# lfda: An R Package for Local Fisher Discriminant Analysis and Visualization

by Yuan Tang and Wenxuan Li

###### Abstract

Local Fisher discriminant analysis is a localized variant of Fisher
discriminant analysis and it is popular for supervised dimensionality
reduction method. lfda is an R package for performing local Fisher
discriminant analysis, including its variants such as kernel local Fisher
discriminant analysis and semi-supervised local Fisher discriminant analysis.
It also provides visualization functions to easily visualize the dimension
reduction results by using either rgl for 3D visualization or ggfortify for 2D
visualization in ggplot2 style.

## Introduction

Fisher discriminant analysis (Scholkopft and Mullert, 1999) is a popular
choice to reduce the dimensionality of the original data set. It maximizes
between-class scatter and minimizes within-class scatter. It works really well
in practice but lacks some considerations for multimodality. Multimodality
exists within many applications, such as disease diagnosis, where there may be
multiple causes for a particular disease. In this situation, Fisher
discriminant analysis cannot capture the multimodal characteristics of the
clusters. To deal with multimodality, local-preserving projection (Niyogi,
2004) preserves the local structure of the data in that it keeps nearby data
pairs in the original data space close in the embedding space. As a result,
multimodal data could be embedded and its local structure will not be lost.

Later on, a new dimensionality reduction method called local Fisher
discriminant analysis (Sugiyama, 2006) was proposed to combine both advantages
of Fisher discriminant analysis and those of local-preserving projection in a
way that between-class separability is maximized while within-class
separability is minimized and its local structure is preserved. Furthermore,
with the help of kernel trick, local Fisher discriminant analysis can also be
extended to deal with non-linear dimensionality reduction situations.

Principal components analysis (Jolliffe, 2002) is another popular choice for
performing dimension reduction. However, in practice sometimes principal
components analysis generates bad principal components that cannot explain a
great amount of variance in the original data set. For example, if the
original data set has six dimensions and we reduce the dimension into three
using PCA, the resulted three principal components might not capture some
important features and variance in the original data features. Using this
result from PCA sometimes lead to very bad accuracy for machine learning
models due to the poor pre-processing that losses a lot of the essential
information in the original data features. On the other hand, metric learning
methods such as LFDA in particular can, surprisingly, enhance the distinctive
characteristics of the original data set and pull data points that have
similar characteristics close to each other. However, both PCA and LFDA have
their own shortcomings and a combined approach called semi-supervised local
Fisher discriminant analysis (Sugiyama et al., 2010) mix the supervised and
unsupervised approaches to provide a more stable result. We will briefly
discuss this in the following sections.

In this paper, we will introduce an R package called lfda (Tang and Deane-
Mayer, 2016) . It’s is an R package for performing local Fisher discriminant
analysis as well as its variants such as kernel local Fisher discriminant
analysis and semi-supervised local Fisher discriminant analysis. It also
provides visualization functions to easily visualize the dimension reduction
results by using either rgl (Adler et al., 2016) for 3D visualization or using
ggfortify (Horikoshi and Tang, 2015) for 2D visualization in ggplot2 (Wickham,
2009) style.

## Theoretical background

### Fisher linear discriminant analysis (FDA)

Fisher linear discriminant analysis is a popular method used to find a linear
combination of features that characterizes or separates two or more classes of
objects and events.

Let $S^{(w)}$ and $S^{(b)}$ be the within-class scatter matrix and the
between-class scatter matrix defined by the following:

$S^{(w)}=\sum_{i=1}^{l}\sum_{j:y_{j}=i}(x_{j}-\mu_{i})(x_{j}-\mu_{i})^{T},$

and

$S^{(b)}=\sum_{i=1}^{l}n_{i}(\mu_{i}-\mu)(\mu_{i}-\mu)^{T},$

where $\mu_{i}$ is the mean of samples in the class $i$ and $\mu$ is the mean
of all samples, such that:

$\mu_{i}=\frac{1}{n_{i}}\sum_{j:y_{j}=i}x_{j},$
$\mu=\frac{1}{n}\sum_{i=1}^{n}x_{i}.$

The FDA transformation matrix, denoted as $T_{FDA}$, is defined as follows:

$T_{FDA}=argmax_{T\in\mathbb{R}^{d\times
m}}((T^{T}S^{(w)}T)^{-1}T^{T}S^{(b)}T).$

In other words, $T$ is found so that between-class scatter is maximized and
within-class scatter is minimized. In order to optimize this, $T_{FDA}$ can be
found by the following:

$T_{FDA}=(\phi_{1}|\phi_{2}|\cdots|\phi_{m}),$

where $\\{\phi_{i}\\}_{i=1}^{d}$ are the generalized eigenvectors associated
to the generalized eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}$ of the generalized
eigenvalue problem:

$S^{(b)}\phi=\lambda S^{(w)}\phi.$

### Locality-preserving projection (LPP)

Locality preserving projections are linear projective maps that arise by
solving a variational problem that optimally preserves the neighborhood
structure of the data set.

Let $A$ be the affinity matrix where its $(i,j)$-th element is the affinity
between two points $x_{i}$ and $x_{j}$. The elements in $A$ ia larger if two
points are close to each other. We define $A_{i,j}=1$ if $x_{j}$ is the
$k$-nearest neighbor of $x_{i}$ or vice versa while $A_{i,j}=0$ otherwise. The
LPP transformation matrix can then be defined as the following:

$T_{LPP}=argmin_{T\in\mathbb{R}^{d\times
m}}\frac{1}{2}\sum_{i,j=1}^{n}A_{i,j}||T^{T}x_{i}-T^{T}x_{j}||^{2}$

subject to

$T^{T}XDX^{T}T=I,$

where $D$ is the diagonal matrix with $i$-th diagonal element being

$D_{i,i}=\sum_{j=1}^{n}A_{i,j}.$

$T_{LPP}$ is found such that nearby data pairs in the original data space are
kept close in the embedding space. $T_{LPP}$ is given by

$T_{LPP}=(\phi_{d-m+1}|\phi_{d-m+2}|\cdots|\phi_{d}),$

where $\\{\phi_{i}\\}_{i=1}^{d}$ are the eigenvectors associated to the
eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}$ of the
following eigenvalue problem

$XLX^{T}\phi=\lambda XDX^{T}\phi,$

where

$L=D-A.$

### Local Fisher discriminant analysis (LFDA)

We combine the concepts for both FDA and LPP to define local Fisher
discriminant analysis. A local within-class scatter matrix

$S^{(w)}=\frac{1}{2}\sum_{i,j=1}^{n}W_{i,j}^{(w)}(x_{i}-x_{j})(x_{i}-x_{j})^{T},$
$S^{(b)}=\frac{1}{2}\sum_{i,j=1}^{n}W_{i,j}^{(b)}(x_{i}-x_{j})(x_{i}-x_{j})^{T},$

where

$W_{i,j}^{(w)}=\left\\{\begin{array}[]{ll}A_{i,j}/n_{l}&\mbox{if
}y_{i}=y_{j}=l,\\\ 0&\mbox{if }y_{i}\neq y_{j},\end{array}\right.$

$W_{i,j}^{(b)}=\left\\{\begin{array}[]{ll}A_{i,j}(1/n-1/n_{l})&\mbox{if
}y_{i}=y_{j}=l,\\\ 1/n&\mbox{if }y_{i}\neq y_{j}.\end{array}\right.$

Intuitively, we use the above equations to weight the values for each pair of
samples that belong to the same class. Namely, for sample pairs that are far
apart from each other in the same class, we give less weight/influence on the
two local scatter matrices.

The LFDA transformation matrix $T_{LFDA}$ is then defined as the following:

$T_{LFDA}=argmax_{T\in\mathbb{R}^{d\times
r}}\left[tr((T^{T}S^{(w)}t)^{-1}T^{T}S^{(b)}T)\right]$

The transformation matrix $T_{LFDA}$ is found so that data pairs that are near
each other in the same class become closer and data pairs in different classes
become far away from each other. Note that data pairs in the same class that
are far away from each other do not come closer.

### Kernel local Fisher discriminant analysis (KLFDA)

LFDA can be non-linearized by the kernel trick, which enables the algorithm to
operate in a high-dimensional feature space but avoids computing the
coordinates of the data in that space. It works by computing the inner
products between all pairs of data in the feature space. Kernel trick also
greatly reduces the costs of computation of the coordinates. This then becomes
kernel local Fisher discriminant analysis (Sugiyama, 2006).

One particular choice of kernel function is the Gaussian kernel defined as the
following:

$K(x,x^{\prime})=exp(-\frac{||x-x^{\prime}||^{2}}{2\sigma^{2}}),$

with $\sigma>0.$ $K$ is now the kernel matrix where its $(i,j)$-th element is
given by

$K_{i,j}=<\phi(x_{i}),\phi(x_{j})>=K(x_{i},x_{j}),$

which we then use to perform the regular LFDA.

### Semi-supervised local Fisher discriminant analysis (SELF)

The performance of supervised dimensionality reduction methods, such as LFDA,
tend to be degraded when only a small number of labeled samples are available.
Thus, the supervised methods overfit embedding spaces to the labeled samples.
On the other hand, unsupervised dimensionality reduction methods, such as
principal components analysis, preserve the global data structure really well.
Semi-supervised local Fisher discriminant analysis smoothly bridges LFDA and
PCA so that our reliance on the global structure of unlabeled samples and
information brought by labeled samples can be controlled.

## Software architecture

This package fully utilizes S3 generic functions
(http://adv-r.had.co.nz/S3.html) to recognize lfda objects obtained from
lfda(), klfda(), or self().

Each lfda object is a list consists of the following items:

  * •

T: the trained transformation matrix

  * •

Z: the transformed original data with reduced dimensionality

The following methods can be applied to lfda objects:

  * •

predict(): generate predictions on new data

  * •

print(): print out summary statistics of the lfda object, including the
trained transformation matrix and the original data after applying the
transformation matrix

  * •

plot(): generate 3D visualization with the help from rgl package

The integration with ggfortify recognizes the lfda class and then plots the
transformed data that’s similar to its PCA visualizations. We will illustrate
this by providing some examples in the next section.

## Illustrations

In the following sub-sections, we will perform LFDA and its variants on the
famous (Fisher’s or Anderson’s) iris data set that gives the measurements in
centimeters of the variables sepal length and width and petal length and
width, respectively, for 50 flowers from each of 3 species of iris, which are
Iris setosa, versicolor, and virginica.

