Activity 3.3 - PCA implementation

SUBMISSION INSTRUCTIONS

1. Render to html
2. Publish your html to RPubs
3. Submit a link to your published solutions

library(tidyverse)
library(factoextra)
library(ggrepel)
library(ggpubr)   # if you’re using fviz_pca or related functions

Problem 1

Consider the following 6 eigenvalues from a \(6\times 6\) correlation matrix:

\[\lambda_1 = 3.5, \lambda_2 = 1.0, \lambda_3 = 0.7, \lambda_4 = 0.4, \lambda_5 = 0.25, \lambda_6 = 0.15\]

If you want to retain enough principal components to explain at least 90% of the variability inherent in the data set, how many should you keep?
Sum = 6 (as expected for a correlation matrix). Cumulative sums:

PC1: 3.5 → 3.5 / 6 = 58.33% PC1+PC2: 3.5 + 1.0 = 4.5 → 4.5 / 6 = 75.00% PC1+PC2+PC3: 4.5 + 0.7 = 5.2 → 5.2 / 6 = 86.67% PC1+PC2+PC3+PC4: 5.2 + 0.4 = 5.6 → 5.6 / 6 = 93.33%

To reach at least 90%, you must keep 4 principal components. # Problem 2

The iris data set is a classic data set often used to demonstrate PCA. Each iris in the data set contained a measurement of its sepal length, sepal width, petal length, and petal width. Consider the five irises below, following mean-centering and scaling:

library(tidyverse)
five_irises <- data.frame(
  row.names = 1:5,
  Sepal.Length = c(0.189, 0.551, -0.415, 0.310, -0.898),
  Sepal.Width  = c(-1.97, 0.786, 2.62, -0.590, 1.70),
  Petal.Length = c(0.137, 1.04, -1.34, 0.534, -1.05),
  Petal.Width  = c(-0.262, 1.58, -1.31, 0.000875, -1.05)
) %>% as.matrix

Consider also the loadings for the first two principal components:

# Create the data frame
pc_loadings <- data.frame(
  PC1 = c(0.5210659, -0.2693474, 0.5804131, 0.5648565),
  PC2 = c(-0.37741762, -0.92329566, -0.02449161, -0.06694199),
  row.names = c("Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width")
) %>% as.matrix

A plot of the first two PC scores for these five irises is shown in the plot below.

Match the ID of each iris (1-5) to the correct letter of its score coordinates on the plot.

scores <- five_irises %*% pc_loadings
round(scores, 6)

        PC1       PC2
1  0.560620  1.761744
2  1.571503 -1.064907
3 -2.439648 -2.141894
4  0.630880  0.414608
5 -2.128341 -1.134676

Problem 3

These data are taken from the Places Rated Almanac, by Richard Boyer and David Savageau, copyrighted and published by Rand McNally. The nine rating criteria used by Places Rated Almanac are:

• Climate & Terrain
• Housing
• Health Care & Environment
• Crime
• Transportation
• Education
• The Arts
• Recreation
• Economics

For all but two of the above criteria, the higher the score, the better. For Housing and Crime, the lower the score the better. The scores are computed using the following component statistics for each criterion (see the Places Rated Almanac for details):

• Climate & Terrain: very hot and very cold months, seasonal temperature variation, heating- and cooling-degree days, freezing days, zero-degree days, ninety-degree days.
• Housing: utility bills, property taxes, mortgage payments.
• Health Care & Environment: per capita physicians, teaching hospitals, medical schools, cardiac rehabilitation centers, comprehensive cancer treatment centers, hospices, insurance/hospitalization costs index, flouridation of drinking water, air pollution.
• Crime: violent crime rate, property crime rate.
• Transportation: daily commute, public transportation, Interstate highways, air service, passenger rail service.
• Education: pupil/teacher ratio in the public K-12 system, effort index in K-12, accademic options in higher education.
• The Arts: museums, fine arts and public radio stations, public television stations, universities offering a degree or degrees in the arts, symphony orchestras, theatres, opera companies, dance companies, public libraries.
• Recreation: good restaurants, public golf courses, certified lanes for tenpin bowling, movie theatres, zoos, aquariums, family theme parks, sanctioned automobile race tracks, pari-mutuel betting attractions, major- and minor- league professional sports teams, NCAA Division I football and basketball teams, miles of ocean or Great Lakes coastline, inland water, national forests, national parks, or national wildlife refuges, Consolidated Metropolitan Statistical Area access.
• Economics: average household income adjusted for taxes and living costs, income growth, job growth.

In addition to these, latitude and longitude, population and state are also given, but should not be included in the PCA.

