多變量分析 HW2
劉昱維, 吳冠瑋, 廖永賦, 謝靖惟, 黃奎鈞
Q1: 8.1
covmt <- matrix(c(5,2,2,2), nrow = 2)
eig <- eigen(covmt)eigval <- eig$values
eigvec <- eig$vectors
percent <- eigval[1]/sum(eigval)First Principle Component: \(\lambda_1 = 6\), \(Y_1 = -0.89X_1 -0.45X_2\)
Second Principle Component: \(\lambda_2 = 1\), \(Y_2 = 0.45X_1 -0.89X_2\)
Proprotion of variance explained by the first PC: \(\frac{\lambda_1}{\Sigma \lambda_i} = 85.71\)%
Q2: 8.2
cormt <- diag(1/sqrt(diag(covmt)))%*%covmt%*%diag(1/sqrt(diag(covmt)))\(\rho = diag(\Sigma)^{\frac{-1}{2}} ~ \Sigma ~ diag(\Sigma)^{\frac{-1}{2}} = \begin{pmatrix} 1.00 & 0.63 \\ 0.63 & 1.00 \\ \end{pmatrix}\)
(a)
eig <- eigen(cormt)
eigval <- eig$values
eigvec <- eig$vectors
percent <- eigval[1]/sum(eigval)First Principle Component: \(\lambda_1 = 1.63\), \(Y_1 = -0.71Z_1 -0.71Z_2\)
Second Principle Component: \(\lambda_2 = 0.37\), \(Y_2 = 0.71Z_1 -0.71Z_2\)
Proprotion of variance explained by the first PC: \(\frac{\lambda_1}{\Sigma \lambda_i} = 81.62\)%
(b)
Principle components from 8.1 & 8.2 are different since they came from two different matrix (covariance matrix vs. correlation matrix). Mathmetically, the difference resulted from two different matrices have different eigenvalues and eigenvectors, hence different principle component. This difference reflects the fact that scaling has an effect on the principle component.
(c)
e_11 <- round(eigvec[1,1],3)
sqrt_eigen_1 <- round(sqrt(eigval[1]), 3)
v11 <- eigvec[1,1]*sqrt(eigval[1])
e_12 <- round(eigvec[2,1],2)
v12 <- eigvec[2,1]*sqrt(eigval[1])
e_21 <- round(eigvec[1,2],2)
sqrt_eigen_2 <- round(sqrt(eigval[2]), 2)
v21 <- eigvec[1,2]*sqrt(eigval[2])\(\rho_{Y_i,~Z_k} = \frac{e_{ik} \sqrt{\lambda_i}} {\sqrt{\sigma_{kk}}}\)
\(\rho_{Y_1,~Z_1} = \frac{(-0.71)(1.28) }{\sqrt{1}} = -0.9\)
\(\rho_{Y_1,~Z_2} = -0.9\)
\(\rho_{Y_2,~Z_1} = 0.43\)
Q3: PCA with Covariance Matrix vs. Correlation Matrix
air<- read_table2("data/T1-5.dat", col_names = FALSE)
colnames(air) <- c("wind", "radiation", "CO", "NO", "NO2", "O3", "HC")PCA_cov <- prcomp(air, scale=F)
PCA_cor <- prcomp(air, scale =T)library(factoextra)
library(cowplot)
p_cov <- fviz_eig(PCA_cov, choice = "eigenvalue", main="PCA with S", geom = "line")
p_cor <- fviz_eig(PCA_cor, choice = "eigenvalue", main="PCA with R", geom = "line")
plot_grid(p_cov, p_cor, nrow = 1)
Principle Components: Covariance Matrix
kable(PCA_cov[["rotation"]], format = "markdown", align = ifelse(knitr::is_html_output(), "c", "l"), digits = 2)| PC1 | PC2 | PC3 | PC4 | PC5 | PC6 | PC7 | |
|---|---|---|---|---|---|---|---|
| wind | -0.01 | 0.08 | -0.03 | 0.92 | 0.34 | -0.01 | -0.17 |
| radiation | 0.99 | 0.12 | -0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| CO | 0.01 | -0.10 | 0.18 | -0.14 | 0.65 | 0.56 | 0.44 |
| NO | 0.00 | 0.01 | 0.13 | -0.33 | 0.64 | -0.50 | -0.46 |
| NO2 | 0.02 | -0.15 | 0.96 | 0.10 | -0.21 | 0.01 | -0.11 |
| O3 | 0.11 | -0.97 | -0.17 | 0.06 | 0.00 | -0.05 | -0.07 |
