1 Introduction

The Internet of Things (IoT) is a network of physical and virtual objects that can communicate with one another and share data wirelessly, eliminating the need for human intervention in the process. Objects in the Internet of Things can be anything that satisfies a demand, whether it is a physical or mechanical entity. Internet of Things (IoT) enables smarter living by connecting devices and delivering smart home gadgets. The Internet of Things (IoT) is a key for companies since it allows for smart procedures and cuts down on labor and time. Internet of Medical Things (IoMT) refers to the network of interconnected electronic devices and systems used in healthcare. According to [1], an IoMT is a collection of medical devices that communicate with one another through a Gateway and a network. The Internet of Medical Things (IoMT) has the potential to develop a wide range of applications, including systems for remote health monitoring, patient care, exercise programs, and the care of the elderly. In patient care management, IoMT has demonstrated promise by cutting down on treatment times and labor expenses. Patient access to treatment equipment can also be efficiently scheduled using the system. These changes in health care have been illustrated in Fig. 1.

Fig. 1
figure 1

Current health care trends in medicine [2]

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The incidence of neurological illnesses is rising at an exponential rate, according to recent studies, as the population ages. In recent years, smart home technologies that utilize the Internet of Things have grown in popularity. A primary focus is providing assistance to the elderly and those with disabilities. Movement problems are a hallmark of several degenerative neurological conditions, including Parkinson's disease (PD). Worldwide, between seven and ten million people suffer from Parkinson's disease (PWP) at this time, according to data compiled by the World Health Organization (WHO) [3]. The elderly have a disproportionately high number of documented cases of advanced Parkinson's disease. One of the most common and debilitating motor symptoms in people with Parkinson's disease is freezing of gait (FoG)[4]. Episodes of freezing of gait (FoG) can occur suddenly and impact a person's ability to walk, leading to a halt or continuation of their stride and possibly accompanied by tremors or unexpected paralysis in their legs [5,6,7,8].

Freezing of gait (FoG) is linked to injuries, decreased mobility, and falls. While wearable sensor-based devices can distinguish frostbite in progress and help a person resume walking, there is a need to prevent freezing by anticipating when an episode will begin and sending a timely signal. Fog prediction systems have been developed using wearable sensors implanted on various parts of the body. Advancements in technologies like 5G, the Internet of Things (IoT), and cloud computing have made data exchange faster and more secure. The Internet of Things (IoT) presents numerous opportunities in the e-Health field [9,10,11].

The exponential rise of smart gadgets and 5G mobile technology has led to the quick proliferation of the Internet of Things (IoT). As a result, issues like urban traffic congestion, pollution control, and the amount of sensory data produced by monitoring equipment have all seen a dramatic uptick. The virtually infinite computer resources made available by cloud computing allow it to handle the comparable workload with ease. However, alternative technologies like fog computing have been created to expedite the processing and management of sensory data in real-world scenarios like the Internet of Things and smart grid, where security and quality of service are paramount in delay-sensitive applications. In mobile edge computing, processing workloads at the network's edge lowers latency but significantly raises sensor power consumption. Enhancing the energy model of fog devices at the network's edge is, therefore, critically necessary. In addition, there is a significant delay and high cost associated with moving large amounts of data to cloud computing classes for precise and sophisticated analysis. Low-consumption, straightforward modeling that achieves great accuracy is necessary for the implementation of big projects in fog devices. Making a real-time processing will be much easier with this [12,13,14].

FoG detection is essentially a classification issue. The goal of solving a classification problem is to use the training data set to establish which classes fresh observations should be placed into. Several classifiers that rely on statistical analysis to deduce meaning from an observation have been discovered for this exact reason. Similar to the probability concept in statistics or the probability assignment in Dempster–Shafer theory, these methods require prior knowledge of the data. Thus, to train the observation samples, a preliminary understanding of the data is required [15,16,17]. However, the Internet of Medical Things (IoMT) generates massive amounts of patient and healthcare data every day. One of the major issues of the medical Internet of Things is the time required to process and evaluate massive data that is collected from edge sensor devices while maintaining the accuracy required for data classification. To address these limitations of current methods, this study presents a decision-making model for FoG detection on the Daphnet dataset that uses a combination of fuzzy logic and basic analytical maps trained using meta-heuristic algorithms. With the goal of simplifying the detection system and enabling real-time operations within the IoMT architecture, the suggested detection system design is presented. Training accuracy and learning time for a vast data set were both improved by this decision-making model that uses a meta-heuristic method coupled with fuzzy logic. After training, the system that is shown is merely a linear map, also known as a Mobius fractional map, which can be modeled and programmed at any time with the least amount of complexity when it comes to fog calculations. This will significantly aid in cutting down on the computation time for FOG detection. The shortcomings of current methods, which typically call for in-depth statistical analysis, are overcome by this approach, which aggregates multiple performance criteria to arrive at a quantitative judgment. Thus, the innovations of this article derive from its high applicability for linear or Mobius mapping-based FOG detection systems, which makes it valuable for high-accuracy edge surface or fog computations in lightweight machine learning systems. Another feature of this article's innovations that can be applied to multi-class applications is the enhancement of classification accuracy for this system for multi-class structures using new combined techniques of Fuzzy-ISSA and TSO algorithms.

