Last updated: 2024-04-20
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Knit directory: muse/
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File | Version | Author | Date | Message |
---|---|---|---|---|
Rmd | 8c5bc97 | Dave Tang | 2024-04-20 | Euclidean vs. Cosine |
Following this excellent Euclidean vs. Cosine Distance article.
c(
c(6.6, 6.2, 1),
c(9.7, 9.9, 2),
c(8.0, 8.3, 2),
c(6.3, 5.4, 1),
c(1.3, 2.7, 0),
c(2.3, 3.1, 0),
c(6.6, 6.0, 1),
c(6.5, 6.4, 1),
c(6.3, 5.8, 1),
c(9.5, 9.9, 2),
c(8.9, 8.9, 2),
c(8.7, 9.5, 2),
c(2.5, 3.8, 0),
c(2.0, 3.1, 0),
c(1.3, 1.3, 0)
) |>
matrix(ncol = 3, byrow = TRUE) |>
as.data.frame() -> my_df
colnames(my_df) <- c('weight', 'length', 'label')
my_df <- dplyr::mutate(my_df, label = factor(label, levels = 0:2))
head(my_df)
weight length label
1 6.6 6.2 1
2 9.7 9.9 2
3 8.0 8.3 2
4 6.3 5.4 1
5 1.3 2.7 0
6 2.3 3.1 0
Plot.
ggplot(my_df, aes(weight, length, colour = label)) +
geom_point() +
theme_minimal()
Example.
my_eg <- my_df[c(1, 2, 5, 15), ]
my_eg
weight length label
1 6.6 6.2 1
2 9.7 9.9 2
5 1.3 2.7 0
15 1.3 1.3 0
Euclidean distance.
\[ \sqrt{\sum^n_{i=1} (x_i - y_i)^2} \] Calculate Euclidean distances.
euclid_dist <- function(x, y){
sqrt(sum((x - y)^2))
}
x <- my_eg[4, -3]
apply(my_eg[-4, -3], 1, function(y) euclid_dist(x, y))
1 2 5
7.218033 12.021647 1.400000
Cosine distance.
\[ \frac{x \cdot y}{\sqrt{x \cdot x} \sqrt{y \cdot y}} \]
Dot product.
\[ (\vec{x}, \vec{y}) = \vec{x} \cdot \vec{y} = \sum_{i=1}^{n}{x_{i}y_{i}} \]
dot_prod()
function versus built-in.
dot_prod <- function(x, y){
sum(x * y)
}
x <- unlist(as.vector(my_eg[4, -3]))
identical(dot_prod(x, x), as.vector(x %*% x))
[1] TRUE
Calculate Cosine distances.
cosine_dist <- function(x, y){
x <- unlist(as.vector(x))
y <- unlist(as.vector(y))
as.vector((x %*% y) / (sqrt(x %*% x) * sqrt(y %*% y)))
}
x <- my_eg[4, -3]
apply(my_eg[-4, -3], 1, function(y) cosine_dist(x, y))
1 2 5
0.9995121 0.9999479 0.9438584
Angles.
ggplot(my_eg, aes(weight, length)) +
geom_point() +
geom_segment(aes(xend=weight, yend = length), x = 0, y=0, lty = 3) +
theme_minimal() +
scale_x_continuous(limits = c(0, max(my_eg[, 1]))) +
scale_y_continuous(limits = c(0, max(my_eg[, 2])))
x0 <- unlist(as.vector(my_eg[1, -3]))
x1 <- unlist(as.vector(my_eg[2, -3]))
euclid_dist(x0, x1)
[1] 4.827007
cosine_dist(x0, x1)
[1] 0.9991413
\(L_1\) norm:
\[ \sum_i{x_i} \]
\(L_2\) norm:
\[ \sqrt{\sum_i{x^2_i}} \]
Distances with \(L_1\) normalisation.
l1_norm <- function(x){
x/sum(x)
}
l2_norm <- function(x){
x/sqrt(sum(x^2))
}
x0_n <- l1_norm(x0)
x1_n <- l1_norm(x1)
euclid_dist(x0_n, x1_n)
[1] 0.02931246
cosine_dist(x0_n, x1_n)
[1] 0.9991413
sessionInfo()
R version 4.3.3 (2024-02-29)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 22.04.4 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so; LAPACK version 3.10.0
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
time zone: Etc/UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4
[5] purrr_1.0.2 readr_2.1.5 tidyr_1.3.1 tibble_3.2.1
[9] ggplot2_3.5.0 tidyverse_2.0.0 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] sass_0.4.9 utf8_1.2.4 generics_0.1.3 stringi_1.8.3
[5] hms_1.1.3 digest_0.6.35 magrittr_2.0.3 timechange_0.3.0
[9] evaluate_0.23 grid_4.3.3 fastmap_1.1.1 rprojroot_2.0.4
[13] jsonlite_1.8.8 processx_3.8.4 whisker_0.4.1 ps_1.7.6
[17] promises_1.3.0 httr_1.4.7 fansi_1.0.6 scales_1.3.0
[21] jquerylib_0.1.4 cli_3.6.2 rlang_1.1.3 munsell_0.5.1
[25] withr_3.0.0 cachem_1.0.8 yaml_2.3.8 tools_4.3.3
[29] tzdb_0.4.0 colorspace_2.1-0 httpuv_1.6.15 vctrs_0.6.5
[33] R6_2.5.1 lifecycle_1.0.4 git2r_0.33.0 fs_1.6.3
[37] pkgconfig_2.0.3 callr_3.7.6 pillar_1.9.0 bslib_0.7.0
[41] later_1.3.2 gtable_0.3.4 glue_1.7.0 Rcpp_1.0.12
[45] highr_0.10 xfun_0.43 tidyselect_1.2.1 rstudioapi_0.16.0
[49] knitr_1.46 farver_2.1.1 htmltools_0.5.8.1 labeling_0.4.3
[53] rmarkdown_2.26 compiler_4.3.3 getPass_0.2-4