# Dance Style Transfer with Cross-modal Transformer

Wenjie Yin  KTH Royal Institute of Technology, Stockholm, Sweden  Hang Yin
KTH Royal Institute of Technology, Stockholm, Sweden  Kim Baraka  Danica
Kragic  KTH Royal Institute of Technology, Stockholm, Sweden  Mårten Björkman
KTH Royal Institute of Technology, Stockholm, Sweden

###### Abstract

We present CycleDance, a dance style transfer system to transform an existing
motion clip in one dance style to a motion clip in another dance style while
attempting to preserve motion context of the dance. Our method extends an
existing CycleGAN architecture for modeling audio sequences and integrates
multimodal transformer encoders to account for music context. We adopt
sequence length-based curriculum learning to stabilize training. Our approach
captures rich and long-term intra-relations between motion frames, which is a
common challenge in motion transfer and synthesis work. We further introduce
new metrics for gauging transfer strength and content preservation in the
context of dance movements. We perform an extensive ablation study as well as
a human study including 30 participants with 5 or more years of dance
experience. The results demonstrate that CycleDance generates realistic
movements with the target style, significantly outperforming the baseline
CycleGAN on naturalness, transfer strength, and content preservation. 111Demo
at https://youtu.be/kP4DBp8OUCk.

##  1 Introduction

Style transfer methods facilitate art creation of a target style for media
such as images [16] and music [4]. Similar methods are promising for creators
to use an existing dance sequence as a starting point to generate variations
across different movement styles. In a video game context, these style
variations may -e.g. be associated to different characters with different
attributes or personalities. In a choreographic context, such a tool may lead
to hybrid human-artificial creative processes, where style transfers are used
to iterate on interesting, unexpected, or complementary variations of an
initial choreographic material.

Existing research on transferring human movement styles largely focuses on
simple locomotive or exercise motions [30, 12, 1] and domain transfers between
adults and children. Technical methods for transferring such sequential data
include cycle-consistent adversarial networks (CycleGAN) [46] and adaptive
instance normalization (AdaIN) [24]. However, a research gap remains for
applying similar techniques to enable style transfer of more complex
movements, such as dance movements. Dance movements usually have no explicit
functional purpose and tend to exhibit considerable richness in posture,
rhythm and their composition. Generation of dance movements can be
particularly challenging since it demands a multi-layer approach that captures
motion qualities such as the coordination of joint dynamics and socio-cultural
factors associated with the production and perception of the movement.
Meanwhile, there exist a variety of such characteristics within different
dance styles, originating from different historical backgrounds. Dance styles
could be thought more generally of as styles of performing certain dance
movements rather than strictly dance genres. This adds another layer of
complexity to the generation of high-quality dance movements of a specific
target style. All these challenges call for computational models that can
capture both high-frequency features and long-term dependencies over time, and
as such generate realistic dance with aesthetic and coherence.

Moreover, dance is commonly accompanied by music which can provide tremendous
clues for understanding and composing movement. Recent works have shown the
effectiveness of music-conditioned dance synthesis [39, 6], which can directly
generate dance motion given music context. However, it is unclear whether
music context will also facilitate style transfer tasks and how such a multi-
modal input should be processed in this context.

In this paper, we propose CycleDance, a multimodal system (see Figure
LABEL:fig:teaser) for dance style transfer. CycleDance adopts a generative
scheme by extending CycleGAN-VC2 [23] to work with unpaired data. To tackle
the challenges identified above, we exploit a cross-modal transformer
architecture [39] that aims to effectively capture relevant features among
different modalities so as to enhance style transfer quality. Specifically, we
design a two pathways transformer-based architecture to extract temporally
aligned motion and music representations in the context of style transfer. We
further propose to train CycleDance progressively with a curriculum learning
scheme inspired by Fu et al. [14]. This alleviates issues of instability in
training large adversarial models and premature convergence that can lead to
inferior performance. We evaluate our framework on the AIST++ [37] dance
database, with the analysis focused on transfer between various dance genres.
Two new metrics based on probabilistic divergence and selected key pose frames
are proposed to quantitatively assess the quality of dance style transfer.
Moreoever, we invite a group of human participants with extensive dancing
experience to provide a subjective evaluation and insights from an expert
perspective. These evaluations show that CycleDance greatly outperforms a
baseline method and its ablative versions. As an illustration, a video with
generated examples can be found at  https://youtu.be/kP4DBp8OUCk.

In summary, our contributions are mainly as follows:

  * •

Our approach is, to the best of our knowledge, the first to combine complex
dance motion and music context in the style transfer task, unlocking potential
applications in choreography, gaming, and animation, as well as in tool
development for artistic and scientific innovations in the field of dance.

  * •

We introduce new metrics based on probabilistic divergence and selected key
pose frames for gauging transfer strength and content preservation in the
context of dance movements.

  * •

We provide an extensive user study of the proposed model. The evaluations and
insights from a group of experienced dance performers reveal essential aspects
of designing such systems.

##  2 Related Work

In this section, we first provide an overview of prior works on general style
transfer in Section 2.1 and focus on motion style transfer in Section 2.2. As
another relevant topic, motion synthesis from multi-modal data will be briefly
reviewed in Section 2.3.

###  2.1 Style Transfer

In recent years, style transfer has achieved impressive progress in computer
vision, speech processing, music processing, natural language processing,
motion animation, etc. In computer vision, the pioneering work of Gatys et al.
[16] introduces the concept of style transfer and uses convolutional neural
networks (CNNs) to merge the style and content between arbitrary images. Huang
et al. [21] propose an adaptive instance normalization (AdaIN) layer to
directly apply arbitrary target styles to an image. Zhu et al. [46] propose
CycleGAN, using a pair of generators and discriminators to translate image
style. The general idea of CycleGAN has been further developed and improved in
StarGAN [7], with domain labels as additional input, so that image styles can
be transformed to multiple corresponding domains, such as facial appearances
and expressions.

In research on voice conversion (VC), Kaneko and Kameoka [22] develop
CycleGAN-VC based on CycleGAN, but with gated CNNs and an identity-mapping
loss. This is further improved by CycleGAN-VC2 [23] which adopts two-step
adversarial losses, a 2-1-2D convolution structure, and PatchGAN. Fu et al.
[14] further incorporate transformers and curriculum learning in voice
conversion. Research has also been conducted to transfer symbolic music
styles, with examples such as Groove2Groove [8], which employs an encoder-
decoder structure and parallel data, and [4] for MIDI music with a CycleGAN-
based approach.

For style transfer in natural language processing (NLP), Mueller et al. [31]
propose recurrent variational auto-encoders (VAE) to revise text sequences. Fu
et al. [15] construct a multi-decoder and a style-embedding model to learn
independent content and style representations with adversarial networks. Dai
et al. [9] propose a Style Transformer network with a special training scheme,
which employs an attention mechanism and makes no assumption about the latent
representation.

Our work focuses on transferring motion data, in particular, dance movements.
We adopt CycleGAN-VC2, previously used for voice conversion, as the basis of
our framework and augment training with an extra music modality.

###  2.2 Motion Style Transfer

Early works on motion style transfer rely on hand-crafted features [2, 38, 42,
3, 20], while most modern studies advocate learning by extracting features
from data [19, 12, 18, 35, 27, 29, 33, 11]. Typical models in use include
convolutional auto-encoders [19], CycleGAN [11], temporal invariant AdaIN
layers [1], autoregressive flows [41] and spatial-temporal graph neural
networks [33]. Some research also concerns efficient generation for real-time
style transfer [43, 35, 29]. All these works target relatively simple human
movements, such as locomotion and exercise, for which the variation in style
is often limited e.g. transferring between adult and children locomotion [11].

Our work handles transfer of dance movements that exhibit substantial richness
in terms of postures, rhythms, transitions and artistic styles and as such may
be of greater empirical value for e.g. video game or film industries. To
handle such complexities, our method significantly differs from the reviewed
work, with transformer and curriculum learning leveraged on top of CycleGAN-
VC2 for more effective training on more complex movement data.

###  2.3 Music-conditioned Motion Synthesis

A plethora of research works have focused on human motion synthesis [5, 44,
17, 45, 28]. Since dance is often combined with music, cross-modal motion
generation, an emerging research topic that explores the association between
different modalities, is often explored for better understanding of human
motion and music-conditioned motion synthesis. Most early works focus on
statistical models [34, 13, 25] and typically generate motions by selection.
To be specific, this means synthesizing motion by selecting the motion
segments in a database whose features (such as rhythm, structure, and
intensity) match each music segment. With the development of deep learning,
learning-based methods have also been explored.

