
\chapter{Breadth of OODA\label{chap1B:Breadth-of-OODA}}

This chapter illustrates the breadth of OODA through relatively brief
overviews of quite diverse applications.

\section{Amplitude and Phase Data Objects\label{sec:Amplitude-and-Phase}}

A\index{data objects!amplitude - phase|(} challenging situation in
FDA\index{Functional Data Analysis|(} is when the curve data objects\index{data objects!curves}
are misaligned, as shown in the top panel of Figure \ref{fig:TIC-spectra}
for a Proteomics data set called the \emph{TIC Curves} here. Many
statistical methods can be strongly impacted by misalignment. An example
of the impact of misalignment on the sample mean in FDA is shown using
the Shifted Betas\index{Shifted Betas data} toy data in Figure \ref{figD-M:ToyShiftsnFRReg}.
A quite different type of impact on PCA\index{Principal Component Analysis|(}
appears in Figure \ref{figH.A:ToyBadPCA}. As noted in \citet{MarronRamsay2014},
FDA approaches to dealing with alignment issues are sometimes called
\emph{\index{curve registration|(}curve registration}, because it
is very useful in situations where the curve data objects are clearly
misaligned. 

There are many approaches to the curve registration challenge, with
an overview provided in the survey paper \citet{MarronRamsay2015}.
Most methods in the area involve tuning parameters that have proven
to be tricky to choose in a fully automatic way, as illustrated in
Figure \ref{figH.H:Comparisons}. This problem has been solved using
the OODA way of thinking, as discussed in Chapter \ref{chapH:FDA-Curve-Registration}.
In particular, that approach is based on unusually deep mathematical
ideas based on the Fisher-Rao metric\index{metric!Fisher-Rao}, which
resulted in a rigorous methodology that is hence fully automatically
useful.

An interesting example of curve registration, from \citet{KochEJS2014}
and \citet{KochEJS2014Rejoinder}, is shown in Figure \ref{fig:TIC-spectra}.
The data objects here are proteomics mass spectrometry profiles from
\citet{Ho2011TICcurves}, a larger study of bio-markers in Acute Myeloid
Leukemia. A detailed description of this data set including a number
of pre-processing steps (such as median smoothing\index{smoothing}
and interpolation to an equally spaced grid) can be found in \citet{KochEJS2014}.
Essentially there are 5 patients, represented as colors, with 3 replicate
curves for each patient, thus 15 curves in all, shown in the top part
of the top panel. Each curve shows Total Ion Counts (TIC), for an
equally spaced grid of 2001 mass to charge ratios (horizontal coordinate).
The TIC curves have many peaks, which correspond to various peptides.
A common goal of mass spectrometry analyses is curve registration,
i.e. finding deformations, sometimes called \emph{warping\index{warping}s}
(intuitively thought of as stretchings and compressings of the horizontal
axis), to properly align the peaks so that they chemically correspond.
In most contexts it is hard to quantitatively assess the performance
of a given registration, but this data set is special because the
locations of several of the actual peptide peaks have been (laboriously)
found for each curve using additional information as detailed in \citet{KochEJS2014}.
These peak locations, for each of the 15 curves, are indicated by
peak numbers (1-14), with colors corresponding to the curves. The
peak numbers are sorted vertically by height of the corresponding
peak and connected with gray line segments to give some visual correspondence\index{correspondence}.
It is hard to see much pattern, showing this to be a challenging curve
registration problem. 

As noted above, there are a number of approaches to this type of data
challenge, with several such analyses of this data set discussed in
\citet{KochEJS2014Rejoinder}. The bottom panel of Figure \ref{fig:TIC-spectra}
shows the results of registration of these same TIC curves using the
Fisher-Rao method proposed in \citet{Srivastava2011FisherRao} and
\citet{KurtekSrivastava2012} (discussed in more detail in Section
\ref{secH.A:Fisher-Rao-Reg}), using only the curves themselves and
not the peak location information. The colored numbers reveal that
this is a particularly challenging problem, because the peaks have
quite different heights across patients. Peak 10 is particularly challenging
as it is quite low for the red patient (especially compared to nearby
very tall peaks), yet is the highest peak for other patients. Note
the alignment is not perfect for every numbered peak, but it is still
of impressively high quality. Roughly comparable quality has been
obtained using a linear registration approach that is integrated with
clustering\index{clustering!integrated with registration} in \citet{Bernardi2014Proteomics},
and by a Bayesian approach in \citet{ChengDryden2014EJS}.

