# Characterizing Correlation Matrices that Admit a Clustered Factor
Representation __

Chen Tong‡ and Peter Reinhard Hansen§

‡ _Department of Finance, School of Economics, Xiamen University_
§ _University of North Carolina & Copenhagen Business School_ __ Chen Tong
acknowledges financial support from the Ministry of Education of China,
Humanities and Social Sciences Youth Fund (22YJC790117).

(_August 28, 2024_)

###### Abstract

The Clustered Factor (CF) model induces a block structure on the correlation
matrix and is commonly used to parameterize correlation matrices. Our results
reveal that the CF model imposes superfluous restrictions on the correlation
matrix. This can be avoided by a different parametrization, involving the
logarithmic transformation of the block correlation matrix.

Keywords: Block correlation matrix, copula, clustering, factor model.

JEL Classification: C38

##  1 Introduction

In empirical models involving high-dimensional covariance matrices, it is
often beneficial to introduce a parsimonious structure to mitigate issues
stemming from overfitting. A commonly used approach is the Clustered Factor
(CF) model, which is characterized by a linear factor structure and group
clustering, where factor loadings are shared within each group.

The clustered factor model emerged from high-dimensional copula models with a
multi-factor structure, see e.g Krupskii and Joe, (2013) and Creal and Tsay,
(2015). Clustering became a popular additional structural component, because
it is easy to interpret and makes the model more parsimonious, see Krupskii
and Joe, (2015), Oh and Patton, 2017a (, 2017b, 2023), Manner et al., (2019),
and Opschoor et al., (2021).

It is well known that the CF model induces a block structure on the
correlation matrix (whenever variables are ordered by group assignments).
However, the CF model is not consistent with all block correlation matrices.
It cannot generate negative within-group correlations. However, this is
unlikely to be problematic in empirical applications, because variables in the
same cluster are expected to share traits and be positively correlated. It is
more problematic that the CF model imposes other restrictions that rules out a
class of block correlation matrices, which seem perfectly reasonable and
ordinary from an empirical viewpoint. Below we fully characterize the class of
block correlation matrices that are coherent with the CF model, and express
the superfluous restrictions the CF model imposes as a testable hypothesis. We
then proceed to highlight an alternative parametrization of block correlation
matrices by Archakov and Hansen, (2023), which is based on the matrix
logarithm of the correlation matrix of Archakov and Hansen, (2021). This
parametrization uniquely parametrizes any non-singular block correlation
matrix, without imposing additional structure. A simple linear factor
structure is also applicable to this parametrization, if a more parsimonious
model is required.

The rest of this paper is organized as follows. In Section 2, we introduce the
required notation and the CF model followed by the theoretical results that
characterizes the block correlation matrices that admit CF model. We express
the CF structure as a testable hypothesis, and detail a testing procedure for
selecting the number of factors. In Section 3, we presents the alternative
parametrization by Archakov and Hansen, (2023) and illustrate it with a simple
example. We conclude in Section 4 with a brief summary.

##  2 Clustered Factor Models and Block Correlation Matrices

Consider an $n$-dimensional random variable, $X\in\mathbb{R}^{n}$, with non-
singular covariance matrix $\Sigma=\mathrm{var}(X)$. Then the corresponding
correlation matrix is given by

$C=\Lambda_{\sigma}^{-1}\Sigma\Lambda_{\sigma}^{-1},$

where
$\Lambda_{\sigma}=\mathrm{diag}(\sqrt{\Sigma_{11}},\ldots,\sqrt{\Sigma_{nn}})$,
and the vector of standardized variables,

$Z=\Lambda_{\sigma}(X-\mu),\qquad\mu=\mathbb{E}X,$

is such that $C=\mathbb{E}(ZZ^{\prime})$.

In the copula literature, it is common to parametrize $C$ with the clustered
factor (CF) model, which is characterized by a linear factor structure for $Z$
and a partitioning of the variables into clusters/groups. The factor structure
is defined by,

$Z_{i}=b_{i}^{\prime}f+\varepsilon_{i},\qquad i=1,\ldots,n,$ (1)

where $f\in\mathbb{R}^{r}$ is a common vector with $r$ orthogonal factors,
$\mathbb{E}f=0$ and $\mathrm{var}(f)=I_{r}$, and the idiosyncratic variables
$(\varepsilon_{1},\ldots,\varepsilon_{n})$ are mutually uncorrelated and
uncorrelated with the factors, $\mathrm{cov}(f,\varepsilon_{i})=0$ and
$\mathrm{cov}(\varepsilon_{i},\varepsilon_{j})=0$ for all $i\neq j$. The
vectors of _factor loadings_ are given by $b_{i}={\rm
corr}(f,Z_{i})\in\mathbb{R}^{r}$, for $i=1,\ldots,n$, and it follows that
$\mathrm{var}(\varepsilon_{i})=1-b_{i}^{\prime}b_{i}$.

A partitioning of the $n$ elements into $K<n$ groups adds additional
structure. It is assumed that the factor loadings are common within each
group, such that

$b_{i}=\beta_{k}\quad\text{for all }i\in{\rm G}_{k},$ (2)

where $({\rm G}_{1},\ldots,{\rm G}_{K})$ is a partition of $\\{1,\ldots,n\\}$,
with ${\rm G}_{k}$ containing $n_{k}$ elements, $k=1,\ldots,K$, such that
$n=\sum_{k}n_{k}$.

The CF model implies that the correlation between two variables is solely
determined by their group classification. For $i\in{\rm G}_{k}$ and $j\in{\rm
G}_{l}$ with $i\neq j$ we have

$C_{ij}={\rm corr}(z_{i},z_{j})=\rho_{kl}\equiv\beta_{k}^{\prime}\beta_{l}.$

By rearranging $X_{1},\ldots,X_{n}$ according to their group assignments, we
obtain a block correlation matrix that can be expresses as

$C=\left[\begin{array}[]{cccc}C_{[1,1]}&C_{[1,2]}&\cdots&C_{[1,K]}\\\
C_{[2,1]}&C_{[2,2]}\\\ \vdots&&\ddots\\\
C_{[K,1]}&&&C_{[K,K]}\end{array}\right],$ (3)

where $C_{[k,l]}$ is an $n_{k}\times n_{l}$ matrix given by

$C_{[k,k]}=\left[\begin{array}[]{cccc}1&\rho_{kk}&\cdots&\rho_{kk}\\\
\rho_{kk}&1&\ddots\\\ \vdots&\ddots&\ddots\\\
\rho_{kk}&&&1\end{array}\right]\quad{\rm and}\quad
C_{[k,l]}=\left[\begin{array}[]{ccc}\rho_{kl}&\cdots&\rho_{kl}\\\
\vdots&\ddots\\\ \rho_{kl}&&\rho_{kl}\end{array}\right]\quad{\rm if}\ k\neq
l.$

The advantage of a block correlation matrix, is that its number of unique
correlations is (at most) $d=K\left(K+1\right)/2$, whereas the number of
distinct correlations in an unrestricted $n\times n$ correlation matrix is
$n\left(n-1\right)/2$. If $K$ is fixed, then $d$ does not increase with $n$,
which makes it possible to model high-dimensional correlation matrices with
relatively few parameters.

###  2.1 When is the Clustered Factor Structure Applicable

An interesting question is whether a give block correlation matrix can be
expressed as a CF model. Any block correlation matrix is clearly coherent with
a group partitioning (2), but additional structure is needed for it to have
the representation in (1). In some applications, it will be relevant to know
if the factor structure rules out empirically relevant correlation matrices.
It is therefore interesting to characterize the set of block correlation
matrices that are compatible with the CF model.

For later use, we define the matrix
$B=\left(\beta_{1},\beta_{2},\ldots,\beta_{K}\right)^{\prime}\in\mathbb{R}^{K\times
r}$, where $\beta_{k}\in\mathbb{R}^{r\times 1}$ is the vector of group-
specific factor loadings, as defined in (1) and (2).

From a block correlation matrix, $C$, we define the following $K\times K$
matrix

$A^{*}=\left[\begin{array}[]{cccc}\rho_{11}&\rho_{12}&\cdots&\rho_{1K}\\\
\rho_{21}&\rho_{22}\\\ \vdots&&\ddots\\\
\rho_{K1}&&&\rho_{KK}\end{array}\right].$ (4)

In this paper we focus, without loss of generality, on the case where $C$ is
nonsingular, which implies that $\rho_{kk}<1$ for all within-group
correlations, $k=1,\ldots,K$.

######  Remark.

If $n_{k}=1$ for some $k$, then $A^{\ast}$ is not fully defined by $C$,
because there is no within-group correlation in the $k$-th group, leaving
$\rho_{kk}$ undefined. This is the reason that we treat the case with
singleton groups separately in the following Theorem.

######  Theorem 1.

Let $C$ be a nonsingular block correlation matrix. If $\min_{k}n_{k}\geq 2$,
then $C$ has a factor representation if and only if $A^{*}$ is positive semi-
definite. If, instead, $n_{k_{1}}=\cdots=n_{k_{m}}=$1 (and $n_{k}\geq 2$ for
all other), then $C$ has a factor representation if and only if there exist
$\rho_{k_{1}k_{1}},\ldots,\rho_{k_{m}k_{m}}<1$ such that $A^{*}$ is positive
semi-definite.

###### Proof.

Suppose that $A^{*}$ is positive semi-definite (psd) with $\rho_{kk}<1$ for
all $k$, then it can be expressed as $A^{*}=BB^{\prime}$, where
$\beta_{k}^{\prime}\beta_{k}=\rho_{kk}<1$. This shows that $C$ admits a factor
representation. (The matrix of factor loadings, $B$, is not unique, but one
valid choice is $B=Q\Lambda^{1/2}$, where $A^{*}=Q\Lambda Q^{\prime}$ is the
eigendecomposition of $A^{\ast}$). That the resulting correlation matrix, $C$,
is nonsingular follows from Lemma 1.

Now suppose $C$ is a nonsingular block correlation matrix that is given from
CF model with factor loadings $B=(\beta_{1},\ldots,\beta_{K})^{\prime}$. Then
it follows that $\rho_{kl}=\beta_{k}^{\prime}\beta_{l}$, such that
$A^{\ast}=BB^{\prime}$ is psd with $\rho_{kk}=\beta_{k}^{\prime}\beta_{k}<1$
for all $k$. This completes the proof. ∎

######  Lemma 1.

Suppose that the matrix $A^{*}$ is positive semi-definite with $\rho_{kk}<1$,
for all $k=1,\ldots,K$. Then the corresponding block correlation matrix, $C$,
is positive definite.