For example, in ChoreoMaster [6], an embedding module is designed to capture
music-dance connections. Sun et al. [36] propose DeepDance, a cross-modal
association system, which correlates dance motion with music, and Lee et al.
[47] a decomposition-to-composition framework that leverages MM-GAN for music
based dance unit organization. In DanceNet [47], a musical context aware
encoder fuses music and motion features, while in DanceFormer [26] kinematics
enhanced transformer guided networks are used for motion curve regression.
More recently, cross-modal transformers have been successfully applied to
model the distribution between music and motion [39].

Music-conditioned dance synthesis aims to generate dance motion sequences
associated to a given music context. Our work explores the dance style
transfer task, focusing on manipulating the style of the existing dance
movements while keeping the contextual information. The music modality is not
mandatory for our style transfer model to work, but can be incorporated to
benefit the generation quality when the data is available.

##  3 Methodology

This section formulates our target problem and establishes notations used
throughout the paper. Preliminaries about CycleGAN and CycleGAN-VC2 are also
given for self-containment. On the basis of these, we present the contributed
technical framework CycleDance.

Figure 1: The CycleDance architecture. In the generator, there is a motion
pathway and a music pathway. Each pathway starts with downsampling blocks,
followed by a 2D-1D block. The motion, music and cross-modal transformer
blocks are standard full-attention transformer encoders. Then the fused path
is followed by a 1D-2D block and upsampling blocks. In the discriminator, like
in Kaneko et al. [23], convolution at used in the last layer.

###  3.1 Problem Formulation

Figure 2: The two step adversarial generative training strategy. The full
objective includes four types of losses; adversarial loss $\mathcal{L}_{adv}$,
cycle-consistency loss $\mathcal{L}_{cyc}$, identity-mapping loss
$\mathcal{L}_{id}$, and second adversarial loss $\mathcal{L}_{adv2}$. See
Section 3.2 for the definition of notations.

Our goal is to learn mapping functions between two domains $X$ and $Y$ without
relying on paired data between these domains. In our scenario, we transfer
dance between two style domains $X$ and $Y$ given dance sample $x\sim P_{X}$
and $y\sim P_{Y}$. The dance samples may be paired with music $m_{x}\in M_{x}$
and $m_{y}\in M_{y}$ with associated styles, although the music modality is
only optional in the transfer task.

###  3.2 Adversarial Training Loss and Strategy

We address the formulated problem with a CycleGAN-like architecture [46], as
illustrated in Figure 2. The architecture includes two discriminators $D_{X}$
and $D_{Y}$ which are used to distinguish the real and generated data, as well
as two mappings $G_{X\rightarrow Y}$ and $G_{Y\rightarrow X}$ for generating
patterns of the target style. The mappings are also cycled such that the
generated patterns can be converted back to the original domains. To this end,
we follow CycleGAN-VC2 [23] and incorporate four types of losses, also see
Figure 2.

Adversarial loss $\mathcal{L}_{adv}^{X\rightarrow Y}$: this loss measures the
discrepancy between the transferred data $G_{X\rightarrow Y}(x,m_{x})$ and the
target $y$, with the discriminator $D_{Y}$ attempts to distinguish the
transferred data from real data:

$\displaystyle\mathcal{L}_{adv}^{X\rightarrow Y}$
$\displaystyle=\mathbb{E}_{y\sim P_{Y}}[\log D_{Y}(y)]$ (1)
$\displaystyle+\mathbb{E}_{x\sim P_{X}}[\log(1-D_{Y}(G_{X\rightarrow
Y}(x,m_{x})))].$

Correspondingly, the adversarial loss $\mathcal{L}_{adv}^{Y\rightarrow X}$ can
be defined for $G_{Y\rightarrow X}$ and discriminator $D_{X}$.

Cycle-consistency loss $\mathcal{L}_{cyc}$: this accounts for the loss of
contextual information by recovering the original $x$ and $y$ from generated
patterns through $G_{X\rightarrow Y}(x,m_{x})$ and $G_{Y\rightarrow
X}(y,m_{y})$:

$\displaystyle\mathcal{L}_{cyc}$ $\displaystyle=\mathbb{E}_{x\sim
P_{X}}[\left\|G_{Y\rightarrow X}(G_{X\rightarrow Y}(x,m_{x}))-x\right\|_{1}]$
(2) $\displaystyle+\mathbb{E}_{y\sim P_{Y}}[\left\|G_{X\rightarrow
Y}(G_{Y\rightarrow X}(y,m_{y}))-y\right\|_{1}].$

Identity-mapping loss $\mathcal{L}_{id}$: this further encourages input
preservation by enforcing an identity transformation when $G_{X\rightarrow Y}$
and $G_{Y\rightarrow X}$ are applied to the other domain:

$\displaystyle\mathcal{L}_{id}$ $\displaystyle=\mathbb{E}_{x\sim
P_{X}}[\left\|G_{Y\rightarrow X}(x,m_{x})-x\right\|_{1}]$ (3)
$\displaystyle+\mathbb{E}_{y\sim P_{Y}}[\left\|G_{X\rightarrow
Y}(y,m_{y})-y\right\|_{1}]$

Two-step adversarial loss $\mathcal{L}_{adv2}$: this is a second adversarial
loss to alleviate the over-smoothing reconstruction statistics in the cycle-
consistency loss [23]. Note this introduces an additional discriminator
$D_{X}^{{}^{\prime}}$ and $\mathcal{L}_{adv2}^{Y\rightarrow X\rightarrow Y}$
can be defined in a similar way:

$\displaystyle\mathcal{L}_{adv2}^{X\rightarrow Y\rightarrow
X}=\mathbb{E}_{x\sim P_{X}}[\log D_{X}^{{}^{\prime}}(x)]$ (4)
$\displaystyle+\mathbb{E}_{x\sim
P_{X}}[\log(1-D_{X}^{{}^{\prime}}(G_{Y\rightarrow X}(G_{X\rightarrow
Y}(x,m_{x}))))]$

The overall objective is finally written as a weighted sum of the above terms

$\displaystyle\mathcal{L}_{full}$
$\displaystyle=\mathcal{L}_{adv}^{X\rightarrow
Y}+\mathcal{L}_{adv}^{Y\rightarrow
X}+\lambda_{cyc}\mathcal{L}_{cyc}+\lambda_{id}\mathcal{L}_{id}$ (5)
$\displaystyle+\mathcal{L}_{adv2}^{X\rightarrow Y\rightarrow
X}+\mathcal{L}_{adv2}^{Y\rightarrow X\rightarrow Y},$

where $\lambda_{cyc}$ and $\lambda_{id}$ trade off the consistency and
identity loss terms.

Besides, we adopt a curriculum learning algorithm as the training scheme. The
intuition is that the training can be more effective by starting with simpler
data and progressively handle more complex data. Such a strategy has been
applied to various applications and scenarios, showing an ability to improve
the convergence rate and generalization capacity, and providing better
training stability [40]. We adopt a length-based curriculum learning strategy
by training data truncation, similar to [14]. The length of input sequences is
increased gradually to allow the model to learn from short samples to long
samples.

###  3.3 Network Architecture

Our CycleDance framework adopts CycleGAN-VC2 as the backbone and extends it
with a cross-modal transformer, as depicted in Figure 1. The cross-modal
transformer concatenates two pathways of motion and music encodings, both of
which are obtained through a sequence of layers including 2D convolution
(purple blocks in Figure 1), 2D-1D reshaping (red), residual convolution
(green) and modality-specific transformers (yellow). The 2D CNN layers are
used to perform downsampling while preserve the original sequential structure.
The downsampled features are reshaped and pass through the residual blocks of
1D CNNs. The reshaped 1D sequences are processed by transformers which adopt a
position embedding and output encodings capturing temporal relations among
timesteps. In these blocks, we adopt gated linear units (GLUs) [10] as a
tunable activation function to learn a sequential and hierarchical structure.

For the discriminator, CycleDance also first downsamples motion data with a 2D
CNN. We only use convolution at the last layer to alleviate training
instability, as is suggested in [23]. The output layer uses sigmoid activation
to predict whether the motion clip is real or generated.

##  4 Experiments and Evaluations

In this section, we first describe the used dataset (Section 4.1), how they
are processed and the concrete experimental setup. We then detail our
assessments, including both objective (Section 4.3) and subjective (Section
4.4) evaluations, and report their results on benchmarking different dance
style transfer methods and ablations.

