d\tau_{1}\int^{\tau_{1}}d\tau_{2}\bar{\tilde{z}}_{1}\varphi_{1}\frac{e^{-i\frac{\tau_{1}}{2}}}{(\tau_{1}-\tau_{2})^{1-2\epsilon}}\xi_{1}\psi(x_{1})$
$\displaystyle=(\alpha_{1}-\alpha_{2})\frac{g^{2}N_{1}}{8\pi\epsilon}\int
d\tau\ i\bar{\tilde{z}}e^{-i\frac{\tau}{2}}\xi\psi\varphi.$ (C.15)

The evaluation of $\Gamma^{\rm\ref{subfig:fermion1b}}_{\text{fermion}}$
proceeds exactly in the same way, the only change being the replacement of
$A_{\mu}$ with $\hat{A}_{\mu}$.

###  C.3 Scalar vertex corrections

Here we compute in details the scalar vertex corrections of figure 9.

Starting from diagram 9 we have

$\displaystyle\Gamma^{\rm\ref{subfig:scalarvertexa}}_{\text{scalar}}$
$\displaystyle=g^{4}M_{I}^{\ K}M_{L}^{\ J}\int d\tau_{1}\int
d\tau_{2}\big{(}\bar{\varphi}C_{K}\bar{C}^{I}\varphi\big{)}(x_{1})\big{(}\bar{\varphi}C_{J}\bar{C}^{L}\varphi\big{)}(x_{2})$
$\displaystyle=-g^{4}N_{1}M_{I}^{\ K}M_{K}^{\
J}\frac{\Gamma(\frac{1}{2}-\epsilon)}{4\pi^{\frac{3}{2}-\epsilon}}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bar{\varphi}_{2}C_{J}(x_{2})\bar{C}^{I}(x_{1})\varphi_{1}(\tau_{12})^{-1+2\epsilon}\,,$
(C.16)

which in the $\tau_{2}\to\tau_{1}$ limit gives

$\Gamma^{\rm\ref{subfig:scalarvertexa}}_{\text{scalar}}=-g^{4}\frac{N_{1}}{8\pi\epsilon}M_{I}^{\
K}M_{L}^{\ J}\int_{-L}^{L}d\tau\,\bar{\varphi}C_{J}\bar{C}^{I}\varphi\,.$
(C.17)

Diagram 9 contributes with

$\displaystyle\Gamma^{\rm\ref{subfig:scalarvertexb}}_{\text{scalar}}$
$\displaystyle=\int d\tau_{1}\int\tau_{2}\int
d^{d}x\big{(}i\bar{\varphi}A_{\mu}\dot{x}^{\mu}\varphi\big{)}(x_{1})\big{(}i\bar{\varphi}A_{\nu}\dot{x}^{\nu}\varphi\big{)}(x_{2})\big{(}A_{\rho}C_{I}\bar{C}^{I}A^{\rho}\big{)}(x)$
$\displaystyle=g^{2}N_{1}\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\,\bigg{[}\epsilon_{\rho\mu\sigma}\epsilon^{\rho\nu\lambda}\dot{x}_{1}^{\mu}\dot{x}_{2\nu}\frac{(x_{1}-x)^{\sigma}(x_{2}-x)_{\lambda}}{|x_{1}-x|^{3-2\epsilon}|x_{2}-x|^{3-2\epsilon}}$
$\displaystyle\times\varphi_{1}\bar{\varphi}_{2}C_{I}(x)C^{I}(x)\bigg{]}\,.$
(C.18)

The $d^{d}x$ integral can be solved by using (C.8). In the
$\tau_{2}\to\tau_{1}$ limit, we obtain

$\displaystyle\Gamma^{\rm\ref{subfig:scalarvertexb}}_{\text{scalar}}$
$\displaystyle=g^{2}N_{1}\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\frac{2\pi^{\frac{3}{2}-\epsilon}(2-2\epsilon)}{(1-2\epsilon)^{2}\Gamma(\frac{1}{2}-\epsilon)}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}(\tau_{12})^{-1+2\epsilon}\varphi_{1}\bar{\varphi}_{1}C_{I}(x_{1})\bar{C}^{I}(x_{1})$
$\displaystyle=g^{2}\frac{N_{1}}{8\pi\epsilon}\int
d\tau\,\bar{\varphi}C_{I}\bar{C}^{I}\varphi\,.$ (C.19)

For diagram in figure 9, we first consider the contribution
$\Gamma^{\rm\ref{subfig:scalarvertexc},1}_{\text{scalar}}$ in (3.35), that is
the one obtained by using the ABJ(M) Yukawa vertex
$2g^{2}C_{I}\bar{C}^{J}\bar{\psi}^{I}\psi_{J}$. To begin with, we write

$\displaystyle\Gamma^{\rm\ref{subfig:scalarvertexc},1}_{\text{scalar}}$
$\displaystyle=-2ig^{2}\int d\tau_{1}\int d\tau_{2}\int
d^{d}x\big{(}\bar{\varphi}\bar{f}\bar{\tilde{z}}\big{)}(x_{1})\big{(}\tilde{z}f\varphi\big{)}(x_{2})(C_{I}\bar{C}^{J}\bar{\psi}^{I}\psi_{J})$
$\displaystyle=4ig^{4}N_{2}\bar{\alpha}^{2}\alpha_{2}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\,\bigg{[}\bar{\varphi}_{1}C_{1}(x)\bar{C}^{1}(x)\varphi_{2}e^{i\frac{\tau_{12}}{2}}e^{i\tau_{2}}u^{\alpha}(\tau_{1})v_{\beta}(\tau_{2})$
$\displaystyle\times\langle\psi^{\beta}(x_{2})\bar{\psi}_{\delta}(x)\rangle\langle\psi^{\delta}(x)\bar{\psi}_{\alpha}(x_{1})\rangle\bigg{]}\,.$
(C.20)

We then proceed as we have done in section C.2 for the fermion vertex
corrections, so obtaining

$\displaystyle\Gamma^{\rm\ref{subfig:scalarvertexc},1}_{\text{scalar}}$
$\displaystyle=-4g^{4}N_{2}\bar{\alpha}^{2}\alpha_{2}\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\frac{2\pi^{\frac{3}{2}-\epsilon}}{(1-2\epsilon)^{2}\Gamma(\frac{1}{2}-\epsilon)}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\bigg{[}\bar{\varphi}_{1}C_{1}(x_{1})\bar{C}^{1}(x_{1})\varphi_{2}$
$\displaystyle\times\cos\frac{\tau_{12}}{2}\bigg{(}(3-2\epsilon)|x_{1}-x_{2}|^{-1+2\epsilon}+(x_{1}-x_{2})^{2}(-1+2\epsilon)|x_{1}-x_{2}|^{-3+2\epsilon}\bigg{)}\bigg{]}\,.$
(C.21)

By computing the $\tau_{2}$-integral in the $\tau_{2}\to\tau_{1}$ limit we
eventually find

$\Gamma^{\rm\ref{subfig:scalarvertexc},1}_{\text{scalar}}=-g^{4}\frac{N_{2}}{2\pi\epsilon}\bar{\alpha}^{2}\alpha_{2}\int
d\tau\,\bar{\varphi}C_{1}\bar{C}^{1}\varphi\,.$ (C.22)

The contribution $\Gamma^{\rm\ref{subfig:scalarvertexc},2}_{\text{scalar}}$ in
(3.35), coming from the ABJ(M) vertex
$-g^{2}C_{I}\bar{C}^{I}\bar{\psi}^{J}\psi_{J}$, corresponds to performing
contractions in the following string

$\Gamma^{\rm\ref{subfig:scalarvertexc},2}_{\text{scalar}}=ig^{2}\int
d\tau_{1}\int d\tau_{2}\int
d^{d}x\,\langle\big{(}\bar{\varphi}\bar{f}\bar{\tilde{z}}\big{)}(x_{1})\big{(}\tilde{z}f\varphi\big{)}(x_{2})(C_{I}\bar{C}^{I}\bar{\psi}^{J}\psi_{J})\rangle\,.$
(C.23)

The computation is analogous to the previous one, so we do not replicate it.
The final result reads

$\Gamma^{\rm\ref{subfig:scalarvertexc},2}_{\text{scalar}}=g^{4}\frac{N_{2}}{4\pi\epsilon}\bar{\alpha}^{2}\alpha_{2}\int
d\tau\,\bar{\varphi}C_{I}\bar{C}^{I}\varphi\,.$ (C.24)

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# MD-split+: Practical Local Conformal Inference in High Dimensions

Benjamin LeRoy  David Zhao

###### Abstract

Quantifying uncertainty in model predictions is a common goal for
practitioners seeking more than just point predictions. One tool for
uncertainty quantification that requires minimal assumptions is conformal
inference, which can help create probabilistically valid prediction regions
for black box models. Classical conformal prediction only provides marginal
validity, whereas in many situations locally valid prediction regions are
desirable. Deciding how best to partition the feature space $\mathcal{X}$ when
applying localized conformal prediction is still an open question. We present
MD-split+, a practical local conformal approach that creates $\mathcal{X}$
partitions based on localized model performance of conditional density
estimation models. Our method handles complex real-world data settings where
such models may be misspecified, and scales to high-dimensional inputs. We
discuss how our local partitions philosophically align with expected behavior
from an unattainable conditional conformal inference approach. We also
empirically compare our method against other local conformal approaches.