| HC | 0.00 | -0.02 | 0.09 | 0.11 | 0.06 | -0.66 | 0.74 |
Principle Components: Correlation Matrix
kable(PCA_cor[["rotation"]], format = "markdown", align = ifelse(knitr::is_html_output(), "c", "l"), digits=2)| PC1 | PC2 | PC3 | PC4 | PC5 | PC6 | PC7 | |
|---|---|---|---|---|---|---|---|
| wind | -0.24 | 0.28 | -0.64 | 0.17 | -0.56 | 0.22 | 0.24 |
| radiation | 0.21 | -0.53 | -0.22 | 0.78 | 0.16 | 0.01 | 0.01 |
| CO | 0.55 | -0.01 | 0.11 | 0.01 | -0.57 | 0.11 | -0.59 |
| NO | 0.38 | 0.43 | 0.41 | 0.29 | 0.06 | 0.45 | 0.46 |
| NO2 | 0.50 | 0.20 | -0.20 | -0.04 | -0.05 | -0.74 | 0.34 |
| O3 | 0.32 | -0.57 | -0.16 | -0.51 | -0.08 | 0.33 | 0.42 |
| HC | 0.32 | 0.31 | -0.54 | -0.14 | 0.57 | 0.27 | -0.31 |
Interpretation
PCA with \(\mathbf{S}\)
The scree plot of PCA using \(\mathbf{S}\) indicates that only one principle component is important, which explains 87.29% of the total variance.
The first principle component obtained by covariance matrix explains nearly all the variation in the data, this is probably due to the large scale of the variable radiation compared to orther variables. A better explanation of PCA should rely on the correlation matrix in this case.
PCA with \(\mathbf{R}\)
According to the unity criterion and the scree plot (a not-so-obvious elbow at the forth PC), three principle components can well summarize the data, which explains 70.38% of the total variance.
The interpretation of the first principle component obtained by the correlation matricies is straightforward: the variable wind constrasts with other variables consist of pollutants, probably because strong wind blows away pollutants, i.e. wind and polutants have opposite effect on air polution.
Interpretation of the remaining two principle components requires extensive knowledge on air pollution.
Q4
sweet <- read_table2("data/T5-1.txt", col_names = FALSE)
colnames(sweet) <- c("sweet_rate", "sodium", "potassium")source("multivariate_fc.R")
PCA <- prcomp(sweet, scale=F)
PCA_cor <- prcomp(sweet, scale=T)
p_cov <- scree_plot(PCA, "variance", addlabels=TRUE, main="Covariance matrix", sub_axis_name="", axis_limits = c(0,100))
p_cor <- scree_plot(PCA_cor, "variance", addlabels=TRUE, main="Correlation matrix",ylab = "", axis_limits = c(0,100))
cowplot::plot_grid(p_cov, p_cor, nrow = 1)
library(mat2tex)
kable(PCA[["rotation"]], format="markdown", align="c", digits = 2)| PC1 | PC2 | PC3 | |
|---|---|---|---|
| sweet_rate | 0.05 | -0.57 | -0.82 |
| sodium | 1.00 | 0.05 | 0.02 |
| potassium | -0.03 | 0.82 | -0.58 |
quantiles <- qnorm(p=seq(0.25,9.75, 0.5)/10)
PC1 <- as.data.frame(PCA[["x"]][,1])
PC1 <- cbind(rownames(PC1), PC1)
colnames(PC1) <- c("ID", "PC1")
PC1 <- PC1 %>%
arrange(PC1) %>% mutate(quantile=quantiles)
PC2 <- as.data.frame(PCA[["x"]][,2])
PC2 <- cbind(rownames(PC2), PC2)
colnames(PC2) <- c("ID", "PC2")
PC2 <- PC2 %>%
arrange(PC2) %>% mutate(quantile=quantiles)
