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Article

A Set Theoretic Framework for Unsupervised Preprocessing and Power Consumption Optimisation in IoT-Enabled Healthcare Systems for Smart Cities

Department of Information Technology, Melbourne Polytechnic, 77 St Georges Rd, Preston, VIC 3072, Australia
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Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(16), 9047; https://doi.org/10.3390/app15169047 (registering DOI)
Submission received: 29 June 2025 / Revised: 10 August 2025 / Accepted: 12 August 2025 / Published: 16 August 2025
(This article belongs to the Special Issue IoT in Smart Cities and Homes, 3rd Edition)

Abstract

The emergence of the Internet of Things (IoT) has brought about a significant technological shift, coupled with the rise of intelligent computing. IoT integrates various digital and analogue devices with the Internet, enabling advanced communication between devices and humans.The pervasive adoption of IoT has transformed urban infrastructures into interconnected smart cities. Here, we propose a framework that mathematically models and automates power consumption management for IoT devices in smart city environments ranging from residential buildings to healthcare settings. The proposed framework utilises set theoretic association-rule mining and combines unsupervised preprocessing with frequent-item set mining and iterative numerical optimisation to reduce non-critical energy consumption. Readings are first converted into binary transaction matrices; then a modified Apriori algorithm is applied to extract high-confidence usage patterns and association rules. Dimensionality reduction techniques compress these transaction profiles, while the Gauss–Seidel method computes control set points that balance energy efficiency. The resulting rule set is deployed through a web portal that provides real-time device status, remote actuation, and automated billing. These associative rules generate predictive control functions, optimise the response of the framework, and prepare the framework for future events. A web portal is introduced that enables remote control of IoT devices and facilitates power usage monitoring, as well as automated billing.

1. Introduction

The Internet of Things (IoT) has evolved from a collection of networked gadgets into a pervasive cyber-physical fabric that underpins today’s smart city infrastructures [1,2]. IoT is defined as the integration of diverse digital and analogue devices with the Internet, such as smartphones, personal digital assistants, computers, and tablets. This integration enables advanced modes of communication among devices and between devices and humans [3]. The fundamental goal of IoT is the efficient and intelligent connectivity and monitoring of physical devices or objects in our environment [4,5]. By extending Internet Protocol (IP) connectivity to sensors, actuators, and embedded controllers, IoT enables continuous machine–to-machine and machine-to-human communication across domains as diverse as transport, industry, and public health [6,7,8,9,10,11].
In high-density urban centres, however, the very scale of connected devices introduces new challenges—most notably, the need to manage aggregate power demand without compromising service quality. This issue is particularly acute in hospitals, where life-critical equipment must remain fully operational even as administrators seek to curtail non-essential energy use [10,12,13,14]. The concept of smart cities underscores this potential and has been attracting interest from diverse socioeconomic groups globally [13,15]. The research community is focused on enhancing IoT technology to improve its effectiveness and reliability [2,9,12,16,17,18,19]. Intelligent algorithms play a crucial role in efficiently managing IoT-based devices and improving their effectiveness and reliability [15,20,21,22]. Existing work on smart hospital energy management typically falls into two streams. The first focuses on device-level metering and manual scheduling, offering fine-grained insights but limited automation [23,24]. The second employs machine learning classifiers for demand prediction, yet often requires labelled data and overlooks the combinatorial patterns that arise when devices operate in concert [25,26]. These gaps motivate a data-driven approach that operates in an unsupervised manner and captures frequent multi-device usage patterns. To address these requirements, this study proposes a framework based on set theory to optimise the power consumption of healthcare-related devices in a smart city, aiming to enhance both efficiency and reliability in managing and supervising energy use. This study introduces a set theoretic framework that unifies four methodological pillars: (1) Unsupervised preprocessing by using raw sensor streams from IoT devices is discretised into a binary transaction profile, enabling downstream pattern discovery without prior labelling. (2) Frequent-item-set mining by using a modified Apriori algorithm extracts high-confidence usage patterns and association rules that characterise typical device co-activation scenarios. (3) Iterative numerical optimisation using a Gauss–Seidel scheme refines control set points to reconcile energy efficiency objectives with clinical constraints. (4) Dimensionality reduction techniques such as Principal Component Analysis (PCA) and Independent Component Analysis (ICA) further compress transaction profiles.
The primary objective of this framework is to enhance the efficiency and reliability of health data collection and supervision. Furthermore, a web portal has been developed based on the proposed framework to facilitate reliable remote supervision and management of IoT-embedded devices at sites from any worldwide location. The portal accurately tracks and optimises power consumption, enabling automatic billings. Furthermore, it allows the classification of frequently used IoT devices at the site based on their usage period.
The rest of the paper is organised as follows: The Section 2 explores the most recent state-of-the-art literature in the optimisation of IoT device usage and the background of the algorithms employed in the proposed framework. Section 3 and Section 4 describe the methodology of this study, the architecture and functionality of the proposed framework, and Section 5 and Section 6 discuss the application of the proposed framework in a healthcare setting with evaluated results. Finally, Section 7 concludes the paper.

