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\author{Hannah Frick\\University College London \And
	Ioannis Kosmidis\\University of Warwick \\ The Alan Turing Institute}
\Plainauthor{Hannah Frick, Ioannis Kosmidis}

\title{\pkg{trackeR}: Infrastructure for Running, Cycling and Swimming Data from
  GPS-Enabled Tracking Devices in \proglang{R}}
\Plaintitle{trackeR: Infrastructure for Running, Cycling and Swimming Data from
  GPS-Enabled Tracking Devices in R}
\Shorttitle{\pkg{trackeR}: Infrastructure for Running, Cycling and Swimming Data in
\proglang{R}}

\Keywords{sports, tracking, work capacity, running, cycling, distribution profiles}
\Plainkeywords{sports, tracking, work capacity, running, cycling, distribution profiles}


\Abstract{

  {\color{blue} This introduction to the \proglang{R} package
    \pkg{trackeR} is a modified version of
    \cite{trackeR:Frick+Kosmidis:2017}, published in the \emph{Journal
      of Statistical Software}. The majority of changes and additions
    in the text are due to work by Ioannis Kosmidis, which involved
    the complete rewrite of most of the \pkg{trackeR} codebase at
    version 1.1.0, in order to make the package faster across the
    board and include support of all running, cycling and swimming and
    sport-specific units and variables, fix bugs, and implement new
    features and enhancements.}

  The use of GPS-enabled tracking devices and heart rate
  monitors is becoming increasingly common in sports and fitness
  activities. The \pkg{trackeR} package aims to fill the gap
  between the routine collection of data from such devices and their
  analyses in \proglang{R}.
  The package provides methods to import tracking data into data structures
  which preserve units of measurement and are organised in sessions.
  The package implements core infrastructure for
  relevant summaries and visualisations, as well as support for handling units
  of measurement. There are also methods for relevant analytic tools such
  as time spent in zones, work capacity above critical power (known as
  $W'$), and distribution and concentration profiles. A case study
  illustrates how the latter can be used to summarise the information
  from training sessions and use it in more advanced statistical
  analyses.}

\Address{
  Hannah Frick\\
  Department of Statistical Science\\
  University College London\\
  Gower Street\\
  London, WC1E 6BT\\
  United Kingdom\\
  E-mail: \email{hannah.frick@gmail.com} \\
  URL: \url{http://www.ucl.ac.uk/~ucakhfr/} \bigskip \\
  %
  Ioannis Kosmidis \\
  Department of Statistics \\
  University of Warwick \\
  Gibbet Hill Road \\
  Coventry, CV4 7AL \\
  United Kingdom \\ \smallskip
  E-mail: \email{ioannis.kosmidis@warwick.ac.uk} \\
  URL: \url{http://www.ikosmidis.com}
}

%% Sweave/vignette information and metadata
%% need no \usepackage{Sweave}
\SweaveOpts{engine = R, eps = FALSE, keep.source = TRUE}
%\VignetteIndexEntry{Infrastructure for Running, Cycling and Swimming Data}
%\VignetteDepends{trackeR}
%\VignetteKeywords{sports, tracking, W', running, cycling, swimming distribution profiles, R}
%\VignettePackage{trackeR}

<<preliminaries, echo=FALSE, results=hide>>=
options(width = 70, prompt = "R> ", continue = "+  ", useFancyQuotes = FALSE)
library("trackeR")
library("ggplot2")
data("runs", package = "trackeR")
set.seed(403)
cache <- FALSE
@

\begin{document}

\section{Introduction} \label{sec:intro}

%% data collection: what/how and why
Recent technological advances allow the collection of detailed data on
fitness activities and on multiple aspects of training and competition
in professional sport. The focus of this paper is on data collected by
GPS-enabled tracking devices and heart rate monitors. Such devices are
routinely used in fitness activities such as running, cycling, and
swimming and also during training in sports like field hockey and
football. Basic questions associated with tracking data include how
often, much, or hard an individual or a group trains, and a more
advanced outlook tries to explain the impact of training on athlete
physiology or performance.

%% tools: commercial, GC
Tools for basic analytics are usually offered by the manufacturers of the
tracking devices, such as Garmin, Polar, and Catapult, and through a wide range
of applications for devices
such as smartphones and smartwatches, e.g., Strava Running and Cycling GPS,
Endomondo -- Running \& Walking, and Runtastic Running GPS Tracker.
A notable open-source effort is Golden Cheetah
(\url{http://www.goldencheetah.org}), which has now, perhaps, become
the gold standard in terms of facilities for importing tracking data
from cycling activities and for associated analytics. However,
Golden Cheetah is not designed to offer general flexibility in the statistical
analysis of such sports tracking data.

%% tools: R in general
The \proglang{R} system for statistical computing \citep{trackeR:R}
with its ecosystem of add-on packages provides a wide range of
possibilities for the handling and analysis of tracking data.

%% other packages for trajectory data
GPS-enabled tracking devices typically record irregularly sampled
spatio-temporal data. Infrastructure for such data is provided in the
\pkg{trajectories} package \citep{trackeR:pkg:trajectories},
which is developed around the \class{STIDF} class of the \pkg{spacetime} package
\citep{trackeR:Pebesma:2012}. However, the \class{STIDF} class does not
accommodate missing values in positional or temporal information. Since this is
commonly the case in data from GPS-enabled tracking devices (e.g., sequences of
missing values in the positional data because the GPS signal is temporarily
lost), a different approach is taken in \pkg{trackeR}
(see Section~\ref{sec:trackeRdata}).
%
Other packages that offer tools for spatio-temporal data include
\pkg{adehabtitatLT} \citep{trackeR:Calenge:2006}, \pkg{trip}
\citep{trackeR:pkg:trip} and \pkg{move} \citep{trackeR:pkg:move}. The main focus
of those packages is on animal tracking, e.g., estimation of habitat choices,
and they are not directly suitable for tracking the various aspects of athlete
activity.


%% Sports-related packages in \proglang{R}
Despite the wide range of \proglang{R} packages available, there is
only a handful of packages specific to sport data and their analysis.
The available packages focus on topics such as sports management
\citep[\pkg{RcmdrPlugin.SM},][]{trackeR:pkg:RcmdrPlugin.SM}, ranking
sports teams \citep[\pkg{mvglmmRank},][]{trackeR:pkg:mvglmmRank}, and
accessing betting odds
\citep[\pkg{pinnacle.API},][]{trackeR:pkg:pinnacle.API}. \linebreak \pkg{SportsAnalytics}
is a package that focuses on the analysis of performance data, and currently
 offers only ``a selection of data sets, functions to fetch sports data,
 examples, and demos'' \citep{trackeR:pkg:SportsAnalytics}.
%
The \pkg{cycleRtools} package \citep{trackeR:pkg:cycleRtools} provides
functionality to import cycling data into \proglang{R} as well as
tools for cycling-specific, descriptive analyses.

%% trackeR
The \pkg{trackeR} package aims to fill the gap between the routine
collection of data from GPS-enabled tracking devices and the analyses
of such data within the \proglang{R} ecosystem. The package provides utilities
to import sports data from GPS-enabled devices, and, after careful processing,
organises them in data objects which are organised in separate sessions/workouts
and carry information about the units of measurement (e.g., distance and speed
units) as well as of any data operations that have been carried out (e.g., smoothing).
The package also implements core infrastructure for the handling of measurement units
and for summarising and visualising tracking data. It also provides functionality
for calculating time in zones \citep[e.g.,][]{trackeR:Seiler+Kjerland:2006},
work capacity $W'$ \citep{trackeR:Skiba+Chidnok+Vanhatalo:2012}, and distribution
and concentration profiles \citep{trackeR:Kosmidis+Passfield:2015}, including
a few methods for the analysis of these profiles.

%% paper structure
Section~\ref{sec:pkgstructure} gives an overview of the package and
introduces the basic objects and the methods that apply to
them. Section~\ref{sec:import} describes the importing utilities, and
Section~\ref{sec:trackeRdata} details the structure and construction
of the \class{trackeRdata} object, which is at the core of
\pkg{trackeR}. Section~\ref{sec:summaries} is devoted to the
calculation of relevant summaries (time in zones, work capacity,
distribution and concentration profiles) and the corresponding methods
for visualisation. Section~\ref{sec:units} and
Section~\ref{sec:smoothing} focus on basic methods for unit
manipulation as well as smoothing and thresholding. The case study in
Section~\ref{sec:casestudy} investigates the key features in
\Sexpr{nsessions(runs)} sessions through a functional principal
components analysis \citep[e.g.,][]{trackeR:Ramsay+Silverman:2005} on
the concentration profiles for speed.