Conformal Inference, Local Conformal Inference, Model Diagnostics, MD-split+,
distribution-free inference, ICML

##  1 Introduction

Quantifying uncertainty in model predictions is a fundamental goal of
predictive inference. In this setting, the practitioner is interested in using
covariates or features ${\mathbf{X}}\in\mathcal{X}\subseteq\mathbb{R}^{d}$ to
learn about a response variable $Y\in\mathbb{R}$. Without assumptions on the
distribution $F$ of the underlying data, it is challenging to create valid
prediction regions for $Y|{\mathbf{X}}={\mathbf{x}}$. One tool for uncertainty
quantification that requires minimal distributional assumptions is conformal
inference, which can be applied to a range of black box models. The classical
version of conformal inference [Vovk et al., 2005] creates marginally valid
prediction regions:

$\mathbb{P}_{({\mathbf{X}},Y)\sim F}(Y\in C({\mathbf{x}}))>1-\alpha\;,$

where $C({\mathbf{x}})\subseteq\mathbb{R}$ denotes the prediction region for
$Y$ given ${\mathbf{X}}={\mathbf{x}}$. $C({\mathbf{x}})$ provides $1-\alpha$
level coverage for $Y$, averaged across the marginal distribution of
${\mathbf{X}}$.

Marginal validity may not be sufficient in practice when working with large,
heterogeneous populations. For example, suppose $Y$ is the effect of a
cholesterol-lowering medication on a patient, and ${\mathbf{X}}$ includes the
patient’s characteristics. Marginally valid prediction regions might be
extremely precise for patients weighing less than 200 lbs., but be correct 0%
of the time for patients weighing more than 200 lbs. Patients in the latter
category are not helped by the guarantee that prediction regions on average
over all patients achieve $1-\alpha$ level coverage.

To address this concern, several papers [Lei and Wasserman, 2014, Barber et
al., 2019, Guan, 2019] have highlighted the desirable properties of local
conformal inference, which provides the stronger guarantee of locally valid
prediction regions. Local validity ensures $1-\alpha$ level coverage not only
on average across ${\mathbf{X}}$, but also within predefined local regions $A$
of the $\mathcal{X}$ space:

$\mathbb{P}_{({\mathbf{X}},Y)\sim F}(Y\in C({\mathbf{x}})|{\mathbf{x}}\in
A)>1-\alpha\,,\ \forall A\in\mathcal{A},$

where $\mathcal{A}$ is a partition of $\mathcal{X}$. Returning to the
cholesterol medication example, a local conformal approach might define bins
based on the patient’s weight (e.g. 100-120 lbs., 120-140 lbs., etc.), and
thus ensure $1-\alpha$ level coverage within each subpopulation. Lei and
Wasserman [2014]’s approach is similar in spirit to this, defining regions $A$
as local segments of $\mathcal{X}$ using Euclidean distance.

In addition to providing validity assurances, prediction regions attempt to be
as efficient as possible; that is, be as small as possible while still valid.
Density level sets, by construction, define the minimum sized region that
contains a certain amount of probability mass [Lei et al., 2013].
Additionally, modern advances in statistics have led to conditional density
estimators (CDEs) $\widehat{f}(y|{\mathbf{x}})$ that can handle complex data
structures [e.g. Izbicki and Lee, 2017]. As such, it should be no surprise
that CDEs have been increasingly used in conjunction with conformal inference
[e.g Lei and Wasserman, 2014, Izbicki et al., 2021].

Izbicki et al. [2021] recently proposed a method that aims to achieve local
conformal coverage even when ${\mathbf{X}}$ is high-dimensional. To avoid the
curse of dimensionality, they define local partitions of $\mathcal{X}$ based
on the structure of the fitted CDEs $\widehat{f}$. This method obtains
asymptotic conditional validity,

$\mathbb{P}_{({\mathbf{X}},Y)\sim F}(Y\in
C({\mathbf{X}})|{\mathbf{X}}={\mathbf{x}})>1-\alpha\,,\
\forall{\mathbf{x}}\in\mathcal{X},$

but requires strong assumptions. These assumptions require relatively strong
convergence of $\widehat{f}$ to the true conditional density for almost all
$X$ values [Izbicki et al., 2021, Assumption 9].

This assumption of a well fit CDE $\widehat{f}$ may be unrealistic for finite-
sample real world examples, especially in fields where models may be
intentionally constrained by fundamental assumptions or selected from a less
flexible model class.

In this work, we propose a novel approach that, while similar in spirit to
Izbicki et al. [2021], we believe gets closer to the heart of local conformal
inference. Our method defines local partitions of $\mathcal{X}$ based on the
local performance of a fitted CDE model. In particular, we define a distance
between ${\mathbf{X}}$ values that reflects the difference between the true
local coverage of CDE’s level sets and expected coverage. This distance
attempts to partition $\mathcal{X}$ into groups that would have similar
conformal score distributions under an unattainable conditional conformal
approach. Our approach is able to evaluate coverage by leveraging tools
presented in Zhao et al. [2021], a recent paper that explores the local
validation of conditional density models.

This paper is organized as follows. In Section 2, we provide relevant
background on conformal inference and model diagnostic tools. In Section 3, we
introduce our proposed method for local conformal inference and describe its
motivation. In Section 4, we present two examples to demonstrate that our
method achieves local validity for well-designed partitions, even in high
dimensions. Finally, in Section 5, we discuss conclusions and future work.

##  2 Related Work

In this section, we provide relevant background on the foundations of
conformal inference, approaches to local conformal inference, and local model
diagnostics.

The field of conformal inference aims to create prediction regions with valid
coverage, under minimal parametric assumptions [Vovk et al., 2005]. Conformal
methods can be seen as a type of “wrapper” around black box models, generating
valid, finite sample prediction regions while requiring only the assumption
that previous and future observations are exchangeable. One creates conformal
prediction regions by evaluating potential candidate values $y$, and including
them if they are not too extreme relative to previous observations. The
measure of extremeness comes from a conformal or non-conformal score. A
conformal score has smaller values indicating more extremeness; a common
example is the conditional density estimate $\widehat{f}(y|{\mathbf{x}})$. A
non-conformal score has larger values indicating more extremeness; a common
example is the quantity $|y-\mu({\mathbf{x}})|$ defined using a regression
function $\mu({\mathbf{x}})$. (Multiplying by -1 would convert a conformal
score to a non-conformal score, and vice-versa.)

To illustrate how conformal inference works in more detail, we describe a
popular method called split-conformal inference [Vovk et al., 2005, pg.
110-111]. Assuming we have already define a conformal score function
$cs(\cdot)$, split-conformal inference decides if a new $y$ should be
contained in a prediction region by following three steps. The first step is
to split the observed data into disjoint training and calibration sets. We
denote the indices for observations $i\in\\{1,...,n\\}$, and let the sets
$\mathcal{I}_{train}$ and $\mathcal{I}_{cal}$ contain the indices of the
training and calibration points, respectively. The second step is to build a
prediction model using the training set, which is used to calculate conformal
scores
($cs\left(y_{i}|{\mathbf{x}}_{i},\\{{\mathbf{x}}_{j}:j\in\mathcal{I}_{train}\\}\right)$)
for observations in the calibration set ($i\in\mathcal{I}_{cal}$). Finally,
the third step is to evaluate potential $\widehat{y}_{n+1}$ values, and
include these values in our prediction region if the conformal score
$cs\left(\widehat{y}_{n+1}|{\mathbf{x}}_{n+1},\\{{\mathbf{x}}_{j}:j\in\mathcal{I}_{train}\\}\right)$
is less extreme than $100\alpha\%$ of the conformal scores for the calibration
set. Algorithm 1 formally presents the split-conformal approach.

Split-conformal inference obtains finite sample validity as a result of the
exchangeability of points in the calibration set and the new point. Given this
exchangeability, the rank of the conformal score for the true $y_{n+1}$ with
respect to the conformal scores from the calibration set is distributed
uniformly between 1 and $1+|\mathcal{I}_{cal}|$. Therefore, including all
$\widehat{y}_{n+1}$ that are less extreme than $100\alpha\%$ of the conformal
scores for the calibration set creates a prediction region that a finite
sample validity of $1-\alpha$ as desired.

Algorithm 1 Split Conformal Inference

Input: Conformal measure $cs$, significance level $1-\epsilon$, previous
observed pairs $\\{z_{i}\\}_{i=1}^{n}$ where $z_{i}=({\mathbf{x}}_{i},y_{i})$,
disjoint index sets for training and calibration sets ($\mathcal{I}_{train}$
and $\mathcal{I}_{cal}$), a new object $x_{n+1}$ and a potential $y$.

Task: Decide if $y$ should be included in a prediction region for $x_{n+1}$ at
significance level $1-\epsilon$.

Algorithm:

  1. 1.

Provisionally view the $z_{n+1}=(x_{n+1},y)$

  2. 2.1.

Conformal scores for the calibration data’s set:

  3. For all $j\in\mathcal{I}_{cal}$ set

$\alpha_{j}=cs(z_{j}|\\{z_{i}:i\in\mathcal{I}_{train}\\})$

  4. 2.2.

Conformal scores for new observation:

  5. $\alpha_{n+1}=cs(z_{n+1}|\\{z_{i}:i\in\mathcal{I}_{train}\\},)$

  6. 3.

Set

$p_{y}=\frac{\\#\\{j\in\mathcal{I}_{cal}:\alpha_{j}\geq\alpha_{n+1}\\}}{|\mathcal{I}_{cal}|+1}$

  7. 4.

Include $y$ in the new prediction region if $p_{y}>\epsilon$.

