Notes from Linear Algebra for Data Science with R
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| Rmd | c2ae4e5 | John Blischak | 2019-03-31 | Notes from Linear Algebra for Data Science with R. |
Notes from the DataCamp course Linear Algebra for Data Science with R.
v <- c(10, 10)
v[1] 10 10Increase x-component:
A <- matrix(c(5, 0, 0, 1), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 5 0
[2,] 0 1A %*% v [,1]
[1,] 50
[2,] 10Decrease y-component:
A <- matrix(c(1, 0, 0, 1/5), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 1 0.0
[2,] 0 0.2A %*% v [,1]
[1,] 10
[2,] 2Reflect about the x-axis:
A <- matrix(c(1, 0, 0, -1), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 1 0
[2,] 0 -1A %*% v [,1]
[1,] 10
[2,] -10Reflect about the y-axis:
A <- matrix(c(-1, 0, 0, 1), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] -1 0
[2,] 0 1A %*% v [,1]
[1,] -10
[2,] 10Identity matrix:
diag(3) [,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0 1Find the inverse with solve.
A <- matrix(c(0, 5, -2, 0), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 0 5
[2,] -2 0solve(A) [,1] [,2]
[1,] 0.0 -0.5
[2,] 0.2 0.0A %*% solve(A) [,1] [,2]
[1,] 1 0
[2,] 0 1solve(A) %*% A [,1] [,2]
[1,] 1 0
[2,] 0 1Compute the determinant:
A <- matrix(c(1, 0, 0, 2), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 1 0
[2,] 0 2det(A)[1] 2A <- matrix(c(1, 0, 2, 0), nrow = 2, byrow = TRUE)
A [,1] [,2]
[1,] 1 0
[2,] 2 0det(A)[1] 0Compute eigenvalues (\(\lambda\)) and eigenvectors (\(\textbf{x}\)):
\[ A\textbf{x} = \lambda\textbf{x}\]
A <- cbind(c(1,-1), c(-1,1))
(E <- eigen(A))eigen() decomposition
$values
[1] 2 0
$vectors
[,1] [,2]
[1,] -0.7071068 -0.7071068
[2,] 0.7071068 -0.7071068A %*% E$vectors[, 1] [,1]
[1,] -1.414214
[2,] 1.414214E$values[1] * E$vectors[, 1][1] -1.414214 1.414214A %*% E$vectors[, 2] [,1]
[1,] 0
[2,] 0E$values[2] * E$vectors[, 2][1] 0 0Principal Component Analysis
combine_url <- "https://assets.datacamp.com/production/repositories/2654/datasets/760dae913f682ba6b2758207280138662ddedc0d/DataCampCombine.csv"
combine <- read.csv(combine_url)
dim(combine)[1] 2885 13head(combine) player position school year height weight forty vertical
1 Jaire Alexander CB Louisville 2018 71 192 4.38 35.0
2 Brian Allen C Michigan St. 2018 73 298 5.34 26.5
3 Mark Andrews TE Oklahoma 2018 77 256 4.67 31.0
4 Troy Apke S Penn St. 2018 74 198 4.34 41.0
5 Dorance Armstrong EDGE Kansas 2018 76 257 4.87 30.0
6 Ade Aruna DE Tulane 2018 78 262 4.60 38.5
bench broad_jump three_cone shuttle
1 14 127 6.71 3.98
2 27 99 7.81 4.71
3 17 113 7.34 4.38
4 16 131 6.56 4.03
5 20 118 7.12 4.23
6 18 128 7.53 4.48
drafted
1 Green Bay Packers / 1st / 18th pick / 2018
2 Los Angeles Rams / 4th / 111th pick / 2018
3 Baltimore Ravens / 3rd / 86th pick / 2018
4
5
6 Minnesota Vikings / 6th / 218th pick / 2018x <- as.matrix(combine[, 5:12])
# Subtract the column means
(column_means <- apply(x, 2, mean)) height weight forty vertical bench broad_jump three_cone
73.992721 252.087348 4.811584 32.598267 21.091854 113.190295 7.305719
shuttle
4.408548 x <- sweep(x, 2, column_means)
apply(x, 2, mean) height weight forty vertical bench
-1.572879e-15 -9.502468e-15 3.977476e-16 -1.041735e-16 1.217401e-15
broad_jump three_cone shuttle
4.524137e-15 2.807089e-16 -2.469472e-17 # Calculate variance-covariance matrix
x_cov <- t(x) %*% x / (nrow(x) - 1)
dim(x_cov)[1] 8 8x_cov height weight forty vertical bench
height 7.1597944 90.788084 0.52676257 -5.065512 6.2780614
weight 90.7880840 2105.176834 13.04832553 -132.887401 188.5644427
forty 0.5267626 13.048326 0.10318906 -1.034265 0.9549273
vertical -5.0655119 -132.887401 -1.03426459 18.232972 -8.7243553
bench 6.2780614 188.564443 0.95492726 -8.724355 40.8989801
broad_jump -11.2472274 -330.417460 -2.55786742 33.083582 -24.1124924
three_cone 0.5955444 15.944407 0.11523591 -1.245058 1.2152061
shuttle 0.3912377 9.644035 0.06894544 -0.811669 0.6913887
