For the purposes of this question, assume we have 10-dimensional data - that is, ignore the Overall column.
A)
Explain why we need to scale this data set before performing PCA. Every variable will not be weighted the same in the PCA without scaling the data.
B)
Use svd() to find the first 2 principal component scores and their loadings. Full credit will only be granted if you use the svd() ingredients u, d, and v. What percent of the overall variability do the first two PCs explain? About 50%
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 3.5.2 ✔ tibble 3.3.0
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.1.0
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
Find and print the loadings. Based on the loadings alone, if the first two PCs are plotted in a 2D plane as shown below, which of the four quadrants will the medalists be in? Explain your reasoning. Quadrant 2; PC1 has positive loadings for almost all races, so the longer it takes to finish them, the more the point will be pushed to the right, and the shortest race times will be the medal winners. For the field events, the loadings are negative, so the shortest distances will be the furthest right. For PC2, most of the significant loadings are field events (discuss, pole vault, shot put), which have positive loadings, so the further the distance, the higher up the point will be pushed.
Add the PCs to the decathlon data set and create a scatterplot of these PCs, with the points labeled by the athletes’ names. Color-code the points on whether or not the athlete was a medalist. Use the ggrepel package for better labeling. Verify that your intuition from C) is correct. It was!
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_point()`).
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_text_repel()`).
E)
Canadian Damian Warner won the gold medal in the decathlon in the 2020 Tokyo games. He began the 2024 decathlon but bowed out after three straight missed pole vault attempts.
Would this have won a medal if it had happened in 2024? To answer this, we will compute his PCs with respect to the 2024 athletes and add it to the plot to see where his 2020 gold-medal performance compares to the 2024 athletes. To do this:
Find the mean vector from the 2024 athletes. Call it mean_vec_24.
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_point()`).
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_text_repel()`).
Do you think his 2020 performance would have won a medal if it had happened in 2024? No, just because the actual winners were much higher for PC2, and while PC1 does explain more variabilty than PC2, I still don’t think it would be enough
Question 2
Below is a screenshot of a conversation between me and chatbot Claude:
After looking at the graphs, I grew skeptical. So I said:
Behold, Claude’s three data sets which I’ve called claudeA, claudeB, and claudeC:
Each data set has an X and a Y column which represent 2-dimensional variables that we need to rotate.
A)
Scale each data set and plot them side-by-side using the patchwork package. Make sure the aspect ratio of each graph is 1 (i.e., make the height and width of each graph equal). At this point, explain why you think I was skeptical. Specifically, do you think the percent variability explained by the first PC of each data set appears to exceed or fall short of the variability I asked it to? Each of them seem like the first PC would have a vast majority of the variability explained by it. Like alot more than 75 and 50%