\begin{figure}[H]
\includegraphics[scale=0.9]{Chapter1/OODAbookChp1FigK}

\caption{Top panel contains raw TIC Curves data\index{TIC Curves data} (\ref{chapH:FDA-Curve-Registration}
with a labeling of certain important peaks in the lower part of the
panel. Bottom panel shows a Fisher-Rao registration of the TIC curves.
Numbers under the curves indicate peak locations, showing that the
registration has been mostly quite effective.\label{fig:TIC-spectra}}
\end{figure}
An important point made in the overview of \citet{MarronRamsay2015}
is that curve registration\index{curve registration|)} methods are
useful more generally than simply to align curves. While in some contexts,
such as that of Figure \ref{fig:TIC-spectra}, the phase component
is merely \emph{nuisance variation\index{nuisance variation}} to
be dealt with but of no intrinsic interest, there are many other situations
where the warps themselves represent useful modes of variation\index{modes of variation|(}.
In such contexts it is insightful to consider different types of data
objects for OODA. In particular, \emph{amplitude\index{amplitude!data objects|(}
data objects}, whose variation is contained in the aligned curves,
and\emph{ phase\index{phase!data objects|(} data objects} which are
the warps used to achieve the alignment. Depending on the context
either or both choices of data object can be of primary interest,
or either could represent just nuisance variation.

The notions of amplitude and phase data objects are illustrated using
the \emph{Bimodal Phase Shift} example in Figure \ref{fig:Sim-Warps}.
The upper left panel shows a simulated functional data set, where
every data object (curve) has two peaks and is a multiple of a beta
mixture probability density\index{probability density}. A rainbow
color scheme is used to distinguish the curves, in order of how separated
the peaks are. The peaks have both different heights showing substantial
amplitude variation, and also quite different locations reflecting
strong phase variation. These two types of mode of variation are decomposed
in a useful way by the \index{warping}warping functions shown in
the bottom right panel, computed using the Fisher-Rao method, described
in Section \ref{secH.A:Fisher-Rao-Reg}. The vertical axis is the
same as in the upper left panel. Rescaling that axis using the magenta
warp function\index{warp function}s moves the magenta peaks inwards,
and using the red warp functions moves the red peaks outwards. The
top right panel shows the amplitude data objects, i.e. aligned curves.
A careful look shows that the random peak heights are linearly related
with the left peak being high when the right peak is low. This set
of amplitude data objects consists of just a single one dimensional
mode of variation. The warps in the lower right panel can be thought
of as the phase data objects, although they are not easy to interpret.
Enhanced interpretation of the variation in the phase data objects
comes from the view in the lower left panel. That is an application
of each of the warps to the Fréchet mean\index{Fréchet mean} (discussed
in Section \ref{eqChpE-F:FrechetMin}) template from the Fisher-Rao
calculation, which nicely reflects the one dimensional phase variation.

\begin{figure}[H]
\includegraphics{Chapter1/OODAbookChp1FigLRawCMYK}\includegraphics{Chapter1/OODAbookChp1FigLAmpCMYK} 

\includegraphics{Chapter1/OODAbookChp1FigLPhaseCMYK}\includegraphics{Chapter1/OODAbookChp1FigLWarpCMYK}

