=Paper=
{{Paper
|id=Vol-3899/paper17
|storemode=property
|title=Hybrid method for restoring missing sensor data with adaptive control based on neuro-fuzzy networks
|pdfUrl=https://ceur-ws.org/Vol-3899/paper17.pdf
|volume=Vol-3899
|authors=Anatoliy Sachenko,Victoria Vysotska,Serhii Vladov,Viktor Vasylenko,Maclej Dobrowolski
|dblpUrl=https://dblp.org/rec/conf/advait/SachenkoVVVD24
}}
==Hybrid method for restoring missing sensor data with adaptive control based on neuro-fuzzy networks==
https://ceur-ws.org/Vol-3899/paper17.pdf
Hybrid method for restoring missing sensor data with
adaptive control based on neuro-fuzzy networks ⋆
Anatoliy Sachenko1,†, Victoria Vysotska2,†, Serhii Vladov3,∗,†, Viktor Vasylenko3,† and Maclej
Dobrowolski4,†
1
West Ukrainian National University, Lvivska Street 11 46009 Ternopil, Ukraine
2
Lviv Polytechnic National University, Stepan Bandera Street 12 79013 Lviv, Ukraine
3
Kharkiv National University of Internal Affairs, L. Landau Avenue 27 61080 Kharkiv, Ukraine
4
Kazimierz Pulaski University of Radom, Department of Informatics, Jacek Malczewski Street 29 26-600 Radom, Poland
Abstract
This research presents a hybrid method for restoring missing sensor data using adaptive control, aimed at
enhancing data recovery accuracy in complex situations. The proposed algorithm involves creating a
training dataset, a control element, and training hybrid models. Key steps include identifying data subsets,
randomly removing elements to simulate gaps, estimating their values, clustering for accuracy, and
developing neuro-fuzzy models tailored to specific data groups. Implemented as a Takagi-Sugeno neuro-
fuzzy network, the method effectively combines fuzzy rules with neural computations, allowing it to handle
variability and noise in real measurements with high adaptability. Computational experiments focused on
restoring gas temperature data from the TV3-117 turboshaft engine demonstrate high accuracy, with
deviations ranging between 0.002 and 0.007, indicating robust performance even under sensor failures. The
model achieves over 99% accuracy, a precision of 0.983, and an F1-score of 0.991, significantly
outperforming traditional methods such as two-layer feedforward networks and ANFIS networks. This
research offers a reliable solution for sensor data recovery and advances adaptive control systems,
improving the reliability and performance of aviation technologies and other dynamic applications.
Keywords
restoring, missing sensor data, hybrid method, Takagi-Sugeno neuro-fuzzy network, helicopter turboshaft
engine, fuzzy rules1
1. Introduction
Restoring missing sensor data is one of the key tasks in technical monitoring and control systems
for complex objects such as aviation engines [1], robotic systems [2], and production lines [3]. The
completeness and reliability of information received from sensors are crucial for these systems
maintaining safe and efficient operation. However, in practice, sensors may fail or transmit distorted
data due to technical malfunctions, external influences, or communication failures [4]. These factors
pose the risk of inaccurate assessments regarding the objects condition, potentially leading to
undesirable outcomes such as accidents or inefficient management.
The research relevance methods for restoring sensor data arises from the need to enhance the
information systems reliability and precision. Modern approaches, including neural networks and
statistical models, enable not only the missing values restoring but also the systematic and random
measurement errors correction. This is particularly important in environments with stringent safety
AdvAIT-2024: 1st International Workshop on Advanced Applied Information Technologies, December 5, 2024, Khmelnytskyi,
Ukraine - Zilina, Slovakia
∗
Corresponding author.
†
These authors contributed equally.
as@wunu.edu.ua (A. Sachenko); victoria.a.vysotska@lpnu.ua (V. Vysotska); serhii.vladov@univd.edu.ua (S. Vladov);
klk.vonrgp@gmail.com (V. Vasylenko); m.dobrowolski@uthrad.pl (M. Dobrowolski)
0000-0002-0907-3682 (A. Sachenko); 0000-0001-6417-3689 (V. Vysotska); 0000-0001-8009-5254 (S. Vladov); 0000-0002-
9313-861X (V. Vasylenko); 0000-0003-0296-9651 (M. Dobrowolski)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
and cost-efficiency requirements, where timely detection and compensation for data anomalies can
extend equipment lifespan and reduce maintenance costs.
2. Related works
Existing research on restoring missing sensor data covers a wide range of approaches, from
traditional interpolation methods to advanced machine learning algorithms and neural networks.
Classical methods like linear interpolation [5] or polynomial approximation [6] are effective for small
gaps in data with predictable patterns but fall short when dealing with nonlinear dependencies,
noise, or dynamic systems, making them less suitable for complex applications.