### Local Fisher discriminant analysis

We pass the original iris samples and its labels into lfda() as x and y,
respectively, along with r that represents the desired dimensionality to
reduce the original data into and the metric that switches the type of
returned metric. The available types of metric are weighted, orthonormalized,
and plain. Once We trained a local Fisher discriminant analysis model, we can
then call predict() to transform new data set using the trained distance
matrix. We can then visualize the LFDA model in 3D by calling plot(). See
Figure 1 for an example visualization.

library(lfda)data(iris)x <\- iris[, -5]y <\- iris[, 5]r <\- 3model <\- lfda(x,
y, r, metric = "plain")transformedData <\- predict(model, iris[, -5])plot(x =
model, labels = iris[, 5])

Figure 1: Local Fisher discriminant analysis 3D visualization.

### Kernel local Fisher discriminant analysis

The following code performs kernel local Fisher discriminant analysis. We
first apply kmatrixGauss() provided in lfda to compute Gaussian kernel matrix
of the original data set, which maps the original data space to non-linear and
higher dimensions. We can then pass this kernel matrix into klfda() to perform
KLFDA. Similarly, we can plot the transformed data in 3D. See Figure 2 for an
example visualization.

library(lfda)data(iris)k <\- kmatrixGauss(iris[, -5])y <\- iris[, 5]r <\-
3model <\- klfda(k, y, r, metric = "plain")plot(x = model, labels = iris[, 5])

Figure 2: Kernel local Fisher discriminant analysis 3D visualization.

Note that for iris data set, the relationships between different classes are
linear. Kernel Local Fisher Discriminant Analysis is only aimed for capturing
non-linear relationships, especially when it comes to many different classes.
In this case, visualization of iris data set is poor because KLFDA is too
strong for capturing linear relationships. If using KLFDA for this kind of
data, later when it comes to classification or clustering tasks, the model
would very likely overfit the transformed data set.

### Semi-supervised local Fisher discriminant analysis

We can perform semi-supervised local Fisher discriminant analysis in a way
similar to LFDA. In addition, we have three more parameters to use:

  * •

beta: the degree of semi-supervisedness with value between 0 and 1 where 0
represents totally supervised (discard all unlabeled samples) and 1 represents
totally unsupervised (discard all labeled samples).

  * •

kNN: the parameter used in local scaling method

  * •

minObsPerLabel: the minimum number observations required for each different
label

Here we discard $10\%$ of the labels and perform semi-supervised local Fisher
discriminant analysis that smoothly bridges supervised LFDA and unsupervised
principal component analysis, by which a natural regularization effect can be
obtained when only a small number of labeled samples are available. See Figure
3 for an example 3D visualization.

library(lfda)data("iris")x <\- iris[, -5]y <\- iris[, 5]r <\- 3model <\-
self(x, y, beta = 0.1, r = 3, metric = "plain")plot(model, iris[, 5])

Figure 3: Semi-supervised local Fisher discriminant analysis 3D visualization.

### Visualize using ggfortify

lfda is integrated with ggfortify so you can use autoplot() to automatically
plot lfda objects in 2D with ggplo2 style. We can specify frame = TRUE to draw
the frame boundary as well as specifying frame.type for shape of the frame and
frame.colour for the column to be used to color the frame. More options can be
found in ggfortify documentation as well as the ggfortify paper on the R
Journal (Tang et al., 2016). All options available in clustering and PCA in
ggfortify can be applied to lfda objects. See Figure 4 for an example.

library(lfda)library(ggfortify)data(iris)model <\- self(iris[, -5], iris[, 5],
beta = 0.1, r = 3, metric = "plain")autoplot(model, data = iris, frame = TRUE,
frame.colour = ’Species’, frame.type = ’norm’)

Figure 4: Semi-supervised local Fisher discriminant analysis visualization
using ggfortify.

## Future development

In the future, users will be able to use methods exported from lfda inside dml
(Tang et al., 2015), which is a collection of distance metric learning
algorithm implementations so users can compare results from different
algorithms very easily. Any suggestions collected from StackOverflow, Github
issues, etc, as well as contributions are welcomed. The package is hosted on
Github at https://github.com/terrytangyuan/lfda.

## Summary

lfda provides a simple interface to conduct local Fisher discriminant analysis
and its various variants, such as kernel local Fisher discriminant analysis
and semi-supervised local Fisher discriminant analysis. It also provides
visualization functions to easily visualize the dimension reduction results by
using either rgl for 3D visualization or ggfortify for 2D visualization in
ggplot2 style.

## Acknowledgement

We sincerely thank all developers for their efforts behind the packages that
lfda depend on, namely, plyr (Wickham, 2011), rARPACK (Qiu et al., 2016), and
rgl (Adler et al., 2016). We thank Zachary Deane-Mayer for helping setting up
the package structure and continuous integration using Travis CI and thank Nan
Xiao for his initial work-in-progress code.

## References

  * Adler et al. (2016) D. Adler, D. Murdoch, and others.  _rgl: 3D Visualization Using OpenGL_ , 2016.  URL https://CRAN.R-project.org/package=rgl.  R package version 0.96.0.
  * Horikoshi and Tang (2015) M. Horikoshi and Y. Tang.  _ggfortify: Data Visualization Tools for Statistical Analysis Results_ , 2015.  URL http://CRAN.R-project.org/package=ggfortify.  R package version 0.1.0.
  * Jolliffe (2002) I. Jolliffe.  _Principal component analysis_.  Wiley Online Library, 2002.
  * Niyogi (2004) X. Niyogi.  Locality preserving projections.  In _Neural information processing systems_ , volume 16, page 153. MIT, 2004.
  * Qiu et al. (2016) Y. Qiu, J. Mei, and authors of the ARPACK library. See file AUTHORS for details.  _rARPACK: Solvers for Large Scale Eigenvalue and SVD Problems_ , 2016\.  URL https://CRAN.R-project.org/package=rARPACK.  R package version 0.11-0.
  * Scholkopft and Mullert (1999) B. Scholkopft and K.-R. Mullert.  Fisher discriminant analysis with kernels.  _Neural networks for signal processing IX_ , 1(1):1, 1999.
  * Sugiyama (2006) M. Sugiyama.  Local fisher discriminant analysis for supervised dimensionality reduction.  In _Proceedings of the 23rd international conference on Machine learning_ , pages 905–912. ACM, 2006.
  * Sugiyama et al. (2010) M. Sugiyama, T. Idé, S. Nakajima, and J. Sese.  Semi-supervised local fisher discriminant analysis for dimensionality reduction.  _Machine learning_ , 78(1-2):35–61, 2010.
  * Tang and Deane-Mayer (2016) Y. Tang and Z. Deane-Mayer.  _lfda: Local Fisher Discriminant Analysis_ , 2016.  URL https://github.com/terrytangyuan/lfda.  R package version 1.1.1.
  * Tang et al. (2015) Y. Tang, G. Tao, and X. Nan.  _dml: Distance Metric Learning in R_ , 2015.  URL https://CRAN.R-project.org/package=dml.  R package version 1.1.0.
  * Tang et al. (2016) Y. Tang, M. Horikoshi, and W. Li.  ggfortify: Unified interface to visualize statistical result of popular r packages.  _R JOURNAL_ , 8(2), 2016.
  * Wickham (2009) H. Wickham.  _ggplot2: elegant graphics for data analysis_.  Springer Science & Business Media, 2009.
  * Wickham (2011) H. Wickham.  The split-apply-combine strategy for data analysis.  _Journal of Statistical Software_ , 40(1):1–29, 2011.  URL http://www.jstatsoft.org/v40/i01/.

_Yuan Tang
Uptake Technologies, Inc.
600 West Chicago Ave, Chicago, IL 60654
United States
_<EMAIL_ADDRESS>
_Wenxuan Li
Department of Agricultural Economics, Purdue University
403 W State Street, West Lafayette, IN, 47907
United States
_<EMAIL_ADDRESS>


$S_{\mu}=\sum_{p=1}^{d-2}\epsilon_{\mu}^{(p)}S^{p},\;F_{\mu}=\sum_{p=1}^{d-2}\epsilon_{\mu}^{(p)}F^{p},\;\tilde{F}_{\mu}=\sum_{p=1}^{d-2}\epsilon_{\mu}^{(p)}\tilde{F}^{p},\;\;\epsilon^{(p)}\cdot
k=\epsilon^{(p)}\cdot\tilde{k}=0,\epsilon^{(p)}\cdot\epsilon^{(q)}=\delta^{pq}$,
$f_{\mu\nu}=\sum_{p,q=1}^{d-2}\epsilon_{\mu}^{(p)}\epsilon_{\nu}^{(q)}f^{pq},$
$\sum_{p=1}^{d-2}f^{pp}=0,$ and $k\cdot k=\tilde{k}\cdot\tilde{k}=0$.

The gauge-invariant fluctuations are
$\psi,\,\tilde{\tilde{E}},\,\tilde{F}_{\mu},\,\Phi=\phi-\partial_{d+1}(B-\partial_{d+1}E/2),{\bf\tilde{B}}=\tilde{B}-\partial_{d+1}\tilde{E},\,{\bf
S_{\mu}}=S_{\mu}-\partial_{d+1}F_{\mu}$ and $f_{\mu\nu}$. The vacuum
linearized equations of motion set $\tilde{\tilde{E}}=0,$
$\Phi=-(d-2)\psi,\tilde{F}=0,\partial_{d+1}^{2}\psi=0,\;\partial_{d+1}{\bf\tilde{B}}=0,\;2(d-1)\partial_{d+1}\psi+(k\cdot\tilde{k}){\bf\tilde{B}}=0,\;\partial_{d+1}{\bf
S_{\mu}}=0,\partial_{d+1}f_{\mu\nu}=0$. The $(d-1)(d-2)/2-1$ degrees of
freedom of the $d+1$ dimensional graviton are parametrized by the transverse
(in a $(d-1)$-spatial sense since $e_{\mu}^{(p)}=(0,\vec{e}_{\mu}^{(p)})$ )
and traceless tensor $f_{\mu\nu}$, the transverse vector ${\bf S}_{\mu}$ and
the scalar $\psi$, all of which are independent of the compact coordinate
$x^{d+1}$. This is what we expect to see when performing a Kaluza-Klein
reduction of $d+1$ gravity, in the massless sector.