As mentioned, classical conformal inference only provides marginal finite
sample validity (that is, $\mathbb{P}_{({\mathbf{X}},Y)\sim F}(Y\in
C({\mathbf{x}}))>1-\alpha$). However, practitioners are often more interested
in conditional validity. It is known that distribution-free methods cannot
achieve conditional coverage, unless one imposes strong assumptions on the
underlying distribution [Vovk, 2013, Lei and Wasserman, 2014, Barber et al.,
2019]. In this paper, we focus on local conformal inference. In local
conformal inference as defined by Barber et al. [2019], validity is obtained
in smaller partitions $A$ of the $\mathcal{X}$ space, such that
$\mathbb{P}_{({\mathbf{X}},Y)\sim F}(Y\in C({\mathbf{x}})|{\mathbf{x}}\in
A)>1-\alpha$ for each partition $A$. When used with split-conformal inference,
this means that in the final step of creating a prediction region, we only
view calibration points in the same $A\subseteq\mathcal{X}$ as the new test
${\mathbf{X}}$ value as exchangeable with that new test observation, and
therefore only use those calibration points’ scores to define our notions of
extremeness. We now describe two different local conformal approaches, both of
which mirror the structure that Barber et al. [2019] describes and seek to
define desirable local regions. Lei and Wasserman [2014] defines local regions
relative to the $X$ space, and creates bins of $X$ values based on Euclidean
distance between points. As the number of observations increases, the size of
the $X$ bins can decrease; in the limit of $n\to\infty$, asymptotic
conditional coverage is achieved. Izbicki et al. [2021]’s CD-split+ method
provides a different way to create localized partitions. Concerned about the
scalability of Lei and Wasserman [2014]’s approach to high dimensional
$\mathcal{X}$ spaces, CD-split+ defines partitions of $\mathcal{X}$ by fitting
a CDE model $\widehat{f}(y|{\mathbf{x}})$ and using a similarity measure of
the predicted conditional density estimates. Specifically, CD-split+ creates
local groups of $X$ values for those that are close relative to a profile
distance:

$d_{\text{profile}}^{2}\left({\mathbf{x}}_{a},{\mathbf{x}}_{b}\right):=\int_{-\infty}^{\infty}\left(\widehat{H}\left(y\mid{\mathbf{x}}_{a}\right)-\widehat{H}\left(y\mid{\mathbf{x}}_{b}\right)\right)^{2}dy\;,$

where $\widehat{H}(y|{\mathbf{X}})$ is the model estimated highest predictive
density (HPD) value:

$\displaystyle{\text{HPD}}(y;{\mathbf{x}})=\int_{{y^{\prime}}:\widehat{f}({y^{\prime}}|{\mathbf{x}})\geq\widehat{f}(y|{\mathbf{x}})}\widehat{f}({y^{\prime}}|{\mathbf{x}})d{y^{\prime}}.$
(1)

This approach avoids the curse of dimensionality. Under the assumption that
the estimated conditional density converges to the true conditional density,
CD-split+ obtains asymptotic conditional validity. Izbicki et al. [2021] also
proposes another technique called HPD-split (which is similar to Gupta et al.
[2020, Appendix D]). HPD-split uses HPD values instead of CDE values as
conformal scores, but still crucially relies on the assumption that
$\widehat{f}$ is well fit.

What happens to local conformal inference techniques when $\widehat{f}$ is not
well fit? We explore this question in our paper, with the help of recently
developed tools for diagnosing issues with CDE models. Zhao et al. [2021] has
recently developed diagnostics for validating the goodness-of-fit of CDEs
locally in the feature space $\mathcal{X}$. The crux of their method involves
fitting a model to smoothly estimate the proportion of time an observation
obtains an HPD value above a certain threshold, at any location
${\mathbf{X}}={\mathbf{x}}$. In particular, they define a set of functions
indexed by $\alpha\in[0,1]$:

$r_{\alpha}({\mathbf{x}})=\mathbb{P}\left(\widehat{HPD}(Y;{\mathbf{x}})<\alpha|{\mathbf{x}}\right)\;,$
(2)

where $\widehat{HPD}(\cdot)$ is the estimated HPD value function from the
given CDE model $\widehat{f}(y|{\mathbf{x}})$. Recall that HPD values would be
uniform if the model were correctly fit and would also imply that
$r_{\alpha}({\mathbf{x}})=\alpha$ for any $\alpha\in[0,1]$. Zhao et al. [2021]
showed that we can estimate $r_{\alpha}$ by fitting a quantile regression
model $\widehat{r}_{\alpha}$ across the feature space. By considering a grid
of values for $\alpha\in[0,1]$, we obtain an informative sequence of
$\\{\widehat{r}_{\alpha}\\}$ estimates, at any given location
${\mathbf{x}}\in\mathcal{X}$ in the feature space. Deviations of
$\widehat{r}_{\alpha}({\mathbf{x}})$ from $\alpha$ for various values of
$\alpha$ indicate that the CDE model $\widehat{f}$ does not fit the true
conditional density $f$ well. Furthermore, the specific pattern of which
values of $\alpha$ are in disagreement, and by how much, form a rough
“signature” of the specific nature of the distributional deviation of
$\widehat{f}$ from $f$.

##  3 Methods

In this section, we propose a novel local conformal inference method, MD-
split+, which synthesizes the above ideas in order to choose a better informed
partition ${\mathcal{A}}$ of the feature space $\mathcal{X}$. Our method is
practical to use and scales to high dimensions. In particular, it enables
practitioners to leverage CDE models to create valid prediction regions, while
being robust to misspecifications of those models.

MD-split+ fits under the split-conformal framework. It utilizes the following
data splits: (i) training set to build a CDE model $\widehat{f}$, (ii) a
validation set to select optimal parameters for $\widehat{f}$, (iii) a
diagnostic set (to be discussed in more detail) and (iv) the standard split-
conformal calibration set.

Our procedure first trains a CDE model with the training and validation sets.
Then, we train a quantile regression function
$\\{\widehat{r}_{\alpha}(\cdot):\alpha\in[0,\delta,2\delta,\ldots,1]\\}$, to
predict quantiles of HPD values for new points in the diagnostic set (as done
in Zhao et al. [2021]). We also tune this quantile regression function using
the validation set. (We imagine that the CDE model and quantile regression
function are tuned in tandem, but one could also use 2 different validation
sets.) Next, we define a model diagnostic distance:

$d_{\text{md}}^{2}\left({\mathbf{x}}_{a},{\mathbf{x}}_{b}\right):=\frac{1}{\lceil\frac{1}{\delta}\rceil+1}\sum_{\begin{subarray}{c}\alpha\in\\\
[0,\delta,2\delta,\ldots,1]\end{subarray}}\left(\widehat{r}_{\alpha}({\mathbf{x}}_{a})-\widehat{r}_{\alpha}({\mathbf{x}}_{b})\right)^{2}\;.$

Using this distance (in a similar fashion to Izbicki et al. [2021]’s CD-
split+), we define $K$ clusters $C_{1},\ldots,C_{K}$ to be the centroids of
the calibration set points $X_{i}$ according to to $d_{\text{md}}^{2}$. These
centroids define group indices for ${\mathbf{X}}$ values, via the index of the
closest $C_{k}$ according to $d_{\text{md}}^{2}$. We can then apply local
conformal inference, using this local structure combined with an HPD conformal
score, to new test points. This procedure is detailed in Algorithm 2.

Algorithm 2 MD-split+ Local Conformal Inference

Input: significance level $1-\epsilon$, previous observed pairs
$\\{z_{i}\\}_{i=1}^{n}$ where $z_{i}=({\mathbf{x}}_{i},y_{i})$, disjoint index
sets for training, validation, diagnostic and calibration sets
($\mathcal{I}_{train}$,
$\mathcal{I}_{val},\mathcal{I}_{diag},\mathcal{I}_{cal}$), and a new object
$x_{n+1}$.

Task: create a prediction region for $y_{n+1}|{\mathbf{x}}_{n+1}$ at
significance level $1-\epsilon$.

Algorithm:

  1. 1.

Train conditional density estimator using
$\\{z_{i}:i\in\mathcal{I}_{train}\\}$

  2. 2.

Train model-fit quantile regression function
$\\{\widehat{r}_{\alpha}(\cdot)\\}_{\alpha}$ using pairs
$z_{i}^{\prime}=\left(\widehat{HPD}(y_{i}|{\mathbf{x}}_{i}),{\mathbf{x}}_{i}\right)$
for $i\in\mathcal{I}_{diag}$.

  3. 3.

Tune conditional density estimator $\widehat{f}(\cdot)$ and quantile
regression function $\\{\widehat{r}_{\alpha}(\cdot)\\}_{\alpha}$ parameters
with $z_{j}$ and $z_{j}^{\prime}$ for $j\in\mathcal{I}_{val}$ respectively.

  4. 4.

Define grouping centroids by using k-means++ on a discretized set of vector
$[r_{\alpha}({\mathbf{x}}_{l})\text{ for }\alpha\in[0,...,1]]$ using
$l\in\mathcal{I}_{cal}$.

  5. 5.

Apply local split-conformal inference using the clustering defined in Step 4,
with conformal scores defined by $\widehat{HPD}(\cdot)$ with significance
level $1-\epsilon$.

###  3.1 Motivation and Intuition

We believe that our approach, while similar in spirit to Lei and Wasserman
[2014] and Izbicki et al. [2021], gets closer to the heart of local conformal
inference.

A conformal prediction region can be viewed as a level set of the conformal
score function on $\mathcal{Y}$. Specifically, the region contains all
$y_{new}^{\prime}$ such that $cs(y_{new}^{\prime}|{\mathbf{x}}_{new})$ is
greater than or equal to some threshold. In this way, a given conformal score
function can be seen to define a collection of nested level sets [Vovk et al.,
2005, pg. 9]. Thus, if a confidence level of $1-\alpha$ is desired, conformal
inference can be viewed as selecting the level set among this collection that
would be expected to contain a probability mass of $1-\alpha$ (at least in a
marginal sense).

Taking this view of conformal inference, if the conformal score were an HPD
function, then conformal prediction regions would be level sets indexed by HPD
values (as thresholds). If the HPD function (derived from the CDE model
$\widehat{f}$) were well fit and we wanted a prediction region with confidence
level $1-\alpha$, then conformal inference would simply return the set defined
by $y$ values with HPD values $\geq 1-\alpha$. That is, we would obtain
$\\{y:HPD(y|{\mathbf{x}})\geq 1-\alpha\\}$. If the HPD function were not well
fit, then conformal inference might select an HPD threshold different from the
nominal $1-\alpha$ level, in order to return a prediction set with achieved
$1-\alpha$ coverage. In this sense, conformal inference serves as a correction
to the HPD function, which may be imperfectly estimated.

