Artificial intelligence-driven ensemble deep learning models for smart monitoring of indoor activities in IoT environment for people with disabilities
This study develops a MOEM-SMIADP model. The proposed model concentrates on detecting and classifying indoor activities using IoT applications for physically challenged people. It encompasses four steps: data normalization, MPA-based feature selection, an ensemble of classification models, and parameter selection using ICOA. Figure 1 illustrates the workflow of the MOEM-SMIADP model.
Workflow of MOEM-SMIADP model.
Min–max normalization
At first, the data preprocessing executes min–max normalization to convert input data into useful format29. This is chosen because it can scale data to a fixed range, usually [0, 1] or [− 1, 1], ensuring uniform contribution from all features during the learning process. This technique is particularly effectual when features have varying scales, preventing dominant features from overshadowing smaller ones. Unlike standardization, which transforms data based on mean and standard deviation, min–max normalization preserves the data’s distribution and relationships, making it ideal for algorithms sensitive to an absolute scale, such as GCNs and LSTMs. Additionally, it improves the convergence speed of optimization algorithms, mitigating training time and improving stability. Its computational efficiency makes it well-suited for large datasets and real-time applications.
Min–max normalization is an effective data preprocessing mode, which measures feature values to a definite range from [0, 1], maintaining the relations within the data. In smart observing of indoor actions utilizing IoT applications, this model certifies consistency across data gathered from numerous sensors, which might have dissimilar ranges or units. For disabled persons, accurate recognition of anomalies and activities is vital, and min–max normalization decreases the impact of outliers and noise, thereby improving the excellence of data. It enhances the performance of ML techniques by delivering standardized inputs, permitting methods to classify refined variations in activity patterns. This ensures adaptive, reliable, and effective monitoring methods, which provide individual requirements, promoting safety and independence.
Feature selection using MPA
Furthermore, the MPA performs the feature selection process30. This technique is chosen because it can effectually identify the most relevant features while discarding noisy or redundant ones. MPA replicates the intelligent foraging strategies of marine predators, effectively balancing exploration and exploitation during optimization. Unlike conventional techniques like correlation-based or wrapper methods, MPA can handle high-dimensional data and intrinsic interactions between features. This mitigates computational overhead and enhances the model’s performance by focusing only on significant features. Its adaptability and robustness in diverse data scenarios make it particularly appropriate for improving classifier accuracy and ensuring improved generalization.
The marine predator’s foraging movements stimulate MPA. Predators often switch between dual motion patterns: Brownian motion (BM), which involves consecutive moves in a similar position that improves exploitation, and Levy motion (LM), which involves short motions followed by higher jumps that increase exploration.
Stage (1). Initialization: the search space is packed using the randomly and uniformly distributed primary solutions.
Stage (2). The prey matrices are upgraded in stage (1), considered by higher-velocity ratios. This upgrade occurs in the first third of iterations while exploring problems.
$$\overrightarrow {{Step_{i} }} = \overrightarrow {{Step_{i} }} \otimes \left( {\overrightarrow {{Elite_{i} }} – \left( {\overrightarrow {{R_{B} }} \otimes \overrightarrow {{Prey_{i} }} } \right)} \right)$$
(1)
$$\overrightarrow {{Prey_{i} }} = \overrightarrow {{Prey_{i} }} + \left( {P \cdot \vec{R} – \left( {\overrightarrow {{R_{B} }} \otimes \overrightarrow {{Step_{i} }} } \right)} \right)$$
(2)
Meanwhile, RB denotes the vector of random numbers according to the standard distribution of BM, \(P=0.5\), and \(R\) signifies a vector of uniform randomly formed integers amongst 0 and 1.
Stage (3). Stage (2) upgrade is identified as the transitional optimizer stage, whereas the model shifts from exploration to exploitation. This procedure occurs in the second third of the iterations.
$$\overrightarrow {{Step_{i} }} = \overrightarrow {{R_{L} }} \otimes \left( {\overrightarrow {{Elite_{i} }} – \left( {\overrightarrow {{R_{L} }} \otimes \overrightarrow {{Prey_{i} }} } \right)} \right)$$
(3)
$$\overrightarrow {{Prey_{i} }} = \overrightarrow {{Prey_{i} }} + \left( {P \cdot \vec{R} \otimes \overrightarrow {{Step_{i} }} } \right)$$
(4)
\(RL\), an arbitrary value vector according to the LM standard distribution, is multiplied through the prey during the step equation. For the second half of populations, it can be upgraded utilizing Eqs. (5) and (6).