\newpage

\section{Package structure}\label{sec:pkgstructure}

Figures~\ref{fig:pkgstructure:read} and~\ref{fig:pkgstructure:rest} show a
schematic overview of the package structure, split into two parts for reading
data and further operations. Squared boxes indicate objects of a particular
class, diamonds indicate files of a particular format,
and boxes with rounded corners represent methods that apply to those
objects. The respective class and method names are given in the boxes.
An arrow from an object/file type to a method indicates that the method
applies to objects of the respective class; an arrow from a method to
an object indicates that the method outputs objects of the respective
class. A bi-directional arrow between a method and an object indicates
that the method's input and output are of the same class, such as the
method \fct{threshold} and objects of class
\class{trackeRdata}. Arrows to or from groups of boxes apply to
each box in the group. For example, the method \fct{changeUnits}
applies to objects of classes \class{trackeRdataZones},
\class{trackeRdataSummary}, \class{trackeRWprime},
\class{distrProfile}, and \class{conProfile}.
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  \caption{Package structure - Functionality to read tracking data.}
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  }
  \caption{Package structure - Functionality to analyse tracking data.}
  \label{fig:pkgstructure:rest}
\end{figure}
%

Data in various formats are imported and stored in the central data
object of \linebreak class~\class{trackeRdata} from which summaries
for descriptive purposes or further analyses can be derived. Methods
for visualisation and data handling are available for data objects and
summary objects.  A list of all functionality is provided in
Tables~\ref{tab:pkg:functions:1} and~\ref{tab:pkg:functions:2}. For
convenience, most camel-case function names in
Tables~\ref{tab:pkg:functions:1} and~\ref{tab:pkg:functions:2} are
aliased to function names were words are separated by underscores,
e.g.~\fct{concentrationProfile} and \fct{concentration\_profile},
and \fct{readContainer} and \fct{read\_container}.


\begin{table}[p]
  \centering
  \begin{tabular}{p{3cm} p{4cm} p{6.5cm}}
    \hline
    Function & Class & Description \\
    \hline
    \fct{readTCX} & TCX file & read TCX file \\
    \fct{readGPX} & GPX file & read GPX file \\
    \fct{readDB3} & DB3 file (SQLite) & read DB3 file \\
    \fct{readJSON} & Golden Cheetah's JSON file & read JSON file \\
    \fct{readContainer} & TCX/GPX/DB3/JSON file & read a TCX/GPX/DB3/JSON file \\
    \fct{readDirectory} & TCX/GPX/DB3/JSON files & read all TCX/GPX/DB3/JSON files in a
    \linebreak directory \\
    \fct{trackeRdata} & \class{data.frame} & construct a \class{trackeRdata}
    object \\
    %
    \fct{c} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata} & combine sessions \\
    \fct{sort} & \class{trackeRdata} & sort sessions by start time \\
    \fct{unique} & \class{trackeRdata} & extract unique sessions \\
    \code{[]} & \class{trackeRdata} & subset sessions \\
    % append
    %
    \fct{plot} & \class{trackeRdata} & plot session profiles \\
    \fct{plotRoute} & \class{trackeRdata} & plot route on a static map \\
    \fct{leafletRoute} & \class{trackeRdata} & plot route on an interactive map \\
    \fct{threshold} & \class{trackeRdata} & apply lower and upper bounds on data range \\
    \fct{smoother} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata} & smooth data by applying a summary
    function such as mean or median to a window \\
    %
    \fct{getUnits} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata}, \class{trackeRdataSummary}, \class{trackeRdataZones}, \class{trackeRWprime} & access units of measurement \\
    \fct{changeUnits} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata}, \class{trackeRdataSummary}, \class{trackeRdataZones}, \class{trackeRWprime}  & change units of measurement \\
    \fct{nsessions} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata}, \class{trackeRdataSummary}, \class{trackeRdataZones}, \class{trackeRWprime} & number of sessions \\
    \fct{fortify} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata}, \class{trackeRdataSummary}, \class{trackeRWprime} & convert object into a data frame for plotting \\
    %
    \hline
  \end{tabular}
  \caption{Functions available in the \pkg{trackeR} package (part 1).}
  \label{tab:pkg:functions:1}
\end{table}

\begin{table}[p]
  \centering
  \begin{tabular}{p{4.1cm} p{3.9cm} p{5.5cm}}
    \hline
    Function & Class & Description \\
    \hline
     \fct{summary} & \class{trackeRdata} & summarise sessions \\
    \fct{print} & \class{trackeRdata}, \class{trackeRdataSummary} & print sessions summaries \\
    \fct{plot} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata}, \class{trackeRdataSummary}, \class{trackeRdataZones}, \class{trackeRfpca}, \class{trackeRWprime} \\
    \fct{timeline} & \class{trackeRdata}, \class{trackeRdataSummary} & plot timeline summary \\
    %
    \fct{zones} & \class{trackeRdata} & time spent in zones \\
    %
     \fct{Wprime} & \class{trackeRdata} & calculate \emph{$W'$ balance} or
    \emph{$W'$ expended} \\
    \fct{plot} & \class{trackeRWprime} & plot \emph{$W'$ balance} or \emph{$W'$
      expended} \\
    \fct{session\_times} & \class{trackeRdata},
                          \class{trackeRdataSummary} & return the start and
                                                       end date and time of its
                                                       session \\
    \fct{session\_duration} & \class{trackeRdata},
                             \class{trackeRdataSummary} & return the duration
                                                          of each session \\
    \fct{get\_sport} & \class{conProfile}, \class{distrProfile},
                       \class{trackeRdata},
                       \class{trackeRdataSummary},
                       \class{trackeRWprime} & return the sports of
                                               each session in the object \\
    \fct{concentrationProfile} & \class{distrProfile}, \class{trackeRdata}  & calculate concentration
                                                       profiles \\
    \fct{distributionProfile} & \class{trackeRdata} & calculate distribution profiles \\
    % append
    \fct{ridges} & \class{conProfile}, \class{distrProfile}, \class{trackeRdata} & ridgeline plots of concentration/distribution  profiles \\
    \fct{profile2fd} & \class{conProfile}, \class{distrProfile} & convert profiles to \class{fd} class \\
    \fct{funPCA} & \class{conProfile}, \class{distrProfile} & functional principal components analysis \\
    %
    \hline
  \end{tabular}
  \caption{Functions available in the \pkg{trackeR} package (part 2).}
  \label{tab:pkg:functions:2}
\end{table}


\section{Import utilities}\label{sec:import}

\pkg{trackeR} provides utilities for data in common formats from
GPS-enabled tracking devices. The family of the supplied reading
functions, \fct{read*}, currently includes functions for reading TCX
(Training Centre XML), GPX (as exported by Strava but other versions
should work, too), DB3 (for SQLite, used, e.g., by devices from
GPSports) and Golden Cheetah's JSON files. These functions read the
tracking data, and return a \code{data.frame} with a specific
structure.

The following code chunk illustrates the use of the \fct{readTCX}
function using a TCX file that ships with the package and shows the
name and type of variables that are present in the resulting data
frame.
%
<<eval = TRUE>>=
filepath <- system.file("extdata/tcx", "2013-06-01-183220.TCX.gz",
                        package = "trackeR")
runDF <- readTCX(file = filepath, timezone = "GMT")
str(runDF)
@
%
Power is not available in the above data frame because the data has
been identified to come from a running training by regex matching its
contents with a sport-determining list of keywords. Times are taken
here to be in GMT. The default for argument \code{timezone} is
\code{""} and is system-specific; see \code{?as.POSIXct} for details.

\pkg{trackeR} can accommodate the addition of extra formats by simply
authoring appropriate import functions. Such functions should take as
input the path of the file to be read and return a data frame with the
same structure and attributes as in the above example.

\section[trackeRdata class]{\class{trackeRdata} class}\label{sec:trackeRdata}

\subsection{Object structure}
The core object of \pkg{trackeR} has class \class{trackeRdata}.
The \class{trackeRdata} objects are
session-based, unit-aware and operation-aware structures, which
organise the data in a list of multivariate \code{zoo} objects
\citep{trackeR:Zeileis+Grothendieck:2005} with one element per session.
The observations within each session are ordered according to
the time stamps as these are read from the GPS-enabled tracking
devices. Each \class{trackeRdata} object has an attribute on the
measurement units of the data it holds, and, if applicable, an
attribute detailing the operations, such as smoothing, it has gone through.

\class{trackeRdata} objects result from the constructor function
\fct{trackeRdata}, which takes as input the output of the \fct{read*}
functions. Apart from the allocation of observations into distinct
sessions, the constructor function also performs some data processing,
including basic sanity checks (for example, removing observations with negative
or missing values for cumulative distance or speed), handling of
measurement units, correction of distances using altitude data if
required, and data imputation, discussed in Section~\ref{sec:imputation}.

\subsection{Constructor function}
The interface of the constructor function for class
\class{trackeRdata} is
\begin{Code}
       trackeRdata(dat, units = NULL, sport = NULL, session_threshold = 2,
       correct_distances = FALSE, from_distances = TRUE, country = NULL,
       mask = TRUE, lgap = 30, lskip = 5, m = 11, silent = FALSE)
\end{Code}

\code{dat} is the data frame containing the tracking data and
\code{units} is used to specify the units of
measurement. Table~\ref{tab:units} shows the currently supported units
and notes the units that are used by default when \code{units =
  NULL}. The argument \code{sport} indicate the sport from which the
data is coming from and must be ``running'', ``cycling'',
``swimming''. This affects the calculation of $W'$ (based on power or
speed for cycling and running, respectively) and the thresholds
applied before plotting the session data. The other arguments are
specific to the data processing operations, which are briefly
described in the following subsections.

