The point is that my line of business requires travel, and sometimes that is a lot of the time, like say almost all of last year. Inevitable comparisons to George Clooney's character in Up in the Air were made (ironically I started to read that book, then left it on a plane in a seatback pocket), requests about favours involving duty free, and of course many observations and gently probing questions about frequent flier miles (FYI I've got more than most people, but a lot less than the entrepreneur I sat next to one time, who claimed to have close to 3 million).
But I digress.
In my case this means flying Delta.
So I happened to notice in one of my many visits to Delta's website that they have data on all of their aircraft in a certain site section. I thought this would be an interesting data set on which to do some analysis, as it has both quantitative and qualitative information and is relatively complex. What can we say about the different aircraft in Delta's fleet, coming at it with 'fresh eyes'? Which planes are similar? Which are dissimilar?
Aircraft data card from Delta.com
The data set comprises 33 variables on 44 aircraft taken from Delta.com, including both quantitative measures on attributes like cruising speed, accommodation and range in miles, as well as categorical data on, say, whether a particular aircraft has Wi-Fi or video. These binary categorical variables were transformed into quantitative variables by assigning them values of either 1 or 0, for yes or no respectively.
data <- read.csv(file="delta.csv", header=T, sep=",", row.names=1) # scatterplot matrix of intermediary (size/non-categorical) variables plot(data[,16:22])
We can see that there are pretty strong positive correlations between all these variables, as all of them are related to the aircraft's overall size. Remarkably there is an almost perfectly linear relationship between wingspan and tail height, which perhaps is related to some principle of aeronautical engineering of which I am unaware.
The exception here is the variable right in the middle which is the number of engines. There is one lone outlier [Boeing 747-400 (74S)] which has four, while all the other aircraft have two. In this way the engines variable is really more like a categorical variable, but we shall as the analysis progresses that this is not really important, as there are other variables which more strongly discern the aircraft from one another than this.
How do we easier visualize a high-dimensional data set like this one? By using a dimensionality reduction technique like principal components analysis.
Principal Components Analysis
Next let's say I know nothing about dimensionality reduction techniques and just naively apply principle components to the data in R:
# Naively apply principal components analysis to raw data and plot pc <- princomp(data) plot(pc)
Taking that approach we can see that the first principal component has a standard deviation of around 2200 and accounts for over 99.8% of the variance in the data. Looking at the first column of loadings, we see that the first principle component is just the range in miles.
# First component dominates greatly. What are the loadings? summary(pc) # 1 component has > 99% variance loadings(pc) # Can see all variance is in the range in miles
Importance of components:
Comp.1 Comp.2 Comp.3 Comp.4
Standard deviation 2259.2372556 6.907940e+01 2.871764e+01 2.259929e+01
Proportion of Variance 0.9987016 9.337038e-04 1.613651e-04 9.993131e-05
Cumulative Proportion 0.9987016 9.996353e-01 9.997966e-01 9.998966e-01
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
Seat.Width..Club. -0.144 -0.110
Seat.Pitch..Club. -0.327 -0.248 0.189
Seat.Width..First.Class. 0.250 -0.160 -0.156 0.136
Seat.Pitch..First.Class. 0.515 -0.110 -0.386 0.112 -0.130 0.183
Seats..First.Class. 0.258 -0.124 -0.307 -0.109 0.160 0.149
Seat.Width..Business. -0.154 0.142 -0.108
Seat.Pitch..Business. -0.514 0.446 -0.298 0.154 -0.172 0.379
Seats..Business. -0.225 0.187
Seat.Width..Eco.Comfort. 0.285 -0.224
Seat.Pitch..Eco.Comfort. 0.159 0.544 -0.442
Seats..Eco.Comfort. 0.200 -0.160
Seat.Width..Economy. 0.125 0.110
Seat.Pitch..Economy. 0.227 0.190 -0.130
Seats..Economy. 0.597 -0.136 0.345 -0.165 0.168
Accommodation 0.697 -0.104 0.233
Cruising.Speed..mph. 0.463 0.809 0.289 -0.144 0.115
Wingspan..ft. 0.215 0.103 -0.316 -0.357 -0.466 -0.665
Tail.Height..ft. -0.100 -0.187
Length..ft. 0.275 0.118 -0.318 0.467 0.582 -0.418
This is because the scale of the different variables in the data set is quite variable; we can see this by plotting the variance of the different columns in the data frame (regular scaling on the left, logarithmic on the right):
# verify by plotting variance of columns mar <- par()$mar par(mar=mar+c(0,5,0,0)) barplot(sapply(data, var), horiz=T, las=1, cex.names=0.8) barplot(sapply(data, var), horiz=T, las=1, cex.names=0.8, log='x') par(mar=mar)
We correct for this by scaling the data using the scale() function. We can then verify that the variances across the different variables are equal so that when we apply principal components one variable does not dominate.
