STAT3064/STAT3406/STAT4067 Statistical Data Science
Computer Lab 1, Semester 2 2020
- Things you may need to know/do.
- Q1
- Q2
- Q3
- Answers
- Q1
- a) Read the data into R, and summarise to understand the inputs. Are there any missing values?
- b) Show separate histogram of Power and log10( Power ). Describe the shape of the histograms.
- c) Construct smoothed histograms of Power and log10( Power ).
- d) Create new variables logPower equal to log10( Power ) and similarly for the remaining variables, apart from Period and Year. Do this by creating a new dataframe with these variables plus Period and Year. Make Period a factor on this new dataframe. Why might the log transformation be a good idea to help in understanding these data? (Note that we have specified log10 rather than the natural log, because this is easier to interpret in the circumstances we have.)
- e) Show pairwise plots of the logged variables. Which pair has the largest absolute correlation?
- e) The code chunk below generates a 3D scatterplot of the variables logPower, Span and Length. Comment on the scatterplots in relation to what it says about the variables, and discuss the shortcomings of this sort of display. Experiment with changing the viewing angles phi and theta.
- f) The code chunk below makes a parallel coordinate plot of the 6 logged variables, coloured by the value of Period, and with boxplots and suitable transparency. It is arranged so that so that the values are not scaled. Comment on what this figure says about the data, and discuss any shortcomings of the plotting method.
- Q2
- a) How many observations belong to each period?
- b) Which years belong to each period?
- c) Which variable has the largest interquartile range? Which has the largest median? For which variables are the means larger than 1.04 times the median? (Leave out Period for this question, as it is a factor, not a numeric variable. Don’t do the second part of this question by just looking at the output from say the summary command. Do it by writing a small amount of code that will tell you the answer directly.)
- d) Divide the scaled data into the three period groups. For all the data and separately for period 1, give the histograms and density plots of logPower. Compare the results for the complete data with those obtained from period 1. Comment.
- Q3
Things you may need to know/do.
- Relates to lecture 1.
- Libraries ggplot2, tidyverse, MASS, GGally may be useful. Others may be mentioned below in the hints.
- You might like to set up a project for each lab if you are using RStudio. Then you can copy a .Rmd file into that directory and write your answers in that file
Q1
Consider the aircraft data (aircraft.csv) with variables Year, Period, Power, Span, Length, Weight, Speed and Range.
Data description (from the R help for package sm)
These data record six characteristics of aircraft designs which appeared during the twentieth century. The variables are: - Year: year of first manufacture - Period: a code to indicate one of three broad time periods - Power: total engine power (kW) - Span: wing span (m) - Length: length (m) - Weight: maximum take-off weight (kg) - Speed: maximum speed (km/h) - Range: range (km)
Source: The data were collected by P. Saviotti and are described in detail in Saviotti (1996), “Technological Evolution, Variety and Economy”, Edward Elgar: Cheltenham.
Read the data into R, and summarise it to understand the inputs. Are there any missing values? (Hint: read.csv, head and summary.) aircraft<-read.csv(“aircraft.csv”) head aircraft
Show separate histogram of Power and log10( Power ). Describe the shape of the histograms. (Hint: truehist (from MASS) or ggplot with geom_histogram.)
Construct smoothed histograms of Power and log10( Power ). (Hint: ggplot + geom_density.)
Create new variables logPower equal to log10( Power ) and similarly for the remaining variables, apart from Period and Year. Do this by creating a new dataframe with these variables plus Period and Year. Make Period a factor on this new dataframe. Why might the log transformation be a good idea to help in understanding these data? (Note that we have specified log10 rather than the natural log, because this is easier to interpret in the circumstances we have.) (Hint: Can be done directly but could use mutate from dplyr, which is part of tidyverse.)
Show pairwise plots of the logged variables. Which pair has the largest absolute correlation? (Hint: pairs or ggpairs.)
The code chunk below generates a 3D scatterplot of the variables logPower, logSpan and logLength. Comment on the scatterplots in relation to what it says about the variables, and discuss the shortcomings of this sort of display. Experiment with changing the viewing angles phi and theta.