2. Background

2.1. Related Work

The authors of a recent article [27] proposed an energy-efficient autonomous smart home application that integrates the power of big data and cloud computing to provide comprehensive monitoring for seniors. This innovative multimedia-based supervisor empowers elderly individuals to oversee a range of energy-efficient systems, control smart home devices through gestures, receive timely alerts about device status changes, and share multimedia information with a specific group. Additionally, an autonomous switch-on/off management device has been developed, utilising speech or gestures to conserve energy. The successful recognition and utilisation of speech-based keywords as control signals highlight the potential for widespread adoption of this architecture, as demonstrated by empirical results.
Recent research [28] studied the latest sustainable energy management practices and thermal processes. It explores AI techniques for improving energy efficiency in smart building settings by analysing various AI models to gain insight into energy consumption patterns. It also discussed the challenges and benefits of integrating these technologies with advanced AI methods in energy management systems. A recent paper [29] explored data-driven building management systems focusing on energy management practices. It studied and demonstrated that the integration of IoT and AI technologies for monitoring and controlling smart buildings supports and improves the efficiency of energy management systems, especially in reducing energy consumption. The paper briefly discussed the use of modern technologies, such as blockchain, to further enhance the efficiency of smart building management systems.
The work in [30] introduced a smart home health monitoring system designed to analyse patients’ blood pressure and glucose readings within the comfort of their homes, promptly notifying healthcare providers when any anomalies are detected. The approach combined conditional decision-making and machine learning techniques to predict the status of hypertension and diabetes separately. The primary objective is to forecast hypertension and diabetes statuses by leveraging patients’ glucose and blood pressure readings. Employing supervised machine learning classification algorithms, the system is trained to predict a patient’s diabetes and hypertension statuses accurately.
A recent study [31] proposed a smart home method with moderate security and minimal development costs in the context of the Internet of Things. This procedure, on the other hand, only examines a sequential approach, which is incapable of handling congestion caused by a high number of devices. Another study [32] examined power usage and conservation in homes, combining IoT devices into a single line. They used an image processing method to detect human functions in this breakthrough. Image processing, on the other hand, takes longer to pick the desired devices. Another study, based on Wi-Fi and GSM technologies, is being conducted simultaneously. The goal of this project is to gather data from sensors and operate equipment based on that data [33]. Despite the fact that various types of studies are being conducted elsewhere, there is still room for development in the home automation system utilising IoT.
The authors of [34] place a greater focus on customer-centric techniques to make the system more scalable and have provided an architecture for an IoT-based last-meter smart grid based on this method. Their design offers four significant benefits, the first of which is seamless integration with smart home apps via a smart grid connection. Second, communication protocols are used to gather and analyse the detected data. Furthermore, only authorised individuals have access to the system. The technology also enables mapping sensors and actuators to the layer of abstraction, which simplifies interaction with non-technical users. The experiment is based on a ZigBee network that connects to an IoT server and communicates with it via a gateway that converts packets from ZigBee to IP gateway. On the other hand, the work in [35] addressed the resource allocation problem for edge computing in healthcare systems that handle large datasets.
Traditional patient data within the healthcare sector were conventionally archived in paper format, incurring substantial expenses and operational inefficiencies. Embracing big data technology presents a superior approach for collecting, preserving, and scrutinising patient information in a digital format. This transformation holds the potential to curtail expenses and enhance the overall effectiveness of healthcare delivery [36]. Recent advancements in the healthcare sector consistently demonstrate that integrating various technologies holds the promise of strengthening healthcare services and aiding healthcare professionals in the effective and efficient delivery of healthcare solutions [37].
A latest study [38] provides a comprehensive review of intelligent IoT-based embedded healthcare systems, exploring the benefits for physicians and patients. It suggests research directions and emphasises the importance of modern safety infrastructure for IoT in healthcare, benefiting governments, health experts, and clinicians. Accordingly, a recent publication [39] developed a framework using multimodal data to classify Alzheimer’s disease, employing graph neural networks for knowledge graph creation and region-based convolutional neural networks for image-to-knowledge graph generation. A study [40] addressed the need for visual representations and efficient navigation of scholarly articles to visualise unstructured data and facilitate insight into dementia risk factors. The model enabled automated knowledge extraction, storage, and visual exploration, showcasing promising results for knowledge discovery in dementia research. Additionally, another study [41] discussed Poland’s healthcare digital transformation and emphasised patient engagement through digital social innovation solutions based on Big Data Analytics (BDA). It identifies BDA’s potential role in developing social innovations in healthcare, providing insights into future research and practical implications for health policy and wearable device manufacturers.
This proposed framework integrates an advanced algorithm for remote monitoring of IoT devices in a healthcare setting by analysing different health parameters, which represents a significant advancement in IoT-based healthcare services. Existing studies focus on energy efficiency, gesture-based control, or monitoring of other health conditions; this proposed framework offers a solution for real-time event management or emergency preparation through systematic data organisation of healthcare IoT devices. Additionally, the combination of our framework and the web portal we have developed for remote supervision, power consumption tracking, and automated billing presents a unique blend of healthcare monitoring and operational efficiency. The Apriori algorithm is based on mathematical set theory and a well-established mathematical method for learning and generating association rules from massive datasets. Association rules are IF–THE statements that aid in the discovery of links between unrelated data in relational databases, as well as other types of data and characteristics. The Apriori method is a relatively advanced preference selection strategy that focuses on frequently occurring item sets and is subsequently combined with association rules. The different types of data items with their preferred properties or attributes are correlated here using the association rules [42].
In mathematical terms, let B = { B 1 , B 2 , B n } represent a collection of binary characteristics known as the item set, and T = { T 1 , T 2 , , T n } a set of transactions known as the dataset. T includes a subset of items from B, and each transaction has its own transaction ID. As a result, a rule is defined as an implication of the type X Y , where X , Y B and X Y = . The antecedent and consequent of the rules are the sets of objects X and Y, respectively. A simple mathematical notation may be used to specify an association rule, which can be defined mathematically as
X Y [ S , C ]
where X and Y are two distinct item sets, S is the support, and C is the confidence. The support count refers to the frequency of rules within the data categorisation, where X and Y are two separate item sets, S represents support, and C represents confidence. Support is counted at several data levels, such as high and low values. A high number implies that the rule has the most influence on the dataset. At the same time, a low value suggests that the rule is of lesser relevance in the creation of association rules and can be represented as
support ( X Y [ S , C ] ) = P ( P Q )
The variable C, which represents confidence, is used for the percentage of transactions in the rule generation and is executed in Algorithm 1. Algorithm 1 is augmented with explanatory comments to illustrate its application in this study, specifically for the minimum support algorithm. Confidence is mathematically given as
confidence ( X Y ) = P X Y = support ( X , Y ) support ( Y )
These support and confidence equations guide the decision-making process for generating association rules and controlling IoT devices. The thresholds for applying the rule are analysed by the mathematical attributes of the Apriori algorithm.
Algorithm 1 Apriori Algorithm for Minimum Support
procedure  A p r i o r i_A l g o r i t h m ( E k , F k )
    Input: E k : Candidate item set of size k //set of electronic device.
    Output: F k : frequent item set of size k //electronic device in candidate item set.
    Initialise:
    •  F 1 = { f r e q u e n t i t e m s } ; // initialise set of frequencies items
    •  m = 1
    REPEAT
    •  E m = c a n d i d a t e g e n e r a t e s f r o m F m //generates a new candidate
     REPEAT
     •  t T // transaction in T, database subgroups of electronic device
     • do begin
         E m + 1 = s u b s e t ( E m , t ) // this function generates candidates in transaction
         e · e E m e E m + 1 // for all candidates
        • do begin
            e . c o u n t + + //determine support
        • end
         F m + 1 = e | e E m e E m + 1 e . c o u n t m i n S u p // new set
      • end
         m = m + 1
      UNTIL  F m ,
    UNTIL  m > k ,
    OUTPUT E k + 1 F k + 1
    END
end procedure

