Exploring the data


Palmer penguins dataset


Data were collected and made available by Dr. Kristen Gorman and the Palmer Station, Antarctica LTER, a member of the Long Term Ecological Research Network. The aim was to explore all the data about ecological sexual dimorphism and environmental variability within a Community of Antarctic penguins of genus Pygoscelis (Gorman, Williams, and Fraser 2014)

Penguins package is available at CRAN - Package palmerpenguins (Horst, Hill, and Gorman 2020)

df = na.omit(penguins) #remove rows with NA values

kable(df) %>% kable_styling(fixed_thead = T, full_width = FALSE) %>%
  scroll_box( height = "600px")
species island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g sex year
Adelie Torgersen 39.1 18.7 181 3750 male 2007
Adelie Torgersen 39.5 17.4 186 3800 female 2007
Adelie Torgersen 40.3 18.0 195 3250 female 2007
Adelie Torgersen 36.7 19.3 193 3450 female 2007
Adelie Torgersen 39.3 20.6 190 3650 male 2007
Adelie Torgersen 38.9 17.8 181 3625 female 2007
Adelie Torgersen 39.2 19.6 195 4675 male 2007
Adelie Torgersen 41.1 17.6 182 3200 female 2007
Adelie Torgersen 38.6 21.2 191 3800 male 2007
Adelie Torgersen 34.6 21.1 198 4400 male 2007
Adelie Torgersen 36.6 17.8 185 3700 female 2007
Adelie Torgersen 38.7 19.0 195 3450 female 2007
Adelie Torgersen 42.5 20.7 197 4500 male 2007
Adelie Torgersen 34.4 18.4 184 3325 female 2007
Adelie Torgersen 46.0 21.5 194 4200 male 2007
Adelie Biscoe 37.8 18.3 174 3400 female 2007
Adelie Biscoe 37.7 18.7 180 3600 male 2007
Adelie Biscoe 35.9 19.2 189 3800 female 2007
Adelie Biscoe 38.2 18.1 185 3950 male 2007
Adelie Biscoe 38.8 17.2 180 3800 male 2007
Adelie Biscoe 35.3 18.9 187 3800 female 2007
Adelie Biscoe 40.6 18.6 183 3550 male 2007
Adelie Biscoe 40.5 17.9 187 3200 female 2007
Adelie Biscoe 37.9 18.6 172 3150 female 2007
Adelie Biscoe 40.5 18.9 180 3950 male 2007
Adelie Dream 39.5 16.7 178 3250 female 2007
Adelie Dream 37.2 18.1 178 3900 male 2007
Adelie Dream 39.5 17.8 188 3300 female 2007
Adelie Dream 40.9 18.9 184 3900 male 2007
Adelie Dream 36.4 17.0 195 3325 female 2007
Adelie Dream 39.2 21.1 196 4150 male 2007
Adelie Dream 38.8 20.0 190 3950 male 2007
Adelie Dream 42.2 18.5 180 3550 female 2007
Adelie Dream 37.6 19.3 181 3300 female 2007
Adelie Dream 39.8 19.1 184 4650 male 2007
Adelie Dream 36.5 18.0 182 3150 female 2007
Adelie Dream 40.8 18.4 195 3900 male 2007
Adelie Dream 36.0 18.5 186 3100 female 2007
Adelie Dream 44.1 19.7 196 4400 male 2007
Adelie Dream 37.0 16.9 185 3000 female 2007
Adelie Dream 39.6 18.8 190 4600 male 2007
Adelie Dream 41.1 19.0 182 3425 male 2007
Adelie Dream 36.0 17.9 190 3450 female 2007
Adelie Dream 42.3 21.2 191 4150 male 2007
Adelie Biscoe 39.6 17.7 186 3500 female 2008
Adelie Biscoe 40.1 18.9 188 4300 male 2008
Adelie Biscoe 35.0 17.9 190 3450 female 2008
Adelie Biscoe 42.0 19.5 200 4050 male 2008
Adelie Biscoe 34.5 18.1 187 2900 female 2008
Adelie Biscoe 41.4 18.6 191 3700 male 2008
Adelie Biscoe 39.0 17.5 186 3550 female 2008
Adelie Biscoe 40.6 18.8 193 3800 male 2008
Adelie Biscoe 36.5 16.6 181 2850 female 2008
Adelie Biscoe 37.6 19.1 194 3750 male 2008
Adelie Biscoe 35.7 16.9 185 3150 female 2008
Adelie Biscoe 41.3 21.1 195 4400 male 2008
Adelie Biscoe 37.6 17.0 185 3600 female 2008
Adelie Biscoe 41.1 18.2 192 4050 male 2008
Adelie Biscoe 36.4 17.1 184 2850 female 2008
Adelie Biscoe 41.6 18.0 192 3950 male 2008
Adelie Biscoe 35.5 16.2 195 3350 female 2008
Adelie Biscoe 41.1 19.1 188 4100 male 2008
Adelie Torgersen 35.9 16.6 190 3050 female 2008
Adelie Torgersen 41.8 19.4 198 4450 male 2008
Adelie Torgersen 33.5 19.0 190 3600 female 2008
Adelie Torgersen 39.7 18.4 190 3900 male 2008
Adelie Torgersen 39.6 17.2 196 3550 female 2008
Adelie Torgersen 45.8 18.9 197 4150 male 2008
Adelie Torgersen 35.5 17.5 190 3700 female 2008
Adelie Torgersen 42.8 18.5 195 4250 male 2008
Adelie Torgersen 40.9 16.8 191 3700 female 2008
Adelie Torgersen 37.2 19.4 184 3900 male 2008
Adelie Torgersen 36.2 16.1 187 3550 female 2008
Adelie Torgersen 42.1 19.1 195 4000 male 2008
Adelie Torgersen 34.6 17.2 189 3200 female 2008
Adelie Torgersen 42.9 17.6 196 4700 male 2008
Adelie Torgersen 36.7 18.8 187 3800 female 2008
Adelie Torgersen 35.1 19.4 193 4200 male 2008
Adelie Dream 37.3 17.8 191 3350 female 2008
Adelie Dream 41.3 20.3 194 3550 male 2008
Adelie Dream 36.3 19.5 190 3800 male 2008
Adelie Dream 36.9 18.6 189 3500 female 2008
Adelie Dream 38.3 19.2 189 3950 male 2008
Adelie Dream 38.9 18.8 190 3600 female 2008
Adelie Dream 35.7 18.0 202 3550 female 2008
Adelie Dream 41.1 18.1 205 4300 male 2008
Adelie Dream 34.0 17.1 185 3400 female 2008
Adelie Dream 39.6 18.1 186 4450 male 2008
Adelie Dream 36.2 17.3 187 3300 female 2008
Adelie Dream 40.8 18.9 208 4300 male 2008
Adelie Dream 38.1 18.6 190 3700 female 2008
Adelie Dream 40.3 18.5 196 4350 male 2008
Adelie Dream 33.1 16.1 178 2900 female 2008
Adelie Dream 43.2 18.5 192 4100 male 2008
Adelie Biscoe 35.0 17.9 192 3725 female 2009
Adelie Biscoe 41.0 20.0 203 4725 male 2009
Adelie Biscoe 37.7 16.0 183 3075 female 2009
Adelie Biscoe 37.8 20.0 190 4250 male 2009
Adelie Biscoe 37.9 18.6 193 2925 female 2009
Adelie Biscoe 39.7 18.9 184 3550 male 2009
Adelie Biscoe 38.6 17.2 199 3750 female 2009
Adelie Biscoe 38.2 20.0 190 3900 male 2009
Adelie Biscoe 38.1 17.0 181 3175 female 2009
Adelie Biscoe 43.2 19.0 197 4775 male 2009
Adelie Biscoe 38.1 16.5 198 3825 female 2009
Adelie Biscoe 45.6 20.3 191 4600 male 2009
Adelie Biscoe 39.7 17.7 193 3200 female 2009
Adelie Biscoe 42.2 19.5 197 4275 male 2009
Adelie Biscoe 39.6 20.7 191 3900 female 2009
Adelie Biscoe 42.7 18.3 196 4075 male 2009
Adelie Torgersen 38.6 17.0 188 2900 female 2009
Adelie Torgersen 37.3 20.5 199 3775 male 2009
Adelie Torgersen 35.7 17.0 189 3350 female 2009
Adelie Torgersen 41.1 18.6 189 3325 male 2009
Adelie Torgersen 36.2 17.2 187 3150 female 2009
Adelie Torgersen 37.7 19.8 198 3500 male 2009
Adelie Torgersen 40.2 17.0 176 3450 female 2009
Adelie Torgersen 41.4 18.5 202 3875 male 2009
Adelie Torgersen 35.2 15.9 186 3050 female 2009
Adelie Torgersen 40.6 19.0 199 4000 male 2009
Adelie Torgersen 38.8 17.6 191 3275 female 2009
Adelie Torgersen 41.5 18.3 195 4300 male 2009
Adelie Torgersen 39.0 17.1 191 3050 female 2009
Adelie Torgersen 44.1 18.0 210 4000 male 2009
Adelie Torgersen 38.5 17.9 190 3325 female 2009
Adelie Torgersen 43.1 19.2 197 3500 male 2009
Adelie Dream 36.8 18.5 193 3500 female 2009
Adelie Dream 37.5 18.5 199 4475 male 2009
Adelie Dream 38.1 17.6 187 3425 female 2009
Adelie Dream 41.1 17.5 190 3900 male 2009
Adelie Dream 35.6 17.5 191 3175 female 2009
Adelie Dream 40.2 20.1 200 3975 male 2009
Adelie Dream 37.0 16.5 185 3400 female 2009
Adelie Dream 39.7 17.9 193 4250 male 2009
Adelie Dream 40.2 17.1 193 3400 female 2009
Adelie Dream 40.6 17.2 187 3475 male 2009
Adelie Dream 32.1 15.5 188 3050 female 2009
Adelie Dream 40.7 17.0 190 3725 male 2009
Adelie Dream 37.3 16.8 192 3000 female 2009
Adelie Dream 39.0 18.7 185 3650 male 2009
Adelie Dream 39.2 18.6 190 4250 male 2009
Adelie Dream 36.6 18.4 184 3475 female 2009
Adelie Dream 36.0 17.8 195 3450 female 2009
Adelie Dream 37.8 18.1 193 3750 male 2009
Adelie Dream 36.0 17.1 187 3700 female 2009
Adelie Dream 41.5 18.5 201 4000 male 2009
Gentoo Biscoe 46.1 13.2 211 4500 female 2007
Gentoo Biscoe 50.0 16.3 230 5700 male 2007
Gentoo Biscoe 48.7 14.1 210 4450 female 2007
Gentoo Biscoe 50.0 15.2 218 5700 male 2007
Gentoo Biscoe 47.6 14.5 215 5400 male 2007
Gentoo Biscoe 46.5 13.5 210 4550 female 2007
Gentoo Biscoe 45.4 14.6 211 4800 female 2007
Gentoo Biscoe 46.7 15.3 219 5200 male 2007
Gentoo Biscoe 43.3 13.4 209 4400 female 2007
Gentoo Biscoe 46.8 15.4 215 5150 male 2007
Gentoo Biscoe 40.9 13.7 214 4650 female 2007
Gentoo Biscoe 49.0 16.1 216 5550 male 2007
Gentoo Biscoe 45.5 13.7 214 4650 female 2007