# Scale data2 <- data.frame(scale(data)) # Verify variance is uniform plot(sapply(data2, var))
|After applying the scale() function the variance is now constant across variables|
Now we can apply principal components to the scaled data. Note that this can also be done automatically in call to the prcomp() function by setting the parameter scale=TRUE. Now we see a result which is more along the lines of something we would expect:
# Proceed with principal components pc <- princomp(data2) plot(pc) plot(pc, type='l') summary(pc) # 4 components is both 'elbow' and explains >85% variance
Great, so now we're in business. There are various rules of thumb for selecting the number of principal components to retain in an analysis of this type, two of which I've read about are:
- Pick the number of components which explain 85% or greater of the variation
- Use the 'elbow' method of the scree plot (on right)
# Get principal component vectors using prcomp instead of princomp pc <- prcomp(data2) # First for principal components comp <- data.frame(pc$x[,1:4]) # Plot plot(comp, pch=16, col=rgb(0,0,0,0.5))
So what were are looking at here are twelve 2-D projections of data which are in a 4-D space. You can see there's a clear outlier in all the dimensions, as well as some bunching together in the different projections.
library(rgl) # Multi 3D plot plot3d(comp$PC1, comp$PC2, comp$PC3) plot3d(comp$PC1, comp$PC3, comp$PC4)
# Determine number of clusters wss <- (nrow(mydata)-1)*sum(apply(mydata,2,var)) for (i in 2:15) wss[i] <- sum(kmeans(mydata, centers=i)$withinss) plot(1:15, wss, type="b", xlab="Number of Clusters", ylab="Within groups sum of squares")
|Clustering without the nstart parameter can lead to variable results for each run|
|Clustering with the nstart and iter.max parameters leads to consistent results, allowing proper interpretation of the scree plot|
# From scree plot elbow occurs at k = 4 # Apply k-means with k=4 k <- kmeans(comp, 4, nstart=25, iter.max=1000) library(RColorBrewer) library(scales) palette(alpha(brewer.pal(9,'Set1'), 0.5)) plot(comp, col=k$clust, pch=16)
# 3D plot plot3d(comp$PC1, comp$PC2, comp$PC3, col=k$clust) plot3d(comp$PC1, comp$PC3, comp$PC4, col=k$clust)
# Cluster sizes sort(table(k$clust)) clust <- names(sort(table(k$clust))) # First cluster row.names(data[k$clust==clust,]) # Second Cluster row.names(data[k$clust==clust,]) # Third Cluster row.names(data[k$clust==clust,]) # Fourth Cluster row.names(data[k$clust==clust,])
 "CRJ 100/200 Pinnacle/SkyWest" "CRJ 100/200 ExpressJet"
 "E120" "ERJ-145"
 "Airbus A330-200" "Airbus A330-200 (3L2)"
 "Airbus A330-200 (3L3)" "Airbus A330-300"
 "Boeing 747-400 (74S)" "Boeing 757-200 (75E)"
 "Boeing 757-200 (75X)" "Boeing 767-300 (76G)"
 "Boeing 767-300 (76L)" "Boeing 767-300 (76T)"
 "Boeing 767-300 (76Z V.1)" "Boeing 767-300 (76Z V.2)"
 "Boeing 767-400 (76D)" "Boeing 777-200ER"
 "Boeing 777-200LR"