###### Proof.

From Archakov and Hansen, (2023) it follows that a block correlation matrix is
positive definite if and only if

$A=\left[\begin{array}[]{cccc}1+\left(n_{1}-1\right)\rho_{11}&\rho_{12}\sqrt{n_{1}n_{2}}&\cdots&\rho_{1K}\sqrt{n_{1}n_{K}}\\\
\rho_{21}\sqrt{n_{1}n_{2}}&1+\left(n_{2}-1\right)\rho_{22}\\\
\vdots&&\ddots\\\
\rho_{K1}\sqrt{n_{K}n_{1}}&&&1+\left(n_{K}-1\right)\rho_{KK}\end{array}\right],$

is positive definite with $|\rho_{kk}|<1$, for all $k$. This matrix can be
expressed as $A=\Lambda_{n}^{1/2}A^{*}\Lambda_{n}^{1/2}+\Psi_{1-\rho}$, where

$\Psi_{1-\rho}=\left[\begin{array}[]{ccc}1-\rho_{11}&\cdots&0\\\
\vdots&\ddots&\vdots\\\
0&\cdots&1-\rho_{KK}\end{array}\right],\qquad\text{and}\qquad\Lambda_{n}=\left[\begin{array}[]{ccc}n_{1}&&0\\\
&\ddots\\\ 0&&n_{K}\end{array}\right].$

If matrix $A^{*}$ is psd with $\rho_{kk}<1$, then $\Psi_{1-\rho}$ is positive
definite and $\Lambda_{n}^{1/2}A^{*}\Lambda_{n}^{1/2}$ is psd. It now follows
that $A$ is positive definite because it is the sum of a positive definite
matrix and a positive semidefinite matrix, and from Archakov and Hansen,
(2023, theorem 1) it follows that the corresponding block correlation matrix
is also positive definite. ∎

Although Lemma 1 shows that a psd $A^{*}$ with $\rho_{kk}<1$ implies that $A$
is positive definite, the converse is not true. We have that
$A^{*}=\Lambda_{n}^{-1/2}\left(A-\Psi_{1-\rho}\right)\Lambda_{n}^{-1/2}$,
which is psd if and only if $A-\Psi_{1-\rho}$ is psd. The latter is not
guaranteed. The simplest example is if a within-group correlation is negative,
since the diagonal elements of $A-\Psi_{1-\rho}$ equal $n_{k}\rho_{kk}$,
$k=1,\ldots,K$. Another, less obvious, example is the following block
correlation matrix:

$C=\left[\begin{array}[]{ccccccccc}1&\bullet&\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\bullet&\bullet&\bullet\\\
0.70&1&\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\bullet&\bullet&\bullet\\\
0.70&0.70&1&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\bullet&\bullet&\bullet\\\
\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&1&\bullet&\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet\\\
\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&0.63&1&\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet\\\
\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&\pagecolor{gray!15}0.58&0.63&0.63&1&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet&\pagecolor{gray!15}\bullet\\\
0.54&0.54&0.54&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&1&\bullet&\bullet\\\
0.54&0.54&0.54&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&0.71&1&\bullet\\\
0.54&0.54&0.54&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&\pagecolor{gray!15}0.19&0.71&0.71&1\end{array}\right],$
(5)

which is a positive definite, it’s smallest eigenvalue is $\lambda_{{\rm
min}}(C)=0.26$. However,

$A^{*}=\left[\begin{array}[]{ccc}0.70&0.58&0.54\\\ 0.58&0.63&0.19\\\
0.54&0.19&0.71\end{array}\right],$

has a negative eigenvalue, $\lambda_{\min}(A^{\ast})=-0.02$, and $C$ can
therefore not be expresses as a CF model. It seems unjustified that the
Clustered Factor (CF) model precludes the correlation matrix in (5) (and
similar matrices) in advance, because there does not appear to be anything
bizarre or unusual about this particular block correlation matrix.

Another question is how the number of factors, $r$, are needed to to generate
a given correlation matrix. We address this in the following Theorem.

######  Theorem 2 (Minimal factor model).

If $C$ admits a CF model, then the minimal number of factors is $r={\rm
rank}\left(A^{\ast}\right)$.

###### Proof.

If $C$ is generated by a factor model with $\tilde{r}$ factors, then
$A^{*}=\tilde{B}\tilde{B}^{\prime}$ where
$\tilde{B}\in\mathbb{R}^{K\times\tilde{r}}$. We have
$r=\mathrm{rank}(A^{\ast})\leq\min(K,\tilde{r})$, and we can express
$A^{*}=BB^{\prime}$ where ${\rm rank}\left(B\right)=r$, which shows that $C$
admits a factor structure with $r$ orthogonal factors. ∎

Theorem 2 shows that increasing the number of factors beyond $K$ does not
broaden the range of attainable correlation matrices. The (maximum) number of
free parameters in the CF model is $K(K+1)/2$, because we can always rotate
the latent factors, $f$, such that $B$ is upper (or lower) triangular.

###  2.2 Testing if $C$ Admits a Factor Representation

Theorem 1 shows that the block correlation matrices that admits a CF
representation, are characterized the smallest eigenvalue of $A^{\ast}$ being
nonnegative, i.e. $\lambda_{\min}(A^{\ast})\geq 0$. Here $A^{\ast}$ is defined
by (4) where we set $\rho_{kk}=1$ whenever $n_{k}=1$.

In practice, it is therefore relatively straight forward to test the CF
representation using the null hypothesis:

$H_{F}:\lambda_{\min}(A^{\ast})\geq 0.$

###  2.3 Determining the Minimal Number of Factors

Given a block correlation matrix with a CF structure, we can proceed to
estimate the minimum number of factors, $r$, as defined by the rank of
$A^{\ast}$. This could be done with the sequential procedure propose in
Pantula, (1989), which is commonly used to determine the cointegration rank in
VAR models, see Johansen, (1988, 1991).111The same testing principle is used
for lag-length selection in time series model, see Ng and Perron, (2001), and
in multiple comparisons to determine the model confidence set, see Hansen et
al., (2011).

Let $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{K}$ denote the
eigenvalues of $A^{\ast}$, and consider the hypothesis
$H_{F,q}:\lambda_{r+1}=\cdots=\lambda_{K}=0$, implying that $A^{\ast}$ is psd
with $\mathrm{rank}(A^{\ast})\leq r$. The sequential procedure begins by
testing $H_{F,0}$ against $H_{F,K}$. In the event of a rejection, we proceed
to test $H_{F,1}$ against $H_{F,K}$, and so forth until a null hypothesis is
not rejected. We can set $\hat{r}=r^{\ast}$, where $r^{\ast}$ is the first
instance (the smallest $r$) where $H_{F,r}$ is not rejected. The degrees of
freedom in $B$ for a model with $r\leq K$ factors is $r[K+(K-r)+1]/2$.

##  3 A Better Parametrization Block Correlation Matrices

A new parametrization of correlation matrices was proposed in Archakov and
Hansen, (2021), and is defined by

$\gamma={\rm vecl}\left(\log C\right)\in\mathbb{R}^{d},\qquad d=n(n-1)/2,$ (6)

where ${\rm vecl}(\cdot)$ extracts and vectorizes the elements below the
diagonal and $\log C$ is the matrix logarithm of the correlation matrix.222For
a nonsingular correlation matrix, we have $\log C=Q\log\Lambda Q^{\prime}$,
where $C=Q\Lambda Q^{\prime}$ is the spectral decomposition of $C$, so that
$\Lambda$ is a diagonal matrix with the eigenvalues of $C$. The identity (6)
defines a one-to-one mapping between $\mathbb{R}^{d}$ and the set of non-
singular $n\times n$ correlation matrices.

Because the matrix logarithm preserves the block structure in $C$, see
Archakov and Hansen, (2023), $\gamma$ will contain many “duplicates”. Thus, we
can parametrize block correlation matrices with $\eta\in\mathbb{R}^{q}$, where
$\eta$ is a subvector of $\gamma$ and $q\leq K(K+1)/2$. The follow example
will serve as an illustration,

$\underbrace{\left[\begin{array}[]{cccccc}1.0&0.4&0.4&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3\\\
0.4&1.0&0.4&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3\\\
0.4&0.4&1.0&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3\\\
\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&1.0&0.6&0.6\\\
\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&0.6&1.0&0.6\\\
\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&\pagecolor{gray!15}0.3&0.6&0.6&1.0\end{array}\right]}_{=C}\quad\underbrace{\left[\begin{array}[]{rrrrrr}-.19&.326&.326&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162\\\
.326&-.19&.326&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162\\\
.326&.326&-.19&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162\\\
\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&-.38&.533&.533\\\
\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&.533&-.38&.533\\\
\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&\pagecolor{gray!15}.162&.533&.533&-.38\end{array}\right]}_{=\log
C}.$

Here we can use
$\eta=(\begin{array}[]{ccc}0.326&\pagecolor{gray!15}0.162&0.533\end{array})^{\prime}$
as the condensed vector parametrization of the block correlation matrix. For a
given block partitioning, $(n_{1},\ldots,n_{K})$, any non-singular block
correlation matrix will map to a unique vector $\eta$, and any vector
$\eta\in\mathbb{R}^{q}$ will map to a unique non-singular block correlation
matrix, see Archakov and Hansen, (2023).

It is straightforward to add additional structure onto the
$\eta$-parametrization, for instance by restricting $\eta$ to be in a subspace
of lower dimension than $q$. For a multivariate GARCH model, these ideas are
explored in Archakov et al., (2020).

Creal and Kim, (2022) recently adopted this parametrization for Bayesian
modeling of block correlation matrices. An attractive feature of this
parametrization, it that it facilitates priors with full support on the entire
set of non-singular block correlation matrices.

##  4 Summary

We have characterized the class of block correlation matrices that can be
expressed as a clustered factor model. While the clustered factor model serves
as a valuable tool for generating clustered correlation structures, it does
introduce unnecessary constraints on the correlation matrix, which may limit
its practical relevance for some empirical problems. The alternative
parametrization, which is based on the matrix logarithm of the correlation
matrix, seems better suited for the modeling of block correlation matrices. It
avoids the imposition of superfluous constraints on the correlation matrix
and, if needed, it provides a flexible framework for further reduction of the
degrees of freedom.

