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Received: 26 November 2024
Revised: 30 December 2024
Accepted: 1 January 2025
Published: 3 January 2025
Citation: Lv, J.; Mao, H.; Wang, Y.;
Yao, Z. Reconstructionand Prediction
of Chaotic Time Series with Missing
Data: Leveraging Dynamical
Correlations Between Variables.
Mathematics 2025,13, 152. https://
doi.org/10.3390/math13010152
Copyright: © 2025 by the authors.
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Article
Reconstruction and Prediction of Chaotic Time Series with
Missing Data: Leveraging Dynamical Correlations
Between Variables
Jingchan Lv 1,2 , Hongcun Mao 1, Yu Wang 1and Zhihai Yao 1,*
1Department of Physics, Changchun University of Science and Technology, Changchun 130022, China;
ljingchan@mails.cust.edu.cn (J.L.); 2020200021@mails.cust.edu.cn (H.M.);
2021200025@mails.cust.edu.cn (Y.W.)
2College of Physics and Electronic Information, Baicheng Normal University, Baicheng 137000, China
*Correspondence: yaozh@cust.edu.cn
Abstract: Although data-driven machine learning methods have been successfully applied
to predict complex nonlinear dynamics, forecasting future evolution based on incomplete
past information remains a significant challenge. This paper proposes a novel data-driven
approach that leverages the dynamical relationships among variables. By integrating Non-
Stationary Transformers with LightGBM, we construct a robust model where LightGBM
builds a fitting function to capture and simulate the complex coupling relationships among
variables in dynamically evolving chaotic systems. This approach enables the reconstruc-
tion of missing data, restoring sequence completeness and overcoming the limitations of
existing chaotic time series prediction methods in handling missing data. We validate the
proposed method by predicting the future evolution of variables with missing data in both
dissipative and conservative chaotic systems. Experimental results demonstrate that the
model maintains stability and effectiveness even with increasing missing rates, particularly
in the range of 30% to 50%, where prediction errors remain relatively low. Furthermore,
the feature importance extracted by the model aligns closely with the underlying dynamic
characteristics of the chaotic system, enhancing the method’s interpretability and reliabil-
ity. This research offers a practical and theoretically sound solution to the challenges of
predicting chaotic systems with incomplete datasets.
Keywords: chaotic time series prediction; missing data reconstruction; multivariate time
series; dynamical correlation; machine learning
MSC: 37M10
1. Introduction
Chaotic time series research is fundamental to scientific computing, as it captures the
nonlinear dynamical behaviors intrinsic to complex systems. Since the 1990s, machine
learning techniques have been increasingly applied to chaotic time series prediction, result-
ing in diverse methods, such as genetic algorithms [
1
], neuro-fuzzy systems [
2
], wavelet
neural networks [
3
], nonlinear autoregressive models [
4
], Hybrid Connected Complex Neu-
ral Networks (HCNNs) [
5
], ant colony optimization [
6
], Monte Carlo cross-validation [
7
],
and scaled UKF-NARX hybrid models [
8
]. Advances in computational power and data
processing have enabled models like LSTM neural networks [
9
], Echo State Networks
(ESNs) [
10
,
11
], Physics-Informed Neural Operators [
12
], Transformers [
13
], DeepVar [
14
],
and Non-Stationary Transformers (NSTs) [
15
] to effectively capture the intricate dynamics
Mathematics 2025,13, 152 https://doi.org/10.3390/math13010152
Mathematics 2025,13, 152 2 of 26
of chaotic time series. Among these, NSTs have shown exceptional capability in address-
ing non-stationarity through series stationarization and de-stationary attention. In recent
years, the prediction of multi-dimensional or high-dimensional chaotic time series has
also garnered widespread attention. For example, Liu [
16
] proposed a hybrid Proper
Orthogonal Decomposition (POD) and Next-Generation Reservoir Computing (NGRC)
method, Xiong [
17
] developed the Dynamic Adaptive Graph Convolutional Transformer
(DGCT) model, and Fu et al. [
18
] introduced the CTF-former model. These approaches
have provided new insights into multi-task prediction of chaotic time series.
In these applications, a common default assumption is that the input data are com-
plete. However, in practice, missing data are frequently encountered due to sensor failures,
storage device malfunctions, network interruptions, or natural disasters, leading to incom-
plete datasets [
19
]. The continuity of time series and the contextual information required
by models are severely disrupted. The loss of information makes it difficult to capture
long-term dependencies, which is detrimental to any model attempting to extract patterns
from the data. When the proportion of missing data exceeds 5%, the model’s predictions
may become biased, and when the missing rate surpasses 15%, the model’s performance
deteriorates significantly [20–22].
Researchers have explored various methods for managing missing data in chaotic
time series prediction. Jerez [
23
] demonstrated the effectiveness of neural network models
in breast cancer diagnosis using three statistical and three machine learning approaches
for missing data. Similarly, Laña [
24
] showed that machine learning-based imputation
methods enhance traditional traffic congestion prediction models. Although random
sampling approximation techniques [
25
] have achieved good results, they require large
sample sizes to ensure accuracy. The FI-GEM network [
26
] performs well with incomplete
time series but suffers from extended training times and high computational demands
in large-scale or high-dimensional datasets. Zhao Pengcheng [
27
] proposed an auxiliary
variable method to address truncated chaotic missing data. Rangel et al. [
28
] utilized
artificial neural networks and k-Nearest Neighbor (k-NN) techniques to reconstruct and
predict incomplete chaotic time series, achieving significant improvements even at missing
rates exceeding 7%.
However, current research primarily relies on statistical and mathematical modeling
to handle missing data. Traditional imputation methods, such as mean or median filling
and simple regression models, leverage statistical properties. Although easy to apply,
these methods may result in information loss or introduce significant bias [
29
]. In contrast,
algorithms such as k-Nearest Neighbor, Expectation-Maximization, Matrix Factorization,
and Multiple Imputation using Chained Equations [
20
,
30
–
32
] consider multiple influencing
parameters of the real system and their interrelationships as comprehensively as possible,
thereby reducing imputation bias. Recently, interpolation models based on Generative
Adversarial Networks (GANs) [
33
] have achieved higher accuracy; however, their training
process may encounter issues such as mode collapse and difficulty in convergence. The
SAITS model [
34
] improved interpolation performance through a self-attention mechanism,
but it exhibits weak capabilities in modeling nonlinear relationships and is highly sensitive
to hyperparameter selection.