Thus, in the zero-mode sector, for the on-shell linearized fluctuation, we can
reach a gauge where all the gauge-dependent terms are zero and write

$\displaystyle h_{\mu\nu}|_{g.i.}=2\psi\eta_{\mu\nu}+f_{\mu\nu}$
$\displaystyle h_{d+1\mu}|_{g.i.}={\bf S}_{\mu},\qquad
h_{d+1\,d+1}|_{g.i.}=2\Phi.$ (188)

which is of the form used in (195). The one caveat is that the transverse
tensors $f_{\mu\nu}$ are transverse in a $d-1$ sense for the zero modes, while
for the massive modes ($k_{d+1}\neq 0$ which implies $k\cdot k\neq 0$) the
tensor fluctuations are transverse in a $d$-sense (and the scalar and vector
fluctuations are zero).

##  Appendix E The $SO(1,2)$ SVT decomposition of the metric fluctuations
about a 4d flat background

In this appendix we want to use the familiarity of 4d gravity to study a less
standard way to decompose the metric fluctuations, namely using the $SO(1,2)$
background isometry rather than the rotational isometry $SO(3)$. As we will
see, unlike the $SO(3)$ case analyzed in Section IV A, the tensor modes are
not the only dynamical ones, and both scalar and vector modes contribute
together with the tensor modes to the gravitational energy-momentum tensor.

We begin by writing the metric perturbation as

$\displaystyle
h_{\mu\nu}=\left(\begin{array}[]{cc}2\psi\eta_{\bar{\mu}\bar{\nu}}+\partial_{\bar{\mu}}\partial_{\bar{\nu}}E+\partial_{\bar{\mu}}F_{\bar{\nu}}+\partial_{\bar{\nu}}F_{\bar{\mu}}+f_{\bar{\mu}\bar{\nu}}&\partial_{\bar{\nu}}B+S_{\bar{\nu}}\\\
\partial_{\bar{\mu}}B+S_{\bar{\mu}}&2\phi\end{array}\right),\,$ (191)

where we denote the 4d indices by $\mu,\nu=0,1,2,3$, while
$\bar{\mu},\bar{\nu}=0,1,2$. The gauge invariant pieces are $\Phi,\Psi,{\bf
S}_{\bar{\mu}}$ and $f_{\bar{\mu}\bar{\nu}}$, where

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\phi-\partial_{3}(B-\tfrac{1}{2}\partial_{3}E)$
$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\psi$ $\displaystyle{\bf
S}_{\bar{\mu}}$ $\displaystyle=$ $\displaystyle
S_{\bar{\mu}}-\partial_{3}F_{\bar{\mu}}\,.$ (192)

As in Abramo:1997hu  we can restrict to the gauge-invariant fluctuations by
going to the gauge:

$\displaystyle
h_{\mu\nu}|_{g.i.}=\left(\begin{array}[]{cc}2\Psi\eta_{\bar{\mu}\bar{\nu}}+f_{\bar{\mu}\bar{\nu}}&{\bf
S}_{\bar{\nu}}\\\ {\bf S}_{\bar{\mu}}&2\Phi\end{array}\right).$ (195)

We apply the same SVT $SO(1,2)$ decomposition to the Einstein equations
$G_{\mu\nu}=T_{\mu\nu}$:

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}2G^{(Y)}\eta_{\bar{\mu}\bar{\nu}}+\partial_{\bar{\mu}}\partial_{\bar{\nu}}G^{(\operatorname{LL})}+\partial_{\bar{\mu}}G^{(L)}_{\bar{\nu}}+\partial_{\bar{\nu}}G^{(L)}_{\bar{\mu}}+G^{(TT)}_{\bar{\mu}\bar{\nu}}&\;\;\;\partial_{\bar{\nu}}G^{(L)}_{3}+G^{(T)}_{\bar{\nu}3}\\\
\partial_{\bar{\mu}}G^{(L)}_{3}+G^{(T)}_{\bar{\mu}3}&\;\;\;2G_{33}\,,\end{array}\right)$
(198) $\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}2T^{(Y)}\eta_{\bar{\mu}\bar{\nu}}+\partial_{\bar{\mu}}\partial_{\bar{\nu}}T^{(\operatorname{LL})}+\partial_{\bar{\mu}}T^{(LT)}_{\bar{\nu}}+\partial_{\bar{\nu}}T^{(LT)}_{\bar{\mu}}+T^{(TT)}_{\bar{\mu}\bar{\nu}}&\;\;\;\partial_{\bar{\nu}}T^{(L)}_{3}+T^{(T)}_{\bar{\nu}3}\\\
\partial_{\bar{\mu}}T^{(L)}_{3}+T^{(T)}_{\bar{\mu}3}&\;\;\;2T_{33}\end{array}\right).$
(201)

The linearized equations of motion for the scalar fluctuations $\Phi$ and
$\Psi$ come from the components $\delta^{(1)}G_{33}$,
$\delta^{(1)}G^{(L)}_{3}$, $\delta^{(1)}G^{(Y)}$and
$\delta^{(1)}G^{(\operatorname{LL})}$:

$\displaystyle 2\delta^{(1)}G_{33}$ $\displaystyle=$ $\displaystyle
2\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}\Psi=2T_{33}$
$\displaystyle\delta^{(1)}G^{(L)}_{3}$ $\displaystyle=$
$\displaystyle-2\partial_{3}\Psi=T^{(L)}_{3}$ $\displaystyle
2\delta^{(1)}G^{(Y)}$ $\displaystyle=$ $\displaystyle
2\partial_{3}^{2}\Psi+\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}\Phi+\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}\Psi=2T^{(Y)}$
$\displaystyle\delta^{(1)}G^{(LL)}$ $\displaystyle=$
$\displaystyle-\Phi-\Psi=T^{(\operatorname{LL})}\,.$ (202)

The equations of motion for the transverse vector ${\bf S}_{\bar{\mu}}$ come
from the components $\delta^{(1)}G^{(T)}_{\bar{\mu}3}$ and
$\delta^{(1)}G^{(LT)}_{\bar{\mu}}$:

$\displaystyle\delta^{(1)}G^{(T)}_{\bar{\mu}3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\beta}{\bf
S}_{\bar{\mu}}=T^{(T)}_{\bar{\mu}3}$
$\displaystyle\delta^{(1)}G_{\bar{\mu}}^{(LT)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{3}{\bf
S}_{\bar{\mu}}=T^{(LT)}_{\bar{\mu}}\,.$ (203)

Lastly, the equation of motion for the transverse traceless tensor
${f}_{\bar{\mu}\bar{\nu}}$ comes from
$\delta^{(1)}G^{(TT)}_{\bar{\mu}\bar{\nu}}$,

$\displaystyle\delta^{(1)}G^{(TT)}_{\bar{\mu}\bar{\nu}}=-\frac{1}{2}\partial_{3}^{2}{f}_{\bar{\mu}\bar{\nu}}-\frac{1}{2}\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}{f}_{\bar{\mu}\bar{\nu}}=T^{(TT)}_{\bar{\mu}\bar{\nu}}\,.$
(204)

Next, we count the degrees of freedom by considering the vacuum equations of
motion:

$\displaystyle\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}\Psi=0\,,\qquad\partial_{3}\Psi=0\,,\qquad\qquad\Phi=-\Psi$
$\displaystyle\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}{\bf
S}_{\bar{\mu}}=0\,,\qquad\partial_{3}{\bf S}_{\bar{\mu}}=0\,,$
$\displaystyle\partial_{3}^{2}{f}_{\bar{\mu}\bar{\nu}}+\eta^{\bar{\alpha}\bar{\beta}}\partial_{\bar{\alpha}}\partial_{\bar{\beta}}{f}_{\bar{\mu}\bar{\nu}}=0\,.$
(205)

Due to the constraints $\partial_{3}\Psi=0$ and $\partial_{3}{\bf
S}_{\bar{\mu}}=0$, it is natural to separately consider the case $p_{3}=0$ and
the case $p_{3}\neq 0$. When $p_{3}=0$: ${\Psi}$ describes a 3d massless
scalar, which has one degree of freedom; ${\bf S}_{\bar{\mu}}$ describes a 3d
massless vector, which has one degree of freedom; ${f}_{\bar{\mu}\bar{\nu}}$
describes a 3d massless graviton, which has zero degrees of freedom. When
$p_{3}\neq 0$: ${\Psi}$ has no solution, hence its degree of freedom is 0;
${\bf S}_{\bar{\mu}}$ has no solution and its degree of freedom is 0;
${f}_{\bar{\mu}\bar{\nu}}$ describes a 3d massive graviton, which has two
degrees of freedom. This again adds up to the correct number of degrees of
freedom of the 4d graviton. However, our analysis was not rigorous. If
$p_{3}=0$ then the equations of motion require that the 3-momentum $p_{\mu}$
is null ($\eta^{\bar{\mu}\bar{\nu}}p_{\bar{\mu}}p_{\bar{\nu}}=0$) which means
that the SVT decomposition starting point (191) is invalid. Nonetheless, as we
showed in Appendix D, the conclusion reached here stands: the two degrees of
freedom of the 4d graviton are the massless scalar $\Psi$ and the massless
vector ${\bf S}_{\bar{\mu}}$.

The point to be made is that unlike the $SO(3)$ SVT decomposition, where only
the $SO(3)$-tensor fluctuation is dynamical, when performing an $SO(1,2)$
decomposition all types of fluctuations (scalar, vector and tensor) are
dynamical. Thus, the energy-momentum tensor can receive contributions from all
three types of fluctuations. If the $x^{3}$ dimension were compact, the
Fourier spectrum along $x^{3}$ is discrete and indeed all three types of
fluctuations do contribute, with the scalar and vector modes part of the
massless sector of a Kaluza-Klein reduction. If on the other hand the $x^{3}$
dimension is non-compact, the Fourier spectrum along $x^{3}$ is continuous and
only the tensor modes contribute to the radiated power as we will see in
Appendix F.

##  Appendix F The $SO(1,2)$ SVT modes sourced by a binary in 4d flat space
and the luminosity of the gravitational waves

In this appendix we derive concrete expressions for the $SO(1,2)$ SVT modes in
4d flat space, asymptotically far away from sources, and verify that we
correctly reproduce known results for the radiated power using the results
from Section II.