###  4.1 Dataset

We generate 3D dance motion samples with paired music from an existing
database called AIST++ Dance Database [27]. AIST++ reconstructs 3D motions
from multi-view videos in the AIST Dance Database in terms of SMPL parameters
[37]. To obtain motion features, we downsample all the motion data to 30
frames per second (fps) and retarget the motion to a skeleton with 21 body
joints using Autodesk MotionBuilder. Similarly to [39], we use exponential map
parametrization of the 3D rotation to represent all the joints (non-root). The
root joint (hip) has four extra features representing the vertical root
position, the ground-projected position changes, and the 2D facing angle
changes. In total, each motion frame of dance sequences is represented by a
63-dim vector. The music features are extracted with the Librosa toolbox in a
similar way to [37]. We combine 20-dim MFCC, 12-dim chroma, 1-dim one-hot
peaks, and 1-dim one-hot beats, resulting in a total 35-dim audio feature.

The selected dance styles are ballet-jazz, locking, waacking, hip-hop, pop and
house dance. As the example of data statistics, the ballet-jazz set and the
locking dance set both contain 141 motion sequences with 6 songs, lasting
1910.8 seconds and 1898.5 seconds respectively.

###  4.2 Baseline Models and Ablations

We implement the proposed CycleDance model and the CycleGAN-VC2 baseline with
PyTorch and train both models on the preprocessed dataset. In order to assess
the significance of contributed design choices, such as the cross-modal
transformer and curriculum learning strategy, three alternative architectures
are also implemented for an ablation study. The first ablated configuration is
CycleTransGAN, which removes the music pathway and the cross-modal transformer
and disables the curriculum learning strategy. We expect to use this
comparison to highlight the utility of the introduced transformer
architecture. The second ablation, CycleTransGAN+CL, applies curriculum
learning to CycleTransGAN. We aim to assess the performance gains by
meticulously modulating the complexities of samples that the model is exposed
to during training. The final ablation, CycleCrossTransGAN, also uses cross-
modal transformers for motion and music information as the encoder. Curriculum
learning is, however, not adopted in this configuration. We aim to see the
impact of having cross-modal transformers by analyzing the differences between
CycleTransGAN and CycleCrossTransGAN.

###  4.3 Objective Evaluation

Table 1: Quantitative objective evaluation: Motion Fréchet distance (MFD) and
pose Fréchet distance (PFD) for the five competing models, includes the
baseline model, our proposed CycleDance, as well as the ablations. BJ2LC
denotes transferring from ballet-jazz to locking dance. Correspondingly, LC2BJ
denotes transferring from locking dance to ballet-jazz. Similarly, WK, HP, PO,
HO denotes waacking, hip-hop, pop, and house dance.

Method | MFD | PFD
---|---|---
BJ2LC | LC2BJ | WK2HP | HP2WK | PO2HO | HO2PO | BJ2LC | LC2BJ | WK2HP | HP2WK | PO2HO | HO2PO
CycleGAN-VC2 | 9.9430 | 3.4063 | 1.4354 | 1.2645 | 2.2841 | 1.9515 | 0.4897 | 0.3499 | 0.4847 | 0.3313 | 0.5212 | 0.5625
CycleTransGAN | 3.5643 | 0.7886 | 1.0564 | 0.9464 | 1.5515 | 1.5354 | 0.4749 | 0.2501 | 0.4754 | 0.2834 | 0.4048 | 0.5215
CycleTransGAN+CL | 2.9188 | 0.5848 | 1.0847 | 0.9847 | 1.4852 | 1.5521 | 0.4897 | 0.2543 | 0.4644 | 0.2882 | 0.4125 | 0.4185
CycleCrossTransGAN | 2.7446 | 0.5819 | 0.9872 | 1.0782 | 1.4254 | 1.5251 | 0.4419 | 0.2244 | 0.4490 | 0.2880 | 0.3841 | 0.4126
CycleDance | 2.6109 | 0.5755 | 0.8752 | 0.9501 | 1.3452 | 1.4855 | 0.4216 | 0.2230 | 0.4485 | 0.2960 | 0.3954 | 0.3827

The main task of all these models is to transfer dance style from a source to
a target dance style. To allow for thorough quality assessment of complex
motion patterns, common in dance, we perform evaluations from both objective
and subjective perspectives. In the objective evaluation, we use 17 dance
sequences per style. We transferred the style for each ablated model and
evaluated two metrics that capture how well the style is transferred and how
well the content is preserved. To this end, we design metrics based on the
Fréchet distance, similar to [39].

Transfer strength. The most important aspect of style transfer is transfer
strength, which measures the degree of conversion from the source style to the
target style. To assess the transfer strength, for one dance style, we use the
Fréchet distance between the true dance motion and the generated dance motion.
Specifically, we use two consecutive raw poses $(x_{i-1},x_{i})$ to convert
the representations of both true and generated motions to joint velocity
$v_{i}$, without normalization. Similarly, we use three consecutive poses
$(x_{i-1},x_{i},x_{i+1})$ to calculate the joint acceleration $a_{i}$. We call
this measure the motion Fréchet distance (MFD) and use it to measure how close
the generated motion is to the true motion of a target style.

Content preservation. Another indispensable evaluation metric for style
transfer is content preservation. For this dimension, for one dance movements,
we use the Fréchet distance between distributions of key poses $x_{k}$.
Keyframes containing such poses are extracted by detecting local maxima in
joint acceleration. To make frames comparable, skeleton poses in keyframes are
normalized to a hip-centric origin. We call this measure the pose Fréchet
distance (PFD) and evaluate to what extent these salient poses are kept after
the transfer.

Table 1 summarises the quantitative results of the proposed model and
ablations. We transfer the dance style between three pairs of dance genres in
both directions, including ’ballet-jazz to locking dance’ (BJ2LC) and ’locking
dance to ballet-jazz’ (LC2BJ), ’waacking to hip-hop dance’ (WK2HP) and ’hip-
hop to waacking dance’ (HP2WK), as well as ’pop to house dance’ (PO2HO) and
’house to pop dance’ (HO2PO).

Figure 3: Example locking dance sequences (top, blue y-bot) transferred to
ballet-jazz dance by CycleGAN-VC2 (mid, red x-bot) and CycleDance (bottom, red
x-bot).  Figure 4: Example ballet-jazz dance sequences (top, red x-bot)
transferred to locking dance by CycleGAN-VC2 (mid, blue y-bot) and CycleDance
(bottom, blue y-bot).

We observe that the baseline model CycleGAN-VC2 struggles in this style
transfer task, evident from the much higher MFD for the baseline model than
for all other ablation methods. The complete framework, CycleDance, achieves
the best performance on both metrics and almost all transfer pairs. This
highlights the necessity of all introduced techniques in this task.

An example of synthesized motion clip is presented in Figure 3, which
illustrates dance style transfer from locking to ballet-jazz dance. The top
keyframe sequence shows the original locking dance. The sequence in the middle
is generated by CycleGAN-VC2 and the bottom one by the proposed CycleDance.
Another example in Figure 4 shows dance style transfer from ballet-jazz to
locking dance. By comparing the poses of each column, it can be observed that
the extracted key gestures are representative to the pose sequences.
CycleDance has a higher similarity to the source gestures and can thus
preserve more content while having better alignment to the target dance style.

In addition, through the ablation study, we observe that CycleTransGAN
(CycleGAN-VC2 and transformer combined) achieves lower MFD, which can be seen
as, with the help of the transformer, the model benefits from capturing richer
intra-relations among frames. By comparing CycleTransGAN and
CycleCrosTransGAN, both MFD and PFD are improved. We take this as evidence
that the music information facilitates accurate generation of the target style
and that this context information is successfully encoded by the cross-modal
transformer. The comparison between of CycleTransGAN and CycleTransGAN+CL
reveals that curriculum learning greatly improves transfer strength, showing
the effectiveness of gradually increasing the difficulty by training with
longer clips.

###  4.4 Subjective Evaluation

In addition to the objective evaluation, we conducted a user study to evaluate
our model and the baseline by scoring three aspects: motion naturalness,
transfer strength, and content preservation. We also ask some open-ended
questions to gather opinions that may not be covered by the above aspects, to
provide suggestions for future work.

Our analysis mainly focuses on ballet-jazz and locking dance, since the
characterics of these are well understood by dance professionals. The user
study was performed through an online survey covering transfer tasks for both
’locking dance to ballet-jazz’ and ’ballet-jazz to locking dance’. We used
Blender to render 8-second video clips with an x-bot character (for ballet-
jazz) and a y-bot character (for locking dance) for each source and target
dance sequence. The participants could play video clips and get acquainted
with the animated dance in an introduction phase. In the actual survey, the
participants were asked to watch a source dance video clip and a corresponding
generated target dance clip. The target dance video clip was generated either
from CycleDance or from the baseline, and the order of target dance clips was
randomly selected and balanced to relieve potential order effects. The
participants could repeatedly play the clips before answering three questions:

  * •

Motion naturalness:  To what extent do you agree with the following statement?
— The generated motion clip looks natural after style transfer. (Likert item
ranging from 1 (strongly disagree) to 5 (strongly agree)).