In summary, the current methods for handling chaotic time series with missing data
still have significant shortcomings, including: (1) insufficient capability for dynamic mod-
eling of missing data, making it difficult to fully capture the temporal dependencies and
coupling relationships between variables; (2) low computational efficiency or excessive sen-
sitivity to model parameters, making them unsuitable for large-scale or high-dimensional
scenarios. These issues significantly limit the broad applicability of chaotic time series
prediction in practical applications.
Mathematics 2025,13, 152 3 of 26
A chaotic system is a dynamic system that evolves over time. Compared to general
time series, the variation of each state variable depends not only on its own values but
also on the influence of other state variables. We argue that when addressing the problem
of missing data in chaotic time series, fully considering the dynamical characteristics of
the chaotic sequence—starting from time-delay embedding vectors and the topological
features of the reconstructed system [
35
–
38
], and leveraging the dynamical relationships
among variables—should yield better results than the aforementioned algorithms, such as
k-Nearest Neighbor.
Based on the above considerations, this study proposes a new framework called
LNS_PM (the integration of LightGBM and Non-Stationary Transformers for the prediction
model) for handling prediction tasks of chaotic time series with random missing data at high
missing rates (30–50%). The framework preprocesses missing data using LightGBM [
39
]
by constructing a fitting function
f(x)
to capture and simulate the complex coupling
relationships among variables within the chaotic system that dynamically evolve over time.
The fitting function
f(x)
is then used to reconstruct variables with missing data, restoring
their completeness and providing a complete dataset for subsequent time series prediction
using NST. This method not only considers the temporal dependencies of chaotic time
series data but also fully leverages the nonlinear interactions between variables, thereby
significantly improving the accuracy and reliability of system behavior predictions. It
effectively addresses the limitations of existing chaotic time series prediction methods in
handling missing data and achieves direct prediction of incomplete chaotic time series,
ensuring consistency and efficiency from input to output.
2. Model
As illustrated in Figure 1, the LNS_PM framework proposed in this paper comprises
two primary modules: LightGBM and NST. These modules are responsible for missing
data imputation and time series prediction. The model exploits the dynamical correlations
between variables, effectively reconstructing the missing data in chaotic time series and
facilitating high-precision predictions.
Figure 1. Framework of the LNS_PM.
By setting a series of discrete time points (time vector
tsim
, of length
p
), we numerically
solve the chaotic dynamical equations at each time step
∆tsim
, obtaining approximate
values of the system states. This process forms a matrix with with
p
time steps and
q−
1 state variables, denoted as
p×(q−
1
)
. The time vector
tsim
is then added as an
Mathematics 2025,13, 152 4 of 26
additional column to this matrix, resulting in a data matrix
Mp×q
, which contains both time
information and state variables, Here, the first column represents the time information,
while the remaining
q−
1 columns correspond to the system’s state variable values at the
respective time points.To simulate missing data, one column of the system’s state variables
is randomly set to be completely missing, with a missing rate ranging from 10% to 50%.
M=
t1x1,1 x1,2 . . . x1,q−1
t2x2,1 x2,2 . . . x2,q−1
.
.
..
.
..
.
.....
.
.
tpxp,1 xp,2 . . . xp,q−1
In the phase dedicated to reconstructing the missing data, the primary objective is
to formulate a fitting function
f(x)
that can precisely predict the target variable based
on the provided input variables. This reconstruction employs LightGBM, which builds
decision trees to model and capture the nonlinear interdependencies intrinsic to the data.
This process initiates with the data matrix decomposition, laying the groundwork for
subsequent modeling and analysis.
We define an indicator vector
I= (I1
,
I2
,
. . .
,
Ip)T
to identify rows containing missing
values. Specifically, if the
i
-th row in
M
has a missing value in the designated column,
then
I[i] =
1; otherwise,
I[i] =
0. Based on this indicator vector, we define two index
sets:
Imiss
and
Iobs
,
Imiss ={i∈ {
1, 2,
. . .
,
p}:Ii=
1
}
, representing the indices of rows
with missing values, and
Iobs ={i∈ {
1, 2,
. . .
,
p}:Ii=
0
}
, representing the indices of
rows without missing values. Using these index sets, we separate the data matrix
M
into
two submatrices:
Mmiss = (M[i
,
:])i∈Imiss
, which contains rows with missing values, and
Mobs = (M[i
,
:])i∈Iobs
, which contains rows without missing values. Here,
Mmiss
consists
of the rows in Mindexed by Imiss , while Mobs consists of the rows in Mindexed by Iobs .
To maintain the continuity of indices in the new matrices, we reindex
Mobs
and assume
it consists of
m
rows and
n+
1 columns. Next,
Mobs
is divided into two matrices:
X
and
y
. The matrix
X
represents the input feature vector matrix, containing
m
samples and
n
features, while ycorresponds to the vector of mlabels.
Mobs =
x11 x12 ··· x1ny1
x21 x22 ··· x2ny2
.
.
..
.
.....
.
..
.
.
xm1xm2··· xmn ym
,X=
x11 x12 ··· x1n
x21 x22 ··· x2n
.
.
..
.
.....
.
.
xm1xm2··· xmn
,y=
y1
y2
.
.
.
ym
We consider a feature vector
xil
, where
i=
1, 2,
. . .