\subsection{Identifying distinct sessions}
The constructor function groups the observations into sessions
according to their time stamps. Specifically, the time stamps in the
data from the \fct{read*} functions are first sorted, and all
consecutive observations whose time stamps are no further apart from
each other than a specified threshold $t^*$ are considered to
belong to a distinct session. The value of $t^*$ is set via
the \code{session_threshold} argument of the \fct{trackeRdata} function
and it defaults to 2 hours.

\subsection{Distance correction using altitude data}
If the distances in the data have been calculated solely based on
latitude and longitude data, without taking into account the altitude,
then the distance covered can be underestimated. The
\code{correct_distances} argument of the \fct{trackeRdata} function
controls whether the distances should be corrected for altitude
changes.

If the uncorrected distance covered at time point $t_{i}$ is
$d_{2,i}$, then setting \code{correct_distances = TRUE} uses the
Pythagorean theorem to correct the distance covered between time point
$t_{i-1}$ and time point $t_{i}$ to
\begin{equation*}
  d_{i} - d_{i-1} = \sqrt{ (d_{2,i} - d_{2,i-1})^2 + (a_i - a_{i -
      1})^2 }\, ,
\end{equation*}
where $d_{i}$ and $a_i$ are the corrected cumulative distance and the
altitude at time $t_i$, respectively.

If no altitude measurements are available, these are extracted
from \emph{SRTM 90m Digital Elevation Data} via the \pkg{raster}
package \citep{trackeR:Hijmans:2015} using the latitude and longitude
measurements. The arguments \code{country} and \code{mask} control the
extraction of altitudes.

\subsection{Imputation process}\label{sec:imputation}
Occasionally, there is a large time difference between consecutive
observations in the same session, sometimes of the order of several
minutes. This can happen, for example, if the device is intentionally
paused by the athlete or if the proprietary algorithm controlling the
operating sampling rate of the device detects no significant change in
position. For example, in the manual of a GPS device,
the Forerunner$^{\copyright}$ 310XT, it is stated that
 \emph{``The Forerunner uses smart recording. It records key points when
   you change direction, speed, or heart rate''} \citep{trackeR:forerunner310XT}.
In both cases, interpolating directly to get the speed or power will lead to
overestimation of the total workload within those intervals.

We assume that such intervals appear only when there is no significant
work happening, and hence impute them with observations with zero
speed (for running) or zero speed and power (for cycling).

%
\begin{figure}[t]
  \centering
  %% \input{imputation_speed_manual}
% Graphic for TeX using PGF
% Title: /home/frick/svn/EISproject/pkg/trackeR/vignettes/imputation_speed.dia
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  \caption{Illustration of the imputation process for speed with $m = 11$.}
  \label{fig:imputation}
\end{figure}
%
Figure~\ref{fig:imputation} shows a schematic representation of the
imputation process for speed. The parameters $l_{gap}$, $m$ and $l_{skip}$
control the imputation, and can be specified via the \code{lgap},
\code{m} and \code{lskip} arguments of the \fct{trackeRdata} function,
respectively.

If the observations at times $t_i$ and $t_{i + 1}$ are more than
$l_{gap}$ seconds apart, then it is assumed that there is no
significant work happening between $t_i$ and $t_{i + 1}$. The number
of imputed records in the interval is $m$, and consists of two 'outer'
records and $m - 2$ 'inner' records. The 'outer' records are
$l_{skip}$ seconds apart from the existing observations forming the
beginning and the end of the interval, respectively. The 'inner'
records are $h = (t_{i+1} - t_i - 2 l_{skip}) / (m - 1)$ seconds apart.

The imputed records between $t_i$ and $t_{i + 1}$ have zero speed or
power, and the latitude, longitude and altitude measurements are set
to their values at time $t_i$. All other variables are set to
\code{NA}.

\fct{trackeRdata} also adds five records at the beginning and five at
the end of a session, based on the assumption that there is no activity before
and after the available records. These observations have zero speed or power,
their latitude, longitude and altitude measurements are as in the first
and last observations, respectively, and all other variables are set
to \code{NA}. The imputed records are one second apart from each other
and from the first and the last observation, respectively.

After the imputation process, the cumulative distances are updated
based on the imputed speeds and the time differences between
consecutive observations, according to
\begin{equation*}
  d_{i+1} = d_{i} + s_i (t_{i+1} - t_i)
\end{equation*}
where $s_i$ and $d_i$ denote the speed and cumulative distance at time
point $t_i$, respectively.

The following code chunk takes as input the raw data in the data frame
\code{runDF} and constructs the corresponding \class{trackeRdata} object.
<<eval = TRUE>>=
runTr0 <- trackeRdata(runDF)
@

The \fct{print} method for \class{trackeRdata} objects displays some
basic summaries about the sessions, including the sports present in
the sessions, the number of sessions, training coverage, total
training duration and the units for each of the variables.

<<eval = TRUE>>=
runTr0
@

The function \fct{readContainer} is a convenience wrapper that calls
the suitable reading function and, then, \fct{trackeRdata} for the
data processing and the organisation of the data in a
\class{trackeRdata} object (see \code{?readContainer} for the
available arguments). For example,
<<eval = TRUE>>=
runTr1 <- readContainer(filepath, type = "tcx", timezone = "GMT")
identical(runTr0, runTr1)
@

The function \fct{readDirectory} allows the user to read all files of
a supported format in a directory, rather than calling, e.g.,
\fct{readContainer} on each file separately. For example, the
following chunk of code will read a directory with one cycling, one
running and, one swimming session, constructs the corresponding
trackeRdata object, and then extracts the sports for each session.
<<eval = TRUE>>=
gpxDir <- system.file("extdata/gpx", package = "trackeR")
workouts <- readDirectory(gpxDir, verbose = FALSE)
get_sport(workouts)
@


Using the argument
\code{aggregate}, the user can decide if all data are first combined in a data
frame and then split into sessions solely based on the time difference between
consecutive observations. This way, e.g., warm-up and cool-down phases are put
into the same session as the central part of training, even if they are recorded
in separate container files. Alternatively, data from different container files
are always stored in separate sessions.


%
<<dataLoad, echo = FALSE>>=
data("run", package = "trackeR")
data("runs", package = "trackeR")
@
%
\pkg{trackeR} ships with two \class{trackeRdata} objects containing
\Sexpr{nsessions(run)} and \Sexpr{nsessions(runs)} running sessions, respectively,
and which can be loaded via
%
<<dataLoad0, eval = FALSE>>=
data("run", package = "trackeR")
data("runs", package = "trackeR")
@
%
We will use those objects for the illustrations throughout the paper.

%% %\pagebreak


\section{Session summaries and visualisation}\label{sec:summaries}

\pkg{trackeR} provides methods for summarising sessions in terms of scalar
summaries, the time spent exercising in specified zones, the concept of work
capacity, and distribution and concentration profiles.


\subsection{Visualisation}
For a first visual inspection of the data, the \fct{plot} method shows
by default the evolution of heart rate and pace over the course of the
selected sessions. For example, Figure~\ref{fig:defaultPlot:1} shows
the evolution of heart rate and pace for the first three sessions
in the \code{runs} object.
%% session 3 is an interval session? mention anything about the content of the plots?
%
<<eval = FALSE, echo = TRUE>>=
plot(runs, session = 1:3)
@
%
\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
%
<<defaultPlot, echo = FALSE, fig = TRUE, width = 7, height = 4.5>>=
plot(runs, session = 1:3)
@
%
\caption{Heart rate and pace over the course of sessions~1--3.}
  \label{fig:defaultPlot:1}
\end{figure}

%\clearpage

The route covered during a session can also be displayed on a static map
via the \fct{plotRoute} method. The \fct{plotRoute} method uses the
\pkg{ggmap} package \citep{trackeR:Kahle+Wickham:2013} and, hence, can
work with the sources and maps supported by \pkg{ggmap}. For example,
Figure~\ref{fig:routePlot} shows the route covered during session~4 in
\code{runs} using a map downloaded from Google.
Interactive maps can be produced with \fct{leafletPlot}, using the
\pkg{leaflet} package \citep{trackeR:pkg:leaflet}.
%
<<eval = FALSE, echo = TRUE>>=
plotRoute(runs, session = 4, zoom = 13)
@
%
\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
  \centering
% %
% <<routePlot, eval = FALSE, echo = TRUE, fig = TRUE, width = 6, height = 6>>=
% plotRoute(runs, session = 4, zoom = 13)
% @
%
  \includegraphics{plotRouteGoogle}
  \caption{Route covered during session~4 on a map from Google.}
  \label{fig:routePlot}
\end{figure}


\subsection{Scalar summaries}
Each session can be summarised through common summary statistics using the
\fct{summary} method. Such a session summary includes estimates of the total
distance covered, the total duration, the time spent moving, and work to rest
ratio. It also includes averages of speed, pace, cadence, power, and heart rate,
calculated based on total duration or the time spent moving.