PC3 <- as.data.frame(PCA[["x"]][,3])
PC3 <- cbind(rownames(PC3), PC3)
colnames(PC3) <- c("ID", "PC3")
PC3 <- PC3 %>%
arrange(PC3) %>% mutate(quantile=quantiles)library(ggplot2)
library(scales)
library(cowplot)
library(latex2exp)
size <- 2.3 # dot label text-size
p1 <- ggplot(PC1, aes(x=quantiles, y=PC1))+
geom_point(color=hue_pal()(3)[1])+
geom_text(aes(label=ID),
hjust=-0.1, vjust=0, size = size)+
scale_x_continuous(breaks = seq(-2,2, 0.5))+
scale_y_continuous(breaks = seq(-35,35, 5))+
labs(x=TeX("$q_{(j)}$"),
y="PC1",
title="PC1 Q-Q Plot")+ theme
p2 <- ggplot(PC2, aes(x=quantiles, y=PC2))+
geom_point(color=hue_pal()(3)[2])+
geom_text(aes(label=ID),
hjust=-0.1, vjust=0, size = size)+
scale_x_continuous(breaks = seq(-2,2, 0.5))+
scale_y_continuous(breaks = seq(-4.5,3.5, 0.5))+
labs(x=TeX("$q_{(j)}$"),
y="PC2",
title="PC2 Q-Q Plot")+ theme
p3 <- ggplot(PC3, aes(x=quantiles, y=PC3))+
geom_point(color=hue_pal()(3)[3])+
geom_text(aes(label=ID),
hjust=-0.1, vjust=0, size = size)+
scale_x_continuous(breaks = seq(-2,2, 0.5))+
scale_y_continuous(breaks = seq(-2,2, 0.5))+
labs(x=TeX("$q_{(j)}$"),
y="PC3",
title="PC3 Q-Q Plot")+ theme
plot_grid(p1, p2, p3, nrow = 1)
Suspect Observations?
From the three Q-Q plots above, there seems no obvious deviation from the straight lines, and threr are no unique points that deviate from the main data in all three Q-Q plots. For detecting suspect observations, more techniques are needed.
Q5
fp <- read_table2("data/FOODP.txt", col_names = FALSE,
skip = 5)[1:24, -1]
colnames(fp) <- c("city", paste("food",1:5,sep=""))(a) Scree Plot, PCs, Interpretation
library(factoextra) # For Visualizing MvA
PCA <- prcomp(fp[,-1], scale=T)
# fviz_eig(PCA)
eigva <- as_data_frame(PCA[["sdev"]]^2)
scree <- cbind(1:5 , eigva)
scree <- cbind(scree, "a")
colnames(scree) <- c("PC", "var", "a")
ggplot(scree, aes(x = PC, y = var, group = a)) +
geom_point() + geom_line() +
scale_y_continuous(
sec.axis = sec_axis(~ . / sum(eigva) * 100,
name = "Variance Accounted (%)")
)+
labs(x="Principle Components",
y=latex2exp::TeX("$\\hat{\\lambda_i}$"),
title="Scree Plot") 
pc <- PCA[["rotation"]]
kable(round(pc,3),"markdown", align = ifelse(knitr::is_html_output(), "c", "l"))| PC1 | PC2 | PC3 | PC4 | PC5 | |
|---|---|---|---|---|---|
| food1 | -0.51 | 0.06 | -0.40 | -0.53 | -0.54 |
| food2 | -0.52 | -0.28 | -0.41 | 0.07 | 0.69 |
| food3 | -0.40 | 0.10 | 0.77 | -0.43 | 0.24 |
| food4 | -0.29 | 0.88 | -0.07 | 0.37 | 0.05 |
| food5 | -0.48 | -0.38 | 0.28 | 0.62 | -0.41 |
Interpretation
By either the “elbow” of the scree plot or the unity criterion, the first principle component is sufficient, which explains 48.79% of the total variance.
The data can be greatly reduced to 1 variable by the first principle component, and the “weight” of the first principle component on each variable have the same sign, indicating the price of each food has similar effect on the variation of the data.
