# Propagating Residual Biases in Cosmic Shear Power Spectra

T. D. Kitching<EMAIL_ADDRESS>Mullard Space Science Laboratory,
University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK  P.
Paykari  Mullard Space Science Laboratory, University College London, Holmbury
St Mary, Dorking, Surrey RH5 6NT, UK  H. Hoekstra  Leiden Observatory, Leiden
University, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands  M. Cropper
Mullard Space Science Laboratory, University College London, Holmbury St Mary,
Dorking, Surrey RH5 6NT, UK

###### Abstract

In this paper we derive a full expression for the propagation of
multiplicative and additive shape measurement biases into the cosmic shear
power spectrum. In doing so we identify several new terms that are associated
with selection effects, as well as cross-correlation terms between the
multiplicative and additive biases and the shear field. The computation of the
resulting bias in the shear power spectrum scales as the fifth power of the
maximum multipole considered. Consequently the calculation is unfeasible for
large $\ell$-modes, and the only tractable way to assess the full impact of
shape measurement biases on cosmic shear power spectrum is through forward
modelling of the effects. To linear order in bias parameters the shear power
spectrum is only affected by the mean of the multiplicative bias field over a
survey and the cross correlation between the additive bias field and the shear
field. If the mean multiplicative bias is zero then second order convolutive
terms are expected to be orders of magnitude smaller.

##  1 Introduction

The statistical properties of the large-scale matter distribution over cosmic
time encodes key information about the late time evolution of the Universe,
and also allows us to improve constraints on the initial conditions. Thanks to
technological advances we can now efficiently survey larger and larger areas
of sky, but the interpretation of galaxy redshift surveys is hampered by the
fact that galaxies are biased tracers of the underlying dark matter
distribution. Fortunately, the distortion of space-time by matter results in
correlations in the ellipticities of distant galaxies that are the result of
the differential deflection of light rays, a phenomenon called gravitational
lensing. The statistics of these correlations can be directly related to those
of the large-scale structure. This in turn enables us to constrain the nature
of dark energy and to test gravity on cosmological scales.

The cosmological lensing signal has now been robustly measured using large
ground-based imaging surveys (e.g. Hildebrandt et al., 2018; Troxel et al.,
2018). However to shed light on the nature of dark energy, the precision needs
to increase significantly. This is the objective of a number of planned
projects that will commence soon. Euclid (Laureijs et al., 2011) aims to image
15 000 deg2 of extragalactic sky from space, while the Large Synoptic Survey
Telescope (LSST) will survey a similar area from the ground. To exploit fully
the potential of these data for cosmology, it is essential that astrophysical
and instrumental sources of biases are accounted for at levels that are small
compared to the statistical uncertainties on measured cosmological parameters.
Accurate measurements of the shapes of small, faint galaxies are therefore
essential.

The observed ellipticities of galaxies used in weak lensing studies are
typically biased with respect to the true ellipticities that would have been
measured given ideal data and an ideal measurement algorithm. The dominant
sources of bias are a result of the convolution by the point spread function
(PSF) and noise in the images. For this reason the performance of shape
measurements has been studied extensively (Heymans et al., 2006; Massey et
al., 2007; Bridle et al., 2010; Kitching et al., 2012; Mandelbaum et al.,
2015).

To first order the biases can be separated into multiplicative and additive
functions that act on the true shear. Additive biases arise from anisotropies
in the data, such as an anisotropic PSF or detector effects. These do not only
affect the measurement of the galaxy shape, but also the detection and
selection of sources. Characterizing and correcting for these sources of bias
is essential, but residual spurious alignments might still be removed through
empirical corrections. For instance, the mean shear when averaged in the
coordinate frame defined by the detector should vanish. The detection of a
coherent signal would thus indicate an imperfect correction, but that signal
could also be fitted for in the cosmological analysis. However we note that
this is only partially effective because sources of additive bias are expected
to introduce multiplicative biases with a similar amplitude. Unfortunately,
multiplicative bias cannot be readily inferred from the imaging data directly.
Instead image simulations are used to calibrate the biases in the shape
measurement algorithms (e.g. Hoekstra et al., 2015; Kannawadi et al., 2018),
although we note that alternative approaches have been recently proposed (Huff
& Mandelbaum, 2017a).

The desired accuracy in cosmological parameter estimates determines the level
at which shape measurement biases can be tolerated. The propagation of biases,
or residual biases after calibration, into the weak lensing power spectra (or
‘cosmic shear’ power spectrum) is not straightforward. Some studies
approximate the full expression (Taylor & Kitching, 2016; Kitching et al.,
2012; Massey et al., 2013) but, as we show in this paper, these results do not
capture the spatially varying sources of biases correctly. This is of
particular importance because the theoretical propagation of such biases into
power spectrum residuals drives the design requirements for experiments that
use weak lensing as a cosmological probe (Cropper et al., 2013).

In this paper we show how multiplicative and additive biases in shape
measurement propagate through the cosmic shear power spectra, discuss how this
formalism relates to previous studies, and discuss the implications for the
assessment of shape measurement biases on cosmological parameter performance
verification of experiments. In Section 2 we present the formalism, in Section
3 we present some simple simulations that demonstrate the accuracy of the
formalism, and in Section 4 we examine the implications of this study;
conclusions are presented in Section 5.

##  2 Method

We begin with the expression for the measured shear in real (angular) space

$\widetilde{\gamma}(\mathbf{\Omega})=[1+m_{0}+m(\mathbf{\Omega})]\gamma(\mathbf{\Omega})+[c_{1,0}+{\rm
i}c_{2,0}+c(\mathbf{\Omega})],$ (1)

where $\widetilde{\gamma}(\mathbf{\Omega})$ is the measured shear as a
function of angle $\mathbf{\Omega}=(\theta,\phi)$ where $\theta$ and $\phi$
are arbitrary spherical coordinates, $m_{0}$ is a constant multiplicative
bias, $m(\mathbf{\Omega})=m^{R}(\mathbf{\Omega})+{\rm
i}m^{I}(\mathbf{\Omega})$ is a position-dependent multiplicative bias term,
$\gamma(\mathbf{\Omega})=\gamma_{1}(\mathbf{\Omega})+{\rm
i}\gamma_{2}(\mathbf{\Omega})$ is the true shear, $c_{1,0}$ and $c_{2,0}$ are
constant additive biases that contribute to the real and imaginary parts of
the additive field, and $c(\mathbf{\Omega})$ is a position-dependent additive
bias. We assume no non-local terms, e.g.
$m(\mathbf{\Omega}^{\prime})\gamma(\mathbf{\Omega})$, since such terms could
be always re-written as a local per galaxy
$m(\mathbf{\Omega})\gamma(\mathbf{\Omega})$ term.

We discuss the choice of the multiplicative bias expression in Appendix A. We
note that the choice to express the multiplicative effect as a product of
complex numbers makes the spin-preserving assumption that any effect only
changes the amplitude and/or angle of the observed ellipse relative to the
unbiased case. There are more general expressions that can be used to capture
biases that can occur in the case of anisotropic systematic effects, however
image simulations suggest that our adopted approach is accurate for residual
systematic effects after calibration with image simulations. In the case where
no rotational change is present this reduces to a multiplication by a single
scalar field $m(\mathbf{\Omega})$. We will revisit these assumptions later in
the analysis.

###  2.1 Spherical Harmonic Representation

We now determine the spherical harmonic representation of a biased shear
field. We adapt the methodology from CMB pseudo-$C_{\ell}$ analysis here for
the general bias case; in particular we follow Lewis et al. (2002); Brown et
al. (2005); Zaldarriaga & Seljak (1997); Grain et al. (2012) but we generalize
their formalism further to include general spin-2 bias functions and additive
terms. In Kitching et al. (2012) a similar adaption was made, but under
simplifying assumptions that did not capture the general case.

Since for cosmic shear the cosmological information is contained within the
E-mode (gradient) component of the field, and not in the B-mode (curl)
component, we work on spherical harmonic coefficients $\gamma^{E}_{\ell m}$
and $\gamma^{B}_{\ell m}$. We can write down the E- and B-mode coefficients in
terms of the shear field as

$\displaystyle\gamma^{E}_{\ell m}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int{\rm
d}\mathbf{\Omega}\,[\gamma(\mathbf{\Omega})\,{}_{2}Y^{*}_{\ell
m}(\mathbf{\Omega})+\gamma^{*}(\mathbf{\Omega})\,{}_{-2}Y^{*}_{\ell
m}(\mathbf{\Omega})]$ $\displaystyle\gamma^{B}_{\ell m}$ $\displaystyle=$
$\displaystyle\frac{-{\rm i}}{2}\int{\rm
d}\mathbf{\Omega}\,[\gamma(\mathbf{\Omega})\,{}_{2}Y^{*}_{\ell
m}(\mathbf{\Omega})-\gamma^{*}(\mathbf{\Omega})\,{}_{-2}Y^{*}_{\ell
m}(\mathbf{\Omega})],$ (2)

where $\ell$ and $m$ are angular wavenumbers111Note that $m$ is used in the
spherical harmonic function, and $m_{0}$ and $m(\mathbf{\Omega})$ as
multiplicative biases; we choose to keep this standard notation for both cases
as the use should be clear from the context., ${}_{2}Y_{\ell
m}(\mathbf{\Omega})$ is the standard spin-weighted spherical harmonic function
for a spin-$2$ field and ∗ denotes a complex conjugate. This expression is
exact for an all-sky unbiased measurement of the shear. This formalism could
be generalised to the pure-mode case (Grain et al., 2012), that would become
important in the presence of masks, but we leave this masked data
generalisation for future work.

To compute the effect of the biases we now replace $\gamma(\mathbf{\Omega})$
in equation (2.1) with $\widetilde{\gamma}(\mathbf{\Omega})$ from equation
(1). This results in the following expressions

$\displaystyle\widetilde{\gamma}^{E}_{\ell m}$ $\displaystyle=$
$\displaystyle\gamma^{E}_{\ell m}+m_{0}\gamma^{E}_{\ell m}$ $\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}m^{\prime}}[\gamma^{E}_{\ell^{\prime}m^{\prime}}W^{+}_{\ell\ell^{\prime}mm^{\prime}}+\gamma^{B}_{\ell^{\prime}m^{\prime}}W^{-}_{\ell\ell^{\prime}mm^{\prime}}]$
$\displaystyle+$ $\displaystyle c^{E}_{\ell m}$
$\displaystyle\widetilde{\gamma}^{B}_{\ell m}$ $\displaystyle=$
$\displaystyle\gamma^{B}_{\ell m}+m_{0}\gamma^{B}_{\ell m}$ (3)
$\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}m^{\prime}}[\gamma^{B}_{\ell^{\prime}m^{\prime}}W^{+}_{\ell\ell^{\prime}mm^{\prime}}-\gamma^{E}_{\ell^{\prime}m^{\prime}}W^{-}_{\ell\ell^{\prime}mm^{\prime}}]$
$\displaystyle+$ $\displaystyle c^{B}_{\ell m}.$

Here we expanded the additive term as $c(\mathbf{\Omega})=\sum_{\ell
m}(c^{E}_{\ell m}+{\rm i}c^{B}_{\ell m})_{2}Y_{\ell m}(\mathbf{\Omega})$. We
also defined