,
m
and
l∈(
1,
n)
, as an example to
illustrate the process of reconstructing missing values. The reconstruction process begins
by initializing f0(xl)as:
f0(xl) = arg min
c
m
∑
i=1
(yi−c)2, (1)
where
c
is a constant representing the mean value of the target variable
y
. Subsequently,
the Gradient Boosted Decision Tree (GBDT) algorithm with a leaf-wise learning strategy is
applied to construct the fitting function. Splits are performed on different values
s
of the
feature vector xil . For a given split point s, the loss function is defined as:
L(s) = ∑
i∈RL(s)
(rti)2+∑
i∈RR(s)
(rti)2, (2)
where
RL(s)
and
RR(s)
represent the subsets of samples split by
s
. The residual
rti
, repre-
senting the negative direction of the gradient, is defined as:
Mathematics 2025,13, 152 5 of 26
rti =−∂(yi−f(xil ))2
∂f(xil )f(xil) = ft−1(xi l )
=2(yi−ft−1(xil )). (3)
Here,
yi
is the target value and
ft−1(xil )
is the prediction value of the model for
the sample in the
t−1
iteration. The residual is calculated as the difference between
yi
and
ft−1(xil )
, reflecting the model’s error at the current iteration. Using
(xil
,
rti)
, the
t
-th regression tree is fitted, and the optimal split point
s∗
is chosen to minimize the loss:
s∗=arg minsL(s)
. Based on
s∗
, each leaf node’s region is denoted as
Rtj
,
j=
1, 2,
. . .
,
J
.
For each leaf region j, the optimal fitting value ctj minimizes the loss within that region:
ctj =arg min
c∑
xil ∈Rtj
(yi−ft−1(xil )−c)2. (4)
where
Rtj
represents the set of samples in leaf node
j
. By minimizing the squared loss for
each leaf node, the model can dynamically adjust the leaf node predictions to accommodate
the local distribution characteristics of missing data. By setting the derivative of Equation (4)
to zero, we find:
ctj =1
|Rtj |∑
xil ∈Rtj
(yi−ft−1(xil )). (5)
Here,
|Rtj |
denotes the number of samples in the leaf node region. This gradient-based
optimization mechanism enables LightGBM to progressively approach the true distribution
of missing values with each iteration. The fitting function for the t-th regression tree is:
ft(xl) = ft−1(xl) +
J
∑
j=1
ctj Il ea f (xl∈Rt j), (6)
where
Ilea f (xl∈Rt j)
is an indicator function that determines whether a sample belongs to
region Rtj. After Titerations, the final model combining all regression trees is:
f(xl) = f0(xl) +
T
∑
t=1
J
∑
j=1
ctj Il ea f (xl∈Rt j). (7)
When dealing with high-dimensional variables that contain multiple sparse features,
using the Exclusive Feature Bundling (EFB) algorithm alongside the histogram algorithm
can significantly reduce memory consumption and improve the efficiency of feature split-
ting. The EFB algorithm works by bundling mutually exclusive features—those where only
one feature can have a non-zero value for any given sample—into a single new feature,
thus reducing the total number of features. For each feature’s split, the histogram algorithm
divides continuous features into a fixed number of buckets, where each bucket represents a
range of continuous values. This mapping of feature values to bucket indices simplifies the
search for split points, reducing computational complexity. Following this approach, we
construct a function to model the relationship between inputs and outputs. By inputting the
feature variables of
Mmiss
, we obtain the corresponding target variable, which represents
the imputed missing values. By merging the imputed data with the original dataset and
sorting them by time, we obtain the complete dataset
Y
. Furthermore, during the decision
tree generation process, the model can assess feature importance based on the performance
of input features during training.
Mathematics 2025,13, 152 6 of 26
The completed data
Y
are passed to the NST module for time series prediction. First,
Y
undergoes a normalization process, transforming the raw data into a standardized form
with mean µYand standard deviation σY, resulting in the normalized sequence Y′:
Y′=Y−1µ⊤
Y
σY
(8)
We begin with the original attention formula [15]:
Attn(Q,K,V) = Softmax σ2
YQ′K′⊤ +1µ⊤
QK⊤
√dk!(9)
Here,
Q
,
K
,
V
represent the query, key, and value matrices computed from the input
sequence
Y
, respectively. The normalized matrices are defined as
Q′=Q−1µ⊤
Q
σY
, where
µQ
is the mean of
Q
along the temporal dimension. Similarly,
K′
and
V′
are computed in the
same manner. The term dkdenotes the dimension of the key vectors.
In addition to the normalized components
Q′
and
K′
derived from the stationary
sequence
Y′
, the attention formula also incorporates non-stationary information, such as
σY
,
µQ
, and the unnormalized
K
, which are partially removed during the normalization
process. To capture the non-stationary properties of the input chaotic time series, the
formula introduces two de-trending factors:
τ=σ2
Y∈R+
and
∆=KµQ∈Rm×1
.
Given that strict linear assumptions rarely hold for deep models, the de-trending factors are
learned directly from the statistical features of the unnormalized sequence
Y
using a simple
yet effective multi-layer perceptron (MLP):
log τ=MLP(σY
,
Y)
,
∆=MLP(µY
,
Y)
. The
final formula for non-stationary attention is as follows:
Attn(Q′,K′,V′,τ,∆) = Softmax τQ′K′⊤ +∆⊤
√dk!V′(10)
The prediction result obtained after processing through the encoder-decoder structure
is represented as the standardized sequence
Z′
. It is then transformed back to the original
time series scale through a de-normalization step, resulting in the final output Z:
Z=σY(Z′+µY). (11)
The advantage of this modular design lies in that the LightGBM module effectively
imputes missing values by using the weighted residuals of decision trees. This approach
allows the model to leverage correlations between different feature variables, selecting the
most predictive features to construct the tree branches. The multi-dimensional feature corre-
lations enable the model to use information from related variables to impute missing values,
thereby reconstructing complete data that adhere to the system’s dynamical rules, ensuring
accuracy and robustness in data imputation. On the other hand, the NST module addresses
complex time series features and non-stationarity through the self-attention mechanism,
avoiding over-smoothing. This mechanism allows the NST to capture dynamic changes in
time series data, thereby enhancing the model’s prediction accuracy and robustness.
3. Experiments
3.1. Chaotic Systems
In this section, we use four chaotic dynamical systems—the Lorenz system [
40
], the
Hyperchaotic Lorenz system [
41
], the Coupled Lorenz system [
42
], and the Conservative
Chaotic system [
43
]—to illustrate the LNS
_
PM method described in Section 2. We also
compare its performance with other classical prediction models to verify its advantages.