The code chunk below makes a parallel coordinate plot of the 6 logged variables, coloured by the value of Period, and with boxplots and suitable transparency. It is arranged so that the values are not scaled. Comment on what this figure says about the data, and discuss any shortcomings of the plotting method.
Code chunk for 3d scatterplots.
Remove the “, eval = FALSE” before running knitting in RStudio.
library( plot3D )
scatter3D (aircraft$logWeight, aircraft$logSpan, aircraft$logLength,
phi = 20, theta = 80,
col = NULL, NAcol = "white", breaks = NULL,
colkey = NULL, panel.first = NULL,
clim = NULL, clab = NULL,
bty = "b", CI = NULL, surf = NULL,
add = FALSE, plot = TRUE)Code chunk for parallel co-ordinate plot.
Remove the “, eval = FALSE” before running knitting in RStudio.
library(GGally)
ggparcoord( aircraft,
groupColumn = "Period",
columns = 3:8,
alpha = 0.1,
scale = "globalminmax",
boxplot = TRUE ) +
theme( legend.position = "bottom")Q2
Continue with the aircraft data of Q1 (using the logged variables).
The data are from three different periods.
(Hints for these: summary, xtabs, sapply, filter.)
- How many observations belong to each period?
- Which years belong to each period?
- Which variable has the largest interquartile range? Which has the largest median? For which variables are the means larger than 1.04 times the median? (Leave out Period for this question, as it is a factor, not a numeric variable. Don’t do the second part of this question by just looking at the output from say the summary command. Do it by writing a small amount of code that will tell you the answer directly.)
- Divide the dataframe into the three period groups. For all the data and separately for period 1, give the histograms and density plots of logPower. Compare the results for the complete data with those obtained from period 1. Comment. (Hint, for interest and because it will be useful in the first assignment: It is fairly easy to get 3 separate density plots. The following on the other hand will generate a plot more useful for comparisons.)
# density plots
ggplot( aircraft, aes( logPower, group = Period, colour = Period ) ) +
geom_density( aes( fill = Period ), alpha = 0.4 ) +
ggtitle("All data"))- Compare the pairs or ggpairs results for the logged variables for the whole data and for period 1. Comment.
Q3
Continue with the aircraft data of Q2.
- Construct contour plots of the 2D smoothed histograms for the pair of variables logLength and logWeight, and also for logSpeed and logSpan for all records.
(Hint: This code chunk does the second one.)
ggplot( aircraft, aes( logSpeed, logSpan ) ) +
geom_density_2d( ) +
geom_point( aes( colour = Period ), alpha = 0.6 ) +
theme( legend.position = "bottom")- Describe the shape of these density plots (eg bimodal, skewed, etc).
- What do the shapes of the contours in these figures say about the variable pairs? Hint: look at the correlations you calculated earlier.
Answers
Start R
library(tidyverse)
library(ggplot2)
library(MASS)
library(GGally)
library( plot3D )Q1
a) Read the data into R, and summarise to understand the inputs. Are there any missing values?
library( ggplot2 )
aircraft0 = read.csv( "aircraft.csv" )
head( aircraft0 )