With the artificial intelligence and machine learning advancement, the neural networks use has
gained significant attention in data restoring. Recurrent neural networks (RNN) [7], particularly long
short-term memory (LSTM) models [8,9], are frequently used to account for temporal dependencies
and can recover information despite extended gaps. Other approaches [10,11] involve autoencoders,
which train data representations and restore missing values. These methods show strong
performance with nonlinear time series and hold promise for real-time applications.
Neuro-fuzzy systems offer another effective approach, combining neural networks with fuzzy
logic to handle uncertainty and noise [12]. Models like the Sugeno approach apply flexible rules to
process fuzzy data and adapt to changing data characteristics, making them especially relevant for
systems with high uncertainty or significant random errors [13].
However, key challenges remain unresolved. One issue is the need for adaptive control in data
restore when system characteristics change [14]. Many models are fine-tuned to specific system
parameters, but their performance degrades with changing conditions. Addressing this requires
models capable of adjusting their parameters dynamically. Neuro-fuzzy systems with adaptive
control elements [15] offer a promising solution for real-time adaptation.
Another challenge is managing multidimensional dependencies between sensors, especially when
parameters are highly interdependent. Current studies [16] often focus on single-sensor data restore,
neglecting correlations across channels. Developing multichannel models that consider sensor
interactions and adapt to their evolving characteristics is critical for improving restore performance.
Thus, adaptive control and neuro-fuzzy systems represent a promising direction for addressing
these challenges, enhancing data restore quality while adapting to changing operational conditions.
3. Proposed method
This research foundation is a general approach to developing a hybrid model, proposed in [29]. The
original dataset portion, containing only complete data, is used for model construction, and certain
elements are randomly removed from it. For each artificially excluded element, values are estimated
using various methods. As a result, a restored set values are generated for each element, which is
then used as the input vector to build the adaptation model. The known value of the removed element
serves as the output:
𝐮𝐮 = {𝑢𝑢1 , 𝑢𝑢2 , 𝑢𝑢3 , 𝑢𝑢4 , 𝑣𝑣 }, (1)
where ui is the value obtained using one method, and v is the element original value. The resulting
training dataset (1) is applied for developing and tuning the adaptation model.
The missing element value estimate in the original dataset, each of the four methods is first
applied. The results are input into the model, producing a final estimate for the missing element
value. A critical aspect in model development is the architecture choice. A linear model in the form
v = w1 ∙ u1 + w2 ∙ u2 + w3 ∙ u3 + w4 ∙ u4 + w0 [17] was examined, along with various nonlinear models
based on artificial neural networks [18], which implement a function in the form v = f(u1, u2, u3, u4).
None of the models analyzed demonstrated an improvement in accuracy compared to the individual
methods included in the model.
To create a more sophisticated model, a cluster analysis is performed on the training datasets (1)
for all employed data arrays. As an analytical tool, presented in [17], it is recommended to utilize the
Fuzzy C-means method with the Xie-Beni criterion [19] for the clusters number assessing. For the
clustering procedure, it is proposed to use modified samples that include not the results generated
by the methods, but the absolute errors related to their performance:
𝐮𝐮 = {|𝑢𝑢1 − 𝑣𝑣|, |𝑢𝑢2 − 𝑣𝑣|, |𝑢𝑢3 − 𝑣𝑣|, |𝑢𝑢4 − 𝑣𝑣|}. (2)
Conducting cluster analysis on the training datasets allows the several main groups identification
within the original data arrays, where a similar pattern is observed regarding the original methods
accuracy. Analysis [19,20] demonstrated that constructing a separate hybrid model for each of these
groups based on a feedforward neural network significantly improved the hybrid method accuracy
compared to the original approaches. The feedforward neural network key limitation lies in its
inability to effectively handle uncertainty and variability in data, particularly under conditions
characterized by complex nonlinear dependencies. This necessity highlights the neuro-fuzzy
networks application, which, unlike standard neural networks, can account for the information
fuzziness and adapt to changing conditions, ensuring higher accuracy in tasks such as clustering and
complex data analysis.
This research proposes a method for restoring missing sensor data, featuring control based on
neuro-fuzzy networks, with the developed scheme illustrated in Figure 1.
Data without Data restoring Missing element
gaps method clustering
Sensor 1
Original data Restored
Sensor 2
array data array
Sensor N
Data with Missing element Neuro-fuzzy
gaps classifier (hybrid) model
Figure 1: The scheme for the proposed hybrid method for restoring missing sensor data, featuring
control based on neuro-fuzzy networks. (author’s research).
Based on this scheme, the following algorithm has been developed:
Step 1. The training dataset formation.
1.1. The subset S ⊆ D identification from the array containing complete data to create a reliable
foundation for training, where ∣S∣ = n represent instances number.