(b) The First Two principal component scores
# PCA[["x"]] returns the "rotated" score(pc score) of the original data
pc_score12 <- as.data.frame(PCA[["x"]][,1:2])
pc_score12 <- cbind(fp$city, pc_score12) %>% rename(city=`fp$city`)
kable(pc_score12, align="c",digits = 2, format ="markdown")| city | PC1 | PC2 |
|---|---|---|
| Anchorage | -4.58 | 0.53 |
| Atlanta | -0.32 | -0.08 |
| Baltimore | -0.36 | -1.10 |
| Boston | -0.57 | -1.62 |
| Buffalo | 2.64 | -1.17 |
| Chicago | -0.51 | 1.29 |
| Cincinnati | -0.27 | -0.79 |
| Cleveland | 0.74 | 0.27 |
| Dallas | -0.38 | 0.92 |
| Detroit | 0.20 | -1.53 |
| Honolulu | -2.80 | 0.91 |
| Houston | 0.46 | 1.34 |
| Kansas_City | 0.42 | -0.39 |
| Los_Angeles | 1.55 | 1.17 |
| Milwaukee | 1.28 | 0.54 |
| Minneapolis | 0.73 | -0.60 |
| New_York | -1.29 | -0.92 |
| Philadelphia | -2.10 | -0.80 |
| Pittsburgh | 0.32 | -1.08 |
| St_Louis | 1.25 | 0.02 |
| San_Diego | 2.12 | 0.68 |
| San_Francisco | 0.71 | 1.32 |
| Seattle | 1.00 | 0.34 |
| Washington | -0.25 | 0.74 |
(c) Scatter Plot: PC1 vs. PC2
ggplot(pc_score12, aes(x=PC1, y=PC2))+
geom_point()+
geom_text(aes(label=city),
hjust=-0.1, vjust=0, size = 3)+
scale_x_continuous(limits = c(-4.6, 2.9), breaks = seq(-4.5,3, 0.5))+
scale_y_continuous(limits = c(-1.7, 1.5), breaks = seq(-1.75,1.5, 0.25))+
geom_segment(x=3,y=1.1,
xend=-4.6, yend=-0.5)+
labs(title="Scatter Plot of PC1 & PC2",
x="Principle Component 1",
y="Principle Component 2")
Two clusters seems to be identified on the scatter plot, ignoring Buffalo.
(d) Food Price in Different Cities
## Max PC score: cheapest food price
cheap <- pc_score12$city[pc_score12$PC1 == max(pc_score12$PC1)]
## min PC score
expense <- pc_score12$city[pc_score12$PC1 == min(pc_score12$PC1)]Using principle component 1 as indicator, since it has the same sign on all “weighted” food price, and it’s the most important principle component in explaining the variability of the data, the x-axis of the scatter above showed that Buffalo has the cheapest food price, and Anchorage has the most expensive food price.
Q6
var_content <- read_delim("data/var_masst.txt",
";", escape_double = FALSE, col_names = FALSE,
trim_ws = TRUE)
colnames(var_content) <- NULL
response <- read.table("C:/Users/user/local_depend/1062_class/Multivariate_Analysis/HW2/data/masst.txt", quote="\"", comment.char="", na.strings=".", stringsAsFactors=FALSE) %>%
drop_na()PCA <- prcomp(response, scale = TRUE)
std_pc <- PCA[["sdev"]]
# std(sqrt(eigenvalue) of corr matrix) of PCs(a)
Weight of the First 5 Pricinple Components on each Variable
# First 5 PCs
pc5 <- as.data.frame(PCA[["rotation"]][,1:5])
styled_pc5 <- round(pc5,3) %>%
mutate(
PC1 = cell_spec(PC1, "html", color = ifelse(PC1 < 0, "red", "blue")),
PC2 = cell_spec(PC2, "html", color = ifelse(PC2 < 0, "red", "blue")),
PC3 = cell_spec(PC3, "html", color = ifelse(PC3 < 0, "red", "blue")),
PC4 = cell_spec(PC4, "html", color = ifelse(PC4 < 0, "red", "blue")),
PC5 = cell_spec(PC5, "html", color = ifelse(PC5 < 0, "red", "blue"))
)
loadings <- kable(styled_pc5, "markdown", escape = F, align="c", row.names = T) %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
loadings %>%
kableExtra::landscape()| PC1 | PC2 | PC3 | PC4 | PC5 | |
|---|---|---|---|---|---|