An athlete is considered to be moving
if the speed is larger than some threshold $s^*$. This threshold can be set via
the \code{moving_threshold} argument of the \fct{summary} method, and the package
assumes that anything between not moving at all and walking with a speed below
that threshold is resting. The default value for \code{moving_threshold} has been
set to 1~meter per second for running, which is just below the speed humans prefer to walk
at on average \citep[1.4~meters per second;
see][]{trackeR:Bohannon:1997}, 0.5~meter per second for swimming and 2
meters per second for cycling.

The ``average speed moving'' is calculated as total distance covered divided by
time moving while ``average speed'' is calculated as total distance divided by
total duration. The average pace~(moving) is calculated
as the inverse of the average speed~(moving). The work to rest ratio
is calculated as time moving divided by (total duration - time
moving).
The averages for cadence, power, and heart rate (total and moving) are weighted
averages with weights depending on the time difference to the next
observation. These averages also need to take into account missingness in the
observations. For a variable of interest $V$, we can calculate a weighted mean
for the total session while accounting for missing values via
\begin{equation*}
  \sum_{i}{v_i \frac{\Delta_i K_i}{\sum_{i}{\Delta_i K_i}}}
  %%\label{eq:wmean}
\end{equation*}
and its counterpart for the part of the session spent in motion via
\begin{equation*}
  \sum_{i}{v_i \frac{\Delta_i K_i I(s_i > s^*)}
                   {\sum_{i}{\Delta_i K_i I(s_i > s^*)}}}
  %%\label{eq:wmean:moving}
\end{equation*}
where $v_i$ is the value of $V$ at time point $t_i$, $K_i$ is 1 if $v_i$
is available, i.e., not missing, and 0 otherwise, $I(\cdot)$ denotes
the indicator function, and $\Delta_i = t_i - t_{i-1}$ the time difference
between observations at $t_i$ and $t_{i-1}$.

The \fct{summary} method for \class{trackeRdata} objects returns a data
frame which can be used for further analysis. The return object is classed as
\class{trackeRdataSummary} for which several methods are available. With the
\fct{print} method, one can set the number of digits printed for the scalar
summary statistics. The following example shows the summaries for sessions~1--2
with the default number of digits of 2 and then the summary of session~1 with
3~digits for comparison.
%
<<summarySessionsPrint>>=
summary(runs, session = 1:2)
runSummary <- summary(runs, session = 1)
print(runSummary, digits = 3)
@
%

The \fct{plot} method shows the evolution of the various summary statistics over
calender time or over the course of the sessions. For example, the following
code chunk produces Figure~\ref{fig:summaryPlot}.
%
<<summarySessions, eval = FALSE>>=
runSummaryFull <- summary(runs)
plot(runSummaryFull, group = c("total", "moving"),
  what = c("avgSpeed", "distance", "duration", "avgHeartRate"))
@
%

\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
%
<<summarySessionsPlot, echo = FALSE, fig = TRUE, width = 7, height = 6.3>>=
runSummaryFull <- summary(runs)
plot(runSummaryFull, group = c("total", "moving"),
  what = c("avgSpeed", "distance", "duration", "avgHeartRate"))
@
%
  \caption{Selected session summaries for all \Sexpr{nsessions(runs)}~sessions.}
  \label{fig:summaryPlot}
\end{figure}



\subsection{Times in zones}\label{subsec:zones}
A common way to summarise and characterise a session is to calculate
how much time was spent exercising in certain zones, e.g., heart rate zones.

The \fct{zones} method for sessions returns an object of class
\class{trackeRdataZones} for which methods \fct{changeUnits} and \fct{plot}
are provided. The user can specify the variables, such as heart rate and speed,
and their respective zones via the arguments \code{what} and \code{breaks},
respectively. Figure~\ref{fig:zonesPlot} shows a graphical representation of the
zones summary, making it easier to see that more (relative) time was spent
training with high speed ($> \, 4 \, m/s$) in sessions~3 and~4 than in
sessions~1 and~2. The following code chunk illustrates three equivalent ways
to specify the zones for a single variable: 1) in the standard way through
arguments \code{what} and \code{breaks} 2) if \code{breaks} is a named list,
argument \code{what} can be left unspecified and 3) if only a single variable is
to be evaluated, \code{breaks} can also be a vector.
%
<<zones, eval = FALSE>>=
runZones <- zones(runs[1:4], what = "speed",
  breaks = list(speed = c(0, 2:6, 12.5)))
runZones <- zones(runs[1:4], breaks = list(speed = c(0, 2:6, 12.5)))
runZones <- zones(runs[1:4], what = "speed", breaks = c(0, 2:6, 12.5))
plot(runZones)
@
%

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
  \centering
%
<<zonesPlot, echo = FALSE, fig = TRUE, width = 7, height = 4>>=
runZones <- zones(runs[1:4], what = "speed",
  breaks = list(speed = c(0, 2:6, 12.5)))
runZones <- zones(runs[1:4], breaks = list(speed = c(0, 2:6, 12.5)))
runZones <- zones(runs[1:4], what = "speed", breaks = c(0, 2:6, 12.5))
plot(runZones)
@
%
  \caption{Zone summaries for speed of sessions~1--4.}
  \label{fig:zonesPlot}
\end{figure}


%%\pagebreak %%% when JSS header is included

\subsection{Quantifying work capacity}\label{subsec:workcapacity}
The critical power model \citep{trackeR:Monod+Scherrer:1965} describes the
relationship between the power output $P$ and the time $t_e$ to exhaustion at that
power output
%
\begin{equation}
  \label{eq:CPmodel}
  P = (W'_0 / t_e) + CP
\end{equation}
%
in terms of two parameters $W'_0$ and CP. The critical power (CP) is defined by
\citet{trackeR:Monod+Scherrer:1965} as ``the maximum rate (of work) that [can be
kept] up for a very long time without fatigue.''
\citet{trackeR:Skiba+Chidnok+Vanhatalo:2012} describe CP as ``a power output
that could \emph{theoretically} be maintained indefinitely on the basis of
principally 'aerobic' metabolism.'' $W'$ (read W~prime) represents a finite
work capacity above CP. \citet{trackeR:Skiba+Chidnok+Vanhatalo:2012} assume that
$W'$ gets depleted during exercise with a power output above CP but also
replenished during exercise with a power output of or below CP.
We denote as $W'$ the general concept of work capacity above CP, and $W'(t)$ is
the state of $W'$ at time $t$. The latter is also sometimes referred to as
\emph{$W'$~balance} at time $t$.  Additionally, the initial state of $W'$ at the start
of an exercise $t = t_0$ is $W'_0 = W'(t_0)$, which is one of the parameters in
the critical power model (Equation~\ref{eq:CPmodel}). Total depletion of $W'_0$
results in the inability to produce a power output above CP. Thus, knowledge of
the current state $W'(t)$, i.e., how much of that finite work capacity $W'_0$ is
left at time $t$, is important to an athlete, particularly in a race.

While this concept is most commonly applied to cycling, where the power output
is routinely measured, \cite{trackeR:Skiba+Chidnok+Vanhatalo:2012} suggest that
it can also be applied to running, substituting power and critical power by
speed and critical speed, respectively.
For running, the model postulates that each runner has a finite capacity in
terms of distance covered above the critical speed.
Depending on how much the runner exceeds this critical speed, the finite
capacity $W'_0$ is being exhausted in shorter times.
Below we describe the models for depletion and replenishment of work capacity
and how they are combined in \pkg{trackeR}.


\subsubsection{Depletion of work capacity}
Assuming constant power for periods of exertion above CP,
\cite{trackeR:Skiba+Fulford+Clarke:2015} assume that $W'$ is depleted at a rate
directly proportional to the difference between the power output and~CP
%
\begin{equation}
  \label{eq:discharge:rate}
  \frac{d}{d t} W'(t) = -(P - CP) \, .
\end{equation}
%
Solving Equation~\ref{eq:discharge:rate} for $W'(t)$ gives
%
\begin{equation}
  \label{eq:discharge}
  W'(t) = -(P - CP) t + D
\end{equation}
%
where $D \in \mathds{R}$ is constant over $t$.