2.2. Principal Component Analysis for Smart Digital Devices

The Principal Component Analysis (PCA) algorithm is a linear algebra-based method used to reduce the complexity of a dataset. This is achieved by performing covariance matrix analysis on the different variables of the dataset. It is intended to address large multidimensional datasets on string sequences [1,2,3] and to identify mapping and formatting digital devices. It constructs linear combinations of the digital devices [4]. The PCA extracts do not change the number of extracted data points from the observed one, but instead reduce the dimensionality of the matrix. For example, measuring 1000 IoT devices for 100 rooms will require a matrix of 100 × 1000 measurements as depicted in Figure 1. This will be an n × n matrix that can be formulated as follows:
P 11 Λ P 1 n M O M P n 1 Λ P n n
where P defines the combination of digital against every room in our study.
The eigenvectors represent the repeated factors in the case of a multidimensional dataset, which are called the principal components and eigenvalues, which are calculated as follows:
a 11 Λ a 1 n M O M a n 1 Λ a n n λ 1 Λ P 1 n M O M P n 1 Λ λ n
where a 11 , , a n 1 , , a n n are the eigenvectors with the corresponding eigenvalues λ 1 , λ 2 , , λ n . According to PCA, digital devices can be represented by Gaussian signals. Singular value decomposition (SVD) is a mathematical technique closely related to PCA, which factorises a matrix into smaller matrices. It can be applied alternatively to the data of digital devices. Assume a matrix A of p rows and q columns; thus, the SVD for the large string digital data is as follows:
A = M P V T
where M is an m × n matrix that contains left singular vector elements and V T is an n × n matrix that contains correct singular vector elements. P is a nonzero value on the diagonal and is called a singular value.

2.3. Independent Component Analysis

A statistical and computational method named Independent Component Analysis (ICA) finds statistically ‘independent’ and ‘non-Gaussian’ components of the multivariate data to separate a multivariate signal into additive, independent components. Information theory is primarily used for data reduction, noise reduction, and pattern recognition. Finding new components that are mutually independent and minimise the dependence between the elements is the primary goal of ICA. This entropy theory-based method is used to obtain a linear representation of non-Gaussian data to obtain independent components.
ICA finds a linear that is a linear transformation [43]. If m random variables are observed, they can be represented as linear combinations by the following mathematical equation:
a i = p i 1 s 1 + p i 1 s 1 + + p i n s n , i = { 1 , 2 , , m } .
where ( a 1 , a 2 , , a m ) is a vector of m random variables that are observed; ( s 1 , s 2 , , s n ) represents n random variables, where every s j is a set of independent components (ICs); and finally, p i j is the unknown coefficients. Using the matrix form, Equation (4) can be written as below:
a = P s
where a and s are the sets of their transpose and P is the n × m matrix for p i j . The observations can be modelled using the independent component matrix S as
A = P S
MacKay [44] proposed an ensemble learning ICA approach for easier decomposition, as follows:
D m i = E m t F t i + K m i
where i refers to an input given as original variables, m is the enumerating samples, K represents the Gaussian noise, and t is the latent variable. Here, K provides the reconstruction error for any particular E and F, which is given by the following equation:
K = D E F
According to the reconstruction error, the data power of latent variables is given by the following equation:
p t = m , i ( e m t f t i ) 2 m , i ( d m i ) 2 = m ( e m t ) 2 i ( f t i ) 2 m , i ( d m i ) 2
Generally, the digital devices’ sequences represent a room of the total rooms of the hospital as a mixture of independent analytical processes. Every process forms a vector that shows the proper shapes of the devices. Mathematically, it represents a linear combination of n digital devices, as given by Equation (4), which is represented in matrix form in Equation (5) above. Thus, Equation (5) for our proposed framework can be written as
a 1 M a m = p 11 Λ m 1 n M O M p m 1 Λ p m n s 1 M s m
It can be observed that all digital devices are represented as a = ( a 1 , , a n ) T , where a cell is governed by n independent device processes s = ( s 1 , , s n ) T , where each of the vectors represents k device levels. ICA works with a vast amount of data, along with reducing facilities, but Pushdown Automata (PDA) bring a new advantage by freeing memory space. It builds as a stack performing push-pop function.