Mathematics 2025,13, 152 7 of 26
The dimensions of the Lorenz system, Hyperchaotic Lorenz system, and Coupled Lorenz
system progressively increase, reflecting the growing complexity of their chaotic behavior.
This gradual progression allows for a comprehensive evaluation of the model’s predictive
capabilities under varying levels of complexity and dimensionality, as well as the influence
of the number of feature variables on the model’s performance. In contrast to relatively
dissipative systems, chaos control in conservative and spatiotemporal systems is more
challenging. Conservative systems, characterized by energy conservation, exhibit more
complex and unpredictable dynamical behavior. In the aerospace field, the principles and
methods of chaos control in conservative systems have direct and practical applications.
1. Lorenz System
The Lorenz system, proposed by the American meteorologist Edward Lorenz in 1963,
is a model describing thermal convection instability. It exhibits key chaotic character-
istics, including sensitivity to initial conditions and non-periodicity, making it an ideal
candidate for evaluating time series prediction models. The system is governed by the
following equations:
˙
x=−σ(x−y),
˙
y=−xz +rx −y,
˙
z=xy −bz.
(12)
where
σ=
10,
b=
8
/
3, and
r=
28. Under these conditions, the system exhibits
chaotic behavior. The Lorenz system is deterministic, and governed by known parameters,
which ensures repeatable dynamics under identical initial conditions. In this study, we
use the classical Runge–Kutta method to solve the equations, with initial values set to
(1.01, 1.01, 0.00), resulting in a dataset comprising a 3 ×105matrix.
2. Hyperchaotic Lorenz System
The Hyperchaotic Lorenz system, due to its multiple positive Lyapunov exponents,
exhibits stronger resistance to interference compared to general chaotic systems, making its
dynamic behavior more difficult to predict. The system can be expressed as follows:
˙
x=a(y−x) + w,
˙
y=cx −y−xz,
˙
z=xy −bz,
˙
w=−yz +rw.
(13)
When the parameters are
a=
10,
b=
8
/
3,
c=
28, and
−
1.52
<r<−
0.06, the
system exhibits hyperchaotic behavior. Specifically, when
r=−
1, the system has four
Lyapunov exponents:
λ1=
0.3381,
λ2=
0.1586,
λ3=
0,
λ4=−
15.1752. In this study, the
initial values are set to (1.1, 2.2, 3.3, 4.4), resulting in a dataset matrix of 4 ×12, 485.
3. Coupled Lorenz System
To verify the model’s predictive capabilities on higher-dimensional nonlinear systems
with missing data, this study uses the Coupled Lorenz system model to simulate inter-
actions among subsystems. The dynamic state of each subsystem
k
is represented by the
following differential equations:
dxk/dt =−10hxk−yk+c∑N
l=1a(x,y)
kl (yl−yk)i,
dyk/dt =28(1+hk)xk−yk−xkzk,
dzk/dt =−(8/3)zk+xkyk.
(14)
Mathematics 2025,13, 152 8 of 26
where
c = 1, N = 10
, and the coupling matrix
a(x,y)
kl
is randomly generated as 0 or 1, rep-
resenting whether there is a coupling between subsystems
k
and
l
. The values of
hk
are
randomly generated between
−
0.25 and 0.25 to simulate the dynamic differences between
each subsystem. The initial state of each subsystem is generated randomly within the range
[0, 0.1], resulting in a dataset of a 30 ×40, 000 matrix.
4. Conservative chaotic system
To validate the generalization capability of the proposed model on Conservative
Chaotic systems, this study uses a four-dimensional Conservative Chaotic System (CCS)
that exhibits infinite-scroll characteristics.
˙
x1=ax2,
˙
x2=bsin(x1)x4−asin(x1),
˙
x3=cx4
˙
x4=−bsin(x1)x2−cx3.
(15)
When appropriate initial values and control parameters are selected, the system ex-
hibits a multi-scroll phenomenon. For example, by setting
(a
,
b
,
c) = (
6, 5.57, 6
)
and initial
values
(
1.9, 1, 1, 1
)
, we conducted simulations at various times:
t=
100
s
,
t=
200
s
,
t=
400
s
,
t=
600
s
, and
t=
6000
s
. The corresponding phase trajectories display 7,
9, 11, 12, and 56 scrolls, respectively. These simulations demonstrate that the number of
scrolls increases with longer simulation times. Theoretically, as time approaches infinity,
the system will exhibit an infinite number of scrolls along the
x1
-axis. From an equilibrium
perspective, the central equilibrium points and unstable saddle points alternate along the
x1
-axis. Despite the increasing number of scrolls, the system’s Hamiltonian energy remains
constant at
H3=
1.823289. This characteristic not only tests the model’s ability to learn
complex dynamic behaviors but also evaluates its effectiveness in handling chaotic phe-
nomena observed in real-world physical systems. In this study, at
t=
200
s
, the equations
were solved using the fourth-order Runge–Kutta method, resulting in a
2×100, 000
matrix.
3.2. Dataset
For each chaotic system, datasets are generated based on the system parameters
specified in Section 3.1. We randomly set the target variable with missing data at rates
of 10%, 20%, 30%, 40%, and 50%. These datasets are classified into training (65% of the
samples), validation (15% of the samples), and test sets (20% of the samples). The training
set allows the optimization of model parameters, such as weights and biases. The validation
set, in contrast, is not directly involved in the optimization process but is used to fine-tune
the model’s hyperparameters. Table 1summarizes the specific hyperparameters for the
LNS
_
PM model. Here, Lr is the learning rate,
λ1
and
λ2
are the regularization coefficients
controlling overfitting, ml is the minimum number of samples required in a leaf node, nl
is the number of leaves used in LightGBM, ep is the number of training epochs, bs is the
batch size for each training iteration, hl is the number of hidden layers in the projector, hd
is the hidden layer dimension of the projector, and es is the early stopping criterion, which
ensures the training process terminates when performance stops improving.