1.2. The elements’ specified number m is randomly removed from the selected subset S, creating
gaps in the data, resulting in a new dataset S′ such that:
S′ = S − {x1, x2, …, xm}, (3)
where xi denotes removed elements.
1.3. The missing element values yi estimation is performed using all available methods M, such
as regression, interpolation, or machine learning techniques.
𝑦𝑦𝑖𝑖 = 𝑓𝑓�𝑥𝑥𝑗𝑗1 , 𝑥𝑥𝑗𝑗2 , … , 𝑥𝑥𝑗𝑗𝑘𝑘 �, (4)
with j1, j2, …, jk being indices of available elements in S′.
1.4. The training dataset T is formed based on the obtained estimates and missing elements
known values, creating input-output pairs for model training, i.e.:
𝑇𝑇 = �(𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 )�𝑥𝑥𝑖𝑖 ∈ 𝑆𝑆 ′ , 𝑦𝑦𝑖𝑖 is estimated or known� (5)
1.5. Steps 1.2–1.4 are repeated until the training dataset T amount reaches a sufficient size,
defined as the instances number exceeding a predetermined threshold Tmin (e.g., 500
instances, i.e., ∣T∣ > 500).
Step 2. The control element creation.
2.1. A clustering method C (for instance, fuzzy C-means) is applied to determine elements groups
Gk ⊆ S in the array exhibiting similar accuracy in method performance.
2.2. A control element is developed and trained is a classifier Cclassifier for missing elements based
on known portions xi and assigned groups Gk. The classifier primary idea is to the elements
group identify (from step 2.1) to which a particular missing value belongs, based on the
instance with gaps known part. Instances from the array with randomly removed elements
(from step 1.2) are used for training. Thus,
Cclassifier: xi ↦ Gk . (6)
Step 3. Training hybrid models.
For each identified group Gk, a hybrid model Hk is constructed based on a neuro-fuzzy network
utilizing a feedforward structure. Training is conducted on the dataset formed in step 1 portion T,
corresponding to the elements specific group Gk, ensuring that the model accounts for each group
characteristics. In this context, it is advisable to employ a Takagi-Sugeno neuro-fuzzy network
architecture [21–23] since it effectively combines fuzzy rules with neural computations, providing a
high degree of adaptability to changing data conditions. This architecture includes fuzzy rules for
managing uncertainty in data, which is crucial in the variability presence and real measurements
noise typical [21, 22]. Moreover, the networks’ neural component enables modeling complex
nonlinear relations between input and output data, leading to the missing values more accurate
restoring [23]. This architecture use also facilitates adaptive learning mechanisms [22, 24] that can
account for each group characteristics, forming flexible and highly effective models tailored to the
data specifics, significantly enhancing prediction accuracy in missing value restore tasks. The
Takagi-Sugeno-type architecture employs fuzzy rules Rj defined as:
Rj: IF “x” IS “Aj” THEN “y = fj(x)”, (7)
where Aj denotes fuzzy sets and fj(x) represents linear combinations for outputs.
Step 4. The missing values restoring.
An instance xmissing containing missing values is input into the control element, realized based on
the classifier. The control element decides dk which hybrid model should be applied for the given
instance, i.e.:
dk = Cclassifier(xmissing). (8)
Apply the selected hybrid model Hk to restore missing values yrestored:
yrestored = Hk(xmissing). (9)
Additionally, a confidence assessment mechanism Cconfidence is implemented, which analyzes the
uncertainty level when selecting a model, considering factors such as the initial data quality and the
outcomes from previous models. This will improve the value restore accuracy, particularly in
complex or atypical situations. Thus,
Cconfidence(dk) = quality(xmissing) + performance(Hk). (10)
The proposed algorithm scientific novelty lies in its comprehensive approach to restoring missing
values, which includes stages for forming a training dataset, clustering elements, and adaptively selecting
hybrid models. Unlike similar methods [17, 19, 20], the algorithm does not simply apply a single universal
model to all data; it first elements groups identifies with similar performance accuracy, allowing for the
specialized hybrid models optimized construction for each group. Furthermore, the control element
implementation in the classifier form, which makes decisions regarding model selection based on the
instance specific characteristics, ensures higher accuracy and reliability in restoring missing values under
data uncertainty and variability conditions.
The Takagi-Sugeno neuro-fuzzy network architecture (Figure 2) for missing value restore is
designed to combine the fuzzy logic strengths for handling uncertainty and neural networks for
modeling complex, nonlinear relations between inputs and outputs.
х1 w1
μ11(х1) *
μ21(х1)
...
μА1(х1) +
х2
μ12(х1)
μ22(х1) w2 y(х)
*
...
μА2(х1)
...
хi +
μ 1i (х1)
μ 2i (х1)
...
μАi (х1) wi
*
Figure 2: The Takagi-Sugeno neuro-fuzzy network proposed architecture [22, 23].