 "Airbus A319" "Airbus A320" "Airbus A320 32-R"
 "Boeing 717" "Boeing 737-700 (73W)" "Boeing 737-800 (738)"
 "Boeing 737-800 (73H)" "Boeing 737-900ER (739)" "Boeing 757-200 (75A)"
 "Boeing 757-200 (75M)" "Boeing 757-200 (75N)" "Boeing 757-200 (757)"
 "Boeing 757-200 (75V)" "Boeing 757-300" "Boeing 767-300 (76P)"
 "Boeing 767-300 (76Q)" "Boeing 767-300 (76U)" "CRJ 700"
 "CRJ 900" "E170" "E175"
 "MD-88" "MD-90" "MD-DC9-50"
Ahhh, that's the way fly (some day, some day...). This is apparently the plane professional sports teams and the American military often charter to fly - this article in the Sydney Morning Herald has more details.
|Top: CRJ100/200. Bottom left: Embraer E120. Bottom right: Embraer ERJ-145.|
I've flown many times in the venerable CRJ 100/200 series planes, in which I can assure you there is only economy seating, and which I like to affectionately refer to as "little metal tubes of suffering."
These are split into two clusters, which seem to again divide the planes approximately by size (both physical and accommodation), though there is crossover in the Boeing craft.
# Compare accommodation by cluster in boxplot boxplot(data$Accommodation ~ k$cluster, xlab='Cluster', ylab='Accommodation', main='Plane Accommodation by Cluster')
# Compare presence of seat classes in largest clusters data[k$clust==clust,30:33] data[k$clust==clust,30:33]
|Airbus A330-200 (3L2)||0||1||1||1|
|Airbus A330-200 (3L3)||0||1||1||1|
|Boeing 747-400 (74S)||0||1||1||1|
|Boeing 757-200 (75E)||0||1||1||1|
|Boeing 757-200 (75X)||0||1||1||1|
|Boeing 767-300 (76G)||0||1||1||1|
|Boeing 767-300 (76L)||0||1||1||1|
|Boeing 767-300 (76T)||0||1||1||1|
|Boeing 767-300 (76Z V.1)||0||1||1||1|
|Boeing 767-300 (76Z V.2)||0||1||1||1|
|Boeing 767-400 (76D)||0||1||1||1|
|Airbus A320 32-R||1||0||1||1|
|Boeing 737-700 (73W)||1||0||1||1|
|Boeing 737-800 (738)||1||0||1||1|
|Boeing 737-800 (73H)||1||0||1||1|
|Boeing 737-900ER (739)||1||0||1||1|
|Boeing 757-200 (75A)||1||0||1||1|
|Boeing 757-200 (75M)||1||0||1||1|
|Boeing 757-200 (75N)||1||0||1||1|
|Boeing 757-200 (757)||1||0||1||1|
|Boeing 757-200 (75V)||1||0||1||1|
|Boeing 767-300 (76P)||1||0||1||1|
|Boeing 767-300 (76Q)||1||0||1||1|
|Boeing 767-300 (76U)||0||1||1||1|
Looking at the raw data, the difference I can ascertain between the largest two clusters is that all the aircraft in the one have first class seating, whereas all the planes in the other have business class instead [the one exception being the Boeing 767-300 (76U)].
If I did this again, I would structure the data differently and see what relationships such analysis could draw out using only select parts of the data (e.g. aircraft measurements only). The interesting lesson here is that it when using techniques like dimensionality reduction and clustering it is not only important to be mindful of applying them correctly, but also what variables are in your data set and how they are represented.
References & Resources