  * •

Transfer strength:  To what extent do you agree with the following statement?
— The generated motion clip looks like the target dance style.  (Likert item
ranging from 1 (strongly disagree) to 5 (strongly agree)).

  * •

Content preservation: Which feature(s) do you think is (are) the most
preserved between the original and the result video? — Orientation through
space; — Shapes of the limbs; — Shape of the body trunk; — Rhythmic patterns —
Other: ___. (One or more of these four aspects could be selected). This list
was based on the most salient features that dance analysts look at when
analyzing expressive movement [32].

In the study, 30 participants with at least 5 (cumulative) years of dance
experience (including training, performing, choreographing, or teaching) were
recruited. Participants were between 20 and 41 years of age (median 30), 37.9%
male, 58.6% female, and 3.4% others. According to the demographic questions,
the participants’ familiarity with the ballet-jazz dance and locking dance
were M=3.93 (SD=1.05) and M=3.03 (SD=1.18) respectively, where 1 meant not at
all and 5 meant very familiar. Since the generated motions were shown using
virtual characters, we also counted the frequency at which participants played
video games, which were 34.5% weekly, 13.8% monthly, 13.8% yearly, and 37.9%
rarely.

Figure 5: Subjective evaluation results on motion naturalness and transfer
strength. Error bars represent standard errors of the averages. Statistical
significance is the result of the Wilcoxon signed-rank test that compares the
medians ($***$ means $p<0.0001$). Figure 6: Subjective evaluation results on
content preservation. The CycleDance outperforms the baseline model on
orientation through space, shape of the limbs, shape of the body trunk, and
rhythmic patterns. Statistical significance represents the results of the
Wilcoxon signed-rank test that compares the medians ($***$ means $p<0.00001$,
$**$ means $p<0.0001$, and $n.s.$ means $p>0.05$). Figure 7: Example ballet-
jazz dance (top, red x-bot) transferred to locking dance (bottom, blue y-bot)
by CycleDance.

We analyzed the subjective responses to provide statistical support for the
results of the user study, and assessed whether the proposed method could be
further improved. Figure 5 demonstrates the responses of the motion
naturalness and transfer strength. On both aspects, the experts rated
CycleDance higher on average compared to the baseline model. The subjective
responses were compared through a Wilcoxon signed-rank test statistical
significance. Both the median value of motion naturalness
$(Z=-9.2262,p<0.0001)$ and transfer strength $(Z=-8.7677,p<0.0001)$ were
significantly higher for CycleDance compared to the baseline model. Thus from
the view of dance experts, CycleDance is favoured for improved naturalness as
well as similarity to the target dance style, consistent with what we observe
from the objective quantitative results (Section 4.3). As for the responses of
the content preservation, Figure 6 summarizes the total statistics on the four
aspects queried. On all four aspects, the experts chose CycleDance more often
than the baseline model, when asked which specific features they believe are
preserved. We ran a McNemar test for assessing the statistical significance of
these gaps. The test revealed no significant statistical differences between
the Baseline model and CycleDance on ’Orientation through space’ $(p=0.1724)$
and ’Shapes of the limbs’ $(p=0.1573)$. In terms of median value of the
’Shapes of the body trunk’ $(p=0.000002)$ and ’Rhythmic patterns’
$(p=0.00004)$, the McNemar test showed a strong significance in support of the
proposed CycleDance model. Among the four aspects, both CycleDance and the
baseline received higher scores on rhythmic patterns and orientation through
space. This implies that it is comparatively easier to keep dance orientation
and rhythm while performing dance style transfer. CycleDance outperforms the
baseline on preserving the shape of the body trunk. Preserving the shape of
the limbs, on the other hand, appears to be more challenging.

Responding to open-ended questions, the dance experts commented that for
’ballet-jazz to locking dance’, both methods have a jerky style that emulates
pop and lock dance. The example shown in Figure 7 is frequently mentioned as a
major indication of ’transfer’ with a visible locking dance style from the
view of experienced dancers. For CycleDance samples of transferring ’locking
dance to ballet-jazz’, the dance experts responded that the character arms are
clearly jazz or ballet and are really good at holding ’traditional’ shapes.
The dance experts also commented on some limitations. One commonly mentioned
point is that some motions look wobbly, which may indicate the need for
applying some filters to smooth the generated results. The experts also
pointed out that ballet-jazz usually requires dancers to point their feet
while the generated motions always show flexed ankle joints. This shows the
limitation of the considered data which currently do not capture fine foot
movements. This caveat also causes some physically unrealistic effects such as
the character appears floating when sometimes its body does not have a contact
point on the floor.

##  5 Discussion of Societal Impact

This work contributes a framework for style transfer that aims to offer
artistic and scientific innovations to the field of dance. In the short term,
we could foresee several impacts on industries and society. The positive
effect would be the progress in choreographic practice and dance research,
which unlock new possibilities in terms of hybrid human-artificial co-creation
of dance material. Certain industry sectors could benefit as well, such as
video game and animation industries (e.g., group dances where each character
has a different motion style). Such effects could lead to displacement of jobs
and a shift towards jobs that relies more on a combination of creativity and
automation, as well as development of new user-friendly interfaces and tools.
We also foresee potentially negative impacts or misuses. This technology could
blur the lines of ownership in creative processes, i.e., who is/are the
creator(s). For movement styles beyond dance, such transfer models, if trained
on non-representative datasets, could reinforce movement stereotypes of
certain societal groups by learning a biased association between group
membership and movement styles, e.g. elderly people or people with
disabilities.

##  6 Conclusion and Future work

This work explores challenging style transfer for sequential data with rich
variations and complex frame dependencies such as dance movements. The
proposed CycleDance manages to alleviate these challenges by exploiting
expressive data encoders, cross-modal contexts and a curriculum based training
scheme. Quantitative results from similarity metrics and human expert
evaluations confirm the effectiveness of CycleDance. To the authors’
knowledge, this is the first work where music context is used for dance or
general motion style transfer. In the future, we plan to extend the backbone
from the CycleGAN-based model to StarGAN or an AdaIN-based model to handle
more than two dance genres. Research is also needed to address identified
limitations on preserving limb shapes. Based on these techniques, we envision
new tools in dance motion design for choreography, film industry, and video
games.

## Acknowledgements

This study has received funding from the European Commission Horizon 2020
research and innovation program under grant agreement number 824160
(EnTimeMent). This work benefited from access to the HPC resources provided by
the Swedish National Infrastructure for Computing (SNIC), partially funded by
the Swedish Research Council through grant agreement no. 2018-05973.

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#  Dynamic Arctic weather variability and connectivity

Jun Meng  School of Science, Beijing University of Posts and
Telecommunications, Beijing 100876, China  Potsdam Institute for Climate
Impact Research, Potsdam 14412, Germany  Jingfang Fan<EMAIL_ADDRESS>School of Systems Science/Institute of Nonequilibrium Systems, Beijing Normal
University, Beijing 100875, China  Potsdam Institute for Climate Impact
Research, Potsdam 14412, Germany  Uma S Bhatt  Geophysical Institute,
University of Alaska Fairbanks, Fairbanks, AK 99775, USA  Jürgen Kurths
Potsdam Institute for Climate Impact Research, Potsdam 14412, Germany
Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775,
USA  Institute of Physics, Humboldt-University, Berlin 10099, Germany

(August 27, 2024)

###### Abstract

The rapidly shrinking Arctic sea ice is changing weather patterns and
disrupting the balance of nature. Dynamics of Arctic weather variability (WV)
plays a crucial role in weather forecasting and is closely related to extreme
weather events. Yet, assessing and quantifying the WV for both local Arctic
regions and its planetary impacts under anthropogenic climate change is still
unknown. Here, we develop a complexity-based approach to systematically
evaluate and analyze the dynamic behaviour of WV. We reveal that the WV within
and around the Arctic is statistically correlated to the Arctic Oscillation at
the intraseasonal time scale. We further find that the variability of the
daily Arctic sea ice is increasing due to its dramatic decline under a warming
climate. Unstable Arctic weather conditions can disturb regional weather
patterns through atmospheric teleconnection pathways, resulting in higher risk
to human activities and greater weather forecast uncertainty. A multivariate
climate network analysis reveals the existence of such teleconnections and
implies a positive feedback loop between the Arctic and global weather
instabilities. This enhances the mechanistic understanding of the influence of
Arctic amplification on mid-latitude severe weather. Our framework provides a
fresh perspective on the linkage of complexity science, WV and the Arctic.