2.4. Pushdown Automaton

Pushdown Automata (PDA) combine formal language theory and automata theory with a stack-based computational model to free frequently used memory space. There are three main components in PDA, namely, the input tape, the control unit, and the infinite size of the stacks [45]. After completing the control unit process, in the case of matching into the anchor, the count unit’s activity appears. It counts the matches of the anchor with the input dataset and remembers the number of matches it returns as the value of the seeds. However, if a mismatch occurs, then it has nothing to count and does not return any seed value. Thus, the control unit removes the symbol from the dataset using finite state control, thereby freeing memory.

2.5. Gauss–Seidel

The Gauss–Seidel method is a numerical method applied to a matrix with nonzero elements to solve a linear system with n equations and unknown variables. It is also known as a successive approximation method as it uses iterative techniques of numerical linear algebra. Other comparable methods to the Gauss–Seidel method that can iteratively adjust model parameters, such as Bayesian inference techniques and Markov chain Monte Carlo (MCMC) methods. Both techniques progressively improve their estimates and align with observed data. However, this study utilises the Gauss–Seidel method because it provides a straightforward, efficient approach for solving linear systems without the need for complex and computationally intensive sampling techniques [46]. Following the application of the Gauss–Seidel method, the current study employs the Apriori algorithm to ascertain frequent item sets within the transaction dataset. This algorithm, as shown in Algorithm 2, systematically structures and identifies frequent items by mining the dataset individually. The presentation of Algorithm 2 is enhanced by mapping each step to our proposed framework, elucidating the relevance of the algorithm to this study.
Algorithm 2 Algorithm for mining the dataset
procedure  U n k n o w n_A l g o r i t h m ( a , n , m , g )
    Input: m , n , g 1 : Candidate set items m of size n, and initial guess g 1
    Output: g: //
     REPEAT
     •  i 1 t o n // transaction in T, database subgroups of electronic device
     •  a = 0 // transaction in T, database subgroups of electronic device
     •  j 1 t o n // transaction in T, database subgroups of electronic device
       •if  i j  then
          a = a + m i j g j //
       • endif
         g i = 1 m i j + ( n i a ) // new set
     • endfor
     • endfor
     UNTIL  c o n v e r g e n c e ,
    OUTPUT g
    END
end procedure

3. Methodology

In this study, the case study utilised the cases of digital devices in Square Hospital, a renowned institution for its exceptional healthcare services in Bangladesh, as an example of IoT devices in a healthcare setting. Five datasets were collected from the hospital for this study. Only data about the device in each room were used in this study, and no patient data were included. Our research methodology involves applying the proposed framework across five datasets, labelled D0, D1, D2, D3, and D4. We will assume that there are several rooms (R1, R2, …, Rn) with several digital devices, such as an Air Conditioner (A), a Light (L), a Fan (F), and a Television (T). If the digital elements are presented in the sample, then we map them to 1; otherwise, 0. This will convert the attributes into binary flags. Table 1 shows a possible configuration.
Figure 2 [47] is based on the taxonomy of the IT-related parts of the healthcare system. This taxonomy will help understand and simplify the complexity of the healthcare system design, implementation, maintenance, and use. It depicts a detailed, hierarchical structure for data management for staff and patient services, infrastructure or business management, and branches extending to various categories, such as data management or warehouse, application integration, security, and analysis, representing specific processes. The proposed framework is based on this taxonomy.

4. Proposed Framework for Power Optimisation in Healthcare Devices

Based on a comprehensive understanding of IT devices within healthcare systems, we introduce a robust supervisory framework designed to streamline data mining processes. Leveraging the recursive Gauss–Seidel method, our framework integrates the mathematical set theory-based Apriori algorithm to identify frequent device sets within the transactional dataset systematically. This algorithm meticulously extracts frequent items from datasets, enabling users to manage digital devices flexibly in healthcare settings. It facilitates remote oversight and control of multiple smart devices, offering real-time updates on equipment status.
Our framework encapsulates a meticulous data mining process tailored to pinpoint frequent device sets and generate association rules from datasets. Our proposed framework is visually represented in Figure 3, providing a comprehensive illustration of its functionality and workflow.
Primarily associated with the Apriori algorithm, this process is pivotal for identifying patterns within extensive datasets efficiently. Initially, the framework scans the training dataset (D) to tally occurrences of each candidate device, thereby establishing their frequency. Subsequently, it compares the support count (SC) of each device against a predefined threshold, termed the minimum support, to identify ‘frequent’ devices.
Following this initial scan, the framework derives an initial set of candidate devices (C) and a set of frequent device sets (L) based on this comparison against the minimum support threshold. Utilising the frequent device sets from L, the framework generates reduced combinations or pairs of candidate devices, subsequently scanning the dataset (D) to determine their support counts. This process results in modified frequent device sets, now representing pairs or combinations of items. Iteration of this process continues until no new frequent device sets can be generated. Figure 4 illustrates the graphical formation of PCA for digital device formation for every 10 devices’ formation.
Once frequent device sets (e.g., in L3) are identified, the framework proceeds to generate association rules. These rules are derived from device sets by evaluating all nonempty subsets against a confidence threshold, typically set at 80%. The PCA for digital device formatting to generate association rules is shown in Figure 5. Mainly, there are two stacks: the input tape and the stack. The required dataset from the digital devices is stored on the stack table. The control unit assesses whether a match is taking place and subsequently dispatches the symbols that correspond to either a game or a mismatch. This method conserves memory by creating available memory space, as illustrated in Figure 5.
The framework selectively retains rules that meet or exceed this confidence threshold while rejecting others. This innovative framework demonstrates how specific device combinations undergo rigorous evaluation based on support count and subsequent rule generation, assessed according to confidence levels. The application of this framework is exemplified through a real-world case study in hospital data, demonstrating its practical utility and effectiveness in healthcare settings, which is detailed in the next section.

