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.matrixActivity 3.3 - PCA implementation
SUBMISSION INSTRUCTIONS
- Render to html
- Publish your html to RPubs
- Submit a link to your published solutions
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?
1st PC: 3.5/6 = 58.3%
2nd PC: 4.5/6 = 75%
3rd PC: 5.2/6 = 86.7%
4th PC: 5.6/6 = 93.3%
So, I need to keep the first 4 PCs.
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:
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.matrixA plot of the first two PC scores for these five irises is shown in the plot below.
scores <- five_irises %*% pc_loadings
round(scores, 2) PC1 PC2
1 0.56 1.76
2 1.57 -1.06
3 -2.44 -2.14
4 0.63 0.41
5 -2.13 -1.13Match the ID of each iris (1-5) to the correct letter of its score coordinates on the plot.
1 - b 2 - d 3 - a 4 - c 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 135282A.
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(dplyr)
library(factoextra)Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBaX <- places %>% select(Climate, Housing, HlthCare, Crime, Transp, Educ, Arts, Recreat, Econ)
places_pca <- prcomp(X, center = TRUE, scale. = TRUE)
fviz_eig(places_pca, geom = "line") +
labs(title = "Places: Scree Plot") +
theme_classic(base_size = 15)
I would use k = 5 PCs and explaining baout 80% of the total variance.
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!
library(dplyr)
library(factoextra)
X <- places %>% select(Climate, Housing, HlthCare, Crime, Transp, Educ, Arts, Recreat, Econ)
places_pca <- prcomp(X, center = TRUE, scale. = TRUE)
fviz_contrib(places_pca, choice = "var", axes = 1) +
theme_classic(base_size = 10) +
labs(x = "Variable", title = "PC1 loadings")
fviz_contrib(places_pca, choice = "var", axes = 2) +
theme_classic(base_size = 10) +
labs(x = "Variable", title = "PC2 loadings")
On PC1 there is more Arts, HlthCare, Transport, recreat, Education, and Economy but less Housing cost and Crime. On PC2 thers is more Education, Economy, Recreat, Crime but less HlthCare, Housing, Climate, Arts, and Transport.
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?
library(dplyr)
library(factoextra)
library(ggplot2)
library(ggrepel)
X <- places %>% select(Climate, Housing, HlthCare, Crime, Transp, Educ, Arts, Recreat, Econ)
places_pca <- prcomp(X, center = TRUE, scale. = TRUE)
PCS_labels <- data.frame(PC1 = places_pca$x[,1],
PC2 = places_pca$x[,2],
Item = places$City)
p1 <- fviz_pca(places_pca, axes = c(1,2),
label = "var", repel = TRUE,
selec.var = list(contrib = 5),
col.var = "darkblue",
col.ind = "gold",
alpha.ind = 0.8) +
geom_text_repel(aes(x = PC1, y = PC2, label = Item),
data = PCS_labels,
size = 2, max.overlaps = 6,
color = "black")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>.p1Warning: ggrepel: 304 unlabeled data points (too many overlaps). Consider
increasing max.overlaps
New-YorkNy because it was low PC2 and strong transport, arts, economy but expensive.
ChicagoIL, PhiladelphiaPA-NJ, WashingtonDC-MD-VA, BostonMA has strong transport, arts, education, health care but less on recreation and climate.
Las-VegasNV, Atlantic-CityNJ, Miami-HialeahFL, Salinas-Seaside-MontereyCA, MidlandTX has strong PC2 and high recreation, tourism, economy, warm climate.
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 0.07613A.
My analysis suggests 3 PCs should be retained. Support or refute this suggestion. What percent of variability is explained by the first 3 PCs?
library(dplyr)
library(factoextra)
X <- bc_cells %>% select(Radius, Texture, Smoothness, Compactness, Concavity, ConcavePts, Symmetry, FracDim)
bc_cells_pca <- prcomp(X, center = TRUE, scale. = TRUE)
fviz_eig(bc_cells_pca, geom = "line") +
labs(title = "Scree Plot") +
theme_classic(base_size = 15)
PC1 = 50%, PC2 = 23%, PC3 = 10% So, about 83% of the variability is explained by the first 3 PCs.
B.
Interpret the first 3 principal components by examining the eigenvectors/loadings. Discuss.
Higher PC1 is bigger, rough, and less smooth cells. Radius, compactness, concavity, concave points, FracDim goes up but smoothness goes down.
Higher PC2 is more textured and less smooth. Texture goes up and the smoothness goes down.
Higher PC3 is more symmetric and lower is more jagged. Symmetry goes up and FracDim and concavity goes down.
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:
library(dplyr)
library(factoextra)
library(ggplot2)
library(ggrepel)
X <- bc_cells %>% select(Radius, Texture, Smoothness, Compactness, Concavity, ConcavePts, Symmetry, FracDim)
bc_cells_pca <- prcomp(X, center = TRUE, scale. = TRUE)
scores <- data.frame(PC1 = bc_cells_pca$x[,1],
PC2 = bc_cells_pca$x[,2],
PC3 = bc_cells_pca$x[,3],
Diagnosis = bc_cells$Diagnosis)
p <- fviz_pca(bc_cells_pca, axes = c(1,2), col.var = "black",
col.ind = "white", label = "var", repel = TRUE)
p + geom_point(data = scores,
aes(PC1, PC2, color = Diagnosis, size = abs(PC3)), alpha = 0.5) +
scale_color_manual(values = c(B = "#EC407A",
M = "#009E73",
labels = c(B = "Benign", M = "Maligant"),
name = "Diagnosis"))
- What characteristics distinguish malignant from benign cells?
Malignant cells are on the right side. They have high radius, texture, concavity and concave points so they are rough and less symmetric.
Benign cells are on the left side so they are small, smoother, more symmetric and less concave.
- Of the 3 PCs, which does the best job of differentiating malignant from benign cells?
PC1 separates the groups most clearly.