1813.973,81.51677057
1.404,1469.152,72.26293904
8.029,1229.84,70.20005539
13,398.452,115.4551749
9.199,426.761,64.27043829
0.609,95.211,43.6310281
10.945,2139.785,123.4443418
8.749,2975.448,155.0207565
6.532,683.892,48.74715772
0.379,1612.036,74.45924606
0.097,2284.824,102.6415913
8.161,1986.237,97.69073771
3.25,343.756,40.40071906
10.079,807.144,77.73382976
3.131,1619.666,74.53294777
8.209,1210.049,68.15675782
0.644,2226.663,100.316249
9.387,2304.88,115.5928024
10.273,1214.883,90.86298872
2.978,1596.795,77.32360979
1.415,2375.327,99.71403403
12.652,1347.051,137.3004576
7.729,2332.801,105.295623
12.135,2740.358,180.4771349
2.804,2934.91,135.7028694
4.181,2418.46,113.5213978
9.529,399.445,61.57369879
4.363,1938.53,83.19556187
8.639,771.437,67.07714487
8.409,555.89,59.67501148
10.516,1211.154,94.31047177
1.83,2628.484,111.0806978
2.628,2530.022,119.1730984
7.085,2197.452,97.52413626
5.505,646.277,45.29033457
6.429,2639.948,117.4679461
12.625,1742.82,137.2526938
4.353,2585.764,120.9337496
2.916,560.596,51.98227409
7.251,1021.818,55.34877794
4.519,535.468,60.6307964
11.953,2157.279,144.5339093
8.268,2213.403,97.82948153
10.388,2943.446,164.6574237
2.389,2280.501,94.19040277
9.292,572.128,70.70991109
7.275,1900.228,99.44974036
5.073,988.154,49.94273898
3.963,610.98,59.86896537
9.708,2431.902,134.6609353
5.639,655.718,44.80511364
4.786,1706.868,70.39317007
7.59,1666.451,79.60862807
6.266,294.084,47.26068722
1.695,2305.535,105.1304953
10.211,1540.278,96.33676542
9.205,2055.065,109.4717415
6.193,667.826,63.60019833
11.884,1896.687,141.1691101
9.165,2986.775,163.8560849
7.648,176.51,48.10801544
11.115,2432.112,150.8943394
4.93,1560.858,68.16533603
2.142,499.779,43.29146588
10.367,896.985,92.32179029
11.095,2543.14,144.719205
8.896,1416.303,80.57001453
2.399,2432.759,113.3201454
8.729,2678.112,131.4506039
10.265,1994.92,108.7202044
12.344,39.706,96.8346326
4.619,1038.275,60.31699493
7.343,2585.033,113.9652385
6.697,889.05,52.43761333
10.935,2773.589,152.5718188
9.169,1876.293,97.11180368
1.94,1730.795,83.19550685
4.071,1446.052,67.26710255
9.039,1018.371,73.90411669
4.819,520.542,59.59775067
11.403,894.721,106.4963625
6.38,1444.717,67.28374586
8.34,1493.251,78.26475282
3.848,1018.204,48.01078837
4.891,580.673,59.48331787
0.752,497.179,41.59311925
1.247,2162.801,93.8649535
10.306,1755.054,113.2611976
10.957,1198.282,104.5255274
11.071,1328.845,100.3928266
5.515,981.968,51.55159695
8.766,1058.242,81.02221406
10.297,746.699,76.61805161
12.786,1187.583,134.4001485
8.429,2480.208,117.272658
0.536,353.053,38.32620687
1.237,748.065,45.53913237
4.121,517.238,57.19663674
3.309,2735.253,125.2320244
7.294,1.878,63.03695551
6.247,18.615,56.06957704
5.607,253.462,42.39190617
2.042,2881.85,131.7507867
6.288,481.721,52.50705628
2.453,2381.396,95.33557149
10.076,1373.528,84.25215658
12.319,682.612,116.0912212
10.541,2543.812,135.6389855
9.572,298.902,77.92023112
3.99,1237.535,55.92729923
7.943,324.745,53.845321
9.531,2831.042,144.2655307
3.113,1227.333,70.87496612
4.694,1782.584,71.50593482
10.598,2510.405,139.0491518
0.626,1530.846,75.48230921
0.708,2179.679,94.85262908
4.572,1009.916,60.69636157
9.173,1662.421,98.18166577
3.372,1384.38,56.70952657
1.411,993.328,49.03892591
0.008,1928.958,92.37732935
2.221,244.12,51.29679643
0.547,2478.343,112.7818419
12.245,1426.653,129.8083674
10.531,2848.025,153.1706361
0.044,467.852,41.927824
12.123,1823.084,135.0232059
12.978,1692.936,152.9133592
6.012,2349.604,96.0204345
8.734,581.696,56.5387151
6.144,2583.784,123.4909698
9.882,1772.457,92.69018454
3.12,1233.029,53.04811574
0.15,2596.808,108.0421554
12.786,557.364,124.3646153
11.12,365.602,86.84090435
8.391,2059.93,102.4640139
3.897,1498.437,77.45239574
11.945,446.075,95.10859742
7.131,84.274,54.19404999
12.401,1688.606,131.2118828
5.477,522.757,61.24422382
8.342,300.94,59.79473555
8.493,877.089,58.6999144
3.067,2219.134,96.79021344
0.272,1060.408,61.38334933
5.184,1050.2,63.05455405
5.944,157.156,53.70919519
0.48,404.96,49.23978983
3.02,1496.282,65.0451129
7.379,2866.742,131.8411653
5.616,638.772,58.34838047
10.782,2579.245,137.6095846
9.886,422.14,72.65220894
10.379,957.226,94.84497541
7.028,1088.771,67.64605835
10.995,2790.008,168.9388078
7.252,1027.476,54.82821968
5.743,1156.982,59.26692581
2.646,1782.798,71.11398736
6.54,813.143,61.38532045
4.247,47.156,53.49541057
7.524,2646.648,118.6075691
6,1529.821,65.76309209
9.262,1312.883,88.9914264
7.393,459.406,54.15969533
12.115,2497.891,161.591116
6.425,1208.261,72.29212788
1.504,2189.05,101.7531727
1.994,18.82,55.60305844
5.791,1067.09,68.15604288
10.377,482.589,82.82190961
9.188,1295.098,74.18403727
11.736,2774.207,180.4713525
10.656,2958.038,170.0285182
10.88,31.467,80.29137561
2.362,71.928,45.40954921
4.082,1571.922,67.80525291
12.191,1521.385,134.5748065
10.376,2968.063,158.1005439
1.669,1116.755,54.25440659
0.293,803.599,49.17485827
3.672,537.786,51.98800287
1.028,665.039,52.92370683
8.669,223.584,54.98337624
6.643,2322.221,102.0367106
11.01,720.164,91.67630882
6.532,1060.127,58.9071045
5.191,2448.949,100.0604139
11.24,1788.403,117.3982271
1.067,1974.585,91.3993765
11.546,2763.371,176.9649514
2.827,232.733,40.02143638
2.583,1605.491,73.77595421
8.568,230.155,65.19700809
8.42,574.17,68.62816194
11.796,2729.533,165.7775705
9.858,1029.625,85.64937073
9.055,788.082,68.66278788
7.566,1930.884,86.21345328
9.61,2853.717,154.8915756
9.526,2076.788,108.2491873
12.58,1050.846,125.7132843
7.62,1221.046,69.57330618
8.79,85.481,70.715182
4.709,2779.192,118.7758786
6.147,2606.414,120.0194433
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