summary( aircraft0 )## Year Period Power Span Length Weight
## Min. :14.00 Min. :1.00 Min. : 30 Min. : 6.68 Min. : 5.69 Min. : 481
## 1st Qu.:34.00 1st Qu.:1.00 1st Qu.: 421 1st Qu.:11.00 1st Qu.: 9.02 1st Qu.: 2386
## Median :42.00 Median :2.00 Median : 1014 Median :14.30 Median :11.96 Median : 5500
## Mean :46.27 Mean :2.02 Mean : 4062 Mean :18.13 Mean :15.51 Mean : 17236
## 3rd Qu.:61.00 3rd Qu.:3.00 3rd Qu.: 3080 3rd Qu.:22.10 3rd Qu.:17.84 3rd Qu.: 14000
## Max. :84.00 Max. :3.00 Max. :80000 Max. :70.10 Max. :75.54 Max. :365000
## Speed Range
## Min. : 105.0 Min. : 80
## 1st Qu.: 259.0 1st Qu.: 850
## Median : 408.0 Median : 1400
## Mean : 561.3 Mean : 2039
## 3rd Qu.: 655.0 3rd Qu.: 2400
## Max. :3219.0 Max. :20120b) Show separate histogram of Power and log10( Power ). Describe the shape of the histograms.
library( MASS )
truehist( aircraft0$Power ) # uses MASS::truehist
library( ggplot2 )
ggplot( aircraft0, aes( Power ) ) + geom_histogram() # example, from ggplot2## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft0, aes( log10( Power) ) ) + geom_histogram() # example, from ggplot2## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft0, aes( Power ) ) + geom_histogram() + scale_x_log10() # example, from ggplot2## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
c) Construct smoothed histograms of Power and log10( Power ).
ggplot( aircraft0, aes( Power ) ) + geom_density(fill = "blue")
ggplot( aircraft0, aes( log10( Power ) ) ) + geom_density(fill = "blue")
d) Create new variables logPower equal to log10( Power ) and similarly for the remaining variables, apart from Period and Year. Do this by creating a new dataframe with these variables plus Period and Year. Make Period a factor on this new dataframe. Why might the log transformation be a good idea to help in understanding these data? (Note that we have specified log10 rather than the natural log, because this is easier to interpret in the circumstances we have.)
# Using dplyr
# names( aircraft0 )
# "Year" "Period" "Power" "Span" "Length" "Weight" "Speed" "Range"
aircraft = mutate( aircraft0,
Period = factor( Period),
logPower = log10( Power ),
logSpan = log10( Span ),
logLength = log10( Length ),
logWeight = log10( Weight ),
logSpeed = log10( Speed ),
logRange = log10( Range )
) %>%
dplyr::select( Year, Period, starts_with("log") )
summary( aircraft )## Year Period logPower logSpan logLength logWeight logSpeed
## Min. :14.00 1:216 Min. :1.477 Min. :0.8248 Min. :0.7551 Min. :2.682 Min. :2.021
## 1st Qu.:34.00 2:263 1st Qu.:2.624 1st Qu.:1.0414 1st Qu.:0.9552 1st Qu.:3.378 1st Qu.:2.413
## Median :42.00 3:230 Median :3.006 Median :1.1553 Median :1.0777 Median :3.740 Median :2.611
## Mean :46.27 Mean :3.084 Mean :1.2038 Mean :1.1265 Mean :3.801 Mean :2.637
## 3rd Qu.:61.00 3rd Qu.:3.489 3rd Qu.:1.3444 3rd Qu.:1.2514 3rd Qu.:4.146 3rd Qu.:2.816
## Max. :84.00 Max. :4.903 Max. :1.8457 Max. :1.8782 Max. :5.562 Max. :3.508
## logRange
## Min. :1.903
## 1st Qu.:2.929
## Median :3.146
## Mean :3.172
## 3rd Qu.:3.380
## Max. :4.304# alternatively (more typing)
aircraft = data.frame( Year = aircraft0$Year,
Period = factor( aircraft0$Period))
aircraft$logPower = log10( aircraft0$Power )
aircraft$logSpan = log10( aircraft0$Span )
aircraft$logLength = log10( aircraft0$Length )
aircraft$logWeight = log10( aircraft0$Weight )
aircraft$logSpeed = log10( aircraft0$Speed )
aircraft$logRange = log10( aircraft0$Range )