Arctic sea ice is declining and thinning at an accelerating rate due to
anthropogenic climate change rothrock_thinning_1999 ; comiso_accelerated_2008
. The warming trend is more prominent in the Arctic and is double of the
global average or even greater regionally rantanen2022arctic , a phenomenon
known as Arctic amplification (AA) miller2010arctic ; serreze_processes_2011 ;
previdi2021arctic . The Arctic sea ice conditions can affect the Arctic
ecosystem, wildlife, hunting and shipping, exploration of nature resources and
more orbaek2007arctic ; peeken_arctic_2018 ; schneider2021consequences . As
one crucial component of the complex Earth system steffen_emergence_2020 ;
assessment2005arctic , changes in Arctic sea ice are found to have statistical
and dynamical connections with regional as well as remote climatic impacts
sevellec_arctic_2017 ; screen_consistency_2018 ; chemke_effect_2019 ;
blackport_minimal_2019  (as shown in Fig. 1) through both large-scale
atmospheric and oceanic circulations rahmstorf_thermohaline_2003 ;
budikova_role_2009 ; kushnir_atmospheric_2002 ; alexander_atmospheric_2004 ;
honda_influence_2009 . The rapid shrinking of the ice cover has attracted much
attention about the Arctic sea ice teleconnections and predictions from
seasonal-to-decadal time scales in recent years francis_evidence_2012 ;
cohen_recent_2014 ; guemas_review_2016 ; olonscheck_arctic_2019 . However, the
understanding about its variability on weather time scales is still in its
infancy smith_sea_2016 ; mohammadi-aragh_predictability_2018 , although it is
crucial for weather forecasting, the safety of commercial and subsistence
maritime activities, the survival of polar mammals and the benefit of polar
economics. The impact of day-to-day Arctic sea ice variations has been
underestimated in most of the climate models dammann_impact_2013 . To fill
this gap, here we adopt complexity-based approaches and the climate network
framework to investigate the daily WV of the Arctic sea ice and its
connections to climate phenomena on different spatio-temporal scales,
including the Arctic Oscillation (AO), climate change and local weather
conditions even in regions faraway.

Complexity science employs the mathematical representation of network science
and provides a powerful tool to study the structure, dynamics and function of
complex systems newman_networks_2010 . The climate system is a typical complex
adaptive system due to its nonlinear interactions and feedback loops between
and within different layers and components. In recent years, network science
has been implemented to the climate system to construct the climate network
(CN) tsonis_architecture_2004 . The CN is a novel tool to unveil and predict
various important climate mechanisms and phenomena fan_statistical_2021 ,
including forecasting of the El Niño Southern Oscillation
ludescher2013improved ; meng2017percolation  and Indian summer monsoon
rainfall stolbova2016tipping ; fan2022network , the global pattern of extreme-
rainfall boers2019complex , the changes of global-scale tropical atmospheric
circulation under global warming fan2018climate , teleconnections among
tipping elements in the Earth system liu_teleconnections_2023 , the Indian
Ocean Dipole lu2022early  and so on.

The AO is one of the major modes of atmospheric circulation over the mid-to-
high latitudes of the Northern Hemisphere (NH) thompson_arctic_1998 , which
influences climate patterns in Eurasia, North America, Eastern Canada, North
Africa, and the Middle East, especially during boreal winter
deser_teleconnectivity_2000 ; rigor_response_2002 ; he_impact_2017 . The AO
index is defined as the leading empirical orthogonal function of NH sea level
pressure anomalies from latitudes $20^{\circ}$ N to $90^{\circ}$ N and is
characterized by the back-and-forth shifting of atmospheric pressure between
the Arctic and the mid-latitudes. During the positive AO phases, the surface
pressure is lower-than-average in the Arctic region and the jet stream shifts
northward accompanied by a poleward shift of the storm track
simmonds_arctic_2008 . Correspondingly, we find that both the sea ice and air
temperature in mid-to-high latitudes of the NH changes more rapidly (i.e.,
with blueshifted frequency spectrum) paired with more stable weather
conditions (i.e., redshifted) in regions further south during the AO positive
phases, in contrast to the AO negative phases when pressure north of the
Arctic Circle is higher than normal. To quantify the blue/red-shift effect and
its geographic distribution indicating increased/reduced WV, here we introduce
two novel mathematical techniques: the advanced autocorrelation function
method, i.e., $W_{ACF}$ and the advanced power spectrum method, i.e., $W_{PS}$
(see Methods). This way enables us to find that the day-to-day variability of
ice cover for a large area of the Arctic is increasing due to the dramatic
melting of the sea ice kwok2018arctic , which indicates more enhanced risks
for severe weather under climate change zhang_climatology_2004 ;
yin_consistent_2005 ; valkonen_arctic_2021 ; parker2022influence . This may
also increase the probability of unstable weather conditions globally through
atmospheric teleconnections between the Arctic and the global climate systems
(see links shown in Fig. 1). Finally, we statistically verify the existence of
such teleconnections between the Arctic sea ice and weather conditions in
remote global regions via a multivariate climate network framework. Such
teleconnections can result in a positive feedback loop of WV between the
Arctic and the rest (see Fig. 1) and contribute to understanding the
mechanisms of linkage between the AA and mid-latitude weather
cohen2020divergent . The presented results and methodology not only facilitate
a quantitative risk assessment of extreme weather events (see Fig. S1), but
also reveal the existence of interaction or synchronization paths among
regional and global climate components.

## Results

### Linkage of the weather variability and the AO

The WV refers to the irregularity/predictability of the climate data at
weather time scales (i.e., hours - days). There are various ways to evaluate
the data variability/irregularity, such as the entropy pincus1995approximate ;
richman2000physiological ; costa2005multiscale , the detrended fluctuation
analysis peng1995quantification ; livina2007modified , the correlation
dimension procacia1983measuring , the lyapunov exponents analysis
wolf1985determining , etc. However, most of them would be problematic, biased
or invalid when dealing with short and noisy data, such as the weather data.
The standard deviation (SD) is an effective technique to quantify the
dispersion of data, but not a good measure for irregularity, e.g., the SD of a
randomly shuffled data is the same as the original. Besides, the auto-
correlation function describes how fast the self-similarity of a variable
decays with time box_time_2015  and the power spectral analysis
stoica2005spectral  allows us to discover periodicity in the data. Yet, a
systematic evaluation of the auto-correlation and the power spectrum as well
as their dynamic evolution for non-stationary climate data are still lacking.

Therefore, here we introduce two mathematical functions: $W_{ACF}$ and
$W_{PS}$ (see Methods for details) to quantify the WV in and around the Arctic
in a given month, as well as its dynamic behavior during the period from Jan.
1980 to Dec. 2019. For a given time series, the physical meanings of these
metrics are: higher values of the $W_{ACF}$ stands for weaker short-term
memory; while higher values of the $W_{PS}$ indicates faster changes. In
particular, to better understand their physical meanings, we construct various
nonlinear time series (as shown in Fig. 2a) via the following dynamical
equations,

$\displaystyle x_{t}=\cos{(2\pi t/20)},$ (1) $\displaystyle y_{t}=\cos{(2\pi
t/10)},$ (2) $\displaystyle z_{t}^{x}=0.2x_{t}+0.8u_{t},$ (3) $\displaystyle
z_{t}^{y}=0.2y_{t}+0.8u_{t},$ (4)

where $t\in[0,1000]$, $u_{t}$ is the nonlinear logistic function as:
$u_{t+1}=\mu u_{t}(1-u_{t})$. Here we set the parameter $\mu=3.8$ and
$u_{0}=0.01$, i.e., it generates a chaotic behavior pomeau_intermittent_1980 .
Mathematically, Eqs. (1) and (2) are two periodic functions but with different
periods $20$ and $10$, respectively; while, Eqs. (3) and (4) consist of a
periodic term and a chaotic term (Fig. 2a). Therefore, strictly speaking, the
value of $W_{ACF}$ for $z_{t}^{x}$ ($z_{t}^{y}$) is higher than $x_{t}$
($y_{t}$), i.e., weaker short-term memory, due to the chaotic term $u_{t}$;
the value of $W_{PS}$ for $y_{t}$ ($z_{t}^{y}$) is higher than $x_{t}$
($y_{t}$), i.e., faster changes, due to the periodic term with different
periods. One should note that a segment of unstable data is usually changing
faster, with both high $W_{ACF}$ and $W_{PS}$, e.g., Eqs. (3) and (4). While a
segment of quickly changing data is not necessarily irregular, such as high
frequency periodic data, with high $W_{PS}$ but low $W_{ACF}$, as Eq. (2). We
extract 31 (i.e., the maximal length of one month in the climate data)
consecutive data points from each of the samples and perform $W_{ACF}$ and
$W_{PS}$ analysis on the extracted subsets. All results presented in Fig. 2b
and c, are consistent with our theory, which indicates that our two functions
can be used as effective tools to describe the variability (both disorder and
frequency) for given time series.