## References

  * Archakov and Hansen, (2021) Archakov, I. and Hansen, P. R. (2021).  A new parametrization of correlation matrices.  Econometrica, 89:1699–1715.
  * Archakov and Hansen, (2023) Archakov, I. and Hansen, P. R. (2023).  A canonical representation of block matrices with applications to covariance and correlation matrices.  Forthcoming in Review of Economics and Statistics.
  * Archakov et al., (2020) Archakov, I., Hansen, P. R., and Lunde, A. (2020).  A multivariate Realized GARCH model.  arXiv:2012.02708, [econ.EM].
  * Creal and Kim, (2022) Creal, D. and Kim, J. (2022).  Bayesian estimation of block covariance matrices.  Working paper.
  * Creal and Tsay, (2015) Creal, D. and Tsay, R. S. (2015).  High dimensional dynamic stochastic copula models.  Journal of Econometrics, 189:335–345.
  * Hansen et al., (2011) Hansen, P. R., Lunde, A., and Nason, J. M. (2011).  The model confidence set.  Econometrica, 79:456–497.
  * Johansen, (1988) Johansen, S. (1988).  Statistical analysis of cointegration vectors.  Journal of Economic Dynamics and Control, 12:231–254.
  * Johansen, (1991) Johansen, S. (1991).  Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models.  Econometrica, 59:1551–1580.
  * Krupskii and Joe, (2013) Krupskii, P. and Joe, H. (2013).  Factor copula models for multivariate data.  Journal of Multivariate Analysis, 120:85–101.
  * Krupskii and Joe, (2015) Krupskii, P. and Joe, H. (2015).  Structured factor copula models: Theory, inference and computation.  Journal of Multivariate Analysis, 138:53–73.
  * Manner et al., (2019) Manner, H., Stark, F., and Wied, D. (2019).  Testing for structural breaks in factor copula models.  Journal of Econometrics, 208:324–345.
  * Ng and Perron, (2001) Ng, S. and Perron, P. (2001).  Lag length selection and the construction of unit root tests with good size an power.  Econometrica, 69:1519–1554.
  * (13) Oh, D. H. and Patton, A. J. (2017a).  Modeling dependence in high dimensions with factor copulas.  Journal of Business & Economic Statistics, 35:139–154.
  * (14) Oh, D. H. and Patton, A. J. (2017b).  Time-varying systemic risk: Evidence from a dynamic copula model of cds spreads.  Journal of Business & Economic Statistics, 36:181–195.
  * Oh and Patton, (2023) Oh, D. H. and Patton, A. J. (2023).  Dynamic factor copula models with estimated cluster assignments.  Journal of Econometrics, Forthcoming.
  * Opschoor et al., (2021) Opschoor, A., Lucas, A., Barra, I., and van Dijk, D. (2021).  Closed-form multi-factor copula models with observation-driven dynamic factor loadings.  Journal of Business and Economic Statistics, 39:1066–1079.
  * Pantula, (1989) Pantula, S. G. (1989).  Testing for unit roots in time series data.  Econometric Theory, 5:256–271.


11institutetext: Department of Biomedical Engineering and Health Systems , KTH
Royal Institute of Technology, Stockholm , Sweden 22institutetext: Department
of Clinical Science, Intervention and Technology, Karolinska Institutet,
Stockholm, Sweden 33institutetext: Department of Oncology Pathology,
Karolinska Institutet, Stockholm, Sweden

# PriorNet: lesion segmentation in PET-CT including prior tumor appearance
information

Simone Bendazzoli 1122 Mehdi Astaraki 1133

###### Abstract

Tumor segmentation in PET-CT images is challenging due to the dual nature of
the acquired information: low metabolic information in CT and low spatial
resolution in PET. U-Net architecture is the most common and widely recognized
approach when developing a fully automatic image segmentation method in the
medical field. We proposed a two-step approach, aiming to refine and improve
the segmentation performances of tumoral lesions in PET-CT. The first step
generates a prior tumor appearance map from the PET-CT volumes, regarded as
prior tumor information. The second step, consisting of a standard U-Net,
receives the prior tumor appearance map and PET-CT images to generate the
lesion mask. We evaluated the method on the 1014 cases available for the
AutoPET 2022 challenge, and the results showed an average Dice score of 0.701
on the positive cases.

###### Keywords:

Tumor segmentation deep-learning PET-CT.

##  1 Introduction

Tumor lesion segmentation is one of the primary tasks performed on PET-CT
scans in oncological practice. The main aim is to identify and delineate the
tumor region, enabling a quantitative assessment, performing feature
extraction and planning the treatment strategy accordingly.

U-Net model [5] training is the most common supervised deep learning approach
that yielded promosing results for different medical image segmentation tasks
[2] [6]. We propose a 2-step approach, where the first step aims at generating
and providing prior tumor appearance information to the second step, a
conventional U-Net employed for the tumor segmentation.

##  2 Method description

The proposed method consists of two main modules. First, inspired by our
recent Normal Appearance Autoencoder (NAA) model [1], the appearance of
healthy anatomies from PET-CT images is learned by training an inpainting
model. In specific, a Partial Convolution Neural Network (PCNN) [4] was
employed to capture the distributions of healthy anatomies. Prior information
regarding the appearance of the tumors was, then, estimated by calculating the
residuals between the reconstructed fake healthy images from the tumoral ones.
Second, the prior information highlighting the presence of candidate tumoral
regions was added as an additional channel into a supervised segmentation
network in order to guide the attention of the model to the candidate regions.

###  2.1 Estimating the tumor appearance

Figure 1: Tumor appearance estimation workflow. An RGB 2D image is initially
generated by stacking the CT and the PET slices [CT, PET, CT]. The RGB image
is then reconstructed with the inpainting model. Finally, the image difference
is estimated, and the first and second channels are taken as CT residual slice
and PET residual slice, respectively.

Estimating the prior information regarding the appearance of tumors can be
achieved by, first, modeling the healthy anatomies and then detecting the
tumors as anomalies. To model the distribution of complicated healthy
anatomies from whole-body PET-CT volumes, we employed a PCNN model as a robust
inpainting network. This inpainting model can replace the pathological regions
with the characteristics of nearby healthy tissues and generate plausible
pathology-free images with realistic-looking and anatomically meaningful
visual patterns. This can be achieved by the following two steps: 1) forcing
the model to learn the appearance of healthy anatomies, and 2) guiding the
model to inpaint only the tumoral regions.
To learn the attributes of healthy anatomies, healthy image slices from the
PET-CT dataset were employed as the training set of the inpainting model. In
specific, more than 30000 healthy image slices were used for the training set,
while the pathological slices were employed for the testing set. Considering
the large diversity in the shape, size, and location of the tumoral regions,
random irregular shapes were synthesized by combining regular geometrical
shapes, including circles, squares, and ellipses, to corrupt the healthy
images. The PCNN model is trained until it fills the random holes and replaces
the gaps with meaningful anatomical and imagery patterns. The objective
function of the PCNN model is constructed based on several loss terms,
including per-pixel loss, perceptual loss, style loss, and total variation
loss. This multi-objective optimization leverage the quality of the inpainted
images and reconstructs high-quality image while preserving the anatomical
details. The performance of the PCNN model was evaluated by quantifying the
following metrics: peak signal-to-noise ratio, mean square error, and
structural similarity index.
During the training step, the model learns to fill the random holes with the
attributes of the nearby healthy tissues. This process enforces the model to
model the distribution of the healthy anatomies. Therefore, in the test phase,
the learned model can be used to replace the tumoral regions with visual
characteristics of healthy tissues. However, such a tumoral removal step
essentially requires tumoral masks. While in the NAA model, a second
autoencoder model was employed to remove the tumors automatically, in this
study, we utilized the learned PCNN model to directly inpaint the tumoral
regions by gaining from the hyperintensity patterns of PET images. In
specific, tumoral regions in PET volumes often appear with higher FDG uptake
with respect to nearby healthy tissues. Therefore, by setting a simple
threshold on the PET volumes, the candidate regions for the inpainting step
can be estimated. The corresponding candidate regions over the PET-CT slices
are then set as a hole to be inpainted the learned PCNN model. As a result,
the tumoral regions are inpainted and replaced with the appearance of learned
healthy tissues.
It should be noted that other normal tissues with a high level of SUV can be
presented in the thresholded images, and their inpainting leads to a large
value of false positive rate. However, the PCNN model already learned the
appearance of healthy anatomies from the training phase. Therefore, in such
cases, it replaces the healthy tissues with the attributes of healthy tissues.
As a result, for every input image slice containing a candidate region(s), a
new inpainted image will be synthesized. If the candidate region represents
healthy anatomies, the inpainted model generates another healthy image, while
for the tumoral regions, a tumor-free image will be synthesized. For each
subject, a stack of volume is then reconstructed from the 2D inpainted slices.
Finally, the residual volumes were calculated by finding the voxel-wise
intensity differences between the original volume and the inpainted one. As
the residual volume contains highlight regions presenting the tumor
characteristics, it is named as the prior image.

Figure 2: Inpainting model residual outputs. Clockwise from top-left: 2D CT
slice, 2D PET slice, 2D residual PET and 2D residual CT.

###  2.2 Prior-aware segmentation model

The second step of our proposed workflow combines the PET-CT volumes with the
residual images obtained as describe in 2.1, in order to perform a
segmentation task and predict the tumoral lesions (Figure 3).

Figure 3: U-Net architecture adopted for the lesion segmentation.

###  2.3 Training procedures

The inpainting model was trained with the same protocols described in [1][4].
In brief, this model was trained for 150 epochs in the first phase and another
150 epochs in the second phase. The differences between the first and second
phases were the batch normalization flag which was deactivated in the encoder
part of the model for the second phase, and the learning rate, which was
decreased from 0.0001 to 0.00005 from the first to the second phase.
The images were corrupted by the random holes with variable sizes, which, on
average, corrupted 25 to 30 percent of the image size. Since the healthy
slices were used for training and the pathological slices were saved for the
testing phase, no cross-validation was needed. Finally, in the test phase, a
circular hole with a radius of 17 pixels was used to mask out the tumoral
region for tumor inpainting.

For the prior-aware segmentation step, we adopted the conventional nnUNet
framework [3], with a 4-channel input (PET, CT, residual-PET and residual-CT),
and a single channel output (tumor lesion mask). We trained the model for 1000
epochs, with 250 iterations per epoch and a batch size of 2. The chosen loss
function is the default combined Dice-Cross Entropy Loss, with a learning rate
of 0.0001 and the Adam optimizer.

##  3 Data preprocessing and augmentation

###  3.1 Tumor appearance estimation

In the first step of our proposed workflow, we aim at estimating the tumor
lesion appearance on the 2D image slices. In order to generate the 2D image
slices, the 3D PET and CT volumes are sliced along the axial orientation and
normalized within the [0-1] range:

$\newline CT_{norm}=\frac{CT_{x,y}-(-1000)}{800-(-1000)}\newline \newline
CT_{norm}=\begin{cases}0&\quad\text{if }CT_{norm}<0\\\ 1&\quad\text{if
}CT_{norm}>1\\\ CT_{norm}&\quad\text{otherwise }\end{cases}\newline \newline
PET_{norm}=\frac{PET_{x,y}}{12}\newline \newline
PET_{norm}=\begin{cases}0&\quad\text{if }PET_{norm}<0\\\ 1&\quad\text{if
}PET_{norm}>1\\\ PET_{norm}&\quad\text{otherwise }\end{cases}\newline $

The normalized 2D slices are then stacked together to form an RGB image (Fig.
1). In detail, the normalized CT slice is used for the red and the blue
channel, while the normalized PET slice is stacked as the green channel.