5. Case Study: IoT-Based Power Optimisation in Healthcare

This section presents a comprehensive application of the proposed framework, accompanied by a detailed case study tailored to a healthcare setting. This application demonstrates the effectiveness and versatility, highlighting the potential benefits of the proposed framework.

5.1. Proposed Framework for Dataset D0

In the first step, we scan the dataset D0 to count each candidate, and then perform a comparison between the support count (SC) and the minimum possible support. Thus, C 1 and L 1 are generated as shown in Figure 6. From L 1 , two frequent patterns of devices, C 2 , are generated. Then, scan the dataset D to obtain the support count, L 2 . After obtaining C 2 and L 2 , generate L 3 ; then classify the different digital device classes. L 3 indicates a classification group that satisfies our considerable support count. Now, apply the association rule considering the desired confidence value.
Next, we generate the association rules for frequent item sets from L 3 . For every item set of L 3 , all nonempty subsets of frequent item sets are generated. Consider N = { L , F , T } ; then its all nonempty subsets are { L } , { F } , { T } , { L , G } , { L , T } , { F , T } . Consider that the minimum confidence threshold is 80%. The resulting association rules are as follows in Table 2:
As a result, the flags of dataset D0 are presented in Table 1, which in turn can be modelled using the following set of equations:
3 a + l + f + t = 6 a + 3 l + f + t = 6 2 ( l + f + t ) = 6 a + l + f + 2 t = 5
which can be simplified into
a = 2 l + f + t 3 l = 2 a + f + t 3 f = 3 l t t = 2.5 0.5 ( a + l + f )
Next, we applied the Gauss–Seidel method on the above set of equations to obtain the following successive iterations, as illustrated in the following Table 3:

5.2. Proposed Framework for Dataset D1

For dataset D1 with segmented data, the following samples will be generated, as shown in Table 4:
In the next step, we scan the dataset D1 to count each candidate and then compare. Thus, C 1 and L 1 are generated. From L 1 , two frequent patterns of sequence, C 2 , are generated. Then scan the dataset, D1, to obtain the support count, L 2 . After obtaining C 2 and L 2 , the minimum support L 3 is generated, and then the different digital device classes are classified. L 3 indicates a classification group that satisfies our considerable support count. Now, apply the association rule considering the desirable confidence value. Figure 7 shows these steps.
L 3 indicates one classification group that satisfies our considerable support count. Next, we apply the association rule considering the desired confidence value. The association rules are generated for frequent item sets from L 3 . For every item set of L 3 , all nonempty subsets of frequent item sets are generated. Consider N = A, L, T; then all its nonempty subsets are {A}, {L}, {T}, {A,L}, {L,T}, {A,T}. Consider that the minimum confidence threshold is 80%. The resulting association rules are shown in the following Table 5:
As a result, the flags of dataset D1, which are presented in Table 4, can be modelled using the following set of equations:
2 a + l + 2 t = 5 a + 2 l + f + t = 5 a + l + 2 f + t = 5 a + l + 3 t = 5
which can be simplified into
a = 2.5 0.5 l t l = 2.5 0.5 ( a f t ) f = 2.5 0.5 ( a l t ) t = 1.67 0.33 ( a l )
Next, the Gauss–Seidel method is applied to the above set of equations to obtain the following successive iterations, as illustrated in the following Table 6:

5.3. Proposed Framework for Dataset D2

For dataset D2, the segmented data are generated in a manner similar to that in Table 7. Support count is also generated using the same method above, as demonstrated in Figure 8 below:
Then, an association rule is applied on the nonempty subsets—{A}, {L}, {T}, {A,L}, {L,T}, {A,T}—considering a minimum confidence threshold of 80%, which results in the association rules shown in Table 8 below:
The flags of dataset D2, which are presented in Table 7, can be modelled similarly using the following set of equations:
3 a + 2 l + t = 6 2 a + 2 l + f + t = 6 a + 2 l + 2 f + t = 6 a + 2 l + f + 2 t = 6
which can be simplified into
a = 2.5 0.67 l 0.33 t l = 3.0 a 0.5 f 0.5 t f = 3.0 0.5 a l 0.5 t t = 3.0 0.5 a l 0.5 f
Next, the Gauss–Seidel method is applied to the above set of equations to obtain the following successive iterations, as illustrated in the following Table 9:

5.4. Proposed Framework for Dataset D3

For dataset D3, the segmented data are generated similarly to those in Table 10. The support count is also generated using the same method as above, as demonstrated in Figure 9 below. The association rule is applied on the nonempty subsets— {A}, {L}, {F}, {T}, {A,L}, {A,F}, {L,T}, {L,F}, {A,T}, {A,L,F}, {A,F,T}, {L,F,T}, {A,L,T}—considering a minimum confidence threshold of 80%, which results in the association rules shown in Table 11.
Next, the Gauss–Seidel method is applied to the above set of equations to obtain the successive iterations, as illustrated in Table 12.
2 a + 2 + f + t = 6 a + 2 l + f + 2 t = 6 a + 2 l + 2 f + t = 6 a + l + 2 f + 2 t = 6
which can be simplified into
a = 3.0 0.5 l f 0.5 t l = 3.0 0.5 a 0.5 f t f = 3.0 0.5 a l 0.5 t t = 3.0 0.5 a 0.5 l f