Gentoo Biscoe 48.4 14.6 213 5850 male 2007
Gentoo Biscoe 45.8 14.6 210 4200 female 2007
Gentoo Biscoe 49.3 15.7 217 5850 male 2007
Gentoo Biscoe 42.0 13.5 210 4150 female 2007
Gentoo Biscoe 49.2 15.2 221 6300 male 2007
Gentoo Biscoe 46.2 14.5 209 4800 female 2007
Gentoo Biscoe 48.7 15.1 222 5350 male 2007
Gentoo Biscoe 50.2 14.3 218 5700 male 2007
Gentoo Biscoe 45.1 14.5 215 5000 female 2007
Gentoo Biscoe 46.5 14.5 213 4400 female 2007
Gentoo Biscoe 46.3 15.8 215 5050 male 2007
Gentoo Biscoe 42.9 13.1 215 5000 female 2007
Gentoo Biscoe 46.1 15.1 215 5100 male 2007
Gentoo Biscoe 47.8 15.0 215 5650 male 2007
Gentoo Biscoe 48.2 14.3 210 4600 female 2007
Gentoo Biscoe 50.0 15.3 220 5550 male 2007
Gentoo Biscoe 47.3 15.3 222 5250 male 2007
Gentoo Biscoe 42.8 14.2 209 4700 female 2007
Gentoo Biscoe 45.1 14.5 207 5050 female 2007
Gentoo Biscoe 59.6 17.0 230 6050 male 2007
Gentoo Biscoe 49.1 14.8 220 5150 female 2008
Gentoo Biscoe 48.4 16.3 220 5400 male 2008
Gentoo Biscoe 42.6 13.7 213 4950 female 2008
Gentoo Biscoe 44.4 17.3 219 5250 male 2008
Gentoo Biscoe 44.0 13.6 208 4350 female 2008
Gentoo Biscoe 48.7 15.7 208 5350 male 2008
Gentoo Biscoe 42.7 13.7 208 3950 female 2008
Gentoo Biscoe 49.6 16.0 225 5700 male 2008
Gentoo Biscoe 45.3 13.7 210 4300 female 2008
Gentoo Biscoe 49.6 15.0 216 4750 male 2008
Gentoo Biscoe 50.5 15.9 222 5550 male 2008
Gentoo Biscoe 43.6 13.9 217 4900 female 2008
Gentoo Biscoe 45.5 13.9 210 4200 female 2008
Gentoo Biscoe 50.5 15.9 225 5400 male 2008
Gentoo Biscoe 44.9 13.3 213 5100 female 2008
Gentoo Biscoe 45.2 15.8 215 5300 male 2008
Gentoo Biscoe 46.6 14.2 210 4850 female 2008
Gentoo Biscoe 48.5 14.1 220 5300 male 2008
Gentoo Biscoe 45.1 14.4 210 4400 female 2008
Gentoo Biscoe 50.1 15.0 225 5000 male 2008
Gentoo Biscoe 46.5 14.4 217 4900 female 2008
Gentoo Biscoe 45.0 15.4 220 5050 male 2008
Gentoo Biscoe 43.8 13.9 208 4300 female 2008
Gentoo Biscoe 45.5 15.0 220 5000 male 2008
Gentoo Biscoe 43.2 14.5 208 4450 female 2008
Gentoo Biscoe 50.4 15.3 224 5550 male 2008
Gentoo Biscoe 45.3 13.8 208 4200 female 2008
Gentoo Biscoe 46.2 14.9 221 5300 male 2008
Gentoo Biscoe 45.7 13.9 214 4400 female 2008
Gentoo Biscoe 54.3 15.7 231 5650 male 2008
Gentoo Biscoe 45.8 14.2 219 4700 female 2008
Gentoo Biscoe 49.8 16.8 230 5700 male 2008
Gentoo Biscoe 49.5 16.2 229 5800 male 2008
Gentoo Biscoe 43.5 14.2 220 4700 female 2008
Gentoo Biscoe 50.7 15.0 223 5550 male 2008
Gentoo Biscoe 47.7 15.0 216 4750 female 2008
Gentoo Biscoe 46.4 15.6 221 5000 male 2008
Gentoo Biscoe 48.2 15.6 221 5100 male 2008
Gentoo Biscoe 46.5 14.8 217 5200 female 2008
Gentoo Biscoe 46.4 15.0 216 4700 female 2008
Gentoo Biscoe 48.6 16.0 230 5800 male 2008
Gentoo Biscoe 47.5 14.2 209 4600 female 2008
Gentoo Biscoe 51.1 16.3 220 6000 male 2008
Gentoo Biscoe 45.2 13.8 215 4750 female 2008
Gentoo Biscoe 45.2 16.4 223 5950 male 2008
Gentoo Biscoe 49.1 14.5 212 4625 female 2009
Gentoo Biscoe 52.5 15.6 221 5450 male 2009
Gentoo Biscoe 47.4 14.6 212 4725 female 2009
Gentoo Biscoe 50.0 15.9 224 5350 male 2009
Gentoo Biscoe 44.9 13.8 212 4750 female 2009
Gentoo Biscoe 50.8 17.3 228 5600 male 2009
Gentoo Biscoe 43.4 14.4 218 4600 female 2009
Gentoo Biscoe 51.3 14.2 218 5300 male 2009
Gentoo Biscoe 47.5 14.0 212 4875 female 2009
Gentoo Biscoe 52.1 17.0 230 5550 male 2009
Gentoo Biscoe 47.5 15.0 218 4950 female 2009
Gentoo Biscoe 52.2 17.1 228 5400 male 2009
Gentoo Biscoe 45.5 14.5 212 4750 female 2009
Gentoo Biscoe 49.5 16.1 224 5650 male 2009
Gentoo Biscoe 44.5 14.7 214 4850 female 2009
Gentoo Biscoe 50.8 15.7 226 5200 male 2009
Gentoo Biscoe 49.4 15.8 216 4925 male 2009
Gentoo Biscoe 46.9 14.6 222 4875 female 2009
Gentoo Biscoe 48.4 14.4 203 4625 female 2009
Gentoo Biscoe 51.1 16.5 225 5250 male 2009
Gentoo Biscoe 48.5 15.0 219 4850 female 2009
Gentoo Biscoe 55.9 17.0 228 5600 male 2009
Gentoo Biscoe 47.2 15.5 215 4975 female 2009
Gentoo Biscoe 49.1 15.0 228 5500 male 2009
Gentoo Biscoe 46.8 16.1 215 5500 male 2009
Gentoo Biscoe 41.7 14.7 210 4700 female 2009
Gentoo Biscoe 53.4 15.8 219 5500 male 2009
Gentoo Biscoe 43.3 14.0 208 4575 female 2009
Gentoo Biscoe 48.1 15.1 209 5500 male 2009
Gentoo Biscoe 50.5 15.2 216 5000 female 2009
Gentoo Biscoe 49.8 15.9 229 5950 male 2009
Gentoo Biscoe 43.5 15.2 213 4650 female 2009
Gentoo Biscoe 51.5 16.3 230 5500 male 2009
Gentoo Biscoe 46.2 14.1 217 4375 female 2009
Gentoo Biscoe 55.1 16.0 230 5850 male 2009
Gentoo Biscoe 48.8 16.2 222 6000 male 2009
Gentoo Biscoe 47.2 13.7 214 4925 female 2009
Gentoo Biscoe 46.8 14.3 215 4850 female 2009
Gentoo Biscoe 50.4 15.7 222 5750 male 2009
Gentoo Biscoe 45.2 14.8 212 5200 female 2009
Gentoo Biscoe 49.9 16.1 213 5400 male 2009
Chinstrap Dream 46.5 17.9 192 3500 female 2007
Chinstrap Dream 50.0 19.5 196 3900 male 2007
Chinstrap Dream 51.3 19.2 193 3650 male 2007
Chinstrap Dream 45.4 18.7 188 3525 female 2007
Chinstrap Dream 52.7 19.8 197 3725 male 2007
Chinstrap Dream 45.2 17.8 198 3950 female 2007
Chinstrap Dream 46.1 18.2 178 3250 female 2007
Chinstrap Dream 51.3 18.2 197 3750 male 2007
Chinstrap Dream 46.0 18.9 195 4150 female 2007
Chinstrap Dream 51.3 19.9 198 3700 male 2007
Chinstrap Dream 46.6 17.8 193 3800 female 2007
Chinstrap Dream 51.7 20.3 194 3775 male 2007
Chinstrap Dream 47.0 17.3 185 3700 female 2007
Chinstrap Dream 52.0 18.1 201 4050 male 2007
Chinstrap Dream 45.9 17.1 190 3575 female 2007
Chinstrap Dream 50.5 19.6 201 4050 male 2007
Chinstrap Dream 50.3 20.0 197 3300 male 2007
Chinstrap Dream 58.0 17.8 181 3700 female 2007
Chinstrap Dream 46.4 18.6 190 3450 female 2007
Chinstrap Dream 49.2 18.2 195 4400 male 2007
Chinstrap Dream 42.4 17.3 181 3600 female 2007
Chinstrap Dream 48.5 17.5 191 3400 male 2007
Chinstrap Dream 43.2 16.6 187 2900 female 2007
Chinstrap Dream 50.6 19.4 193 3800 male 2007
Chinstrap Dream 46.7 17.9 195 3300 female 2007
Chinstrap Dream 52.0 19.0 197 4150 male 2007
Chinstrap Dream 50.5 18.4 200 3400 female 2008
Chinstrap Dream 49.5 19.0 200 3800 male 2008
Chinstrap Dream 46.4 17.8 191 3700 female 2008
Chinstrap Dream 52.8 20.0 205 4550 male 2008
Chinstrap Dream 40.9 16.6 187 3200 female 2008
Chinstrap Dream 54.2 20.8 201 4300 male 2008
Chinstrap Dream 42.5 16.7 187 3350 female 2008
Chinstrap Dream 51.0 18.8 203 4100 male 2008
Chinstrap Dream 49.7 18.6 195 3600 male 2008
Chinstrap Dream 47.5 16.8 199 3900 female 2008
Chinstrap Dream 47.6 18.3 195 3850 female 2008
Chinstrap Dream 52.0 20.7 210 4800 male 2008
Chinstrap Dream 46.9 16.6 192 2700 female 2008
Chinstrap Dream 53.5 19.9 205 4500 male 2008
Chinstrap Dream 49.0 19.5 210 3950 male 2008
Chinstrap Dream 46.2 17.5 187 3650 female 2008
Chinstrap Dream 50.9 19.1 196 3550 male 2008
Chinstrap Dream 45.5 17.0 196 3500 female 2008
Chinstrap Dream 50.9 17.9 196 3675 female 2009
Chinstrap Dream 50.8 18.5 201 4450 male 2009
Chinstrap Dream 50.1 17.9 190 3400 female 2009
Chinstrap Dream 49.0 19.6 212 4300 male 2009
Chinstrap Dream 51.5 18.7 187 3250 male 2009
Chinstrap Dream 49.8 17.3 198 3675 female 2009
Chinstrap Dream 48.1 16.4 199 3325 female 2009
Chinstrap Dream 51.4 19.0 201 3950 male 2009
Chinstrap Dream 45.7 17.3 193 3600 female 2009
Chinstrap Dream 50.7 19.7 203 4050 male 2009
Chinstrap Dream 42.5 17.3 187 3350 female 2009
Chinstrap Dream 52.2 18.8 197 3450 male 2009
Chinstrap Dream 45.2 16.6 191 3250 female 2009
Chinstrap Dream 49.3 19.9 203 4050 male 2009
Chinstrap Dream 50.2 18.8 202 3800 male 2009
Chinstrap Dream 45.6 19.4 194 3525 female 2009
Chinstrap Dream 51.9 19.5 206 3950 male 2009
Chinstrap Dream 46.8 16.5 189 3650 female 2009
Chinstrap Dream 45.7 17.0 195 3650 female 2009
Chinstrap Dream 55.8 19.8 207 4000 male 2009
Chinstrap Dream 43.5 18.1 202 3400 female 2009
Chinstrap Dream 49.6 18.2 193 3775 male 2009
Chinstrap Dream 50.8 19.0 210 4100 male 2009
Chinstrap Dream 50.2 18.7 198 3775 female 2009