This motivates how our proposed MD-split+ method partitions $\mathcal{X}$. MD-
split+ defines local partitions of $\mathcal{X}$ based on the local
performance of a fitted CDE model. Effectively, what we aim to do is to group
together ${\mathbf{X}}={\mathbf{x}}$ values that would get similar
“corrections” to their HPD function, if we were able to run a conditional
conformal method at every ${\mathbf{X}}={\mathbf{x}}$. In reality, of course,
we cannot create a conditional conformal method. But we can estimate what
those “corrections” would look like, at any location
${\mathbf{x}}\in\mathcal{X}$, thanks to the diagnostic framework of Zhao et
al. [2021]. We use these estimates to form better local partitions of
$\mathcal{X}$, pairing together ${\mathbf{X}}$ values that would see similar
conformal score distributions under a conditional conformal method.

##  4 Examples

###  4.1 Simple Linear Regression Conditional Density Estimation

Our first example has one-dimensional $\mathcal{X}$ and $\mathcal{Y}$ spaces,
and we specifically developed it to be juxtaposed against Izbicki et al.
[2021, Figure 3]. We define the underlying distribution of the data by
$X\sim\text{Unif}(-4,4)$ and a varying $Y|X=x$ distribution centered at $x$.
The conditional density distribution varies from a scaled $t$-distribution
($df=3$), to a scaled normal distribution, to a scaled truncated normal
distribution, and back as $x$ varies between $-4$ and $4$. More specifically,
the underlying distribution is defined as:

$\displaystyle X$ $\displaystyle\sim\text{Unif}(-4,4)$

If $X=x$ and $|x|\leq 2$:

$\displaystyle\quad(Y-x)$
$\displaystyle\sim\begin{array}[]{l}\text{trunc\\_norm}\Big{(}\sigma\\!:\sigma(|x|-2),\\\
\quad\text{low}\\!:-\sigma(|x|-2)\cdot\text{trunc\\_max}(|x|-2),\\\
\quad\text{up}\\!:\sigma(|x|-2)\cdot\text{trunc\\_max}(|x|-2)\Big{)}\end{array}$

If $X=x$ and $|x|>2$:

$\displaystyle\quad(Y-x)$
$\displaystyle\sim\frac{1}{2}\sigma(|x|-2)\cdot\text{student\\_t}(\text{df}:\text{df}(|x|-2))$

where

$\displaystyle\sigma(x)$ $\displaystyle:=1+1.5|x|$
$\displaystyle\text{trunc\\_max}(x)$ $\displaystyle:=.5+\log(2/|x|)$
$\displaystyle\text{df}(x)$ $\displaystyle:=(3-|x|)^{3}+2\,.$

Figure 1’s top subplot presents a set of random draws from this distribution.

We then fit a Gaussian with varying mean and spread to estimate the
conditional density function for this $Y|X$. This is a modeling choice that a
practitioner might reasonably try when presented with this data. However, this
CDE model is inherently misspecified.

We compare the performance of our MD-split+ with Izbicki et al. [2021]’s CD-
split+ on this example, and also discuss how Lei and Wasserman [2014]’s local
approach would have performed.

First, we note that MD-split+ uses data differently than CD-split+ and Lei and
Wasserman [2014]’s local conformal approach would. In this example, MD-split+
places 25% in a training set, 25% in validation set, 25% in a diagnostic set
and 25% in calibration set, while CD-split+ splits the data 33% in training,
33% in validation, and 33% in the calibration set.

For both procedures, we separately fit a Gaussian conditional density
estimator, where the conditional mean $\widehat{\mu}({\mathbf{x}})$ comes from
a linear regression, and the conditional variance
$\widehat{\sigma}^{2}({\mathbf{x}})$ comes from a smoothing spline fit to the
squared residuals of the linear regression. Both fits (whether using 25% data
splits in MD-split+ or 33% data splits in CD-split+) are basically the same,
and we visualize these two functions from MD-split+ in Figure 1, top panel.
For MD-split+, we also fit a smooth quantile regression function on HPD values
computed using the CDE model, with a small quantile regression neural network
with Adam optimization [Kingma and Ba, 2014] with learning rate $10^{-3}$, and
2 hidden layers with 10 nodes each. One can imagine that a set of smoothing
spline binary regressors would perform similarly across this simple space.

Figure 1: This figure relates to the example discussed in Section 4.1. The top
subplot presents the underlying data distribution, as well as the fit of the
mean and standard deviation that defines the CDE model. The lower two subplots
display the local groupings for the local conformal approaches CD-split+ and
MD-split+ with 5 clusters each. Notice that CD-split+ groups $X$ values with
$Y$ that have a truncated normal together with $X$ values with $Y$ that have
$t$-distributions with low degrees of freedom, and in general groups by
$\sigma(x)$ values ($|X|-2$). This is not optimal, given that they have
largely differing conformal score distributions. MD-split+ focuses more on
grouping similar underlying distributions together.

For both procedures we create 5 partitions of the $\mathcal{X}$ space, and
these can be seen in Figure 1’s lower two subplots This figure shows that the
CD-split+ clusters utilizes the estimated $\hat{\sigma}(x)$ values (visually
clustering by $|{\mathbf{x}}|$), whereas our approach uses the relative fit of
the CDE model (visually clustering by $|{\mathbf{x}}|-2$). If we had also
applied Lei and Wasserman [2014]’s approach, we would have seen clusters based
solely on ${\mathbf{X}}$ value. Compared to CD-split+, our MD-split+ algorithm
clusters with respect to the underlying distributional differences the data
has. Compared to Lei and Wasserman [2014]’s, MD-split+ clusters more similar
underlying residual distributions, by not being constrained to only cluster
linearly on contiguous ${\mathbf{X}}$ values. This simple example highlights a
drawback of clustering on model conditional density estimation when model fit
is imperfect.

Figure 2: This figure captures the difference in empirical coverage and
expected coverage for small bins on the $X$ space related to the generative
distribution for example 1 and the clusters visualized in Figure 1. The top
subplot relates to the CD-split+ approach with 5 clusters and the lower
subplot relates to our proposed approach with 5 clusters as well.

In addition, this misspecified CDE model (Gaussian with varying mean and
variance) means that as the number of observations ($n$) and the number of
clusters ($k$) increases, we will not see asymptotic conditional validity
emerge under CD-split+. Figure 2 captures the difference in local validity for
a grid of small bins of $X$ values relative to both clustering approaches,
with stronger colors highlighting large differences. As expected, CD-split+
performs worse on this localized level than our MD-split+ approach. In other
words, a prediction region made using CD-split+ will more often have a
different amount of conditional coverage than desired, and this difference
will often be larger than that seen with prediction regions made using MD-
split+.

###  4.2 Convolutional Neural Density Estimation for Galaxy Images

In this example, we consider the problem of estimating prediction regions for
synthetic “redshift” values $Z$ (the response), which are assigned to
photometric or “photo-z” galaxy images ${\mathbf{X}}$ (the features). Here,
${\mathbf{X}}$ represents a $20\times 20$-pixel image of an elliptical galaxy
generated by GalSim, an open-source toolkit for simulating realistic images of
astronomical objects [Rowe et al., 2015]. In GalSim, we can vary the axis
ratio $\lambda$, defined as the ratio between the minor and major axes of the
projection of the elliptical galaxy, as well as the galaxy’s rotational angle
$\theta$ with respect to the x-axis. We create 9 equally sized populations of
galaxies, representing every combination of $\lambda\in\\{0.1,0.4,0.75\\}$ and
$\theta\in\\{\pi/4,2\pi/3,5\pi/6\\}$111Note that these $\theta$ values are not
equally spaced., with uniform noise added to $\lambda$ and $\theta$ for each
individual image. See Figure 3 for representative examples of each galaxy
image type.

Figure 3: We create 9 equally sized populations of galaxy images, representing
combinations of axis ratios $\lambda\in\\{0.1,0.4,0.75\\}$ and angles
$\theta\in\\{\pi/4,2\pi/3,5\pi/6\\}$. Individual images have a small amount of
noise in both $\lambda$ and $\theta$. This grid shows a representative image
from each group. Figure 4: We define 9 conditional density models for the
“redshift” $Z$, corresponding to the 9 galaxy image populations in Figure 3.
Galaxy images with the same $\lambda$ value have densities $f(z|{\mathbf{x}})$
that have the same variance, while those with the same $\theta$ value have
densities $f(z|{\mathbf{x}})$ that have the same distributional shape.

We then assign a response variable $Z$ with different bimodal mixture
distributions as follows:

$\displaystyle Z|\lambda,\theta\sim
a_{\lambda}\left[\frac{1}{2}U_{\theta}+\frac{1}{2}V_{\theta}\right]$

where

$\displaystyle U_{\theta}=\begin{cases}\mathcal{N}(2,1),&\theta=\pi/4\\\
\text{Exponential}(1)\text{ centered at 2},&\theta=2\pi/3\\\
\frac{1}{2}\mathcal{N}(2.7,0.25)+\frac{1}{2}\mathcal{N}(1.3,0.25),&\theta=5\pi/6\end{cases}$
$\displaystyle V_{\theta}=\begin{cases}\mathcal{N}(-2,1),&\theta=\pi/4\\\
\text{Exponential}(1)\text{ centered at -2},&\theta=2\pi/3\\\
\frac{1}{2}\mathcal{N}(-2.7,0.25)+\frac{1}{2}\mathcal{N}(-1.3,0.25),&\theta=5\pi/6\end{cases}$

and

$\displaystyle a_{\lambda}=\begin{cases}0.5,&\lambda=0.1\\\ 1,&\lambda=0.4\\\
2,&\lambda=0.75\end{cases}$

See Figure 4 for a plot comparing these distributions.