The hyperparameter hd represents the hidden layer dimension of the MLP-based pro-
jector responsible for capturing these factors. To balance the efficiency of hyperparameter
tuning and model performance, we fixed the number of hidden layers at 2 and varied the
hidden layer dimension within 64, 128, 256. Each experiment was repeated using different
random seeds to ensure robustness, with lower RMSE/MAPE values indicating better
performance. This design allows the projector to effectively model non-stationary dynamics
while maintaining computational efficiency.
Mathematics 2025,13, 152 9 of 26
Table 1. Hyperparameters of LNS_PM model components.
Module Lr es λ1λ2ml nl ep bs
LightGBM 0.01 50 0.5 0.5 20 24- -
NST 0.0008 100 - - - -
100 256
Additional NST Hyperparameters:
hl 2 (number of hidden layers in projector)
hd 128 (hidden layer dimension of the projector)
In model identification, the test set is not used for training the model but is exclu-
sively employed to evaluate the model’s performance in inference mode. We included
the initial state data generated by the chaotic equations as part of the training data to
account for the influence of the transient on model training. This approach ensures that the
training data capture the system’s long-term and transient behavior, enabling the model
to learn richer dynamic characteristics. We set random seeds for tree iteration, split-point
selection, and global settings to enhance the stability and reproducibility of the results. This
procedure ensures that data partitioning, feature selection, and other random processes
remain consistent each time the code runs. Additionally, the model incorporates an early
stopping mechanism. The training process ends when the model’s performance on the
validation set does not significantly improve over consecutive iterations. This method
avoids unnecessary iterations and prevents overfitting.
Using these datasets, the performance differences between LNS
_
PM and other models,
such as NST, DeepVar, LSTM, and ESN, can be investigated when handling complex,
chaotic systems with varying degrees of missing data. Table 2presents the parameter
settings for these models. In this table, rs refers to the reservoir size, sr denotes the spectral
radius, sp represents sparsity, and ts corresponds to Teacher Forcing, which we used
during the training phase. Appropriate scaling of these parameters can help stabilize the
training process.
Table 2. Hyperparameters of the comparison models.
Lr ep bs hd/hs rs sr/sp/ts
NST 0.0001 500 256 128 – –
DeepVar 0.01 100 128 30 – –
LSTM 0.001 100 32 30 – –
ESN – – – – 500 1/0.45/1.12
3.3. Performance Metrics
We use the Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error
(MAPE) as metrics to evaluate the effectiveness of the LNS
_
PM model. RMSE measures
the absolute error between predicted and actual values, while MAPE quantifies the relative
error, making it suitable for comparing data of different scales. Lower values of RMSE
and MAPE indicate better model performance. To further assess whether the performance
differences among various prediction models across multiple datasets are statistically
significant, we employ analysis of variance (ANOVA). This method evaluates the statistical
significance of performance differences between the models.
RMSE =r1
nΣn
i=1(zi−ˆ
zi)2, MAPE =100%
nΣn
i=1
zi−ˆ
zi
zi
.(16)
In addition, to quantify the uncertainty of the predictive performance of different
models under varying data missing rates, we also calculated the confidence interval (CI) of
Mathematics 2025,13, 152 10 of 26
the prediction mean at a 95% confidence level. The calculation of the confidence interval
helps to understand the stability and reliability of the model’s prediction results, where a
narrower confidence interval indicates more stable predictive performance. The formula
for calculating the confidence interval is as follows:
CI =¯
x±tα/2,d f ·s
√n(17)
where
¯
x
is the sample mean;
tα/2,d f
is the
t
-value corresponding to the chosen confidence
level (e.g., 95%), with
d f =n−
1 as the degrees of freedom;
s
is the sample standard
deviation;
n
is the sample size. By using this formula, the confidence interval at the 95%
confidence level can be calculated, which facilitates statistical validation of the predictive
model’s performance.
3.4. Results Analysis
We analyzed the predictive performance of five models (LNS
_
PM, NST, DeepVar,
LSTM, and ESN) on the chaotic systems described in Section 3.1, evaluating their perfor-
mance under missing data rates ranging from 10% to 50%. For greater clarity, we plotted
only the last 50 points of the predicted data (see Figure 2). Each system’s results are dis-
played in five subplots corresponding to the different missing rates (10%, 20%, 30%, 40%,
and 50%). Each subplot includes six curves representing the true values and the predictions
of the LNS_PM, NST, LSTM, ESN, and DeepVar models.
From these curves, it is evident that as the missing rate increases, the discrepancy
between the predicted results of all models and the actual observed values becomes more
pronounced. Specifically, DeepVar demonstrates significant bias at some time points, while
LSTM and ESN show particularly large deviations from the observed values. In contrast,
LNS
_
PM and NST generally track the trends of the actual data more closely. However, at
missing rates between 30% and 50%, NST exhibits a noticeable bias between its predictions
and the true values. Conversely, LNS
_
PM consistently maintains lower fluctuations and
bias, producing smoother curves that align closely with the observed values, even in regions
of frequent small fluctuations.
Figure 2also reveals that under identical missing rate conditions, reducing the amount
of training data decreases prediction accuracy while increasing the number of feature
variables improves predictive performance. Overall, LNS
_
PM demonstrates superior
predictive accuracy across chaotic systems of varying dimensions and missing rates. It
effectively captures the data trends and exhibits a distinct advantage in chaotic prediction
with missing data.
Another perspective for evaluating the results is through scatter plots, which depict
the relationship between each pair of predicted and actual values. The closer the points are
to the 45-degree line (where predicted values equal actual values), the more accurate the
model’s predictions.
Figures 3–6present scatter plots for the predictions from five different models for
the Lorenz, Hyperchaotic Lorenz, Coupled Lorenz, and Conservative Chaotic systems,
respectively. Figure 3shows that testing different prediction models on the same dataset
with identical missing rates reveals that LNS
_
PM delivers highly stable prediction results.
Most points align closely with a straight line, indicating a strong linear relationship between
the predicted and actual values and suggesting that the model effectively captures the
underlying data trends. NST also demonstrates relatively stable predictions but with
slightly larger discrepancies between the predicted and actual values compared to LNS
_
PM,
which may indicate a degree of bias. DeepVar exhibits significant volatility in its predictions,
with notable deviations between the predicted and actual values. Similarly, predictions
Mathematics 2025,13, 152 11 of 26
from LSTM and ESN show fluctuations, with substantial discrepancies in certain cases,
reflecting instability in their performance.