5.5. Proposed Framework for Dataset D4

For dataset D4, the segmented data are generated similarly to those in Table 13. The support count is also generated using the same method as above, as demonstrated in Figure 10 below. The association rules are applied with a minimum confidence threshold of 80%, which results in the association rules shown in Table 14 below.
2 a + l + + 2 f = 5 a + 3 l + t = 5 a + l + 2 f + t = 5 a + l + f + 2 t = 5
which can be simplified into
a = 2.5 0.5 f l = 1.67 0.33 a 0.33 t f = 2.5 0.5 a 0.5 l 0.5 t t = 2.5 0.5 a 0.5 l 0.5 f
Next, the Gauss–Seidel method is applied to the above set of equations in order to obtain the successive iterations, as illustrated in the following Table 15:

6. Result and Discussion

Based on the previous findings, it is evident that the Gauss–Seidel method iteratively simulates the datasets of digital devices utilised in Square Hospital, enabling the identification of common segments. These device sequences represent the most prevalent device patterns. The iterative simulation using Gauss–Seidel on training digital data ensures the acquisition of the desired device sequences. The utilisation of association rules facilitates the automation of data handling in both new testing and training datasets. These rules are generated to address future datasets and issues. PCA and ICA play a crucial role in mapping the entire digital device segments. According to the analysis conducted, ICA demonstrates superior performance compared with PCA. For evaluating results, unsupervised learning has been considered for handling a large number of digital devices in the hospital, and the evaluation criterion is time in nanoseconds. Table 16 shows the results of the PCA and ICA before and after using PDA. The first column of the table reflects the data size of random digital devices for the system. Datasets have been collected from a modernly equipped large hospital (Square Hospital, Bangladesh), and all the devices used are correctly addressed. Moreover, our proposed framework can handle a random number of datasets. The first table depicts the impact before the application of the PDA approach.
Table 16 shows that, before applying PDA to the data samples, the data take 1276 ns using PCA, whereas it takes 734 ns using ICA. Consequently, ICA needs less time than PCA to generate the result for these data. Similarly, ICA takes 899 ns, which is 475 ns less than PCA for the second data size. The third data point in Table 3 shows a difference of 656 ns. Moreover, the differences of PCA and ICA for the exact data sizes are 961, 1689, 2198, 2922, 3500, 6423, 8277, 10,150, and so on. When checking for large datasets, the differences are increasing. PCA consumes more time than ICA.
According to Table 17, the first data take 598 ns for PCA, whereas ICA takes 545 ns after attaching the PDA. Hence, ICA requires around 9% less time than PCA. Similarly, for the second dataset, ICA requires around 8% less time than PCA. The differences in execution time between the two algorithms indicate that ICA is more efficient.
The proposed framework boasts several notable advantages. First, it offers a user-friendly interface centred around a dataset that simplifies the task of turning home automation devices on or off. With a straightforward click, users can easily control various devices, ensuring convenience and ease of use. Additionally, the system is highly scalable, thanks to the use of a scale-based indicator that can accommodate a range of devices, including the regulation of fan speeds. This adaptability ensures that the system can effectively cater to the diverse needs of users.
In addition, the proposed framework can provide necessary data acquisition tools to collect, aggregate, and analyse relevant information. Users can access monitored devices based on the data received and analysed, empowering them to make informed choices to handle and respond to critical situations. Moreover, the framework’s performance leverages the enhanced association rules generated by the Apriori algorithm. These rules enable the system to adapt effectively to various scenarios and environments, ensuring optimal functionality. In addition to its core functionalities, the proposed framework is practically adaptable to support emergencies, particularly in small cities where healthcare infrastructure may be limited and power infrastructure less resilient. By extending the rule-based prioritisation, the proposed framework can continuously monitor the power consumption patterns of critical medical devices or equipment for any anomalies, such as ventilators or emergency lights. The framework can trigger an alert or invoke predefined emergency protocols, which may include prioritising the power delivery if a deviation from regular use is identified. The incoming critical health device data streams are converted into binary vectors and reduced using PCA/ICA; support values from association rules assign priority weights, and an event-driven controller recomputes Gauss–Seidel power balancing under emergency constraints to shed non-essential loads while prioritising supply to life-critical device circuits. The modular nature of the framework enables the continuous operation of crucial support devices while shedding non-essential loads, providing a resource-efficient solution for small municipalities with restricted infrastructure resources.
Hence, the proposed supervisory smart hospital system presents a reliable solution for collecting and analysing big data in the health domain, with several features that provide users with an easy-to-use system while maintaining control over data. Figure 11 is a snapshot of the prototype of the web portal dashboard, which demonstrates how aggregate energy consumption can be monitored and individual IoT-enabled devices can be controlled directly within a room. The central panel displays the total energy usage for a room, while the device control card lists each device, such as the Air Conditioner, Light, Fan, and Television, with on/off toggles. This demonstrates that our proposed framework not only tracks power consumption in real-time but also enables users to remotely control medical and support equipment.