$\displaystyle W^{+}_{\ell\ell^{\prime}mm^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(_{2}W^{R,mm^{\prime}}_{\ell\ell^{\prime}}+_{-2}W^{R,mm^{\prime}}_{\ell\ell^{\prime}})$
$\displaystyle+$ $\displaystyle{\rm
i}(_{2}W^{I,mm^{\prime}}_{\ell\ell^{\prime}}-_{-2}W^{I,mm^{\prime}}_{\ell\ell^{\prime}})],$
$\displaystyle W^{-}_{\ell\ell^{\prime}mm^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2}[(_{2}W^{R,mm^{\prime}}_{\ell\ell^{\prime}}-_{-2}W^{R,mm^{\prime}}_{\ell\ell^{\prime}})$
(4) $\displaystyle+$ $\displaystyle{\rm
i}(_{2}W^{I,mm^{\prime}}_{\ell\ell^{\prime}}+_{-2}W^{I,mm^{\prime}}_{\ell\ell^{\prime}})],$

and

${}_{s}W^{R,mm^{\prime}}_{\ell\ell^{\prime}}=\int{\rm
d}\mathbf{\Omega}\,_{s}Y^{*}_{\ell^{\prime}m^{\prime}}(\mathbf{\Omega})m^{R}(\mathbf{\Omega})_{s}Y_{\ell
m}(\mathbf{\Omega});$ (5)

and similarly for $m^{I}(\mathbf{\Omega})$. In this derivation we have
expressed the real and imaginary parts of the multiplicative bias as
$m(\mathbf{\Omega})=m^{R}(\mathbf{\Omega})+{\rm i}m^{I}(\mathbf{\Omega})$. We
note that when considering _residual_ systematic effects, i.e. when any
amplitude and rotational changes caused by multiplicative systematic effects
are small (see Appendix A), that $m^{R}(\mathbf{\Omega})\simeq
m(\mathbf{\Omega})$ and $m^{I}(\mathbf{\Omega})\simeq 0$222We note that
$m^{R}(\mathbf{\Omega})\simeq m(\mathbf{\Omega})$ and
$m^{I}(\mathbf{\Omega})\simeq 0$ implies that, if one expresses the
multiplicative biases as
$m_{1}(\mathbf{\Omega})\gamma_{1}(\mathbf{\Omega})+{\rm
i}m_{2}(\mathbf{\Omega})\gamma_{2}(\mathbf{\Omega})$ (where $1$ and $2$ denote
the ellipticity components measured parallel to Cartesian axes in a
measurement frame, and measured at $45$ degrees to these axes),
$m(\mathbf{\Omega})=m_{1}(\mathbf{\Omega})=m_{2}(\mathbf{\Omega})$. This is
found to be the case in state-of-the-art methods, e.g. Pujol et al. (2019).
When considering residual systematic effects, after calibration with
simulations, this is also expected to be the case..

Already from the expressions in equation (2.1) it can be seen that
multiplicative biases in general mix E and B-modes together, both from the
underlying shear field and the multiplicative bias field, and the propagation
of such terms is in the form of a convolution represented as a sum over
wavenumbers. Furthermore the window function caused by multiplicative biases
is $\ell$ and $m$-mode dependent since in general these are not isotropic on
the celestial sphere.

We note that in this case the constant additive biases $c_{1,0}$ and $c_{2,0}$
do not appear in equation (2.1). This is because a constant term only affects
the $\ell=0$ mode, but shear is a spin-2 field where the spherical harmonic
transform is not defined for $\ell<2$; because ${}_{s}Y_{\ell
m}(\mathbf{\Omega})=0$ for $\ell<|s|$. Therefore a constant additive bias
cannot affect the cosmic shear power spectrum.

###  2.2 Biased Cosmic Shear Power Spectra

We now compute the expressions for the biased cosmic shear power spectra by
taking the correlation of the expressions in equation (2.1). The full
expression can be written as a series of terms that pertain to multiplicative,
additive and cross-terms, and depend on the true EE, EB and BB power spectra.
The power spectra estimates are computed by taking the correlation of the
spherical harmonic coefficients from equation (2.1) where

$\displaystyle\widetilde{C}^{GH}_{\ell}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}\widetilde{\gamma}^{G}_{\ell
m}\widetilde{\gamma}^{H,*}_{\ell m}$ (6)

for $G=(E,B)$ and $H=(E,B)$.

We provide the full expanded expression for the biased power spectra in
Appendix B. If we assume that $C^{EB}_{\ell}=0$, which is the case in all but
the most exotic dark energy models, then the three estimated power spectra
(EE, BB and EB) are:

$\displaystyle\widetilde{C}^{EE}_{\ell}$ $\displaystyle=$
$\displaystyle(1+2m_{0}+m_{0}^{2})C^{EE}_{\ell}$ $\displaystyle+$
$\displaystyle(1+m_{0})({\mathcal{N}}^{+}_{\ell}+{\mathcal{N}}^{+,*}_{\ell})C^{EE}_{\ell}$
$\displaystyle+$ $\displaystyle
2(1+m_{0})C^{c_{E}E}_{\ell}+C^{c_{E}c_{E}}_{\ell}$ $\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{M}}^{++}_{\ell\ell^{\prime}}C^{EE}_{\ell^{\prime}}+{\mathcal{M}}^{--}_{\ell\ell^{\prime}}C^{BB}_{\ell^{\prime}}]$
$\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{B}}^{+EE}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+EE}_{\ell\ell^{\prime}})^{*}+{\mathcal{B}}^{-BE}_{\ell\ell^{\prime}}+({\mathcal{B}}^{-BE}_{\ell\ell^{\prime}})^{*}]$
$\displaystyle\widetilde{C}^{BB}_{\ell}$ $\displaystyle=$
$\displaystyle(1+2m_{0}+m_{0}^{2})C^{BB}_{\ell}$ $\displaystyle+$
$\displaystyle(1+m_{0})({\mathcal{N}}^{+}_{\ell}+{\mathcal{N}}^{+,*}_{\ell})C^{BB}_{\ell}$
$\displaystyle+$ $\displaystyle
2(1+m_{0})C^{c_{B}B}_{\ell}+C^{c_{B}c_{B}}_{\ell}$ $\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{M}}^{--}_{\ell\ell^{\prime}}C^{EE}_{\ell^{\prime}}+{\mathcal{M}}^{++}_{\ell\ell^{\prime}}C^{BB}_{\ell^{\prime}}]$
$\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{B}}^{+BB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+BB}_{\ell\ell^{\prime}})^{*}-{\mathcal{B}}^{-EB}_{\ell\ell^{\prime}}-({\mathcal{B}}^{-EB}_{\ell\ell^{\prime}})^{*}]$
$\displaystyle\widetilde{C}^{EB}_{\ell}$ $\displaystyle=$
$\displaystyle-(1+m_{0}){\mathcal{N}}^{-,*}_{\ell}C^{EE}_{\ell}+(1+m_{0}){\mathcal{N}}^{-}_{\ell}C^{BB}_{\ell}$
$\displaystyle+$ $\displaystyle
2(1+m_{0})C^{c_{E}B}_{\ell}+C^{c_{E}c_{B}}_{\ell}$ $\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{M}}^{-+}_{\ell\ell^{\prime}}C^{BB}_{\ell^{\prime}}-{\mathcal{M}}^{+-}_{\ell\ell^{\prime}}C^{EE}_{\ell^{\prime}}]$
$\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}[{\mathcal{B}}^{+EB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+BE}_{\ell\ell^{\prime}})^{*}+{\mathcal{B}}^{-BB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{-EE}_{\ell\ell^{\prime}})^{*}].$

The various terms in the full expression are

$\displaystyle{\mathcal{M}}^{XY}_{\ell\ell^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{mm^{\prime}}W^{X}_{\ell\ell^{\prime}mm^{\prime}}(W^{Y}_{\ell\ell^{\prime}mm^{\prime}})^{*}$
$\displaystyle{\mathcal{N}}^{X}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}W^{X}_{\ell\ell mm}$
$\displaystyle{\mathcal{B}}^{XGH}_{\ell\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{mm^{\prime}}W^{X}_{\ell\ell^{\prime}mm^{\prime}}\gamma^{G}_{\ell^{\prime}m^{\prime}}(c^{H}_{\ell
m})^{*},$ (8)

where $X=(+,-)$, $Y=(+,-)$, $G=(E,B)$ and $H=(E,B)$. The power spectra in the
full expression are labelled in their superscripts as either correlations
between shear coefficients ($EE$, $EB$, $BB$), correlations between the
additive bias terms ($c_{E}c_{E}$, $c_{E}c_{B}$, $c_{B}c_{B}$), or cross
correlations between shear and additive bias terms ($c_{E}E$, $c_{E}B$,
$c_{B}B$). Equation (2.2) should be defined as the measured power spectrum (on
the left hand sides), compared to the power spectrum that would have been
measured with no systematic effects (the $C^{GH}_{\ell}$’s on the right hand
sides). However we note that the terms convolved with the window function
($W^{X}_{\ell\ell^{\prime}mm^{\prime}}$ in ${\mathcal{M}}$ and
${\mathcal{N}}$) in equation (2.2) are derived by taking the ensemble-average
of equation (6), and making use of the statistical rotational invariance of
the ensemble-averaged harmonic modes. Therefore equation (2.2) is a hybrid of
ensemble-averaged terms and un-averaged terms which may be non-zero only for a
given realisation (as is the case in the examples shown in Section 3).

It can be shown that the ${\mathcal{N}}$ terms are simply the mean of the
spatially varying multiplicative bias field. If we consider
${\mathcal{N}}^{+}_{\ell}$ we find

$\displaystyle{\mathcal{N}}^{+}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\ell+1)}\sum_{m}\int{\rm
d}\mathbf{\Omega}\,m^{R}(\mathbf{\Omega})_{2}Y^{*}_{\ell
m}(\mathbf{\Omega})_{2}Y_{\ell m}(\mathbf{\Omega})$ (9)

We can simplify this expression further by using the generalised addition
theorem for spin-weighted spherical harmonics Grain et al. (2012)

$\sum_{m}{}_{s}Y^{*}_{\ell
m}(\mathbf{\Omega}){}_{s^{\prime}}Y_{\ell^{\prime}m^{\prime}}(\mathbf{\Omega}^{\prime})=\left[\frac{(2\ell+1)}{4\pi}\right](-1)^{s-s^{\prime}}D^{\ell}_{ss^{\prime}}(\alpha,\beta,\gamma)\,{\rm
e}^{-2{\rm i}s\gamma},$ (10)

where $(\alpha,\beta,\gamma)$ are Euler angles between $\mathbf{\Omega}$ and
$\mathbf{\Omega}^{\prime}$ which in our case are zero, and
$D^{\ell}_{ss^{\prime}}(\alpha,\beta,\gamma)$ are the Wigner rotation matrices
which for $D^{\ell}(0,0,0)=\delta^{K}_{ss′}$. This leads to

$\displaystyle{\mathcal{N}}^{+}_{\ell}=\frac{1}{4\pi}\int{\rm
d}\mathbf{\Omega}\,m^{R}(\mathbf{\Omega})=\langle
m^{R}(\mathbf{\Omega})\rangle,$ (11)

and similarly ${\mathcal{N}}^{-}_{\ell}=\langle
m^{I}(\mathbf{\Omega})\rangle$. We choose to keep $m_{0}$ and the mean of
$m(\mathbf{\Omega})$ separate since these could have different physical
origins i.e. one is a true constant, the other the mean of a spatially varying
field. We note that the sum of $m_{0}$ and $\langle m(\mathbf{\Omega})\rangle$
is similar to the bias $b_{m}$ term in Taylor & Kitching (2016).