Looking for simple correlations


We can try to understand for example if there’s any correlation between body mass and flipper length of all the penguins, no matter what species they belong to.

We can perform for istance a simple linear regression model between those two variables.

reg = lm(df$flipper_length_mm ~ df$body_mass_g) %>% summary()
x_coeff = reg$coefficients[2] %>% round(digits= 4)
intercept = reg$coefficients[1] %>% round(digits = 0)
reg_formula = paste('y = ', intercept, '+', x_coeff, 'x', sep = '')
r_squared = reg$r.squared %>% round(digits = 3)

reg
## 
## Call:
## lm(formula = df$flipper_length_mm ~ df$body_mass_g)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.698  -4.983   1.056   5.101  13.933 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.370e+02  1.999e+00   68.56   <2e-16 ***
## df$body_mass_g 1.520e-02  4.667e-04   32.56   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.847 on 331 degrees of freedom
## Multiple R-squared:  0.7621, Adjusted R-squared:  0.7614 
## F-statistic:  1060 on 1 and 331 DF,  p-value: < 2.2e-16


Just take a look at the graph below


ggplot(df, aes(x=body_mass_g, y=flipper_length_mm))+geom_point(color= 'deepskyblue3')+
  geom_smooth(method = 'lm', se=FALSE, color= 'grey')+
  annotate('text', x=3200, y=230, label = reg_formula, size= 3.5)+
  annotate('text', x=3200, y=226, label = paste('R² =',r_squared), size= 3.5)+
  theme_bw()


Go deep…


There’s certainly a correlation between the two variables, but we can go deeper in our exploration and ask ouselves if there is any kind of difference among the species.


ggplot(df, aes(x=body_mass_g, y=flipper_length_mm, colour=species))+
  geom_point()+
  theme_bw()+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  geom_smooth(method = 'lm', se=FALSE, color= 'grey')+
  annotate('text', x=3200, y=230, label = reg_formula, size= 3.5)+
  annotate('text', x=3200, y=226, label = paste('R² =',r_squared), size= 3.5)+
  stat_ellipse()


…and deeper


We can now add more levels of complexity (thus more information) to our data. We may further group the data acoording to another categorical variable, e.g., the sex of penguins.


ggplot(df, aes(x=body_mass_g, y=flipper_length_mm, colour=species, shape= sex))+
  geom_point()+
  theme_bw()+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  annotate('text', x=3200, y=230, label = reg_formula, size= 3.5)+
  annotate('text', x=3200, y=226, label = paste('R² =',r_squared), size= 3.5)+
  stat_ellipse()


We already collect a lot of information so far: we saw a statistical correlation between body mass and flipper length and, based on our original dataset, that penguins of the gentoo species have greater mass and longer flippers. Males, in particular, have recorded higher values than females.


