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How to generate orthogonal polynomials in R?
We can say that orthogonal is a synonym of perpendicular. If the inner product (inner product is generalization of dot product) of two polynomials is zero then we call them orthogonal polynomials. In R, we can find the orthogonal product by using poly function as shown in the below examples.
Example1
> x<-rnorm(10) > x
Output
[1] 1.53798786 -0.85463326 2.39444451 0.82559418 -2.22197322 -1.04243823 [7] -0.04693054 -0.68691236 -1.63040923 -1.42408865
Example
> orthogonal_x<-poly(x,degree=2) > orthogonal_x
Output
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"
Example2
> orthogonal_x_degree3<-poly(x,degree=3) > orthogonal_x_degree3
Output
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"
Example3
> orthogonal_x_degree5<-poly(x,degree=5) > orthogonal_x_degree5
Output
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"
Example4
> orthogonal_y_degree3<-poly(y,degree=3) > orthogonal_y_degree3
Output
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"
Example5
> orthogonal_y_degree5<-poly(y,degree=5) > orthogonal_y_degree5
Output
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"
Updated on: 19-Nov-2020
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