We discuss further simplifications of these expressions below. In Appendix C
we show the generalisation of this to the case of multiple tomographic bins.

case 1 case 2 case 3

Figure 1: The real part of the multiplicative field $m^{R}(\mathbf{\Omega})$,
in the three example cases investigated. Shown is a simulated celestial sphere
in a Mollweide Projection with $\theta=\phi=0$ at the North pole. The colour
scale represents the amplitude of the biases.

####  2.2.1 Discussion of the terms

We can now discuss each term in the full expression and its physical meaning.

  * •

$m_{0}$ and $m_{0}^{2}$: These terms are the normal contribution from the
constant multiplicative bias terms. These arise from the limitations with
which shape measurement algorithms can be calibrated (see e.g. Hoekstra et
al., 2017). As we are concerned with the residual biases _after_ such a
calibration, $m_{0}^{2}\ll m_{0}$ as $m_{0}\lesssim 2\times 10^{-3}$ (Cropper
et al., 2013)

  * •

${\bf\mathcal{N}}$: These terms represent multiplicative biases, and are the
mean of the multiplicative bias field.

  * •

${\bf\mathcal{M}}$: These terms represent multiplicative terms of order
$m^{2}$. The rows and columns show how the E and B-mode power have mixed
terms, where the $++$ terms pick up contributions from the real part of the
multiplicative bias field, the $--$ terms pick up contributions from the
imaginary part, and $+-$ or $-+$ are mixed terms.

  * •

${\bf\mathcal{B}}$: These terms represent third-order, bispectrum-like,
correlations between the position-dependent multiplicative bias
$m(\mathbf{\Omega})\gamma(\mathbf{\Omega})$ and the position-dependent
additive bias $c(\mathbf{\Omega})$. Such effects are likely since areas in a
survey, or particular pointings, that have detector, telescope or background
effects that cause additive biases will also lead to multiplicative biases.
This is because any anisotropic change in the quadrupole moments will modify
the size, and thus the multiplicative bias. We note that note that in this
term the multiplicative and shear terms are always spatially coupled and this
combination is correlated with the spatially varying additive term.

  * •

$c_{E}E$, $c_{E}B$, and $c_{B}B$: these terms capture the correlations between
the underlying shear field and the additive bias terms. Such terms are
expected to be caused by selection effects in a real survey, where for example
blending in high shear regions (e.g. about clusters) could cause an additive
bias contribution.

  * •

$c_{E}c_{E}$, $c_{E}c_{B}$, and $c_{B}c_{B}$: These are the power spectra of
the position-dependent additive biases. We note again that constant additive
bias terms do not contribute to cosmic shear power spectra.

###  2.3 Linear Expressions

Here we show the linearised expressions of the biased cosmic shear power, that
only include terms that are linear in the bias parameters. We find that

$\displaystyle\widetilde{C}^{EE}_{\ell}$ $\displaystyle\approx$
$\displaystyle(1+2m_{0})C^{EE}_{\ell}+2\langle m^{R}(\mathbf{\Omega})\rangle
C^{EE}_{\ell}+2C^{c_{E}E}_{\ell},$ $\displaystyle\widetilde{C}^{EB}_{\ell}$
$\displaystyle\approx$ $\displaystyle-\langle m^{I}(\mathbf{\Omega})\rangle
C^{EE}_{\ell}+\langle m^{I}(\mathbf{\Omega})\rangle
C^{BB}_{\ell}+2C^{c_{E}B}_{\ell},$ $\displaystyle\widetilde{C}^{BB}_{\ell}$
$\displaystyle\approx$ $\displaystyle(1+2m_{0})C^{BB}_{\ell}+2\langle
m^{R}(\mathbf{\Omega})\rangle C^{BB}_{\ell}+2C^{c_{B}B}_{\ell}.$ (12)

We have included $B$-mode power since as shown in Schneider et al. (2002)
source redshift clustering can cause a small $B$-mode component.

We see that the impact of spatially varying biases will be, to linear order,
captured by the mean of the multiplicative bias and the additive-shear cross
correlation power spectrum, but in the presence of intrinsic B-modes,
$\widetilde{C}^{EB}_{\ell}$ now includes a term $2\langle
m^{I}(\mathbf{\Omega})\rangle C^{BB}_{\ell}$.

We note that if $\langle m^{R}(\mathbf{\Omega})\rangle=\langle
m^{I}(\mathbf{\Omega})\rangle$, then (twice) the EB power spectrum could be
added to the EE power spectrum to cancel out multiplicative effects; however
this is not expected to be the case in general, or for small biases.

##  3 Simple Simulations

To test that the above formalism can indeed capture the propagation of general
position-dependent multiplicative and additive bias terms into the cosmic
shear power spectrum we generate several toy examples and investigate the
contributions of each term to the overall change. For each case we define a
multiplicative constant and field, $m_{0}$ and $m(\mathbf{\Omega})$, and an
additive constant and field, $c_{0}=c_{1,0}=c_{2,0}$ and $c(\mathbf{\Omega})$,
although the choice for $c_{0}$ has no impact on cosmic shear power spectra by
definition. We normalise these fields such that $\langle
m_{0}+m(\mathbf{\Omega})\rangle=2\times 10^{-3}$ and $\langle
c_{0}+c(\mathbf{\Omega})\rangle=1\times 10^{-4}$, which represent the overall
requirements for a _Euclid_ -like experiment (Cropper et al., 2013); however
we note that the amplitude of $\langle c_{0}+c(\mathbf{\Omega})\rangle$ will
have no effect on the power spectrum as discussed previously. For each case we
compare the computation of the analytic expression in equation (2.2) and a
numerical case where we compute the real space shear field
$\widetilde{\gamma}(\mathbf{\Omega})=[1+m_{0}+m(\mathbf{\Omega})]\gamma(\mathbf{\Omega})+[c_{0}+c(\mathbf{\Omega})]$
and then compute the power spectra of this directly using a spherical harmonic
transform. In all cases we compute the original $\gamma(\mathbf{\Omega})$
field using a Gaussian random field generated by using a cosmic shear power
spectrum based on the Planck $\Lambda$CDM cosmology (Planck Collaboration et
al., 2018), using the massmappy code (Wallis et al., 2017). In all cases we
use SSHT McEwen et al. (2013) to compute the spin-weighted spherical
harmonics, which sample the sphere using the sampling scheme of McEwen & Wiaux
(2011).

The cases we consider are shown below. Note that we express these in terms of
an arbitrary amplitude $A$ since these are all normalised to have $\langle
m_{0}+m(\mathbf{\Omega})\rangle=2\times 10^{-3}$ and $\langle
c_{0}+c(\mathbf{\Omega})\rangle=1\times 10^{-4}$. The cases are simple
examples but nonetheless are approximations of realistic spatial variations
that could occur:

  1. 1.

Case 1, Simple Galactic Plane:

     * •

$m^{R}(\mathbf{\Omega})=A[\pi-|\phi-\pi|]$, $m^{I}(\mathbf{\Omega})=0$,

     * •

$c^{R}(\mathbf{\Omega})=c^{I}(\mathbf{\Omega})=A[\pi-|\phi-\pi|]$,

     * •

$c_{0}=A$, $m_{0}=A$;

  2. 2.

Case 2, Simple Patch Pattern:

     * •

$m^{R}(\mathbf{\Omega})=10A\sin(100|\phi-\pi|)\sin(100|\theta-\pi|)$,
$m^{I}(\mathbf{\Omega})=0$,

     * •

$c^{R}(\mathbf{\Omega})=c^{I}(\mathbf{\Omega})=10A\sin(10|\phi-\pi|)\sin(10|\theta-\pi|)$,

     * •

$c_{0}=A$, $m_{0}=A$;

  3. 3.

Case 3, Simple Scanning Pattern:

     * •

$m^{R}(\mathbf{\Omega})=Ai$, where i is an iterative pixel number count, which
is reset when $i=10$, $m^{I}(\mathbf{\Omega})=0$,

     * •

$c^{R}(\mathbf{\Omega})=c^{I}(\mathbf{\Omega})=Ai^{2}$, where i is an
iterative pixel number count, which is reset when $i=10$,

     * •

$c_{0}=A$, $m_{0}=A$.

Figure 2: The residual power spectrum $\delta
C_{\ell}=\widetilde{C}^{EE}_{\ell}-C^{EE}_{\ell}$ for the three cases
considered. The left plots show the comparison between the numerical case
computed by transforming the modified shear field and performing a spherical
harmonic transform (i.e. a forward model), and the analytic case computed
using equation (2.2). The blue band shows the $1$-sigma scatter about the mean
of forward model, the red lines show the analytic prediction. The right plots
show the contribution to the analytic case from each of the components in
equation (2.2). In all cases the legends label the coloured lines. The mean
multiplicative terms (green and pink lines) have the same value due to the
scaling, and hence are over-plotted.

The first case approximates a Galactic plane dependency, the second case
approximates a patch-dependent systematic effect, and the third case is a non-
analytic case that approximates a scanning sequence of exposures. To
demonstrate the complexity of the spatial variation of the cases we show in
Figure 1 the real part of the multiplicative field for each of the cases (we
do not show all the fields associated with the systematic effects since they
are largely similar in form).

In Figure 2 we show the residual power spectra $\delta
C_{\ell}=\widetilde{C}^{EE}_{\ell}-C^{EE}_{\ell}$ for each of the cases
considered. We compute the error on the forward model power spectrum as
$\sigma(\delta
C_{\ell})=[(\widetilde{C}^{EE}_{\ell})^{2}+(C^{EE}_{\ell})^{2}]^{1/2}$
(Joachimi & Bridle, 2010; Hu & Jain, 2004).

We note that in all cases we use a maximum angular multipole of $L=32$. This
is because the calculations are particularly numerically demanding. The
$W^{\pm}_{\ell\ell^{\prime}mm^{\prime}}$ calculations have dimension $L^{4}$,
and for each of these spin-weighted spherical harmonic functions must be
computed each of which scale like $L^{2}\log L$ at best (McEwen et al., 2013).
This point is discussed further in Section 4.1. In all cases the analytic
formula given in equation (2.2) accurately captured the form of the residual
power spectrum; the very small differences are due to the numerical stability
of the spin-weighted spherical harmonic transform calculations.

Figure 3: Timing of the forward model and analytic calculations using a 2016
Macbook Pro, 3.3 GHz Intel Core i7, 16 GB 2133 MHz LPDDR3. The blue line shows
the forward model scaling, the orange line shows the analytic scaling, the
thin red and green lines are proportional to $L^{2}$ and $L^{5}$ respectively
where $L$ is the maximum $\ell$-mode.

With regard to the different terms we find in all cases that the
${\mathcal{N}}C^{EE}_{\ell}$ term is dominant, which is expected since it is
of linear order in $m(\mathbf{\Omega})$, followed by the $C^{c_{E}E}_{\ell}$
cross-correlation term and the $(m_{0}+m_{0}^{2})C^{EE}_{\ell}$ terms. The
convolutive ${\mathcal{M}}$, ${\mathcal{B}}$ and $C^{c_{E}c_{E}}_{\ell}$ terms
are all at least an order of magnitude lower in all cases. Therefore the
linearised expression in equation (11) $\delta C_{\ell}\approx
2m_{0}C^{EE}_{\ell}+2\langle m^{R}(\mathbf{\Omega})\rangle
C^{EE}_{\ell}+2C^{c_{E}E}_{\ell}$ is a good approximation in these simple
examples. In the case that the mean of $m(\mathbf{\Omega})$ and
$c(\mathbf{\Omega})$ are both zero, all of the remaining terms at second and
third order in bias would become important at approximately the same level.
Note that we plot the absolute value of the residual power spectrum
contributions, since some terms can be negative depending on the nature of
spatial pattern used in the simulations.