Based on the marginal distribution of the response, a practitioner might
reasonably decide to model conditional density estimates as a mixture of
Gaussians. In particular, here we fit a Gaussian convolutional mixture density
network (ConvMDN, [D’Isanto and Polsterer, 2018]), which can effectively train
on the image feature space.

This is a situation where the data structure is more complex, so local
conformal inference is challenging. Lei and Wasserman [2014]’s partitioning
method does not scale well here because ${\mathbf{X}}$ is high-dimensional.
Furthermore, we will demonstrate that even well-tuned, sophisticated
conditional density models trained on this dataset with reasonably large
sample sizes may be misspecified in subtle but fundamental ways. Thus, the
methods of Izbicki et al. [2021] may not lead to sensible local subgroups. On
the other hand, our proposed MD-split+ method provides a practical way to do
local conformal inference successfully, even on complex data with imperfectly
fit CDEs. In this example, we compare MD-split+ to the related approach CD-
split+ [Izbicki et al., 2021].

For this complex data structure, we dedicate more data to training and
validating the ConvMDN model. Specifically, for our MD-split+ implementation,
we dedicate 27,000 observations to the training and validation sets (70/30
split), 4,500 observations for our diagnostic set (to train our HPD quantile
functions), and 4,500 for the calibration set. For CD-split+ we add what was
the diagnostic set above to the training and validation sets, thus giving us
31,500 observations (again with 70/30 split). We also evaluate both approaches
on 4,500 test observations.

For both approaches, we fit a Gaussian convolutional mixture density network
(ConvMDN, [D’Isanto and Polsterer, 2018]), with two convolutional and two
fully connected layers with ReLU activations [Glorot et al., 2011]. We train
these models using the Adam optimizer [Kingma and Ba, 2014] with learning rate
$10^{-3}$, $\beta_{1}=0.9$, and $\beta_{2}=0.999$. We tune several
hyperparameters in search of the best fitting ConvMDN model. We allow $M$, the
number of mixture components, to vary from 2 to 10. We also tune the number of
hidden units in the penultimate layer and whether to include dropout
[Srivastava et al., 2014] after the convolutional layers. For both approaches,
we found that the best ConvMDN model with the lowest KL loss had parameters
$M=2$, 10 hidden layers, and a 50% dropout layer. For MD-split+ we also train
a quantile regression model (with $\alpha\in\\{0,0.02,....,1\\}$) estimating
$\widehat{HPD}(Y|{\mathbf{X}})$ values given ${\mathbf{X}}$, where
$\widehat{HPD}$ is defined relative to the ConvMDN.

We then define three clusters of the calibration points using $d_{md}$ and
$d_{profile}$ for MD-split+ and CD-split+ respectively. Next, we construct
prediction regions on a test set of 4500 new observations (galaxy images +
redshifts), and verify their coverage (whether the prediction region includes
the true redshift). By construction, even this “best” ConvMDN model is
misspecified, because the true conditional densities $f(z|{\mathbf{x}})$ are
not all mixtures of 2 Gaussians. Therefore, simply using a distance that
compares fitted CDE models $\widehat{f}$, as CD-split+ does, can lead to
undesirable clusters that fail to achieve local conformal coverage on the true
subpopulations of interest. In contrast, our method partitions the feature
space into sensible regions, as the HPD distributions are the same across the
scalings defined by $\lambda$. See Figure 5 for a comparison of the clusters
achieved by these two methods.

Figure 5: Pie charts showing the fraction of points assigned to different
clusters. Top: Our method MD-split+ obtains reasonable clusters. For the most
part, points with similarly shaped conditional densities (see Figure 4) are
grouped together. Intuitively, such points will have similar HPD coverage
profiles, and thus should belong to the same local conformal partition.
Bottom: CD-split+ forms unintuitive clusters. This method clusters together
distributions with the same variance, regardless of how differently shaped
those distributions might be.

Next, we form local conformal prediction regions using the two sets of
clusters and their associated conformal scores. Recall that CD-split+ uses
conditional density estimate values, while MD-split+ uses $\widehat{HDP}$
values. Of course, by construction one will obtain validity on average over
each cluster, for any arbitrary clusters one wishes to define. But we are
interested in whether validity is achieved over the 9 true subpopulations in
the dataset. Figure 6 shows the empirical coverage relative to the conformal
confidence levels across the 9 true subpopulations. We see that, across the
range of confidence levels, our method obtains empirical coverage very close
to what is expected. On the other hand, CD-split+’s empirical coverage for
these 9 true subpopulations varies drastically. Figure 7 shows this
information for four specific confidence levels (0.2,0.4,0.6, and 0.8), again
highlighting the performance differences between these two methods.

Figure 6: We check nominal vs. achieved coverage of local conformal sets
within each of the 9 true subpopulations defined earlier. Top: Using the 3
clusters obtained from our method MD-split+, we achieve correct local
conformal coverage across the range of $\alpha\in[0,1]$. Bottom: Using the 3
clusters obtained from CD-split+, we see clear deviations from correct local
conformal coverage in many of the 9 true subpopulations.

Figure 7: In this figure we select 4 particular confidence levels and check
the achieved local conformal coverage by group, for nominal coverage levels of
$\alpha=0.2,0.4,0.6,0.8$. Top: Our method MD-split+ achieves approximately
correct coverage in every true subpopulation. Bottom: CD-split+ leads to large
deviations from the correct coverage in most subpopulations. CD-split+
clusters together groups (1,2,3), (4,5,6), and (7,8,9) in the figure. While
coverage is correct when averaged across these clusters, it is incorrect for
the true groups of interest.

In this example, we have seen how a complex CDE model $\widehat{f}$ can remain
misspecified, despite careful tuning. Moreover, this kind of misspecification
would not be obvious to the practitioner. In theory, one can always simply
“find a better model”, but in reality, models are often imperfect, especially
in complex data settings. It is in this realistic context that MD-split+,
unlike previous known methods, can still achieve valid, scalable local
conformal inference.

##  5 Discussion

We have developed a novel local conformal inference method that is both
scalable to high dimensions and extendable to complex data settings where
conditional density estimators may be poorly estimated. As such, our method
has great practical utility. If CDE models were always well fit, then the
estimated conditional density level sets would always be close to the true
level sets, and there would be no reason to use conformal inference at all.
Instead, one could simply use the CDE model to select the level set with the
probability mass equal to the desired confidence level.

In comparison with other local conformal inference techniques that use CDE or
HPD values as their conformal scores, our approach does require an additional
data split. However, the example in Section 4.2 shows that this added split
will often not be too burdensome for the practitioner. In developing our
approach, we observed the impact of having the CDE model or HPD quantile
regression model not being well tuned. A poorly tuned CDE model (specifically
one that is overfit), can make it harder for the HPD quantile regression
function to correctly cluster ${\mathbf{X}}$ observations based on model
performance. Similarly, a poorly tuned HPD quantile regression function can
reduce the ability to obtain optimal clusters.

There is a lot of potential for future work in understanding the theoretical
aspects and more nuanced properties of our approach. Specifically, we are
interested in exploring whether our method achieves asymptotic conditional
coverage with meaningfully weaker assumptions than, say, Izbicki et al.
[2021]. We expect this to require understanding the smoothness in
$\mathcal{X}$ of the true and estimated conditional density, as well as the
smoothness in $\mathcal{X}$ of the oracle versus estimated HPD quantile
regression function. Additionally, we seek to better understand (potentially
empirically) how comparisons of our MD-split+ (with less data to train the
estimated CDE model) to Izbicki et al. [2021]’s CD-split+ might change if,
instead of fundamentally misspecified models, we had CDE and HPD quantile
regression functions that were poorly fit due to finite sample sizes, but
eventually converge to the true conditional density and oracle HPD quantile
regression functions, respectively.

Overall, we see MD-split+ as a new tool in the practitioner’s toolkit for
performing local conformal inference using conditional density
estimation$-$one that is especially useful when $\mathcal{X}$ is high
dimensional or the CDE is misspecified. We hope that our procedure will appeal
to a wide range of practitioners, and that they will appreciate the
philosophical foundations of our approach as one of its strengths.

##  6 Acknowledgements

BL and DZ are grateful for fruitful discussions with Aaditya Ramdas about this
topic. BL would like to thank his advisor Chad Schafer for discussions on
local conformal inference literature and potential extensions. DZ would like
to thank his advisor Ann B. Lee and collaborator Rafael Izbicki, with whom he
developed the conditional density model diagnostic tools leveraged in this
paper.

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# Generation of kink-antikink pairs through the excitation of an oscillon in
the $\phi^{4}$ model

Fabiano C. Simas1,3 and E. da Hora2,3<EMAIL_ADDRESS><EMAIL_ADDRESS>1
Centro de Ciências de Chapadinha (CCCh), Universidade Federal do Maranhão
(UFMA), 65000-000 Chapadinha, MA, Brazil
2 Coordenação do Curso de Bacharelado Interdisciplinar em Ciência e
Tecnologia, Universidade Federal do Maranhão, 65080-805, São Luís, Maranhão,
Brazil.
3 Programa de Pós-Graduação em Física, Universidade Federal do Maranhão
(UFMA), Campus Universitário do Bacanga, 65085-580, São Luís, Maranhão, Brazil

###### Abstract

We examine the oscillon decay in a model with a single scalar field and two
asymmetric fundamental states. Specifically, we verify how the evolution of an
initially excited oscillon leads to the formation of antikink-kink pairs in
the presence of a centered long-lived pulse. In such a context, the magnitude
of the original perturbation stands for the most important factor influencing
the process. For intermediary values of this magnitude, we find a transitory
behavior which suggests that the energy exchange between translational and
vibrational modes begins to play an important role in the overall evolution.
As the magnitude increases, the centered pulse intermediates an interaction
between the inner structures which compose the pairs whose behavior mimics
that of a standard antikink-kink collision and therefore reinforces the role
played by the energy transfer mechanism. We depict the resulting resonant
structure and note that it mimics that of an usual kink-antikink scattering.
We also reproduce some of our results now via an initial lumplike perturbation
proposed as a kink-antikink pair. For different values of width, an almost
pure antikink-kink pair escapes to the infinity after multiple bounces.