###  3.2 Prior-aware segmentation

In the second step of the proposed workflow, the voxel intensities for both CT
and SUV PET volumes are clipped within the percentile range [0.5-99.5],
followed by a z-score normalization:

$CT_{x,y,z}=\begin{cases}CT_{0.5p}&\quad\text{if }CT_{x,y,z}<CT_{0.5p}\\\
CT_{99.5p}&\quad\text{if }CT_{x,y,z}>CT_{99.5p}\\\
CT_{x,y,z}&\quad\text{otherwise }\end{cases}\newline
PET_{x,y,z}=\begin{cases}PET_{0.5p}&\quad\text{if }PET_{x,y,z}<PET_{0.5p}\\\
PET_{99.5p}&\quad\text{if }PET_{x,y,z}>PET_{99.5p}\\\
PET_{x,y,z}&\quad\text{otherwise }\end{cases}\newline \newline \newline
CT_{norm}=\frac{CT_{x,y,z}-\mu_{CT}}{\sigma_{CT}}\newline \newline
PET_{norm}=\frac{PET_{x,y,z}-\mu_{PET}}{\sigma_{PET}}$

As visually explained in Fig. 1 both PET and CT residual volumes have
intensity values in the range [-1,1]; for this reason, no normalization is
performed for these volumes.

The default nnUNet data augmentation setting, presented in [3], is adopted;
the default configuration consist of random rotations, random scaling, random
elastic deformations and gamma correction augmentation.

##  4 Results

In order to evaluate the performance of our proposed approach, we performed a
5-fold cross-validation on the 1014 PET-CT volumes of the AutoPET 2022
challenge.

The presented evaluation metrics are the same used in the challenge to
determine the participant’s rankings: Dice score, False Positive Volume and
False Negative Volume (expressed in liters).

To better understand the lesion segmentation performances in relation to the
tumor type (lung cancer, lymphoma, melanoma), in Table 1 we provide the class-
wise average scores, alongside the average negative (no lesions in the ground
truth mask) and positive classes.

Table 1: PriorNet validation: class-wise average metrics. Metric | Global | Positive | Negative | Lung Cancer | Lymphoma | Melanoma
---|---|---|---|---|---|---
Dice | - | 0.701 | - | 0.743 | 0.734 | 0.638
False Positive Volume [ L ] | 12.5 | 9.28 | 15.6 | 5.67 | 10.5 | 11.55
False Negative Volume [ L ] | - | 6.8 | - | 9.31 | 4.91 | 6.03

In Figure 4 the Dice score distribution for the different tumor classes is
represented.

Figure 4: PriorNet validation: class-wise Dice score

##  5 Public Code Repository

The original code of PriorNet can be found at the following GitHub public
repository: https://github.com/SimoneBendazzoli93/PriorNet

## References

  * [1] Astaraki, M., Smedby, Ö., Wang, C.: Prior-aware autoencoders for lung pathology segmentation. Medical Image Analysis p. 102491 (2022)
  * [2] Bukhari, S.T., Mohy-ud Din, H.: E1d3 u-net for brain tumor segmentation: Submission to the rsna-asnr-miccai brats 2021 challenge (2021). https://doi.org/10.48550/ARXIV.2110.02519, https://arxiv.org/abs/2110.02519
  * [3] Isensee, F., Jaeger, P.F., Kohl, S.A.A., Petersen, J., Maier-Hein, K.H.: nnU-net: a self-configuring method for deep learning-based biomedical image segmentation. Nature Methods 18(2), 203–211 (Dec 2020). https://doi.org/10.1038/s41592-020-01008-z, https://doi.org/10.1038/s41592-020-01008-z
  * [4] Liu, G., Reda, F.A., Shih, K.J., Wang, T.C., Tao, A., Catanzaro, B.: Image inpainting for irregular holes using partial convolutions. In: Proceedings of the European conference on computer vision (ECCV). pp. 85–100 (2018)
  * [5] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation (2015)
  * [6] Zhao, W., Jiang, D., Peña Queralta, J., Westerlund, T.: Mss u-net: 3d segmentation of kidneys and tumors from ct images with a multi-scale supervised u-net. Informatics in Medicine Unlocked 19, 100357 (2020). https://doi.org/https://doi.org/10.1016/j.imu.2020.100357, https://www.sciencedirect.com/science/article/pii/S2352914820301969


$(P^{\phi}_{+a},{\mathfrak{n}},\widehat{a}-a)$ is normal. Replacing $H$,
$(P,{\mathfrak{m}},\widehat{a})$ by $H^{\phi}$,
$(P^{\phi}_{+a},{\mathfrak{n}},{\widehat{a}-a})$, we arrange that
$(P,{\mathfrak{m}},\widehat{a})$ itself is normal. Then
$(P,{\mathfrak{m}},\widehat{a})$ has an ultimate refinement by Proposition
IV.4.14, and applying Corollary III.3.35 to this refinement and using Lemma
IV.4.10, we obtain an ultimate refinement
$(P_{+a},{\mathfrak{n}},\widehat{a}-a)$ of $(P,{\mathfrak{m}},\widehat{a})$
and an active $\phi$ in $H$ with $0<\phi\preccurlyeq 1$ such that the
$Z$-minimal slot $(P_{+a}^{\phi},{\mathfrak{n}},\widehat{a}-a)$ in $H^{\phi}$
is deep, normal, and ultimate. Again replacing $H$,
$(P,{\mathfrak{m}},\widehat{a})$ by $H^{\phi}$,
$(P^{\phi}_{+a},{\mathfrak{n}},{\widehat{a}-a})$, we arrange that
$(P,{\mathfrak{m}},\widehat{a})$ is deep, normal, and ultimate. Corollary
III.3.47 yields a deep and strictly normal refinement
$(P_{+a},{\mathfrak{m}},{\widehat{a}-a})$ of $(P,{\mathfrak{m}},\widehat{a})$;
this refinement is still ultimate by Lemma IV.4.10. Hence
$(P_{+a},{\mathfrak{m}},{\widehat{a}-a})$ is a refinement of
$(P,{\mathfrak{m}},\widehat{a})$ as required, with $\phi=1$. ∎

Combining Lemmas III.2.26 and IV.5.41 with Corollary IV.5.39 yields:

######  Corollary IV.5.42.

Assume $H$ is Liouville closed, $\upomega$-free, and of Hardy type, and
$\operatorname{I}(K)\subseteq K^{\dagger}$. Then every $Z$-minimal slot in $H$
of order $r=1$ has a refinement $(P,{\mathfrak{m}},\widehat{a})$ such that
$(P^{\phi},{\mathfrak{m}},\widehat{a})$ is eventually deep, ultimate, and
strongly repulsive-normal.

In the next subsection we show how minimal holes of degree $>1$ in $K$ give
rise to deep, ultimate, strongly repulsive-normal, $Z$-minimal slots in $H$.

#### Achieving strong repulsive-normality

Let $H$ be an $\upomega$-free Liouville closed $H$-field with small derivation
and constant field $C$, and $(P,{\mathfrak{m}},\widehat{a})$ a minimal hole of
order $r\geqslant 1$ in $K:=H[\imag]$. Other conventions are as in the
subsection Achieving repulsive-normality. Our goal is to prove a version of
Theorem IV.5.28 with “repulsive-normal” improved to “strongly repulsive-normal
+ ultimate”:

######  Theorem IV.5.43.

Suppose $C$ is archimedean, $\operatorname{I}(K)\subseteq K^{\dagger}$, and
$\deg P>1$. Then one of the following conditions is satisfied:

  1. $\mathrm{(i)}$

$\widehat{b}\notin H$ and some $Z$-minimal slot
$(Q,{\mathfrak{m}},\widehat{b})$ in $H$ has a special refinement
${(Q_{+b},{\mathfrak{n}},\widehat{b}-b)}$ such that
$(Q^{\phi}_{+b},{\mathfrak{n}},\widehat{b}-b)$ is eventually deep, strongly
repulsive-normal, and ultimate;

  2. $\mathrm{(ii)}$

$\widehat{c}\notin H$ and some $Z$-minimal slot
$(R,{\mathfrak{m}},\widehat{c})$ in $H$ has a special refinement
${(R_{+c},{\mathfrak{n}},\widehat{c}-c)}$ such that
$(R^{\phi}_{+c},{\mathfrak{n}},\widehat{c}-c)$ is eventually deep, strongly
repulsive-normal, and ultimate.

The proof of this theorem rests on the following two lemmas, where the
standing assumption that $H$ is Liouville closed can be dropped.

######  Lemma IV.5.44.

Suppose $\widehat{b}\notin H$ and $(Q,{\mathfrak{m}},\widehat{b})$ is a
$Z$-minimal slot in $H$ with a refinement
${(Q_{+b},{\mathfrak{n}},\widehat{b}-b)}$ such that
$(Q^{\phi}_{+b},{\mathfrak{n}},\widehat{b}-b)$ is eventually deep and
repulsive-normal. Then $(Q,{\mathfrak{m}},\widehat{b})$ has a refinement
${(Q_{+b},{\mathfrak{n}},\widehat{b}-b)}$ such that
$(Q^{\phi}_{+b},{\mathfrak{n}},\widehat{b}-b)$ is eventually deep and almost
strongly repulsive-normal.

###### Proof.

We adapt the proof of Lemma IV.3.34. Let
${(Q_{+b},{\mathfrak{n}},\widehat{b}-b)}$ be a refinement of
$(Q,{\mathfrak{m}},\widehat{b})$ and let $\phi_{0}$ be active in $H$ such that
$0<\phi_{0}\preccurlyeq 1$ and
$(Q^{\phi_{0}}_{+b},{\mathfrak{n}},\widehat{b}-b)$ is deep and repulsive-
normal. Then Corollary IV.5.37 yields a refinement

$\big{(}(Q^{\phi_{0}}_{+b})_{+b_{0}},{\mathfrak{n}}_{0},(\widehat{b}-b)-b_{0}\big{)}$

of $(Q^{\phi_{0}}_{+b},{\mathfrak{n}},\widehat{b}-b)$ which is deep and almost
strongly repulsive-normal. Hence

$\big{(}(Q_{+b})_{+b_{0}},{\mathfrak{n}}_{0},(\widehat{b}-b)-b_{0}\big{)}\ =\
\big{(}Q_{+(b+b_{0})},{\mathfrak{n}}_{0},\widehat{b}-(b+b_{0})\big{)}$

is a refinement of $(Q,{\mathfrak{m}},\widehat{b})$, and
$\big{(}Q^{\phi}_{+(b+b_{0})},{\mathfrak{n}}_{0},\widehat{b}-(b+b_{0})\big{)}$
is eventually deep and almost strongly repulsive-normal by Corollary IV.5.39.
∎

In the same way we obtain:

######  Lemma IV.5.45.

Suppose $\widehat{c}\notin H$ and $(R,{\mathfrak{m}},\widehat{c})$ is a
$Z$-minimal slot in $H$ with a refinement
${(R_{+c},{\mathfrak{n}},\widehat{c}-c)}$ such that
$(R^{\phi}_{+c},{\mathfrak{n}},\widehat{c}-c)$ is eventually deep and
repulsive-normal. Then $(R,{\mathfrak{m}},\widehat{c})$ has a refinement
${(R_{+c},{\mathfrak{n}},\widehat{c}-c)}$ such that
$(R^{\phi}_{+c},{\mathfrak{n}},\widehat{c}-c)$ is eventually deep and almost
strongly repulsive-normal.

Theorem IV.5.28 and the two lemmas above give Theorem IV.5.28 with “repulsive-
normal” improved to “almost strongly repulsive-normal”. We now upgrade this
further to “strongly repulsive-normal + ultimate” (under an extra assumption).