We begin with a known form of the 4d metric perturbation far away from a
binary source (see for example eqn. (5.24) in Du:2020rlx ). Keeping only the
time-dependent parts, $h_{\mu\nu}$, which are the relevant pieces for the
computation of the radiated power, we write

$h_{\mu\nu}=\tilde{h}_{\mu\nu}-\tfrac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta}\tilde{h}_{\alpha\beta}=\left(\begin{array}[]{cccc}\frac{1}{2}\widetilde{h}_{00}&-n_{1}\tilde{h}_{11}-n_{2}\tilde{h}_{12}&-n_{1}\tilde{h}_{12}-n_{2}\tilde{h}_{22}&0\\\
-n_{1}\tilde{h}_{11}-n_{2}\tilde{h}_{12}&\tilde{h}_{11}+\frac{1}{2}\tilde{h}_{00}&\tilde{h}_{12}&0\\\
-n_{1}\tilde{h}_{12}-n_{2}\tilde{h}_{22}&\tilde{h}_{12}&\tilde{h}_{22}+\frac{1}{2}\tilde{h}_{00}&0\\\
0&0&0&\frac{1}{2}\tilde{h}_{00}\end{array}\right)\,,$ (206)

where $\tilde{h}_{\mu\nu}$ is the so-called trace-reversed metric fluctuation,
and $n_{i}=x^{i}/r$ are the Cartesian components of a radial pointing, unit
vector $\vec{n}=\vec{r}/r$. In writing (206) we have used that for the binary
solution $\tilde{h}_{3\mu}=0$ and $\tilde{h}_{11}+\tilde{h}_{22}=0$. Note that
the $x^{3}$ direction is perpendicular to the plane of the binary. This metric
perturbation satisfies the harmonic gauge
$\eta^{\mu\nu}\partial_{\mu}\tilde{h}_{\nu\rho}=0$, which implies that
$\tilde{h}_{00}$ can be written as

$\displaystyle\tilde{h}_{00}=(n_{1})^{2}\tilde{h}_{11}+2n_{1}n_{2}\tilde{h}_{12}+(n_{2})^{2}\tilde{h}_{22}\,.$
(207)

Each of the components $\tilde{h}_{00}$, $\tilde{h}_{11}$, $\tilde{h}_{12}$,
$\tilde{h}_{22}$ are in the form of spherical waves. For example,
$\tilde{h}_{12}=\mu r_{12}^{2}\Omega^{2}\sin(2\Omega(t-r))/(2\pi r)$, where
$\mu$ is the reduced binary mass, $r_{12}$ is the separation distance between
the binary components, $\Omega$ is the angular frequency of the binary, and we
set $8\pi G=1$.

Next let us consider the $\operatorname{SO}(1,2)$ decomposition of the
perturbation (191) and the gauge-invariant fluctuations given in (192). We
recall that $\bar{\mu},\bar{\nu}=0,1,2$. Later we will also use indices
$\bar{i},\bar{j}=1,2$. From (206), we can see that $B=0$ and
$S_{\bar{\mu}}=0.$ In terms of the SVT decomposed fields, the harmonic gauge
condition
$\partial_{\mu}\tilde{h}^{\mu\bar{\nu}}=\partial_{\bar{\mu}}\tilde{h}^{\bar{\mu}\bar{\nu}}=\partial_{\bar{\mu}}(h^{\bar{\mu}\bar{\nu}}+(1/2)\eta^{\bar{\mu}\bar{\nu}}\tilde{h})=0$
becomes

$\displaystyle\partial_{\bar{\nu}}(\Box_{\text{3d}}E+2\Psi-2\phi)+\Box_{\text{3d}}F_{\bar{\nu}}=0,$
(208)

where
$\Box_{\text{3d}}=-(\partial_{0})^{2}+(\partial_{1})^{2}+(\partial_{2})^{2}$.
We also used that

$\displaystyle
2\phi=\tfrac{1}{2}\tilde{h}_{00},\qquad\tilde{h}=\tilde{h}_{\mu\nu}\eta^{\mu\nu}=-\tilde{h}_{00}\,.$
(209)

By tracing $h_{\bar{\mu}\bar{\nu}}$ with $\eta^{\bar{\mu}\bar{\nu}}$ we learn

$\displaystyle\Box_{\text{3d}}E+6\Psi=2\phi\,.$ (210)

From (208) and (210) we infer that
$\Box_{\text{3d}}F_{\bar{\nu}}=4\partial_{\bar{\nu}}\Psi$, which given the
transversality of $F_{\bar{\nu}}$ implies $\Box_{\text{3d}}\Psi=0$. However,
given the spherical symmetry of the problem we are led to conclude $\Psi=0$,
and $F_{\bar{\nu}}=0$.

Thus, the $\operatorname{SO}(1,2)$ decomposition becomes

$\displaystyle h_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\partial_{\bar{\mu}}\partial_{\bar{\nu}}E+f_{\bar{\mu}\bar{\nu}}&0\\\
0&2\phi\end{array}\right)\,.$ (213)

Next we solve for $E$ and $f_{\bar{\mu}\bar{\nu}}$ by further assuming $E$ is
in the form of a spherical wave. We can verify our assumption later by
checking the consistency of our solution. This method works when the solution
is supposed to be unique. With this assumption for $E$ and keeping everything
to leading order in $1/r$, we have

$\displaystyle\Box_{\text{3d}}E\simeq(n_{3})^{2}\omega^{2}E\simeq\frac{1}{2}\tilde{h}_{00}\,.$
(214)

Thus, to leading oder in $1/r$

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{2(n_{3})^{2}\omega^{2}}\tilde{h}_{00}\,,$
$\displaystyle f_{\bar{\mu}\bar{\nu}}$ $\displaystyle=$ $\displaystyle
h_{\bar{\mu}\bar{\nu}}-\partial_{\bar{\mu}}\partial_{\bar{\nu}}E=h_{\bar{\mu}\bar{\nu}}+\frac{n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}\tilde{h}_{00}\,,$
(215)

where we have defined

$\displaystyle n_{\bar{\mu}}=(-1,n_{1},n_{2})\,.$ (216)

Since $(n_{1})^{2}+(n_{2})^{2}+(n_{3})^{2}=1$ we have

$\displaystyle\eta^{\bar{\mu}\bar{\nu}}n_{\bar{\mu}}n_{\bar{\nu}}=-(n_{3})^{2}\,.$
(217)

We are now checking that the gauge-invariant tensor fluctuation
$f_{\bar{\mu}\bar{\nu}}$ is indeed transverse and traceless. For
tracelessness, we have

$\displaystyle\eta^{\bar{\mu}\bar{\nu}}f_{\bar{\mu}\bar{\nu}}$
$\displaystyle=$
$\displaystyle\eta^{\bar{\mu}\bar{\nu}}h_{\bar{\mu}\bar{\nu}}+\frac{\eta^{\bar{\mu}\bar{\nu}}n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}\tilde{h}_{00}$
(218) $\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{h}_{00}-\frac{1}{2}\tilde{h}_{00}=0\,.$

For transversality,

$\displaystyle\partial^{\bar{\mu}}f_{\bar{\mu}\bar{\nu}}$ $\displaystyle=$
$\displaystyle\partial^{\bar{\mu}}h_{\bar{\mu}\bar{\nu}}+\frac{n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}\partial^{\bar{\mu}}\tilde{h}_{00}$
(219) $\displaystyle=$
$\displaystyle-n^{\bar{\mu}}\dot{h}_{\bar{\mu}\bar{\nu}}-\frac{n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}n^{\bar{\mu}}\dot{\tilde{h}}_{00}$
$\displaystyle=$
$\displaystyle-\dot{h}_{0\bar{\nu}}-n^{\bar{i}}\dot{h}_{\bar{i}\bar{\nu}}+\frac{1}{2}n_{\bar{\nu}}\dot{\tilde{h}}_{00}$

must vanish. Substituting $\bar{\nu}=0$ in (219) yields

$\displaystyle\partial^{\bar{\mu}}f_{\bar{\mu}0}$ $\displaystyle=$
$\displaystyle-\dot{h}_{00}-n^{\bar{i}}\dot{h}_{\bar{i}0}+\frac{1}{2}n_{0}\dot{\tilde{h}}_{00}$
(220) $\displaystyle=$
$\displaystyle-\frac{1}{2}\dot{\tilde{h}}_{00}-n^{\bar{i}}\dot{\tilde{h}}_{0\bar{i}}-\frac{1}{2}\dot{\tilde{h}}_{00}=0\,,$

where in the last step we used the transversality of $\tilde{h}_{\mu\nu}$ and
the spherical wave nature of the fluctuations. Similarly, if
$\bar{\nu}=\bar{j}$ we find

$\displaystyle\partial^{\bar{\mu}}f_{\bar{\mu}\bar{j}}$ $\displaystyle=$
$\displaystyle-\dot{h}_{0\bar{j}}-n^{\bar{i}}\dot{h}_{\bar{i}\bar{j}}+\frac{1}{2}n_{\bar{j}}\dot{\tilde{h}}_{00}$
(221) $\displaystyle=$
$\displaystyle-\dot{\tilde{h}}_{0\bar{j}}-n^{\bar{i}}\left(\dot{\tilde{h}}_{\bar{i}\bar{j}}+\frac{1}{2}\delta_{\bar{i}\bar{j}}\dot{\tilde{h}}_{00}\right)+\frac{1}{2}n_{\bar{j}}\dot{\tilde{h}}_{00}=0\,.$

This concludes the check on our asymptotic solution for
$f_{\bar{\mu}\bar{\nu}}$.

We compute next the gauge-invariant scalar $\Phi$. We find that it vanishes

$\displaystyle\Phi=\phi-\partial_{3}(B-\tfrac{1}{2}\partial_{3}E)=\phi+\tfrac{1}{2}(\partial_{3})^{2}E=0\,.$
(222)

We see that the relation which the two gauge-invariant scalars obey in vacuum
$\Phi=-\Psi$ is satisfied by our asymptotic solution. The vanishing of scalar
and vector modes may not be a coincidence but could be a general feature of
uncompactified flat spacetime. Since in the analysis performed in this
appendix the extra dimension ($x^{3}$) is non-compact, the spectrum of Fourier
modes of the gauge-invariant fluctuations is continuous. Because the scalar
and vector modes consist of only zero modes of the 3d d’Alembertian, they
vanish in such cases.