\caption{\index{Bimodal Phase Shift data}Bimodal Phase Shift data (top left
panel) showing decomposition into amplitude (top right panel) and
phase (bottom left panel) modes of variation. Decomposition is based
on the warping functions (bottom right panel). Rainbow color scheme
highlights the phase mode, with red for closest peaks through magenta
for farthest peaks.  \label{fig:Sim-Warps}}
\end{figure}
As clearly demonstrated in Figures \ref{figH.A:ToyBadPCA}, \ref{figH.B:ToyBadScores},
\ref{figH.I:ToyVertPCA}, and \ref{figH.K:ToyHorizPCA}, decompositions
of the type shown in Figure \ref{fig:Sim-Warps} can be much more
useful than a standard PCA in FDA, which tends to both mix the amplitude
and phase components, and also to spread the variation of the phase
component over a large number of components, because it is a nonlinear
mode of variation from that viewpoint. As discussed in \citet{MarronRamsay2014,MarronRamsay2015},
amplitude-phase decomposition is useful in many FDA\index{Functional Data Analysis|)}
applications. As noted above, for some of these, such as the TIC data
shown in Figure \ref{fig:TIC-spectra}, the amplitude data objects
are the focus of the analysis, and the phase data objects can be viewed
as nuisance parameters. However in other situations, for example when
analyzing neural spike train data (as discussed in \citet{WuSrivastava2014})
the phase data objects are of primary interest, and the amplitude
data objects are the nuisance component. In still other situations,
both amplitude and phase data objects are important, as is their joint
variation. These include the variation in the AneuRisk65 artery shape
data in \citet{SangalliSecchi2014}, and in the juggling data\index{juggling data}
discussed in \citet{RamsayKurtek2014}.

Figure \ref{fig:Juggling} shows some of the analysis of the \emph{Juggling}
data from \citet{LuEJS2014juggling}. The starting point was positional
recordings of location over time of the hand of a juggler, which were
reduced to time series of acceleration curves, as discussed in \citet{RamsayKurtek2014}.
These traces were cut into cycles and time registered, to obtain the
113 curves shown in the far left panel of Figure \ref{fig:Juggling}.
Thus the data objects in this OODA are time registered 1-d acceleration
curves. Figure 3 of \citet{LuEJS2014juggling} shows a variety of
PCA type scores plots. Most of these seem to indicate a homogeneous
population. The middle left panel of Figure \ref{fig:Juggling} shows
the version based on the method of Principal Nested Spheres\index{Principal Nested Spheres}
(PNS), from \citet{JungPNS2012}. As further described in Sections
\ref{figG.D:PNSintro} and \ref{secH.B:PNSonFisherRao}, PNS makes
special use of the fact that Fisher-Rao warp data objects naturally
lie on a high dimensional sphere. The value added of using this method
which takes the curvature of the sphere properly into account, is
that it shows two clear clusters, which are highlighted using the
graphical technique of \emph{brush\index{brushing}ing}, i.e. visually
separating the cluster through the use of colors and symbols. See
Section \ref{secH.B:PNSonFisherRao} for more discussion (based on
\citet{YuLuMarron2017}) of how and why PNS provides enhanced statistical
analysis of Fisher-Rao phase data objects. The clusters shown in the
center left panel of Figure \ref{fig:Juggling} represent important
underlying structure in the data. This is clear from the two right
hand panels of Figure \ref{fig:Juggling}, which show actual vertical
and horizontal locations of the paths (orthographic projections) corresponding
to these clusters, using the same colors. These are clearly two quite
different types of motions present in the data, which correspond to
``better controlled'' and ``less well controlled'' cycles.

\begin{figure}[H]
\includegraphics[scale=0.4]{Chapter1/OODAbookChp1FigM1}\qquad{}\includegraphics[scale=0.4]{Chapter1/OODAbookChp1FigM2}\qquad{}\includegraphics[scale=0.4]{Chapter1/OODAbookChp1FigM3}\qquad{}\includegraphics[scale=0.4]{Chapter1/OODAbookChp1FigM4}

\caption{Analysis of the Juggling data. Far left panel shows the input acceleration
curves. Center left is the Principal Nested Spheres scatterplot\index{scores!scatterplot},
revealing two distinct clusters, highlighted by brushing. Right panels
verify these clusters represent two different types of cycles.\label{fig:Juggling}}
\end{figure}
Figure \ref{fig:Juggling} uses parts of Figures 2, 3 and 4 from \citet{LuEJS2014juggling}.\index{data objects!amplitude - phase|)}

