### Visualizing Data Using the Functional Boxplot

```
library(fda)
data(growth)
girls=growth$hgtf
age=growth$age
```

The Berkley growth data found in the fda package is an excellent example of functional data. Here we consider each growth curve to be one observation. With data of this type there are a couple of plotting avenues we might take: over plotting or sequential boxplots.

These methods aren’t inherently “incorrect”, but limitations to their interpretability exist especially when encountered with larger data sets. The functional boxplot approach helps visualize these larger data sets while also providing an interpretation familiar to us as box plots are. One can easily see in the output below the shape of the median observation, inner quartile range, and the maximal non-outlying envelope.

The fbplot command is straightforward receiving a matrix of observed values, a vector of corresponding x values, and standard base graphics arguments. The fbplot is based on a calculation of band depth for each observation, which is a measure of central tendency. Output consists of the band depth, identification of the median observation, and any observations that are considered to be outliers.

```
fbplot(girls,x=age,
xlim=c(0,20),ylim=c(50,200),
xlab="Age (Years)",ylab="Height (cm)",
main="Functional Boxplot: Height of Girls")
```

```
## $depth
## girl01 girl02 girl03 girl04 girl05 girl06
## 0.40919952 0.49664683 0.30156556 0.43267735 0.47162485 0.42960596
## girl07 girl08 girl09 girl10 girl11 girl12
## 0.23571944 0.03703704 0.48985032 0.21643448 0.43925407 0.40958274
## girl13 girl14 girl15 girl16 girl17 girl18
## 0.08502401 0.44806812 0.34402967 0.18687022 0.42103988 0.18413133
## girl19 girl20 girl21 girl22 girl23 girl24
## 0.49167061 0.42719393 0.34362390 0.48044453 0.42615698 0.28134510
## girl25 girl26 girl27 girl28 girl29 girl30
## 0.30034264 0.20578323 0.48245644 0.48132369 0.08928451 0.47829738
## girl31 girl32 girl33 girl34 girl35 girl36
## 0.45609883 0.39549943 0.38087509 0.48449652 0.45554090 0.44654652
## girl37 girl38 girl39 girl40 girl41 girl42
## 0.29858434 0.18224341 0.38718131 0.34355628 0.44220712 0.11380492
## girl43 girl44 girl45 girl46 girl47 girl48
## 0.22408196 0.45660603 0.47345641 0.39476680 0.36635784 0.17495097
## girl49 girl50 girl51 girl52 girl53 girl54
## 0.37891391 0.48951218 0.37859832 0.47450463 0.27684791 0.33681049
##
## $outpoint
## [1] 8
##
## $medcurve
## girl02
## 2
```

Additional information can be found in the publication authored by Ying Sun and Marc Genton or in the fda package R documentation.