$$\overrightarrow {{Step_{i} }} = \overrightarrow {{R_{B} }} \otimes \left( {\left( {\overrightarrow {{R_{B} }} \otimes \overrightarrow {{Elite_{i} }} } \right) – \overrightarrow {{Prey_{i} }} } \right)$$
(5)
$$\overrightarrow {{Prey_{i} }} = \overrightarrow {{Elite_{i} }} + \left( {P \cdot CF \otimes \overrightarrow {{Step_{i} }} } \right)$$
(6)
Determine that RB and the matrix of Elite in Eq. (5) are multiplied to mimic the BM. CF denotes the parameter for controlling the step dimensions and can be upgraded with Eq. (7).
$$CF= [1 – (1 / {MaxIter})] ^ {(2{Iter}/ {MaxIter})}$$
(7)
Stage (4). Stage 3, the last third of iterations, is measured as the last phase of the optimizer procedure. The predator moves to utilize LM, and subsequently, the prey matrix can be upgraded using Eqs. (8) and (9).
$$\overrightarrow {{Step_{i} }} = \overrightarrow {{R_{L} }} \otimes \left( {\left( {\overrightarrow {{R_{L} }} \to \otimes \overrightarrow {{Elite_{i} }} } \right) – \overrightarrow {{Prey_{i} }} } \right)$$
(8)
$$\overrightarrow {{Prey_{i} }} = \overrightarrow {{Elite_{i} }} + \left( {P \cdot CF \otimes \overrightarrow {{Step_{i} }} } \right).$$
(9)
The \(RL\) and \(Elite\) matrix multiplication mimics the predator’s motion in Lévy’s tactic.
Stage (5). Finalization Process: The optimal solutions are continually included in the matrix of the Elite succeeding all iterations. After achieving the maximal iteration counts, the last solution with the top fitness function (FF) will become apparent.
The FF imitates the accuracy of the classifier and the sum of the chosen features. It exploits the classifier’s accuracy and diminishes the set dimensions of the chosen features. Consequently, the FF is used to assess individual solutions, which is given in Eq. (10).
$$Fitness = \alpha *ErrorRate + \frac{{\left( {1 – \alpha } \right)*SF}}{{All_{F} }}$$
(10)
Here, \(ErrorRate\) refers to the classification rate of error using the chosen features. \(It\) has been computed as the ratio of improper classifications to the number of classifications between 0 and 1. \(SF\) refers to the quantity of chosen features, and \({All}_{F}\) means the total amount of features. \(\alpha\) is applied for controlling the import of classifier excellence and sub-set length.
Indoor activities detection using ensemble models
For the detection of indoor activities, the proposed MOEM-SMIADP model applies an ensemble of three classifiers: the GCN model, the LSTM-seq2seq method, and the CAE classifier. This ensemble technique is chosen for its ability to capture diverse and complementary data features. GCN outperforms in learning spatial and structural patterns, LSTM-seq2seq effectually handles temporal sequences and long-term dependencies, and CAE identifies complex latent features. This integration outperforms single-model approaches by utilizing the merits of each classifier, improving detection accuracy and robustness. Unlike standalone methods, the ensemble approach mitigates the risk of overfitting and adapts better to complex indoor activity patterns. Its versatility makes it specifically effective in handling multimodal and dynamic datasets.
GCN model
GCN is based on CNN, which concentrates on the convolutional process of operation31. It applies functional mapping to make novel node information by combining the neighbouring and the present node data. The space-based GCN straight aggregates the handling of graph-structured data based on nodes and edges, significantly decreasing the calculation sum, and is currently frequently utilized in networks. Here, a GCN‐based model is presented for learning to carry out the generator’s positions by removing power grid graph data. Applying the topologic architecture and bus attributes of the power grid, a graphical representation of the power grid \(G\) can be originated, \({N}_{B}=\left\{1,\dots,{n}_{B}\right\}\) denotes a collection of buses in \(G,\) \({N}_{F}=\left\{1,\dots,{n}_{F}\right\}\) represents a collection of bus feature sizes. Bus within the power grid is separated into dual types according to the absence or presence of the generator. This attribute is about loaded data and generator data related to limitations. The message concerning transmission lines related to limits also needs to be considered. Hence, the capability of transmission lines among buses must be combined into the feature matrix. The feature matrix construction \(X\) for GCN can be established as shown.