Suppose that the exercise over time $t_0 = 0$ to $t_n = T$ can be split into $n$
intervals with breakpoints $t_0, t_1, \ldots, t_n$ such that the power output
within each interval is constant, that is $P(t) = P_i$ for $t \in
[t_{i-1}, t_i)$, $i \in \{ 1, \ldots, n \}$. Then, using
Equation~\ref{eq:discharge}, the change in $W'(t)$ over the interval can be
expressed as
%
\begin{align}
  W'(t_i) - W'(t_{i-1}) %&= - (P_i - CP) t_i + D - \left( -(P_i - CP) t_{i-1} + D \right) \nonumber \\
   &= - (P_i - CP) (t_i - t_{i-1}) \, .   \label{eq:discharge:interval}
\end{align}
%


\subsubsection{Replenishing of work capacity}
\cite{trackeR:Skiba+Fulford+Clarke:2015} assume that the periods with a
power output at or below CP are periods of recovery during which $W'$ is
replenished with a rate that depends on the difference between CP and the power
output, and the amount of $W'_0$ remaining, as follows:
%
\begin{equation}
  \label{eq:recharge:rate}
  \frac{d}{d t} W'(t) = \left( 1 - \frac{W'(t)}{W'_0} \right) (CP - P) \, .
\end{equation}
%
Equation~\ref{eq:recharge:rate} assumes that recovery slows down as $W'(t)$
approaches the initial capacity $W'_0$. Employing the substitution rule for
integrals while solving Equation~\ref{eq:recharge:rate} and
reexpressing in terms of $W'(t_{i-1})$ (see Appendix~\ref{sec:appendix} for details) gives
%
\begin{equation}
  \label{eq:recharge}
   W'(t_i) = W'_0 - \left( W'_0 - W'(t_{i-1}) \right) \exp \left( \frac{P_i -
      CP}{W'_0} (t_i - t_{i-1}) \right) \, .
\end{equation}
%

%% Link to 2015 paper and their equation
Since $W'(t_{i-1})$ is the amount of $W'_0$ remaining at the start of the
interval $[t_{i-1}, t_i)$, \linebreak $W'_0 - W'(t_{i-1})$ is the amount of
$W'_0$ which has been depleted prior to $t_{i-1}$ and not yet been
replenished. \cite{trackeR:Skiba+Chidnok+Vanhatalo:2012} refer to this as
\emph{$W'$~expended}. \cite{trackeR:Skiba+Fulford+Clarke:2015} describe the
replenishing of $W'$ indirectly by describing how \emph{$W'$~expended} is
reduced over the course of such a recovery interval. The exponential decay
factor used in Equation~\ref{eq:recharge} here is the same as their Equation~4
with only different notation. \cite{trackeR:Skiba+Fulford+Clarke:2015} use $t$
to describe the length of the interval, $D_{CP} = CP - P_i$ for the difference
between critical power and power output, and $W'_{exp}$ for the amount of $W'$
previously expended. For $P_i < CP$, as is required for replenishment, $-D_{CP}$
and $P_i - CP$ are negative and thus the exponential factor is smaller than 1,
leading to an exponential decay as described.

%% link to 2012 paper and their equation
\cite{trackeR:Skiba+Chidnok+Vanhatalo:2012} also assume an exponential decay
of previously expended $W'$ to describe replenishing $W'$, albeit
with a different decay factor. Instead of $(P_i - CP) / W'_0$, they use
$1/\tau_{W'}$. The relationship between the time constant of replenishing $\tau_{W'}$
and the difference between critical power and recovery power $\bar{P}$ is
estimated based on experimental data as
%
\begin{equation*}
  %\label{eq:tau}
  \tau_{W'} = 546 \exp \left(-0.01 (CP - \bar{P}) \right) + 316
\end{equation*}
%
with recovery power $\bar{P}$ estimated by the mean of all power outputs below CP.

Using Equation~\ref{eq:recharge}, i.e., the formulation of
\cite{trackeR:Skiba+Fulford+Clarke:2015}, the change in $W'$ over the corresponding
interval $[t_{i-1}, t_i)$ can be described through
%
\begin{align}
  W'(t_i) - W'(t_{i-1}) %% &= W'_0 - \left( W'_0 - W'(t_{i-1}) \right) \exp \left(
%%    \frac{P_i - CP}{W'_0} (t_i - t_{i-1}) \right)  - W'(t_{i-1}) \nonumber \\
  &= (W'_0  - W'(t_{i-1})) \left( 1 - \exp \left(
    \frac{P_i - CP}{W'_0} \Delta_i \right) \right) \, . \label{eq:recharge:interval}
\end{align}
%


\subsubsection{Work capacity at time $t_j$}
Equation~\ref{eq:discharge:interval} describes the depletion of $W'$ (when $P_i
> CP$) and Equation~\ref{eq:recharge:interval} describes replenishment of $W'$
(when $P_i \le CP$) over an interval $[t_{i-1}, t_i)$. These two aspects can be
combined to describe the change over the interval as
%
\begin{align}
   W'(t_i) - W'(t_{i-1}) = {} & -(P_i - CP) \Delta_i I(P_i > CP) \, + \nonumber \\
   &
    (W'_0  - W'(t_{i-1})) \left( 1 - \exp \left(
    \frac{P_i - CP}{W'_0} \Delta_i \right) \right) (1 - I(P_i > CP))
   \, . \nonumber
\end{align}
%

The amount of $W'$ left at time point $t_j$, $j \in \{ 1, \ldots, n \}$, can thus be
described through the initial amount $W'_0$ and the changes happening in the $j$
intervals of constant power previous to $t_j$:
%
\begin{align}
  W'(t_j) = {} & W'_0 + \sum_{i=1}^{j}{ ( W'(t_i) - W'(t_{i-1}) ) }  \nonumber \\
  = {} & W'_0 - \sum_{i=1}^{j}{(P_i - CP) \Delta_i I(P_i > CP)} \, + \nonumber \\
  & \sum_{i=1}^{j}{
    (W'_0  - W'(t_{i-1})) \left( 1 - \exp \left(
    \frac{P_i - CP}{W'_0} \Delta_i \right) \right) (1 - I(P_i > CP))
  } \, . \label{eq:Wbalance}
\end{align}
%

\emph{$W'$~expended} at time $t_j$ is then $W'_0 - W'(t_j)$.

Function \fct{Wprime} can be used to calculate \emph{$W'$~expended} by setting
argument \code{quantity} to \code{"expended"}. If \code{quantity} is set to
\code{"balance"}, \fct{Wprime} calculates the current state $W'(t)$
(Equation~\ref{eq:Wbalance}). \fct{Wprime} contains implementations for
\cite{trackeR:Skiba+Chidnok+Vanhatalo:2012} and
\cite{trackeR:Skiba+Fulford+Clarke:2015}, which can be selected via the
\code{version} argument.
For example, session~11 of the example data is an interval training with a
warm-up and cool-down phase. Assuming a critical speed of 4~meters per second,
the following code chunk produces Figure~\ref{fig:WprimePlots}, which shows
\emph{$W'$~expended}, based on the specification of
\cite{trackeR:Skiba+Chidnok+Vanhatalo:2012}, along with the corresponding speed
profile.
%
<<Wprime, echo = TRUE, eval = FALSE>>=
wexp <- Wprime(runs, session = 11, quantity = "expended",
  cp = 4, version = "2012")
plot(wexp, scaled = TRUE)
@
%
During the warm-up phase speed rarely exceeds 4 meters per second and
\emph{$W'$~expended} remains low. Over the course of the interval training,
\emph{$W'$~expended} rises during the high-intensity phases and drops during the
recovery phases. In the last part of the session, speeds are mostly below 4
meters per second and \emph{$W'$~expended} drops again.

\setkeys{Gin}{width=\textwidth}
\begin{figure}
  \centering
%
<<WprimePlot2, eval = TRUE, echo = FALSE, fig = TRUE, width = 7, height = 3.5>>=
wexp <- Wprime(runs, session = 11, quantity = "expended", cp = 4, version = "2012")
plot(wexp, scaled = TRUE)
@
%
    \caption{$W'$ expended in session~11.}
    \label{fig:WprimePlots}
\end{figure}



\subsection{Distribution and concentration profiles}\label{subsec:profiles}
\cite{trackeR:Kosmidis+Passfield:2015} introduce the concept of distribution
profiles for which the \pkg{trackeR} package provides an implementation. These
profiles are motivated by the need to compare sessions
%% in terms of quantities such as heart rate and speed
and use information on such variables as heart rate or speed during a session
for further modelling.

For a session lasting $t_n$ seconds, the distribution profile is defined as the
curve $\{ v, \Pi(v) | v \geq 0\}$ where
%
\begin{equation*}
  %%\label{eq:distributionProfile}
  \Pi(v) = \int_{0}^{t_n}{I(v(t) > v) dt} \, .
\end{equation*}
%
The function $\Pi(v)$ is monotone decreasing and describes the time
spent exercising above a threshold $v$ for a variable $V$ under consideration
(e.g., heart rate or speed).

On the basis of observations $v_0, \ldots, v_n$ for $V$, at respective time
points $t_0, \ldots, t_n$, the observed version of  $\Pi(v)$ can be calculated as
%
\begin{equation*}
  %%\label{eq:distributionProfile:observed}
  P(v) = \sum_{i = 1}^{n}{(t_{i} - t_{i-1}) I(v_{i} > v)} \, .
\end{equation*}
%
This can subsequently be smoothed respecting the positivity and monotonicity of
$\Pi(v)$, e.g., via a shape constraint additive model with Poisson responses
\citep{trackeR:Pya+Wood:2015}.