7. Conclusions and Future Work

In conclusion, this paper presents a comprehensive framework for exploring mathematical methods to automate and monitor power consumption optimisation and health data collection from electronic devices in a healthcare setting. Using mathematical algorithms such as the Apriori algorithm, PCA, ICA, and the Gauss–Seidel method, the proposed framework efficiently organises sensor data from devices to perform dynamic analysis and predictions, thereby enhancing the efficiency and reliability of supervising and managing health data. The proposed framework enables the collection, aggregation, and analysis of data for users within different hospital settings, utilising a combination of mathematical theory-based algorithms. This framework provides accessible features to enable monitoring various health parameters, generating necessary health alerts, and storing and sharing health data when necessary. For example, in healthcare settings of a small city, the adaptive framework and the rule-based controls can be configured to prioritise critical medical equipment during emergencies.
In the future, our research will focus on several key areas to further enhance the capabilities of IoT-enabled healthcare systems. First, we plan to conduct an extended evaluation of the framework’s performance through a comprehensive simulation model that considers a broader range of parameters. This will provide deeper insights into the framework’s efficiency and areas for improvement. In parallel, we plan to extend the framework to include advanced emergency management capabilities. For example, we will explore the integration of machine learning (ML) algorithms with real-time data analysis to enhance the accuracy of decision-making processes. ML techniques offer the potential to identify hidden patterns and gain data insights, enabling more precise predictions of healthcare events and emergency preparations. Lastly, we aim to explore and integrate additional algorithms based on modern mathematical theory into the framework further to enhance the processing and optimisation of data records. It will improve the performance of the proposed framework in monitoring and responding to health-related issues in smart healthcare settings. Therefore, the framework will contribute to the development of more intelligent, efficient, and reliable healthcare IoT devices.

