Activity 3.3 - PCA implementation

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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?

You would want to keep 4 of them.

3.5 + 1 + .7 + .4 + .25 + .15 = 6

6 * .9 = 5.4

3.5 + 1 + .7 + .4 = 5.6

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.

pc_scores <- five_irises %*% pc_loadings
pc_scores

         PC1        PC2
1  0.5606200  1.7617440
2  1.5715031 -1.0649071
3 -2.4396481 -2.1418936
4  0.6308802  0.4146079
5 -2.1283408 -1.1346763

plot 1: b

plot 2: d

plot 3: a

plot 4: c

plot 5: e

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('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.

criteria <- places %>% select(Climate, Housing, HlthCare, Crime, Transp, Educ, Arts, Recreat, Econ)

criteria_scaled <- scale(criteria)

pca_criteria <- prcomp(criteria_scaled, center = TRUE, scale. = TRUE)

plot(pca_criteria, type = 'lines', main = 'Scree Plot')

summary(pca_criteria)

Importance of components:
                          PC1    PC2    PC3    PC4    PC5     PC6     PC7
Standard deviation     1.8462 1.1018 1.0684 0.9596 0.8679 0.79408 0.70217
Proportion of Variance 0.3787 0.1349 0.1268 0.1023 0.0837 0.07006 0.05478
Cumulative Proportion  0.3787 0.5136 0.6404 0.7427 0.8264 0.89650 0.95128
                           PC8     PC9
Standard deviation     0.56395 0.34699
Proportion of Variance 0.03534 0.01338
Cumulative Proportion  0.98662 1.00000

I would use 6 principal components. The reason for this is because with 6 principal components 89% of the variability is explained.

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!

criteria_loadings <- pca_criteria$rotation

criteria_loadings[, 1:2]

               PC1        PC2
Climate  0.2064140  0.2178353
Housing  0.3565216  0.2506240
HlthCare 0.4602146 -0.2994653
Crime    0.2812984  0.3553423
Transp   0.3511508 -0.1796045
Educ     0.2752926 -0.4833821
Arts     0.4630545 -0.1947899
Recreat  0.3278879  0.3844746
Econ     0.1354123  0.4712833

library(ggplot2)

pc_loadings_df <- as.data.frame(criteria_loadings[, 1:2])
pc_loadings_df$Variable <- rownames(pc_loadings_df)

ggplot(pc_loadings_df, aes(x = PC1, y = PC2, label = Variable)) +
  geom_segment(aes(x = 0, y = 0, xend = PC1, yend = PC2),
               arrow = arrow(length = unit(0.25, "cm"))) +
  geom_text(hjust = 0.5, vjust = -0.5, size = 4) +
  theme_minimal() +
  labs(title = "Loadings Plot for PC1 and PC2",
       x = "PC1 Loadings",
       y = "PC2 Loadings")

The PC1 separates cities by arts, transportation, and health care versus economic strength. Categories such as art, transportation, and health care have higher loadings and lower loadings in economic strength. Looking at PC2 cities with positive loadings in the economic strength variable have higher crime and more recreational activities. Then looking at the negative loadings the cities with strong education will also have strong health care systems. This shows that cities with strong economics and more leisure active are more socially unstable because of the higher crime.

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?

places_scores <- as.data.frame(pca_criteria$x[, 1:2])

places_and_scores <- cbind(places, places_scores)

library(ggrepel)
ggplot(places_and_scores, aes(x = PC1, y = PC2, label = City)) +
  geom_point(color = "blue") +
  geom_text_repel(size = 3) +
  labs(
    title = "Biplot of the First Two Principal Components",
    x = "PC1",
    y = "PC2"
  ) +
  theme_minimal()

Warning: ggrepel: 304 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

There are three cities that I would like to talk about. The first one being Las Vegas. The variable that is most associated to Las Vegas is economic strength which makes sense with all the casinos and recreational activities. The second city is New York. The variable that is most associated with New York is the arts variable. This makes sense because that area is well known for art and museums. The third city is Chicago and the variable that is most associated with Chicago is education which also makes sense with all the prestigious universities and schools in that area.