\section{Tree Structured Data Objects\label{sec:Tree-Structured-Data}}

A\index{data objects!tree structured|(} very different example of
OODA is \emph{\index{tree|(}trees},\index{tree structured data|(}
in the sense of graph\index{graph|(} theory, as data objects. An
interesting data set, where each data object is a representation of
the set of arteries in one person's brain, was collected by \citet{bullitt2002volume,AylwardBullitt2002}.
While a long term goal is to study pathologies, including stroke tendency
or brain cancer, such cases were deliberately screened out of this
data set, to focus on normal variation within the population. Interesting
quantities that are useful for various comparisons below are age and
gender. Section \ref{secI-A:ArterialTrees} gives an overview of various
analytic approaches that feature improving abilities to distinguish
age and gender. 

These data objects were acquired using a modality of Magnetic Resonance
Imaging called Magnetic Resonance Angiography (MRA). This modality
flags motion as white, so the flow of blood through the arteries shows
up very well. This is seen in Figure \ref{fig:MRA} as the white spots,
where the different panels show adjacent horizontal slices of the
3-d image.

\begin{figure}[H]
\includegraphics[scale=0.7]{Chapter1/OODAbookChp1FigG}

\caption{Three adjacent slices of an MRA image for a single subject. Arteries
show up as white dots and curves.\label{fig:MRA}}
\end{figure}
A major contribution of \citet{AylwardBullitt2002} was the development
of a 3-d tube tracking algorithm which was used to generate reconstruction
of a given artery tree. At this point the data object is the union
of many small spheres, whose centers follow the central curve of each
arterial branch, and whose radii are the branch radius at that point.
This tree representation, from the MRA shown in Figure \ref{fig:MRA}
can be seen in Figure \ref{fig:ArtTree-3views}. The three panels
show different rotations of the same set of arteries. The left panel
is a fairly large rotation and the right panel is a small one, with
the closest vessels moved to the left and right respectively.

\begin{figure}[H]
\includegraphics[scale=0.7]{Chapter1/OODAbookChp1FigH}

\caption{Three views of the arterial tree for the subject in Figure \ref{fig:MRA},
showing the 3-d structure through somewhat different rotations.\label{fig:ArtTree-3views}}
\end{figure}
Such data object representations have been computed for approximately
100 people, in the \index{Brain Artery data}Brain Artery data set.
For example, three more of these for three different subjects are
shown in Figure \ref{fig:3ArtTrees}. The original study was a little
larger, but some were deleted due to MRA acquisition problems.

\begin{figure}[H]
\includegraphics[scale=0.6]{Chapter1/OODAbookChp1FigI}

\caption{Artery tree data objects for three additional subjects.\label{fig:3ArtTrees}}
\end{figure}
Data objects of this type present major challenges to doing statistical
analysis. For example, it is really not clear how to define even the
sample mean of such a set of objects. Understanding variation about
the mean, e.g. as done by PCA in Section \ref{sec:Curves-as-Data},
is a further challenge. A number of approaches to this are discussed
in Chapter \ref{chapI:Tree-Structured-Data}, which studies these
trees\index{tree structured data} in the more general context of
graph\index{graph|)}s as data objects.\index{data objects!tree structured|)}

\section{Sounds as Data Objects\label{sec:Sounds-as-Data}}

Another example of OODA is \emph{sounds} as data objects\index{data objects!sounds|(},
which have been studied in a particularly deep way in a series of
papers analyzing human speech based on digital recordings. \citet{HadjipantelisAston2012,HadjipantelisAston2015}
investigated Mandarin Chinese using a mixed effect model to develop
relations between dialects which were consistent with linguistic ideas.
\citet{ColemanAstonPigoli2015} used these methods to extrapolate
back in time to estimate how archaic languages may have sounded. \citet{PigoliAston2015}
analyze the relationships between modern romance languages, yielding
insights well beyond those available from classical textual linguistic
analysis (such as studied in Section \ref{secI-B:NetworkData}). In
addition a transformation\index{transformation} is proposed that
provides an estimated reconstruction of how a given speaker would
sound speaking a different language. \citet{TavakoliPigoliAston2016}
combined these analyses with spatial smoothing\index{smoothing}\index{}
to produce a dialectic map of the United Kingdom. \citet{ShiersAston2017}
developed a sound based evolutionary tree for romance languages and
dialects. 