summary( aircraft )## Year Period logPower logSpan logLength logWeight logSpeed
## Min. :14.00 1:216 Min. :1.477 Min. :0.8248 Min. :0.7551 Min. :2.682 Min. :2.021
## 1st Qu.:34.00 2:263 1st Qu.:2.624 1st Qu.:1.0414 1st Qu.:0.9552 1st Qu.:3.378 1st Qu.:2.413
## Median :42.00 3:230 Median :3.006 Median :1.1553 Median :1.0777 Median :3.740 Median :2.611
## Mean :46.27 Mean :3.084 Mean :1.2038 Mean :1.1265 Mean :3.801 Mean :2.637
## 3rd Qu.:61.00 3rd Qu.:3.489 3rd Qu.:1.3444 3rd Qu.:1.2514 3rd Qu.:4.146 3rd Qu.:2.816
## Max. :84.00 Max. :4.903 Max. :1.8457 Max. :1.8782 Max. :5.562 Max. :3.508
## logRange
## Min. :1.903
## 1st Qu.:2.929
## Median :3.146
## Mean :3.172
## 3rd Qu.:3.380
## Max. :4.304e) Show pairwise plots of the logged variables. Which pair has the largest absolute correlation?
pairs( aircraft[ , 3:8 ] )
cor( aircraft[ , 3:8 ])## logPower logSpan logLength logWeight logSpeed logRange
## logPower 1.0000000 0.57436142 0.8556267 0.9477214 0.80368463 0.6835478
## logSpan 0.5743614 1.00000000 0.8496270 0.7699374 0.04056524 0.6200422
## logLength 0.8556267 0.84962700 1.0000000 0.9534497 0.46575591 0.7185541
## logWeight 0.9477214 0.76993744 0.9534497 1.0000000 0.63055999 0.7463050
## logSpeed 0.8036846 0.04056524 0.4657559 0.6305600 1.00000000 0.4645887
## logRange 0.6835478 0.62004216 0.7185541 0.7463050 0.46458873 1.0000000library( GGally )
ggpairs( aircraft, columns = 3:8, progress = FALSE )
e) The code chunk below generates a 3D scatterplot of the variables logPower, Span and Length. Comment on the scatterplots in relation to what it says about the variables, and discuss the shortcomings of this sort of display. Experiment with changing the viewing angles phi and theta.
library( plot3D )
scatter3D (aircraft$logWeight, aircraft$logSpan, aircraft$logLength, phi = 20, theta = 80,
col = NULL, NAcol = "white", breaks = NULL,
colkey = NULL, panel.first = NULL,
clim = NULL, clab = NULL,
bty = "b", CI = NULL, surf = NULL,
add = FALSE, plot = TRUE)
f) The code chunk below makes a parallel coordinate plot of the 6 logged variables, coloured by the value of Period, and with boxplots and suitable transparency. It is arranged so that so that the values are not scaled. Comment on what this figure says about the data, and discuss any shortcomings of the plotting method.
library(GGally)
ggparcoord( aircraft, groupColumn = "Period", columns = 3:8, alpha = 0.1, scale = "globalminmax", boxplot = TRUE ) + theme( legend.position = "bottom")
Q2
a) How many observations belong to each period?
summary( aircraft$Period ) ## 1 2 3
## 216 263 230b) Which years belong to each period?
xtabs( ~ Period + Year, data = aircraft )## Year
## Period 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
## 1 3 4 12 15 8 2 4 3 5 9 5 7 10 8 12 10 7 8 18 18 23 25 0 0 0 0 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 24 22 23 25 14 20 6 25 13 3 7 9
## 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## Year
## Period 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 2 11 3 5 9 9 11 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 16 6 5 14 4 12 13 9 8 15 9 13 7 10 9 10 10 3 9 6 6 8 11 5 3 2 5 1
## Year
## Period 84
## 1 0
## 2 0
## 3 1c) Which variable has the largest interquartile range? Which has the largest median? For which variables are the means larger than 1.04 times the median? (Leave out Period for this question, as it is a factor, not a numeric variable. Don’t do the second part of this question by just looking at the output from say the summary command. Do it by writing a small amount of code that will tell you the answer directly.)