Figure 2. Prediction results of five forecasting models (colored lines) for the Lorenz system, Hyper-
chaotic Lorenz system, Coupled Lorenz system, and Conservative Chaotic system (columns 1 to 4),
with missing rates ranging from 10% to 50% (raws 1 to 5).
Mathematics 2025,13, 152 12 of 26
Figure 3. Scatter plots of observed versus forecasted values for the Lorenz system, with different
predictors (columns 1 to 5) and increasing missing rates (rows 1 to 5, ranging from 10% to 50%).
Perfect predictions align along the diagonal.
Mathematics 2025,13, 152 13 of 26
Figure 4. Scatter plots of observed versus forecasted values for the Hyperchaotic Lorenz system, with
different predictors (columns 1 to 5) and increasing missing rates (rows 1 to 5, ranging from 10% to
50%). Perfect predictions align along the diagonal.
Mathematics 2025,13, 152 14 of 26
Figure 5. Scatter plots of observed versus forecasted values for the Coupled Lorenz system, with
different predictors (columns 1 to 5) and increasing missing rates (rows 1 to 5, ranging from 10% to
50%). Perfect predictions align along the diagonal.
Mathematics 2025,13, 152 15 of 26
Figure 6. Scatter plots of observed versus forecasted values for the Conservative Chaotic system,
with different predictors (columns 1 to 5) and increasing missing rates (rows 1 to 5, ranging from 10%
to 50%). Perfect predictions align along the diagonal.
Testing the same prediction model on datasets with different missing rates reveals that
LNS
_
PM produces consistently stable predictions across various missing rates, demonstrat-
ing its robustness across different samples. NST also shows a certain level of stability, but its
prediction errors become larger at higher missing rates (30–50%). DeepVar’s performance is
inconsistent across the different missing rates. LSTM and ESN exhibit significant volatility
Mathematics 2025,13, 152 16 of 26
in their predictions, with large discrepancies between predicted and actual values. The
scatter plot results for the Hyperchaotic Lorenz (Figure 4), Coupled Lorenz (Figure 5), and
Conservative Chaotic systems (Figure 6) align with this analysis. Specifically, in Figure 6,
the prediction results for the Conservative Chaotic system show that LNS_PM and NST
have their predicted values well distributed near the diagonal across all missing rates, indi-
cating good performance. On the other hand, as the missing rate increases, the predicted
values of DeepVar and LSTM gradually deviate from the diagonal, suggesting that these
models perform poorly at higher missing rates. ESN, on the other hand, has predicted
values concentrated near the horizontal axis at all missing rates, with little variation with
the observed values, indicating that this model is unsuitable for predicting chaotic time
series with missing data. Overall, LNS
_
PM consistently delivers a high level of accuracy,
with predicted values closely matching the actual ones, while the other models show larger
prediction errors.
Figures 7and 8present the variation in the root mean square error (RMSE) and
mean absolute percentage error (MAPE) for different prediction models under four chaotic
systems (Lorenz, Hyperchaotic Lorenz, and Coupled Lorenz). From the figures, it can
be observed that the LNS_PM model performs the best across all three Lorenz systems,
particularly at higher missing rates (30% to 50%), where both RMSE and MAPE remain
at relatively low values, demonstrating the model’s strong robustness to missing data. In
comparison, the NST model performs similarly to LNS_PM but exhibits slightly higher
errors in low-dimensional systems (e.g., the Lorenz system). Moreover, as the missing rate
increases, the errors for NST also increase. Traditional LSTM and ESN models are more
sensitive to missing data; with increasing missing rates, their RMSE and MAPE values rise
significantly, particularly in the Hyperchaotic Lorenz system, where their performance is
notably poor.
Figure 7. The bar charts display the RMSE values with error bars for four chaotic systems: Lorenz (a),
Hyperchaotic Lorenz (b), Conservative Chaotic system (c), and Coupled Lorenz (d), under varying
missing rates (10–50%). The results are compared across five predictors: LNS
_
PM (orange), NST
(gray), LSTM (yellow), ESN (blue), and DeepVar (green). The error bars indicate the consistency of
the RMSE results.
Mathematics 2025,13, 152 17 of 26
Figure 8. The bar charts display the MAPE values with error bars for four chaotic systems: Lorenz (a),
Hyperchaotic Lorenz (b), Conservative Chaotic system (c), and Coupled Lorenz (d), under varying
missing rates (10–50%). The results are compared across five predictors: LNS
_
PM (orange), NST
(gray), LSTM (yellow), ESN (blue), and DeepVar (green). The error bars indicate the consistency of
the MAPE results.
Furthermore, the performance of the LNS_PM model varies across systems of different
dimensions. As shown in Figure 8, for the Lorenz system, where sufficient training data
are available, the LNS_PM model achieves smaller prediction errors. For the Hyperchaotic
Lorenz system, the increased system complexity is offset by the availability of additional
feature inputs, enabling the LNS_PM model to better capture useful features and achieve su-
perior prediction performance. For the Coupled Lorenz system, although the training data
are relatively limited, the increased system dimension introduces more features, allowing
the model to construct more complex mapping relationships and thereby improve predic-
tion accuracy. The consistently low errors across all missing rates in the Coupled Lorenz
system further highlight the superiority of the LNS_PM model in multi-variable systems.
In summary, the LNS_PM model demonstrates high prediction accuracy and robust-
ness across all cases, particularly under high missing rates. This is primarily attributed to
the interpolation method proposed in this study, which effectively reconstructs the missing
data and enables the model to capture the temporal evolution patterns of system states. In
contrast, traditional LSTM and ESN models tend to overfit incomplete data, preventing
them from learning truly useful patterns. Consequently, their performance deteriorates
significantly when applied to new datasets with similarly high missing rates.