A typical first step in those analyses is to decompose the raw digital
recording of the sound into a spectrogram, which is a moving window
version of the Fourier\index{Fourier!transform} transform, giving
a frequency representation in time, as shown in Figure \ref{fig:Spectrogram},
from the study of \citet{PigoliAston2015}, kindly provided by Davide
Pigoli. The top panel is the raw recording of one person saying the
word ``deux'' (two) in French.

\begin{figure}[H]
\includegraphics[scale=0.25]{Chapter1/OODAbookChp1FigSSpectrogram}

\caption{Summarization of raw recording of a human speech sound of ``deux''
in French, top panel, into a corresponding spectrogram showing time
and frequency information with color coding height, shown in the bottom
panel. \label{fig:Spectrogram}}
\end{figure}
Frequently, the focus is on human speech from the viewpoint that aspects
such as pitch and timing are nuisances to be removed from the analysis.
For that choice of data objects, those effects are removed by reducing
the spectrogram to appropriately defined time and frequency \index{data objects!covariance matrices}covariance
matrices (which are finite representations of covariance functions).
Mean vectors also sometimes play an important role. Color \index{heat-map}heat-map
representation summaries (as discussed in Section \ref{secF.A:Heatmap-Views})
of five covariance matrices (with entries colored according to the
bars on the right, all using the same scale\index{scale} to facilitate
comparison) from \citet{PigoliAston2015} are shown in Figure \ref{fig:Sound-Covs},
also from Davide Pigoli. For each language these summaries are based
on aggregating sounds for the spoken digits (1-10). An exploratory\index{exploratory analysis}
visual comparison of these suggest some similarities (e.g. American
and Castilian Spanish) and also some stark contrasts such as Portuguese
from the others. Confirmatory analysis\index{confirmatory analysis!sounds}
of these points and a number of others using permutation testing methods
can be found in \citet{PigoliAston2015}.

\begin{figure}[H]
\includegraphics[scale=0.25]{Chapter1/OODAbookChp1FigTfeq_covariances}

\caption{Covariance representation summaries of speech sounds from five different
languages/dialects. Note strong differences between them, with potentially
interesting historical and geographical connections.\label{fig:Sound-Covs}}
\end{figure}
In the overall area of sounds as data objects, there is another interesting
parallel to the phenomenon noted in Section \ref{sec:Amplitude-and-Phase},
that depending on the context either phase\index{phase!data objects|)}
or amplitude\index{amplitude!data objects|)} data objects could be
the major focus of the analysis with the other considered to be nuisance
variation\index{nuisance variation}. In particular, the above work
focused on a particular type of analysis of sounds as data objects,
where the goal was to study human speech, by a variety of speakers.
As the human brain does when parsing speech, they deliberately chose
data objects which focused on aspects of the sound that are about
meaning of the words, which means generally treating issues such as
pitch, volume and timing as nuisances, to be mathematically ignored.
This a strong contrast with the area of Music Data Analysis, which
has been deeply studied in \citet{weihs2016music} where timing, volume
and pitch are actually of keen interest as the data objects.\index{data objects!sounds|)}

Statistical analysis of covariance matrices as data objects\index{data objects!covariance matrices}
is particularly challenging, and discussed further in Section \ref{secG-B:CovMatDataObjects}.

\section{Images as Data Objects\label{sec:Images-as-Data} }

The\index{data objects!images|(} field of image analysis is very
large. Statistics has traditionally appeared there in several ways.
Early work, with famous papers including \citet{GemanGeman1984} and
\citet{Besag1986}, tended to focus on aspects of mostly a single
image, with tasks such as denoising\index{denoising}, segmentation\index{segmentation}
and registration being predominant. However, those fields are now
relatively mature, so a currently more important role for statistical
ideas comes at the population level which yields a very rich source
of potential data objects. For example, the shapes studied in Section
\ref{sec:Shapes-as-Data} and the tree\index{tree|)}s considered
in Section \ref{sec:Tree-Structured-Data} are two types of data objects
derived from images. 