Other correlations


Similarly we may explore the other variables looking for further information. Let’s say we want to know if there are differences regarding the size of the beak among the observed samples.


Artwork by allison_horst
Artwork by allison_horst


ggplot(df, aes(x=bill_length_mm, y=bill_depth_mm, colour=species, shape= sex))+
  geom_point()+
  theme_bw()+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))


In this case, for these particular attributes, differences are more obvious, even if there’s not a clear correlation between the two variables.


Too many variables!


We can keep looking for any kind of correlations. For example splitting the graphs and separate the samples according to others categorical variables.


ggplot(df, aes(x=bill_length_mm, y=bill_depth_mm, colour=species, shape= sex))+
  geom_point()+
  theme_bw()+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  facet_grid(~island)+
  stat_ellipse()


We may wonder at this point if there is a tool of some kind with which we can perform a complete analysis of all the variables and generate a 2-dimensional plot which summarize the relationship between each samples and each variable.


PCA


Principal component analysis


Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data.

Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. (“Principal Component Analysis” 2022)



So, we want to perform PCA in order to better understand all statistical relationships among the variables, in our case all the penguins characteristics.


Scaling data


First step is scaling data, which consist in subtracting for all values within each column their mean and then divide by their standard deviation


\[ X^*_{i,j} = \frac{X_{i,j} - \overline{X_i}}{S_{X_i}} \]

This operation it’s called autoscaling and allows you to have all the data in the same order of magnitude.


df_scaled = df[,3:6] %>% scale() %>% data.frame()