The main contributions of this work are as follows:

  • Creating a minimally complex intelligent architecture to enable real-time FoG gait freezing detection for individuals with Parkinson's disease.

  • Utilizing TSO and Fuzzy-ISSA algorithms to enhance the efficiency of the suggested model training system and raise classification accuracy because of the abundance of training data.

  • Generating a multi-level output for the FoG detection system using fuzzy logic of the Takagi Sugeno type in order to classify the system for various inputs.

  • Using simple linear and Mobius maps to improve classification performance.

  • Improving the performance of SSA algorithm with the help of Mamdani type fuzzy logic system.

2 Related Works

New technologies like the Internet of Medical Things (IoMT), Wireless Body Area Networks (WBAN), and cloud computing are becoming increasingly important in the health industry and many other industries due to the prevalence of most human disorders. Also, because to these innovations, billions of gadgets may now link up to the web and exchange data with one another. Using fog technologies and cloud computing, the authors of [18] construct an IoMT architecture that includes wireless body area networks (WBANs) and analyze massive health data derived from these networks. In contrast to cloud computing, which can handle complicated and time-consuming analyses, fog computing is best suited for simple and fast analyses. A diabetes prediction scenario is used to introduce the proposed IoMT system. On the cloud, using machine learning techniques such as support vector machine (SVM), random forest (RF), and artificial neural network (ANN), the fuzzy logic decision-making process for diabetes prediction is carried out. Big data analysis using fuzzy logic and machine learning algorithms makes use of the datasets produced by WBANs. For diabetes prediction, SVM achieved an accuracy of 89.5% in fog, RF an accuracy of 88.4%, and ANN an accuracy of 87.2% in the cloud. In contrast, fuzzy logic achieved a 64% accuracy performance in fog. In the WBAN scenario, which was set up using the AODV routing protocol and the IEEE 802.15.6 standard, the analysis also includes the throughput and latency consequences of different nodes with different priorities.

The study [19] suggests a gray relation analysis-based FoG detection technique that makes use of sensor information to forecast whether a patient will experience freezing or not. Furthermore, an ensemble learning method for FoG detection is also demonstrated, using gray relational analysis as the foundation classification model. To confirm their viability, the suggested methods have been used on a benchmark gait freezing dataset. The outcomes of the simulation indicate that the suggested approaches are more accurate than other machine learning strategies already in use.

The paradigm suggested in [20] makes advantage of state-of-the-art methods and services, including distributed storage, information services at the network's periphery, and embedded data mining. The fog layer employs the event-based data transmission method to manage patient data in real-time. The patient's Time Health Index (THI) is calculated using the time mining concept, which allows for the analysis of adverse occurrences. The system's validity was tested for 30 days using health data routinely gathered in an Internet of Things (IoT) smart home setting on 67 patients. When compared to other classification methods, the results demonstrate that the model based on the suggested BBN classifier has a high response time and accuracy in determining the status of an event. Decisions based on healthcare data in real-time also expand the system's potential uses.

According to [21], the LAS model produces few false positives while maintaining performance comparable to the two-way model in fog prediction. People with PD may be able to tolerate the higher false positive rate, considering the benefits of single-sensor systems. Hence, a fog forecasting system can be developed using a single foot pressure sensor mounted on the LAS, creating results comparable to a two-way system.

Health services can be enhanced and new innovations can be spurred by IoT technology. Cloud computing and the Internet of Things can greatly enhance patient monitoring in this procedure. One noteworthy innovative aspect of article [22] is the use of a prioritization system to prioritize sensitive information in the Internet of Things. Additionally, the article discusses how LSTM deep neural networks are used in cloud computing for patient condition classification and remote monitoring. In an IoT setting, the data collected by sensors is transmitted to the cloud via the fifth-generation Internet. Deep neural networks, namely the LSTM (long short-term memory) approach, are fundamental to cloud computing. Results obtained from comparing the proposed technique's simulations to those of other methods show that, on average, the offered method improves upon the other methods by 10.41%, leading to an accuracy of 97.13%.

In the paper [23], an integrated healthcare monitoring solution utilizing distributed computing and the Internet of Things (IoT) was presented for soldiers deployed in harsh weather circumstances. Each soldier's health parameters should be tracked in real time, and a subsequent analysis of the data set should be carried out to provide the necessary medical treatment as soon as possible. This study proposes a three-layer service-based Internet of Things architecture where computational operations are shared across all layers. Two levels of redundant information belonging to secure soldiers have been implemented by the proposed distributed computing mechanism. Both the end node and the middle node employ different methods for filtering; the end node uses fuzzy classification and the middle node uses time series pattern analysis. When data flooding and cloud computing load are reduced through this layered filtration procedure, system response times in accordance with emergency plans are enhanced.