Next, we apply $W_{ACF}$ and $W_{PS}$ to quantify the Arctic sea ice WV based
on the sea ice cover dataset (daily, 1979-2019, see DATA for details). Our
results are shown in Figs. 2d-h. A positive value of $r$ denoted by blue in
Figs. 2d or e, indicates positive correlation between the annual mean of
$W_{ACF}$ or $W_{PS}$ with the AO index. We observe that both $W_{ACF}$ and
$W_{PS}$ tend to be higher, i.e., indicating faster and more irregular day-to-
day changes of ice cover, during the AO positive phases than AO negative
phases, in some parts of the Arctic region, such as, the Canadian Archipelago,
Beaufort Sea, and the Central Arctic. To illustrate the effect of the AO on
$W_{PS}$, we show that the power spectrum of Arctic sea ice during the AO
positive phase, e.g., Jan. 1989, is significantly blueshifted comparing to
that during the AO negative phase, e.g., Jan. 2010 (see Fig. 2f). To
illustrate the effect of the AO on $W_{ACF}$, we show that the timeseries of
AO index and $W_{ACF}$ are significantly synchronized during the period
1980-2019 (as shown in Fig. 2g and h). Moreover, we uncover that the climatic
effects of the AO are more prominent in winter-spring than in summer-autumn
(see Figs. S2 and S3).

The underlying physical mechanism is related to the typical atmospheric
character of the AO, as well as the close interactions between the Arctic sea
ice and the surface atmosphere. During the positive phases of AO, the jet
stream shifts northward and the storm tracks are located farther north than
during the AO negative phases thompson_regional_2001 , see Fig. S4. This
results in more unstable regional weather in mid-to-high latitudes of the NH,
and yields higher $W_{ACF}$ and $W_{PS}$ of the air temperature data, see Fig.
3 and Figs. S5-S8. In contrast, the $W_{ACF}$ and $W_{PS}$ of the air
temperature in the mid-latitudes of the NH increase with more outbreaks of
significant weather events (e.g., cold events, frozen precipitation and
blocking days) thompson_regional_2001  as the zonal wind weakens during the
negative AO phases, see Fig. 3 and Figs. S5-S8. In particular, as shown in
Figs. S5-S8, there are even significant connections between the AO and the WV
in some regions of the Southern Hemisphere.

The $W_{ACF}$ and $W_{PS}$ analysis provides an additional way to describe the
quantitative response of both the Arctic sea ice and the atmosphere to the AO,
thus could be used to assess the risk of extreme events in mid-to-high
latitudes of the NH.

### Increased irregularity of Arctic sea ice cover

In the following, our results shown in Fig. 4 indicate that the sea ice cover
in a large area of the Arctic, including the East Siberian, the Beaufort Sea
and the Central Arctic, where the ice thickness decrease is dramatic (as shown
in Fig. S9), has changed more rapidly and irregularly over the past 40 years
(1980-2019). That is since both values of the $W_{ACF}$ and $W_{PS}$ are
significantly increasing. The observed enhancing trend of WV may be attributed
to the following two reasons: One is related to the development of remote
sensing and data analyzing technology, resulting in better data resolution and
accuracy over the data record; the other reason is the rapid decline of multi-
year ice cover, due to the dramatic increase of air temperature
comiso_large_2012 . The multi-year sea ice has been defined as the ice that
survives at least one summer melt and represents the thick sea ice cover,
while the first-year ice refers to the ice that has no more than one-year’s
growth. As more of perennial ice cover is replaced by younger and thinner ice
cover, the regional ice cover becomes more fragile and vulnerable to
fluctuations of air temperature or some other forces kwok2018arctic .
Therefore, local interactions between the sea ice and atmosphere would be
enhanced and the weather in the Arctic and remote global regions may affect
each other more easily through potential tele-connected pathways (e.g., Fig.
5), which may increase the WV associated with the short-term weather
predictability.

In addition, we observe relatively more areas with a significant trend of
enhanced instability in the melt season under global warming (see Fig. 4a).
This is because during the melt season (Apr.-Aug.), the sea ice declines and
fluctuates more dramatically than in other seasons when the monthly average
ice cover extent (the area of ocean with at least $15\%$ sea ice, marked by
the blue curve in Fig. 4a) reaches its maximum/minimum. An intensification of
the summer Arctic storm activity is also likely to happen as the land-sea
thermal contrast increases under global warming day2018growing ;
kenigson_arctic_2021 ; peng_role_2021 , which can increase the WV both in the
ocean and atmosphere.

### Arctic-global teleconnection patterns

Next, we propose the multivariate climate network approach to statistically
reveal the potential teleconnection patterns between the Arctic sea ice (Fig.
S10a) and the global air temperature field (Fig. S10b), see more details in
the Methods. Different from the classical climate network approach with only
one climate variable, see Ref. donges_unified_2015 ; dijkstra2019networks ;
fan_statistical_2021  and references therein, we construct climate networks
where each link connects one node located in the Arctic (Fig. S10a) and the
other in the globe (Fig. S10b). In particular, the link weight quantifies the
similarity of temporal evolution between two different climate variables,
i.e., the Arctic sea ice and the global air temperature. By comparing to a
Null-model (see Methods), we observe the dynamic behavior of network
connectivity (as shown in Fig. S11a), which is defined as the ratio of
significant links for each month’s network. The statistical significance for
each link is defined by comparing to the null-model, see details in Method
section. A value of above $5\%$ connectivity indicates statistically
significant synchronization of weather between the Arctic and areas outside,
such as Feb. 2010 (see Fig. S11b and c), when the AO is in a strong negative
phase and the cold polar air plunged into lower latitudes of the NH and result
in extreme weather conditions in a large area of the globe cohen2010winter ;
liu2012impact . We identify the significant Arctic-global teleconnection
patterns by using climate network node degree fields, which are defined as the
number of significant links that connect to the Arctic for each global node,
for two specific periods, Feb. 2010 (AO negative phase) and Mar. 2019 (AO
positive phase) in Fig. 5a and c, respectively.

Moreover, two typical links presented in Fig. 5 indicate strong
synchronizations between the daily sea ice cover for one Arctic node and the
air temperature for another remote global node (their time series are shown in
Figs. S12 and S13). As shown in Fig. S12b, changes in the sea ice for node $i$
($77.5\hbox{${}^{\circ}$}$ N, $160\hbox{${}^{\circ}$}$ E) in the Arctic are
two days ahead of the air temperature variations for node $j$
($30\hbox{${}^{\circ}$}$ N, $105\hbox{${}^{\circ}$}$ E) in the Sichuan
Province of Southwest China, i.e., evolution of the Arctic sea ice could
affect the anomalies of air temperature in Southwest China. To better
understand how sea ice affects air temperature variability faraway, we
identify the most probable teleconnection propagation path through the
shortest path method (see Methods for more details). We show a potential
propagation path for this teleconnection (marked by yellow in Fig. 5b) and
find that it seems to be roughly a straight line from the Arctic to Southwest
China through Eastern Russia and Mongolia. The path length is close to $6400$
km. From a meteorological perspective, this path can be well explained by the
main large-scale atmospheric circulation. A negative phase of the AO leads to
a stronger Siberian High and extends farther southeastward. This results in
repeated cold air outbreaks into South China he_impact_2017 . Our analysis is
highly consistent with the wind climatology, see the background information of
Fig. 5b.

In addition, its feedback is also considered. However, we observe a relatively
weaker connection in the opposite direction, i.e., from Southwest China to the
Arctic. We find that changes in the air temperature at the same location in
Southwest China influence that of the sea ice for the same Arctic node $11$
days later, as shown in Fig. S12c. Correspondingly, we identify its potential
propagation path (marked by orange in Fig. 5b) and find it corresponds to
negative wind anomalies from Southwest China to the Arctic. These two tele-
connected paths form an interaction loop that suggests a large-scale
atmospheric feedback of WV between the Arctic and Southwest China.

In a contrast, during a positive phase of the AO, we show another
teleconnection and its path in Fig. 5c and d, which indicates that the
fluctuations of air temperature in California can affect the Arctic sea ice
through the upper atmospheric circulations. Meanwhile, changes in the Arctic
sea ice can also influence the temperature fluctuations in California along
upper wind routes in an opposite direction, however, at a weaker strength (see
more details in Fig. S13c). This is because during the positive phase of the
AO, low pressure dominates the Arctic regions, leading to a northward and
intensified jet stream that blocks the outbreaks of frigid polar air into
lower latitudes and reduces storm activity in California stroeve_arctics_2012
. The uncovered teleconnection loop between the Arctic and California suggests
that Arctic sea ice decline may drive more California droughts and wildfires
cvijanovic_future_2017 .