At last, we can compute the radiated power, substituting the gauge-invariant
fluctuations in the formula for the gravitational energy-momentum tensor. We
need ${\cal T}_{0\bar{i}}$ and ${\cal T}_{03}$ which can be easily extracted
from (28) substituting

$\displaystyle
h_{\mu\nu}\bigg{|}_{g.i.}=\left(\begin{array}[]{cc}f_{\bar{\mu}\bar{\nu}}&0\\\
0&0\end{array}\right).$ (225)

To compute the radiated power we start by integrating over a 3d sphere as in
(23) and use the same simplifications of turning spatial-derivatives into
time-derivatives (to leading order in $1/r$) when acting on spherical waves at
spatial infinity. Explicitly, we write:

$\displaystyle\langle P\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}n^{{i}}\mathcal{T}_{0{i}}|_{r=R_{\infty}}$ (226)
$\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}\frac{1}{4}\dot{f}_{\bar{\mu}\bar{\nu}}\dot{f}^{\bar{\mu}\bar{\nu}}$
$\displaystyle=$ $\displaystyle\frac{1}{4T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}\left(\dot{h}_{\bar{\mu}\bar{\nu}}+\frac{n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}\dot{\tilde{h}}_{00}\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}\left(\dot{\tilde{h}}_{\bar{\mu}\bar{\nu}}+\frac{1}{2}\eta_{\bar{\mu}\bar{\nu}}\dot{\tilde{h}}_{00}+\frac{n_{\bar{\mu}}n_{\bar{\nu}}}{2(n_{3})^{2}}\dot{\tilde{h}}_{00}\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}\left(\dot{\tilde{h}}_{\bar{\mu}\bar{\nu}}\dot{\tilde{h}}^{\bar{\mu}\bar{\nu}}+\frac{3}{4}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}+\frac{1}{4}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}+\eta^{\bar{\mu}\bar{\nu}}\dot{\tilde{h}}_{\bar{\mu}\bar{\nu}}\dot{\tilde{h}}_{00}+\frac{n^{\bar{\mu}}n^{\bar{\nu}}}{(n_{3})^{2}}\dot{\tilde{h}}_{\bar{\mu}\bar{\nu}}\dot{\tilde{h}}_{00}-\frac{1}{2}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{4T}\int_{0}^{T}\operatorname{dt}\int
d\Omega_{2}R_{\infty}^{2}\left(\frac{1}{2}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}-2\dot{\tilde{h}}_{0\bar{i}}\dot{\tilde{h}}_{0\bar{i}}+\dot{\tilde{h}}_{\bar{i}\bar{j}}\dot{\tilde{h}}_{\bar{i}\bar{j}}\right)$
$\displaystyle=$
$\displaystyle\frac{R_{\infty}^{2}}{20G}\langle\dot{\tilde{h}}_{\bar{i}\bar{j}}\dot{\tilde{h}}_{\bar{i}\bar{j}}\rangle=\frac{32}{5}G\mu^{2}\Omega^{6}(r_{12})^{4}\,,$

where we used the transversality of $\tilde{h}_{\mu\nu}$ and that the
fluctuations are spherical waves to set
$n^{\bar{\mu}}\dot{\tilde{h}}_{\bar{\mu}\bar{\nu}}=0$. In the last step we
reintroduced the dependence on Newton’s constant (recall that we have been
working with $8\pi G=1$). We have thus recovered a well known 4d result (see
for example footnote 10, with $D=4$, leading further to eqn. (6.11) in
Du:2020rlx ).

##  Appendix G The $SO(1,3)$ SVT modes sourced by a binary in 5d flat space
with compact $x^{5}$ and the luminosity of the gravitational waves

In this appendix we aim to recover the power radiated away by gravitational
waves in a 5d flat space, with one compact dimension $x^{5}\sim x^{5}+l$
Du:2020rlx . As discussed in Du:2020rlx , for a small extra dimension, the
contribution from the 5d graviton modes with $p_{5}\neq 0$ can be safely
ignored far away from the sources, since it is exponentially suppressed. This
fact can be easily understood from a 4d perspective where these modes appear
massive, with the 4d mass proportional to the $p_{5}$ momentum. Thus, the
radiated power receives its dominant contribution from 5d graviton modes with
$p_{5}=0$. This translates in 5d fluctuations which are independent of
$x^{5}$. Far away from a binary source, the metric fluctuations are given in
the equation (5.24) in Du:2020rlx , and the radiated power, which was computed
using Isaacson’s averaging scheme, is given in the equation (6.8) and (6.10)
in Du:2020rlx .

Given the symmetry of the problem, we proceed with performing an $SO(1,3)$ SVT
decomposition of the metric fluctuations.

$\displaystyle h_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\psi\eta_{\mu\nu}+\partial_{\mu}\partial_{\nu}E+\partial_{\mu}F_{\nu}+\partial_{\nu}F_{\mu}+f_{\mu\nu}\,,$
$\displaystyle h_{\mu 5}$ $\displaystyle=$
$\displaystyle\partial_{\mu}B+S_{\mu}\,,$ $\displaystyle h_{55}$
$\displaystyle=$ $\displaystyle 2\phi\,.$ (227)

The analog of the 4d trace-reversed fluctuations in 5d is $\tilde{h}_{MN}$
where

$\displaystyle\tilde{h}_{MN}$ $\displaystyle=$ $\displaystyle
h_{MN}-\frac{1}{2}\eta_{MN}h\,,\qquad h=h_{MN}\eta^{MN}\,,$ (228)
$\displaystyle h_{MN}$ $\displaystyle=$
$\displaystyle\tilde{h}_{MN}-\frac{1}{3}\eta_{MN}\tilde{h}\,\qquad\tilde{h}=\tilde{h}_{MN}\eta^{MN}\,.$
(229)

The 5d solution in Du:2020rlx  is of the form

$\displaystyle\tilde{h}_{MN}$ $\displaystyle\sim$
$\displaystyle\left(\begin{array}[]{cc}\tilde{h}_{\mu\nu}&0\\\
0&0\end{array}\right)\sim\frac{3}{4}\text{}\left(\begin{array}[]{cc}\tilde{h}^{(4d)}_{\mu\nu}&0\\\
0&0\end{array}\right)\,,$ (234)

where we used the squiggle line to indicate that we only take into account
components that are explicitly time dependent and drop the static (Coulombic)
metric fluctuation, just as we did in the previous Appendix F. The time-
independent terms are irrelevant in the computation of the radiated power.

Assuming no $x^{5}$-coordinate dependence, the harmonic gauge condition

$\displaystyle\partial^{M}\tilde{h}_{MN}$ $\displaystyle=$ $\displaystyle 0$
(235)

reduces to

$\displaystyle\partial^{\mu}h_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\nu}h\,.$ (236)

After comparing (227) with the solution in Du:2020rlx  we identify

$\displaystyle\Phi=\phi=\tfrac{1}{2}h_{55}=-\tfrac{1}{6}\tilde{h}\,.$ (237)

The harmonic gauge condition becomes

$\displaystyle\partial_{\nu}\left(\frac{1}{2}\Box_{\text{4d}}E-\frac{1}{2}h_{55}-2\psi\right)+\Box_{\text{4d}}F_{\nu}$
$\displaystyle=$ $\displaystyle 0\,.$ (238)

Next, while remaining in the harmonic gauge, we use the residual gauge freedom
to set $\Box_{\text{4d}}E$ to be 0. We use the gauge parameter

$\displaystyle\xi_{M}$ $\displaystyle=$ $\displaystyle(\xi_{0},0,0,0,0)\,.$
(239)

Since the harmonic gauge constrains $\xi^{M}$ by $\Box_{\text{4d}}\xi^{M}=0$,
then far away from the sources we take with $\xi_{0}$ to be of the form of a
(4d) spherical wave. After performing this gauge transformation, the new
metric is $h_{MN}^{\operatorname{new}}$:

$\displaystyle h^{(\operatorname{new})}_{MN}$ $\displaystyle=$ $\displaystyle
h_{MN}+\partial_{M}\xi_{N}+\partial_{N}\xi_{M}\,.$ (240)

We consider the trace of the 4d part of the metric perturbation

$\displaystyle\eta^{\mu\nu}h^{(\operatorname{new})}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta^{\mu\nu}h_{\mu\nu}-2\partial_{0}\xi_{0}=8\psi^{(\operatorname{new})}+\Box_{\text{4d}}E^{(\operatorname{new})}$
(241)

and we require that $\Box E^{(\operatorname{new})}=0$. We recall that since
$\psi$ is gauge-independent then $\psi^{(\operatorname{new})}=\psi=\Psi$. We
use the vacuum equations for the scalar fluctuations to relate $\Psi$ and
$\Phi$: $\Psi=-(1/2)\Phi$ and solve for the gauge-parameter from (241)

$\displaystyle 2\partial_{0}\xi_{0}$ $\displaystyle=$
$\displaystyle\eta^{\mu\nu}h_{\mu\nu}+2h_{55}=\eta^{\mu\nu}\tilde{h}_{\mu\nu}+2\tilde{h}_{55}-2\tilde{h}=-\tilde{h}\,.$
(242)

We proceed to compute the gauge-transformed metric perturbation:

$\displaystyle h^{(\operatorname{new})}_{00}$ $\displaystyle=$ $\displaystyle
h_{00}+2\partial_{0}\xi_{0}=\tilde{h}_{00}-\frac{2}{3}\tilde{h}\,,$
$\displaystyle h_{0i}^{(\operatorname{new})}$ $\displaystyle=$ $\displaystyle
h_{0i}+\partial_{i}\xi_{0}=\tilde{h}_{0i}+\frac{n_{i}}{2}\tilde{h}\,,$
$\displaystyle h_{ij}^{(\operatorname{new})}$ $\displaystyle=$ $\displaystyle
h_{ij}=\tilde{h}_{ij}-\frac{1}{3}\delta_{ij}\tilde{h}\,,$ $\displaystyle
h^{(\operatorname{new})}_{5\mu}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle h^{(\operatorname{new})}_{55}$ $\displaystyle=$ $\displaystyle
h_{55}=-\frac{1}{3}\tilde{h}\,.$ (243)