One of the most debilitating and concerning symptoms of Parkinson's disease (PD) is freezing of gait (FoG). Fog is caused by movement problems and decreased brain control, both of which make it very difficult to move ahead. The use of the 5G spectrum at 4.8 GHz, a possible frequency band for the Internet of Things in China, is presented in the research [24] as a means to identify freezing episodes experienced by PD patients. Antennas, radio frequency (RF) signal generators, network interface cards (NICs), and other wireless equipment can be used to extract variance domain information from wireless channel characteristics and incorporate them into the 5G communication system. Five distinct human actions were carried out: sitting in a chair, strolling slowly, walking quickly, pausing voluntarily, and FoG episodes. In order to identify the data and assess the efficacy of the suggested system, a completely multi-layer softmax neural network with several layers was built using the collected information. In comparison to other advanced detection systems, the aforementioned tasks attained a very high classification accuracy of 99.3 percent.

To reduce energy consumption and latency in secure IoT systems with multi-sensor frameworks, the research [25] tries to change the energy model of fog devices at the edge of the network by employing the idea of green energy. The suggested approach employs a genetic algorithm (GA) to manage the high volume of requests while adhering to quality and security requirements. When compared to a baseline technique, the simulation results demonstrate that the suggested way can concurrently decrease power consumption and latency of edge devices.

In [26], the primary objective was to modify and enhance the existing robust feature extraction and inference methods so that they could incorporate more features than what is now accessible. Using the current DAPHNet dataset, they proceed to the next stage by applying feature selection in an effort to achieve maximum recognition results. This dataset was gathered utilizing a wearable health aid system that measured the patient's mobility using a 3-axis accelerometer. They chose ten people with PD and had them walk through a series of tests in a controlled environment. Using the recommended algorithms for feature extraction, feature selection, and classification, overall performance was evaluated using both subject-dependent and subject-independent methodologies. With an AUC, sensitivity, and maximum accuracy (AUC) of around 99%, the results demonstrated that the suggested machine learning algorithms can detect FoG.

The goal of this paper is to decrease response time (RT) and bandwidth use in a fog computing infrastructure by combining feature selection (FS) and deep learning (DL) to increase data categorization accuracy [27]. Using the Internet of Things (IoT), cloud computing, and fog computing, the primary objective is to remotely analyze and categorize patients' illnesses. By using Imperial Competitive Algorithm (ICA) and Particle Swarm Optimization (PSO) in fog computing to identify prominent features, information processing time is decreased. By using a deep neural network (DNN) model to categorize the new data, the classification accuracy is increased. Based on the simulation results, the accuracy of patient classification and remote monitoring is 98.54%. This is an improvement of 4.5% compared to not using the PSO-ICA algorithm and generally about 10% better than other methods such as Bayesian Belief Network (BBN), Neural Network (NN), K nearest neighbors (KNN), and linear regression (LR).

To identify FoG episodes in people with PD, the authors of the article [28] suggest a deep learning approach. To train the model, they use a new way of representing spectral data that takes into account data from both the past and the present signal windows. A lumbar inertia measurement device was used to assess their method in 21 PD patients who presented with bouts of FLG. A comparison study with the FOG monitoring system was conducted using these data, which were also utilized to replicate the advanced approaches. This study's findings demonstrate that the suggested strategy achieves better outcomes than the current best practices for automatic FoG identification. To be more exact, while state-of-the-art approaches fell short of 83% on the geometric mean between sensitivity and specificity, the deep learning model managed 90%.

Sections from [29], which discussed the introduction of acceleration sensor site selection for PD patient monitoring, were used to construct a patient-specific model for FoG detection. These sensors quantify patient mobility. To differentiate between freezing and non-freezing events, the ranking features were subjected to the suggested classification, which is based on fog detection using linear support vector machines (SVMs). This was done using the improved feature selection (IFS) approach. Both the proposed IFS-based detection model and the eigenvector feature selection showed comparable results when evaluated in terms of performance rating of features extracted from all acceleration signals from different sensors. Nevertheless, the outcomes demonstrated that the suggested patient-dependent model utilizing IFS ranking features for FoG identification was more effective, suggesting that it could be utilized to enhance the precision of PD monitoring systems.

A real-time architecture for FoG detection in the IoMT network was provided in this research, which is both simple and comprehensive. This is the basis for Table 1, which summarizes the research on FoG diagnosis from the published studies.

Table 1 A review of the work done to detect FoG
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3 Concepts

Here we present the modified concepts behind the algorithms and methodologies employed in this study.