The concentration profile is defined in \cite{trackeR:Kosmidis+Passfield:2015}
as the negative derivative of a distribution profile and is suitable for
revealing concentrations of time around certain values of the variable under
consideration.

Distribution profiles can be calculated using the \fct{distributionProfile}
function which returns an object of class \class{distrProfile}.
Concentration profiles can be derived from distribution profiles using
\fct{concentrationProfile}, which returns an object of class
\class{conProfile}. Table~\ref{tab:pkg:functions:2} includes an overview of
constructor functions and available methods for distribution and concentration
profiles.

By default, distribution profiles are calculated for speed and heart rate on
grids inferred from the data.
% covering the ranges of $[0, 12.5]$ meters per second and $[0, 250]$ beats
% per minute, respectively.
The following code chunk illustrates the use of
\fct{distributionProfile} and shows how users can specify the variables for
which to calculate profiles and the respective grids.
%
<<distrProfiles, eval = FALSE>>=
dProfile <- distributionProfile(runs, session = 1:4,
  what = c("speed", "heart_rate"),
  grid = list(speed = seq(0, 12.5, by = 0.05), heart_rate = seq(0, 250)))
plot(dProfile, multiple = TRUE)
@
%

\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
%
<<dprofilePlot, echo = FALSE, fig = TRUE, width = 8, height = 4>>=
dProfile <- distributionProfile(runs, session = 1:4,
  what = c("speed", "heart_rate"),
  grid = list(speed = seq(0, 12.5, by = 0.05), heart_rate = seq(0, 250)))
plot(dProfile, smooth = FALSE, multiple = TRUE)
@
%
  \caption{Distribution profiles for sessions~1--4.}
  \label{fig:dprofile:smooth}
\end{figure}

The \code{multiple} argument of the \fct{plot} method determines whether to plot
the profiles in separate panels (\code{FALSE}) or overlay them in a common
panel (\code{TRUE}), as in Figure~\ref{fig:dprofile:smooth}. The different
session lengths are clearly visible in the height of the curves at 0. Amongst
the distribution profiles for speed, the descent of the profile
for session~3 is slower than for the other sessions. This difference is most
apparent in the concentration profiles, which are shown in
Figure~\ref{fig:cprofile:smooth} and are produced by the following code chunk.
%
<<concentrationProfiles, eval = FALSE>>=
cProfile <- concentrationProfile(dProfile, what = "speed")
plot(cProfile, multiple = TRUE, smooth = TRUE)
@
%
The profile for session~3 has a mode at around 3.5~meters per second and another
one at 5~meters per second, showing that this session involved training at a
combination of low and high speeds.

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
  \centering
%
<<cprofilePlot, echo = FALSE, fig = TRUE, width = 6, height = 4>>=
cProfile <- concentrationProfile(dProfile, what = "speed")
plot(cProfile, multiple = TRUE, smooth = TRUE)
## cProfileS <- concentrationProfile(dProfileS)
## plot(cProfileS, what = "speed", smooth = FALSE, multiple = TRUE)
@
%
  \caption{Concentration profiles for sessions~1--4.}
  \label{fig:cprofile:smooth}
\end{figure}

\section{Handling units of measurement}\label{sec:units}

Data objects of class \class{trackeRdata} and all objects derived from these
(\class{trackeRdataSummary}, \class{trackeRdataZones}, \class{trackeRWprime}, %\linebreak
\class{distrProfile}, and \class{conProfile}) carry an attribute with the
relevant units of measurement. The \fct{getUnits} method returns the units of
measurement for each variable and the \fct{changeUnits} method can be
used to change one or more variables from one set of units to another.
The following code chunk displays the current units of \code{run} for running, changes the
unit for speed for running to miles per hour, and displays the changed units.
%
<<eval = TRUE>>=
subset(getUnits(runs), sport == "running")
runTr2 <- changeUnits(run, variable = "speed", unit = "mi_per_h", sport = "running")
subset(getUnits(runTr2), sport == "running")
@
%
Table~\ref{tab:units} shows the variables and the corresponding
units that are currently supported in \pkg{trackeR}.

\begin{table}[t]
  \centering
  \begin{tabular}{l p{12.5cm}}
    \hline
    Measurement & Unit(s) \\
    \hline
    %% initial measurements/variables in trackeRdata objects
    latitude & degrees (\code{degree}, default) \\
    longitude & degrees (\code{degree}, default) \\
    altitude & meters~(\code{m}, default), kilometres~(\code{km}), miles~(\code{mi}),
    feet~(\code{ft}) \\
    distance & meters~(\code{m}, default), kilometres~(\code{km}), miles~(\code{mi}),
    feet~(\code{ft}) \\
    speed & meters per second~(\code{m_per_s}, default), kilometres per
    hour~(\code{km_per_h}), feet per minute~(\code{ft_per_min}), feet per
    second~(\code{ft_per_s}), miles per hour~(\code{mi_per_h}) \\
    cadence\_running & steps per minute~(\code{steps_per_min}, default) \\
    cadence\_cycling & revolutions per minute~(\code{rev_per_min}, default) \\
    power & Watts~(\code{W}, default), kilowatts~(\code{kW}) \\
    heart rate & beats per minute~(\code{bpm}, default) \\
    %% quantities derived for session summaries
    pace & minutes per kilometre (\code{min_per_km}, default), minutes per mile~(\code{min_per_mi}), seconds per meter~(\code{s_per_m}) \\
    duration & seconds~(\code{s}), minutes~(\code{min}), hours~(\code{h}) --
    default is the largest possible unit for which the duration is
               larger than 1 \\
    temperature & degrees Celcius~(\code{C}, default), degrees
                  Fahrenheit~(\code{F}) \\
    \hline
  \end{tabular}
  \caption{Supported units of measurement.}
  \label{tab:units}
\end{table}

If objects with different units are \fct{c}ombined in one object,
the units of the first session are applied to all other sessions.
Furthermore, the \fct{changeUnits} method uses name matching to figure
out which conversion needs to be done. This allows the user to easily
add support for converting from \code{unitOld} to \code{unitNew} by
authoring a function named \code{unitOld2unitNew}.


If we wish to report the speed summaries for session~1 in \code{runSummary} in
feet per hour (not currently supported) instead of meters per second, we need to
simply provide the appropriately named conversion function as illustrated
below. Note that the conversion applies to all speed summaries, i.e., to
``average speed'' and ``average speed moving''.
%
<<summarySessionsUnits>>=
m_per_s2ft_per_h <- function(x) x * 3937/1200 * 3600
changeUnits(runSummary, variable = "speed", unit = "ft_per_h")
@
%


\section{Thresholding and smoothing}\label{sec:smoothing}

There are instances where the data include artefacts due to
inaccuracies in the GPS measurements. These can be handled with the
\fct{threshold} method for objects of class \class{trackeRdata}, which
replaces values outside the specified thresholds with \code{NA}. The variables
and the (lower and upper) thresholds which should be applied for each variable
can be specified through the arguments \code{variable}, \code{lower}, and
\code{upper}, respectively. An example is given in \code{?threshold}. The
default thresholds are listed in Table~\ref{tab:defaultThresholds} and, if
necessary, are converted to the units of measurement used for the
\class{trackeRdata} object.

\begin{table}[t]
  \centering
  \begin{tabular}{llrrccc}
    \hline
    Variable & Unit & Lower threshold & Upper threshold & R &
                                                                    C
    & S \\
    \hline
    latitude & degrees & -90 &  90 & x & x & x\\
    longitude & degrees & -180 &  180 & x & x & x \\
    altitude & meter & -500 & 9000 & x & x & x \\
    distance & meter & 0 &  $\infty$ & x & x & x \\
    heart rate & beats per minute & 0 & 250 & x & x & x \\
    speed & meters per second & 0 & $\infty$ & x & x & x \\
    cadence\_running & steps per minute & 0 & $\infty$ & x &  &  \\
    cadence\_cycling & revolutions per minute & 0 & $\infty$ & x & x & x\\
    power & Watts & 0 & $\infty$ &  & x & \\
    pace & minutes per kilometre & 0 & $\infty$ & x & x & x \\
    duration & seconds & 0 & $\infty$ & x & x & x \\
    temperature & degrees Celsius & -20 & 60 & x & x & x\\
    \hline
  \end{tabular}
  \caption{Default thresholds for running, cycling and swimming
    data. R stands for running, C stands for cycling and S for
    swimming, and an x indicates that the corresponding unit applies
    to that sport.}
  \label{tab:defaultThresholds}
\end{table}