Recall from Lemma IV.1.3 that $v(\widehat{b}-H)\subseteq v(\widehat{c}-H)$ or
$v(\widehat{c}-H)\subseteq v(\widehat{b}-H)$. Thus the next two lemmas finish
the proof of Theorem IV.5.43.

######  Lemma IV.5.46.

Suppose $C$ is archimedean, $\operatorname{I}(K)\subseteq K^{\dagger}$, $\deg
P>1$, and

$v(\widehat{b}-H)\ \subseteq\ v({\widehat{c}-H}).$

Let $Q\in Z(H,\widehat{b})$ have minimal complexity. Then the $Z$-minimal slot
$(Q,{\mathfrak{m}},\widehat{b})$ in $H$ has a special refinement
$(Q_{+b},{\mathfrak{n}},{\widehat{b}-b})$ such that
$(Q^{\phi}_{+b},{\mathfrak{n}},{\widehat{b}-b})$ is eventually deep, strongly
repulsive-normal, and ultimate.

###### Proof.

Here are two ways of modifying $(Q,{\mathfrak{m}},\widehat{b})$. First, let
$(Q_{+b},{\mathfrak{n}},\widehat{b}-b)$ be a refinement of
$(Q,{\mathfrak{m}},\widehat{b})$. Lemma IV.1.3 gives $c\in H$ with
$v(\widehat{a}-a)=v(\widehat{b}-b)$ with $a:=b+c\imag$, and so the minimal
hole $(P_{+a},{\mathfrak{n}},\widehat{a}-a)$ in $K$ is a refinement of
$(P,{\mathfrak{m}},\widehat{a})$ that relates to
$(Q_{+b},{\mathfrak{n}},\widehat{b}-b)$ as $(P,{\mathfrak{m}},\widehat{a})$
relates to $(Q,{\mathfrak{m}},\widehat{b})$. So we can replace
$(P,{\mathfrak{m}},\widehat{a})$ and $(Q,{\mathfrak{m}},\widehat{b})$ by
$(P_{+a},{\mathfrak{n}},\widehat{a}-a)$ and
$(Q_{+b},{\mathfrak{n}},\widehat{b}-b)$, whenever convenient. Second, let
$\phi$ be active in $H$ with $0<\phi\preccurlyeq 1$. Then we can likewise
replace $H$, $K$, $(P,{\mathfrak{m}},\widehat{a})$,
$(Q,{\mathfrak{m}},\widehat{b})$ by $H^{\phi}$, $K^{\phi}$,
$(P^{\phi},{\mathfrak{m}},\widehat{a})$,
$(Q^{\phi},{\mathfrak{m}},\widehat{b})$.

In this way we first arrange as in the proof of Corollary IV.5.30 that
$(Q,{\mathfrak{m}},\widehat{b})$ is special. Next, we use Proposition III.3.36
likewise to arrange that $(Q,{\mathfrak{m}},\widehat{b})$ is also normal. By
Propositions IV.4.14 (where the assumption $\operatorname{I}(K)\subseteq
K^{\dagger}$ comes into play) and III.3.25 we arrange that
$(Q,{\mathfrak{m}},\widehat{b})$ is ultimate as well. The properties “special”
and “ultimate” persist under further refinements and compositional
conjugations.

Now Corollary IV.5.30 and Lemma IV.5.44 give a refinement
$(Q_{+b},{\mathfrak{n}},{\widehat{b}-b})$ of the slot
$(Q,{\mathfrak{m}},\widehat{b})$ in $H$ and an active $\phi_{0}$ in $H$ with
$0<\phi_{0}\preccurlyeq 1$ such that the slot
$(Q^{\phi_{0}}_{+b},{\mathfrak{n}},{\widehat{b}-b})$ in $H^{\phi_{0}}$ is deep
and almost strongly repulsive-normal. Corollary III.3.47 then yields a deep
and strictly normal refinement

$\big{(}(Q^{\phi_{0}}_{+b})_{+b_{0}},{\mathfrak{n}},(\widehat{b}-b)-b_{0}\big{)}$

of $\big{(}Q^{\phi_{0}}_{+b},{\mathfrak{n}},{\widehat{b}-b}\big{)}$. This
refinement is still almost strongly repulsive-normal by Lemma IV.5.35, and
therefore strongly repulsive-normal by Lemma IV.5.33. Corollary IV.5.39 then
gives that
$\big{(}Q_{+(b+b_{0})},{\mathfrak{n}},{\widehat{b}-(b+b_{0})}\big{)}$ is a
special refinement of our slot $(Q,{\mathfrak{m}},\widehat{b})$ such that
$\big{(}Q^{\phi}_{+(b+b_{0})},{\mathfrak{n}},{\widehat{b}-(b+b_{0})}\big{)}$
is eventually deep and strongly repulsive-normal. ∎

Likewise:

######  Lemma IV.5.47.

Suppose $C$ is archimedean, $\operatorname{I}(K)\subseteq K^{\dagger}$, $\deg
P>1$, and

$v(\widehat{c}-H)\ \subseteq\ v(\widehat{b}-H).$

Let $R\in Z(H,\widehat{c})$ have minimal complexity. Then the $Z$-minimal slot
$(R,{\mathfrak{m}},\widehat{c})$ in $H$ has a special refinement
$(R_{+c},{\mathfrak{n}},\widehat{c}-c)$ such that
$(R^{\phi}_{+c},{\mathfrak{n}},{\widehat{c}-c})$ is eventually deep, strongly
repulsive-normal, and ultimate.

###  IV.6. The Main Theorem

We prove here the Normalization Theorem from the Introduction, as a corollary
of Theorem IV.5.43. It is accordingly less detailed than the latter, but more
user-friendly. It is what will get used at a key stage in [6].

######  Corollary IV.6.1.

Let $H$ be an $\upomega$-free Liouville closed $H$-field with small
derivation, archimedean ordered constant field $C$, and $1$-linearly newtonian
algebraic closure $H[\imag]$. Suppose $H$ is not newtonian. Then for some
$Z$-minimal special hole $(Q,1,\widehat{b})$ in $H$ with
$\operatorname{order}Q\geqslant 1$ and some active $\phi>0$ in $H$ with
$\phi\preccurlyeq 1$, the hole $(Q^{\phi},1,\widehat{b})$ in $H^{\phi}$ is
deep, strongly repulsive-normal, and ultimate.

###### Proof.

By Proposition I.7.28, $K:=H[\imag]$ is $1$-linearly surjective and
$\operatorname{I}(K)\subseteq K^{\dagger}$. As $H$ is not newtonian, neither
is $K$, by (0.13), so Lemma III.2.1 and subsequent remarks give a minimal hole
$(P,{\mathfrak{m}},\widehat{a})$ in $K$ of order $r\geqslant 1$, where
${\mathfrak{m}}\in H^{\times}$. Then $\deg P>1$ by Corollary III.2.8. By Lemma
IV.2.15 we arrange that ${\widehat{a}\in\widehat{K}:=\widehat{H}[\imag]}$
where $\widehat{H}$ is an immediate $\upomega$-free newtonian $H$-field
extension of $H$. Now $\widehat{a}=\widehat{b}+\widehat{c}\imag$ with
$\widehat{b},\widehat{c}\in\widehat{H}$. By Theorem IV.5.43 there are two
cases:

  1. (1)

$\widehat{b}\notin H$ and some $Z$-minimal slot
$(Q,{\mathfrak{m}},\widehat{b})$ in $H$ has a special refinement
${(Q_{+b},{\mathfrak{n}},\widehat{b}-b)}$ such that
$(Q^{\phi}_{+b},{\mathfrak{n}},\widehat{b}-b)$ is eventually deep, strongly
repulsive-normal, and ultimate;

  2. (2)

$\widehat{c}\notin H$ and some $Z$-minimal slot
$(R,{\mathfrak{m}},\widehat{c})$ in $H$ has a special refinement
${(R_{+c},{\mathfrak{n}},\widehat{c}-c)}$ such that
$(R^{\phi}_{+c},{\mathfrak{n}},\widehat{c}-c)$ is eventually deep, strongly
repulsive-normal, and ultimate.

Assume $Q$, ${\mathfrak{m}}$, $\widehat{b}$ are as in Case (1). (Case (2) goes
the same way.) Lemma IV.3.7 gives $1\leqslant\operatorname{order}Q\leqslant
2r$. Multiplicatively conjugating by ${\mathfrak{n}}$ and renaming
$Q_{+b,\times{\mathfrak{n}}}$ and $\frac{\widehat{b}-b}{{\mathfrak{n}}}$ as
$Q$ and $\widehat{b}$ we arrange that $(Q,1,\widehat{b})$ is a $Z$-minimal
special slot such that $(Q^{\phi},1,\widehat{b})$ is eventually deep, strongly
repulsive-normal, and ultimate (using Lemma IV.4.11 to preserve ultimate).
With Lemma III.2.14, changing $\widehat{b}$ if necessary, we arrange that
$(Q,1,\widehat{b})$ is a hole in $H$, not just a slot in $H$. ∎