The synchronization of day-to-day weather between the Arctic and other regions
can favor positive feedbacks of WV, where increasing WV/instability of the
Arctic sea ice may cause a higher risk of extreme weather conditions in remote
global regions. Meanwhile, impacts from global regions may also induce
unstable weather conditions in the Arctic.

## Discussion

In summary, we have introduced the mathematical $W_{ACF}$ and $W_{PS}$
functions to quantify the short-term dynamic WV relating to the irregularity
and frequency of the day-to-day changes of climate data. By adapting $W_{ACF}$
and $W_{PS}$, we are able to identify significant effects of the AO on day-to-
day changes of the Arctic sea ice as well as the WV in mid-to-high latitudes
of the NH. We attribute the physical mechanism to the shifts of north-to-south
location of jet stream and storm-steering associated with different phases of
the AO. Furthermore, we found that during the past 40 years, the Arctic sea
ice variability on weather time scales is substantially increasing due to the
melting of the thick perennial sea ice. Finally, in order to analyze the
dynamic Arctic weather connectivity, we have constructed multivariable climate
networks, i.e., between the Arctic sea ice and the global air temperature
field. By applying the shortest path method, we are able to identify
teleconnections paths as well as positive feedback loops of WV. We also
proposed a possible physical mechanism underlying these paths. The reduction
of Arctic sea ice stability may increase the risk of unstable weather
conditions and lead to reduced skill of weather forecasts
petropoulos_forecasting_2022  globally through the Arctic-global teleconnected
feedback loops. Our new findings can help to understand the physical
mechanisms linking the AA and the global climate, and implies prominent global
impacts of the Arctic WV on human and natural systems under climate change
cohen2020divergent ; previdi2021arctic .

As the Arctic is considered to be a barometer of global climatic change, in
particular, Arctic sea ice loss is approaching a tipping point and is
extremely crucial for the whole Earth’s climate lenton_climate_2019 . Besides
the immediate utility of being able to quantitatively analyze the dynamics of
WV for local Arctic regions and its global impacts, our framework would be
also applied to study and reveal the short-term synchronizations of
connectivity among remote global regions, sea ice forecasting, as well as
systemic risk induced by the interdependency among other complex subsystems
and cascading of adverse consequences, which is particularly important for a
systemic risk-informed global governance.

Figure 1: The arctic system as a crucial component of the Earth climate
system. a, Schematic view of a climate network. Links indicate interactions
between different regional climate systems in the globe. Golden links
represent teleconnections between the Arctic and regions outside. b,
Illustration of the complex Arctic system. It contains the cryosphere,
biosphere, hydrosphere, and atmosphere as well as the interactions among them.
A change in one component often triggers changes and feedbacks in numerous
interconnected processes (e.g., Arctic sea ice decline). The circular arrow
suggests a positive feedback of the WV between the Arctic and the rest of the
climate system.

Figure 2: Blueshift effect of the Arctic Oscillation on the Arctic weather
variability. a, Sample nonlinear time series generated based on Eqs. (1-4). b,
The auto-correlation functions and values $W_{ACF}$ of each sample time series
shown in a. c, The power spectrum density and values $W_{PS}$ of each sample
time series shown in a. d, The correlations between the annual mean of the AO
index and the $W_{ACF}$ for the Arctic sea ice. The “x” marks represent the
nodes with correlations significant at the 95% confidence level (Student’s t
test). e, The same as d for $W_{PS}$. f, The power spectrum of the sea ice for
all nodes marked by symbol “x” in e in Jan. 1989 with a positive AO phase
comparing to that in Jan. 2010 with a negative AO phase. g, The AO index (pink
solid line for monthly and pink dashed line for annual) versus the $W_{ACF}$
index (dark blue solid line for monthly and dark blue dashed for annual)
averaged over all nodes marked by symbol “x” in d. h, The scatter plots of
annual indexes (dashed lines in g) of the AO versus $W_{ACF}$, the $r$ value
between these two indexes is $0.65$, with a $p$ value of $5.5\times 10^{-6}$.

Figure 3:  The relationships between the AO and weather variability. a,b, The
correlation maps between the annual mean of the AO index and $W_{ACF}$ of the
air temperature at $850hPa$ pressure level during the period of 1980–2019.
c,d, The same as a and b, but for $W_{PS}$. The symbol “x” in each panel
represents the region with correlation significant at the 95% confidence level
(Student’s t-test).

Figure 4: The dynamic weather variability of the Arctic daily sea ice cover
during June. a, The ratio of nodes that has statistically significant
increasing trend for the $W_{ACF}$ (gray) and $W_{PS}$ (purple); the Sea Ice
Index, i.e. the area with at least $15\%$ ice cover (blue) for the same months
during 1980–2019. b, Changes per decade as multiple of one standard deviation
($\sigma$), for each Arctic node’s $W_{ACF}$ during June. c, the same as b for
$W_{PS}$. The symbol “x” in panels b and c represents the region with trend
significant at the 95% confidence level (Student’s t-test).

Figure 5:  Diagram of climate network teleconnection paths. a, Heatmap of the
node degree defined as the number of significant links for each node (see
Methods) in the climate network of Feb. 2010. The blue line indicates the
teleconnection between one Arctic node and one node located in Sichuan
province of China. b, The propagation pathway of the teleconnection marked by
blue in a. c, the same as a for Mar. 2019. The blue line indicates the
teleconnection link between one Arctic node and one node in California of
United States. d, The propagation pathway of the teleconnection marked by blue
in c. The colors and white arrows depict the magnitudes and directions of the
850 (500)-hPa winds in b (d).

## Data and Methods

### Data

The data used in the current work is the $0$ hr (UTC) daily sea ice cover and
the air temperature at $850hPa$ pressure level from the ERA5 era5
(https://apps.ecmwf.int/datasets/) reanalysis, with a spatial (zonal and
meridional) resolution of $2.5^{\circ}\times 2.5^{\circ}$. The searching
principle for $850hPa$ pressure level is, since it is just above the boundary
layer to avoid direct interactions between the sea ice and surface atmosphere
olonscheck_arctic_2019 . We select $8040$ grids from the dataset of air
temperature which approximately equally cover the globe (see Fig. S10b). There
are $377$ grids located in the ocean of the Arctic region that with non-zero
sea ice cover at least for one day (see Fig. S10a). Then, for each calendar
year $y$ and for each node, we calculate the anomalous value for each calendar
day $t$ by using the original value minus the climatological average, then
divided by the climatological standard deviation. The calculations of the
climatological average and standard deviation are based on data from the year
of 1979 to 2019. For simplicity, leap days are excluded.

The AO index was downloaded from:
https://www.cpc.ncep.noaa.gov/products/precip/CWlink/dailyaoindex/monthly.ao.index.b50.current.ascii.
[Accessed in Sep. 2021].

The Arctic Sea Ice Extent was downloaded from :
https://nsidc.org/data/g02135/versions/3. [Accessed in Jan. 2021].

### Assessing Weather Variability Functions

#### Advanced autocorrelation function method

The autocorrelation function (ACF) is widely used to measure the memory of a
time series and reveals how the correlation between any two values of the
signal changes as their time-lag box_time_2015 . Generally, for a given time
series, $x_{t}$, the ACF is defined as,

$C(\mathrm{\tau})=\frac{\operatorname{Cov}\left(\mathrm{x}_{\mathrm{t}},\mathrm{x}_{\mathrm{t}+\mathrm{\tau}}\right)}{\sqrt{\operatorname{Var}\left(\mathrm{x}_{\mathrm{t}}\right)\operatorname{Var}\left(\mathrm{x}_{\mathrm{t}+\mathrm{\tau}}\right)}},$
(5)

where
$\operatorname{Cov}(\mathbb{X},\mathbb{Y})=\mathrm{E}[(\mathbb{X}-\mathrm{E}[\mathbb{X}])(\mathbb{Y}-\mathrm{E}[\mathbb{Y}])]$
and
$\operatorname{Var}(\mathbb{X})=\mathrm{E}[\mathbb{X^{2}}]-\mathrm{E}[\mathbb{X}]^{2}$.
If the $x_{t}$ are completely uncorrelated, for example, a white noise
process, $C(\mathrm{\tau})$ is zero at all lags except a value of unity at lag
zero ($\tau=0$). A correlated process on the other hand, has non-zero values
at lags other than zero to indicate a correlation between different lagged
observations. In particular, short-range memory of the $x_{t}$ are described
by $C(\mathrm{\tau})$ declining exponentially