3.1 Mapping

Consider the function w = f (z) as a transformation that takes any point in the z plane (x and y plane) and transforms it into points in the w plane (u and v plane). We refer to this kind of change as a mapping.

We first give definitions before introducing the maps utilized in this investigation:

Fixed points on the map: these are the points on the map that remain the same after mapping. The equation f (z) = z must be solved in order to determine the fixed points of the mapping if the mapping function is defined as w = f (z).

Isometric mapping: If an angle with the vertex z0 is transferred without changing, the mapping w = f (z) is said to be isometric at that point. It implies that alterations to its size and direction should be avoided.

Theorem: the mapping W = f(z) will be isometric at the position z0 if the function f(z) is analytic and f'(z) is opposite to zero at this location.

3.1.1 Linear Mapping

This mapping becomes w = f(z) = a.z + b, where arbitrary real values a and b are involved. If a value is non-zero, then this function is total. Everywhere, this mapping will be the same.

3.1.2 Fractional Linear Mapping or Möbius Mapping \(w= \frac{az+b}{cz+d}\)

This mapping is also called bilinear mapping. For Mobius mapping, it is assumed that c ≠ 0 because if c = 0 the mapping will become a linear mapping. Also, another assumption governs this mapping: so that the function does not become a numerical constant, ad-bc ≠ 0 is considered. The derivative of this mapping is \(w^{\prime} = \frac{\left( {ad - bc} \right)}{\left( {cz + d} \right)^2 }\).Therefore, this mapping is isometric everywhere except z = − d/c. Combining linear mapping, inverse mapping, and additional linear mapping yields Möbius mapping. Without following the order, this mapping changes a line or circle into another line or circle.

3.2 Fuzzy Logic System

As a type of multi-valued logic, fuzzy logic allows variables to have a truth value between zero and one. A value for correctness could range from entirely true to entirely false; this logic is employed to deal with the idea of partial correctness [38]. In contrast, the only valid values for variables in Boolean logic are 0 and 1.

In 1965, the theory of fuzzy sets was proposed by the Azerbaijani mathematician Lotfizadeh, who also brought the term fuzzy logic into use [39, 40]. Although Łukasiewicz and Tarski [41] were the most prominent researchers in the field, fuzzy logic has been explored as infinite-valued logic since the 1920s.

The idea behind fuzzy logic is that individuals tend to base their conclusions on non-numerical and imprecise information. The term "fuzzy" refers to mathematical techniques for representing inaccurate and ambiguous data, such as fuzzy sets or models. These models are capable of identifying, representing, manipulating, interpreting, and applying unclear and ambiguous information and data [42, 43]. Fuzzy logic has been used in a variety of domains, including artificial intelligence and control theory, as demonstrated by the two varieties of Mamdani and Takagi Sugeno.

3.2.1 Mamdani

Mamdani's rule-based method [44] is the most well-known and employs the following rules:

Fuzzification: in fuzzy membership functions, all input values are fuzzified.

Fuzzy Rule: to compute fuzzy output functions, execute all relevant rules in the rule base.

Defuzzification: to achieve distinct output values, defuzzify fuzzy output functions.

3.2.2 Takagi–Sugeno-Kang (TSK)

The TSK system [45] is similar to Mamdani, but the fuzzification process is included in the implementation of fuzzy rules. These are also modified such that a polynomial function—typically constant or linear—represents the rule's outcome instead. The following is an illustration of a rule with a fixed output:

If the temperature is too cold, it means 2. In this case, the output will be equal to the resulting constant eq. (2). In most scenarios, we will have a full rule base, with 2 or more rules. In this instance, the average outcome of each i-th rule (Yi), which is specified by \(\frac{{\sum }_{i}({h}_{i}. {Y}_{i})}{{\sum }_{i}{h}_{i}}\) and weighted according to its prior membership value (hi), will be the entire output of the rule base.

3.3 Salp Swarm Algorithm (SSA)

Mirjalili created the Salp Swarm Algorithm (SSA), which is categorized as an algorithm inspired by the environment [46]. The direction and search of salps in the oceans served as the foundation for the algorithm's development. Leaders and followers are the two categories that make up SSA. The position of salps in navigation and foraging serves as the basis for the solution candidate, which is typically thought of as a two-dimensional matrix called x for optimization problems. The position of the salps leader, x1j, is updated in accordance with the following expressions, and the current best solution of the food source is designated Fj.