##  4 Discussion

We have shown in general how constant and position-dependent shape measurement
biases propagate through to cosmic shear power spectra.

The multiplicative bias terms shown here are similar to those that result in
CMB polarisation pseudo-$C_{\ell}$ analyses, where masking of the data results
in expressions that also include temperature power spectra (Lewis et al.,
2002; Zaldarriaga & Seljak, 1997; Grain et al., 2012; Brown et al., 2005). The
difference here is that instead of a mask we have a multiplicative bias field
that is in general spin-dependent. We note that this formalism equally applies
to the case of masked cosmic shear data where $m(\mathbf{\Omega})$ may be zero
in some regions, and that such a case would lead to further mode mixing.

In Kitching et al. (2012) a pseudo-$C_{\ell}$ formalism was used to assess
position-dependent shape measurement errors. However in that study the linear
terms ${\mathcal{N}}$, bispectrum and additive terms were not included and the
non-linear convolution term ${\mathcal{M}}$ used a simpler form based on a
flat-sky approximation. In Taylor & Kitching (2016) the propagation of shape
measurement biases was generalised to the convolutive case but the linear
terms, bispectrum and E/B mode mixing terms were ignored.

In Massey et al. (2013) and Cropper et al. (2013) requirements were set on
weak lensing experiments using an approximation for position-dependent shape
measurement biases. In that case a form of propagation was determined for the
constant case
$\widetilde{C}^{EE}_{\ell}=(1+2m_{0}+m_{0}^{2})C^{EE}_{\ell}+c^{2}_{c}$, which
was then replaced with a ‘position-dependent’ formulation proposed by Amara &
Réfrégier (2008),
$C^{EE}_{\ell}=(1+{\mathcal{M}}_{\ell})C^{EE}_{\ell}+{\mathcal{A}}_{\ell}$. We
find that such an expression is similar to the full case when only linear
terms in biases are assumed, whereas the relationship to the underlying
position-dependent bias fields is much more complex. Furthermore in Massey et
al. (2013) a worst-case scenario in sensitivity was assumed for
${\mathcal{M}}_{\ell}$ in which multiplicative biases mimicked the scale-
dependent behaviour of dark energy. These worst-case assumptions are
conservative when designing an experiment and lead to requirements that will
guarantee performance, but when assessing the actual performance of a survey
they are not adequate.

We note that if the mean multiplicative bias is zero – as may be expected if
pre-experiment simulations can determine any mean effect – then only second
and third order convolutive terms remain. The power spectrum residuals caused
by these terms, and the impact on cosmology, are expected to be much lower
than than the mean terms for two reasons. First, because the terms are second
order and so for $m(\mathbf{\Omega})\ll 1$ these are smaller. Second, because
they are convolutions it is unlikely that a functional form will result that
matches the cosmic shear power spectrum; therefore the impact on cosmological
parameter inference is expected to be lower (Taylor & Kitching, 2016).

###  4.1 Scaling

In Figure 3 we show how the analytic and forward modelling cases scale as a
function of $L$, the maximum $\ell$-mode . To compute the full case requires
evaluations of terms that scale like $L^{4}\times L^{2}\log L$, where $L$ is
the maximum multipole; this is because of the
$W_{\ell\ell^{\prime}mm^{\prime}}$ terms that have $\sim L^{4}$ summations,
and the spherical harmonic transforms that scale like $L^{2}\log L$. We find a
slightly better scaling due to pre-computation of the spin-weighted spherical
harmonic functions, but nonetheless the analytic calculation scales like
$\propto L^{5}$ compared to the forward modelling that scales like $\propto
L^{2}$.

We evaluate simple examples for $L=32$, but scaling to a reasonable value of
$L>1000$ would result in prohibitively long calculations. On the other hand
the forward modelling of systematic effects i.e. the evaluation of equation
(1) in real space and a direct spherical harmonic transform to produce a power
spectrum is tractable for $L>1000$ and we therefore advocate this approach in
Taylor et al. (2019) and Paykari et al., (in prep).

We note that one could perform a spherical sky analysis and supplement this by
a fast Fourier transform on small scales. However this approach would require
the sphere to be divided into patches upon which a flat sky could be run with
overlap between patches to capture all angular modes, and a transition from
all-sky to flat-sky computed. This is feasible, but this is unneeded
complexity given that a forward model is very simple to compute.

##  5 Conclusions

In this paper we derive a complete expression for the impact of constant and
spatially varying multiplicative and additive shape measurement biases on the
cosmic shear power spectrum. In doing so we find several terms that have thus
far been overlooked, in particular terms relating to cross-correlations
between biases and shear.

In performing the full calculation we find that to linear order spatially
varying biases are well approximated by the sum of the product of the the
power spectrum and the mean of the multiplicative bias field, and the cross-
correlation term between the additive bias field and the shear field. We note
that the cosmic shear power spectrum is not sensitive to constant or mean
additive biases.

We compare the computation of the full analytic expression with that obtained
using a forward modelling approach using simplified simulations and find good
agreement. Furthermore we use these simplified simulations to demonstrate how
each term in the full expression contributes to the total. However, in
performing the full calculation we also find that its computation scales as
the maximum multipole $\propto L^{5}$. This means that its evaluation for
large $L>1000$ is unfeasible. Therefore we recommend that any assessment of
the impact of biases on cosmic shear power spectrum must be performed using a
forward modelling approach.

###### Acknowledgements.

We thank the developers of SSHT and massmappy, Jason McEwen, Chris Wallis and
Boris Leistedt for making their code publicly available. We thank Dipak Munshi
for useful discussions. We thank Peter Schneider for comments on an early
draft. PP is supported by the UK Science and Technology Facilities Council. TK
is supported by a Royal Society University Research Fellowship. HH
acknowledges support from Vici grant 639.043.512 financed by the Netherlands
Organization for Scientific Research (NWO). We thank an anonymous referee
whose comments improved this manuscript.

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##  Appendix A Bias Propagation

The true shear field $\gamma(\mathbf{\Omega})$ is a spin-2 quantity, and any
impact of imperfect shape measurement should preserve this spin-2 nature. In
general we consider that there are two ways that an imperfect shape
measurement can impact an observed spin-2 field either 1) there can be a
incorrect estimate of the ratio of the semi-major axes, expressed as an
amplitude change of the shear, 2) there can be an incorrect estimation of the
observed angle of the ellipse i.e. a rotation.

###  A.1 Multiplicative Bias

For a multiplicative systematic effect these possibilities can be expressed as

$\displaystyle[1+m(\mathbf{\Omega})\gamma(\mathbf{\Omega})]=[1+m(\mathbf{\Omega}){\rm
e}^{{\rm i}\phi_{m}(\mathbf{\Omega})}]\,|\gamma(\mathbf{\Omega})|{\rm e}^{{\rm
i}2\Phi(\mathbf{\Omega})}$ (13)

where we have expressed $\gamma(\mathbf{\Omega})=|\gamma(\mathbf{\Omega})|{\rm
e}^{{\rm i}2\Phi(\mathbf{\Omega})}$ where $\Phi(\mathbf{\Omega})$ is the angle
between the orientation of the elliptical shape induced by the shear and the
$x$-axis of local Cartesian frame in which measurement has been made. To this
expression we apply an amplitude change $m(\mathbf{\Omega})$ corresponding to
an incorrect measurement of the ratio of the semi-major axes, and a small
rotation $\phi_{m}(\mathbf{\Omega})$. This preserves the spin-2 nature of the
measured field. When we express the product of two complex numbers
$m(\mathbf{\Omega})\gamma(\mathbf{\Omega})$, it should be understood that the
multiplicative bias fields take the form

$\displaystyle m(\mathbf{\Omega})$ $\displaystyle=$ $\displaystyle
m^{R}(\mathbf{\Omega})+{\rm i}m^{I}(\mathbf{\Omega})$ $\displaystyle
m^{R}(\mathbf{\Omega})$ $\displaystyle=$ $\displaystyle
m(\mathbf{\Omega})\cos[\phi_{m}(\mathbf{\Omega})]$ $\displaystyle
m^{I}(\mathbf{\Omega})$ $\displaystyle=$ $\displaystyle
m(\mathbf{\Omega})\sin[\phi_{m}(\mathbf{\Omega})],$ (14)

where $m^{R}(\mathbf{\Omega})$ and $m^{I}(\mathbf{\Omega})$ in the first
equation are the real and imaginary parts respectively which are coupled as
expressed in the subsequent equations. We note that we do not label these
$m_{1}(\mathbf{\Omega})$ and $m_{2}(\mathbf{\Omega})$ since they do not map
solely to the $\gamma_{1}(\mathbf{\Omega})$ and $\gamma_{2}(\mathbf{\Omega})$
components of the shear field where
$\gamma(\mathbf{\Omega})=\gamma_{1}(\mathbf{\Omega})+{\rm
i}\gamma_{2}(\mathbf{\Omega})$.

We refer to Pujol et al. (2019); Huff & Mandelbaum (2017b) for further
discussion of the propagation of more complex multiplicative biases. We note
that if the amplitude of the (residual) biases are small, and that the
rotation angle is random, then it is reasonable to assume that $\langle
m^{R}(\mathbf{\Omega})\rangle=\langle m^{I}(\mathbf{\Omega})\rangle$; if one
assumes instead small residual biases and applies the small angle
approximation then $m^{R}(\mathbf{\Omega})\approx m(\mathbf{\Omega})$ and
$m^{I}(\mathbf{\Omega})\approx
m(\mathbf{\Omega})\phi_{m}(\mathbf{\Omega})\approx 0$. In the case that
$m^{R}(\mathbf{\Omega})=m(\mathbf{\Omega})$ and $m^{I}(\mathbf{\Omega})=0$
then this would result in $m_{1}(\mathbf{\Omega})=m(\mathbf{\Omega})$ and
$m_{1}(\mathbf{\Omega})=m_{2}(\mathbf{\Omega})$ if expressed in this way.

We note that if one applies a bias of the form
$m_{1}(\mathbf{\Omega})\gamma_{1}(\mathbf{\Omega})+{\rm
i}m_{2}(\mathbf{\Omega})\gamma_{2}(\mathbf{\Omega})$ (i.e. a different
independent scalar multiplicative biases applied to each of the shear
components) then this cannot in general be expressed as an amplitude change
with a rotation, and therefore can result in a change in the spin properties
of the measured field. We note that one could create such an expression by
$m^{\prime}(\mathbf{\Omega})\gamma(\mathbf{\Omega})+\delta
m(\mathbf{\Omega})\gamma^{*}(\mathbf{\Omega})$ where,
$m^{\prime}(\mathbf{\Omega})=[m_{1}(\mathbf{\Omega})+m_{2}\mathbf{\Omega})]/2$
and $\delta
m(\mathbf{\Omega})=[m_{1}(\mathbf{\Omega})-m_{2}(\mathbf{\Omega})]/2$, however
the second term would represent a parity change/mislabelling in the
$\gamma_{2}$ component which would be a very large systematic effect. We
explored the expected size of $\delta m/m$ using image simulations that
resemble _Euclid_ based on Hoekstra et al. (2017) and found that $\delta
m/m\sim 0.1$ for both PSF anisotropy and for a simple of charge trailing
between pixels. Hence in practice it appears from these initial studies that
on can typically ignore $\delta m$.

