true 

# load the libraries
library('ggplot2')
library('ggthemes')
library('RColorBrewer')
library('Hmisc')

Load Data

We begin by loading the data.

# load the data
inputFile<-"C:/Users/dgn2/Documents/R/IS607/Project_2/project2_data.csv"
# load data
data <- read.csv(inputFile, header=TRUE)

The data is displayed in the following table:

X and Y by Quarter
I X I Y II X II Y III X III Y IV X IV Y
10 8.04 10 9.14 10 7.46 8 6.58
8 6.95 8 8.14 8 6.77 8 5.76
13 7.58 13 8.74 13 12.74 8 7.71
9 8.81 9 8.77 9 7.11 8 8.84
11 8.33 11 9.26 11 7.81 8 8.47
14 9.96 14 8.10 14 8.84 8 7.04
6 7.24 6 6.13 6 6.08 8 5.25
4 4.26 4 3.10 4 5.39 19 12.50
12 10.84 12 9.13 12 8.15 8 5.56
7 4.82 7 7.26 7 6.42 8 7.91
5 5.68 5 4.74 5 5.73 8 6.89

Exploratory Data Analysis

Data Description

Once the data is loaded, we take a quick look at the attributes of the data:

str(data)
## 'data.frame':    44 obs. of  3 variables:
##  $ quarter: Factor w/ 4 levels "I","II","III",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ x      : int  10 8 13 9 11 14 6 4 12 7 ...
##  $ y      : num  8.04 6.95 7.58 8.81 8.33 ...

We can see that there are 44 rows and 3 columns in our data set (i.e., \(quarter\), \(x\), and \(y\)).

Data Description
variable description
quarter is a factor with 4 levels (I, II, III, & IV)
x is an int
y is a decimal number

A quick summary of the data by the \(quarter\) factor does not provide much insight.

# create the data summary
summaries <- aggregate(data$y, by=list(data$quarter),FUN=summary)
# relabel the data summary
colnames(summaries)<-c('Group','')
summaries
##   Group  .Min. .1st Qu. .Median  .Mean .3rd Qu.  .Max.
## 1     I  4.260    6.315   7.580  7.501    8.570 10.840
## 2    II  3.100    6.695   8.140  7.501    8.950  9.260
## 3   III  5.390    6.250   7.110  7.500    7.980 12.740
## 4    IV  5.250    6.170   7.040  7.501    8.190 12.500

Graphical Exploration

First, we create a single scatter plot varying the color by quarter.

# graph the data
p <- ggplot(data, aes(x, y))+theme_economist() + 
  scale_colour_economist() + ggtitle("Scatterplots of X vs. Y") + 
  geom_rug()
p + geom_point(aes(colour = quarter))

The patterns associated with each quarter are difficult to see on a single scatter plot, so we split out the quarters into separate scatter plots.

qplot(x,y, data=data,color=quarter,facets=~quarter,
      xlab="x", ylab="y",main="Scatterplots of X vs. Y") + 
  geom_rug() + scale_fill_brewer() + 
  theme_economist() + scale_colour_economist()

The patterns in each data set are clearer. The I, III and IV data appear to have linear relationships. The II data is clearly not linear. Both III and IV appear to have outliers that obscure the data.

p <- ggplot(data, aes(quarter,y))
p + geom_boxplot(outlier.colour = "blue",aes(fill = quarter)) + 
  coord_flip() + scale_fill_brewer() + ggtitle('Y Summary By Quarter')+
  theme_economist() + scale_colour_economist()

We now examine the distribution of Y, first by creating a histogram where color denotes the contribution by each quarter

# create a histogram where color denotes contribution by
# each quarter
dplot <- ggplot(data, aes(y, fill = quarter))
dplot + geom_bar(position = "stack",binwidth=1) + 
  theme_economist() + scale_colour_economist() + 
  scale_fill_brewer() + ggtitle('Y By Quarter')

We continue to examine the distribution of Y, focusing next on kernel smoothed densities for each respective quarter.

# plot the kernal smoothed density by quarter
qplot(y, data=data, geom="density",position="stack", fill=quarter,
      alpha=I(.6), main="Distribution of Y BY Quarter",
      xlab="y", ylab="density") + scale_fill_brewer() + 
  theme_economist() + scale_colour_economist()

The shape of the distribution of Y for each quarter is more clear once we look at the distribution for each quarter separately.

# plot the kernal smoothed density by quarter
qplot(y, data=data, geom="density",position="stack", fill=quarter,
      alpha=I(.6), main="Distribution of Y BY Quarter",
      xlab="y", ylab="density") + scale_fill_brewer() + 
  theme_economist() + scale_colour_economist() + 
  facet_grid(. ~ quarter) +  coord_flip()

The density differs significantly by quarter. The mode of the distribution shifts.