##  I Introduction

In the context of classical field models, topological defects are usually
described as the solutions to highly nonlinear equations of motion. In this
case, the nonlinearity is introduced via the potential which describes the
self-interaction of the scalar matter and, as a consequence, also determines
the vacuum structure of the corresponding theory manton .

In such a scenario, localized configurations connect two adjacent minima of
the potential, with both the solution and its anti-version having the very
same total energy, but topological charges with opposite signs.

Under special circumstances, these profiles can be also obtained via a set of
first-order differential equations which emerges from the minimization of the
energy of the model through the implementation of the so-called BPS technique
prasom ; bogo . In this case, it is possible to verify that these first-order
equations lead to the standard second-order equations of motion, from which
one concludes that the BPS configurations are legitimate solutions of the
effective theory indeed.

In this overall context, localized structures have been studied in connection
to a wide variety of physical systems throughout the last few years, ranging
from the subatomic to the cosmological scale dauxois ; vacha . Furthermore,
topological defects are particularly common in high-energy issues giblin ,
optical communication mollgordon  and DNA dna .

The topological structures which emerge in a $(1+1)$-dimensional scalar field
theory are known as kinks. In such a relativistic ambient, these originally
time-independent BPS solutions acquire dynamics via the implementation of a
Lorentz boost, from which the evolution of the resulting travelling profiles
must be studied via the numerical treatment of the effective equations of
motion.

The evolution of travelling kinks have been widely investigated. Among some of
the fields of research, we can highlight the study of kink scattering, which
revealed a surprisingly complex structure of bounces and annihilation. The
final result of the scattering process is intimately influenced by the value
of the initial velocity of the colliding structures, therefore giving rise to
resonance windows campbell ; anninos ; dorey . Additional discussions in the
context of most diverse models can also be found in takyi ; limacarlos ;
henoli ; weigel ; gani1 ; dio1 ; simas1 ; dio2 ; dio3 ; diojoaza ; alonso.2020
; halava.2012 ; Lima.JHEP.2019 ; ivan ; alonso.2021 ; adam.2019 ; adam.2021 ;
adam.2019_1 ; joao.2023 ; joao.2021.2 .

At the same time, the study of the so-called oscillons and spharelons has been
the subject of recent investigations. The sphaleron stands for a static,
finite and unstable solution which appears in Skyrme models and electroweak
theories manton2 . It was demonstrated in the Ref. adamromawere  that the
spharelon serves as the source for the internal mode which gives rise to the
fractal structures in a kink-antikink collision. Oscillons, on the other hand,
are spatially localized oscillatory solutions which present a long lifetime.
Their stability can not assured because due to the lack of a well-defined
topological charge gleiser1 ; gleiser2 . These solutions naturally occur from
the collapse of bubbles cope  and can be also formed in the context of a
domain wall network hind .

Interestingly, connections between spharelons and oscillons have been
discovered recently. In this sense, the Ref. manroman  has examined a
spharelon which, when slightly perturbed, decays into a large amplitude
oscillon. In addition, the Ref. olquromwere  has also investigated the
spharelon-oscillon relation, with the authors showing that the presence of the
internal mode of the spharelon is not a necessary ingredient in the dynamics
of the oscillon.

A recent research has revealed the creation of kink-antikink pairs as a result
of collisions between two identical wave trains. In that work, the authors
have considered the $\phi^{4}$ model, from which they have identified that the
resonant excitation plays an important role in the formation of the new pair,
with the process being determined by the amplitude roman_osc . Most recently,
a scenario with only one symmetric vacuum was proposed. In such a case, the
formation of a fractal structure from the decay of excited oscillons osc_weres
was verified. Here, the most intriguing aspect is that the effective model
does not support any static solutions. However, the resonant window structure,
which is based on the energy exchange between the translational and
vibrational modes, is realized. Examine also the Refs. vacha1 ; dele ; dele1 ;
carpe , which address pair production.

The aim of the present manuscript is to contribute with such a research. In
this sense, we study the production of antikink-kink pairs as a result of the
decay of an oscillon initially excited around a vacuum state in the $\phi^{4}$
theory. In order to present our developments, this work is organized as
follows: in the next Section II, we introduce the particular model and comment
on its general aspects. In the sequence, we define the initial condition (a
Gaussian-like oscillatory perturbation) which depends on two parameters, i.e.,
its amplitude and width. We are particularly interested in the effects brought
about by a perturbation of different intensities. Still in the Sec. II, we
observe that the corresponding solution for the field at the center of mass
mimics the profile inherent to a multiple-bounce window with nontrivial order,
which therefore suggests that the exchange of energy between the translational
and vibrational modes via the energy transfer mechanism begins to play a
relevant role. We then study the corresponding resonant structure and show
that it reveals a fractal pattern which surprisingly mimics that of a standard
kink-antikink scattering. Motivated by such a coincidence, we dedicate the
Section III to the reproduction of the effects caused by a Gaussian
perturbation now via a lumplike profile proposed as a kink-antikink pair, from
which we argue that such an approach holds well in the limit of small
amplitudes. Finally, the Sec. IV brings a brief summary and our perspectives
regarding future investigations.

##  II The model and its results

In this manuscript, we consider a (1+1)-dimensional theory which contains a
single real scalar field $\phi(x,t)$ only, its Lagrangian density being given
by

$\displaystyle{\mathcal{L}}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi),$
(1)

where both the field, coordinates and coupling constants were made
dimensionless via the implementation of an appropriate rescaling defined in
terms of the mass scale of the model. As usual, the Greek index $\mu$ runs
from 0 to 1 and therefore counts the spacetime coordinates. Moreover, we have
chosen to work with $\eta^{\mu\nu}=(+-)$ as the metric signature of the
Minkowski spacetime.

In such a context, the potential $V(\phi)$ determines the self-interaction
inherent to the real field via the specification of the vacuum structure of
the corresponding theory. In the present case, we consider the well-known
$\phi^{4}$ model, whose potential reads

$\displaystyle V(\phi)=\frac{1}{2}(1-\phi^{2})^{2},$ (2)

which has many applications in various physics scenarios. For instance, in
condensed matter physics, a one-dimensional Ginzburg-Landau theory has been
proposed to describe the phase transition in shape memory alloys falk .
Moreover, in Ref. grirub , the authors found that temperature has a
significant influence on the formation of kink-antikink pairs.

The resulting nonintegrable theory is known to support energetically stable
time-independent solutions which connect the two adjacent vacuum predicted by
the $\phi^{4}$ potential above. Specifically, while the resulting kink is
given by $\phi_{K}(x)=\tanh{(x)}$ and interpolates between the states
$\phi_{K}(-\infty)=-1$ and $\phi_{K}(\infty)=1$, the corresponding antikink
can be promptly obtained based on the $Z_{2}$ symmetry inherent to the
original model and therefore connects the adjacent vacuum in an opposite way,
i.e. $\phi_{\bar{K}}(-\infty)=1$ and $\phi_{\bar{K}}(\infty)=-1$. These two
profiles can be verified to have the very same energy, but topological charges
with opposite signs. Furthermore, the stability analysis reveals the existence
of two distinct modes - translational and vibrational ones - inherent to the
kink profile. In such a case, the evolution of a kink-antikink collision is
directly affected by the existence of the vibrational mode (related to
$\omega_{1}=\sqrt{3}$) via the so-called energy transfer mechanism. As a
result, two-bounce windows appear and a typical fractal structure can be
observed.

In this manuscript, instead of colliding kinks, we focus our attention on the
generation of oscillating pulses and kinks as a result of oscillons decay.
With such a purpose in mind, we consider as the initial condition

$\displaystyle\phi(x,0)=1+A_{0}e^{-x^{2}/\sigma},$ (3)

where the parameter $A_{0}$ represents the amplitude and $\sigma$ stands for
the width of a perturbation (with a Gaussian profile) around the positive
vacuum $\phi_{0}=+1$ of the $\phi^{4}$ potential. As we demonstrate below,
depending on the values of these parameters, we find more or less excited
oscillons, which eventually give rise to different final states of the field
$\phi(x,t)$. Here, it is worthwhile to point out that such an initial data is
comparable to that very recently used in the Ref. osc_weres , where the
authors have successfully studied oscillons decay in a model with a symmetric
vacuum only and whose potential can be approximated by the sine-Gordon one.

In what follows, we investigate the evolution of the field configuration in
the full space for different values of $A_{0}$ and with an initially fixed
$\sigma$. At this first moment, we are interested in small perturbations
around the vacuum state, from which we consider very small values of the
amplitude.

The Figure 1 (top) shows the evolution of $\phi(x,t)$ for $\sigma=10$ and
$A_{0}=0.05$ (left) and $0.15$ (right). In such a context, it is important to
note how the original (small) perturbation produces an excited oscillon around
the vacuum which decays and annihilates. In this sense, the field profile at
the center of mass as a function of $t$ is displayed in the Fig. 1 (bottom),
which reveals how the amplitude of the oscillations around $\phi_{0}=+1$
decreases. It is therefore reasonable to conclude that, when submitted to a
relatively small Gaussian-like perturbation around its vacuum value, the
scalar sector tends to return to its fundamental state. In other words, the
corresponding perturbation are not intense enough to remove the field from its
vacuum value and then generate a nontrivial configuration. Remarkably, unlike
what is seen in the Ref. osc_weres , the original perturbation in the present
$\phi^{4}$ scenario does not decompose into two (or more) constituent
oscillons.