### Index

  * active Newtonianity
  * adjoint §II.3
  * almost strongly
    * repulsive-normal Definition IV.5.32
    * split-normal Definition IV.3.21
  * asymptotic couple Asymptotic fields
    * asymptotic integration Asymptotic fields
    * gap Asymptotic fields
    * grounded Asymptotic fields
    * of Hardy type §I.2
    * rational asymptotic integration Asymptotic fields
  * asymptotic field Asymptotic fields
    * asymptotic integration Asymptotic fields
    * $\operatorname{d}$-valued Asymptotic fields
    * grounded Asymptotic fields
    * $H$-asymptotic Asymptotic fields
    * $\uplambda$-free Asymptotic fields
    * newtonian Definition 0.1
    * $\upomega$-free Asymptotic fields
    * pre-$\operatorname{d}$-valued Asymptotic fields
    * $r$-linearly newtonian Newtonianity
    * $r$-newtonian Newtonianity
  * asymptotic integration Asymptotic fields
    * asymptotic field Asymptotic fields
    * rational Asymptotic fields
  * asymptotically surjective §I.4
  * closed
    * $H$-closed Closed $H$-fields
    * $H$-field Closed $H$-fields
    * linearly Linear differential operators
    * Liouville Ordered differential fields
    * $r$-linearly §I.7
    * Schwarz Ordered differential fields
    * trigonometrically §I.2
    * under powers §I.2
    * weakly differentially Linear differential operators
  * closure
    * trigonometric §I.2
    * trigonometric-Liouville §I.3
  * completion §I.6
  * complexity
    * differential polynomial Complexity and the separant
    * hole §III.2
    * slot Definition III.2.12
  * conjugate
    * compositional §III.2
    * multiplicative §III.2
  * decomposition
    * into homogeneous parts §III.1
    * spectral §II.2, §II.4
  * deep §III.3
  * degree
    * differential polynomial Differential polynomials
    * dominant Valued differential fields
    * Newton Newtonianity
  * derivation
    * logarithmic Differential rings and fields
    * small Valued differential fields
  * $\operatorname{d}$-henselian Valued differential fields
  * differential algebra §II.2
    * exponential extension §II.2
    * universal exponential extension §II.2
  * differential field Differential rings and fields
    * $\operatorname{d}$-algebraic extension Differential polynomials
    * differentially algebraic extension Differential polynomials
    * linearly closed Linear differential operators
    * linearly surjective Linear differential operators
    * ordered Ordered differential fields
    * ordered valued Ordered differential fields
    * $r$-linearly closed §I.7
    * $r$-linearly surjective §I.4
    * valued Valued differential fields
    * weakly differentially closed Linear differential operators
  * differential polynomial
    * complexity Complexity and the separant
    * degree Differential polynomials
    * dominant degree Valued differential fields
    * dominant multiplicity Valued differential fields
    * dominant weight Valued differential fields
    * homogeneous part Differential polynomials
    * $K$-external zero §I.7
    * linear part Linear differential operators
    * minimal §I.5
    * multiplicity Differential polynomials
    * Newton degree Newtonianity
    * Newton multiplicity Newtonianity
    * newton position §I.7, §I.7
    * Newton weight Newtonianity
    * order Differential polynomials
    * proper §I.7
    * quasilinear Newtonianity
    * Riccati transform §I.1
    * separant Complexity and the separant
    * strongly split-normal §IV.3
    * weight Differential polynomials
  * eigenvalue §II.4
    * linear differential operator §II.3
  * element
    * $\widehat{a}$-repulsive §IV.5
    * active Newtonianity
    * almost special §I.5
    * attractive §IV.5
    * exponential over §II.2
    * $\gamma$-repulsive Definition IV.5.1
    * repulsive §IV.5
    * special §I.5
    * spectral decomposition §II.2
  * eventually Newtonianity
  * exceptional values Valued differential fields
    * eventual §I.4
    * ultimate §II.5
  * exponential
    * element §II.2
    * extension §II.2
  * extension
    * $\operatorname{d}$-algebraic Differential polynomials
    * differentially algebraic Differential polynomials
    * exponential §II.2
    * gaussian §II.1, Valued differential fields
    * immediate Valued fields
    * spectral §II.5
    * strict §I.5
    * trigonometric §I.2
    * universal exponential §II.2
  * gap Asymptotic fields
  * gaussian extension §II.1, Valued differential fields
  * grounded Asymptotic fields
    * asymptotic couple Asymptotic fields
    * asymptotic field Asymptotic fields
  * group of logarithmic derivatives §I.2
    * complement §II.2
  * group ring §II.1
    * gaussian extension of a valuation §II.1
    * norms §II.1
    * trace §II.1
  * $H$-asymptotic field Asymptotic fields
    * flattening Flattening
    * strongly $r$-newtonian §I.7
  * $H$-field Ordered differential fields
    * closed Closed $H$-fields
    * $H$-closed Closed $H$-fields
    * Liouville closed Ordered differential fields
    * Schwarz closed Ordered differential fields
    * trigonometric closure §I.2
    * trigonometrically closed §I.2
  * Hardy field
    * asymptotic couple §I.2
  * Hardy type §I.2
  * hole §III.2
    * complexity §III.2
    * minimal §III.2
    * $Z$-minimal §III.2
  * irreducible §I.1
  * $K$-external zero §I.7
  * $\uplambda$-free Asymptotic fields
  * least common left multiple §I.1
  * linear differential operator
    * $\widehat{a}$-repulsive splitting §IV.5
    * adjoint §II.3
    * asymptotically surjective §I.4
    * dominant weight Valued differential fields
    * dominant weighted multiplicity Valued differential fields
    * eigenvalue §II.3
    * eventual exceptional values §I.4
    * exceptional values Valued differential fields
    * irreducible §I.1
    * kernel Linear differential operators
    * least common left multiple §I.1
    * monic Linear differential operators, Linear differential operators
    * multiplicity §II.3
    * neatly surjective §I.5
    * order Linear differential operators
    * real splitting Definition I.1.5
    * Riccati transform §I.1
    * $S$-repulsive splitting §IV.5
    * span §III.1
    * spectral decomposition §II.4
    * spectrum §II.3
    * splits Linear differential operators
    * splitting §I.1
    * steep §I.4
    * strong splitting §IV.2
    * terminal §II.5
    * twist §II.3, Linear differential operators
    * type §II.3
    * ultimate exceptional values §II.5
  * linear part
    * differential polynomial Linear differential operators
    * slot §III.2
  * linearly
    * closed Linear differential operators
    * surjective Linear differential operators
  * logarithmic derivative Differential rings and fields
  * minimal
    * annihilator §I.5
    * differential polynomial §I.5
    * hole §III.2
  * multiplicity
    * differential polynomial Differential polynomials
    * dominant Valued differential fields
    * dominant weighted Valued differential fields
    * linear differential operator §II.3
    * Newton Newtonianity
  * neatly surjective §I.5
  * newton
    * degree Newtonianity
    * multiplicity Newtonianity
    * position §I.7, §I.7
    * weight Newtonianity
  * newtonian Definition 0.1
  * normal
    * slot §III.3
    * strictly §III.3
  * $\upomega$-free Asymptotic fields
  * order
    * differential polynomial Differential polynomials
    * linear differential operator Linear differential operators
  * ordered differential field Ordered differential fields
    * $H$-field Ordered differential fields
  * ordered valued differential field Ordered differential fields
    * pre-$H$-field Ordered differential fields
  * pre-$H$-field Ordered differential fields
  * proper §I.7
    * differential polynomial §I.7
    * slot §III.5
  * quasilinear
    * differential polynomial Newtonianity
    * slot §III.2
  * $r$-$\operatorname{d}$-henselian §I.5, Valued differential fields
  * $r$-linearly closed §I.7
  * $r$-linearly newtonian Newtonianity
  * $r$-linearly surjective
    * differential field §I.4
    * valuation ring §I.5
  * $r$-newtonian Newtonianity
    * strongly §I.7
  * refinement §III.2
  * repulsive
    * element §IV.5
  * repulsive-normal Definition IV.5.21
    * almost strongly Definition IV.5.32
    * strongly Definition IV.5.32
  * Riccati transform
    * differential polynomial §I.1
    * linear differential operator §I.1
  * $S$-repulsive
    * element Definition IV.5.1
    * splitting §IV.5
  * separant Complexity and the separant
  * slot Definition III.2.12
    * almost strongly repulsive-normal Definition IV.5.32
    * almost strongly split-normal Definition IV.3.21
    * balanced §III.5
    * complexity Definition III.2.12
    * compositional conjugate §III.2
    * deep §III.3
    * equivalence §III.2
    * isolated Definition III.4.1
    * linear Definition III.2.12
    * linear part §III.2
    * multiplicative conjugate §III.2
    * normal §III.3
    * proper §III.5
    * quasilinear §III.2
    * refinement §III.2
    * repulsive-normal Definition IV.5.21
    * special §III.2
    * split-normal Definition IV.3.3
    * steep §III.3
    * strictly normal §III.3
    * strongly repulsive-normal Definition IV.5.32
    * strongly split-normal Definition IV.3.21
    * ultimate §IV.4, Definition IV.4.9
    * $Z$-minimal Definition III.2.12
  * slowly varying function §I.4
  * small derivation Valued differential fields
  * span §III.1
  * special §I.5
    * almost §I.5
    * slot §III.2
  * spectral decomposition
    * of a differential operator §II.4
    * of an element §II.2
  * spectral extension §II.5
  * spectrum §II.3
  * split-normal Definition IV.3.3
    * almost strongly Definition IV.3.21
    * strongly §IV.3, Definition IV.3.21
  * splitting §I.1
    * $\widehat{a}$-repulsive §IV.5
    * induced by a factorization §I.1
    * real Definition I.1.5
    * $S$-repulsive §IV.5
    * strong §IV.2
  * steep
    * linear differential operator §I.4
    * slot §III.3
  * strictly normal §III.3
  * strongly
    * $r$-newtonian §I.7
    * repulsive-normal Definition IV.5.32
    * split-normal §IV.3, Definition IV.3.21
  * surjective
    * asymptotically §I.4
    * linearly Linear differential operators
    * neatly §I.5
  * terminal §II.5
  * trace §II.1
  * transseries Ordered differential fields
    * oscillating §II.4
  * trigonometric
    * closed §I.2
    * closure §I.2
    * extension §I.2
    * Liouville closure §I.3
  * twist §II.3
  * type §II.3
  * ultimate
    * exceptional values §II.5
    * slot §IV.4, Definition IV.4.9
  * universal exponential extension §II.2, §II.2
    * spectral decomposition §II.2
  * valuation
    * coarsening Valued fields
    * flattening Flattening
    * gaussian extension §II.1, Valued differential fields
    * immediate extension Valued fields
    * specialization Valued fields
    * spectral extension §II.5
  * valued differential field Valued differential fields
    * asymptotic Asymptotic fields
    * completion §I.6
    * $\operatorname{d}$-henselian Valued differential fields
    * differential-henselian Valued differential fields
    * $r$-$\operatorname{d}$-henselian §I.5, Valued differential fields
    * small derivation Valued differential fields
  * values
    * eventual exceptional §I.4
    * exceptional Valued differential fields
    * ultimate exceptional §II.5
  * weight
    * differential polynomial Differential polynomials
    * dominant Valued differential fields, Valued differential fields
    * Newton Newtonianity
  * $Z$-minimal
    * hole §III.2
    * slot Definition III.2.12
  * zero
    * $K$-external §I.7