The input layer receives the available data from an instance x ∈ R, which contains some missing
elements. The input layer goal is to feed the known values of the instance into the subsequent layers,
where x1, x2, …, xk are known elements and y1, y2, …, ym are the missing elements. For each instance,
the input is:
x = [x1, x2, …, xk, _, _, …, _]. (11)
In fuzzification layer, each input is mapped to fuzzy sets. The fuzzification process uses
membership functions (typically Gaussian or triangular) to represent uncertainty in the input values.
Each input xi is associated with a fuzzy set Aj, where the membership degree is computed:
2
�𝑥𝑥𝑖𝑖 − 𝑐𝑐𝑗𝑗 �
𝜇𝜇𝐴𝐴𝑗𝑗 (𝑥𝑥𝑖𝑖 ) = exp �− �, (12)
2 ∙ 𝜎𝜎𝑗𝑗2
where cj is the fuzzy set center and σj is its width.
The rule layer consists of fuzzy IF-THEN rules in the Takagi-Sugeno form [21, 23]. Each rule Rj
takes the form:
Rj: IF (x1 is Aj1) AND (x2 is Aj2) AND … THEN yj = fj(x), (13)
where fj(x) is the inputs linear function:
fj(x) = w1 ∙ x1 + w2 ∙ x2 + … + wk ∙ xk + bj. (14)
The rule layer aggregates multiple rules, each corresponding to different fuzzy regions of the
input space.
In the inference layer, the network evaluates the degree to which each rule applies to the given
input. This is done by calculating the product of the membership degrees for each rule:
𝑘𝑘
𝜇𝜇𝑗𝑗 = � 𝜇𝜇𝐴𝐴𝑗𝑗 (𝑥𝑥𝑖𝑖 ). (15)
𝑖𝑖=1
This layer output is the firing strength μj of each rule, indicating how well the input matches the
rule fuzzy conditions.
In the defuzzification layer, the network merges the results from all rules to generate the final
output by applying a modified center of gravity equation [25]:
∑𝑗𝑗 𝑤𝑤𝑗𝑗 ∙ 𝑓𝑓𝑗𝑗 (𝑥𝑥 ) ∙ 𝜇𝜇𝑗𝑗
𝑦𝑦 = . (16)
∑𝑗𝑗 𝑤𝑤𝑗𝑗 ∙ 𝜇𝜇𝑗𝑗
Here, fj(x) represents the linear function output corresponding to the j-th rule, and μj is the firing
strength of the j-th rule.
The output layer produces the restored missing values yrestored. The result is the estimated values
set for the missing elements based on the weighted contribution from the fuzzy rules. Table 1
presents the Takagi-Sugeno neuro-fuzzy network training algorithm.
The Takagi-Sugeno neuro-fuzzy network's architecture and training process allows for flexible,
adaptive modeling of complex relations in data, making it highly effective for restoring missing
values in datasets with uncertainty and variability. By clustering elements and assigning specialized
models, this approach enhances accuracy and robustness in challenging restoring tasks.
Table 1
The Takagi-Sugeno neuro-fuzzy network training algorithm (author’s research).
Step Step name Description
number
Step 1 Initialization The membership function parameters cj, σj and the linear
function weights w1, w2, …, wk, and biases bj are initialized
randomly.
Step 2 Forward For each training instance x the membership degree in the
Propagation fuzzification layer is computed. The rule activations (firing
strengths) in the inference layer are computed. The network
output is computed by aggregating the rule outputs using a
weighted average in the defuzzification layer.
Step 3 Error Calculation The error as the difference between the network output
yrestored and the true value ytrue for the missing element is
defined as:
𝑚𝑚
1 2
𝐸𝐸 = ∙ ��𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟,𝑖𝑖 − 𝑦𝑦𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖 � .
2
𝑖𝑖=1
Step 4 Backpropagation The error gradient with respect to the parameters wj, bj of
the linear functions in the rule layer is computed. For each
weight wj, the gradient is calculated as:
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
= ∙ .
𝜕𝜕𝑤𝑤𝑗𝑗 𝜕𝜕𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝜕𝜕𝑤𝑤𝑗𝑗
The gradients for the membership function parameters wj,
bj are also computed using the chain rule, adjusting the
centers and widths of the fuzzy sets:
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝜕𝜕𝜇𝜇𝑗𝑗
= ∙ ∙ .