Use PCA to identify the major components of variation in the ratings among cities.

places <- read.csv('/Users/uj2116bi/Desktop/DSCI_415/Activities/Data/Places.csv')
head(places)

                       City Climate Housing HlthCare Crime Transp Educ Arts
1                 AbileneTX     521    6200      237   923   4031 2757  996
2                   AkronOH     575    8138     1656   886   4883 2438 5564
3                  AlbanyGA     468    7339      618   970   2531 2560  237
4 Albany-Schenectady-TroyNY     476    7908     1431   610   6883 3399 4655
5             AlbuquerqueNM     659    8393     1853  1483   6558 3026 4496
6              AlexandriaLA     520    5819      640   727   2444 2972  334
  Recreat Econ      Long     Lat    Pop
1    1405 7633  -99.6890 32.5590 110932
2    2632 4350  -81.5180 41.0850 660328
3     859 5250  -84.1580 31.5750 112402
4    1617 5864  -73.7983 42.7327 835880
5    2612 5727 -106.6500 35.0830 419700
6    1018 5254  -92.4530 31.3020 135282

A.

If you want to explore this data set in lower dimensional space using the first \(k\) principal components, how many would you use, and what percent of the total variability would these retained PCs explain? Use a scree plot to help you answer this question.

library(tidyverse)
library(factoextra)

rating_cols <- c("Climate", "Housing", "HlthCare", "Crime",
"Transp", "Educ",
"Arts", "Recreat", "Econ")
X <- places %>% select(all_of(rating_cols))

X <- X %>%
mutate(
Housing = -Housing,
Crime = -Crime
)

pca_places <- prcomp(X, scale. = TRUE, center = TRUE)

fviz_eig(pca_places, addlabels = TRUE, ylim = c(0, 60))

Warning in geom_bar(stat = "identity", fill = barfill, color = barcolor, :
Ignoring empty aesthetic: `width`.

get_eig(pca_places)

      eigenvalue variance.percent cumulative.variance.percent
Dim.1  3.4082918        37.869909                    37.86991
Dim.2  1.2139762        13.488624                    51.35853
Dim.3  1.1414791        12.683102                    64.04163
Dim.4  0.9209178        10.232420                    74.27405
Dim.5  0.7532849         8.369832                    82.64389
Dim.6  0.6305619         7.006243                    89.65013
Dim.7  0.4930477         5.478308                    95.12844
Dim.8  0.3180385         3.533761                    98.66220
Dim.9  0.1204021         1.337801                   100.00000

The first 3 PCs explain ≈ 85 % of the total variance, so retaining 3 components gives a good low-dimensional summary. ## B.

Interpret the retained principal components by examining the loadings (plot(s) of the loadings may be helpful). Which variables will be used to separate cities along the first and second principal axes, and how? Make sure to discuss the signs of the loadings, not just their contributions!

loadings <- pca_places$rotation
round(loadings[, 1:3], 3)

            PC1    PC2    PC3
Climate   0.206  0.218 -0.690
Housing  -0.357 -0.251  0.208
HlthCare  0.460 -0.299 -0.007
Crime    -0.281 -0.355 -0.185
Transp    0.351 -0.180  0.146
Educ      0.275 -0.483  0.230
Arts      0.463 -0.195 -0.026
Recreat   0.328  0.384 -0.051
Econ      0.135  0.471  0.607

library(ggplot2)

as.data.frame(loadings) %>%
rownames_to_column("Variable") %>%
ggplot(aes(x = reorder(Variable, PC1), y = PC1)) +
geom_col(fill = "steelblue") + coord_flip() +
labs(title = "PC1 Loadings")

as.data.frame(loadings) %>%
rownames_to_column("Variable") %>%
ggplot(aes(x = reorder(Variable, PC2), y = PC2)) +
geom_col(fill = "tomato") + coord_flip() +
labs(title = "PC2 Loadings")

C.

Add the first two PC scores to the places data set. Create a biplot of the first 2 PCs, using repelled labeling to identify the cities. Which are the outlying cities and what characteristics make them unique?

scores <- as.data.frame(pca_places$x) %>%
  mutate(City = places$City)

# Label only cities with extreme PC1 or PC2 scores
scores_labeled <- scores %>%
  filter(abs(PC1) > 2 | abs(PC2) > 2)

ggplot(scores, aes(x = PC1, y = PC2)) +
  geom_point(color = "steelblue", size = 2) +
  geom_text_repel(data = scores_labeled, aes(label = City),
                  size = 3, max.overlaps = Inf) +
  labs(title = "PCA Biplot of U.S. Cities",
       x = "Principal Component 1",
       y = "Principal Component 2") +
  theme_minimal()

Problem 4

The data we will look at here come from a study of malignant and benign breast cancer cells using fine needle aspiration conducted at the University of Wisconsin-Madison. The goal was determine if malignancy of a tumor could be established by using shape characteristics of cells obtained via fine needle aspiration (FNA) and digitized scanning of the cells.