Moreover, the visualization results of feature importance (Figure 9) further emphasize
the critical role of the underlying dynamic features in the model’s predictions. Through the
feature importance analysis of the Lorenz, Hyperchaotic Lorenz, Coupled Lorenz, and Con-
servative Chaotic systems, we observe that “x” plays a critical role in predictions due to its
coupling relationships with other variables (such as “y” and “z”), which significantly influ-
ence the prediction results. This finding aligns with previous studies, which concluded that
variables with higher frequencies of appearance in one or more equations tend to perform
better when used to reconstruct other variables [
36
]. In the Hyperchaotic Lorenz system,
the introduction of the variable “w” not only expanded the system’s dimensionality but
also significantly altered its dynamic properties, driving the system to exhibit hyperchaotic
Mathematics 2025,13, 152 18 of 26
behavior. According to the feature importance comparison in Figure 9b, the importance of
“w” follows closely after “x”. This result not only highlights the pivotal role of “w” in the
system but also confirms its significant contribution to the overall dynamic behavior.
Figure 9. (a) Lorenz System, (b) Hyperchaotic Lorenz system, (c) Conservative Chaotic system,
(d) Coupled Lorenz system—Visualization of the feature’s impact on the model’s output.
Based on the results of the statistical tests (Table 3), we observe significant differences
among the models across varying dimensions. For Lorenz data, the F-statistic is 139.19,
with a p-value of 2.68
×
10
−14
. For Hyperchaotic Lorenz data, the F-statistic increases to
763.04, with a p-value of 1.50
×
10
−21
. For Conservative Chaotic system data, the F-statistic
rises further to 1447.28, accompanied by a p-value of 2.57
×
10
−24
. Finally, for Coupled
Lorenz data, the F-statistic is 857.41, with a p-value of 4.70 ×10−22.
At a significance level of
α=
0.05, all p-values are well below the threshold, con-
firming that the performance differences among models are statistically significant across
all dimensions. The extremely low p-values strongly suggest that as data dimensionality
increases, the performance disparities among models become more pronounced. This
result underscores the impact of data dimensionality on model behavior and highlights the
importance of considering dimensional effects when evaluating model performance. Simi-
larly, based on the CI in Table 4, we can observe that the confidence intervals of LNS_PM
do not overlap with those of ESN, DeepVar, and LSTM, indicating that the differences in
predictive performance among these models are statistically significant. Combined with
the RMSE and MAPE results, LNS_PM not only maintains relatively low errors under
different missing rates but also demonstrates a relatively stable confidence interval width.
We experimentally validated the proposed framework on datasets ranging from 3 to
30 dimensions. In higher-dimensional datasets, the increased diversity and redundancy
of features provide additional information for feature inference, allowing the model to
achieve greater accuracy during the completion phase. The experimental results show that
the prediction accuracy of the proposed framework is comparable to classical models (such
as NST and DeepVar) when the missing data ratio is low. However, as the missing data
ratio increases to 50%, the proposed framework significantly outperforms these classical
models. This advantage stems from the LightGBM module’s strong ability to represent the
complex distribution of high-dimensional data and its robustness in sparse data scenarios.
Mathematics 2025,13, 152 19 of 26
Table 3. Statistical results for different chaotic systems.
Data F-Statistic p-Value Significance Level (α=0.05)
Lorenz 139.19 2.68 ×10−14 Significant
Hyperchaotic
Lorenz 763.04 1.50 ×10−21 Significant
Coupled Lorenz 857.41 4.70 ×10−22 Significant
Conservative
Chaos 1447.28 2.57 ×10−24 Significant
Table 4. Comparison of predictors with 95% confidence intervals for different missing rates (10%,
20%, 30%, 40%, 50%).
Data LNS_PM NST LSTM DeepVar ESN
Lorenz
(23.583, 23.782) (23.424, 23.621) (20.242, 20.420) (21.738, 21.906) (26.553, 26.610)
(23.668, 23.867) (23.490, 23.689) (20.256, 20.435) (22.346, 22.523) (26.469, 26.525)
(23.640, 23.841) (23.648, 23.843) (21.375, 21.544) (24.902, 25.056) (26.601, 26.659)
(23.746, 23.943) (23.723, 23.914) (21.962, 22.100) (22.753, 22.948) (26.586, 26.643)
(23.669, 23.870) (23.525, 23.696) (20.679, 20.804) (20.589, 20.791) (26.624, 26.681)
Hyperchaotic Lorenz
(24.507, 25.053) (24.479, 25.017) (22.436, 22.861) (23.522, 24.043) (39.756, 40.127)
(24.355, 24.899) (23.973, 24.494) (20.174, 20.625) (23.515, 24.049) (39.877, 40.253)
(24.070, 24.617) (24.026, 24.532) (19.714, 20.209) (21.302, 21.796) (38.838, 39.212)
(24.039, 24.579) (24.595, 25.167) (20.230, 20.673) (23.899, 24.442) (38.881, 39.253)
(24.178, 24.725) (24.614, 25.086) (20.003, 20.309) (24.591, 25.161) (38.994, 39.356)
Coupled Lorenz
(17.261, 17.360) (17.249, 17.348) (16.576, 16.654) (18.440, 18.535) (16.915, 17.001)
(17.298, 17.396) (17.311, 17.410) (16.391, 16.466) (18.182, 18.279) (17.094, 17.179)
(17.271, 17.370) (17.271, 17.369) (15.582, 15.653) (17.081, 17.171) (17.132, 17.216)
(17.277, 17.376) (17.229, 17.327) (15.336, 15.413) (17.987, 18.080) (17.044, 17.133)
(17.269, 17.368) (17.279, 17.377) (15.876, 15.941) (17.884, 17.974) (17.042, 17.125)
Conservative Chaos
(−0.007, 0.011) (−0.006, 0.012) (0.048, 0.066) (0.008, 0.025) (−0.646, −0.645)
(−0.006, 0.012) (−0.005, 0.012) (0.119, 0.138) (−0.066, −0.047) (−0.644, −0.643)
(−0.006, 0.011) (−0.007, 0.011) (0.070, 0.088) (0.069, 0.087) (−0.646, −0.646)
(−0.006, 0.012) (−0.007, 0.011) (0.048, 0.067) (−0.101, −0.083) (−0.645, −0.644)
(−0.006, 0.011) (−0.007, 0.011) (0.156, 0.174) (−0.162, −0.143) (−0.641, −0.639)
Based on these results and the theoretical analysis, the proposed framework demon-
strates excellent scalability, making it well suited for real-world complex systems with
high-dimensional datasets and large missing data ratios. Furthermore, its modular design
ensures high flexibility and adaptability when handling datasets of varying scales and
structures.