###  A.2 Additive Bias

In the additive bias case one can add a field with spin-2 properties such that

$\displaystyle\gamma(\mathbf{\Omega})+c(\mathbf{\Omega})=\gamma(\mathbf{\Omega})+|c(\mathbf{\Omega})|{\rm
e}^{{\rm i}\phi_{c}(\mathbf{\Omega})}{\rm e}^{{\rm i}2\Phi(\mathbf{\Omega})}$
(15)

where $|c(\mathbf{\Omega})|$ and $\phi_{c}(\mathbf{\Omega})$ are systematic
changes in the amplitude and rotation angle of the measurements respectively.
In this case the two additive components can be expressed like

$\displaystyle c(\mathbf{\Omega})$ $\displaystyle=$ $\displaystyle
c_{1}(\mathbf{\Omega})+{\rm i}c_{2}(\mathbf{\Omega})$ $\displaystyle
c_{1}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle|c(\mathbf{\Omega})||\cos(2\Phi(\mathbf{\Omega})+\phi_{c}(\mathbf{\Omega}))$
$\displaystyle c_{2}(\mathbf{\Omega})$ $\displaystyle=$
$\displaystyle|c(\mathbf{\Omega})|\sin(2\Phi(\mathbf{\Omega})+\phi_{c}(\mathbf{\Omega})),$
(16)

where in the additive case the real and imaginary parts will add to the
respective $\gamma_{1}$ and $\gamma_{2}$ parts of the shear and hence we label
them as such (which is not the case for the multiplicative biases). In the
constant case one can write $c_{0}=c_{1,0}+{\rm i}c_{2,0}$ where $c_{1,0}$ and
$c_{2,0}$ are constants.

##  Appendix B The Two Dimensional Case

The full expanded expression for the two dimensional case can be written in
matrix form as

$\displaystyle\left(\begin{array}[]{c}\widetilde{C}^{EE}_{\ell}\\\
\widetilde{C}^{EB}_{\ell}\\\ \widetilde{C}^{BB}_{\ell}\end{array}\right)$
$\displaystyle=$
$\displaystyle(1+2m_{0}+m_{0}^{2})\left(\begin{array}[]{c}C^{EE}_{\ell}\\\
C^{EB}_{\ell}\\\ C^{BB}_{\ell}\end{array}\right)$ (23) $\displaystyle+$
$\displaystyle(1+m_{0})\left(\begin{array}[]{ccc}({\mathcal{N}}^{+}_{\ell}+{\mathcal{N}}^{+,*}_{\ell})&{\mathcal{N}}^{-}_{\ell}&0\\\
-{\mathcal{N}}^{-,*}_{\ell}&({\mathcal{N}}^{+}_{\ell}+{\mathcal{N}}^{+,*}_{\ell})&{\mathcal{N}}^{-}_{\ell}\\\
0&-{\mathcal{N}}^{-,*}_{\ell}&({\mathcal{N}}^{+}_{\ell}+{\mathcal{N}}^{+,*}_{\ell})\end{array}\right)\left(\begin{array}[]{c}C^{EE}_{\ell}\\\
C^{EB}_{\ell}\\\ C^{BB}_{\ell}\end{array}\right)$ (30) $\displaystyle+$
$\displaystyle 2(1+m_{0})\left(\begin{array}[]{c}C^{c_{E}E}_{\ell}\\\
C^{c_{E}B}_{\ell}\\\
C^{c_{B}B}_{\ell}\end{array}\right)+\left(\begin{array}[]{c}C^{c_{E}c_{E}}_{\ell}\\\
C^{c_{E}c_{B}}_{\ell}\\\ C^{c_{B}c_{B}}_{\ell}\end{array}\right)$ (37)
$\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}\left(\begin{array}[]{ccc}{\mathcal{M}}^{++}_{\ell\ell^{\prime}}&({\mathcal{M}}^{-+}_{\ell\ell^{\prime}}+{\mathcal{M}}^{+-}_{\ell\ell^{\prime}})&{\mathcal{M}}^{--}_{\ell\ell^{\prime}}\\\
-{\mathcal{M}}^{+-}_{\ell\ell^{\prime}}&({\mathcal{M}}^{++}_{\ell\ell^{\prime}}-{\mathcal{M}}^{--}_{\ell\ell^{\prime}})&{\mathcal{M}}^{-+}_{\ell\ell^{\prime}}\\\
{\mathcal{M}}^{--}_{\ell\ell^{\prime}}&-({\mathcal{M}}^{-+}_{\ell\ell^{\prime}}+{\mathcal{M}}^{+-}_{\ell\ell^{\prime}})&{\mathcal{M}}^{++}_{\ell\ell^{\prime}}\\\
\end{array}\right)\left(\begin{array}[]{c}C^{EE}_{\ell^{\prime}}\\\
C^{EB}_{\ell^{\prime}}\\\ C^{BB}_{\ell^{\prime}}\end{array}\right)$ (44)
$\displaystyle+$
$\displaystyle\left(\begin{array}[]{c}{\mathcal{B}}^{+EE}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+EE}_{\ell\ell^{\prime}})^{*}+{\mathcal{B}}^{-BE}_{\ell\ell^{\prime}}+({\mathcal{B}}^{-BE}_{\ell\ell^{\prime}})^{*}\\\
{\mathcal{B}}^{+EB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+BE}_{\ell\ell^{\prime}})^{*}+{\mathcal{B}}^{-BB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{-EE}_{\ell\ell^{\prime}})^{*}\\\
{\mathcal{B}}^{+BB}_{\ell\ell^{\prime}}+({\mathcal{B}}^{+BB}_{\ell\ell^{\prime}})^{*}-{\mathcal{B}}^{-EB}_{\ell\ell^{\prime}}-({\mathcal{B}}^{-EB}_{\ell\ell^{\prime}})^{*}\end{array}\right).$
(48)

The ${\mathcal{M}}$, ${\mathcal{N}}$ and ${\mathcal{B}}$ terms are defined in
equation (2.2) in the main body of the text.

##  Appendix C The Tomographic Case

In equation (2.2) we consider the case of a single population of galaxies,
however in reality one typically will define several populations labelled as
tomographic bins normally delineated in redshift. In this case equation (1) is
labelled with a tomographic bin $i$ such that

$\displaystyle\widetilde{\gamma}_{i}(\mathbf{\Omega})=[1+m_{0,i}+m_{i}(\mathbf{\Omega})]\gamma_{i}(\mathbf{\Omega})+[c_{0,i}(1+{\rm
i})+c_{i}(\mathbf{\Omega})]$ (49)

and the power spectra are defined as

$\displaystyle\widetilde{C}^{EE}_{\ell,ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}\widetilde{\gamma}^{E}_{\ell
m,i}\widetilde{\gamma}^{E,*}_{\ell m,j}$
$\displaystyle\widetilde{C}^{BB}_{\ell,ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}\widetilde{\gamma}^{B}_{\ell
m,i}\widetilde{\gamma}^{B,*}_{\ell m,j}$
$\displaystyle\widetilde{C}^{EB}_{\ell,ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(2\ell+1)}\sum_{m}\widetilde{\gamma}^{E}_{\ell
m,i}\widetilde{\gamma}^{B,*}_{\ell m,j}$
$\displaystyle\widetilde{C}^{BE}_{\ell,ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(2\ell+1)}\sum_{m}\widetilde{\gamma}^{B}_{\ell
m,i}\widetilde{\gamma}^{E,*}_{\ell m,j},$ (50)

where $i$ and $j$ label tomographic bins. We use notation where
$C^{XY}_{\ell,ij}$ means that field $X$ is associated with $i$, and field $Y$
is associated with $j$. We note that in this case there is a difference
between the $EB$ power spectrum and the $BE$ power spectrum for tomographic
bins $ij$. The full expression in expanded form is then:

$\displaystyle\left(\begin{array}[]{c}\widetilde{C}^{EE}_{\ell,ij}\\\
\widetilde{C}^{EB}_{\ell,ij}\\\ \widetilde{C}^{BE}_{\ell,ij}\\\
\widetilde{C}^{BB}_{\ell,ij}\end{array}\right)$ $\displaystyle=$
$\displaystyle(1+m_{0,i}+m_{0,j}+m_{0,i}m_{0,j})\left(\begin{array}[]{c}C^{EE}_{\ell,ij}\\\
C^{EB}_{\ell,ij}\\\ C^{BE}_{\ell,ij}\\\ C^{BB}_{\ell,ij}\end{array}\right)$
(59) $\displaystyle+$
$\displaystyle(1+m_{0,j})\left(\begin{array}[]{cccc}{\mathcal{N}}^{+}_{\ell,i}&0&{\mathcal{N}}^{-}_{\ell,i}&0\\\
0&-{\mathcal{N}}^{+}_{\ell,i}&0&{\mathcal{N}}^{-}_{\ell,i}\\\
-{\mathcal{N}}^{-}_{\ell,i}&0&{\mathcal{N}}^{+}_{\ell,i}&0\\\
0&-{\mathcal{N}}^{-}_{\ell,i}&0&{\mathcal{N}}^{+}_{\ell,i}\\\
\end{array}\right)\left(\begin{array}[]{c}C^{EE}_{\ell,ij}\\\
C^{EB}_{\ell,ij}\\\ C^{BE}_{\ell,ij}\\\ C^{BB}_{\ell,ij}\end{array}\right)$
(68) $\displaystyle+$
$\displaystyle(1+m_{0,i})\left(\begin{array}[]{cccc}{\mathcal{N}}^{+,*}_{\ell,j}&{\mathcal{N}}^{-,*}_{\ell,j}&0&0\\\
-{\mathcal{N}}^{-,*}_{\ell,j}&{\mathcal{N}}^{+,*}_{\ell,j}&0&0\\\
0&0&{\mathcal{N}}^{+,*}_{\ell,j}&{\mathcal{N}}^{-,*}_{\ell,j}\\\
0&0&-{\mathcal{N}}^{-,*}_{\ell,j}&{\mathcal{N}}^{+,*}_{\ell,j}\\\
\end{array}\right)\left(\begin{array}[]{c}C^{EE}_{\ell,ij}\\\
C^{EB}_{\ell,ij}\\\ C^{BE}_{\ell,ij}\\\ C^{BB}_{\ell,ij}\end{array}\right)$
(77) $\displaystyle+$
$\displaystyle(1+m_{0,j})\left(\begin{array}[]{c}C^{c_{E}E}_{\ell,ij}\\\
C^{c_{E}B}_{\ell,ij}\\\ C^{c_{B}E}_{\ell,ij}\\\
C^{c_{B}B}_{\ell,ij}\end{array}\right)+(1+m_{0,i})\left(\begin{array}[]{c}C^{Ec_{E}}_{\ell,ij}\\\
C^{Ec_{B}}_{\ell,ij}\\\ C^{Bc_{E}}_{\ell,ij}\\\
C^{Bc_{B}}_{\ell,ij}\end{array}\right)+\left(\begin{array}[]{c}C^{c_{E}c_{E}}_{\ell,ij}\\\
C^{c_{E}c_{B}}_{\ell,ij}\\\ C^{c_{B}c_{E}}_{\ell,ij}\\\
C^{c_{B}c_{B}}_{\ell,ij}\end{array}\right)$ (90) $\displaystyle+$
$\displaystyle\sum_{\ell^{\prime}}\left(\begin{array}[]{cccc}{\mathcal{M}}^{++}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{+-}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{-+}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{--}_{\ell\ell^{\prime},ij}\\\
-{\mathcal{M}}^{+-}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{++}_{\ell\ell^{\prime},ij}&-{\mathcal{M}}^{--}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{-+}_{\ell\ell^{\prime},ij}\\\
-{\mathcal{M}}^{-+}_{\ell\ell^{\prime},ij}&-{\mathcal{M}}^{--}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{++}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{+-}_{\ell\ell^{\prime},ij}\\\
{\mathcal{M}}^{--}_{\ell\ell^{\prime},ij}&-{\mathcal{M}}^{++}_{\ell\ell^{\prime},ij}&-{\mathcal{M}}^{+-}_{\ell\ell^{\prime},ij}&{\mathcal{M}}^{++}_{\ell\ell^{\prime},ij}\\\
\end{array}\right)\left(\begin{array}[]{c}C^{EE}_{\ell^{\prime},ij}\\\
C^{EB}_{\ell^{\prime},ij}\\\ C^{BE}_{\ell^{\prime},ij}\\\
C^{BB}_{\ell^{\prime},ij}\end{array}\right)$ (99) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{c}{\mathcal{B}}^{+EE}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{+EE}_{\ell\ell^{\prime},ji})^{*}+{\mathcal{B}}^{-BE}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{-BE}_{\ell\ell^{\prime},ji})^{*}\\\
{\mathcal{B}}^{+EB}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{+BE}_{\ell\ell^{\prime},ji})^{*}+{\mathcal{B}}^{-BB}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{-EE}_{\ell\ell^{\prime},ji})^{*}\\\
({\mathcal{B}}^{+EB}_{\ell\ell^{\prime},ji})^{*}+{\mathcal{B}}^{+BE}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{-BB}_{\ell\ell^{\prime},ji})^{*}+{\mathcal{B}}^{-EE}_{\ell\ell^{\prime},ij}\\\
{\mathcal{B}}^{+BB}_{\ell\ell^{\prime},ij}+({\mathcal{B}}^{+BB}_{\ell\ell^{\prime},ji})^{*}-{\mathcal{B}}^{-EB}_{\ell\ell^{\prime},ij}-({\mathcal{B}}^{-EB}_{\ell\ell^{\prime},ji})^{*}\end{array}\right).$
(104)