𝜕𝜕𝑐𝑐𝑗𝑗 𝜕𝜕𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝜕𝜕𝜇𝜇𝑗𝑗 𝜕𝜕𝑐𝑐𝑗𝑗
Step 5 Parameter Update The parameters using RMSProp optimization method is
updated as:
(𝑡𝑡+1) (𝑡𝑡) 𝜂𝜂 (𝑡𝑡)
𝜃𝜃𝑗𝑗 = 𝜃𝜃𝑗𝑗 − ∙ 𝑔𝑔𝑗𝑗 ,
�𝐺𝐺𝑗𝑗(𝑡𝑡) + 𝜖𝜖
(𝑡𝑡) 𝜕𝜕𝜕𝜕 (𝑡𝑡) (𝑡𝑡−1) (𝑡𝑡) 2
where 𝑔𝑔𝑗𝑗 = (𝑡𝑡) , 𝐺𝐺𝑗𝑗 = 𝜌𝜌 ∙ 𝐺𝐺𝑗𝑗 + (1 − 𝜌𝜌) ∙ �𝑔𝑔𝑗𝑗 � , ρ
𝜕𝜕𝜃𝜃𝑗𝑗
(𝑡𝑡)
is the decay rate (usually set to 0.9), and 𝐺𝐺𝑗𝑗 is the squared
gradients running average, η is the training rate, ϵ is a small
value (e.g., 10−8) added for numerical stability, θj represent
each parameter (which can be wj, bj, cj, or σj).
Step 6 Repeat The steps 2–5 are repeated until the network converges, i.e.,
the error falls below a predefined threshold
4. Results
The research presents the results from a computational experiment focused on restoring missing gas
temperature before the compressor turbine 𝑇𝑇𝐺𝐺∗ data in the TV3-117 TE [26], which is part of the Mi-
8MTV helicopter’s power plant. The parameter values are recorded by a dual sensor comprising 14
T-101 thermocouples [27] at 1-second intervals (Figure 3). Data recording was carried out over a 256-
second time interval. The application of temporal discretization and adaptive quantization with a
dynamic range to the reconstructed 𝑇𝑇𝐺𝐺∗ signal diagram (Figure 3) resulted in its discrete form (Figure
4).
Figure 3: Diagrams of the gas temperature before the compressor turbine 𝑇𝑇𝐺𝐺∗ parameter recorded
onboard the helicopter during the 256-second study interval. (author’s research).
Figure 4: Diagrams of the gas temperature before the compressor turbine 𝑇𝑇𝐺𝐺∗ reconstructed
parameter recorded onboard the helicopter during the 256-second study interval. (author’s research).
The generated diagrams (Figures 3 and 4) play a key role in creating the training dataset.
According to the Nyquist theorem, the sampling rate must be at least twice the maximum frequency
for accurate signal reconstruction [28]. With 256 readings over 256 seconds, the sampling rate is 256
/ 256 = 1.0 readings per second, which corresponds to a time interval between readings of
approximately 7 seconds. This frequency is well-suited for tracking slowly varying parameters and
provides the necessary detail without excessive data amount [27, 28]. Thus, selecting 256 readings
represents an optimal compromise between sampling rate, data volume, and computational costs,
ensuring a balance between accuracy and data processing efficiency. As shown in Figure 4, the
training dataset for the 𝑇𝑇𝐺𝐺∗ parameter has been created and is presented in Table 2. After
normalization, it appears as shown in Table 3.
Table 2
The training dataset fragment of the gas temperature before the compressor turbine 𝑇𝑇𝐺𝐺∗ parameter
values (author’s research).
Number 1 … 38 … 84 … 173 … 256
Value 1106 … 1118 … 1130 … 1122 … 1104
Table 3
The training dataset fragment of the gas temperature before the compressor turbine 𝑇𝑇𝐺𝐺∗ parameter
normalized values (author’s research).
Number 1 … 38 … 84 … 173 … 256
Value 0.971 … 0.979 … 0.989 … 0.984 … 0.968
Thus, the normalized values for the 𝑇𝑇𝐺𝐺∗ parameter range from 0.95 to 0.99. Let’s assume there are
missing values for the 𝑇𝑇𝐺𝐺∗ parameter between 56 and 65 seconds, 137 and 144 seconds, and 215 and 223
seconds.
According to the developed algorithm, at the initial stage, we select a subset S ⊆ D from the
complete dataset, containing all 𝑇𝑇𝐺𝐺∗ parameter values over the entire period without missing data. Let’s
assume, for example, that ∣S∣ = 256 records.