Now, we have the gauge invariant pieces

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\tilde{h}=\frac{1}{6}\tilde{h}_{00}\,,$ (244)
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\frac{1}{12}\tilde{h}=-\frac{1}{12}\tilde{h}_{00}\,,$ (245)
$\displaystyle{\bf S}_{\mu}$ $\displaystyle=$ $\displaystyle 0\,,$ (246)
$\displaystyle f_{\mu\nu}$ $\displaystyle=$ $\displaystyle
h_{\mu\nu}^{(\operatorname{new})}-\frac{1}{6}\tilde{h}\eta_{\mu\nu}\,.$ (247)

We can explictly check that $f_{\mu\nu}$ is indeed transverse and
traceless131313Since we are looking at zero modes, i.e. gravitational waves
with a null 4-momentum, the results of Appendix D apply. The tensor mode
$f_{\mu\nu}$ given in (247) contains a 4d transverse traceless gauge-dependent
term $\partial_{\mu}F_{\nu}+\partial_{\nu}F_{\mu}$, with $F_{\mu}$ such that
$\Box_{\text{4d}}F_{\mu}=0$ which follows from (238). To obtain the gauge-
invariant tensor, which is transverse and traceless in a 3d sense as explained
in Appendix D, we ought to remove this term. Nonetheless in computing the
radiated power such terms are harmless and yield no contribution, so we are
free to leave them packaged in $f_{\mu\nu}$ given in (247).. Finally we can
compute the power radiated by gravitational waves far away, at a distance
$R_{\infty}$, from a binary source using (128):

$\displaystyle\langle P\rangle$ $\displaystyle=$
$\displaystyle\frac{lR_{\infty}^{2}}{T}\int_{0}^{T}dt\int{d\Omega_{2}}\left(6\dot{\Psi}\dot{\Psi}+\frac{1}{4}\dot{f}_{\mu\nu}\dot{f}^{\mu\nu}\right)$
$\displaystyle=$
$\displaystyle\frac{lR_{\infty}^{2}}{4T}\int_{0}^{T}dt\int{d\Omega_{2}}\bigg{[}\left(\frac{4\cdot
6}{144}\,\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}\right)+\left(\dot{\tilde{h}}_{00}-\frac{1}{2}\dot{\tilde{h}}\right)\left(\dot{\tilde{h}}_{00}-\frac{1}{2}\dot{\tilde{h}}\right)$
$\displaystyle-2\left(\dot{\tilde{h}}_{0i}+\frac{n_{i}}{2}\dot{\tilde{h}}\right)\left(\dot{\tilde{h}}_{0i}+\frac{n_{i}}{2}\dot{\tilde{h}}\right)+\left(\dot{\tilde{h}}_{ij}-\frac{1}{2}\delta_{ij}\dot{\tilde{h}}\right)\left(\dot{\tilde{h}}_{ij}-\frac{1}{2}\delta_{ij}\dot{\tilde{h}}\right)\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{lR_{\infty}^{2}}{4T}\int_{0}^{T}dt\int{d\Omega_{2}}\bigg{[}+\left(\frac{1}{6}+\frac{9}{4}\right)\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}$
$\displaystyle+\left(-2\dot{\tilde{h}}_{0i}\dot{\tilde{h}}_{0i}-2\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}-\frac{1}{2}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}\right)+\left(\dot{\tilde{h}}_{ij}\dot{\tilde{h}}_{ij}+\frac{3}{4}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}\right)\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{lR_{\infty}^{2}}{4T}\int_{0}^{T}dt\int{d\Omega_{2}}\left(\frac{2}{3}\dot{\tilde{h}}_{00}\dot{\tilde{h}}_{00}-2\dot{\tilde{h}}_{0i}\dot{\tilde{h}}_{0i}+\dot{\tilde{h}}_{ij}\dot{\tilde{h}}_{ij}\right)$
$\displaystyle=$
$\displaystyle\frac{19\,l\,R_{\infty}^{2}}{360\,G_{\text{5d}}}\langle\dot{\tilde{h}}_{ij}\dot{\tilde{h}}_{ij}\rangle\,,$
(249)

where we recall that $l$ is the length of the compact dimension and where in
the last step we have restored the dependence on the 5d gravitational constant
$G_{\text{5d}}$ and used the transversality of the trace-reversed metric
$\tilde{h}_{\mu\nu}$. After accounting for an overall negative sign which we
introduced in our earlier definition of the radiated power (23), we have thus
recovered the previous 5d result given in the equation (6.8) in Du:2020rlx .
We also notice that the contributions from scalar and tensor fluctuation match
the corresponding parts in Einstein-Maxwell-dilaton theory respectively
Julie:2017rpw .

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# Earable and Wrist-worn Setup for Accurate Step Counting Utilizing Body-Area
Electrostatic Sensing

Sizhen Bian  0000-0001-6760-5539 PBL, D-ITET, ETH ZürichZürichSwitzerland
<EMAIL_ADDRESS>,  Rakita Strahinja  PBL, D-ITET, ETH
ZürichZürichSwitzerland<EMAIL_ADDRESS>,  Philipp Schilk  PBL, D-ITET, ETH
ZürichZürichSwitzerland<EMAIL_ADDRESS>,  Clénin Marc-André  PBL, D-ITET,
ETH ZürichZürichSwitzerland<EMAIL_ADDRESS>,  Silvano Cortesi  PBL,
D-ITET, ETH ZürichZürichSwitzerland<EMAIL_ADDRESS>,  Elio
Reinschmidt  PBL, D-ITET, ETH ZürichZürichSwitzerland
<EMAIL_ADDRESS>,  Kanika Dheman  PBL, D-ITET, ETH
ZürichZürichSwitzerland<EMAIL_ADDRESS>and  Michele Magno  PBL,
D-ITET, ETH ZürichZürichSwitzerland<EMAIL_ADDRESS>
(20xx)

###### Abstract.

Step-counting has been widely implemented in wrist-worn devices and is
accepted by end users as a quantitative indicator of everyday exercise.
However, existing counting approach (mostly on wrist-worn setup) lacks
robustness and thus introduces inaccuracy issues in certain scenarios like
brief intermittent walking bouts and random arm motions or static arm status
while walking (no clear correlation of motion pattern between arm and leg).
This paper proposes a low-power step-counting solution utilizing the body area
electric field acquired by a novel electrostatic sensing unit, consuming only
87.3 µW of power, hoping to strengthen the robustness of current dominant
solution. We designed two wearable devices for on-the-wrist and in-the-ear
deployment and collected body-area electric field-derived motion signals from
ten volunteers. Four walking scenarios are considered: in the parking
lot/shopping center with/without pushing the shopping trolley. The step-
counting accuracy from the prototypes shows better accuracy than the
commercial wrist-worn devices (e.g.,96% of the wrist- and ear-worn prototype
vs. 66% of the Fitbit when walking in the shopping center while pushing a
shopping trolley). We finally discussed the potential and limitations of
sensing body-area electric fields for wrist-worn and ear-worn step-counting
and beyond.

step counting, electric field, body capacitance, Qvar, wearable, earable

††copyright: acmlicensed††journalyear: 20xx††doi: XXXXXXX.XXXXXXX††conference:
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email; xx–xx, 20xx; Woodstock, NY††isbn: 978-1-4503-XXXX-X/18/06††ccs: Do Not
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##  1\. Introduction

Table 1. Sensing modalities explored for step counting Works |  Sensing Modality |  Deploy- ment |  Algorithm |  Accuracy |  Advantage |  Limitation
---|---|---|---|---|---|---
(Moore et al., 2020) |  accelerometer |  wrist, waist, thigh |  filter, peak detector |  7-11%(wrist) |  compact, low-power, robust |  imperfect in accuracy(wrist)
(Ozcan and Velipasalar, 2015) |  phone camera |  hold in hand |  image edge detection |  Avg. 96.94% |  more accurate and robust |  lack of practicality caused by privacy issue, power consumption, etc.
(Ngueleu et al., 2019) |  pressure |  insole |  cumulative sum of FSR |  95.5-98.5%(indoor), 96.5-98.0%(outdoor) |  high accuracy |  cost and wearability
(Łaś and Wiśniowski, 2021) |  magnetic |  wrist |  extrema detector, comparator |  98% |  high accuracy, low cost, compact |  arm swing dependant(like accelerometer), environment dependant
(Xu et al., 2018) |  WIFI |  environ- ment |  CSI time frequency analysis |  90.2%(laboratory), 87.59%(classroom) |  device-free |  environment dependant
This works |  Body-area electric field |  wrist/head |  filter, peak detector |  Avg. 93%(wrist), 94%(head) |  deployment-place free, compact, low-power |  environment dependant

Daily steps have been proven to have an inverse dose-response relationship
with important health outcomes like all-cause mortality and cardiovascular
events (Kraus et al., 2019; Jayedi et al., 2022), thus becoming an important
factor in promoting a healthy daily routine(Hallam et al., 2018). The
functionality of step counting has been widely implemented by wearable
devices, such as fitness trackers and smartwatches, to provide users with
timely feedback on their steps. Accurate step counting is crucial for
individuals who rely on wearable devices to quantify their exercise and
achieve fitness goals. However, existing step-counting methods based solely on
inertial measurement units can be non-robust as they primarily detect the
swinging motion of the device attached to a certain body part, such as the
wrist, rather than directly measuring the leg lift and drop action. This
reliance on the swinging motion of the device can lead to false positive
errors, especially in scenarios where wrist movement is not directly
correlated with leg movement, e.g., when users wave their arms without doing a
step action. A second error source of the current inertial sensor-based
approach is a result of brief intermittent walking bouts, e.g. when users walk
slowly or when they stop and start walking again, the inertial unit-based
counting will be unreliable as the wrist movement is irregular during such
period. In (Toth et al., 2019), the authors evaluated the effects of brief
intermittent walking bouts on the step-counting accuracy of wearable devices
with commercial trackers worn on the wrist (four), hip (four), and ankle
(two). They concluded that most methods required stepping bouts of ¿6–10
consecutive steps to record steps. Rest intervals of 1–2 seconds were
sufficient to break up walking bouts in many methods. The requirement for
several consecutive steps in some methods causes an underestimation of steps
(false negative) in brief, intermittent bouts.