summary( aircraft )## Year Period logPower logSpan logLength logWeight logSpeed
## Min. :14.00 1:216 Min. :1.477 Min. :0.8248 Min. :0.7551 Min. :2.682 Min. :2.021
## 1st Qu.:34.00 2:263 1st Qu.:2.624 1st Qu.:1.0414 1st Qu.:0.9552 1st Qu.:3.378 1st Qu.:2.413
## Median :42.00 3:230 Median :3.006 Median :1.1553 Median :1.0777 Median :3.740 Median :2.611
## Mean :46.27 Mean :3.084 Mean :1.2038 Mean :1.1265 Mean :3.801 Mean :2.637
## 3rd Qu.:61.00 3rd Qu.:3.489 3rd Qu.:1.3444 3rd Qu.:1.2514 3rd Qu.:4.146 3rd Qu.:2.816
## Max. :84.00 Max. :4.903 Max. :1.8457 Max. :1.8782 Max. :5.562 Max. :3.508
## logRange
## Min. :1.903
## 1st Qu.:2.929
## Median :3.146
## Mean :3.172
## 3rd Qu.:3.380
## Max. :4.304sapply( aircraft[,-2], IQR ) ## Year logPower logSpan logLength logWeight logSpeed logRange
## 27.0000000 0.8642686 0.3029996 0.2961883 0.7684576 0.4029415 0.4507923lapply( aircraft[,-2], IQR ) ## $Year
## [1] 27
##
## $logPower
## [1] 0.8642686
##
## $logSpan
## [1] 0.3029996
##
## $logLength
## [1] 0.2961883
##
## $logWeight
## [1] 0.7684576
##
## $logSpeed
## [1] 0.4029415
##
## $logRange
## [1] 0.4507923sapply( aircraft[,-2], median ) ## Year logPower logSpan logLength logWeight logSpeed logRange
## 42.000000 3.006038 1.155336 1.077731 3.740363 2.610660 3.146128sapply( aircraft[,-2], mean) > sapply( aircraft[,-2], median ) ## Year logPower logSpan logLength logWeight logSpeed logRange
## TRUE TRUE TRUE TRUE TRUE TRUE TRUEsapply( aircraft[,-2], mean) > 1.04 * sapply( aircraft[,-2], median ) ## Year logPower logSpan logLength logWeight logSpeed logRange
## TRUE FALSE TRUE TRUE FALSE FALSE FALSEd) Divide the scaled data into the three period groups. For all the data and separately for period 1, give the histograms and density plots of logPower. Compare the results for the complete data with those obtained from period 1. Comment.
aircraft.per1 = filter( aircraft, Period == "1" )
aircraft.per2 = filter( aircraft, Period == "2" )
aircraft.per3 = filter( aircraft, Period == "3" )
ggplot(data = aircraft.per1, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.01) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 1")
ggplot(data = aircraft.per2, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.01) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 2")
ggplot(data = aircraft.per3, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.01) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 3")
ggplot(data = aircraft.per1, mapping = aes(x = logLength, y=..density..)) +
geom_histogram(binwidth=0.05) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 1")
ggplot(data = aircraft.per2, mapping = aes(x = logLength, y=..density..)) +
geom_histogram(binwidth=0.05) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 2")
ggplot(data = aircraft.per3, mapping = aes(x = logLength, y=..density..)) +
geom_histogram(binwidth=0.05) +
geom_density(colour = "blue", alpha=0.5 )+ ggtitle("Period 3")
aircraft.per1 = filter( aircraft, Period == "1" )
aircraft.per2 = filter( aircraft, Period == "2" )
aircraft.per3 = filter( aircraft, Period == "3" )
ggplot(data = aircraft, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.05) +
geom_density(colour = "blue", alpha=0.5 )
ggplot(data = aircraft.per1, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.1) +
geom_density(colour = "blue", alpha=0.5 )
ggplot(data = aircraft.per2, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.1) +
geom_density(colour = "blue", alpha=0.5 )