Next, we remove data from S corresponding to the periods from 56 to 65 seconds, 137 to 144 seconds,
and 215 to 223 seconds. Thus, S′ is the subset S with the removed values, and its size becomes ∣S′∣ = 256
– 28 = 228 values. The missing values will be restored using various methods (regression, interpolation,
machine learning). It is assumed that the implementation of fuzzy rules by a neuro-fuzzy network
(Figure 2) for restoring the missing values is carried out according to expressions such as:
𝑇𝑇55 + 𝑇𝑇66 𝑇𝑇56 + 𝑇𝑇67 𝑇𝑇64 + 𝑇𝑇66
𝑦𝑦56 = , 𝑦𝑦57 = , … , 𝑦𝑦65 = . (17)
2 2 2
Similarly, interpolation is performed for the other intervals with missing data. At the next stage,
a training set T is created, including pairs of inputs xi and their corresponding restored values yi,
using the missing and known data. The dataset size is ∣T∣ > 500. After that, a fuzzy clustering method
(such as fuzzy C-means) is applied to divide the data into groups Gk with similar restoring
characteristics. For example, let the data be divided into 3 groups:
𝐺𝐺1 : [0.95 … 0.96], 𝐺𝐺2 : [0.96 … 0.97], 𝐺𝐺3 : [0.97 … 0.99]. (18)
The classifier is trained on the data to determine which group Gk the missing values belong to,
based on the data known portion. For example, if the missing 𝑇𝑇𝐺𝐺∗ parameter value is 0.955, the
classifier will assign it to group G1. For each group Gk, a hybrid model Hk is constructed based on the
Takagi-Sugeno neuro-fuzzy network (Figure 2). A separate model H1 will be built for group G1, model
H2 for group G2, and so on. These models are trained on the corresponding data from the training set
T according to the proposed training algorithm (see Table 1).
For the missing value at the 56th second, using the known data, for example, T55 = 0.956, the
classifier Cclassifier determines that this value belongs to group G1. To restore the temperature value at
the 56th second, model H1 (corresponding to group G1) is applied, which reconstructs the missing
value y56 = H1(T55, T57, …). The confidence score Cconfidence is calculated based on the quality of the input
data and the performance of model H1, helping to improve the restoration accuracy. Let the
neighboring parameter values for the adjacent seconds appear as follows (in normalized form): T54 =
0.960, T55 = 0.955, T66 = 0.957, T67 = 0.958. For the 56th second, using the neuro-fuzzy network (Figure
2), the missing value is restored as:
𝑇𝑇55 + 𝑇𝑇66 0.955 + 0.957
𝑦𝑦56 = = = 0.956. (19)
2 2
Similarly, other missing values are restored by applying hybrid models and regression methods
for more accurate restoring. Table 4 presents the results evaluating the quality of solving the missing
𝑇𝑇𝐺𝐺∗ parameter values restoring task. The experimental results show that for most values, the
deviations are only 0.002…0.005, confirming the high accuracy achieved by the restoration models.
The maximum deviation of 0.007 also falls within acceptable error limits, proving the stability and
reliability of the proposed approach. Thus, hybrid models and regression methods demonstrate
excellent performance in restoring critical parameters, ensuring accurate system operation even in
the event of sensor failures.
Table 4
The results evaluating the quality of solving the missing 𝑇𝑇𝐺𝐺∗ parameter values restoring task
Number The true value with a The restored value in Discrepancies between
functioning sensor case of a sensor failure the true and restored
values
56 0.959 0.956 0.003
57 0.960 0.957 0.003
58 0.960 0.955 0.005
59 0.961 0.956 0.005
60 0.960 0.956 0.004
61 0.960 0.957 0.003
62 0.960 0.958 0.002
63 0.961 0.958 0.003
64 0.961 0.957 0.004
65 0.961 0.957 0.004
137 0.985 0.982 0.003
138 0.987 0.984 0.003
139 0.988 0.984 0.004
140 0.987 0.983 0.004
141 0.987 0.982 0.005
142 0.985 0.979 0.006
143 0.985 0.980 0.005
144 0.985 0.978 0.007
215 0.973 0.969 0.004
216 0.971 0.966 0.005
217 0.972 0.970 0.002
218 0.970 0.969 0.001
219 0.973 0.969 0.004
220 0.975 0.970 0.005
221 0.974 0.970 0.004
222 0.976 0.971 0.005
223 0.978 0.974 0.004
The neuro-fuzzy network's performance (Figure 2) evaluation was conducted using traditional
metrics, including Accuracy, Loss, Precision, Recall, and F1-score [29]. In the context of restoring
missing sensor values on helicopters, Accuracy reflects the proportion of correctly restored values
compared to the actual sensor data. Loss measures the prediction error, indicating how far the
restored values deviate from the true ones [30]. Precision and Recall evaluate the model's
effectiveness in correctly identifying and recovering missing values, while F1-score provides a
balance between Precision and Recall, ensuring both completeness and accuracy in the restoration
process [29, 30]. These metrics are determined using the formulas provided in [29]:
𝑁𝑁
1 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑐𝑐𝑦𝑦 = ∙ � 𝟏𝟏(𝑦𝑦𝑖𝑖 = 𝑦𝑦�𝑖𝑖 ) , 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = , 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = ,
𝑁𝑁 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹
𝑖𝑖=1
1
(20)
2 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
𝐹𝐹1 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = , 𝐴𝐴𝐴𝐴𝐴𝐴 − 𝑅𝑅𝑅𝑅𝑅𝑅 = � 𝑇𝑇𝑇𝑇𝑇𝑇 ∙ �𝐹𝐹𝐹𝐹𝐹𝐹−1 (𝑡𝑡)� 𝑑𝑑𝑑𝑑.