## List of Symbols

|  Asymptotic differential algebra |
---|---|---
$a^{\dagger}$ |  logarithmic derivative $a^{\dagger}=a^{\prime}/a$ of nonzero $a$ |  Differential rings and fields
$\operatorname{wr}(a,b)$ |  the Wronskian $ab^{\prime}-a^{\prime}b$ of $(a,b)$ |  I.2
$\omega$ |  the function $z\mapsto-(2z^{\prime}+z^{2})$ |  Asymptotic fields
$\sigma$ |  the function $y\mapsto\omega(-y^{\dagger})+y^{2}$ |  Asymptotic fields
$\mathcal{O}_{K}$, $\smallo_{K}$ |  valuation ring of a valued field $K$ and its maximal ideal |  Valued fields
$\Gamma_{K}$, $\operatorname{res}(K)$ |  value group and residue field of a valued field $K$ |  Valued fields
$\preccurlyeq$, $\prec$, $\asymp$, $\sim$ |  asymptotic relations on a valued field |  Valued fields
$v_{\Delta}$, $\dot{v}$ |  $\Delta$-coarsening of the valuation $v$ |  Valued fields
$\preccurlyeq_{\Delta}$, $\prec_{\Delta}$, $\asymp_{\Delta}$, $\sim_{\Delta}$, $\dot{\preccurlyeq}$, $\dot{\prec}$, $\dot{\asymp}$, $\dot{\sim}$ |  asymptotic relations on a valued field coarsened by $\Delta$ |  Valued fields
$\dot{K}$ |  specialization of a valued field $K$ |  Valued fields
$a_{\rho}\leadsto a$ |  the sequence $(a_{\rho})$ pseudoconverges to $a$ |  Valued fields
$[\gamma]$ |  archimedean class of $\gamma$ |  I.2
$[{\mathfrak{m}}]$ |  archimedean class of $v{\mathfrak{m}}$ |  III
$\preccurlyeq^{\flat}$, $\prec^{\flat}$, $\asymp^{\flat}$, $\sim^{\flat}$ |  flattened asymptotic relations on an $H$-asymptotic field |  Flattening
$\preccurlyeq^{\flat}_{\phi}$, $\prec^{\flat}_{\phi}$, $\asymp^{\flat}_{\phi}$, $\sim^{\flat}_{\phi}$ |  flattened asymptotic relations on the compositional conjugate by $\phi$ |  Flattening
$\operatorname{dv}(K)$ |  $\operatorname{d}$-valued hull of the pre-$\operatorname{d}$-valued field $K$ |  Asymptotic fields
$\operatorname{I}(K)$ |  special definable $\mathcal{O}$-submodule of the asymptotic field $K$ |  I.2
$\Upgamma(H)$, $\Uplambda(H)$, $\Updelta(H)$ |  special definable subsets of the pre-$H$-field $H$ |  Ordered differential fields
$H^{\operatorname{trig}}$, $H^{\operatorname{tl}}$ |  trigonometric closure of $H$, trigonometric-Liouville closure of $H$ |  I.2, I.3
|  Linear differential operators |
$R[\der]$ |  ring of linear differential operators over the differential ring $R$ |  Linear differential operators
$\ker A$ |  kernel of $A$ |  Linear differential operators
$A_{\ltimes u}$ |  twist of $A$ by $u$ |  Linear differential operators
$A^{*}$ |  adjoint of the linear differential operator $A$ |  II.3
$\operatorname{Ri}(A)$ |  Riccati transform of $A$ |  I.1
$\operatorname{mult}_{a}(A)$ |  multiplicity of $A$ at $a\in K$ |  II.3
$\operatorname{mult}_{\alpha}(A)$ |  multiplicity of $A$ at ${\alpha\in K/K^{\dagger}}$ |  II.3
$\Sigma(A)$ |  spectrum of $A$ |  II.3
$\operatorname{dwm}(A)$ |  dominant weighted multiplicity of $A$ |  Valued differential fields
$\operatorname{dwt}(A)$ |  dominant weight of $A$ |  Valued differential fields
$\operatorname{dwm}_{A}(\gamma)$ |  dominant weighted multiplicity of $Ay$ where $\gamma=vy$ |  Valued differential fields
$\operatorname{dwt}_{A}(\gamma)$ |  dominant weight of $Ay$ where $\gamma=vy$ |  Valued differential fields
$\operatorname{nwt}_{A}(\gamma)$ |  eventual value of $\operatorname{dwt}_{A^{\phi}}(\gamma)$ |  Valued differential fields
$v_{A}^{\operatorname{e}}(\gamma)$ |  eventual value of $v_{A^{\phi}}(\gamma)-\operatorname{nwt}_{A}(\gamma)v\phi$ |  I.4.5
${\mathscr{E}}(A)$ |  set of exceptional values of $A$ |  Valued differential fields
${\mathscr{E}}^{\operatorname{e}}(A)$ |  set of eventual exceptional values of $A$ |  I.4
${\mathscr{E}}^{\operatorname{u}}(A)$ |  set of ultimate exceptional values of $A$ |  II.5
${\mathfrak{v}}(A)$ |  span of $A$ |  III.1
|  Differential polynomials |
$R\\{Y\\}$ |  ring of differential polynomials over the differential ring $R$ |  Differential polynomials
$P_{+a}$ |  additive conjugate of $P$ by $a$ |  Differential polynomials
$P_{\times a}$ |  multiplicative conjugate of $P$ by $a$ |  Differential polynomials
$P^{\phi}$ |  compositional conjugate of $P$ by $\phi$ |  Differential polynomials
$\operatorname{order}P$ |  order of $P$ |  Differential polynomials
$\deg P$ |  (total) degree of $P$ |  Differential polynomials
$\operatorname{mul}P$ |  multiplicity of $P$ (at $0$) |  Differential polynomials
$\operatorname{wt}P$ |  weight of $P$ |  Differential polynomials
$\operatorname{ddeg}P$ |  dominant degree of $P$ |  Valued differential fields
$\operatorname{dmul}P$ |  dominant multiplicity of $P$ (at $0$) |  Valued differential fields
$\operatorname{dwt}P$ |  dominant weight of $P$ |  Valued differential fields
$\operatorname{ndeg}P$ |  Newton degree of $P$ |  Newtonianity
$\operatorname{nmul}P$ |  Newton multiplicity of $P$ (at $0$) |  Newtonianity
$\operatorname{nwt}P$ |  Newton weight of $P$ |  Newtonianity
$v^{\operatorname{e}}(P)$ |  eventual value of $v(P^{\phi})-\operatorname{nwt}(P)v\phi$ |  I.7
$P_{d}$ |  homogeneous part of degree $d$ of $P$ |  Differential polynomials
$S_{P}$ |  separant of the differential polynomial $P$ |  Complexity and the separant
$\operatorname{c}(P)$ |  complexity of $P$ |  Complexity and the separant
$L_{P}$ |  linear part of $P$ |  Linear differential operators
$\operatorname{Ri}(P)$ |  Riccati transform of $P$ |  I.1.13
$Z(K,\widehat{a})$ |  set of all $P\in K\\{Y\\}^{\neq}$ that vanish at $(K,\widehat{a})$ |  III.2.1
|  Universal exponential extension |
$K^{\dagger}$ |  group of logarithmic derivatives of $K$ |  I.2
$\Lambda$ |  complement of $K^{\dagger}$ |  II.2
${\operatorname{U}}_{K}$ |  universal exponential extension of $K$ |  II.2
$v_{\operatorname{g}}$ |  gaussian extension of the valuation of $K$ to $K[G]$ |  II.1
$\preccurlyeq_{\operatorname{g}}$, $\prec_{\operatorname{g}}$, $\asymp_{\operatorname{g}}$ |  dominance relations associated to $v_{\operatorname{g}}$ |  II.1
|  Slots |
$(P,{\mathfrak{m}},\widehat{a})$ |  slot in $K$ |  III.2, III.2.12
$\Delta({\mathfrak{m}})$ |  convex subgroup of all $\gamma\in\Gamma$ with $[\gamma]<[{\mathfrak{m}}]$ |  III

## Errata and Comments to [ADH]

The changes below apply to the edition published by Princeton University
Press, and are already reflected in the versions posted on the arXiv and on
our personal web pages (as of February 2024). We thank Allen Gehret for
pointing out most of the errors left in that edition. Linguistic slips like
missing commas or articles are not listed below unless they might mislead.
Citations are to the bibliography of [ADH].

Acknowledgments:

The date of September 2015 on p. xiv indicates when the manuscript was first
submitted to Princeton University Press. The published version incorporates
some changes and additions made since then.

Dramatis Personae:

In the item for “$\upomega$-free” under the heading “Asymptotic Fields”,
$f-\omega(g^{\dagger\dagger})\succcurlyeq g^{\dagger}$ should be
$f-\omega(g^{\dagger\dagger})\succcurlyeq(g^{\dagger})^{2}$.

Chapter 1:

  1. (1)

The first sentence of the subsection Irreducibility in 1.1 should be:
Let $X$ and $Y$ be topological spaces.

  2. (2)

In the subsection Localization of modules in 1.4 the formula for addition
should have $s_{2}x_{1}+s_{1}x_{2}$ in the numerator.

  3. (3)

In the subsection Tensor products in 1.7, the $H$ in the 4th line should be a
$B$.

  4. (4)

In the subsection Rational rank in 1.7, in the line following the display:
${\mathbb{Q}}\otimes_{{\mathbb{Z}}}N$ should be
${\mathbb{Q}}\otimes_{{\mathbb{Z}}}M$.

  5. (5)

In the 4th line of the proof of 1.8.12, the second “$:=$” should be “$=$”.

  6. (6)

In the 6th line of the proof of 1.8.13, “$(a,b)\to$” should be
“$(a,b)\mapsto$”.

  7. (7)

In 1.9.6 one should add the assumption that $L$ is separably generated over
$K$, that is, $L$ is separably algebraic over an intermediate field $K(B)$
with $B\subseteq L$ algebraically independent over $K$. This assumption is
satisfied if $\operatorname{char}K=0$. Corollary 1.9.7 is still correct as
stated, but its proof requires for positive characteristic a variant of 1.9.6,
namely: $L$ is separably algebraic over $K$ iff every derivation on $L$
extending the trivial derivation on $K$ is trivial. (This variant with a
proof, as in [249, pp. 370–371] is now included in the arXiv version.) Lemma
1.9.8 should be restricted to the case $\operatorname{char}K=0$.

Chapter 2:

  1. (1)

In the fourth paragraph of 2.3, replace “valued subgroup of $(G,S,v)$” by
“valued subgroup of $(G^{\prime},S^{\prime},v^{\prime})$”.

Chapter 3:

  1. (1)

In the second sentence of the proof of 3.1.21, one can omit “with
$\mathfrak{q}\cap A=\mathfrak{q}^{\prime}\cap A=\mathfrak{m}$” since this
condition is automatically satisfied.

  2. (2)

F.-V. Kuhlmann pointed out that in the “Notes and comments” to 3.2 we
misattribute 3.2.26 to Krull [229]. An early source for a result of this kind
is Theorem 11 in O. Schilling’s book,

> The Theory of Valuations, Mathematical Surveys, no. 4, American Mathematical
> Society, New York, 1950.

This book refers for this theorem to I. Kaplansky’s unpublished Ph.D. thesis

> Maximal Fields with Valuations, Harvard University, 1941.

  3. (3)

Replace “theorem” by “proposition” in the sentence following the statement of
3.4.22.

  4. (4)

Marcus Tressl alerted us to an error in the proof of 3.6.11: replace the
condition $\bf{K}\preccurlyeq\bf{F}$ in the first sentence of the proof by
$\bf{K}\subseteq\bf{F}$, so that Zorn’s lemma can be applied as indicated in
the next sentence.

  5. (5)

Right after 3.7.6, replace

“open ball of the form $\\{y:v(y-f)>vf\\}$ where $f\in K^{\times}$” by

“open ball of the form $\\{y:v(y-f)>vg\\}$ where $f,g\in K^{\times}$,
$f\succcurlyeq g$”.

Chapter 4:

  1. (1)

In the first sentence of the proof of 4.1.10, omit be.

  2. (2)

The last three sentences of the proof of 4.6.12 can be shortened to: Then by
Lemma 1.3.10, $a$ is algebraic over $K$, so $a$ is algebraic over $C$ by Lemma
4.1.2.

Chapter 5:

  1. (1)

In line 5 of 5.5, replace $K[\der]$ by $R[\der]$.