Figure 1: Evolution of the scalar sector $\phi(x,t)$ in the full space (top)
and the field profile at the center of mass (bottom) for $A_{0}=0.05$ (left)
and $0.15$ (right). Here, $\sigma=10$. The oscillon annihilates and the field
returns to its fundamental state.

In the sequence, in order to explore the impact of parameter values on the
oscillon development process, we consider also the case of not so small
perturbations, from which we solve the numerical problem for additional values
of $A_{0}$. The corresponding solutions appear in the figures below.

The Figure 2 brings the results for intermediary $A_{0}$, from which it is
possible to infer the existence of a range of values for which the solutions
present a transitory behavior. In particular, for $A_{0}=0.30$ and $0.40$, the
perturbation produces a long-lived oscillon which, in view of the
nonintegrable nature of the theory, annihilates after an extremely long time
and induces the real field to finally return to its vacuum state. On the other
hand, the solution for $A_{0}=0.50$ reveals the occurrence of a perturbed
pulse (i.e. with anharmonic oscillations) whose existence might be connected
to the exchange of energy between the translational and vibrational modes
through the energy transfer mechanism.

Figure 2: Evolution of the scalar sector $\phi(x,t)$ in the full space (top)
and the field profile at the center of mass (bottom) for $A_{0}=0.30$ (left),
$0.40$ (center) and $0.50$ (right). Here, $\sigma=10$. The solutions present a
transitory behavior, with the anharmonic pulse possibly indicating the
exchange of energy between the translational and vibrational modes.

The suggestion above is corroborated by the solutions which appear in the
Figure 3. Here, for $A_{0}=0.60$, the Gaussian perturbation oscillates and,
through the emission of radiation, rapidly decays into a antikink-kink pair
with a central long-lived small oscillon around the negative vacuum
$\phi_{0}=-1$. In addition, the evolution for $A_{0}=0.70$ shows the
scattering of two almost pure antikink-kink pairs, from which we conclude that
an increasing amplitude generates an even more excited oscillon whose decay
leads to the creation of such pairs followed by the escape of the radiation.
Similar phenomenon was studied in the Ref. fab_lump , where the scattering of
two lumps was verified to produce kink-antikink pairs, while multiple
antikink-kink pairs were obtained in a double sine-Gordon model fab_DsG . An
antikink-kink pair can also be obtained via the collision of two identical
wave trains, see the Ref. roman_osc . Still regarding our solution for
$A_{0}=0.70$, we highlight that, once the time evolution begins, a centered
oscillon emerges and then intermediates a “collision” between the inner
structures whose corresponding pattern mimics the one typically related to a
four-bounce window, see the resulting profile for $\phi(0,t)$. In particular,
between the first two bounces, the scalar field performs two complete
oscillations in a clear manifestation of the energy exchange between the
translational and vibrational modes, such a behavior reinforcing the role
played by the energy transfer mechanism. Moreover, when $A_{0}=0.75$, the
perturbation produces a pair of bions and a central oscillon around
$\phi_{0}=+1$.

Figure 3: Evolution of the scalar sector $\phi(x,t)$ in the full space (top)
and the field profile at the center of mass (bottom) for $A_{0}=0.60$ (left),
$0.70$ (center) and $0.75$ (right). Here, $\sigma=10$. The Gaussian
perturbation rapidly decays into kink-antikink pairs.

The final configurations obtained for high amplitudes are shown in the Figure
4. In general, the initial condition oscillates and then generates two
antikink-kink pairs which propagate without considerable radiation escape,
while the central region is distinguished by the existence of oscillons which
travel around while emitting a certain quantity of radiation. Interestingly,
there are regions where the initial excitation produces two oscillons which
escape to the infinity, again irradiating. In particular, two antikink-kink
pairs appear for $A_{0}=0.80$ and $0.90$. However, there are now two basic
differences in comparison to the previous occurrence: long-lived oscillations
around $\phi_{0}=+1$ take place at the center of mass, and the arising
antikink-kink pairs do not bounce. Note that three non-bouncing pairs emerge
for $A_{0}=0.95$.

Figure 4: Evolution of the scalar sector $\phi(x,t)$ in the full space (top)
and the field profile at the center of mass (bottom) for $A_{0}=0.80$ (left),
$0.90$ (center) and $0.95$ (right). Here, $\sigma=10$. The initial condition
generates antikink-kink pairs which propagate without considerable radiation
escape. In particular, three non-bouncing pairs emerge for $A_{0}=0.95$.

It sounds relevant to study the resonant structure related to the evolution of
the scalar field in view of the Gaussian perturbation. In this sense, the
Figure 5 summarizes the main behaviors involving the oscillon decay, as well
as the appearance of antikink-kink pairs. Specifically, this plot shows the
temporal evolution of the field value at the center of mass for fixed
$\sigma=10$ and different values of $A_{0}$, from which we identify well-
defined segments which alternate between blue and red colors. Here, in view of
the previous figures which bring the evolution of $\phi(x,t)$ in the full
space, it is possible to understand that the blue regions represent a field
oscillation around $\phi_{0}=+1$, whilst the red portions means the emergence
of an antikink-kink pair at the origin. For instance, small values of $A_{0}$
lead to either an annihilation of the original perturbation or the generation
of a long-lived oscillon. On the other hand, when $0.5<A_{0}<0.7$, there are
stripes in red which represent a single antikink-kink pair.

Notably, we also note a fractal structure which resembles the outcome of kink-
antikink collisions. In this case, in order to recognize the details of the
resonant structure, we have included a zoom, see the plots in the center and
in the left of the Figure 5. It is important to make clear that in kink-
antikink scatterings, the resonant structure reflects the number of
interactions between the pair. In our case, however, the graphic depicts the
alternation between the emergence of an oscillation and an antikink-kink pair.

Figure 5: The resonant structure related to the evolution of the scalar field
in view of a Gaussian perturbation. Blue regions represent a field oscillation
around $\phi_{0}=+1$, whilst the red portions means the emergence of an
antikink-kink pair at the origin. The resulting fractal structure resembles
that of a kink-antikink collision. Here, we fixed $\sigma=10$.

Next, a new blue band appears as the value of the amplitude increases. This
area comprises either the formation of oscillatory pulses in the central
region or the emergence of two antikink-kink pairs which escape to infinity.
For those values of $A_{0}$ greater than $0.90$, an intriguing aspect is
revealed, i.e. this region presents multiple red peaks, as can be seen in the
Fig. 5 (left), which suggests the formation of two pairs which scatter
symmetrically and an additional pair located at $x=0$.

We end this section by exploring the effects related to an initial oscillation
with a different width. The numerical results are depicted in the Figure 6 for
$\sigma=20$ and different values of $A_{0}$. In this case, a single antikink-
kink pair is generated by the original perturbation whose amplitude now
determines the number of bounces performed by the pair. For instance, when
$A_{0}=0.55$, the initial configuration oscillates and briefly separates.
However, it lacks sufficient kinetic energy and hence the pair collides. Only
after this first collision, the process of complete separation of the pair
begins. Interestingly, we see that the pair formation occurs only after two
collisions (which corresponds to a first-order two-bounce window) for
$A_{0}=0.53$. In the last case, the solution formed when $A_{0}=0.5346$
remains trapped for a long time, i.e., it performs three complete collisions
(with a second-order three-bounce window near to a seventh-order two-bounce
one) and only then the antikink-kink pair escapes to the infinity. Again,
these results emphasize the role played by the energy transfer mechanism
during the generation of antikink-kink pairs due to an original Gaussian
perturbation.

Figure 6: Evolution of the scalar sector $\phi(x,t)$ in the full space (top)
and the field profile at the center of mass (bottom) for $A_{0}=0.55$ (left),
$0.53$ (center) and $0.5346$ (right). Here, $\sigma=20$. The amplitude
determines the number of bounces performed by the single pair, with two
(three) collisions occurring for $A_{0}=0.53$ ($A_{0}=0.5346$).

##  III The perturbation itself as a kink-antikink pair

As we have pointed out previously, the $\phi^{4}$ model is known to support
the existence of time-independent kinks with finite energy and whose
scattering process has been intensively studied, see the Refs. campbell ;
anninos . Interestingly, the temporal evolution of the real field $\phi(x,t)$
due to a Gaussian perturbation applied to its fundamental state not only
reveals a fractal pattern which is similar to that of a standard kink-antikink
collision, but it is also suggested to occur according to the energy exchange
between the translational and vibrational modes related to the aforementioned
kink profile. In view of these facts, it sounds reasonable to conjecture that
the evolution of $\phi(x,t)$ when submitted to a Gaussian perturbation can
eventually be mimicked by a lumplike profile.

The present section is devoted to the study of such a possibility. In this
sense, we first highlight that the original Gaussian perturbation can be
approximated by a static lump constructed as a kink-antikink pair. Therefore,
instead of $\phi(x,0)=1+A_{0}e^{-x^{2}/\sigma}$, we now adopt the initial
condition as

$\phi\left(x,0\right)=1+\mathcal{A}_{0}\left[\tanh\left(x+\left|x_{0}\right|\right)-\tanh\left(x-\left|x_{0}\right|\right)\right]\text{,}$
(4)

where $\mathcal{A}_{0}$ represents the magnitude of the lumplike perturbation,
while $2x_{0}$ stands for the distance between the kink and the antikink which
compose the pair (therefore representing the “width” of the corresponding
lump, whose center is assumed to be positioned at $x=0$). As we argue below,
this general form can be used to map the results which arise from the Gaussian
perturbation in terms of the properties inherent to a canonical kink-antikink
interaction.