$C(\tau)\sim\exp\left(-\tau/\tau^{*}\right),$ (6)

with a characteristic time scale, $\tau^{*}$. For long-range memory, $C(\tau)$
declines as a power-law

$C(\tau)\propto\tau^{-\gamma},$ (7)

with an exponent $0<\gamma<1$. However, a direct calculation of $C(\tau)$,
$\tau^{*}$ and $\gamma$ is usually not appropriate due to noise superimposed
on the collected data $x_{t}$ and due to underlying trends of unknown origin
kantelhardt_detecting_2001 . In order to overcome the problems described
above, here, we develop an advanced autocorrelation function method to
quantify the memory (both short and long range) strength $W_{ACF}$ of a time
series as,

${W_{ACF}}=\frac{\text{max}\left(|C(\tau)|\right)-\text{mean}\left(|C(\tau)|\right)}{\sqrt{\operatorname{Var}\left(|C(\tau)|\right)}}\equiv\frac{1-\text{mean}\left(|C(\tau)|\right)}{\sqrt{\operatorname{Var}\left(|C(\tau)|\right)}},$
(8)

where ‘max’ and ‘mean’ are the maximum and mean values of the absolute ACF,
i.e., $|C(\tau)|$, respectively. $\tau\in[-\tau_{max},\tau_{max}]$ is the time
lag. In the present work, we take $\tau_{max}=10$ days, since we are
considering the day-to-day changes of data at the time scale of weather
forecasting, i.e., within two weeks. Equation (8) describes the fluctuations
of the ACF and its values reveal the strength of memory, i.e., higher
(smaller) $W_{ACF}$ indicates a weaker (stronger) correlation and results in a
low (strong) memory. For example, white noise has a maximum value
$W_{ACF}=(2\tau_{max}+1)\sqrt{\frac{2\tau_{max}}{2\tau_{max}+1}}$. Other
examples are described in Fig. 2. Another big advancement of our method is
eliminating the problematic nonstationarities.

#### Advanced power spectrum method

The advanced autocorrelation function $W_{ACF}$ quantify well the strength of
memory for an arbitrary time series, but does not reveal any information about
the frequency content. For example, Eqs. (1) and (2) are two functions with
different periods. Their $W_{ACF}$ values are almost the same, as shown in
Fig. 2. To fill this gap, we further develop an advanced power spectrum (PS)
method. Based on the Welch’s method welch_use_1967  we define the advanced
power spectral density $W_{PS}$ as,

$W_{PS}=\int_{f}P(f)\times fdf,$ (9)

where $P(f)$ is the normalized spectral density and $f$ stands for the
corresponding frequency, which can be obtained by Fourier transform. $W_{PS}$
is indeed the weighted mean of $f$, thus has the same unit as frequency.
Notably, a relatively higher value of the $W_{PS}$ indicates a larger ratio of
the high frequency components (i.e., blueshift), see examples shown in Fig. 2.

### Climate Networks

#### Nodes

Different from the classical climate network with only one node
classification, see Ref. dijkstra2019networks ; fan_statistical_2021  and
references therein, here, we define two types of nodes: globe nodes $i$ with
air temperature variable $T_{i}(t)$; Arctic nodes $j$ with Arctic sea ice
cover variable $I_{j}(t)$. We thus have $8040$ globe nodes (as shown in Fig.
S10b) and $377$ Arctic nodes (as shown in Fig. S10a).

#### Links

We construct a sequence of multivariate climate networks. For obtaining the
strength of the links between each pair of nodes $i$ and $j$, we compute, for
each month $m$, the time-delayed, cross-correlation function

$C_{i,j}^{m}(\tau)=\frac{\left\langle
T_{i}^{m}(t)I_{j}^{m}(t-\tau)\right\rangle-\left\langle
T_{i}^{m}(t)\right\rangle\left\langle
I_{j}^{m}(t-\tau)\right\rangle}{\sqrt{\operatorname{Var}(T_{i}^{m}(t))\operatorname{Var}(I_{j}^{m}(t-\tau))}},$
(10)

and

$C_{i,j}^{m}(-\tau)=\frac{\left\langle
T_{i}^{m}(t-\tau)I_{j}^{m}(t)\right\rangle-\left\langle
T_{i}^{m}(t-\tau)\right\rangle\left\langle
I_{j}^{m}(t)\right\rangle}{\sqrt{\operatorname{Var}(T_{i}^{m}(t-\tau))\operatorname{Var}(I_{j}^{m}(t))}},$
(11)

where the bracket $\left\langle\right\rangle$ denotes an average over
consecutive days during a given month $m$, and $\tau\in[0,\tau_{max}]$ is the
time lag. Since we mainly focus on the dynamic Arctic WV, here we chose the
maximal time lag $\tau_{max}=20$ days for Eqs. (10) and (11).

We identify the time lag $\theta$ at which the absolute value of the cross-
correlation function $|C_{i,j}^{m}(\tau)|$ reaches its maximum. The weight of
link $(i,j)^{m}$ is defined as the corresponding value of the cross-
correlation function, i.e. $C_{i,j}^{m}=C_{i,j}^{m}(\tau=\theta)$. Therefore,
the weight of each link could be either positive or negative, but with the
maximum absolute value. The sign of $\theta$ indicates the direction of each
link; that is, when the time lag is positive $\left(\theta>0\right)$, the
direction of this link is from $j$ to $i$, and vice versa fan_network_2017 .

#### Null-model

Next, we investigate the statistical significance of the link weights in the
real networks by comparing to the shuffled surrogate network. In the surrogate
network, to calculate link weight for each pair of nodes, we use two segment
of data, each is corresponding to $30$ consecutive days starting from the
first day of a month that is randomly selected from the period Jan. 1980-Dec.
2019, so that to destroy real correlations between two nodes in the temporal
dimension. Then we define the significant threshold $q$ as the $95\%$ highest
value of the absolute weights for all links in the surrogate network. The link
$(i,j)^{m}$ in the real network for a specific month $m$ is defined as
significant if it is higher than $q$ or lower than $-q$, i.e.,
$|C_{i,j}^{m}|>q$. We find that the number of significant links for each
month’s network are dynamically changing with time as shown in Fig. S11.

#### Node degrees

We define the degree for each global node as the number of significant links
that connect to the Arctic nodes. We show heatmaps of node degrees for two
specific months, i.e., the Feb. 2010 (Fig. 5a) and the Mar. 2019 (Fig. 5b). We
observe higher node degrees in many regions, even in low latitudes, of the NH
for Feb. 2010, comparing to that for Mar. 2019. We suppose it is related to
the different phases of the AO.

### Teleconnection path mining

To identify the teleconnection path, we perform the shortest path method of
complex networks to find the optimal paths in our climate networks. A path is
a sequence of nodes in which each node is adjacent to the next one,
especially, in a directed network, the path can follow only the direction of
an arrow. Here, our climate network is based on only one climate variable–air
temperature at $850hPa$ pressure level, and we select 726 nodes from the
$10512$ nodes fan2022network ; liu_teleconnections_2023 . For each climate
network link $(i,j)^{m}$, we define its cost function value as

$E_{i,j}^{m}=\frac{1}{|C_{i,j}^{m}|}.$ (12)

The Dijkstra algorithm dijkstra_note_1959  was used to determine the directed
optimal path between a source node $i$ and a sink node $j$ with the following
constraints zhou_teleconnection_2015 ; liu_teleconnections_2023 : (i) the
distance for every step is shorter than 1000km; (ii) link time delay
$\theta\geq 0$; (iii) the sum cost function value for all collection of links
through path $i\longrightarrow j$ is minimal. In this way, we identify the
optimal paths for information/energy/matter spreading in the two-dimensional
space.

## Data availability

The data represented in Figs. 2–5 are available as Source Data. All other data
that support the plots within this paper and other findings of this study are
available from the corresponding author upon reasonable request.

## code availability

The C++ and Python codes used for the analysis are available on GitHub:
(https://github.com/fanjingfang/DAWV).

## Acknowledgments

The authors wish to thank T. Liu for his helpful suggestions. We acknowledge
the support by the National Natural Science Foundation of China (Grant No.
12205025, 12275020, 12135003).

## Author Contributions

J.M and J.F designed the research. J.M performed the analysis, J.M, J.F, U.S.B
and J.K generated research ideas and discussed results, and contributed to
writing the manuscript.

## Additional information

Supplementary Information is available in the online version of the paper.

## Competing interests

The authors declare no competing interests.

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