Some specific studies have been published to evaluate how valid current
wearable devices are for measuring steps. In (An et al., 2017), the authors
used ten commercial wrist-worn activity trackers and collected walking data
from thirty-five healthy individuals under three conditions: walking/jogging
on a treadmill, walking over-ground on an indoor track, and a 24-hour free-
living condition. The result shows that the mean absolute percentage error
(MAPE) score for all devices and speeds on a treadmill was 8.2% against
manually counted steps, and the MAPE value was higher for over-ground walking
(9.9%) and even higher for the 24-hour free-living period (18.48%) on step
counts. A similar evaluation was carried out in (Tophøj et al., 2018), where
four commercial activity trackers, Fitbit Surge (FS), Fitbit Charge HR (FC),
Microsoft Band 2 (MB), and AD 101NFC Activity Monitor (AD), were evaluated at
2, 4, 4.5, and 5.5km/h. The experiment of twenty healthy participants walked
on a treadmill in two trials of 100 steps each showed that the MAPE levels
were between 8% and 6% for FS, 15% and 0% for MB, 7% and 21% for FC, and 53%
and 1% for AD. The biggest inaccuracies were seen at 2kmh, where AD was
underestimated by 53%. MB, FS, and AD accurately counted steps when
participants walked with velocities corresponding to a brisk walk. Walking at
lower speeds was not counted accurately.

To achieve more robust step counting with the wearables and address the gap
between inertial sensor readings and real steps, extra step-sensing methods
need to be implemented to supply alternative information on steps when the
inertial sensor reading fails to capture a real step or distinguish between
false positives and true positives.

##  2\. Related Work

Robustness study for well-being has been impressively growing in recent years,
especially along with the advanced machine learning models (Meegahapola et
al., 2024), e.g., by personalizing the models (Meegahapola et al., 2023). In
this paper, we tried to strengthen the robustness of the current IMU-based
step-counting method by introducing an effective complementary sensing
approach. Several studies using different sensing approaches other than the
dominant inertial sensors for step counting have been explored in the past
decade, as Table 1 lists. In (Ozcan and Velipasalar, 2015), the authors
present a robust and reliable method for counting footsteps using videos
captured with a Samsung Galaxy S4 smartphone. With a comparative study, the
proposed method achieved an impressive step counting performance of 3.064% in
average error. In (Ngueleu et al., 2019), an insole equipped with pressure
sensors was introduced for more accurate step detection. With twelve healthy
participants at self-selected and maximal walking speeds in indoor and outdoor
settings, the system accurately detected steps with success rates ranging from
95.5% to 98.5% (indoor) and from 96.5% to 98.0% (outdoor) for self-selected
walking speeds. Despite high accuracy, the implementation cost of such a
sensing system limits its usage scenarios and makes it fall behind compared to
the current inertial sensor-based solution. A cost-comparative method was also
proposed by Patryk et al. in (Łaś and Wiśniowski, 2021), where a magnetic
field sensor-based approach to detect and count steps was described. Outdoor
experiments showed that the proposed detection method achieves up to 98%
accuracy in step counting. Compared with the inertial sensor solution, this
magnetic field-based solution also depends on the arm swing during walking;
thus, errors from the brief intermittent walking bouts still exist. In (Xu et
al., 2018), the authors leveraged ubiquitous WiFi signals and proposed a
device-free step-counting system named WiStep, realizing an overall accuracy
of 90.2% in the laboratory and 87.59% in the classroom. The presented WiStep
enjoys the advantage of device-free sensing but is limited by the usage space
where efficient CSI information must be extracted from available wifi signals.

In this work, we proposed a novel sensing modality based on the body surface
differential electrostatic charge for accurate step counting, trying to
address errors of the inertial sensor-based solution, like the false positive
during leg-static activities or false negative during intermittent walking.
Electrostatic charge on the human body, a fundamental principle of physics,
has found intriguing applications for wearables (Dheman et al., 2022; Irrera
et al., 2024). When a person moves, especially during physical activities, the
body generates electrostatic charges due to friction between surfaces. These
charges can be harnessed and measured by specialized sensors (Bian et al.,
2024). As the body is a perfect conductive object, the surface charge flow can
be observed anywhere on the body, enabling cross-body part sensing. Some
previous studies have presented some preliminary studies using body-area
electric field for human activity recognition (Hu and Yan, 2022; Bian et al.,
2019b), but their explorations were limited to indoor environments and lacked
a dedicated and comprehensive evaluation in step counting. Leveraging the
principles of electrostatic charge promises to revolutionize the capabilities
of wearable devices, ushering in a new era of personalized and accurate
activity tracking.

The paper showcases how cross-body part sensing functionality revolutionizes
step sensing by eliminating the need for specific deployment positioning, thus
offering a novel method for strengthening robustness and achieving advanced
accuracy in step counting. Overall, we bring the following contributions in
this work:

  1. (1)

We designed the body-area electric field sensing module based on an
electrostatic sensor, being able to sense the static electric field variation
during body motion. The module is then integrated into two wrist-worn and ear-
worn prototypes for body motion-sensing exploration.

  2. (2)

We collected the body-area electric field variation data set from ten
volunteers walking in/outside a shopping center with/without the shopping
trolley and evaluated the step-counting performance using the wrist-worn and
ear-worn prototypes. Results show that the static body-area electric field
sensing modality supplies an averaged accuracy of 93%(wrist-worn) and
94%(head/ear-worn) step counting, outperforming two commercial products, which
infers the feasibility of electric field-based step counting. We further
discussed the limitations of the current work and proposed future work in
sensing fusion.

Figure 1. Left: Capacitor-like modeling of the human body describes the Body-
area electric field, capacitive coupling through the air, shoe soles, and
ground. $U$ = human body surface potential, $Q$ = human body surface charge,
$C$ = human body capacitance. Right: Data collection in/out of a shopping
center while pushing a shopping trolley

##  3\. Methods

###  3.1. Principle

Body-area electric field, on a narrow scale, infers the passive static
electric field between the human body and surroundings (especially the
ground), also defined as human body capacitance (Jonassen, 1998). Given the
fact that approximately sixty percent of the human body comprises water, it
inherently exhibits conductive properties, being able to store charges. Being
insulated by the clothing (especially the shoes), a surface potential related
to the environment arises as a result of the charge distribution difference
between the body and the environment, thus forming a static, passive body-area
electric field (Jonassen, 1998; Bian et al., 2022b). Fig. 1 (left) depicts a
capacitor-like model of the human body, where $U$ = human body surface
potential, $Q$ = human body surface charge, and $C$ = human body capacitance.
During motion actions, like steps, the relative distance/overlapping area
change (between the body and surrounding) will cause the variation of the
human body capacitance, which results in the on-body surface charge flow, also
represented as body-area electric field variation. By sensing the body charge
flow pattern, certain body activities could be inferred. Recent years have
shown an increasing trend in such passive body-area static electric field
sensing exploration, like random motion sensing (Cohn et al., 2012), indoor
positioning (Tang and Mandal, 2019), exercise recognition (Bian et al., 2019a,
2022a), etc., based on the fact that such sensing is ultra-low power, low
cost, noninvasive, and enjoying the advantage of deployment position-free
compared to traditional inertial unit, being a promising approach for wearable
devices to accomplish certain tasks in human activity recognition and human-
computer interaction.

Figure 2. Hardware implementation. 1, The main board, including the key
components of a Qvar sensing unit, a low-power MCU, a lithium battery, and a
wireless charging module. 2, In-ear deployment where the main board is in the
case, and the electrode is underneath the cap. 3, Wrist-worn deployment where
the main board is in the case and the electrode is underneath the band. 4/5,
the real device-wearing scenarios. Figure 3. Raw signal in the four sub-
sessions from a volunteer’s single whole session. Top left: at the parking lot
without pushing the shopping trolley; top right: at the parking lot while
pushing the shopping trolley; bottom left: in the shopping center while
pushing the shopping trolley: bottom right: in the shopping center without
pushing the shopping trolley. The unit of Y-axis is 1.8/216(volt)

###  3.2. Hardware Implementation

Fig. 2 shows the hardware implemented in this study. The key components on the
main board include a Qvar sensing unit, a low-power u-blox ANNA-B112 module, a
lithium battery, and a wireless charging module. Qvar (STMicroelectronics, [n.
d.]), stands for electric charge (= Q) variation (= var), a newly released
electrostatic sensor from STMicroelectronics, has an electrical potential
sensing channel able to measure the quasi-electrostatic potential changes,
enabling applications such as contact and no-contact human motion detection,
human presence sensing, and user interface. Researchers have proved its
effectiveness in hand gesture recognition (Reinschmidt et al., 2022) with a
wrist-worn band composing multiple Qvar units and inertial units in a sensor
fusion way, and cardiac monitoring (Dheman et al., 2022) incorporated into a
wearable chest strap that can be integrated seamlessly under clothes. Qvar is
an ultra-low power unit, consuming only 87.3 µW of power and lasting multiple
weeks of operation using a coin cell battery. The processing unit on the
board, ANNA-B112, is a System-in-package (SiP) module around the ultra-low
power and high-performance nRF52832 System-on-Chip (SoC), featuring a state-
of-the-art BLE 5.4 interface, as well as an ARM Cortex-M4F MCU. A similar but
miniaturized design of the main board is designed to fit in a 3D-printed ear-
worn case, where the sensing electrode is put underneath the cap of the case,
forming a contactless in-the-ear sensing prototype, as 3/4 in Fig. 2 show.
Likewise, a wrist-worn prototype is designed with the sensing electrode
underneath the band and covered by insulating tape so that the wrist-worn
prototype senses the body surface potential variation also in a contactless
way. The Qvar signal data was sampled at 240 Hz and transmitted wirelessly
through the BLE module to a nearby personal computer.