But in other situations the images themselves can be treated as data
objects. An example of this is the \emph{Faces} data\emph{\index{Faces data}}
that appears in \citet{Benito2017distance}, which studies a data
set of $n=108$ images (actually $248\times186$ gray level photographs)
of students from the University of Carlos III in Madrid. There is
quite a lot of variation among the faces, yet the human perceptual
system is good at distinguishing gender. In that paper, male vs. female
classification of these data is carefully studied. As discussed in
Section \ref{secD3:Preprocess-Register} of \citet{Benito2017distance},
manual affine registration was used to put each face into a common
location in its image. Then the gray level pixel\index{pixel}s of
the images were \emph{vectorize\index{vectorize}d} (by stacking columns,
an operation sometimes denoted as \emph{vec}) into a single long vector,
and various classification methods were used to try to understand
the difference between males and females. \emph{Classification\index{classification!male vs. female}},
also sometimes called \emph{discrimination}\index{discrimination},
is an important OODA topic discussed in Section \ref{chapJ:Classification}.
The classification methods used on this face data set were linear
methods, as those yield the best interpretation of the results.

Particularly good results came from \emph{\index{Distance Weighted Discrimination}Distance
Weighted Discrimination} (DWD) (proposed by \citet{MarronToddAhn2007}
and studied here in Section \ref{secJ-D:DWD}) as shown in Figure
\ref{fig:Faces-DWD}. DWD is discussed in more detail and compared
with other classification methods in Chapter \ref{chapJ:Classification}.
The lower panel of Figure \ref{fig:Faces-DWD} shows the DWD scores\index{scores!DWD},
i.e. the projections of the data onto the DWD separation direction
(the normal vector to the DWD separating hyperplane) using a format
similar to that of the right panel of Figure \ref{fig:Mortality-PC1}.
The red plus signs correspond to the females and the blue circles
are the males, which are completely separable using DWD. Also shown
are three kernel density estimates\index{kernel density estimate}.
That for the full population appears in black (mostly underneath the
others). Female and Male sub-densities (i.e. rescaled according to
sub-sample size) are shown as and red and blue respectively. The top
panel gives insight into what DWD is doing with the images, by showing
a representative set of 8 reconstructions (i.e. the vectors are converted
back into an image) from 8 equally spaced points (locations shown
as the 8 equally spaced black bars in the bottom panel) along the
DWD separating vector.

\begin{figure}[H]
\includegraphics[scale=0.4]{Chapter1/OODAbookChp1FigV}

\caption{Results of DWD discrimination between males and females in the Faces
data. Bottom panel shows distribution of DWD scores. Top panel contains
8 reconstructions of faces, corresponding to the 8 points along the
DWD separating vector shown as vertical bars in the bottom. Shows
clear insight as to how DWD separates males from females.\label{fig:Faces-DWD}}
\end{figure}
The array of faces in the top panel is quite compelling. They look
clearly very female on the left side, quite androgynous in the middle,
and clearly male on the right. Also apparent in perhaps the second
and third panels from either end is the idea from \citet{LangloisRoggman1990attractive}
that average faces tend to be more beautiful. In addition, note that
farther to the right corresponds to stronger masculinity. Note that
this collection of faces represents yet another type of mode of variation\index{modes of variation|)},
which focuses on the female - male differences, instead of on maximal
variation as in the PCA\index{Principal Component Analysis|)} analysis
of the Spanish Mortality data discussed in Section \ref{sec:Curves-as-Data}.

Another example of images as data objects is the Cornea Curvature
data\index{Cornea Curvature data} studied in Section \ref{secO.B.A:CorneaData}.
That analysis uses a much different data object representation.\index{data objects!images|)}