  2. (2)

In 5.7.3, replace

“${\mathbb{Q}}[\phi,\dots,{}^{n}(\phi)]={\mathbb{Q}}[\phi,\dots,{}^{n}(\phi)]$”
by

“${\mathbb{Q}}[\phi,\dots,{}^{n}(\phi),\phi^{-1}]={\mathbb{Q}}[\phi,\dots,{}^{n}(\phi),\phi^{-1}]$”.

Chapter 6:

  1. (1)

In the second to last line of the proof of 6.1.9, replace $C$ by $D_{0}$.

  2. (2)

In the second line before the first display in the proof of 6.3.2 there is a
misplaced parenthesis in $K[Y,\dots,Y^{(r-1)}]$.

  3. (3)

In the last line of the proof of 6.6.5, replace (ii) by (iii).

Chapter 7:

  1. (1)

In the third line of the proof of 7.5.6, replace $E$ by $E^{\times}$.

Chapter 8:

  1. (1)

In the proof of 8.3.2, $(E,\Gamma,{{\boldsymbol{k}}}_{E})$ should be
$(E,{{\boldsymbol{k}}}_{E},\Gamma)$.

  2. (2)

A few lines before 8.3.3, $\theta_{\operatorname{v}}(v_{1},\dots,v_{k},y)$
should be $\theta_{\operatorname{v}}(v_{1},\dots,v_{k},z)$.

  3. (3)

In the proof of 8.4.12, third line from the bottom,
“$\Gamma_{K_{3}}=\Gamma_{K_{3}}$” should be “$\Gamma_{K_{2}}=\Gamma_{K_{3}}$”.

Chapter 9:

  1. (1)

Two lines before 9.1.10, (3) should be (2).

  2. (2)

Replace “Lemma” in the last line of the proof of 9.2.17 by “Corollary”.

  3. (3)

The correction following 3.7.6 leads to a corresponding correction in
describing the condition $z\in G_{i}$ when $s_{i}\neq 0$, in the proof of
9.7.3.

  4. (4)

Verifying (AC3) in proof of 9.8.2 can be shortened using

$\max\big{\\{}\psi^{\alpha}(\gamma+k\alpha):\ \gamma\in\Gamma,\
k\in{\mathbb{Z}},\ \gamma+k\alpha\neq 0\big{\\}}\ =\ \beta-\alpha.$

  5. (5)

In proof of 9.9.3, insert right after “$v$-slow on the right” the phrase “,
where $v$ is the standard valuation of $\Gamma$”.

Chapter 10:

  1. (1)

In 10.5.12, add “If $K$ is an $H$-field, then so is $K(y)$ with that ordering,
and $C_{K(y)}=C$” and in its proof refer to the remarks after 10.2.3.

  2. (2)

In the last sentence of the third paragraph in the “Notes and comments” to
10.6, “not not” should be “not”.

Chapter 11:

  1. (1)

In the last display before 11.1.4, the expression $\\{\gamma:\
\gamma<(\Gamma^{>})^{\prime}\\}$ should be replaced by $\\{\gamma\in\Gamma:\
\gamma<(\Gamma^{>})^{\prime}\\}$.

  2. (2)

In 11.2.3(ii), complete to “$\operatorname{nmul}P=\operatorname{nmul}P_{+a}$”
at the end.

  3. (3)

In proof of 11.6.3, replace $v(s-a^{\dagger})\in(\Gamma_{F}^{>})^{\prime}$ by
$v(s-a^{\dagger})\in(\Gamma_{F}^{>})^{\prime}\cup\\{\infty\\}$.

  4. (4)

In last sentence of proof of 11.6.14, replace $\sim sf$ by $\sim-sf$.

  5. (5)

In proof of 11.6.17, end of the fourth paragraph, replace $\lambda$ by
$\uplambda$.

  6. (6)

In second part of 11.8.5, omit the assumption that $K$ has asymptotic
integration and replace $=$ at end of proof by $\subseteq$.

  7. (7)

Omit the proof of 11.8.13; it has an erroneous forward reference.

Chapter 12:

  1. (1)

In the statement of 12.6.3, the last part should be
$[g]^{\prime}=[g^{\prime}]$.

Chapter 13:

  1. (1)

In the Notes and comments to 13.3, replace “$n_{0}=2\operatorname{dwm}(P)$” by
“$n_{0}=2\operatorname{dwm}(P)+m+1$ where $m$ is such that
$P{\uparrow^{m}}\in{\mathbb{T}}_{\exp}\\{Y\\}$”. (We thank Julian Ziegler-
Hunts for pointing this out.)

Chapter 14:

  1. (1)

In the line following the statement of 14.0.1, it would be better to refer to
11.7.13 than to 11.7.10.

  2. (2)

In the third line of the proof of 14.1.8, replace $K$ by $K^{\times}$.

  3. (3)

In the last line of the last display preceding 14.2.18, replace
$Y^{\prime\prime}$ by $Y^{\prime\prime}Y$.

  4. (4)

In 14.3.2(iii), replace  at newton position by in newton position.

  5. (5)

In line 6 of the proof of 14.3.2, after
“$\operatorname{nmul}P_{+b}=\operatorname{nmul}P_{+a}=1$” add “by Lemma
11.2.3” (referring to the addition to 11.2.3(ii) made above).

  6. (6)

After 14.5.11, replace  In Section 16.1 by In Section 16.2.

Appendix A:

  1. (1)

In the sixth line before the subsection “Representing ${\mathbb{T}}$ …” on p.
719, replace $[v(\ell_{n-1}]$ by $[v(\ell_{n-1})]$.

Appendix B:

  1. (1)

In the example after B.5.15, replace “Then $V\setminus W$ is infinite …” by
“If ${V\neq W}$, then $V\setminus W$ is infinite …”

  2. (2)

In B.6.1(4) add the axiom $\forall x\forall y(x\leqslant y\vee y\leqslant x)$
to $\operatorname{Or}$.

  3. (3)

In the remark following the definition of “proper filter on $\Lambda$” in B.7
omit “either”.

  4. (4)

In the displayed equivalences in the proof of B.7.7 replace $\mathcal{F}$ by
$\mathcal{U}$.

  5. (5)

In B.10.4(ii) replace “some $\kappa>|N|$” by “some infinite $\kappa>|N|$”, and
in the proof of that corollary replace “$|N|$-saturated” by
“$\kappa$-saturated where $\kappa>|N|$ is infinite”.

  6. (6)

In B.10.15 replace “embeddings $f_{i}\colon A\to M_{i}$ ($i=1,2$) of abelian
groups” by “embeddings $f_{i}\colon A\to M_{i}$ ($i=1,2$) of torsion-free
abelian groups”.

  7. (7)

Replace the last sentence in the remark before B.12.14 by “Then
$\operatorname{RCF}^{\prime}$ is model complete, but does not have QE:
$\\{a\in{\mathbb{R}}:\ a\geqslant 0\\}$ is neither finite nor cofinite and
hence is not definable in the field ${\mathbb{R}}$ by a quantifier-free
$\mathcal{L}_{\operatorname{R}}$-formula.”

  8. (8)

In B.12.15, replace “singletons” by “singletons $\\{a\\}$ where $a\in K$”.

  9. (9)

Add to the “Notes and comments” of B.12 that B.12.9 and B.12.11, with
different proofs, are from:

> A. H. Lightstone, A. Robinson, On the representation of Herbrand functions
> in algebraically closed fields, J. Symb. Logic 22 (1957), 187–204.

### References

  * [1] M. Aschenbrenner, L. van den Dries, Liouville closed $H$-fields, J. Pure Appl. Algebra 197 (2005), 83–139.
  * [2] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Differentially algebraic gaps, Selecta Math. 11 (2005), 247–280.
  * [3] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Maximal immediate extensions of valued differential fields, Proc. London Math. Soc. 117 (2018), no. 2, 376–406.
  * [4] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, _On numbers, germs, and transseries_ , in: B. Sirakov et al. (eds.): Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018, vol. 2, pp. 19–42, World Scientific Publishing Co., Singapore, 2018.
  * [5] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Maximal Hardy Fields, manuscript, arXiv:2304.10846, 2023.
  * [6] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, The theory of maximal Hardy fields, in preparation.
  * [7] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Linear differential equations over Hardy fields, in preparation.
  * [8] M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Linear differential equations over Hardy fields, II, in preparation.
  * [9] M. Boshernitzan, An extension of Hardy’s class L of “orders of infinity”, J. Analyse Math. 39 (1981), 235–255.
  * [10] N. Bourbaki, Fonctions d’une Variable Réelle, Chapitre V, Étude Locale des Fonctions, Hermann, Paris, 1951.
  * [11] J. Écalle, Les Fonctions Résurgentes, Tome I. Les Algèbres de Fonctions Résurgentes, Publications Mathématiques d’Orsay 81, vol. 5, Université de Paris-Sud, Département de Mathématiques, Orsay, 1981.
  * [12] J. Écalle, Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac, Actualités Mathématiques, Hermann, Paris, 1992.
  * [13] A. Gehret, A tale of two Liouville closures, Pacific J. Math. 290 (2017), no. 1, 41–76.
  * [14] G. H. Hardy, Divergent Series, 3rd ed., Oxford, at the Clarendon Press, 1963.
  * [15] J. van der Hoeven, Transseries and Real Differential Algebra, Lecture Notes in Math., vol. 1888, Springer-Verlag, New York, 2006.
  * [16] J. van der Hoeven, Transserial Hardy fields, Astérisque 323 (2009), 453–487.
  * [17] S. Lang, Algebra, 3rd ed., Addison-Wesley Publishing Company, Reading, MA, 1993.
  * [18] C. Miller, Basics of o-minimality and Hardy fields, in: C. Miller et al. (eds.), Lecture Notes on O-minimal Structures and Real Analytic Geometry, pp. 43–69, Fields Institute Communications, vol. 62, Springer, New York, 2012.
  * [19] G. Pólya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 24 (1922), no. 4, 312–324.
  * [20] M. van der Put, M. Singer, Galois Theory of Linear Differential Equations, Grundlehren Math. Wiss., vol. 328, Springer-Verlag, Berlin, 2003.
  * [21] N. Pynn-Coates, Newtonian valued differential fields with arbitrary value group, Comm. Algebra 47 (2019), no. 7, 2766–2776.
  * [22] N. Pynn-Coates, Differential-henselianity and maximality of asymptotic valued differential fields, Pacific J. Math. 308 (2020), no. 1, 161–205.
  * [23] M. Rosenlicht, Differential extension fields of exponential type, Pacific J. Math. 57 (1975), 289–300.
  * [24] M. Rosenlicht, On the value group of a differential valuation, II, Amer. J. Math. 103 (1981), 977–996.
  * [25] M. Rosenlicht, Hardy fields, J. Math. Analysis and Appl. 93 (1983), 297–311.
  * [26] G.-C. Rota, Combinatorics, representation theory and invariant theory: the story of a ménage à trois, Discrete Math. 193 (1998), no. 1–3, 5–16.