| 1 | 0.147 | -0.092 | 0.024 | -0.429 | 0.28 |
| 2 | -0.26 | 0.126 | -0.084 | 0.214 | -0.311 |
| 3 | -0.315 | 0.03 | -0.059 | 0.12 | -0.123 |
| 4 | 0.055 | -0.116 | -0.17 | -0.375 | 0.15 |
| 5 | 0.38 | -0.071 | -0.133 | -0.184 | 0.085 |
| 6 | -0.122 | -0.519 | -0.235 | 0.002 | -0.005 |
| 7 | 0.013 | -0.218 | 0.3 | -0.216 | -0.447 |
| 8 | -0.072 | 0.234 | -0.393 | -0.027 | -0.195 |
| 9 | 0.24 | 0.055 | -0.221 | -0.324 | -0.271 |
| 10 | 0.283 | -0.13 | -0.106 | 0.185 | -0.128 |
| 11 | -0.045 | 0.123 | -0.498 | 0.012 | 0.073 |
| 12 | -0.341 | 0.146 | 0.063 | -0.345 | -0.101 |
| 13 | -0.363 | 0.134 | -0.045 | -0.298 | 0.079 |
| 14 | 0.027 | 0.225 | -0.358 | -0.117 | -0.335 |
| 15 | 0.271 | 0.019 | -0.007 | -0.206 | -0.231 |
| 16 | 0.177 | -0.045 | -0.122 | 0.127 | -0.185 |
| 17 | -0.209 | -0.463 | -0.188 | 0.024 | 0.016 |
| 18 | -0.256 | 0.088 | -0.017 | -0.259 | 0.226 |
| 19 | -0.181 | -0.47 | -0.222 | -0.019 | -0.09 |
| 20 | -0.093 | -0.117 | 0.318 | -0.213 | -0.413 |
Percentage of Variance Accounted
\(\hat{\lambda_1}\) = 2.94, \(\hat{\lambda_2}\) = 2.1, \(\hat{\lambda_3}\) = 2.04, \(\hat{\lambda_4}\) = 1.66, \(\hat{\lambda_5}\) = 1.19.
per_5pc <- sum(std_pc[1:5]^2)/sum(std_pc^2)The percentage of total variance accounted by the first 5 PCs is 49.64%.
(b)
t <- var_content %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X8 | Congress has done all it can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X10 | Conserving electricity will save me money in the long run. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X12 | My electricity bill would be the same no matter what I did. |
| X13 | There is not much an average citizen can do to save electricity. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
| X20 | The electric companies have not done all they can to solve the energy problem. |
BrBG <- colorRampPalette(c("purple","blue","white", "red", "orange"))
hm <- function(data) {
heatmaply::heatmaply(round(cor(data),2), margins = c(50, 50),
k_col = 2, k_row = 2,
colors = BrBG,
dendrogram = "none",
label_names=c("row", "column", "Corr."),
column_text_angle = 60,
fontsize_row=5, fontsize_col=5,
limits = c(-1,1))
}factoextra::fviz_eig(PCA, choice = "variance", main="First 5 PCs", geom = "line", ncp=8, addlabels=T)+ scale_y_continuous(limits = c(4, 16))
First PC
h <- hm(response[,c(which(pc5$PC1>0), which(pc5$PC1<0))])
if (knitr::is_html_output()) {h}t <- var_content[which(pc5$PC1>0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X10 | Conserving electricity will save me money in the long run. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
contrasts
t <- var_content[which(pc5$PC1<0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X8 | Congress has done all it can to solve the energy problem. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X12 | My electricity bill would be the same no matter what I did. |
| X13 | There is not much an average citizen can do to save electricity. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
| X20 | The electric companies have not done all they can to solve the energy problem. |
The first PC group variables as:
The variables 1, 4, 5, 7, 9, 10, 14, 15, 16 seem to reflect the trade-offs between cutting down energy usage and solving energy crysis.
