Data Visualization

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# Data Visualization
## PCA: Principal Component Analysis
### Andrew Irwin, <a href="mailto:a.irwin@dal.ca" class="email">a.irwin@dal.ca</a>
### Math & Stats, Dalhousie University
### 2021-03-01 (updated: 2021-03-09)

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# Plan

* What is PCA and why do we use it?

* Demonstration with gapminder data in 3 dimensions

* More demonstrations

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### What is PCA?

* A tool for simplifying data sets with many variables down to a small number

* Works best with quantitative variables that are partially correlated

* Usually used in combination with one or more categorical variables to identify subgroups

* Main results are
  * Spatial pattern of points
  * Directions of principal components along original axes (loadings)
  * Amount of variation along each PC axis

* Be careful with scaling and units

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### Example data

[Interactive version](plotly-1.html)

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### Pairs plot

```r
my_gapminder %>% select(-year) %>% ggpairs(aes(color = continent))
```

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### PCA: 2007 data only

```r
pca1 <- prcomp(my_gapminder2 %>% select(-continent), scale=TRUE)
autoplot(pca1, data = my_gapminder2, loadings=TRUE, loadings.label = TRUE,
         colour = 'continent')
```

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### PCA: All years

```r
pca1 <- prcomp(my_gapminder %>% select(-continent, -year), 
               scale=TRUE)
autoplot(pca1, data = my_gapminder, loadings=TRUE, 
         loadings.label = TRUE, colour = 'continent')
```

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### Penguins

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```r
pca2 %>% tidy(matrix="loadings") %>%
  filter(PC < 3) %>%
  pivot_wider(values_from="value", 
              names_from="PC", names_prefix="PC_") %>% kable()
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> column </th>
   <th style="text-align:right;"> PC_1 </th>
   <th style="text-align:right;"> PC_2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Culmen Length (mm) </td>
   <td style="text-align:right;"> 0.2871721 </td>
   <td style="text-align:right;"> 0.6602934 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Culmen Depth (mm) </td>
   <td style="text-align:right;"> -0.4102740 </td>
   <td style="text-align:right;"> 0.1879624 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Flipper Length (mm) </td>
   <td style="text-align:right;"> 0.5008652 </td>
   <td style="text-align:right;"> 0.2207240 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Body Mass (g) </td>
   <td style="text-align:right;"> 0.4846484 </td>
   <td style="text-align:right;"> 0.2033560 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Delta 15 N (o/oo) </td>
   <td style="text-align:right;"> -0.4084370 </td>
   <td style="text-align:right;"> 0.3686830 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Delta 13 C (o/oo) </td>
   <td style="text-align:right;"> -0.3108644 </td>
   <td style="text-align:right;"> 0.5501663 </td>
  </tr>
</tbody>
</table>

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# Summary

* PCA to reduce the number of dimensions (variables) in your data

* Interpreting loadings (arrows and numeric vectors)

* Interpreting proportion of variance along each principal component

* Consequences of scaling variables (or not scaling them)

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# Further reading

* Course notes

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## Task

* Bonus task: Practice the PCA skills in this lesson

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### Penguin PCA code

```r
my_penguins_raw = penguins_raw %>% select(-`Sample Number`) %>% 
  select(Species, where(is.numeric) ) %>% na.omit() %>%
  mutate(Species = str_remove(Species, "\\(.*\\)"))
pca2 <- my_penguins_raw %>% select(-Species) %>% prcomp(scale=TRUE)
autoplot(pca2, data = my_penguins_raw, 
         loadings=TRUE, loadings.label = TRUE,
         loadings.label.colour = "black", loadings.colour = "black",
         colour = 'Species') + xlim(-0.15, 0.15) + ylim(-0.15, 0.15)
```