Last updated: 2024-04-20

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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