kable(df_scaled) %>% kable_styling(fixed_thead = T, full_width = FALSE) %>%
  scroll_box( height = "500px")
bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
-0.8946955 0.7795590 -1.4246077 -0.5676206
-0.8215515 0.1194043 -1.0678666 -0.5055254
-0.6752636 0.4240910 -0.4257325 -1.1885721
-1.3335592 1.0842457 -0.5684290 -0.9401915
-0.8581235 1.7444004 -0.7824736 -0.6918109
-0.9312674 0.3225288 -1.4246077 -0.7228585
-0.8764095 1.2365891 -0.4257325 0.5811398
-0.5289757 0.2209665 -1.3532595 -1.2506673
-0.9861254 2.0490872 -0.7111254 -0.5055254
-1.7175649 1.9983060 -0.2116878 0.2396164
-1.3518452 0.3225288 -1.1392148 -0.6297157
-0.9678394 0.9319023 -0.4257325 -0.9401915
-0.2729719 1.7951815 -0.2830361 0.3638067
-1.7541369 0.6272156 -1.2105630 -1.0954294
0.3670377 2.2014306 -0.4970807 -0.0087642
-1.1324133 0.5764344 -1.9240453 -1.0022867
-1.1506993 0.7795590 -1.4959559 -0.7539060
-1.4798471 1.0334646 -0.8538219 -0.5055254
-1.0592694 0.4748722 -1.1392148 -0.3192400
-0.9495534 0.0178420 -1.4959559 -0.5055254
-1.5895630 0.8811212 -0.9965183 -0.5055254
-0.6204057 0.7287778 -1.2819112 -0.8160012
-0.6386916 0.3733099 -0.9965183 -1.2506673
-1.1141273 0.7287778 -2.0667417 -1.3127624
-0.6386916 0.8811212 -1.4959559 -0.3192400
-0.8215515 -0.2360636 -1.6386524 -1.1885721
-1.2421292 0.4748722 -1.6386524 -0.3813351
-0.8215515 0.3225288 -0.9251701 -1.1264770
-0.5655477 0.8811212 -1.2105630 -0.3813351
-1.3884171 -0.0837202 -0.4257325 -1.0954294
-0.8764095 1.9983060 -0.3543843 -0.0708593
-0.9495534 1.4397136 -0.7824736 -0.3192400
-0.3278299 0.6779967 -1.4959559 -0.8160012
-1.1689853 1.0842457 -1.4246077 -1.1264770
-0.7666936 0.9826835 -1.2105630 0.5500922
-1.3701311 0.4240910 -1.3532595 -1.3127624
-0.5838337 0.6272156 -0.4257325 -0.3813351
-1.4615611 0.6779967 -1.0678666 -1.3748576
0.0196039 1.2873702 -0.3543843 0.2396164
-1.2787012 -0.1345014 -1.1392148 -1.4990479
-0.8032655 0.8303401 -0.7824736 0.4879971
-0.5289757 0.9319023 -1.3532595 -0.9712391
-1.4615611 0.3733099 -0.7824736 -0.9401915
-0.3095439 2.0490872 -0.7111254 -0.0708593
-0.8032655 0.2717477 -1.0678666 -0.8780964
-0.7118356 0.8811212 -0.9251701 0.1154261
-1.6444210 0.3733099 -0.7824736 -0.9401915
-0.3644018 1.1858080 -0.0689914 -0.1950496
-1.7358509 0.4748722 -0.9965183 -1.6232382
-0.4741178 0.7287778 -0.7111254 -0.6297157
-0.9129815 0.1701854 -1.0678666 -0.8160012
-0.6204057 0.8303401 -0.5684290 -0.5055254
-1.3701311 -0.2868448 -1.4246077 -1.6853334
-1.1689853 0.9826835 -0.4970807 -0.5676206
-1.5164190 -0.1345014 -1.1392148 -1.3127624
-0.4924037 1.9983060 -0.4257325 0.2396164
-1.1689853 -0.0837202 -1.1392148 -0.7539060
-0.5289757 0.5256533 -0.6397772 -0.1950496
-1.3884171 -0.0329391 -1.2105630 -1.6853334
-0.4375458 0.4240910 -0.6397772 -0.3192400
-1.5529910 -0.4899693 -0.4257325 -1.0643818
-0.5289757 0.9826835 -0.9251701 -0.1329545
-1.4798471 -0.2868448 -0.7824736 -1.4369527
-0.4009738 1.1350269 -0.2116878 0.3017116
-1.9187108 0.9319023 -0.7824736 -0.7539060
-0.7849795 0.6272156 -0.7824736 -0.3813351
-0.8032655 0.0178420 -0.3543843 -0.8160012
0.3304657 0.8811212 -0.2830361 -0.0708593
-1.5529910 0.1701854 -0.7824736 -0.6297157
-0.2181139 0.6779967 -0.4257325 0.0533310
-0.5655477 -0.1852825 -0.7111254 -0.6297157
-1.2421292 1.1350269 -1.2105630 -0.3813351
-1.4249891 -0.5407504 -0.9965183 -0.8160012
-0.3461159 0.9826835 -0.4257325 -0.2571448
-1.7175649 0.0178420 -0.8538219 -1.2506673
-0.1998280 0.2209665 -0.3543843 0.6121874
-1.3335592 0.8303401 -0.9965183 -0.5055254
-1.6261350 1.1350269 -0.5684290 -0.0087642
-1.2238432 0.3225288 -0.7111254 -1.0643818
-0.4924037 1.5920570 -0.4970807 -0.8160012
-1.4067031 1.1858080 -0.7824736 -0.5055254
-1.2969872 0.7287778 -0.8538219 -0.8780964
-1.0409834 1.0334646 -0.8538219 -0.3192400
-0.9312674 0.8303401 -0.7824736 -0.7539060
-1.5164190 0.4240910 0.0737051 -0.8160012
-0.5289757 0.4748722 0.2877498 0.1154261
-1.8272808 -0.0329391 -1.1392148 -1.0022867
-0.8032655 0.4748722 -1.0678666 0.3017116
-1.4249891 0.0686231 -0.9965183 -1.1264770
-0.5838337 0.8811212 0.5017944 0.1154261
-1.0775553 0.7287778 -0.7824736 -0.6297157
-0.6752636 0.6779967 -0.3543843 0.1775213
-1.9918547 -0.5407504 -1.6386524 -1.6232382
-0.1449700 0.6779967 -0.6397772 -0.1329545
-1.6444210 0.3733099 -0.6397772 -0.5986682
-0.5472617 1.4397136 0.1450533 0.6432349
-1.1506993 -0.5915315 -1.2819112 -1.4059052
-1.1324133 1.4397136 -0.7824736 0.0533310
-1.1141273 0.7287778 -0.5684290 -1.5921906
-0.7849795 0.8811212 -1.2105630 -0.8160012
-0.9861254 0.0178420 -0.1403396 -0.5676206
-1.0592694 1.4397136 -0.7824736 -0.3813351
-1.0775553 -0.0837202 -1.4246077 -1.2817149
-0.1449700 0.9319023 -0.2830361 0.7053301
-1.0775553 -0.3376259 -0.2116878 -0.4744778
0.2938937 1.5920570 -0.7111254 0.4879971
-0.7849795 0.2717477 -0.5684290 -1.2506673
-0.3278299 1.1858080 -0.2830361 0.0843786
-0.8032655 1.7951815 -0.7111254 -0.3813351
-0.2363999 0.5764344 -0.3543843 -0.1640021
-0.9861254 -0.0837202 -0.9251701 -1.6232382
-1.2238432 1.6936193 -0.1403396 -0.5365730
-1.5164190 -0.0837202 -0.8538219 -1.0643818
-0.5289757 0.7287778 -0.8538219 -1.0954294
-1.4249891 0.0178420 -0.9965183 -1.3127624
-1.1506993 1.3381514 -0.2116878 -0.8780964
-0.6935496 -0.0837202 -1.7813488 -0.9401915
-0.4741178 0.6779967 0.0737051 -0.4123827
-1.6078490 -0.6423127 -1.0678666 -1.4369527
-0.6204057 0.9319023 -0.1403396 -0.2571448
-0.9495534 0.2209665 -0.7111254 -1.1575245
-0.4558318 0.5764344 -0.4257325 0.1154261
-0.9129815 -0.0329391 -0.7111254 -1.4369527
0.0196039 0.4240910 0.6444909 -0.2571448
-1.0044114 0.3733099 -0.7824736 -1.0954294
-0.1632560 1.0334646 -0.2830361 -0.8780964
-1.3152732 0.6779967 -0.5684290 -0.8780964
-1.1872713 0.6779967 -0.1403396 0.3327592
-1.0775553 0.2209665 -0.9965183 -0.9712391
-0.5289757 0.1701854 -0.7824736 -0.3813351
-1.5347050 0.1701854 -0.7111254 -1.2817149
-0.6935496 1.4904948 -0.0689914 -0.2881924
-1.2787012 -0.3376259 -1.1392148 -1.0022867
-0.7849795 0.3733099 -0.5684290 0.0533310
-0.6935496 -0.0329391 -0.5684290 -1.0022867
-0.6204057 0.0178420 -0.9965183 -0.9091439
-2.1747146 -0.8454372 -0.9251701 -1.4369527
-0.6021197 -0.0837202 -0.7824736 -0.5986682
-1.2238432 -0.1852825 -0.6397772 -1.4990479
-0.9129815 0.7795590 -1.1392148 -0.6918109
-0.8764095 0.7287778 -0.7824736 0.0533310
-1.3518452 0.6272156 -1.2105630 -0.9091439
-1.4615611 0.3225288 -0.4257325 -0.9401915
-1.1324133 0.4748722 -0.5684290 -0.5676206
-1.4615611 -0.0329391 -0.9965183 -0.6297157
-0.4558318 0.6779967 0.0023568 -0.2571448
0.3853236 -2.0134031 0.7158391 0.3638067
1.0984771 -0.4391881 2.0714554 1.8540905
0.8607593 -1.5563730 0.6444909 0.3017116
1.0984771 -0.9977806 1.2152767 1.8540905
0.6596135 -1.3532485 1.0012320 1.4815195
0.4584676 -1.8610598 0.6444909 0.4259019
0.2573217 -1.3024673 0.7158391 0.7363777
0.4950396 -0.9469994 1.2866249 1.2331389
-0.1266840 -1.9118409 0.5731427 0.2396164
0.5133256 -0.8962183 1.0012320 1.1710438
-0.5655477 -1.7594975 0.9298838 0.5500922
0.9156173 -0.5407504 1.0725802 1.6678050
0.2756077 -1.7594975 0.9298838 0.5500922
0.8059014 -1.3024673 0.8585356 2.0403759
0.3304657 -1.3024673 0.6444909 -0.0087642
0.9704752 -0.7438749 1.1439285 2.0403759
-0.3644018 -1.8610598 0.6444909 -0.0708593
0.9521892 -0.9977806 1.4293214 2.5992323
0.4036096 -1.3532485 0.5731427 0.7363777
0.8607593 -1.0485617 1.5006696 1.4194244
1.1350491 -1.4548107 1.2152767 1.8540905
0.2024638 -1.3532485 1.0012320 0.9847583
0.4584676 -1.3532485 0.8585356 0.2396164
0.4218956 -0.6930938 1.0012320 1.0468535
-0.1998280 -2.0641843 1.0012320 0.9847583
0.3853236 -1.0485617 1.0012320 1.1089486
0.6961854 -1.0993428 1.0012320 1.7919953
0.7693294 -1.4548107 0.6444909 0.4879971
1.0984771 -0.9469994 1.3579732 1.6678050
0.6047555 -0.9469994 1.5006696 1.2952341
-0.2181139 -1.5055919 0.5731427 0.6121874
0.2024638 -1.3532485 0.4304462 1.0468535
2.8539319 -0.0837202 2.0714554 2.2887566
0.9339033 -1.2009051 1.3579732 1.1710438
0.8059014 -0.4391881 1.3579732 1.4815195
-0.2546859 -1.7594975 0.8585356 0.9226631
0.0744619 0.0686231 1.2866249 1.2952341
0.0013179 -1.8102786 0.5017944 0.1775213
0.8607593 -0.7438749 0.5017944 1.4194244
-0.2363999 -1.7594975 0.5017944 -0.3192400
1.0253332 -0.5915315 1.7147143 1.8540905
0.2390357 -1.7594975 0.6444909 0.1154261
1.0253332 -1.0993428 1.0725802 0.6742825
1.1899071 -0.6423127 1.5006696 1.6678050
-0.0718260 -1.6579352 1.1439285 0.8605680
0.2756077 -1.6579352 0.6444909 -0.0087642
1.1899071 -0.6423127 1.7147143 1.4815195
0.1658918 -1.9626220 0.8585356 1.1089486
0.2207498 -0.6930938 1.0012320 1.3573292
0.4767536 -1.5055919 0.6444909 0.7984728
0.8241873 -1.5563730 1.3579732 1.3573292
0.2024638 -1.4040296 0.6444909 0.2396164
1.1167631 -1.0993428 1.7147143 0.9847583
0.4584676 -1.4040296 1.1439285 0.8605680
0.1841778 -0.8962183 1.3579732 1.0468535
-0.0352541 -1.6579352 0.5017944 0.1154261
0.2756077 -1.0993428 1.3579732 0.9847583
-0.1449700 -1.3532485 0.5017944 0.3017116
1.1716211 -0.9469994 1.6433661 1.6678050
0.2390357 -1.7087164 0.5017944 -0.0087642
0.4036096 -1.1501239 1.4293214 1.3573292
0.3121797 -1.6579352 0.9298838 0.2396164
1.8847746 -0.7438749 2.1428037 1.7919953
0.3304657 -1.5055919 1.2866249 0.6121874
1.0619052 -0.1852825 2.0714554 1.8540905
1.0070472 -0.4899693 2.0001072 1.9782808
-0.0901120 -1.5055919 1.3579732 0.6121874
1.2264791 -1.0993428 1.5720178 1.6678050
0.6778994 -1.0993428 1.0725802 0.6742825
0.4401816 -0.7946560 1.4293214 0.9847583
0.7693294 -0.7946560 1.4293214 1.1089486
0.4584676 -1.2009051 1.1439285 1.2331389
0.4401816 -1.0993428 1.0725802 0.6121874
0.8424733 -0.5915315 2.0714554 1.9782808
0.6413275 -1.5055919 0.5731427 0.4879971
1.2996230 -0.4391881 1.3579732 2.2266614
0.2207498 -1.7087164 1.0012320 0.6742825
0.2207498 -0.3884070 1.5720178 2.1645662
0.9339033 -1.3532485 0.7871873 0.5190446
1.5556268 -0.7946560 1.4293214 1.5436147
0.6230415 -1.3024673 0.7871873 0.6432349
1.0984771 -0.6423127 1.6433661 1.4194244
0.1658918 -1.7087164 0.7871873 0.6742825
1.2447650 0.0686231 1.9287590 1.7299002
-0.1083980 -1.4040296 1.2152767 0.4879971
1.3361950 -1.5055919 1.2152767 1.3573292
0.6413275 -1.6071541 0.7871873 0.8295204
1.4824829 -0.0837202 2.0714554 1.6678050
0.6413275 -1.0993428 1.2152767 0.9226631
1.5007689 -0.0329391 1.9287590 1.4815195
0.2756077 -1.3532485 0.7871873 0.6742825
1.0070472 -0.5407504 1.6433661 1.7919953
0.0927478 -1.2516862 0.9298838 0.7984728
1.2447650 -0.7438749 1.7860625 1.2331389
0.9887612 -0.6930938 1.0725802 0.8916156
0.5316115 -1.3024673 1.5006696 0.8295204
0.8059014 -1.4040296 0.1450533 0.5190446
1.2996230 -0.3376259 1.7147143 1.2952341
0.8241873 -1.0993428 1.2866249 0.7984728
2.1773504 -0.0837202 1.9287590 1.7299002
0.5864695 -0.8454372 1.0012320 0.9537107
0.9339033 -1.0993428 1.9287590 1.6057098
0.5133256 -0.5407504 1.0012320 1.6057098
-0.4192598 -1.2516862 0.6444909 0.6121874
1.7202007 -0.6930938 1.2866249 1.6057098
-0.1266840 -1.6071541 0.5017944 0.4569495
0.7510434 -1.0485617 0.5731427 1.6057098
1.1899071 -0.9977806 1.0725802 0.9847583
1.0619052 -0.6423127 2.0001072 2.1645662
-0.0901120 -0.9977806 0.8585356 0.5500922
1.3727670 -0.4391881 2.0714554 1.6057098
0.4036096 -1.5563730 1.1439285 0.2085689
2.0310625 -0.5915315 2.0714554 2.0403759
0.8790453 -0.4899693 1.5006696 2.2266614
0.5864695 -1.7594975 0.9298838 0.8916156
0.5133256 -1.4548107 1.0012320 0.7984728
1.1716211 -0.7438749 1.5006696 1.9161856
0.2207498 -1.2009051 0.7871873 1.2331389
1.0801912 -0.5407504 0.8585356 1.4815195
0.4584676 0.3733099 -0.6397772 -0.8780964
1.0984771 1.1858080 -0.3543843 -0.3813351
1.3361950 1.0334646 -0.5684290 -0.6918109
0.2573217 0.7795590 -0.9251701 -0.8470488
1.5921988 1.3381514 -0.2830361 -0.5986682
0.2207498 0.3225288 -0.2116878 -0.3192400
0.3853236 0.5256533 -1.6386524 -1.1885721
1.3361950 0.5256533 -0.2830361 -0.5676206
0.3670377 0.8811212 -0.4257325 -0.0708593
1.3361950 1.3889325 -0.2116878 -0.6297157
0.4767536 0.3225288 -0.5684290 -0.5055254
1.4093389 1.5920570 -0.4970807 -0.5365730
0.5498975 0.0686231 -1.1392148 -0.6297157
1.4641969 0.4748722 0.0023568 -0.1950496
0.3487517 -0.0329391 -0.7824736 -0.7849536
1.1899071 1.2365891 0.0023568 -0.1950496
1.1533351 1.4397136 -0.2830361 -1.1264770
2.5613561 0.3225288 -1.4246077 -0.6297157
0.4401816 0.7287778 -0.7824736 -0.9401915
0.9521892 0.5256533 -0.4257325 0.2396164
-0.2912579 0.0686231 -1.4246077 -0.7539060
0.8241873 0.1701854 -0.7111254 -1.0022867
-0.1449700 -0.2868448 -0.9965183 -1.6232382
1.2081931 1.1350269 -0.5684290 -0.5055254
0.4950396 0.3733099 -0.4257325 -1.1264770
1.4641969 0.9319023 -0.2830361 -0.0708593
1.1899071 0.6272156 -0.0689914 -1.0022867
1.0070472 0.9319023 -0.0689914 -0.5055254
0.4401816 0.3225288 -0.7111254 -0.6297157
1.6104848 1.4397136 0.2877498 0.4259019
-0.5655477 -0.2868448 -0.9965183 -1.2506673
1.8664886 1.8459627 0.0023568 0.1154261
-0.2729719 -0.2360636 -0.9965183 -1.0643818
1.2813370 0.8303401 0.1450533 -0.1329545
1.0436192 0.7287778 -0.4257325 -0.7539060
0.6413275 -0.1852825 -0.1403396 -0.3813351
0.6596135 0.5764344 -0.4257325 -0.4434303
1.4641969 1.7951815 0.6444909 0.7363777
0.5316115 -0.2868448 -0.6397772 -1.8716188
1.7384867 1.3889325 0.2877498 0.3638067
0.9156173 1.1858080 0.6444909 -0.3192400
0.4036096 0.1701854 -0.9965183 -0.6918109
1.2630510 0.9826835 -0.3543843 -0.8160012
0.2756077 -0.0837202 -0.3543843 -0.8780964
1.2630510 0.3733099 -0.3543843 -0.6607633
1.2447650 0.6779967 0.0023568 0.3017116
1.1167631 0.3733099 -0.7824736 -1.0022867
0.9156173 1.2365891 0.7871873 0.1154261
1.3727670 0.7795590 -0.9965183 -1.1885721
1.0619052 0.0686231 -0.2116878 -0.6607633
0.7510434 -0.3884070 -0.1403396 -1.0954294
1.3544810 0.9319023 0.0023568 -0.3192400
0.3121797 0.0686231 -0.5684290 -0.7539060
1.2264791 1.2873702 0.1450533 -0.1950496
-0.2729719 0.0686231 -0.9965183 -1.0643818
1.5007689 0.8303401 -0.2830361 -0.9401915
0.2207498 -0.2868448 -0.7111254 -1.1885721
0.9704752 1.3889325 0.1450533 -0.1950496
1.1350491 0.8303401 0.0737051 -0.5055254
0.2938937 1.1350269 -0.4970807 -0.8470488
1.4459109 1.1858080 0.3590980 -0.3192400
0.5133256 -0.3376259 -0.8538219 -0.6918109
0.3121797 -0.0837202 -0.4257325 -0.6918109
2.1590644 1.3381514 0.4304462 -0.2571448
-0.0901120 0.4748722 0.0737051 -1.0022867
1.0253332 0.5256533 -0.5684290 -0.5365730
1.2447650 0.9319023 0.6444909 -0.1329545
1.1350491 0.7795590 -0.2116878 -0.5365730