The various matrices in the above expression are

$\displaystyle{\mathcal{M}}^{XY}_{\ell\ell^{\prime},ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{mm^{\prime}}W^{X}_{\ell\ell^{\prime}mm^{\prime},i}(W^{Y}_{\ell\ell^{\prime}mm^{\prime},j})^{*}$
$\displaystyle{\mathcal{N}}^{X}_{\ell,i}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{m}W^{X}_{\ell\ell mm,i}$
$\displaystyle{\mathcal{B}}^{XGH}_{\ell\ell^{\prime},ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\sum_{mm^{\prime}}W^{X}_{\ell\ell^{\prime}mm^{\prime}_{i}}\gamma^{G}_{\ell^{\prime}m^{\prime},i}(c^{H}_{\ell
m,j})^{*},$ (105)

where $X=(+,-)$, $Y=(+,-)$, $G=(E,B)$ and $H=(E,B)$. In this case the
linearised expressions, assuming no underlying B-modes, or EB power, are

$\displaystyle\widetilde{C}^{EE}_{\ell,ij}$ $\displaystyle\approx$
$\displaystyle(1+m_{0,i}+m_{0,j})C^{EE}_{\ell,ij}+\langle m^{R}_{i}\rangle
C^{EE}_{\ell,ij}+\langle m^{R}_{j}\rangle
C^{EE}_{\ell,ij}+C^{c_{E}E}_{\ell,ij}+C^{Ec_{E}}_{\ell,ij},$
$\displaystyle\widetilde{C}^{EB}_{\ell,ij}$ $\displaystyle\approx$
$\displaystyle-\langle m^{I}_{i}\rangle
C^{EE}_{\ell,ij}+C^{Ec_{B}}_{\ell,ij},$
$\displaystyle\widetilde{C}^{BE}_{\ell,ij}$ $\displaystyle\approx$
$\displaystyle-\langle m^{I}_{i}\rangle
C^{EE}_{\ell,ij}+C^{c_{B}E}_{\ell,ij},$
$\displaystyle\widetilde{C}^{BB}_{\ell,ij}$ $\displaystyle\approx$
$\displaystyle 0.$ (106)


Academia Sinica (2014-2020). Dr. Hsu received his MS in Computer Science and
Information Engineering from the National Taiwan Normal University (2010) and
Ph.D. in Electrical Engineering from the National Taiwan University (2019).
---|---
|  Ming-Ching Chang is an Associate Professor at the Department of Computer
Science, University at Albany, SUNY. His expertise includes AI, video
analytics, computer vision, and machine learning. His research projects are
funded by DARPA, IARPA, NIJ, VA, GE Global Research. Dr. Chang frequently
serves the program chair, area chair, and referee of leading journals and
conferences. He chairs the steering committee of the IEEE AVSS Conference
since 2022, which he serves as the committee member since 2017. He has
authored more than 100 peer-reviewed journal and conference publications, 7 US
patents and 15 disclosures. He is a senior member of IEEE and member of ACM.
---|---
|  Wei-Chao Chen is the Chief Digital Officer at Inventec Corporation and the
Chairman at Skywatch Inc. Dr. Chen is also a Visiting Professor at the
National Taiwan University. His research interests include graphics hardware,
computational photography, augmented reality, and computer vision. Dr. Chen
was the Chief AI Advisor at Inventec (2018-2020), a senior research scientist
in Nokia Research Center at Palo Alto (2007-2009), and a 3D Graphics Architect
in NVIDIA (2002-2006). Dr. Chen received his MS in Electrical Engineering from
National Taiwan University (1996) and Ph.D. in Computer Science from the
University of North Carolina at Chapel Hill (2002).
---|---


# Extending Word-Level Quality Estimation for Post-Editing Assistance

Yizhen Wei† Takehito Utsuro† Masaaki Nagata‡
†Deg. Prog. Sys.&Inf. Eng., Grad. Sch. Sci.&Tech., University of Tsukuba
‡NTT Communication Science Laboratories, NTT Corporation, Japan

###### Abstract

We define a novel concept called extended word alignment in order to improve
post-editing assistance efficiency. Based on extended word alignment, we
further propose a novel task called refined word-level QE that outputs refined
tags and word-level correspondences. Compared to original word-level QE, the
new task is able to directly point out editing operations, thus improves
efficiency. To extract extended word alignment, we adopt a supervised method
based on mBERT. To solve refined word-level QE, we firstly predict original QE
tags by training a regression model for sequence tagging based on mBERT and
XLM-R. Then, we refine original word tags with extended word alignment. In
addition, we extract source-gap correspondences, meanwhile, obtaining gap
tags. Experiments on two language pairs show the feasibility of our method and
give us inspirations for further improvement.

##  1 Introduction

Post-editing refers to the process of editing a rough machine-translated
sentence (referred to as MT) into a correct one. Compared with conventional
statistical machine translation (Koehn et al., 2003), neural machine
translation (Cho et al., 2014; Sutskever et al., 2014; Vaswani et al., 2017)
can generate translations with high accuracy. However, Yamada (2019) suggested
that there is no significant difference in terms of cognitive load for one to
post-edit an MT even it has high quality. Therefore, post-editing assistance
is profoundly needed.

Traditional post-editing assistance methods leave room for improvement. A
typical method is word-level QE (Specia et al., 2020) that predicts tags
expressed in the form of OK or BAD. However, such a dualistic judgement is not
efficient enough because meaning of BAD is ambiguous.

(a) Original word-level QE

(b) Refined word-level QE. Correspondences between REP are drawn in red and
that between INS are drawn in purple.

Figure 1: A comparison between original word-level QE and our proposal.

Word alignment is also proved to be helpful for post-editing assistance.
Schwartz et al. (2015) demonstrated that displaying word alignment
statistically significantly improves post-editing quality. However, unlike QE
tags, word alignment cannot tell where translation errors are. Besides, it is
non-trivial to extract word alignment between source sentence and MT. Schwartz
et al. (2015) used a built-in function of Moses (Koehn et al., 2007), a
decoder for statistical machine translation that is no longer suitable for
neural models.

In this paper, we propose a novel concept called extended word alignment. In
extended word alignment, we include incorrect word translations and null
alignment between a source sentence and MT. We adopt a supervised method based
on pre-trained language models to extract it. Based on extended word
alignment, we further propose a novel task called refined word-level QE which
outputs refined tags including REP, INS, and DEL along with word-level
correspondences. By referring to those information, post-editors could
immediately realize what operations (replacement, insertion, and deletion
towards MT) to do. Thus, we believe that refined word-level QE can
significantly improve post-editing assistance efficiency. Methodologically, we
firstly predict original word tags by training regression models for sequence
tagging based on architectures such as multilingual BERT (Devlin et al., 2019)
(mBERT) and XLM-RoBERTa (Conneau et al., 2020) (XLM-R). Then, we refine the
original word tags by incorporating extended word alignment in a rule-based
manner. In addition, we adopt a method similar to the one for extended word
alignment to extract source-gap correspondences and then determine gap tags.

Experiments on En-De and En-Zh datasets are conducted. Results show that our
method significantly outperforms the baseline. For En-De, our best performance
outperforms the baseline by 12.9% and 6.0% respectively in terms of mean F1
scores for Source and MT word refined tags. For En-Zh, the gap reaches 48.9%
and 16.9%. Further more, we discuss the effectiveness and limitations of our
method with specific cases.

##  2 Related Work

Word Alignment Extraction. Methods based on statistical models (Brown et al.,
1993; Och and Ney, 2003; Dyer et al., 2013) were dominant methods for word
alignment extraction. In recent years, neural-based methods developed quickly.
Garg et al. (2019) tried to obtain word alignment based on attention inside a
transformer (Vaswani et al., 2017), but their method perform just as well as
statistical tools like GIZA++ (Och and Ney, 2003). Dou and Neubig (2021)
utilized multilingual BERT to extract embeddings of all words conditioned on
context, aligning them under the restriction of optimal transport (Kusner et
al., 2015). Nagata et al. (2020) utilized the pre-trained language model in a
supervised manner and achieved a significant improvement against previous
studies with only around 300 parallel sentence pairs for fine-tuning. In our
work, we adapt their approach from ordinary word alignment to extended word
alignment. Details will be introduced in Section 4.1.

Word-Level QE. One of the conventional architectures for word-level QE is
LTSM-based predictor-estimator (Kim and Lee, 2016; Zhang and Weiss, 2016; Kim
et al., 2017). Recent researches (Wang et al., 2020) adopted new architectures
such as transformer (Vaswani et al., 2017).

For moderner methods, a typical example is QE BERT (Kim et al., 2019). They
built a mBERT for classification with explicit gap tokens in the input
sequence, but we find that regression models with adjustable threshold
consistently outperform classification models and explicit gap tokens harm
final performance. A newer research (Lee, 2020) adopted XLM-R rather than
mBERT, but they did not explain their strategy to determine a threshold.