The other option for data handling is the \fct{smoother} method for
\class{trackeRdata} objects. This applies a summarising function, such as the
mean or median, over a rolling window. Both operations \fct{threshold} and
\fct{smoother} are used in the \fct{plot} method for
\class{trackeRdata} objects. The default settings for \fct{plot} are to apply
the thresholds specified in Table~\ref{tab:defaultThresholds} but not to smooth
the data. The top left panel in Figure~\ref{fig:thresholdPlots} gives an example
where no thresholds are applied and the top right panel uses default
settings. The spike to over 20~meters per second in the top left panel is
clearly an error in the data; the current world record for 100 meters (by Usain
Bolt, August 16, 2009) is 9.58 seconds which translates to an average speed of
10.44 meters per second. The bottom panels show the effect of first applying the
default thresholds and then smoothing the data through a rolling median with a
window width of 20 observations, either done within the \fct{plot} method
(bottom left) or explicitly via the \fct{threshold} and \fct{smoother} methods
(bottom right). The following code chunk produces the four plots in
Figure~\ref{fig:thresholdPlots}.
%
<<thresholdPlots, eval = FALSE>>=
plot(runs, session = 4, what = "speed", threshold = FALSE)
plot(runs, session = 4, what = "speed")
plot(runs, session = 4, what = "speed", smooth = TRUE, fun = "median",
  width = 20)
run4 <- threshold(runs[4])
run4S <- smoother(run4, what = "speed", fun = "median", width = 20)
plot(run4S, what = "speed", smooth = FALSE)
@
%

\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
    \begin{subfigure}[b]{0.45\textwidth}
%
<<thresholdPlots1, eval = TRUE, echo = FALSE, fig = TRUE, width = 3.7, height = 2.9>>=
plot(runs, session = 4, what = "speed", threshold = FALSE)
@
%
    \end{subfigure}
    %~
    \begin{subfigure}[b]{0.45\textwidth}
%
<<thresholdPlots2, eval = TRUE, echo = FALSE, fig = TRUE, width = 3.7, height = 2.9>>=
plot(runs, session = 4, what = "speed") + ggplot2::expand_limits(y = c(0, 21))
@
%
    \end{subfigure}
    %~
    \begin{subfigure}[b]{0.45\textwidth}
%
<<thresholdPlots3, eval = TRUE, echo = FALSE, fig = TRUE, width = 3.7, height = 2.9>>=
plot(runs, session = 4, what = "speed", smooth = TRUE, fun = "median", width = 20, cores = 2) + ggplot2::expand_limits(y = c(0, 12.5))
@
%
    \end{subfigure}
    %~
    \begin{subfigure}[b]{0.45\textwidth}
%
<<thresholdPlots4, eval = TRUE, echo = FALSE, fig = TRUE, width = 3.7, height = 2.9>>=
run4 <- threshold(runs[4])
run4S <- smoother(run4, what = "speed", fun = "median", width = 20, cores = 2)
plot(run4S, what = "speed", smooth = FALSE) + ggplot2::expand_limits(y = c(0, 12.5))
@
%
    \end{subfigure}
    \caption{Speed profile of session~4 without thresholding~(top left),
      with the default settings~(top right), and with default
      thresholds as well as smoothing through a rolling median over a window of
      20 observations done within the plot function~(bottom left) and
      separately~(bottom right).}
    \label{fig:thresholdPlots}
\end{figure}

The method \fct{smoother} is also available for distribution and
concentration profiles. Smoothing a distribution profile requires a smoothing technique
which respects the positivity and monotonicity of the distribution profile. This
can be achieved by fitting a shape constrained additive model with Poisson responses
as implemented in the \pkg{scam} package \citep{trackeR:pkg:scam}.
%
When smoothing concentration profiles, the raw profiles are transformed to
distribution profiles which are subsequently smoothed preserving the positivity
and monotonicity. The smooth concentration profiles are then derived from the
smoothed distribution profiles.
%
% The \fct{plot} methods for \class{distrProfile} and \class{conProfile} smooth
% the profiles prior to plotting by default.

If smooth concentration profiles are all that is required, these can
also be computed by using the \fct{concentrationProfile} method
direction on the \class{trackeRdata} object, which will then use an
appropriately scaled kernel density estimator on the values of the
variables in \code{what}. This is a more computationally efficient
operation, that computing the distribution profiles, smoothing them,
and then taking finite differences of the smoothed distribution
profiles. The following code chunk illustrates this and plots the
concentration profiles using a ridgeline plot
\citep{trackeR:pkg:ggridges}. The resulting plot is shown in
Figure~\ref{fig:ridges}.
<<cP, echo = TRUE, eval = FALSE>>=
cProfile1 <- concentrationProfile(runs,  what = "speed",
                                  limits = list(speed = c(0, 12.5)))
ridges(cProfile1)
@

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
  \centering
<<cpPlot, echo = FALSE, eval = TRUE, fig = TRUE, width = 7, height = 4.5>>=
cProfile1 <- concentrationProfile(runs,  what = "speed",
                                  limits = list(speed = c(0, 12.5)))
ridges(cProfile1)
@
  \caption{Ridgeline plot of concentration profiles.}
  \label{fig:ridges}
\end{figure}


% \clearpage

\section{Case study}\label{sec:casestudy}

<<AppData, echo = FALSE>>=
library("trackeR")
data("runs", package = "trackeR")
runsSummary <- summary(runs)
@

The example data set included in the package contains
\Sexpr{nsessions(runs)}~sessions of a single male runner in June 2013. A
visualisation of scalar summaries for the sessions can be found in
Figure~\ref{fig:summaryPlot}. The distance covered in those sessions ranges from
\Sexpr{round(min(runsSummary$distance) / 1000, 2)} km to
\Sexpr{round(max(runsSummary$distance) / 1000, 2)} km, and most sessions were
spent moving almost the entire time.

The code chunk below loads the data, applies thresholds, and calculates the
smoothed distribution profiles for the \Sexpr{nsessions(runs)}~sessions. The
corresponding concentration profiles are shown in Figure~\ref{fig:app:cpsmooth}.
%
<<casestudyData, eval = FALSE, echo = TRUE>>=
library("trackeR")
data("runs", package = "trackeR")
runsT <- threshold(runs)
dpRuns <- distributionProfile(runsT, what = "speed")
dpRunsS <- smoother(dpRuns)
cpRuns <- concentrationProfile(dpRunsS)
plot(cpRuns, multiple = TRUE, smooth = FALSE)
@
%

\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
%
<<AppCPplot, echo = FALSE, fig = TRUE, width = 8, height = 4>>=
library("trackeR")
data("runs", package = "trackeR")
runsT <- threshold(runs)
dpRuns <- distributionProfile(runsT, what = "speed")
dpRunsS <- smoother(dpRuns)
cpRuns <- concentrationProfile(dpRunsS)
plot(cpRuns, multiple = TRUE, smooth = FALSE) #+ theme(legend.position = "none")
@
%
  \caption{Smoothed speed concentration profiles for all \Sexpr{nsessions(runs)} sessions.}
  \label{fig:app:cpsmooth}
\end{figure}

\pagebreak

The majority of the profiles for speed concentrate around 4~meters per
second. However, the curves differ in their shape (unimodal or multimodal),
height, and location (revealing concentrations at higher or lower speeds).
Functional PCA \citep[e.g.,][]{trackeR:Ramsay+Silverman:2005} can be used to
explain those differences ensuring that the profiles are treated directly as
functions.
\pkg{trackeR} contains a convenience function \fct{funPCA} which converts
concentration/distribution profiles to the required functional data format
and performs a functional PCA. \pkg{trackeR} can also be viewed as a stepping
stone to further analysis of tracking data with other
\proglang{R}~packages. For example, it contains a conversion
function, \fct{profile2fd}, that transforms concentration and
distribution profiles to class \class{fd} so that users have direct access to
the facilities of the \pkg{fda} package \citep{trackeR:pkg:fda} for functional
data analysis.

The following code chunk shows the conversion to the required functional data
format and the fitting of a functional PCA in separate steps. The PCA has four
components and the share of variance is displayed in the last step.
%  %% Alternatively: sppca <- funPCA(cpRuns, what = "speed", nharm = 4)
<<AppFunPrep>>=
library("fda")
cpFd <- profile2fd(cpRuns, what = "speed")
sppca <- pca.fd(cpFd, nharm = 4)
varprop <- round(sppca$varprop * 100); names(varprop) <- 1:4
varprop
@
%

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
  \centering
%
<<AppHarmonicsPlotSpeed, echo = FALSE, fig = TRUE, width = 5, height = 5>>=
## refit fPCA so that trackeR's plot method is used
sppca <- funPCA(cpRuns, what = "speed", nharm = 4)
pp <- FALSE ## dotted = -, dashed = +
plot(sppca, harm = 1:2, pointplot = pp)
@
%
  \caption{Harmonics 1--2 for the speed concentration profiles. Mean
    function~(solid line) with suitable multiples of the harmonic added~(dashed
    line) and subtracted~(dotted line).}
  \label{fig:app:harmonics:speed}
\end{figure}


The first two harmonics capture \Sexpr{round(cumsum(varprop)[2])}\% of the
variation between curves. Since further harmonics capture considerably less
variation, only the first two are chosen for further inspection.