Correlation matrix

Now we can generate the correlation matrix which tell us the degree of correlation between each pair of variables.


df_cor = cor(df_scaled)
kable(df_cor) %>% kable_styling(full_width = FALSE)
bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
bill_length_mm 1.0000000 -0.2286256 0.6530956 0.5894511
bill_depth_mm -0.2286256 1.0000000 -0.5777917 -0.4720157
flipper_length_mm 0.6530956 -0.5777917 1.0000000 0.8729789
body_mass_g 0.5894511 -0.4720157 0.8729789 1.0000000 
library(ggcorrplot)

ggcorrplot(df_cor,  
           type = 'full',
           method = 'square',
           lab = TRUE, 
           lab_size = 5,  
           colors = c("tomato2", "white", "springgreen3"),
           title="Correlogram", 
           ggtheme=theme_bw)


General formula for correlation between two variables \(X\) and \(Y\) is


\[ corr(X,Y) = \frac{cov(X,Y)}{\sigma_x \sigma_y} \]


Eigenvalues and eigenvectors


We can now calculate the eigenvalues and eigenvectors of this matrix.

Between a generic matrix \({\bf C}\), a scalar \(\lambda\) (eigenvalue) and a vector \({\bf v}\) (eigenvector) exists the following relationship


\[ {\bf Cv} = {\bf \lambda v} \]


For our correlation matrix eigenvalues are

eig = eigen(df_cor)
eig$values
## [1] 2.7453557 0.7781172 0.3686425 0.1078846


You may notice that the sum of these eigenvalues is equal to the number of variables of our original data matrix.