All methods above require third-party large-scale parallel data for pre-
training. In contrast, our method introduced in Section 4.2 achieves
acceptable performance with small cost.

Post-Editing User Interface. Nayek et al. (2015) depicted an interface where
words that need editing are displayed with different colors. Schwartz et al.
(2015) emphasized the importance of displaying the word alignment. Both
interfaces do not tell the correctness of the translation of the MT words.
Compared to them, the interface we envisaged provides information about
translation quality (correctness) as well as suggestions of specific post-
editing operations.

There are also some other studies(Herbig et al., 2020; Jamara et al., 2021)
tried to introduce multi-modalities including touching, speech, hand gestures
into post-editing user interface, improving efficiency from another
perspective.

##  3 Refined Word-Level QE for Post-Editing Assistance

###  3.1 Original Word-Level QE

According to Specia et al. (2020), word-level QE shown in Figure 1(a) is a
task that takes a source sentence and its machine-translated counterpart (MT)
as input. It then outputs tags for source words, MT words and gaps between MT
words (MT gaps).111For convenience, source tags and MT word tags are
collectively known as word tags. MT word tags and MT gap tags are collectively
known as MT tags. All those tags are expressed either as OK or BAD. BAD
indicates potential translation errors that post-editors should correct. We
refer to such a task as original word-level QE.

Original word-level QE is not efficient enough for post-editing assistance
because BAD is ambiguous. For example, in Figure 1(a), tag of “white”
indicates a replacement of the mistranslation “黑” (black), but tag of “dogs”
indicates an insertion into the gap between “猫” and “吗”. It is impossible to
distinguish between these indications unless one attend to both entire
sentences, which makes post-editing assistance meaningless.

###  3.2 Extended Word Alignment

We formally define a novel concept called extended word alignment between
source sentence and MT. Ordinary word alignment indicates word-to-word
relations between a pair of semantically equivalent sentences in two
languages. Any word can be theoretically aligned with another semantically
equivalent word on the other side. In contrast, in extended word alignment,
translation errors in MT are considered. Specifically, a source word is
allowed to be aligned with its mistranslation (wrong word choice) and a word
is allowed to be aligned with nothing, namely null-aligned.

###  3.3 Refined Word-Level QE

Extended word alignment can disambiguate BAD tags, overcoming the disadvantage
of original word-level QE. When a BAD-tagged source word is aligned with a
BAD-tagged MT word, it is clear that a replacement is needed. Likewise, a
null-aligned BAD-tagged source word indicates an insertion and a BAD-tagged MT
word is a deletion.

To make our idea more user-friendly, we formally propose a novel task called
refined word-level QE by incorporating extended word alignment with original
word-level QE. Besides extended word alignment, following refined tags are
also included as the objectives.

  * •

REP is assigned to a source word and its mistranslation (wrong word choice) in
MT, indicating a replacement.

  * •

INS is assigned to a source word and the gap where translation should be
inserted in, indicating an insertion.

  * •

DEL is assigned to a redundant MT word, indicating a deletion.

In addition, we include correspondences between INS-tagged source words and MT
gaps to express the insertion points. Those source-gap correspondences along
with extended word alignment are collectively referred to as word-level
correspondences.

Figure 1(b) is an example of our proposal. Compared with Figure 1(a), Figure
1(b) points out the replacement of “黑” (black), the insertions of “and” and
“dogs” to the insertion point, and the deletion of “吗” (an interrogative voice
auxiliary).

##  4 Methodology222Besides the current method, we have also tried to use a
unified model based on architectures like XLM-R to directly predict refined
tags (OK/REP/INS/DEL) and word-level correspondences. However, due to lack of
training data and complexity of the problem, direct approach did not work
well. Therefore, we decided to adopt this multiple-phase approach.

###  4.1 Extended Word Alignment Extraction

Figure 2: Extracting extended word alignment by mBERT. The word will be
aligned with [CLS] token if it is null-aligned.

Extracting extended word alignment is non-trivial. Traditional unsupervised
statistical tools(Och and Ney, 2003; Dyer et al., 2013) cannot work well
because they expect semantically equivalent sentence pair as input. After
trying several neural methods (Garg et al., 2019; Dou and Neubig, 2021), we
empirically adopt the supervised method proposed by Nagata et al. (2020).

Specifically, extended word alignment extraction is regarded as a cross-
lingual span prediction problem similar to the paradigm that utilizes BERT
(Devlin et al., 2019) for SQuAD v2.0 (Rajpurkar et al., 2018). mBERT is used
as the basic architecture. Given a source sentence with one word marked
$S=[s_{1},s_{2},...,M,s_{i},M,...,s_{m}]$ ($M$ stands for a special mark
token) and the MT $T=[t_{1},t_{2},...,t_{n}]$, mBERT is trained to identify a
span $T_{(j,k)}=[t_{j},...,t_{k}](1\leq j\leq k\leq n)$ that is aligned with
the marked source word $s_{i}$. Cross entropy loss is adopted during training.

$\mathcal{L}^{align}_{s_{i}}=-\frac{1}{2}[\mathrm{log}(p_{j}^{start})+\mathrm{log}(p_{k}^{end})]$

Because of the symmetry of word alignment, similar operations will be done
again in the opposite direction. During testing, following Nagata et al.
(2020), we recognize word pairs whose mean probability of both directions is
greater than 0.4 as a valid word alignment. The image of the model is
illustrated in Figure 2.

Nagata et al. (2020) demonstrated that the mBERT-based method significantly
outperforms statistical methods in ordinary word alignment extraction.
According to them, extracting word alignment for each word independently is
the key to outperform other methods. Traditional methods model word alignment
on a joint distribution, so that an incorrect previous alignment might cause
more incorrect alignments like dominos. Our experiments prove that their
method consistently works for extended word alignment.

###  4.2 Original Word Tag Prediction

Figure 3:  Determining original word tags with pre-trained language models.

For Original tags, we conduct sequence tagging with multilingual pre-trained
language models including mBERT and XLM-R. Figure 3 shows the image. Input
sequence is organized in the format of “[CLS] source sentence [SEP] MT [SEP]”
without any mark tokens. Two linear layers followed by Sigmoid function
transform output vectors into scalar values as respective probabilities of
being BAD for each token. Formally, for a source sentence
$S=[s_{1},s_{2},...,s_{i},...,s_{m}]$ and an MT
$T=[t_{1},t_{2},...,t_{j},...,t_{n}]$, the total loss is the mean of binary
cross entropy of all word tags.

$\mathcal{L}^{tag}_{s_{i}}=-[y_{s_{i}}\mathrm{log}(p_{s_{i}})+(1-y_{s_{i}})\mathrm{log}(1-p_{s_{i}})]$
$\mathcal{L}^{tag}_{t_{j}}=-[y_{t_{j}}\mathrm{log}(p_{t_{j}})+(1-y_{t_{j}})\mathrm{log}(1-p_{t_{j}})]$
$\mathcal{L}^{tag}=\frac{1}{m+n}(\sum\limits_{i=1}^{m}\mathcal{L}_{s_{i}}+\sum\limits_{j=1}^{n}\mathcal{L}_{t_{j}})$

We have also implemented our models with classification top-
layers444Classification top-layers refers to a binary classification linear
layer with Softmax., but we find that regression models are consistently
better since we can adopt flexible threshold to offset the bias caused by
imbalance of reference tags.

###  4.3 Word Tag Refinement and Gap Tag Prediction

Figure 4: Refining the word tags by using extended word alignment.

We use extended word alignment to refine the original word tags. Following the
rules described in Section 3.3, we can refine word tags as Figure 4 shows. In
practical situation, some BAD-tagged words are likely to be aligned with OK-
tagged words. In that case, we change OK into BAD encouraging more generation
of REP.

Figure 5:  Determining gap tags by extracting source-gap alignments with
mBERT.

For gap tags, we adopt a method similar to the one described in Section 4.1.
Specifically, we model source-gap correspondences as alignment between source
words and MT gaps. We train a model that aligns an INS-tagged source word with
a two-word span in MT where corresponding gap is surrounded. Figure 5
illustrates our idea. During testing, when a valid source-gap correspondence
is confirmed, we tag the MT gap as INS555As for source words involved, we do
not change their tags and trust the refinement based on extended word
alignment because we believe extended word alignment is easier to model..

It is natural if we use such a method to determine gap tags based on the INS-
tagged source words prediction from previous workflow. However, in experiment,
we notice that absolute value of accuracy of INS-tagged source words is not
high. In order not to be influenced by the previous wrong predictions, instead
of treating this task as a downstream one, we conducted it independently.

##  5 Experiment

| En-De | En-Zh
---|---|---
| Source MCC | MT MCC | Source MCC | MT MCC
OpenKiwi | 0.266 | 0.358 | 0.248 | 0.520
mBERT-cls | 0.314 | 0.419 | 0.309 | 0.555
mBERT | 0.340 | 0.457 | 0.357 | 0.570
XLM-R-cls | 0.326 | 0.446 | 0.330 | 0.579
XLM-R | 0.345 | 0.453 | 0.354 | 0.592
WMT20 Top | 0.523(Wang et al., 2020) | 0.597(Lee, 2020) | 0.336(Rubino, 2020) | 0.610(Hu et al., 2020)

Table 1:  MCC of original tags. All MT gap tags of our systems are set to OK.
For En-De, unlike top systems that employs large-scale third-party resources,
we achieve acceptable performance only using QE dataset.  Figure 6: ROC curve
and AUC of the baseline and our systems(* indicates that the model outperforms
the baseline (OpenKiwi) with statistical significance (p<0.01)).

###  5.1 Data and Experimental Setups

We make full advantage of the En-De and En-Zh datasets of the shared task of
original word-level QE in WMT20666http://www.statmt.org/wmt20/quality-es
timation-task.html. There are 7,000, 1,000, and 1,000 sentence pairs with
annotation of tags respectively for training, development, and test set. Since
the original datasets do not contain refined objectives, we additionally
annotate the original development sets with all the objectives for refined
word-level QE. Those annotated 1,000 pairs are further divided into 200 pairs
for evaluation and 800 pairs for fine-tuning.

Extended Word Align. | Source-Gap Corr. | En-De (F1/P/R) | En-Zh (F1/P/R)
---|---|---|---
FastAlign | mBERT | 0.828/0.812/0.844 | 0.739/0.773/0.709
AWESoME | 0.891/0.915/0.868 | 0.814/0.871/0.764
mBERT | 0.895/0.917/0.875 | 0.836/0.888/0.790
ft-mBERT | ft-mBERT |  0.916/0.913/0.918 |  0.888/0.887/0.889
Table 2:  Evaluation of word-level correspondences. “mBERT” indicates mBERT
trained with 7,000-pair pseudo data and “ft-mBERT” indicates mBERT further
fine-tuned with 800-pair data.

All the experiments are conducted with modified scripts from
transformers-v3.3.1777https://github.com/huggingface/transfo
rmers on an NVIDIA TITAN RTX (24GB) with CUDA 10.1. For pre-trained models, we
use bert-base-multilingual-cased for mBERT and xlm-roberta-large for XLM-R
from Huggingface.

To train the model for original tags described in Section 4.2, we use the
7,000-pair training set provided by WMT20. 800 pairs of manually annotated
data whose refined tags are degenerated into original tags are used for
further training. Learning rate is set to 3e-5 and 1e-5 for mBERT and XLM-R
respectively and both models are trained for 5 epochs. All the other
configurations are remained unchanged as the default.