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('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

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?

cells_numeric <- bc_cells %>% select(Radius, Texture, Smoothness, Compactness, Concavity, ConcavePts, Symmetry, FracDim)

cells_pca <- prcomp(cells_numeric, center = TRUE, scale. = TRUE)

summary(cells_pca)

Importance of components:
                          PC1    PC2    PC3     PC4     PC5     PC6    PC7
Standard deviation     2.0705 1.3504 0.9087 0.70614 0.61016 0.30355 0.2623
Proportion of Variance 0.5359 0.2279 0.1032 0.06233 0.04654 0.01152 0.0086
Cumulative Proportion  0.5359 0.7638 0.8670 0.92937 0.97591 0.98743 0.9960
                           PC8
Standard deviation     0.17837
Proportion of Variance 0.00398
Cumulative Proportion  1.00000

The percentage of variability using the first 3 PCs is 86.7%. I support this suggestions because it is recommended to at least explain 80% of the variability. 86.7% is a good amount of explained variability. If a fourth PC is used it would only give 6.2 more explained variability.

B.

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

cells_pca$rotation[, 1:3]

                   PC1         PC2         PC3
Radius      -0.3003952  0.52850910  0.27751200
Texture     -0.1432175  0.35378530 -0.89839046
Smoothness  -0.3482386 -0.32661945  0.12684205
Compactness -0.4584098 -0.07219238 -0.02956419
Concavity   -0.4508935  0.12707085  0.04245883
ConcavePts  -0.4459288  0.22823091  0.17458320
Symmetry    -0.3240333 -0.28112508 -0.08456832
FracDim     -0.2251375 -0.57996072 -0.24389523

library(factoextra)

Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

library(patchwork)
c1 <- fviz_contrib(cells_pca, choice='var', axes = 1)
c2 <- fviz_contrib(cells_pca, choice='var', axes = 2)
c3 <- fviz_contrib(cells_pca, choice='var', axes = 3)

(c1+c2)/(c3)

For PC1 the biggest contributing variables are compactness, concavity, and concavepts. Higher PC1 values with correspond with cells that are more compact and concave.

For PC2 the biggest contributing variables are fracdim, radius, and texture. High PC2 values will indicate cells that are larger and rougher. Smaller PC2 values will indicate smaller and smoother cells.

For PC3 the biggest contributing variable is texture. Higher values will indicate cells that are more rougher and lower values will indicate cells that are smoother.

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_cells <- as.data.frame(cells_pca$x)
scores_cells$Diagnosis <- bc_cells$Diagnosis

plot_cells <- fviz_pca_biplot(
  cells_pca,
  geom.ind = "point",
  col.ind = "white",
  col.var = "black",
  repel = TRUE
)

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
ℹ The deprecated feature was likely used in the ggpubr package.
  Please report the issue at <https://github.com/kassambara/ggpubr/issues>.

plot_cells + 
  geom_point(
    data = scores_cells,
    aes(x = PC1, y = PC2, color = Diagnosis, size = abs(PC3)), 
    alpha = 0.7
  ) +
  scale_color_manual(values = c("B" = "blue", "M" = "red")) +
  labs(
    title = "Biplot of Breast Cancer Cells (PC1 vs PC2, size = PC3)",
    x = "PC1",
    y = "PC2"
  ) +
  theme_minimal()

When looking at the plot negative PC1 values are going to be more associated with malignant cells. Positive PC1 values are going to be associated with benign cells.

More of the malignant cells have a positive PC2 value. More benign cells have a negative PC2 value.

PC1 does the best job at differentiating between cells. The reason for this is because there can almost be a vertical line that can be drawn to separate the two cells types.

Overall malignant cells are going to be bigger, more concavity, and be rougher. Benign cells are going to smaller, less concavity, and less compact.