$$x_j^1 = \left\{ {\begin{array}{*{20}c} {F_j + c_1 \left( {\left( {ub_j - lb_j } \right)c_2 + lb_j } \right), c_3 \ge r} \\ {F_j - c_1 \left( {\left( {ub_j - lb_j } \right)c_2 + lb_j } \right), c_3 < r} \\ \end{array} } \right.$$
(1)

where the parameters are uniform random values between 0 and 1 contained in SSA, and the upper and lower limits of the search space are represented by the variables ubj, lbj, r, c2, and c3. In accordance with [37], the c1 parameter is computed as follows:

$$C_1 = 2.e^{ - \left( \frac{4t}{T} \right)^2 }$$
(2)

where T is the maximum number of iterations that should be established at first and t is the current iteration. Salps is computed for each follower's position as follows:

$$x_j^i = \frac{1}{2}.\left( {x_j^i + x_j^{i - 1} } \right), i \ge 2$$
(3)

3.4 Improved Salp Swarm Algorithm with Fuzzy Logic (Fuzzy-ISSA)

In the original SSA, the difference between the upper and lower bounds of the identified control variables is used to determine the viable solution area, which is used to direct the salp followers in their quest for the best outcome. When the problem's dimensions are deemed tiny, it can function effectively in this scenario. However, the exploitation process becomes inefficient if a large-scale search is conducted for a workable solution. Equation (3) should be carefully used to identify the appropriate balance between exploration and exploitation while addressing optimization issues. It is evident that SSA has very little capacity for exploration. An enhanced fuzzy model will be employed to boost the initial SSA’s exploration and exploitation potential. The process's ability to use resources is enhanced by removing the lower bound setting in Eq. (4), which results in the first improved version that is stated as follows:

$$x_j^1 = \left\{ {\begin{array}{*{20}c} {F_j + c_1 \left( {\left( {ub_j - lb_j } \right)c_2 } \right), c_3 \ge r} \\ {F_j - c_1 \left( {\left( {ub_j - lb_j } \right)c_2 } \right), c_3 < r} \\ \end{array} } \right.$$
(4)

In order to increase the discovery capability of the original SSA, another improvement has been implemented, where the follower expression in Eq. (3) is as follows:

$$x_j^i = \left( {ub_j - lb_j } \right) \times fuzzySSA\left( {x_j^i , x_j^{i - 1} } \right) + lb_j , i \ge 2$$
(5)

where the fuzzy system defined in Fig. 2 is used to generate the Mamdani-type Fuzzy-SSA system as the fuzzy average function in the range [0, 1]. The fuzzy average model of two prior values is presented as the foundation for the definition of fuzzy rules. Normalization is followed by calculation and application of fuzzy input. A fuzzy system overview, input/output membership functions, and input/output characteristics are shown in Fig. 2. Fuzzy rules are mentioned in Table 2.

Fig. 2
figure 2

A- View of Mamdani's fuzzy logic system. B- Displaying input and output membership functions. C- Characteristics of input and output of fuzzy averaging system

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Table 2 Rules governing the fuzzy logic system
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3.5 Transient Search Optimization (TSO) Algorithm

A novel astrophysically motivated meta-heuristic optimization approach called transit search (TS) is suggested, which is based on a popular exoplanet detection technique. The transit method has been used by the Space Telescope Database to identify almost 3800 planets. With 915 planets found by March 2022, radial velocity is the second most successful approach now in use. Transit, on the other hand, has demonstrated greater potential. Because planets are small in relation to the universe, it is challenging to detect them. The optimization strategy for this study was developed using the transit method, which has been found to be highly effective in astrophysics. The transit method checks for variations in brightness by analyzing the light received from the stars at specific intervals. If a decrease in light received is noticed, it suggests that a planet is passing in front of the star [47].

The following approach is used to implement TSOA:

  1. (A)

    Initialization:

  • To store locations and costs, an empty structure is first defined.

  • After that, the objective function is used to determine the cost of the produced galaxy centre structure, which is placed at random.

  • By calculating the costs associated with each zone and adding random deviations to their placements, regions surrounding the galactic centre are formed.

  • After that, the lowest-cost regions are chosen, and SN livable regions are established around each of them. SN planets with randomly chosen locations are created for each zone, and the objective function is used to calculate their costs.

  • After that, the chosen planets are adapted to the best starting planets.

  1. (B)

    Galaxy phase:

  • The classification and distance to the telescope are used to determine the brightness of each star.

  • The telescope's position is generated at random.

  • The brightness of each planet and the telescope's position are used to determine the transit probability.

  • The planet with the highest probability is selected as the new galactic centre after the planets are arranged in descending order.

  1. (C)

    Transit phase:

  • Random differences are added to the current positions of each planet to create new ones, and the objective function is used to calculate the costs of each new location.

  • Each star's brightness is assessed according to its distance from the telescope and rank.

  • The transit probability for each planet is determined by taking into account both its brightness and the telescope's location.

  • The planets are arranged in decreasing order of likelihood, and the galaxy's new centre is selected from the finest planets.

  1. (D)

    The optimal search regions are kept, and Steps 2 and 3 are repeated up to the maximum number of iterations.