ggplot(data = aircraft.per3, mapping = aes(x = logPower, y=..density..)) +
geom_histogram(binwidth=0.1) +
geom_density(colour = "blue", alpha=0.5 )
# histograms
ggplot( aircraft, aes( logPower ) ) + geom_histogram() + ggtitle("All data")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per1, aes( logPower ) ) + geom_histogram() + ggtitle("Period 1")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per2, aes( logPower ) ) + geom_histogram() + ggtitle("Period 2")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per3, aes( logPower ) ) + geom_histogram() + ggtitle("Period 3")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per1, aes( logLength ) ) + geom_histogram() + ggtitle("Period 1")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per2, aes( logLength ) ) + geom_histogram() + ggtitle("Period 2")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot( aircraft.per3, aes( logLength ) ) + geom_histogram() + ggtitle("Period 3")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# density plots
ggplot( aircraft, aes( logPower, group = Period, colour = Period ) ) + geom_density( aes( fill = Period ), alpha = 0.4 ) + ggtitle("All data")
# pairwise plots
pairs( aircraft.per1[ , 3:8 ] )
ggpairs( aircraft.per1, columns = 3:8, progress = FALSE )
pairs( aircraft.per2[ , 3:8 ] )
ggpairs( aircraft.per2, columns = 3:8, progress = FALSE )
pairs( aircraft.per3[ , 3:8 ] )
ggpairs( aircraft.per3, columns = 3:8, progress = FALSE )
#
scatter3D (aircraft.per1$logWeight, aircraft.per1$logSpan, aircraft.per1$logLength, phi = 20, theta = 80,
col = NULL, NAcol = "white", breaks = NULL,
colkey = NULL, panel.first = NULL,
clim = NULL, clab = NULL,
bty = "b", CI = NULL, surf = NULL,
add = FALSE, plot = TRUE)
#
# scatter3D (aircraft.per2$logWeight, aircraft.per2$logSpan, aircraft.per2$logLength, phi = 20, theta = 80,
# col = NULL, NAcol = "white", breaks = NULL,
# colkey = NULL, panel.first = NULL,
# clim = NULL, clab = NULL,
# bty = "b", CI = NULL, surf = NULL,
# add = FALSE, plot = TRUE)
#
# scatter3D (aircraft.per3$logWeight, aircraft.per3$logSpan, aircraft.per3$logLength, phi = 20, theta = 80,
# col = NULL, NAcol = "white", breaks = NULL,
# colkey = NULL, panel.first = NULL,
# clim = NULL, clab = NULL,
# bty = "b", CI = NULL, surf = NULL,
# add = FALSE, plot = TRUE)
ggparcoord( aircraft.per1, columns = c(1,3:8), alpha = 0.3, scale = "uniminmax" ) + ggtitle("Period 1")
ggparcoord( aircraft.per2, columns = c(1,3:8), alpha = 0.3, scale = "uniminmax" ) + ggtitle("Period 2")
# ggparcoord( aircraft.per3, columns = c(1,3:8), alpha = 0.3, scale = "uniminmax" ) + ggtitle("Period 3")
#
ggparcoord( aircraft.per1, columns = c(3:8), groupColumn = "Year", alpha = 0.3, scale = "globalminmax", boxplot = TRUE ) + ggtitle("Period 1")
ggparcoord( aircraft.per2, columns = c(3:8), groupColumn = "Year", alpha = 0.3, scale = "globalminmax", boxplot = TRUE ) + ggtitle("Period 2")
# ggparcoord( aircraft.per3, columns = c(3:8), groupColumn = "Year", alpha = 0.3, scale = "globalminmax", boxplot = TRUE ) + ggtitle("Period 3")Q3
a) Construct contour plots of the 2D smoothed histograms for the pair of variables logLength and logWeight, and also for logSpeed and logSpan for all records.
ggplot( aircraft, aes( logPower, logWeight )) + geom_density_2d( ) + geom_point( aes( colour = Period ), alpha = 0.6 ) + theme( legend.position = "bottom") 
ggplot( aircraft, aes( logSpeed, logLength )) + geom_density_2d( )+ geom_point( aes( colour = Period ), alpha = 0.6 ) + theme( legend.position = "bottom") 