###  3.3. Data Collection

To verify the capability of step counting with the developed prototypes in the
wild, we chose a general scenario of visiting a shopping center as a practical
everyday scenario. Ten young adults (five females and five males, ages 21 to
32) participated in the data collection activity, and each of them performed
one session of data collection. During the data collection, the subject wore
our prototypes on the wrist and in the ear and two additional commercial
activity trackers (Fitbit Inspire 3 and Xiaomi Smart Band 8, both the latest
in their series) on the other wrist. This setup is simply for user convenience
and will not influence the general counting result, as each submission lasts
around two minutes, and random left and right swing differences can be
ignored. Each session was composed of four sub-sessions. Starting from the
parking lot outside the shopping center, the subject walked normally for
around two minutes. Then, a shopping trolley was picked up and used in the
shopping center; this period also lasted around two minutes. The third sub-
session was normal walking without the shopping trolley in the shopping
center. Finally, the subject had another around two minutes of normal walking
outside the shopping center in the parking lot. Such a configuration aims to
test the sensor’s ability to count steps when, first, the wrist swings
normally during walking and relatively static during pushing a shopping
trolley; second, the environment changes, as this electric field-based sensing
modality describes the static electric relations between human body and
environment, thus, the environment plays a role in signal quality. During each
submission, a second accompanying subject used a smartphone to record the leg
actions, aiming to provide the ground truth. At the start and end of each sub-
session, the step readings of the commercial products were recorded for
comparison. Fig. 1(right) are the screenshots of the recorded data collection
video, showing the inside/outside normal walking while pushing a shopping
trolley. In the parking lot, the ground is composed of concrete bricks, and in
the shopping center, it is composed of wood. Fig. 3 depicts one session of the
Qvar signal composed of the four sub-sessions. As can be seen, the amplitude
of the sensed surface potential variation in the parking lot and shopping
center differs a lot (as the body-area electric field is an interactive signal
between the body and environment), but the step waves remain, and we extract
the step numbers from those wave signals.

Figure 4. One sub-session of Qvar signal and the corresponding power spectral
density describing the distribution of power into frequency components, which
indicates the dominant frequency during walking

##  4\. Evaluation

Like the commonly used algorithms dealing with inertial sensor data for step
counting (Khedr and El-Sheimy, 2017; Yang and Huang, 2015; Brajdic and Harle,
2013), here we use a bandpass filter and a peak detection algorithm for step
extraction. A lot of studies have concluded that the typical walking frequency
is below 2.0 Hz during a normal walk (Yang and Huang, 2015; Heinemann and
Kasperski, 2017; Nguyen et al., 2011). For example, in (Nguyen et al., 2011),
the authors described a biomedical research program that used the GAITRite
electronic walkway system with pressure-activated sensors to investigate the
basic spatial and temporal gait measures. The participants completed a series
of walks at self-selected free (normal), fast and slow gait speeds across the
GAITRite walkway. The obtained mean step frequency for the normal walk is 1.98
Hz with a standard deviation of 0.13 Hz, which compares well with values
reported by Bachmann and Ammann(1987) (Bachmann and Ammann, 1987) and the
European Commission(2006) (Commission et al., 2006). The frequency feature of
our collected step signal based on the body-area electric field also matches
previous findings. Fig 4 shows the power spectral density (PSD) of a one-
subsession Qvar signal, depicting the distribution of power with frequency
components and indicating the dominant frequency during normal walking. We set
the Butterworth bandpass filter with a lower cutoff frequency of 0.5Hz, a
higher cutoff frequency of 2.5Hz, and the order with 5. Following the filter,
we applied the peak detection algorithm, which is implemented by using the
$find\\_peaks$ in the scipy package (Virtanen et al., 2020), which has
multiple parameters, and we only use the prominence (vertical difference
between the peaks height itself and its lowest contour line) and distance
(required minimal horizontal distance in samples between neighboring peaks).
The tested prominence ranges from 200 to 500 with a step of 50, and the tested
distance ranges from 50 to 200 with a step of 50. A grid searching method is
used for each session to find the best parameter set and finally choose the
best set with major voting. Such a subject-undependable fine-tuning will
maximally guarantee the algorithm’s fairness, which has been a growing concern
in recent well-being studies (Yfantidou et al., 2023; Meegahapola et al.,
2024, 2023). Such a straightforward data mining pipeline will lead to a
window-based real-time step counting. Fig 5 shows the depicted steps with red
stars along with the filtered and raw signal in a parking lot sub-session. The
amplitude variation during the sub-session indicates the surrounding
instability from the environment. Since the step action (distance between feet
and ground) strongly influences the capacitance in between, signals of steps
still dominate the changes in sensor reading.

Table 2 listed the step counting result of each sub-session, including the
result of the two commercial devices (Fitbit Inspire 3 and Xiaomi Smart Band
8). The ear-worn Qvar unit from the first, seventh, and eighth subject in the
first submission (normal walking in the parking lot without pushing the
shopping trolley) is missed because of an unexpected break in the device-
computer BLE connection. The accuracy is calculated by the following equation:

(1) $Accuracy=1-\frac{abs(Measurement-Truth)}{Truth}$

Finally, the accuracies of all subjects are averaged as the final result of
each prototype and product. As can be seen, the Qvar-based prototype
outperforms in the scenarios when the subject is pushing the shopping trolley,
especially in the shopping center, with an average accuracy of 96% from both
the wrist-worn and ear-worn prototypes. The Xiaomi Band basically lost the
step counting accuracy when subjects are pushing the shopping trolley, this is
probably because that their step counting algorithm neglect the tiny regular
waves when the wrist is in a near-static status. However, it shows the best-
averaged accuracy during normal walking without the shopping trolley. The
wrist-worn prototype gives better counting accuracy in the shopping center,
while the ear-worn prototype performs better when pushing the shopping
trolley, but overall, they always win the Fitbit with all accuracies above
90%, which proves, first, the deployment location-free character (cross-body
part) of body-area electric field-based motion sensing, as the body is a
complete object, it doesn’t matter where the sensing unit is deployed on body;
The close accuracies between the two prototypes also support this principle.
Second, the body-area electric field-based sensing can be used for accurate
step counting in certain scenarios, especially when the inertial unit-based
solution fails, like during a very slight wrist swing.

Figure 5. Filtered signal marked with the detected peaks Table 2. Step counting result from commercial devices and introduced module deployed on wrist and head (step number / absolute accuracy) Subject |  Truth |  Xiaomi |  Fitbit |  WristQvar |  EarQvar
---|---|---|---|---|---
In parking lot without shopping trolley
1 |  160 |  169/0.94 |  164/0.98 |  160/1.00 |  xx/xx
2 |  160 |  164/0.98 |  174/0.91 |  170/0.94 |  165/0.97
3 |  159 |  152/0.96 |  180/0.87 |  177/0.89 |  115/0.72
4 |  164 |  167/0.98 |  179/0.91 |  171/0.96 |  160/0.98
5 |  159 |  160/0.99 |  181/0.86 |  169/0.94 |  159/1.00
6 |  215 |  211/0.98 |  247/0.85 |  160/0.74 |  176/0.82
7 |  204 |  206/0.99 |  233/0.86 |  210/0.97 |  xx/xx
8 |  205 |  228/0.89 |  226/0.90 |  150/0.73 |  xx/xx
9 |  176 |  173/0.98 |  195/0.89 |  178/0.99 |  177/0.99
10 |  166 |  166/1.00 |  184/0.89 |  168/0.99 |  165/0.99
Avg. |  |  0.97 |  0.89 |  0.91 |  0.92
In parking lot with shopping trolley
1 |  165 |  83/0.50 |  166/0.99 |  168/0.98 |  167/0.99
2 |  167 |  23/0.14 |  200/0.80 |  183/0.90 |  177/0.94
3 |  154 |  0/0.00 |  131/0.85 |  176/0.86 |  159/0.97
4 |  158 |  9/0.06 |  135/0.85 |  163/0.97 |  157/0.99
5 |  167 |  0/0.00 |  170/0.98 |  175/0.95 |  171/0.98
6 |  194 |  64/0.33 |  213/0.90 |  156/0.80 |  161/0.83
7 |  202 |  71/0.35 |  234/0.84 |  167/0.83 |  181/0.90
8 |  198 |  77/0.39 |  224/0.87 |  193/0.97 |  170/0.86
9 |  165 |  30/0.18 |  197/0.81 |  171/0.96 |  169/0.98
10 |  167 |  11/007 |  183/0.90 |  158/0.95 |  173/0.96
Avg. |  |  0.20 |  0.88 |  0.92 |  0.94
In shopping center with shopping trolley
Avg. |  |  0.01 |  0.66 |  0.96 |  0.96
In shopping center without shopping trolley
Avg. |  |  0.95 |  0.83 |  0.94 |  0.93

##  5\. Discussion

This work adopted a shopping activity composed of four subsections to explore
the potential of body-area electric field-based step counting and demonstrate
its feasibility. Nevertheless, it must be acknowledged that this sensing
modality is contingent upon the environment, as evidenced by the amplitude
disparities observed inside and outside the shopping center. The presence of
numerous electric appliances within the center amplifies the variations in the
electric field, rendering them more pronounced. We have observed that in some
places, for instance, in the huge empty training yard, the explored signal is
too weak to lose the motion sensing ability. This indicates that such a
sensing module might lead to issues in robustness. Thus, future work will
focus on explorations towards the robustness by, first, designing a front end
for Qvar or a new front end for electric field variation sensing so that the
sensing unit has a larger input impedance to sense very slight electric field
variation signal and increase the signal-to-noise ratio at the meantime;
second, developing a fusion solution combining the inertial unit and body-area
electric field sensing unit, as they are both ultra-low power, low-cost ideal
wearable components and shares different advantages in motion sensing, a
fusion method will potentially supply the best capability of human activity
recognition like step counting.

##  6\. Conclusion

This work proposed the body-area electric field-based sensing for step
counting. The main goal of the paper is to evaluate this emerging sensing
feature to improve step-counting accuracy in wearable devices. Two prototypes
(a wrist-worn and an ear-worn) were designed to evaluate the step-counting
proposal and verify the cross-body part motion sensing technique. Experimental
results have demonstrated counting accuracies of 91% to 96% in the shopping
activity composed of four subsessions(inside/outside a shopping center and
with/without pushing a shopping trolley), which outperforms the Fitbit Inspire
3 in all four subsessions and Xiaomi Smart Band 8 in two submissions. The
counting result indicates the feasibility of using body-area electric field
sensing for accurate step counting, considering the false positive and false
negative errors of traditional inertial sensor-based, as well as other
activity sensing. Future work will focus on the sensing front-end improvement
and sensor fusion way to increase the robustness of this sensing modality. All
materials from this work will be open-sourced aiming to promote further
exploration in body-area electric field-based motion sensing.

###### Acknowledgements.

This work was supported by the CHIST-ERA project ReHab(20CH21-203783).

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