It is useful to write down the Taylor series expansions for both the Gaussian
and the lumplike perturbations. In this case, whether we compare the relevant
terms in both series, one gets that $\mathcal{A}_{0}$ and $x_{0}$ can be
expressed in terms of $A_{0}$ and $\sigma$ as

$\mathcal{A}_{0}=\frac{A_{0}}{2\sqrt{1-\frac{1}{\sigma}}}\text{,}$ (5)

and

$\left|x_{0}\right|=\text{arctanh}\left[\sqrt{1-\frac{1}{\sigma}}\right]\text{,}$
(6)

via which we promptly rewrite the initial condition in the form

$\phi\left(x,0\right)=1+\frac{A_{0}}{2\sqrt{1-\frac{1}{\sigma}}}\left[\tanh\left(x+\text{arcsech}\left(\sigma^{-\frac{1}{2}}\right)\right)-\tanh\left(x-\text{arcsech}\left(\sigma^{-\frac{1}{2}}\right)\right)\right]\text{,}$
(7)

where we have assumed $\sigma>1$ in order to satisfy $\left|x_{0}\right|>0$.

The expression above can be used as a good approximation to describe the
results obtained in the limit of small $A_{0}$. For instance, the Figure 7
brings a comparison between the profiles for $\phi(0,t)$ obtained via the
Gaussian (blue line) and lumplike (red line) perturbations. Here, we have
fixed $\sigma=10$ and again solved the problem for $A_{0}=0.05$, $0.15$ and
$0.25$. It is possible to see that, despite some differences (of order
$10^{-2}$ and which decrease with time) in the corresponding amplitudes, the
solutions behave in the very same qualitative way. In particular, they suggest
that the central long-lived pulse which emerges for small values of the
original amplitude can be eventually treated as a kink-antikink pair which
perform several collisions without getting enough kinetic energy to escape
from each other. Due to the nonintegrable nature of the overall scenario, the
colliding pair continuously emits radiation and finally disappears, forcing
the field to return to its fundamental state, therefore mimicking the very
same [qualitative] picture which arises from a Gaussian perturbation.

Figure 7: Field profile at the center of mass for $A_{0}=0.05$ (left),
$A_{0}=0.15$ (center) and $A_{0}=0.25$ (right). Here, we have fixed
$\sigma=10$. The results obtained via the Gaussian (lumplike) perturbation
appears as the blue (red) line. For small $A_{0}$, the solutions behave in the
very same qualitative way.

As the value of $A_{0}$ increases, the resulting solutions assume different
profiles (both in magnitude and frequency). As a consequence, the
aforementioned correspondence loses its accuracy and can not be applied
anymore, see the profiles for $A_{0}=0.25$, for instance.

It is always possible to improve the approximation above by fitting the
Gaussian and the lumplike profiles via a rescaling applied on $x$ in the
expression for the last one. As the interested reader can verify, such a
procedure gives rise to even better results (in comparison to the ones
displayed in the previous Figure 7). However, the corresponding accuracy is
still limited by the values of $A_{0}$.

Here, in order to improve the accuracy of the Taylor series based approach, we
adopt an alternative way, i.e. we assume small values of the width $\sigma$.
In this case, the Gaussian profile is even more localized around $x=0$, from
which the series expansion approximation consequently applies in a even better
way. The results for $\sigma=2$ and $A_{0}=0.30$, $0.60$ and $0.90$ appear in
the Figure 8, from which it is possible to see how accurate [both
qualitatively and quantitatively] the results provided by the lumplike
perturbation are. It is however important to reinforce that the corresponding
final configurations dramatically depend on the values of $\sigma$ itself, see
the results displayed in the Figure 6 for $\sigma=20$, for instance. In this
sense, we clarify that, when $\sigma=2$, the final structure solution is a
short-lived (long-lived) centered pulse for $A_{0}=0.30$ ($A_{0}=0.60$), while
it represents the scattering of a single antikink-kink pair (with a central
pulse around the negative vacuum) for $A_{0}=0.90$ (i.e. multiple antikink-
kink pairs were not observed).

Figure 8: Field profile at the center of mass for $A_{0}=0.30$ (left),
$A_{0}=0.50$ (center) and $A_{0}=0.90$ (right). Here, we have fixed
$\sigma=2$. The Gaussian profile is even more localized around the origin,
from which the series expansion provides better results.

In all these cases, the original Gaussian perturbation was approached via a
lumplike profile which was proposed as a static kink-antikink pair. The
results obtained via the Gaussian configuration were then mimicked via those
related to a kink-antikink interaction. We believe that such an approach
contributes to the comprehension on the role played by the energy transfer
mechanism in the formation of the final structures. In addition, the lumplike
perturbation approach is expected to provide some insight about the occurrence
of a fractal pattern which resembles that of a kink-antikink collision.

##  IV Summary and perspectives

In this manuscript, we have studied some of the results caused by a
perturbation applied to a fundamental state of a classical field.

With this aim in mind, we have considered a model with a single real scalar
field $\phi(x,t)$ only whose self-interaction potential $V(\phi)$ has the
usual $\phi^{4}$ profile. The resulting nonintegrable scenario is known to
support energetically stable time-independent solutions which connect the two
adjacent minima of the potential. In addition, these solutions can be verified
to have the very same energy, but topological charges with opposite signs.
Moreover, the stability analysis reveals that such a kink profile admits two
well-defined internal modes - translational and vibrational - which can be
applied to explain the temporal evolution of an scattering in such a classical
ambient via the so-called energy transfer mechanism.

Motivated by the prominent developments recently presented in the Ref.
osc_weres , we have here implemented a Gaussian-like perturbation around the
vacuum state $\phi_{0}=+1$. The perturbation is controlled by two distinct
parameters, namely, $A_{0}$ and $\sigma$, which rotulate its amplitude and
width, respectively.

We have then focused our attention on how different values of $A_{0}$ give
rise to distinct configurations of the real sector $\phi(x,t)$, from which we
have considered some $\sigma$ values in our numerical investigations.

Initially, we have investigated the impact of small amplitudes. These
solutions have revealed that the perturbation produces an excited oscillon
which decays and annihilates. We have also explored the scenario of
intermediate values of $A_{0}$, where the dynamics has indicated the creation
of long-lived oscillons. As the amplitude increases, we have observed the
formation of antikink-kink pairs followed by the eventual emission of
radiation. In particular, we have identified the appearance of a centered
short-lived oscillon which intermediates the interaction between the inner
structures which compose the emergent pairs. Curiously, we have noted that the
corresponding solution for $\phi(0,t)$ mimics the pattern inherent to a four-
bounce window, with the scalar field performing two complete oscillations
between the first two bounces, from which we have argued that such a behavior
reinforces the role played by the energy transfer mechanism. For large
amplitudes, the initial condition oscillates and then generates two pairs
which propagate in opposite directions, followed by an oscillon at the center
of mass. Additionally, we have found that a new antikink-kink pair was formed.

Motivated by these results, we have obtained the resonant structure inherent
to the evolution of the scalar field in view of an initial Gaussian
perturbation, see the Fig. 5. We have identified field oscillations around
$\phi_{0}=+1$ (blue regions) and the emergence of an antikink-kink pair
positioned at $x=0$ (red portions). Notably, we have noted that the resulting
fractal structure mimics that of a kink-antikink scattering. Moreover, we have
clarified that, while the resonant structure related to a standard kink-
antikink collision reflects the number of interactions between the pair, the
structure which we have obtained in the present manuscript depicts the
alternation between the emergence of an oscillatory pulse and an antikink-kink
pair.

We have also investigated the impact of an initial oscillation with a
different width, from which we have fixed $\sigma=20$. In these circumstances,
we have observed that a single antikink-kink pair is generated by the
perturbation, with the amplitude now determining the number of bounces
performed by the pair before its escape to the infinity. We have pointed out
that these multiple bounces can be rotulated as usual, i.e., based on the
order of the window they belong. As a result, we have highlighted that the
existence of these windows with nontrivial order can be understood as a clear
consequence of the energy exchange between the translational and vibrational
modes.

The coincidence between the fractal patterns has inspired us to study the
reproduction of the effects caused by the Gaussian perturbation via a lumplike
profile constructed as a kink-antikink pair. We have connected these two
scenarios via the comparison between the respective Taylor series expansions,
from which we have written the parameters which characterize the lumplike
perturbation in terms of the $A_{0}$ and $\sigma$. We have then depicted the
corresponding results, which have revealed that such a Taylor series based
approach holds well for small values of $A_{0}$, its accuracy decreasing as
the values of $A_{0}$ increase. In particular, when $\sigma$ is small, the
results practically coincide with each other, from what we have pointed out
that such a successful mapping may contribute to the understanding not only of
the coincidence between the fractal pattern, but also of the fundamental role
played by the energy transfer mechanism in the definition of the final
configurations due to a Gaussian perturbation.

Perspectives regarding future investigations include the study of the results
related to an initial perturbation around the fundamental states inherent to
both the $\phi^{6}$ and the sine-Gordon potentials. In this context, the last
one sounds particularly interesting due to the integrable nature of the
corresponding ambient, from which emission of radiation is not expected to
occur. Also, we plan to depict the resonant structures of the resulting
configurations, from which we intend to verify whether the corresponding
fractal patterns can be eventually compared to the well-established ones or,
on the other hand, they reveal a novel profile to be explored. These issues
are currently under investigation, and positive results will be eventually
reported in a future contribution.

## Acknowledgements

The authors thank João Campos and Azadeh Mohammadi for their useful comments
on the present manuscript. EH also thanks the partial financial support
received from the Conselho Nacional de Desenvolvimento Científico e
Tecnológico - CNPq (Brazillian agency) via the Grant N° 309604/2020-6. This
study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001.

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