broad_jump three_cone shuttle
height -11.247227 0.59554442 0.39123769
weight -330.417460 15.94440664 9.64403467
forty -2.557867 0.11523591 0.06894544
vertical 33.083582 -1.24505770 -0.81166895
bench -24.112492 1.21520610 0.69138865
broad_jump 90.079934 -3.11504850 -1.89762575
three_cone -3.115048 0.18612283 0.09866407
shuttle -1.897626 0.09866407 0.07283010# The diagonal entries are the variance of the original columns
diag(x_cov) height weight forty vertical bench broad_jump
7.1597944 2105.1768336 0.1031891 18.2329720 40.8989801 90.0799335
three_cone shuttle
0.1861228 0.0728301 apply(x, 2, var) height weight forty vertical bench broad_jump
7.1597944 2105.1768336 0.1031891 18.2329720 40.8989801 90.0799335
three_cone shuttle
0.1861228 0.0728301 # The non-diagonal entries are the covariance between the two columns
cov(x[, 1], x[, 5])[1] 6.278061x_cov[1, 5][1] 6.278061x_cov[5, 1][1] 6.278061# Calculate eigenvalues of covariance matrix
x_eig <- eigen(x_cov)
x_eigeigen() decomposition
$values
[1] 2.187628e+03 4.403246e+01 2.219205e+01 5.267129e+00 2.699702e+00
[6] 6.317016e-02 1.480866e-02 1.307283e-02
$vectors
[,1] [,2] [,3] [,4] [,5]
[1,] -0.042047079 -0.061885367 0.1454490039 -0.1040556410 0.980792060
[2,] -0.980711529 -0.130912788 0.1270100265 0.0193388930 -0.066908382
[3,] -0.006112061 0.012525260 0.0025260713 -0.0021291637 -0.004096693
[4,] 0.062926466 -0.333556369 0.0398922845 0.9366594549 0.074901137
[5,] -0.088291423 -0.313533433 -0.9363461471 -0.0745692157 0.107188391
[6,] 0.156742686 -0.876925849 0.2904565302 -0.3252903706 -0.126494599
[7,] -0.007468520 0.014691994 0.0009057581 0.0003320888 -0.020902644
[8,] -0.004518826 0.009863931 0.0023111814 -0.0094052914 -0.004010629
[,6] [,7] [,8]
[1,] -0.020679696 -6.155636e-03 0.0008055445
[2,] 0.008423587 6.988341e-04 0.0036087841
[3,] -0.152469227 -2.539868e-01 -0.9549983725
[4,] -0.012214516 7.045063e-03 -0.0070051256
[5,] -0.009167322 -8.604309e-05 -0.0048308793
[6,] -0.013753112 -2.187651e-03 -0.0076907609
[7,] -0.894560357 -3.743559e-01 0.2427137770
[8,] -0.419039274 8.917710e-01 -0.1700673446# Percentage of variance explained by each principal component
barplot(x_eig$values / sum(x_eig$values) * 100)
| Version | Author | Date |
|---|---|---|
| 16cb14b | John Blischak | 2019-04-01 |
# Using prcomp (and also scaling the columns)
scaled <- scale(combine[5:12])
pca <- prcomp(scaled)
summary(pca)Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7
Standard deviation 2.3679 0.9228 0.78904 0.61348 0.46811 0.37178 0.34834
Proportion of Variance 0.7009 0.1064 0.07782 0.04704 0.02739 0.01728 0.01517
Cumulative Proportion 0.7009 0.8073 0.88514 0.93218 0.95957 0.97685 0.99202
PC8
Standard deviation 0.25266
Proportion of Variance 0.00798
Cumulative Proportion 1.00000
sessionInfo()R version 4.3.3 (2024-02-29 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 11 x64 (build 22631)
Matrix products: default
locale:
[1] LC_COLLATE=English_United States.utf8
[2] LC_CTYPE=English_United States.utf8
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C
[5] LC_TIME=English_United States.utf8
time zone: America/New_York
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] workflowr_1.7.1
loaded via a namespace (and not attached):
[1] jsonlite_1.8.8 compiler_4.3.3 highr_0.10 promises_1.2.1
[5] Rcpp_1.0.12 stringr_1.5.1 git2r_0.33.0 callr_3.7.5
[9] later_1.3.2 jquerylib_0.1.4 yaml_2.3.8 fastmap_1.1.1
[13] R6_2.5.1 knitr_1.45 tibble_3.2.1 rprojroot_2.0.4
[17] bslib_0.6.1 pillar_1.9.0 rlang_1.1.3 utf8_1.2.4
[21] cachem_1.0.8 stringi_1.8.3 httpuv_1.6.14 xfun_0.42
[25] getPass_0.2-4 fs_1.6.3 sass_0.4.8 cli_3.6.2
[29] magrittr_2.0.3 ps_1.7.6 digest_0.6.34 processx_3.8.3
[33] rstudioapi_0.15.0 lifecycle_1.0.4 vctrs_0.6.5 evaluate_0.23
[37] glue_1.7.0 whisker_0.4.1 fansi_1.0.6 rmarkdown_2.26
[41] httr_1.4.7 tools_4.3.3 pkgconfig_2.0.3 htmltools_0.5.7