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
0
In the context of restoring missing sensor-registered parameters on board the helicopter, TP (True
Positives) refers to correctly restored values that match the actual sensor data when the sensor is
functioning. TN (True Negatives) represents cases where missing data were correctly identified as
non-recoverable or not requiring restoration. FP (False Positives) indicates instances where data were
incorrectly restored, leading to discrepancies between the restored and actual values. FN (False
Negatives) refers to cases where the system failed to restore missing data correctly, leaving a gap in
the sensor readings.
Figures 5 and 6 display the Accuracy and Loss metric diagrams, demonstrating the network's high
effectiveness. Specifically, the Accuracy exceeds 99 %, while the Loss metric decreases from 2.5% to
0.5% over 180 training epochs, indicating significant improvement in model quality and training
efficiency. This confirms the network's capability for high-precision classification tasks with
minimal error.
Figure 5: The accuracy metric diagram. (author’s research).
Figure 6: The loss metric diagram. (author’s research).
Specifically, the Accuracy (see Figure 5) exceeds 99 %, while the Loss (see Figure 6) metric decreases
from 2.5 to 0.5 % over 180 training epochs, indicating significant improvement in model quality and
training efficiency. This confirms the network's capability for high-precision classification tasks with
minimal error.
These results highlight the robustness and reliability of the neuro-fuzzy network in achieving high
classification accuracy with minimal loss. The steady reduction in the Loss metric demonstrates the
model's ability to generalize well to new data, further supporting its applicability in real-world scenarios
requiring precise decision-making and control.
In the context of restoring missing sensor-registered parameters on board the helicopter, obtained
Precision is 0.983 means that 98.3 % of the values identified as successfully restored are indeed
correct, indicating a very low false positive rate. Recall is 1.0 means that the method was able to
correctly restore all the missing values (no false negatives), achieving 100 % completeness in the
restoration process. The F1-score is 0.991 combines Precision and Recall, indicating a balance
between accuracy and completeness, with a near-perfect performance in restoring the missing sensor
data.
A comparison (Table 5) of the proposed approaches for restoring missing sensor-registered
parameters on board the helicopter was conducted by replacing the developed neuro-fuzzy network
with a two-layer feedforward network (alternative approach 1) and the ANFIS neuro-fuzzy network
(alternative approach 2), using a traditional training algorithm. This comparison aimed to evaluate
the performance of different neural network structures in the context of the restoration task.
Table 5
The comparative analysis results (author’s research).
Number Proposed approach Alternative approach 1 Alternative approach 2
Accuracy 0.991 0.932 0.976
Precision 0.983 0.924 0.971
Recall 1.0 0.817 1.0
F1-score 0.991 0.867 0.985
The comparative analysis of the proposed neuro-fuzzy network against alternative approaches
revealed superior performance in restoring missing sensor-registered parameters on board the
helicopter. The proposed approach achieved an accuracy of 0.991, significantly higher than the 0.932
obtained by the two-layer feedforward network and the 0.976 from the ANFIS network. Furthermore,
the proposed method demonstrated exceptional precision (0.983) and an F1-score (0.991), indicating its
effectiveness in accurately identifying and restoring missing values compared to the alternative
methods, which showed lower precision and recall rates.
5. Discussion
This research lies foundation in the development of a hybrid method for restoring missing sensor
data with adaptive control (see Figure 1). The proposed method is based on an algorithm that involves
forming a training dataset, creating a control element, and training hybrid models. The restoration
of missing values includes identifying a data subset, randomly removing elements to create gaps,
estimating their values using various methods, clustering to identify groups with similar accuracy,
developing and training a classifier, constructing neuro-fuzzy models for each group, and utilizing a
control element to select the model and assess the classifier's confidence. This collective approach
enhances the accuracy of restoring missing values in complex situations. Unlike similar methods,
this algorithm does not use a single universal model for all data; instead, it first identifies groups of
elements with similar accuracy, allowing for the development of specialized hybrid models optimized
for each specific group.
The developed algorithm is implemented as a Takagi-Sugeno neuro-fuzzy network (see Figure 2),
as it effectively combines fuzzy rules and neural computations (see Table 1), ensuring high adaptability
to changing data conditions and allowing for the processing of uncertainty. This is particularly
important in environments with variability and noise in real measurements, as well as modeling
complex nonlinear dependencies between input and output data, facilitating more accurate restoration
of missing values. A computational experiment has been conducted to restore missing gas temperature
data before the compressor turbine (see Figures 3 and 4, Tables 2 and 3) for the TV3-117 TE. The results
confirm the high accuracy of the restoration models, with deviations for most values ranging from
0.002 to 0.005, and the maximum deviation of 0.007 falling within acceptable error limits (see Table 4).