Regarding computational cost, using a dataset with 100 feature dimensions and 100K
samples as an example, the training time of the LightGBM module within the proposed
framework is only a few minutes, with inference time in the millisecond range. In contrast,
the NST model requires several hours of training and seconds for inference. Therefore,
the LightGBM module has a low impact on the overall computational cost, offering high
computational efficiency. Additionally, with the high-performance computing capabilities
of modern GPUs, the proposed framework is less affected by computational resource
limitations in practical applications, demonstrating strong applicability and scalability.
To further validate the superiority of the proposed data imputation method within
the framework designed in this study, we conducted additional comparative experiments
with two classical methods (KNN and GAN [
44
]). The prediction performance of LNS_PM,
KNN-NST, and GAN-NST was analyzed, and the results were presented using prediction
curves, scatter plots, and error plots. As shown in Figure 10, both the LNS_PM and
Mathematics 2025,13, 152 20 of 26
KNN-NST methods effectively capture the trend of the curve, with their prediction results
distributed along the diagonal in the scatter plots (Figures 11–14), indicating that both
methods achieve satisfactory data imputation results. However, analysis of the error plots
(Figures 15 and 16) reveals that LNS_PM achieves lower prediction errors across all datasets,
demonstrating higher prediction accuracy that better meets the requirements of chaotic
system forecasting. This further highlights the advantages of the proposed method in
handling missing data.
Figure 10. Prediction results of three forecasting models (colored lines) for the Lorenz system,
Hyperchaotic Lorenz system, Coupled Lorenz system, and Conservative Chaotic system (columns 1
to 4), with missing rates ranging from 10% to 50% (raws 1 to 5).
Mathematics 2025,13, 152 21 of 26
Figure 11. Scatter plots of observed versus forecasted values for the Lorenz system, with different
predictors (LNS_PM, KNN-NST, GAN-NST, rows 1 to 3) and increasing missing rates (columns 1 to
5, ranging from 10% to 50%). Perfect predictions align along the diagonal.
Figure 12. Scatter plots of observed versus forecasted values for the Hyperchaotic Lorenz system,
with different predictors (LNS_PM, KNN-NST, GAN-NST, rows 1 to 3) and increasing missing rates
(columns 1 to 5, ranging from 10% to 50%). Perfect predictions align along the diagonal.
Mathematics 2025,13, 152 22 of 26
Figure 13. Scatter plots of observed versus forecasted values for the Conservative Chaotic system,
with different predictors (LNS_PM, KNN-NST, GAN-NST, rows 1 to 3) and increasing missing rates
(columns 1 to 5, ranging from 10% to 50%).
Figure 14. Scatter plots of observed versus forecasted values for the Coupled Lorenz system, with
different predictors (LNS_PM, KNN-NST, GAN-NST, rows 1 to 3) and increasing missing rates
(columns 1 to 5, ranging from 10% to 50%).
Mathematics 2025,13, 152 23 of 26
Figure 15. The bar charts display the RMSE values with error bars for four chaotic systems: Lorenz (a),
Hyperchaotic Lorenz (b), Conservative Chaotic system (c), and Coupled Lorenz (d), under varying
missing rates (10–50%). The results are compared across three predictors: LNS_PM (Blue), KNN-NST
(Orange), GAN-NST (Gray). The error bars indicate the consistency of the RMSE results.
Figure 16. The bar charts display the MAPE values with error bars for four chaotic systems: Lorenz (a),
Hyperchaotic Lorenz (b), Conservative Chaotic system (c), and Coupled Lorenz (d), under varying
missing rates (10–50%). The results are compared across three predictors: LNS_PM (Blue), KNN-NST
(Orange), GAN-NST (Gray). The error bars indicate the consistency of the MAPE results.
Mathematics 2025,13, 152 24 of 26
4. Conclusions
We presented a general model, LNS
_
PM, designed to address the challenge of re-
constructing and predicting chaotic time series with missing data. This model integrates
LightGBM, a gradient-boosting machine learning algorithm, into NST. We compared the
prediction accuracy and robustness of five different models (LNS
_
PM, NST, LSTM, ESN,
and DeepVar) on complex, chaotic system time series with missing data, specifically assess-
ing their performance under missing rates of 10% to 50%. Additionally, we examined the
impact of system dimensionality on prediction accuracy. The experimental results show
that the proposed model demonstrates a significant advantage with larger missing rates,
offering high robustness in predicting chaotic time series with missing data. Compared
to NST without LightGBM, LNS
_
PM substantially reduces the prediction error across
chaotic systems of varying complexity. Furthermore, the feature importance analysis aligns
with the underlying dynamic relationships within the chaotic system equations, further
validating the effectiveness and reliability of our approach.
Future research can explore several directions. First, integrating other advanced ma-
chine learning algorithms into LNS
_
PM could enhance its predictive performance. Second,
applying the model to real-world chaotic datasets, such as price fluctuations in financial
markets, stock market prediction, weather forecasting, and wind power generation, which
often involve more 422 complex noise and nonlinear characteristics, to further validate
its applicability. Future studies should also focus on developing specialized optimiza-
tion techniques to address these complexities while evaluating the model’s scalability
and robustness.
Author Contributions: Conceptualization, H.M.; methodology, Y.W.; software, J.L.; validation, J.L.;
formal analysis, J.L.; investigation, J.L.; resources, J.L.; data curation, J.L.; writing—original draft
preparation, J.L.; writing—review and editing, Z.Y. All authors have read and agreed to the published
version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: The data will be made available by the authors upon request.
Conflicts of Interest: The authors declare no conflicts of interest.
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