Figure~\ref{fig:app:harmonics:speed} shows the mean function (solid line) and
the variation captured in the two harmonics (between the dashed and dotted
lines). %(between the $+$ and $-$ signs).
The first harmonic (top panel) illustrates that the most important
characteristic of the concentration profiles is the relative value, which is
closely related to the overall session duration. The left panel of
Figure~\ref{fig:app:score:speed} shows the score on the first harmonic versus
``duration moving'' which is calculated as part of the scalar session
summaries. The second harmonic in the bottom panel of
Figure~\ref{fig:app:harmonics:speed} shows variation along the speed
thresholds in the centre of the curve. This variation can be explained well by
the scalar measure ``average speed moving'' as shown in the right panel of
Figure~\ref{fig:app:score:speed}.

The concentration profiles and a functional PCA thus indicate that the two
scalar summaries ``duration moving'' and ``average speed moving'' provide a
good summary of the speed information in the sessions and can be used, for
example, in order to incorporate speed as explanatory information in regression
analyses.

\setkeys{Gin}{width=\textwidth}
\begin{figure}[t]
  \centering
  \begin{subfigure}[b]{0.45\textwidth}
%
<<AppScoresPlot1, echo = FALSE, fig = TRUE, width = 4, height = 4>>=
## plot scores vs summary statistics
scoresSP <- data.frame(sppca$scores)
names(scoresSP) <- paste0("speed_pc", 1:4)
d <- cbind(runsSummary, scoresSP)

library("ggplot2")
## pc1 ~ session duration (moving)
d$durationMoving <- as.numeric(d$durationMoving)
ggplot(d) + geom_point(aes(x = durationMoving, y = speed_pc1)) + theme_bw() + labs(x = "duration moving [min]", y = "PC1")
@
%
  \end{subfigure}
  % ~
  \begin{subfigure}[b]{0.45\textwidth}
%
<<AppScoresPlot2, echo = FALSE, fig = TRUE, width = 4, height = 4>>=
## pc2 ~ avg speed (moving)
ggplot(d) + geom_point(aes(x = avgSpeedMoving, y = speed_pc2)) + theme_bw() + labs(x = "average speed moving [m/s]", y = "PC2")
@
%
  \end{subfigure}
  \caption{PC1 score vs.\ ``duration moving'' (left) and PC2 score vs.\ ``average
    speed moving''~(right).}
  \label{fig:app:score:speed}
\end{figure}

% \section{Concluding remarks} \label{sec:outro}

% The \pkg{trackeR} package provides classes and methods for importing and
% handling the irregularly sampled spatio-temporal data that arise from
% GPS-enabled tracking devices in sports. Currently it supports TCX and db3
% formats, but the package structure allows the easy inclusion of other
% formats. After careful processing, the data are stored in session-based,
% unit-aware, and operation-aware objects of class \class{trackeRdata}. Associated
% methods handle conversion between units of measurement, calculation of relevant
% summaries and the visualisation of tracking data. Time in zones, work capacity,
% and distribution and concentration profiles can also be readily calculated and
% used for further statistical analyses. An example of such analysis is presented
% in Section~\ref{sec:casestudy}, where functional principal components analysis
% is used to characterise the apparent variability between speed concentration
% profiles.

% Infrastructure for irregularly sampled spatio-temporal data in general is also
% provided in the \pkg{trajectories} package \citep{trackeR:pkg:trajectories},
% which is developed around the \class{STIDF} class of the \pkg{spacetime} package
% \citep{trackeR:Pebesma:2012}. In order to provide users with access to an as
% wide range of data-analytic methods as is possible, we aim to provide a sensible
% coercion function from \class{trackeRdata} to \class{STIDF}. One of the
% challenges we are facing is that the \class{STIDF} class does not
% accommodate missing values in positional or temporal information, which is
% commonly the case in data from GPS-enabled tracking devices (e.g., sequences of
% missing values in the positional data because the GPS signal is temporarily
% lost). Other packages that offer tools for spatio-temporal data include
% \pkg{adehabtitatLT} \citep{trackeR:Calenge:2006}, \pkg{trip}
% \citep{trackeR:pkg:trip} and \pkg{move} \citep{trackeR:pkg:move}. The main focus
% of those packages is on animal tracking (e.g., estimation of habitat choices)
% and they are not directly suitable for tracking the various aspects of athlete activity.

% Areas for further development include the extension of the reading capabilities
% to more file formats and the implementation of more analytic tools, such as
% record power profiles \citep{trackeR:Pinot+Grappe:2011}. The case study in
% Section~\ref{sec:casestudy} draws some links to functional data analysis which
% is another area of further exploration for analytic tools for the package.

%\newpage \clearpage
\section*{Acknowledgements}

We are thankful to Victoria Downie, Andy Hudson, Louis Passfield, Ben
Rosenblatt, and Achim Zeileis for helpful feedback and discussions as well as
providing the data that are used for the illustrations and examples in the package.

\pagebreak
\clearpage

\bibliography{trackeR}


%%check journal and publisher names according to \url{http://www.jstatsoft.org/style}

%% Journals:
%%     The American Statistician (not: American Statistician)
%%     The Annals of Statistics (not: Annals of Statistics)
%%     Journal of the Royal Statistical Society B (not: Journal of the Royal Statistical Society, Series B)

%% Publishers:
%%     Springer-Verlag (not: Springer)
%%     John Wiley & Sons (not: Wiley, John Wiley & Sons Inc.)


%\newpage
\begin{appendix}

\section[Replenishment of W']{Replenishment of $W'$}
\label{sec:appendix}

Assuming that power is constant, the solution of the differential equation
describing the rate of replenishment in Equation~\ref{eq:recharge:rate} with respect
to $W'(t)$ gives
%
\begin{equation}
  \label{eq:recharge:2}
  1 - \frac{W'(t)}{W'_0} = \exp \left( \frac{P- CP}{W'_0}  t + D / W'_0 \right)
  \, .
\end{equation}
%
Using Equation~\ref{eq:recharge:2} over an interval $[t_{i-1}, t_i)$ of constant
power gives
%
\begin{equation*}
  1 - \frac{W'(t_i)}{W'_0} = \exp \left( \frac{P_i - CP}{W'_0}  (t_i -
    t_{i-1}) \right) \left( 1 - \frac{W'(t_{i-1})}{W'_0} \right) \, .
\end{equation*}
%
Hence, $W'(t_i)$ can be expressed in terms of $W'(t_{i-1})$ as
%
\begin{equation*}
   W'(t_i) = W'_0 - \left( W'_0 - W'(t_{i-1}) \right) \exp \left( \frac{P_i -
      CP}{W'_0}  (t_i - t_{i-1}) \right) \, .
\end{equation*}
%


%% , via the substitution rule for integrals in the
%% step from Equation~\ref{eq:recharge:substitution1}
%% to~\ref{eq:recharge:substitution2}, gives the following
%% %
%% \begin{align}
%%   \int{ \frac{d}{dt} W'(t) \frac{1}{1 -  W'(t)/W'_0} } \, dt &= \int{ (CP
%%     - P)} \, dt \label{eq:recharge:substitution1} \\
%%   \int{ \frac{1}{1 -  W'(t)/W'_0} } \, d W'(t) & = \int{(CP - P)} \, dt \label{eq:recharge:substitution2}\\
%%   \log \left( 1 - \frac{W'(t)}{W'_0} \right) (-W'_0) &= (CP- P)
%%   t + D  \nonumber \\
%%   1 - \frac{W'(t)}{W'_0} &= \exp \left( \frac{P- CP}{W'_0}  t + c \right)  \label{eq:recharge:2}
%% \end{align}
%% %
%% with $c = D / W'_0$.
%% Note that power is again assumed to be constant, as it is in the case of
%% depletion of $W'$. Applying this to an interval $[t_{i-1}, t_i)$ and setting
%% Equation~\ref{eq:recharge:2} for time points $t_{i-1}$ and $t_i$ in relation to
%% each other allows expressing $W'(t_i)$ in terms of $W'(t_{i-1})$:
%% %
%% \begin{align*}
%%   \frac{1 - \frac{W'(t_i)}{W'_0}}{1 - \frac{W'(t_{i-1})}{W'_0}} &= \frac{\exp
%%     \left( \frac{P_i - CP}{W'_0}  t_i + c \right) }{\exp \left( \frac{P_i -
%%         CP}{W'_0}  t_{i-1} + c \right) } \\
%%   1 - \frac{W'(t_i)}{W'_0} &= \exp \left( \frac{P_i - CP}{W'_0}  (t_i -
%%     t_{i-1}) \right) \left( 1 - \frac{W'(t_{i-1})}{W'_0} \right) \\
%%   W'_0 - W'(t_i) &= \exp \left( \frac{P_i - CP}{W'_0}  (t_i -
%%     t_{i-1}) \right) \left( W'_0 - W'(t_{i-1}) \right) \\
%%   W'(t_i) &= W'_0 - \left( W'_0 - W'(t_{i-1}) \right) \exp \left( \frac{P_i -
%%       CP}{W'_0}  (t_i - t_{i-1}) \right) \, .
%% \end{align*}
%% %


\end{appendix}


\end{document}