Author Contributions

Conceptualization, S.P.; Methodology, K.F.; Writing—original draft, S.P.; Writing—review & editing, K.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The authors express their sincere gratitude to Amjad Gawanmeh for his insightful contributions and support, which significantly improved the quality of this research.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Mapping of 100 × 1000 measurements PCs in the two-dimensional space.
Figure 1. Mapping of 100 × 1000 measurements PCs in the two-dimensional space.
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Figure 2. Taxonomy of the IT-related components of the healthcare system [47].
Figure 2. Taxonomy of the IT-related components of the healthcare system [47].
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Figure 3. Proposed framework for power optimisation in healthcare devices.
Figure 3. Proposed framework for power optimisation in healthcare devices.
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Figure 4. Principal Component Analysis for digital device formatting.
Figure 4. Principal Component Analysis for digital device formatting.
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Figure 5. Digital device readings to generate association rules.
Figure 5. Digital device readings to generate association rules.
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Figure 6. Reducing set of items using support count and minimum support for dataset D0.
Figure 6. Reducing set of items using support count and minimum support for dataset D0.
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Figure 7. Reducing set of items using support count and minimum support for D1.
Figure 7. Reducing set of items using support count and minimum support for D1.
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Figure 8. Reducing the set of items using support count and minimum support for D2.
Figure 8. Reducing the set of items using support count and minimum support for D2.
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Figure 9. Reducing the set of items using support count and minimum support for D3.
Figure 9. Reducing the set of items using support count and minimum support for D3.
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Figure 10. Reducing set of items using support count and minimum support for D4.
Figure 10. Reducing set of items using support count and minimum support for D4.
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Figure 11. Snapshot of web portal.
Figure 11. Snapshot of web portal.
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Table 1. Device sequence and mapping in four hospital rooms.
Table 1. Device sequence and mapping in four hospital rooms.
RIDDevices SequenceALFT
R 1 ALFTAA3111
R 2 AFLLLT1311
R 3 LFFTTL0222
R 3 TALTF1112
Table 2. Successive iterations for Dataset D0 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
Table 2. Successive iterations for Dataset D0 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
RulesConfidenceResult Status
R 1 : L F T 4/4 = 100%Selected
R 2 : F T L 4/5 = 80%Selected
R 3 : L T F 4/5 = 80%Selected
R 4 : F L T 4/5 = 80%Selected
R 5 : L F T 4/7 = 57.14%Rejected
R 6 : T L F 4/6 = 66.67%Rejected
Table 3. Association rules confidence for D0.
Table 3. Association rules confidence for D0.
Iterationalft
12.000001.340001.660000.00000
21.010001.118901.881100.49500
30.846650.936491.568510.82418
40.901370.912961.262860.96140
50.964720.947641.090960.99834
60.997810.981251.020401.00027
71.009371.000090.999650.99545
Table 4. Devices sequence and mapping for dataset D1.
Table 4. Devices sequence and mapping for dataset D1.
RIDDevices SequenceALFT
R 1 ALTTA2102
R 2 TALLF1211
R 3 AFLFT1121
R 3 TALTT1103
Table 5. Successive iterations for D1 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
Table 5. Successive iterations for D1 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
RulesConfidenceResult Status
R 1 : A L T 4/4 = 100%Selected
R 2 : A T T 4/5 = 80%Selected
R 3 : L T A 4/4 = 100%Selected
R 4 : A T T 4/5 = 80%Selected
R 5 : L T T 4/5 = 80%Selected
R 6 : T L L 4/7 = 57.14%Rejected
Table 6. Association rules confidence for D1.
Table 6. Association rules confidence for D1.
IterationALFT
12.500001.250000.625000.43250
21.442501.250000.937500.78147
31.093531.093751.015630.94820
41.004931.015631.015631.00322
50.988970.996091.005861.01493
60.987020.996091.000981.01557
Table 7. Devices sequence and mapping for dataset D2.
Table 7. Devices sequence and mapping for dataset D2.
RIDDevices SequenceALFT
R 1 AALLTA3201
R 2 LAFLFT1221
R 3 LTALAF2211
R 3 LFTTAL1212
Table 8. Successive iterations for D2 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
Table 8. Successive iterations for D2 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
RulesConfidenceResult Status
R 1 : A L T 4/6 = 66.67%Rejected
R 2 : A T L 4/4 = 100%Selected
R 3 : L T A 4/5 = 80%Selected
R 4 : A L T 4/7 = 57.14%Rejected
R 5 : L A T 4/8 = 50%Rejected
R 6 : T A L 4/5 = 80%Selected
Table 9. Association rules confidence for D2.
Table 9. Association rules confidence for D2.
Iterationalft
12.000001.000001.000000.50000
21.165001.085001.082500.79125
31.011941.051191.047220.91923
40.992361.024421.019790.96951
50.993701.011651.006750.98813
60.996111.006451.001430.99478
70.997401.004490.999420.99710
80.997951.003800.998680.99789
Table 10. Devices sequence and mapping for dataset D3.
Table 10. Devices sequence and mapping for dataset D3.
RIDDevices SequenceALFT
R 1 TLAAFF2121
R 2 LLFTAT1212
R 3 FLFTLA1221
R 3 AFLFTT1122
Table 11. Successive iterations for D3 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
Table 11. Successive iterations for D3 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
RulesConfidenceResult Status
R 1 : A L F T 4/4 = 100%Selected
R 2 : A F T L 4/4 = 100%Selected
R 3 : A L T F 4/4 = 100%Selected
R 4 : L F T A 4/4 = 100%Selected
R 5 : A L F T 4/5 = 80%Selected
R 6 : F A L T 4/7 = 57.14%Rejected
R 7 : L A F T 4/6 = 66.67%Rejected
R 8 : T A L F 4/6 = 66.67%Rejected
Table 12. Association rules confidence for D3.
Table 12. Association rules confidence for D3.
Iterationalft
13.000001.500000.000000.75000
21.875001.312500.375001.03125
31.453131.054690.703131.04297
41.248050.981450.873051.01221
51.130130.986210.942630.99921
61.064670.997150.970920.99818
71.031421.000650.984550.99942
81.015421.000600.991981.00001
91.007711.000140.995991.00008
Table 13. Devices sequence and mapping for dataset D4.
Table 13. Devices sequence and mapping for dataset D4.
RIDDevices SequenceALFT
R 1 AFAFL2120
R 2 ALLLT1301
R 3 AFLFT1121
R 3 LFTAT1112
Table 14. Successive iterations for D4 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
Table 14. Successive iterations for D4 (where bold entries indicate rules meeting the 80% minimum confidence threshold and thus selected).
RulesConfidenceResult Status
R 1 : A L T 4/6 = 66.67%Rejected
R 2 : A T L 4/4 = 100%Selected
R 3 : L T A 4/5 = 80%Selected
R 4 : A L T 4/7 = 57.14%Rejected
R 5 : L A T 4/8 = 50%Rejected
R 6 : T A L 4/5 = 80%Selected
Table 15. Association rules confidence for D4.
Table 15. Association rules confidence for D4.
Iterationalft
12.500000.845000.827500.41375
21.250001.120961.107640.76070
30.831881.144451.131490.94609
40.796291.095021.081301.01370
50.871191.047991.033561.02363
60.942441.021201.006371.01500
70.983041.010650.995661.00533
80.999021.008570.993540.99944
91.002171.009470.994460.99695
Table 16. Time comparisons between PCA and ICA before applying PDA.
Table 16. Time comparisons between PCA and ICA before applying PDA.
Digital Device SizeTime for PCA (ns)Time for ICA (ns)
10001276734
20001374899
30001654998
400020371076
500027871198
600036871489
700047981876
800054542054
900075122189
10,00087652588
11,00010,8712821
12,00013,3453972
13,00016,6434065
14,00017,7544198
15,00021,0984778
16,00026,6545076
17,00029,7659654
18,00037,452120,876
Table 17. Time comparisons between PCA and ICA after applying PDA.
Table 17. Time comparisons between PCA and ICA after applying PDA.
Digital Device SizeTime for PCA (ns)Time for ICA (ns)
1000598545
2000612567
3000745643
40001043754
50001409943
600019641078
700023751112
800030751399
900034781445
10,00043751787
11,00052761967
12,00059762297
13,00071822412
14,00089542587
15,00099122634
16,00010,1232876
17,00010,6543088
18,00010,9123378
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Parvin, S.; Fahd, K. A Set Theoretic Framework for Unsupervised Preprocessing and Power Consumption Optimisation in IoT-Enabled Healthcare Systems for Smart Cities. Appl. Sci. 2025, 15, 9047. https://doi.org/10.3390/app15169047

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Parvin S, Fahd K. A Set Theoretic Framework for Unsupervised Preprocessing and Power Consumption Optimisation in IoT-Enabled Healthcare Systems for Smart Cities. Applied Sciences. 2025; 15(16):9047. https://doi.org/10.3390/app15169047

Chicago/Turabian Style

Parvin, Sazia, and Kiran Fahd. 2025. "A Set Theoretic Framework for Unsupervised Preprocessing and Power Consumption Optimisation in IoT-Enabled Healthcare Systems for Smart Cities" Applied Sciences 15, no. 16: 9047. https://doi.org/10.3390/app15169047

APA Style

Parvin, S., & Fahd, K. (2025). A Set Theoretic Framework for Unsupervised Preprocessing and Power Consumption Optimisation in IoT-Enabled Healthcare Systems for Smart Cities. Applied Sciences, 15(16), 9047. https://doi.org/10.3390/app15169047

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