- The variables 2, 3, 6, 8, 11, 12, 13, 17, 18, 19, 20 seem to reflect 2 attitudes:
- Little can be done to change the energy crysis.
- Favor solving energy crysis at the cost of pollution.
Second PC
h <- hm(response[,c(which(pc5$PC2>0), which(pc5$PC2<0))])
if (knitr::is_html_output()) {h}t <- var_content[which(pc5$PC2>0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X8 | Congress has done all it can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X12 | My electricity bill would be the same no matter what I did. |
| X13 | There is not much an average citizen can do to save electricity. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
contrasts
t <- var_content[which(pc5$PC2<0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X10 | Conserving electricity will save me money in the long run. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
| X20 | The electric companies have not done all they can to solve the energy problem. |
The second PC group variables as:
The variables 2, 3, 8, 9, 11, 12, 13, 14, 15, 18 mostly reflect the attitude that little can be done to solve the energy crysis.
The variables 1, 4, 5, 6, 7, 10, 16, 17, 19, 20 seem to reflect that the energy crysis is changable despite some sacrifices such as pollution.
Third, Forth, and Fifth PC
The third, forth, and fifth PC are not obvious and easy to interpret, and in fact, they account for only 10%, 8%, and 6% of total variance, respectively.
Third PC
h <- hm(response[,c(which(pc5$PC3>0), which(pc5$PC3<0))])
if (knitr::is_html_output()) {h}t <- var_content[which(pc5$PC3>0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X12 | My electricity bill would be the same no matter what I did. |
| X20 | The electric companies have not done all they can to solve the energy problem. |
contrasts
t <- var_content[which(pc5$PC3<0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X8 | Congress has done all it can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X10 | Conserving electricity will save me money in the long run. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X13 | There is not much an average citizen can do to save electricity. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
Forth PC
h <- hm(response[,c(which(pc5$PC4>0), which(pc5$PC4<0))])
if (knitr::is_html_output()) {h}t <- var_content[which(pc5$PC4>0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X10 | Conserving electricity will save me money in the long run. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
contrasts
t <- var_content[which(pc5$PC4<0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X8 | Congress has done all it can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X12 | My electricity bill would be the same no matter what I did. |
| X13 | There is not much an average citizen can do to save electricity. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
| X20 | The electric companies have not done all they can to solve the energy problem. |
Fifth PC
h <- hm(response[,c(which(pc5$PC5>0), which(pc5$PC5<0))])
if (knitr::is_html_output()) {h}t <- var_content[which(pc5$PC5>0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X1 | If the energy shortage gets any worse, the country will be in bad shape. |
| X4 | Saving energy requires you to make major sacrifices. |
| X5 | The energy crisis is for real. |
| X11 | The natural gas companies have done all they can to solve the energy problem. |
| X13 | There is not much an average citizen can do to save electricity. |
| X17 | We should forget about reducing pollution until our energy problems are solved. |
| X18 | My personal conservation efforts have little impact on total consumption of energy. |
contrasts
t <- var_content[which(pc5$PC5<0),] %>%
kable(format = "html", align="l") %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"), full_width = F)
if (knitr::is_html_output()) {t}| X2 | The worst of the energy crisis has passed. |
| X3 | Science and technology will be able to resolve the energy crisis without conservation. |
| X6 | Utility companies should be allowed to burn cheaper fuel even though this would cause more pollution. |
| X7 | The petroleum companies have not done all they can to solve the energy problem. |
| X8 | Congress has done all it can to solve the energy problem. |
| X9 | Rationing of energy resources will be necessary for at least the next five years. |
| X10 | Conserving electricity will save me money in the long run. |
| X12 | My electricity bill would be the same no matter what I did. |
| X14 | President Carter has done all he can to solve the energy problem. |
| X15 | At this point in time our traditional energy resources (coal, oil, natural gas, etc.) are insufficient to continue energy consumption at the present rate. |
| X16 | It would be easy for me to cut down on the use of electricity in my home. |
| X19 | Because of the abundance of coal, industries should be encouraged to switch to coal as a fuel despite the air pollution it causes. |
| X20 | The electric companies have not done all they can to solve the energy problem. |