This indicates the stability and reliability of the approach, ensuring precise system operation even in
cases of sensor failures.
The quality of the proposed Takagi-Sugeno neuro-fuzzy network (see Figure 2) has been assessed
based on traditional quality metrics: Accuracy, Loss, Precision, Recall, and F1-score. The model's
accuracy exceeds 99 % (see Figure 5), while the Loss value decreases from 2.5 to 0.5 % (see Figure 6)
over 180 training epochs, demonstrating a significant improvement in model quality and training
efficiency. These results confirm the capability of the neuro-fuzzy network to perform high-precision
classification tasks with minimal errors, highlighting its reliability and robustness. In the context of
restoring missing values recorded by sensors on board the helicopter, the achieved Precision of 0.983
indicates that 98.3 % of the values identified as successfully restored are indeed correct. The Recall
of 1.0 confirms the complete restoration of all missing values, while the F1-score of 0.991 indicates a
balance between accuracy and completeness, demonstrating nearly perfect performance in restoring
missing sensor data.
The comparative analysis of the proposed approaches for restoring missing sensor-registered
values on board the helicopter showed superior results for the developed neuro-fuzzy network
compared to alternative methods, such as a two-layer feedforward network and the ANFIS neuro-
fuzzy network (see Table 6). The proposed approach achieved an accuracy of 0.991, significantly
higher than 0.932 for the two-layer network and 0.976 for the ANFIS network. Furthermore, the
method demonstrated exceptional precision (0.983) and an F1-score (0.991), indicating its high
efficiency in identifying and restoring missing values compared to alternative methods, which
exhibited lower accuracy and completeness metrics.
Despite the successes achieved in developing the hybrid method for restoring missing data, the
research faces certain limitations. The effectiveness of the algorithm depends on the quality and
completeness of the original data, as insufficient representation in the training dataset may
negatively impact the accuracy of the restored values. The method may encounter challenges when
working with data containing high levels of noise or anomalies, which could lead to a decrease in
overall reliability. Although the proposed algorithm has demonstrated high accuracy, its
performance may vary based on the specifics of different sensor types and helicopter operating
conditions.
Prospects for further research include expanding the developed method to process data from
other sensor types and in various operating conditions. It would also be beneficial to consider
integrating the algorithm with machine learning methods, such as deep neural networks, to enhance
adaptability and improve restoration quality. Additionally, developing more complex strategies for
handling noise and anomalies in the data could significantly increase the algorithm's reliability,
allowing for effective application in real-world scenarios that require high precision and resilience
to failures.
6. Conclusions
This research successfully presents a hybrid method for restoring missing sensor data through
adaptive control, emphasizing the development of specialized hybrid models tailored to specific
groups of data. The algorithm incorporates multiple stages, including data subset identification,
random removal of elements to simulate missing values, and various estimation methods,
culminating in the construction of a Takagi-Sugeno neuro-fuzzy network. This approach not only
enhances the accuracy of missing value restoration but also demonstrates adaptability to changing
data conditions and the ability to handle uncertainty in real-world measurements.
The computational experiments conducted on the gas temperature data before the compressor
turbine of the TV3-117 turboshaft engine confirm the efficacy of the proposed method. The results
show deviations in the restored values ranging from 0.002 to 0.005, with a maximum deviation of
0.007, all within acceptable error limits. This indicates the method's stability and reliability in
operational scenarios, even when sensor failures occur. The ability to maintain precision within
these limits underscores the method's potential for practical applications in critical systems, such as
helicopter avionics.
Furthermore, the performance metrics indicate that the model's accuracy exceeds 99%, with a
Precision of 0.983 and an F1-score of 0.991, validating the robustness of the Takagi-Sugeno neuro-
fuzzy network. These metrics demonstrate that the proposed method effectively balances accuracy
and completeness, achieving near-perfect performance in the restoration of missing sensor data. The
comparative analysis with alternative methods, such as a two-layer feedforward network with an
accuracy of 0.932 and an ANFIS network with an accuracy of 0.976, illustrates the superiority of the
developed approach.
This research contributes significantly to the field by addressing the challenges of missing data
recovery in sensor systems. The proposed hybrid method, achieving an accuracy of 0.991,
demonstrates substantial improvements over traditional approaches, paving the way for further
advancements in the adaptive restoration of missing values. Future research could explore the
extension of this method to different sensor types and operational conditions, potentially integrating
advanced machine learning techniques to enhance adaptability and further improve restoration
quality.
Acknowledgements
This paper is supported by the EU Erasmus+ programme within the Capacity Building Project
“WORK4CE” (619034-EPP-1–2020-1-UA-EPPKA2-CFHE-JP).
Declaration on Generative AI
The authors have not employed any Generative AI tools.
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