Each of them rapresents the amount of variance explained by every principal component.

Scree plot


We can also express them as a percentage of total variance and put them in a graph also known as scree plot.


values_prc = round((eig$values/sum(eig$values)), digits = 4)

pc = prcomp(df_scaled)

ggscreeplot(pc)+
  geom_col(fill = 'cornflowerblue') + 
  geom_line()+
  theme_bw()+
  geom_text(label=(paste0(values_prc*100,'%')), nudge_y = 0.05, nudge_x = 0.15)

pc = pca(df_scaled)


Looking at this chart it’s clear that most of the total variance is explained by the first principal component (the 68.63%).

We can therefore represent our original data using a 2-dimensional graph with PC1 and PC2 as x and Y axis.


Loading vectors


Loadings are the eigenvectors of the correlation matrix and they are normalized, which means that for a \(p\) loading, for \(m\) variables and \(a\) principal component:


\[ \sum_{i=1}^m p_{a,j}^2 = 1 \]


For our dataset loading vectors will be 

load1 = eig$vectors[,1]
load2 = eig$vectors[,2]
load_df = data.frame(load1,load2, row.names = colnames(df[3:6]))

kable(load_df) %>% kable_styling(full_width = FALSE)
load1 load2
bill_length_mm 0.4537532 -0.6001949
bill_depth_mm -0.3990472 -0.7961695
flipper_length_mm 0.5768250 -0.0057882
body_mass_g 0.5496747 -0.0764637


Score vectors


We can project the original observation values (objects) onto the principal components and call them scores. To do this properly we have to multiply each loading vectors component for every value in the original data matrix.

The loadings can be understood then as the weights for each original variable when calculating the principal component.

In our dataset, the first two score vectors are


score_pc1 = as.matrix(df_scaled) %*% load1
score_pc2 = as.matrix(df_scaled) %*% load2

score = data.frame(score_pc1, score_pc2)

kable(score) %>% kable_styling(full_width = FALSE, fixed_thead = TRUE) %>% scroll_box(height = '500px')
score_pc1 score_pc2
-1.8508078 -0.0320212
-1.3142762 0.4428603
-1.3745366 0.1609882
-1.8824555 0.0123327
-1.9170957 -0.8163696
-1.7703561 0.3656727
-0.8172664 -0.5004899
-1.7962546 0.2450252
-1.9532096 -0.9967828
-1.5671647 -0.5772133
-1.7453746 0.6093273
-1.5734059 -0.0867052
-0.8035110 -1.2916122
-2.3466466 0.6442216
-1.0034763 -1.9694587
-2.4046297 0.3085044
-2.1105221 0.1362880
-1.8542668 0.1089801
-1.5027490 0.2886935
-1.5787620 0.6030250
-1.9255694 0.2969481
-1.7603015 -0.1380520
-1.7010535 0.1875201
-2.7100962 0.2008041
-1.6798002 -0.2851133
-1.8771248 0.7814051
-1.9079424 0.4060840
-1.6543430 0.3277930
-1.5161213 -0.3259178
-1.4442933 0.9862011
-1.4384594 -1.0575044
-1.6322051 -0.5473996
-1.7307465 -0.2719852
-2.4040413 -0.0672440
-1.1359380 -0.3572722
-2.2931199 0.5929089
-0.9703884 -0.1173334
-2.3054372 0.4487289
-0.5775328 -1.0530018
-2.0076586 0.9957725
-0.8789399 -0.2117605
-1.9263569 -0.3423663
-1.7803061 0.6564231
-1.4072836 -1.4360998
-1.5715639 0.3390821
-1.1448211 -0.2777526
-1.8632794 0.7661747
-0.7855517 -0.7100785
-2.4442139 0.7936569
-1.2622829 -0.2434011
-1.5466876 0.4810458
-1.2186145 -0.2467827
-2.2553712 1.1878354
-1.5213032 -0.0344841
-2.0131274 1.1242055
-1.1347103 -1.3113099
-1.5685531 0.8325149
-0.9260382 -0.0824033
-2.2415225 0.9954197
-0.9122878 -0.0469222
-1.3397906 1.4060466
-1.2389045 -0.4493729
-1.7982276 1.2309775
-0.5911361 -0.6848560
-2.1082476 0.4718237
-1.2674362 0.0054582
-1.0245570 0.5323563
-0.4038710 -0.8928092
-1.5700758 0.8492803
-0.5857811 -0.4105031
-0.9390163 0.5392216
-1.9244427 -0.1219889
-1.4541603 1.3539626
-0.9361074 -0.5525192
-1.9664363 1.1172411
-0.0467625 -0.1007499
-1.7891428 0.1837263
-1.5234947 0.0762845
-1.6792857 0.5632595
-1.5939995 -0.9067374
-1.8407143 -0.0566247
-1.8545020 0.2702989
-1.5527346 -0.1686678
-1.6196639 -0.0399740
-1.2633326 0.6344664
-0.2000928 -0.0710817
-2.0240494 1.2061822
-1.0041096 0.0871482
-1.8679898 0.8925381
-0.2636310 -0.3628382
-1.5772501 0.1191920
-0.6837945 -0.1460332
-2.5254941 1.7596336
-0.7784545 -0.4389207
-1.5932417 0.7392346
-0.3855951 -0.8678161
-1.7983134 1.2765238
-1.5103855 -0.4661362
-1.9994265 0.2134977
-1.8546142 -0.1609797
-0.8475354 0.6218768
-1.7161212 -0.4768007
-1.9818114 0.8196492
-0.2132138 -0.7072358
-0.7371308 0.9530563
-0.6439060 -1.4771387
-1.4799713 0.3537043
-0.7388288 -0.7521560
-1.7006517 -0.9138785
-0.6318573 -0.3024621
-1.8399634 0.7879967
-1.6070488 -0.5720229
-1.7322412 1.0631311
-1.6254768 -0.1740395
-1.9501221 0.9472126
-1.6608932 -0.3063837
-1.8256180 0.5651217
-0.6698465 -0.2241316
-1.8790822 1.5924682
-0.8756815 -0.3491134
-1.5654959 0.4866150
-0.6189860 -0.1917133
-1.6011755 0.6881827
0.0700754 -0.3334827
-1.6582033 0.3939143
-1.1324087 -0.6560469
-1.6779135 0.3200526
-0.7073229 0.1481622
-1.6858025 0.5508489
-0.9688971 0.2156795
-1.8790104 0.8877464
-1.1076862 -0.7479860
-1.6535452 1.1195099
-0.8037246 0.1731350
-1.1803717 0.5224187
-1.3631810 0.4334435
-1.9729321 2.0935936
-1.0202285 0.4783501
-1.6744145 1.0003866
-1.7627476 -0.0132019
-1.1105260 -0.0537630
-2.0617091 0.3885241
-1.5542648 0.6947886
-1.3432233 0.3482825
-1.5709992 0.9573650
-0.6173743 -0.2465638
1.5911740 1.3397795
2.8877082 -0.4633927
1.5492403 0.6957130
2.6167477 -0.0137027
2.2312012 0.5624408
1.5565478 1.1702527
1.4541886 0.8220921
2.0225060 0.3551143
1.1677457 1.5765451
1.8117853 0.3101087
1.2842555 1.6928527
2.1666905 -0.2527546
1.6659325 1.1879955
2.5021941 0.3923029
1.0366368 0.8355807
2.5185870 -0.1528596
0.9101111 1.7021189
3.0834210 0.0158834
1.4585204 0.7755471
2.4548433 0.2009890
2.8157190 0.3282205
1.7507110 0.8748039
1.3749771 0.7789539
1.6209782 0.2127590
1.8518669 1.6822828
1.7803641 0.5129740
2.3171362 0.3145985
1.5696219 0.6554839
2.5763981 -0.0407150
2.2298884 0.2832764
1.1689393 1.2794897
1.4555996 0.8733597
3.7813279 -1.8332565
2.3299854 0.2981976
2.1386038 -0.2551723
1.5889474 1.4781999
1.4605183 -0.2058128
1.1100112 1.4240192
1.7570828 -0.0358117
0.7088248 1.5642501
2.7095339 -0.2961360
1.2457911 1.2448339
1.8932651 0.2020971
2.5786112 -0.3389990
1.7618822 1.2906837
1.1535934 1.1515189
2.5996811 -0.3259939
1.9632386 1.3732488
1.7003683 0.3097456
1.6277895 0.8477767
2.5244464 0.6328171
1.1556123 0.9742755
2.4758113 0.1197644
1.9011843 0.7702522
1.7999464 0.5150927
0.9984922 1.3294264
1.8883572 0.6266865
0.9295203 1.1384510
2.7742092 -0.0862675
1.0749519 1.2147255
2.2126508 0.5613897
1.4713383 1.1089245
3.3731009 -0.6884070
1.8294135 0.9461052
2.7697932 -0.6435943
2.8935945 -0.3771696
1.6797304 1.1981208
2.8187379 0.0025112
1.7356159 0.4106251
1.8826041 0.2849148
2.1002203 0.0778659
2.0249208 0.5800425
1.5936185 0.5580501
2.9006022 -0.1979454
1.4906493 0.7731534
2.7722172 -0.6084778
1.7301962 1.1705816
2.3517452 0.0021352
1.7031467 0.4726468
2.6959302 -0.4273020
1.6100924 0.6092980
2.4829069 -0.2659571
1.5818379 1.2047460
2.6008710 -0.9451758
1.4803298 1.1385572
2.6541965 0.2859083
1.8423705 0.8266611
2.8177071 -0.9626396
1.9378607 0.4127573
2.6210331 -0.9989751
1.4897733 0.8558823
2.6056849 -0.3204302
1.5168471 0.8744511
2.5697281 -0.2594795
1.8340203 -0.1160138
2.0825565 0.6457999
1.2949305 0.5936200
2.4254842 -0.6201831
1.9937251 0.3120888
3.0848267 -1.3836176
1.7052481 0.2423958
2.8576258 0.1807968
1.9088618 -0.0061402
1.0175038 1.1976515
2.6818992 -0.6108612
1.1244683 1.3177577
1.9724351 0.2579645
2.0980735 -0.0012802
3.0816751 -0.3030479
1.1548695 0.8014558
2.8756395 -0.6090279
1.5786971 0.9743231
3.4740604 -0.9160785
2.6839537 -0.3164447
1.9947138 0.9753037
1.8298973 0.7833311
2.7473705 -0.2661552
1.7112784 0.7247844
2.0155037 -0.3360480
-0.7926440 -0.5015423
-0.3887839 -1.5721950
-0.5142535 -1.5686019
-1.1935828 -0.7049808
-0.3038554 -1.9736103
-0.3261233 -0.3636449
-1.6334624 -0.5494111
-0.0787267 -1.1754459
-0.4695872 -0.9139336
-0.4161926 -1.8584275
-0.5181344 -0.5009882
-0.5774831 -2.0695198
-0.7811325 -0.3299370
0.3690332 -1.2419817
-0.7114281 -0.1185443
-0.0593877 -1.6838101
-0.8336425 -1.7507091
-0.1343689 -1.7377042
-1.0592328 -0.7680059
0.1084363 -1.0058660
-1.3956955 0.1860681
-0.6550610 -0.5494149
-1.4183857 0.4452740
-0.5104665 -1.5868806
-0.7891116 -0.5057394
0.0902991 -1.6136993
-0.3010921 -1.1365082
-0.2325927 -1.3073232
-0.6853042 -0.4687159
0.5563376 -2.1470924
-1.4044313 0.6692145
0.1751051 -2.5987957
-1.1895418 0.4389375
0.2606545 -1.4208168
-0.4772475 -1.1464950
0.0743792 -0.2074346
-0.4200384 -0.8184656
0.7245484 -2.3681089
-1.0421360 0.0561205
0.6005508 -2.1787401
0.1385512 -1.4729731
-0.8398605 -0.3190745
-0.4719766 -1.4760137
-0.5286189 -0.0295692
-0.1434775 -1.0027192
0.4614661 -1.3099855
-0.6445155 -0.8863259
0.4395229 -1.5474656
-0.9163282 -1.3479382
-0.0308528 -0.6402361
-0.1873002 -0.0569617
0.0686083 -1.5305082
-0.6280185 -0.1810677
0.0192537 -1.7470168
-1.3111262 0.1963552
-0.3304281 -1.4883165
-0.8488925 0.1908829
-0.1374369 -1.6742254
-0.0516724 -1.3041144
-1.0719040 -1.0124216
0.2145518 -1.7896008
-0.5051250 0.0185525
-0.4507833 -0.0653506
0.5526429 -2.3440840
-0.7388017 -0.2477821
-0.3673370 -0.9895904
0.4916198 -1.4826181
-0.2130962 -1.2596581