To train the models extracting extended word alignment described in Section
4.1, we utilize AWESoME (Dou and Neubig, 2021) to generate pseudo alignment
data based on 7,000-pair WMT20 training set. We also use extra 800 sentence-
pair annotated alignment data for fine-tuning. Models are pre-trained for 2
epochs and fine-tuned for 5 epochs with a learning rate of 3e-5. Most
configurations are remained unchanged as the default but max_seq_length and
max_ans_length are set to 160 and 15 following Nagata et al. (2020).

To train the model extracting source-gap correspondences described in Section
4.3, similar to what described above, we firstly adopt 7,000 sentence-pair
WMT20 training set, generating pseudo data by randomly dropping out some
target words in PE888Provided [P]ost-[E]dited sentence from MT in WMT20
dataset. It is regarded as the correct translations.. Then we link gaps where
words are dropped with their source counterparts according to the source-PE
alignment extracted by AWESoME. Also, 800 sentence-pair of gold source-gap
correspondences are used for fine-tuning. All model configurations and
training settings are kept identical as those of the model for extended word
alignment extraction.

###  5.2 Experimental Results

####  5.2.1 Evaluation of Original Tags

We firstly compare our performance with other participants of WMT20 Therefore,
we use identical test sets to evaluate and only use data from the original
training set of WMT20 to train our models here. Following WMT20 (Specia et
al., 2020), we adopt the Matthews correlation coefficient (MCC) as the metric.
From the perspective of competition, we make every effort to boost the
performance. Thus we set all gap tags as OK rather than predicting them as we
find such a strategy leads to the best MCC. The results are shown in Table 1.

In general, pre-trained language models consistently outperform the baseline
which is a LSTM-based predictor-estimator implemented with OpenKiwi. For En-
De, our best source and MT MCC would have ranked sixth on the leaderboard of
WMT20. For En-Zh, our best source and MT MCC would have ranked first and
second on the leaderboard of WMT20.

It is also noteworthy that regression models consistently outperform
classification models with the suffix “-cls”. For regression models, we search
an optimized threshold that maximize sum of source and MT MCC on the
development set and adopt it on the test set to determine tags. To exclude
errors caused by single optimized threshold, we further draw the ROC curves
and AUC in Figure 6. The results demonstrate that our regression models based
on mBERT and XLM-R statistically significantly outperform the baseline.

For En-De, Wang et al. (2020) and Lee (2020) both used large-scale third-party
data999Wang et al. (2020) used parallel data from WMT20 news translation task
to pre-train a predictor and Lee (2020) generated 11 million pairs of pseudo
QE data with 23 million pairs of sentences.. Besides the top-2, the third
system (Rubino, 2020) is also pre-trained with 5 million sentence pairs but
got 0.357 and 0.485 respectively. Therefore, we believe that we achieve
acceptable performance with very small cost.

####  5.2.2 Evaluation of Word-Level Correspondences

We evaluate extended word alignment and source-gap correspondences jointly as
word-level correspondences. The results are shown in Table 2. Two baselines
(“FastAlign” and “AWESoME”) cannot predict source-gap correspondences since
they are designed for ordinary word alignment. We combine their extended word
alignment with prediction of source-gap correspondences by “mBERT” for fair
comparison. All predictions are evaluated by F1 score as well as precision and
recall.

Neural-based methods significantly outperform statistical “FastAlign”. The gap
of 0.4% for En-De and 2.2% for En-Zh between “AWESoME” and “mBERT” is not
significant. But it might implies that pre-trained language models like mBERT
is able to filter noises in pseudo data and produce high-quality word-level
correspondences. Additionally, a better performance of “fine-tuned mBERT”
indicates that the upper bound could be higher if more annotated data is
available.

####  5.2.3 Evaluation of Refined Tags

| Extended
---
Word Alignment
| Original
---
QE Tags
| Source F1 Scores
---
Mean (OK/REP/INS)
| MT F1 Scores
---
Mean (OK/REP/DEL/INS)
FastAlign | OpenKiwi | 0.626 (0.696/0.492/0.174) | 0.767 (0.847/0.477/0.124/0.156)
AWESoME | 0.708 (0.781/0.549/0.373) | 0.807 (0.879/0.548/0.395/0.156)
mBERT | mBERT | 0.739 (0.825/0.540/0.421) | 0.820 (0.895/0.544/0.389/0.156)
XLM-R | 0.709 (0.781/0.548/0.410) | 0.809 (0.879/0.522/0.415/0.156)
ft-mBERT | rt-mBERT |  0.755 (0.850/0.538/0.400) |  0.827 (0.904/0.535/0.347/0.175)
rt-XLM-R | 0.685 (0.748/0.544/0.431) | 0.805 (0.871/0.538/0.580/0.175)
(a) En-De Results

| Extended
---
Word Alignment
| Original
---
QE Tags
| Mean Source F1 Scores
---
(OK/REP/INS)
| Mean MT F1 Scores
---
(OK/REP/DEL/INS)
FastAlign | OpenKiwi | 0.360 (0.379/0.280/0.071) | 0.728 (0.781/0.276/0.173/0.042)
AWESoME | 0.371 (0.391/0.285/0.066) | 0.733 (0.786/0.280/0.202/0.042)
mBERT | mBERT | 0.836 (0.914/0.446/0.020) | 0.891 (0.947/0.441/0.316/0.042)
XLM-R | 0.843 (0.929/0.410/0.018) | 0.895 (0.955/0.402/0.275/0.042)
ft-mBERT | rt-mBERT | 0.848 (0.929/0.447/0.034) | 0.897 (0.954/0.441/0.284/0.042)
rt-XLM-R |  0.849 (0.928/0.451/0.028) |  0.897 (0.955/0.446/0.289/0.042)
(b) En-Zh Results

Table 3:  Evaluation of refined tags. Main metric is a weighted mean of F1
scores according to ratio of each type of tags in reference. “ft-” indicates
that the model is fine-tuned with the extra 800-pair annotated alignment data.
“rt-” indicates that the model is further trained with extra 800-pair
annotated tag data.

As introduced, we combine prediction of extended word alignment and original
word tags101010While predicting the original tags, we did not directly use the
optimized threshold determined in Section 5.2.1 since test set here originates
from the original development set. Instead, we take the original test set of
WMT20 for development purposes and re-searched an optimized threshold on it.
to get refined word tags. Moreover, we deduce gap tags from source-gap
correspondences. Origin of source-gap correspondences used is kept consistent
with Table 2 according to extended word alignment. For baseline, combinations
of FastAlign/AWESoME and OpenKiwi is adopted. As for metric, we use F1 score
of each type of tags along with a weighted mean of all those F1 scores, taking
the proportion of each tag in reference as weight. The results are shown in
Table 3.

Our best model outperforms the baseline by 12.9% and 6.0% respectively on
source and MT refined tags in terms of mean F1 scores in En-De experiments. As
for the En-Zh experiments, mean F1 scores are significantly improved by 48.9%
and 16.9%.

We also notice that though fine-tuned mBERT extracts extended word alignment
with good accuracy, the absolute value of refined tag accuracy is still
unsatisfactory (especially that of INS and DEL). We will discuss that in the
next section.

##  6 Discussion on Specific Cases

(a) An En-Zh case with correct refined word tag prediction.

(b) An En-Zh case with incorrect refined word tag prediction.

(c) An En-De case with correct prediction of source-gap correspondence and gap
tag.

(d) An En-Zh case with incorrect prediction of source-gap correspondences and
gap tags.

Figure 7: Specific cases. For visual neatness, most OK tags are omitted and
some continuous spans are merged.

###  6.1 Discussion on Refined Word Tags

In Figure 7(a), our system basically succeeds in detecting errors caused by
incorrect use of punctuations. Our system correctly suggests the replacements
for the second comma and the half-width period. As for the first comma, the
translation is still natural and acceptable if we delete the comma following
the system’s suggestion. Moreover, our system successfully detects
mistranslations of “passes” and “touchdowns”. In MT, those football
terminologies are respectively translated as “通行证” (a pass to enter somewhere)
and “摔倒” (falling down). It is noteworthy that those two mistranslations are
not revised in post-edited corpus provided by WMT21. It implies that our
system performs surprisingly well as it even succeeds in detecting
mistranslations that is not noticed by human annotators.

In Figure 7(b), our system still works well in detecting incorrect use of
half-width punctuations. However, compared with the reference, “abdominal
aneurysm” is mistranslated and our model failed to detect it because both are
tagged as OK during the prediction of original tags. A premature prediction of
OK prevents a word from being refined into REP/INS/DEL later.

We believe that an inappropriate threshold mainly leads to such an issue.
Predicted probabilities of “腹部” and “动脉瘤” are respectively 0.103 and 0.134,
but the optimized threshold used is 0.88 as we searched it to maximize the MCC
on the whole set. Meanwhile, probabilities of all other OK-tagged MT words are
actually smaller than 0.01. As a result, if we set a threshold between
0.01-0.10 for this sentence pair, we could have obtained the perfect result.
In the future, we plan to investigate into methods that can determine fine-
grained optimized threshold for each sentence pair.

###  6.2 Discussion on Gap Tags

Figure 7(c) shows a typical En-De case that our model handles well. In German,
it is more natural to indicate actions took place in the past in perfect tense
rather than past tense. In this case, English verb “drafted” should be
modified to “haben … ausgewählt”. Our model correctly suggests a
correspondence between “drafted” and the MT gap in front of the period. As
there are many cases need similar modification that inserts a particular word
(like particle or infinitive for clause) before the period in MT, it is easier
for our model to learn such laws. It probably explains the relatively good
accuracy of INS in En-De experiments.

In contrast, Figure 7(d) is an En-Zh example showing that our model tends to
align many source words with the gap right before or after their translation
in MT even the translation is correct and needs no extra insertions. The word
“dissected” is unnecessarily aligned with the gap around its translation “解剖”.
Two human names are also unnecessarily aligned with gaps. As a result, four
gaps are incorrectly tagged as INS. We observed the annotated dataset and
noticed that many Chinese words in MT are slightly modified by adding prefixes
and suffixes during post-editing. For example, “成年 海龟” (adult sea turtle) is
modified to “成年的海龟” (adding “的” as a suffix for adjective). “演讲” (the speech)
is modified to “这一演讲” (emphasizing “this” speech). Generally, those
modifications are not necessary because of the free Chinese grammar. However,
existence of those modifications might mislead the model into preferring to
unnecessarily align a word with the gap around its translation like Figure
7(d). To address this issue, we plan to restrict the annotation rules to
exclude meaningless modification in En-Zh training data in the future.

##  7 Conclusion and Future Work

To improve post-editing assistance efficiency, we define a novel concept
called extended word alignment. By incorporating extended word alignment with
original word-level QE, we formally propose a novel task called refined word-
level QE. To solve the task, we firstly adopt a supervised method to extract
extended word alignment and then predict original tags with pre-trained
language models by conducting sequence tagging. We then refine word tags with
extended word alignment. Additionally, we extract source-gap correspondences
and determine gap tags. We perform experiments and a discussion on specific
cases.

In the future, we would like to polish our work in the following perspectives.
Firstly, we want to develop methods that determines fine-grained threshold as
elaborated in Section 6. Moreover, we prepare to conduct a human-evaluated
experiment to prove the superiority of refined word-level QE in terms of post-
editing assistance efficiency.

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