4 Proposed Method

Freezing of gait (FoG) is one of the most incapacitating motor signs of Parkinson's disease (PD). Falls and a decline in quality of life are potential outcomes of FoG episodes. In addition to allowing nonpharmacological support based on rhythmic signals, an accurate FoG evaluation gives neurologists objective information regarding patient state and symptom features [48,49,50]. When looking for fog around the house, wearable sensors are a great tool to have on hand. The length of FoG episodes has also been found to be reduced with the use of real-time feedback. An easy-to-implement, robust real-time FoG detection technique is proposed in this study for use by standalone devices in unsupervised environments. An architecture in the IoMT network can be provided by the upcoming difficulties, which are necessary due to the extremely high volume of data received for processing in order to detect FoG at any given instant. Consequently, our goal in this work is to develop a simple but intelligent classification system that can identify FoG in various individuals. This will allow us to alert Parkinson's patients to the possibility of freezing as soon as possible, preventing them from suffering a catastrophic fall. From this, one of the aims of this suggested approach is to construct a real-time model. The suggested design in Fig. 3 is a block diagram based on learning with meta-heuristic algorithms. The methods implemented for this study are detailed here.

Fig. 3
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Diagram of the proposed design

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4.1 Information Mapping

Redundant data can be found in large datasets. Methods for selecting the most important variables from a dataset often ignore the possibility of interdependencies between them. By applying data mapping techniques, we may extract the most relevant information from a dataset by transforming it into a new dimension. The importance of established methodologies to improve accuracy and reproducibility of outcomes is underscored by this shift in information. Although Möbius data mapping can help with some of the aforementioned issues, there is still much to learn about using them for machine learning applications. Through examination of over 150,000 samples from the human motion dataset, with the help of 10 individuals' profiles in Daphnet, we want to optimize the effect of various data translations on precision, generalizability, and feature selection. Our results show that it is feasible to differentiate between different modalities of FoG incidence in Parkinson's patients using the IoMT network's minimal complexity training data. Importantly, precise categorization relies on finding the Möbius mapping coefficients for the dataset under consideration.

Data classification and clustering is a key component of the information processing challenge [51, 52]. Information can be better classified and demarcated using this method in a homogeneous mapping technique. This information transmission effect is introduced in Fig. 4. We can improve the classification by achieving the maximum information divergence by doing this. The parameters for the suggested mapping in this study, which is a linear and Mobius mapping, must now be determined. These parameters have been computed using meta-heuristic techniques. This work has made use of three algorithms, which we shall go into more detail about later.

Fig. 4
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The impact of successful mappings for information classification

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4.2 Training and Determining Mobius and Linear Mapping Coefficients with the Help of Meta-Heuristic Algorithms

To achieve high classification accuracy, we will classify the test data set using the fuzzy optimized SSA algorithm and the TSO algorithm after two linear and Mobius mappings. By specifying the coefficients and parameters of the mappings for every variable in the test data—which consists of 10 input variables in this Daphnet set—these two algorithms attempt to accomplish a two-class classification. A systematic training procedure for the proposed mapping network scheme is shown in Fig. 5.

Fig. 5
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Flowchart of the proposed FoG detection architecture training (classification into two categories)

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Figure 5 shows how a suggested algorithm is used to modify the mapping coefficient until the greatest classification accuracy of the two categories is reached, following the loading of the parameters as variables of the objective function to minimize the function. Figure 6 displays the pseudocode for the network function that was acquired for the FoG estimation method using Mobius mapping. The linear mapping-based FoG estimate system is also shown in Fig. 7. The interesting thing to note about this study is that an attempt has been made to employ a nested iteration loop to generate multifaceted mappings in order to provide a satisfactory Möbius mapping result. This will allow us to have a suitable encryption for the data transmitted over the IoMT network. Information processing is another skill we may successfully use.

Fig. 6
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Mobius target mapping system pseudocode

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Fig. 7
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Linear mapping target system pseudo code

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The fact that information has been divided into two groups in the impending approach presents a serious and important issue with the proposed categorization method. However, the system under study comprises several clusters. We shall employ a logical approach to classification as the quantity of information classes rises. For the Daphnet dataset, this approach yields three output classes: 0, 1, and 2. To tackle this issue, we divide the real-time data of Parkinson's patients into two groups using data analysis. Table 3 categorizes the output of this data set into groups. Following their individual valuation and training, these data sets are combined to create a final output, which is determined by applying Eq. 6 to the output results of these trained systems. To demonstrate this work, Fig. 8 displays the spatial representation of this technique for a two-dimensional data set.

$${\text{Final system}} = \frac{{({\text{system}}1 + {\text{system}}2 + 2)}}{2}$$
(6)
Table 3 Recommended systematic performance for different systems
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Fig. 8
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Strategies for increasing the number of floors

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4.3 Fuzzy Classification of FoG

While this method of logical categorization has its uses, adding more classification classes will make the task more difficult and take longer to estimate. Due to the fact that it necessitates a number of independent networks estimating techniques. Additionally, more executions of this estimation using a multi-system technique will be needed for each level independently. To accomplish a multi-class mapping based on this suggested fuzzy logic system, we have therefore proposed a Takagi–Sugeno type fuzzy classification method in this work at the end of the objective function to replace the SGN(.) sign function. For three distinct classes to examine this work, Fig. 9 shows the construction of the fuzzy FoG detection system. This diagram shows the input and output characteristics as well as the structure of the input membership functions. Table 4 provides the regulating fuzzy rules.