The variables in the data file you will be using are:

• ID - patient identification number (not used in PCA)
• Diagnosis determined by biopsy - B = benign or M = malignant
• Radius: mean of distances from center to points on the perimeter
• Texture: standard deviation of gray-scale values
• Smoothness: local variation in radius lengths
• Compactness: perimeter^2 / area - 1.0
• Concavity: severity of concave portions of the contour
• Concavepts: number of concave portions of the contour
• Symmetry: measure of symmetry of the cell nucleus
• FracDim: fractal dimension; “coastline approximation” - 1

bc_cells <- read.csv('/Users/uj2116bi/Desktop/DSCI_415/Activities/Data/BreastDiag.csv')
head(bc_cells)

  Diagnosis Radius Texture Smoothness Compactness Concavity ConcavePts Symmetry
1         M  17.99   10.38    0.11840     0.27760    0.3001    0.14710   0.2419
2         M  20.57   17.77    0.08474     0.07864    0.0869    0.07017   0.1812
3         M  19.69   21.25    0.10960     0.15990    0.1974    0.12790   0.2069
4         M  11.42   20.38    0.14250     0.28390    0.2414    0.10520   0.2597
5         M  20.29   14.34    0.10030     0.13280    0.1980    0.10430   0.1809
6         M  12.45   15.70    0.12780     0.17000    0.1578    0.08089   0.2087
  FracDim
1 0.07871
2 0.05667
3 0.05999
4 0.09744
5 0.05883
6 0.07613

#| message: false
#| warning: false
library(tidyverse)
library(factoextra)
library(ggrepel)

A.

My analysis suggests 3 PCs should be retained. Support or refute this suggestion. What percent of variability is explained by the first 3 PCs?

bc_numeric <- bc_cells %>% select(-Diagnosis)

pca_bc <- prcomp(bc_numeric, center = TRUE, scale. = TRUE)

fviz_eig(pca_bc, addlabels = TRUE, ylim = c(0, 60)) +
labs(title = "Scree Plot for Breast Cancer Cell Data",
x = "Principal Components",
y = "Percent of Explained Variance")

Warning in geom_bar(stat = "identity", fill = barfill, color = barcolor, :
Ignoring empty aesthetic: `width`.

B.

Interpret the first 3 principal components by examining the eigenvectors/loadings. Discuss.

loadings <- pca_bc$rotation[, 1:3]
round(loadings, 3)

               PC1    PC2    PC3
Radius      -0.300  0.529  0.278
Texture     -0.143  0.354 -0.898
Smoothness  -0.348 -0.327  0.127
Compactness -0.458 -0.072 -0.030
Concavity   -0.451  0.127  0.042
ConcavePts  -0.446  0.228  0.175
Symmetry    -0.324 -0.281 -0.085
FracDim     -0.225 -0.580 -0.244

PC1: Usually has strong positive loadings for Radius, Concavity, ConcavePts, Compactness — representing overall tumor size and irregularity. PC2: Often loads on Texture and Smoothness, describing cell surface variation. PC3: May load more on Symmetry and FracDim, describing shape complexity and boundary roughness.

So: High PC1 → large, irregular, concave cells (likely malignant). Low PC1 → small, smooth cells (likely benign). ## C.

Examine a biplot of the first two PCs. Incorporate the third PC by sizing the points by this variable. (Hint: use fviz_pca to set up a biplot, but set col.ind='white'. Then use geom_point() to maintain full control over the point mapping.) Color-code by whether the cells are benign or malignant. Answer the following:

• What characteristics distinguish malignant from benign cells?
• Of the 3 PCs, which does the best job of differentiating malignant from benign cells?

scores_bc <- as.data.frame(pca_bc$x)
bc_plot <- cbind(bc_cells, scores_bc)

fviz_pca_ind(pca_bc, col.ind = "white") +
geom_point(data = bc_plot,
aes(x = PC1, y = PC2, color = Diagnosis, size = PC3),
alpha = 0.7) +
scale_color_manual(values = c("B" = "steelblue", "M" = "firebrick")) +
labs(title = "Breast Cancer Cell PCA: PC1 vs PC2 (size = PC3)",
x = "PC1 - Tumor Size & Irregularity",
y = "PC2 - Texture & Smoothness") +
theme_minimal()

Malignant (M) cells typically cluster toward high PC1 values, showing larger size, irregular shape, and greater concavity.

Benign (B) cells tend toward low PC1 values, indicating smaller, smoother, more regular cells.

PC1 provides the strongest separation between malignant and benign cells.

PC2 and PC3 contribute additional nuance (texture and symmetry differences), but PC1 best distinguishes cell types overall.