我们可以说正交是vertical的同义词。如果两个多项式的内积（内积是点积的泛化）为零，则称它们为正交多项式。在R中，我们可以通过使用poly函数来找到正交积，如以下示例所示。
> x<-rnorm(10) > x
输出结果
[1] 1.53798786 -0.85463326 2.39444451 0.82559418 -2.22197322 -1.04243823 [7] -0.04693054 -0.68691236 -1.63040923 -1.42408865
> orthogonal_x<-poly(x,degree=2) > orthogonal_x
输出结果
1 2 [1,] 0.41743651 -0.01687537 [2,] -0.12158589 -0.21414848 [3,] 0.61038362 0.54027924 [4,] 0.25694468 -0.28489882 [5,] -0.42962751 0.57201520 [6,] -0.16389559 -0.14490046 [7,] 0.06037767 -0.37137903 [8,] -0.08380084 -0.26556522 [9,] -0.29635683 0.15165057 [10,] -0.24987583 0.03382235 attr(,"coefs") attr(,"coefs")$alpha [1] -0.3149359 0.5267254 attr(,"coefs")$norm2 [1] 1.00000 10.00000 19.70309 32.71111 attr(,"degree") [1] 1 2 attr(,"class") [1] "poly" "matrix"
> orthogonal_x_degree3<-poly(x,degree=3) > orthogonal_x_degree3
输出结果
1 2 3 [1,] 0.41743651 -0.01687537 -0.41882840 [2,] -0.12158589 -0.21414848 0.23310682 [3,] 0.61038362 0.54027924 0.42176833 [4,] 0.25694468 -0.28489882 -0.44362203 [5,] -0.42962751 0.57201520 -0.48208095 [6,] -0.16389559 -0.14490046 0.25573226 [7,] 0.06037767 -0.37137903 -0.07651475 [8,] -0.08380084 -0.26556522 0.19221655 [9,] -0.29635683 0.15165057 0.11263775 [10,] -0.24987583 0.03382235 0.20558443 attr(,"coefs") attr(,"coefs")$alpha [1] -0.3149359 0.5267254 -0.1157637 attr(,"coefs")$norm2 [1] 1.00000 10.00000 19.70309 32.71111 59.69013 attr(,"degree") [1] 1 2 3 attr(,"class") [1] "poly" "matrix"
> orthogonal_x_degree5<-poly(x,degree=5) > orthogonal_x_degree5
输出结果
1 2 3 4 5 [1,] 0.41743651 -0.01687537 -0.41882840 -0.53824181 -0.4545569 [2,] -0.12158589 -0.21414848 0.23310682 0.05839481 -0.2859929 [3,] 0.61038362 0.54027924 0.42176833 0.21609548 0.0994605 [4,] 0.25694468 -0.28489882 -0.44362203 0.06801947 0.5307426 [5,] -0.42962751 0.57201520 -0.48208095 0.33629533 -0.1868526 [6,] -0.16389559 -0.14490046 0.25573226 -0.09590512 -0.1828276 [7,] 0.06037767 -0.37137903 -0.07651475 0.48513908 0.1252141 [8,] -0.08380084 -0.26556522 0.19221655 0.19189132 -0.3055896 [9,] -0.29635683 0.15165057 0.11263775 -0.37914934 0.4409641 [10,] -0.24987583 0.03382235 0.20558443 -0.34253923 0.2194384 attr(,"coefs") attr(,"coefs")$alpha [1] -0.3149359 0.5267254 -0.1157637 0.1206771 -0.1404103 attr(,"coefs")$norm2 [1] 1.00000 10.00000 19.70309 32.71111 59.69013 67.13315 68.51457 attr(,"degree") [1] 1 2 3 4 5 attr(,"class") [1] "poly" "matrix"
> orthogonal_y_degree3<-poly(y,degree=3) > orthogonal_y_degree3
输出结果
1 2 3 [1,] 0.58967559 0.71092333 0.3049628 [2,] -0.07288125 -0.11174051 0.2422385 [3,] -0.07288125 -0.11174051 0.2422385 [4,] 0.05963012 -0.15968259 0.0224123 [5,] 0.19214148 -0.10138725 -0.2628119 [6,] 0.19214148 -0.10138725 -0.2628119 [7,] 0.19214148 -0.10138725 -0.2628119 [8,] -0.07288125 -0.11174051 0.2422385 [9,] -0.20539262 0.04243901 0.2150832 [10,] -0.07288125 -0.11174051 0.2422385 [11,] 0.05963012 -0.15968259 0.0224123 [12,] -0.33790399 0.30285595 -0.2406372 [13,] -0.07288125 -0.11174051 0.2422385 [14,] -0.33790399 0.30285595 -0.2406372 [15,] 0.19214148 -0.10138725 -0.2628119 [16,] 0.05963012 -0.15968259 0.0224123 [17,] -0.33790399 0.30285595 -0.2406372 [18,] -0.20539262 0.04243901 0.2150832 [19,] 0.05963012 -0.15968259 0.0224123 [20,] 0.19214148 -0.10138725 -0.2628119 attr(,"coefs") attr(,"coefs")$alpha [1] 4.550000 5.352546 6.177915 attr(,"coefs")$norm2 [1] 1.0000 20.0000 56.9500 354.4091 1091.8162 attr(,"degree") [1] 1 2 3 attr(,"class") [1] "poly" "matrix"
> orthogonal_y_degree5<-poly(y,degree=5) > orthogonal_y_degree5
输出结果
1 2 3 4 5 [1,] 0.58967559 0.71092333 0.3049628 0.06126607 0.01071962 [2,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433 [3,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433 [4,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992 [5,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731 [6,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731 [7,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731 [8,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433 [9,] -0.20539262 0.04243901 0.2150832 -0.46796052 0.37518656 [10,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433 [11,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992 [12,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808 [13,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433 [14,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808 [15,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731 [16,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992 [17,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808 [18,] -0.20539262 0.04243901 0.2150832 -0.46796052 0.37518656 [19,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992 [20,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731 attr(,"coefs") attr(,"coefs")$alpha [1] 4.550000 5.352546 6.177915 4.717763 4.126941 attr(,"coefs")$norm2 [1] 1.0000 20.0000 56.9500 354.4091 1091.8162 982.6517 729.7255 attr(,"degree") [1] 1 2 3 4 5 attr(,"class") [1] "poly" "matrix"