Fig. 9
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a View of Mamdani's fuzzy logic system. b Displaying input and output membership functions. c Characteristics of input and output of fuzzy averaging system

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Table 4 Rules governing the fuzzy logic system
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5 Experimental Analysis

Experimental investigation on the Daphnet frozen data set of gait has demonstrated the validity of the suggested approaches, which are based on relational analysis of linear mapping or trained Mobius using the provided optimization algorithms and classification using fuzzy classification. MATLAB R2016b was used for the analysis. The outcome is compared with popular machine learning methods like random forest (RF) and support vector machine (SVM).

5.1 Description of the FoG Dataset

Using wearable accelerometers on the legs and hips of ten patients, the dataset measures the identification of gait freezing [53,54,55]. They performed genuine actions such as opening doors, getting coffee, and walking in a straight manner. The ETH Zurich, Switzerland's Wearable Computing Laboratory collaborated to create the dataset [56]. The final result is separated into three classes.

For each sample: 1 for unfrozen, 2 for frozen, and 0 for samples not used in the study. All three classes are taken into account for testing; however, only classes 1 and 2 are tested for prediction accuracy. Additionally, the analysis is provided for two data sets that were created by picking 8516 samples at random from the data set of various patients and 17,063 samples from the data set of 150,000 samples, respectively.

5.2 Performance Evaluation

The training and test data for subject-dependent studies come from the same patient group. Ten-variable cross-validation plots were generated using each of the important features from the previous section's classification solutions. In both training and testing phases, the data are randomly reviewed in order to assess the mutual validity of ten variables. A subset of the data is designated as the test set, and the remaining portion is designated as the training set. In order to complete the separable mapping of the system, the evaluation for the mapping is then repeated three times. Each component of the data is evaluated once, and based on the pseudo code in Fig. 6, the ultimate accuracy is determined after three repetitions.

One benefit of the validation examined in this work is that every data set is utilized for both training and validation, with each segment being used precisely once for validation. The purpose of this task is to calculate the classification criteria more precisely.

The classifiers were trained using specific features from different subjects in subject-independent testing, and then their performance was assessed on the rest of the subjects. Classification accuracy (ACC), sensitivity (SE), specificity (SP), and area under the curve (AUC), as specified in Eqs. 7, 8, 9, 10, and 11, were utilized to examine the performance of the suggested technique for subject-independent tenfold cross-validation. Table 5 displays the confusion matrix that was used to create these four performance metrics.

$$ACC = \frac{TP + TN}{{TP + TN + FP + FN}} \times 100$$
(7)
$$SE = \frac{TP}{{TP + FP}} \times 100$$
(8)
$$SP = \frac{TN}{{TN + FP}} \times 100$$
(9)
$${\text{AU}}RO{\text{C}} = { }\frac{SE + SP}{2}$$
(10)
Table 5 Confusion matrix
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F1-Score: determined using the following formula, the F1-Score is an additional metric used in the statistical analysis of binary classification to assess a test's accuracy.

$$F1 - Score = \frac{2 \times TP}{{2 \times TP + FP + FN}}$$
(11)

The total of true positives that correctly indicate FoG as FoG constitutes TP in this confusion matrix. FP, the sum of all non-FoGs that were mistakenly classified as FoGs. A non-FoG that is correctly detected as such is represented by TN, the sum of true negatives. FN stands for total false negatives, or FoG that were incorrectly classified as non-FoG. In contrast, area under the receiver operating characteristic (ROC) curve, which displays the true negative and true positive rates, is called area under the curve: AUC.

The key feature of this suggested strategy is its ability to classify non-experimental data in state 0. The criteria are then calculated by checking only states 1 and 2. In other words, while identification is verified for every state in the system, the specified criteria are verified only for freezing-occurrence or non-occurrence states Table 6.

Table 6 Description of the dataset
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5.3 Simulation Results

Table 7 displays the values of several criteria for the suggested approaches in comparison to other techniques, SVM, and RF for various data sets displayed in Table 6.