Score plot


Let’s have a look to what happens when we now plot these values onto a scatter plot


score_df = data.frame(df[,c(1,2,7,8)], score)

ggplot(data=score_df, aes(score_pc1, score_pc2, color=species, shape=sex))+
  geom_point()+
  theme_bw()+
  stat_ellipse(level = 0.9)+
  xlab(paste('PC1 - ', values_prc[1]*100, '%'))+
  ylab(paste('PC2 - ', values_prc[2]*100, '%'))+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
  geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)

It’s immediately clear how the data are grouped into two major clusters and penguins of species Gentoo seem to have different characteristics than the others.

A similar result could have been achieved through a classical cluster analysis. With PCA however we’re able to see which variables are most responsible for the variation.

Loading plot


Let’s now take the first two loading vectors we calculate before and plot them one against the other.

ggplot(load_df, aes(load1, load2))+
  geom_point()+
  theme_bw()+
  geom_label(label= rownames(load_df),color='white', fill= 'darkolivegreen4', fontface = 'bold')+
  xlim(-1,1)+ylim(-1,0.2)+
  xlab(paste('PC1 - ', values_prc[1]*100, '%'))+
  ylab(paste('PC2 - ', values_prc[2]*100, '%'))+
  geom_segment(xend=0, yend=0, color= 'grey')+
  geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
  geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)


This graph show us the direction of each variable along the principal components.

Vectors of flipper_length_mm and body_mass_g are almost parralel to each other, this means they are highly correlated (as we’re already seen before with linear regression).

On the other hand bill_depth_mm and bill_length_mm are located on the opposite sides of the first principal component (the x axis). This suggest there is a weak negative correlation between them.


Biplot


Here’s what happens when we plot together the two previous graphs 

ggplot(data= score_df, aes(score_pc1, score_pc2, color=species))+
  geom_point(size=1)+
  theme_bw()+
  xlab(paste('PC1 - ', values_prc[1]*100, '%'))+
  ylab(paste('PC2 - ', values_prc[2]*100, '%'))+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  geom_label(data=load_df, aes(load1*3,load2*3), label=rownames(load_df), size=3, color='white', fill= 'darkolivegreen4', fontface = 'bold' )+
  geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
  geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)


ggplot(data= score_df, aes(score_pc1, score_pc2, color=sex))+
  geom_point(size=1)+
  theme_bw()+
  stat_ellipse(level = 0.9)+
  xlab(paste('PC1 - ', values_prc[1]*100, '%'))+
  ylab(paste('PC2 - ', values_prc[2]*100, '%'))+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  geom_label(data=load_df, aes(load1*3,load2*3), label=rownames(load_df), size=3, color='white', fill= 'darkolivegreen4', fontface = 'bold' )+
  geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
  geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)


ggplot(data= score_df, aes(score_pc1, score_pc2, color=island))+
  geom_point(size=1)+
  theme_bw()+
  xlab(paste('PC1 - ', values_prc[1]*100, '%'))+
  ylab(paste('PC2 - ', values_prc[2]*100, '%'))+
  scale_color_manual(values=c('cornflowerblue', 'darkorange', 'brown3'))+
  geom_label(data=load_df, aes(load1*3,load2*3), label=rownames(load_df), size=3, color='white', fill= 'darkolivegreen4', fontface = 'bold' )+
  geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
  geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)









References

Gorman, Kristen B., Tony D. Williams, and William R. Fraser. 2014. “Ecological Sexual Dimorphism and Environmental Variability Within a Community of Antarctic Penguins (Genus Pygoscelis).” Edited by André Chiaradia. PLoS ONE 9 (3): e90081. https://doi.org/10.1371/journal.pone.0090081.
Horst, Allison Marie, Alison Presmanes Hill, and Kristen B Gorman. 2020. “Palmerpenguins: Palmer Archipelago (Antarctica) Penguin Data.” https://doi.org/10.5281/zenodo.3960218.
“Principal Component Analysis.” 2022. https://en.wikipedia.org/w/index.php?title=Principal_component_analysis&oldid=1111924861.