Table 7 Results for the reference sequence set for the studied data set
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Table 7 shows that for the dataset (Table 6), MT-FISSA offers the best accuracy. Additionally, the AUROC value demonstrates that MT-FISSA and LT-FISSA outperform other classifiers. Because of its segmentation-based ensemble learning technique, RF performs better than other SVM machine learning models currently in use. To fix the issue of missing variables in Table 6, we can use the classification accuracy coefficient (ACC) calculation, which considers the confusion matrix's class ratios. Both MT-FISSA and MT-TSO are confirmed to be the best classifiers by looking at the ACC value. Results are inconsistent, though, when additional performance metrics are taken into account. Based on Table 7, it is evident that SVM and RF outperform MT-FISSA and MT-TSO when it comes to sensitivity. However, keep in mind that while support vector machines (SVMs) provide a sensitivity result of 100%, their specificity measurement is 0%, rendering the result practically unattainable. Consequently, the idea of the area under a receiver operating characteristic (ROC) curve is taken into consideration in order to ascertain which classifier best predicts the classes. The receiver operating characteristic area under the curve (AUROC) is a summary statistic used to evaluate a classifier's performance in a binary classification task. This outcome eliminates any ambiguity that could arise from using several criteria. The ROC curves, which represent the classifiers' performance at all conceivable thresholds, are displayed in Fig. 10. Curves are generated by plotting the sensitivity, or true positive rate, against the specificity, or false positive rate.

Fig. 10
figure 10

ROC detection of FOG for the Daphnet dataset

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It is evident from the AUROC value data presented in Tables 6 that the suggested approaches (MT-FISSA, MT-TSO, and LT-FISSA) perform more competitively than other advanced classifiers.

In terms of AUROC values, MT-FISSA shows an improvement of 12.4% over RF, 7.137% over MT-TSO, 78.32% over SVM and 1.25% over LT-FISSA for FoG detection dataset for Parkinson patients, while that LT-FISSA has an improvement of 11.22% compared to RF. 7.51% over MT-TSO, 78.94% over SVM and 1.6% over MT-TSO. Figure 11 shows the bar chart for the studied systems.

Fig. 11
figure 11

Display the bar graph related to the results of Table 7

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5.4 Discussion

One noteworthy finding from the empirical study is that the suggested MT-FISSA model outperforms the MT-TSO model. Without a doubt, the ensemble method boosts the baseline model's efficiency. The idea of mapping and substitution is intriguing from a practical standpoint due to the ever-changing nature of environmental elements. The main issue, though, is how the model reacts depending on the total number of output categories. It appears that the suggested method is unable to deliver satisfactory accuracy even when the number of output classes for multi-class data is increased. However, by utilizing the logical classification method as shown in Table 3 and expanding it to higher classes, this accuracy can be further enhanced.

The main benefit of this suggested strategy is that, once the suggested algorithms are trained, the hardware implementation of mapping systems and system functions can be implemented simply. It is clear that implementing a genuine system (using the suggested model) is successful in terms of implementation requirements, as it eliminates the need for extensive statistical analysis or thresholding in order to achieve a conclusion. It is also not contingent on any potential assignment. The primary benefit of the suggested model is its balanced performance. In contrast to existing approaches, the proposed model does not display a notable disparity between sensitivity and specificity. Furthermore, body sensors and smart objects are the only primary supplementary systems needed by the IoT-based approach to gather data for gait freeze detection. Recall that the necessary hardware model is likewise reasonably priced. It's not always expensive to install acceleration sensors and Internet of Things gateways to process incoming data via a reliable Internet connection. It also doesn't cost much to use a basic headphone to deliver the audio signals that are returned for patient alertness and gait therapy.

It is critical to make decisions through the classifier in real-time for an effective real-world gait recognition model implementation. It is clear from the complexity study that the suggested set model (MT-FISSA, which demonstrates the highest performance) has a relatively short execution time. Sending data to the fog computing centre and processing time for linear and Mobius mapping are the sole primary elements influencing the suggested model's performance. The processing speed of a very basic system for conducting real-time tasks makes this time quite possible. There is little effect on the execution time of the algorithm because a moderate number of weak learners are required for better model performance. As a result, the implementation is also efficient in terms of time. Additionally, the MT-FISSA method requires far less training time than the MT-TSO method.

6 Conclusion

The purpose of the present study is to present a classification approach for detecting and classifying the existence of freezing of gait (FoG) in an Internet of Things environment that is based on linear or Mobius mapping and its set version. The method would be used to patients with Parkinson's disease. The suggested approaches do not rely on complicated statistical analysis or probabilistic assignment to determine each new reference sequence. Each relative sample class (0 for no test, 1 for no ice, and 2 for frozen) is used by the suggested methods to classify a fresh sample via mapping transformations. According to the findings of the AUROC study, the proposed methods have better overall performance, ACC, F1 score, and accuracy (> 90%) than the current machine learning techniques. The most crucial factor is how easily the system's final model can be applied to real-time systems due to its simplicity after training. To further enhance the outcomes of the suggested FoG detection method, the authors hope to create a hybrid classifier in the future that combines existing non-parametric machine learning techniques with clustering algorithms.