$$\left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {x_{j} = \left[ {x_{{\grave{j}}}^{LF} ,x_{{\grave{j}}}^{G} } \right]} \hfill \\ {x_{{\grave{j}}}^{LF} = \left[ {p_{{\grave{j}},1}^{L} , \ldots ,p_{{{\grave{j}},n_{T} }}^{L} ,f_{{\grave{j}},1}^{L} , \ldots ,f_{{{\grave{j}}n_{B} }}^{L} } \right]} \hfill \\ {x_{{\grave{j}}}^{G} = g_{j} \left[ {P_{i}^{max} ,P_{i}^{min} , \ldots ,T_{i}^{on} ,T_{i}^{off} } \right]} \hfill \\ \end{array} } \hfill & {\forall i \in N_{G} ,\forall j \in N_{B} } \hfill \\ \end{array} } \right.$$
(11)
$$X={\left[{X}_{1},\dots,{X}_{nB}\right]}^T$$
(12)
When there are no lines of transmission among nodes \(j\) and \(j,{f}_{j,k}^{L}\) is equivalent to \(0.\) \({g}_{j}\) denotes binary variable to designate if a generator lies in bus \(j\); when a generator occurs at node \(j,{g}_{j}\) is equivalent to \(one\), or else it becomes \(zero.\)
Owing to the distinctive measurement elements related to the various features within the input feature data, straight calculations and evaluations are not possible. To tackle this problem, a normalization method is used to preprocess \(X\). Essentially, this process is not utilized for the complete dataset. Still, it is performed distinctly for every feature. This method can improve the efficiency of the training. The succeeding equation is applied for the normalization procedure.
$${\grave{x}} = \frac{{x – x_{min} }}{{x_{max} – x_{min} }}$$
(13)
The GCN input is \({\grave{X}}\), and the initial hidden feature vector \({H}_{1}\) is gained after the initial graph convolution layer (GCL). This method has been executed by aggregating the neighbouring bus features and then passing over a linear transformation. Afterward numerous convolution layers, the last output outcome is gained over the fully connected layer (FCL). Meanwhile, \(A\) is not normalized; handling \(A\) would alter the feature vector scales. To resolve these issues, A and the matrix of degree \(I\) are included and then standardized over the node matrix of degree \(D.\)
$${\grave{A}} = D^{{\frac{ – 1}{2}}} \left( {A + I} \right)D^{\frac{1}{2}}$$
(14)
Formerly, the computation of every GCL is stated as follows:
$$\left\{ {\begin{array}{*{20}l} {H_{l + 1} = h\left( {A,H_{l} } \right)} \hfill \\ {h\left( {H_{l} ,{\grave{A}}} \right) = \sigma \left( {{\grave{A}}*H_{l} *W_{l} } \right)} \hfill \\ \end{array} } \right.$$
(15)
While \(\sigma \left( \cdot \right)\) denotes the activation function. Figure 2 illustrates the infrastructure of the GCN model.
As the preprocessed \({\grave{X}}\) aids as the input to GCN, each bus’s novel \({n}_{T}\) dimensional feature vectors are gained over the transformation of GCL and FCL, and the attainment of the forecast promise is characterized as shown.
$$U^{p} = F\left( {G\left( {{\grave{X}}} \right)} \right)$$
(16)
While \(G\left( \cdot \right)\) specifies the GCN process, \(F\left( \cdot \right)\) specifies the removal of consistent likelihood vectors.
GCN is applied to discover the binary variable patterns, so the learning task corresponds to a multi-class binary classification task.
$$Loss = BCELoss\left( {U^{p} ,{\grave{U}}} \right) = \mathop \sum \limits_{{i \in N_{G} }} \mathop \sum \limits_{{t \in N_{T} }} \left[ { – \left( {u_{i,t}^{p} \cdot log\left( {{\grave{u}}_{i,t} } \right) + \left( {1 + u_{i,t}^{p} } \right) \cdot log\left( {1 – {\grave{u}}_{i,t} } \right)} \right)} \right]$$
(17)
\(Ontheotherhand,loss\) represents the variance between the target and the predicted values.
LSTM-seq2seq model
The LSTM-seq2seq framework contains a decoder and encoder32. In encoding NN, the input sequence \(\left\{ {x_{1} ,x_{2} , \ldots ,x_{T} } \right\}\) with the time step counts \(T\) is read a single time step at a period. Finally, the hidden layer \({h}_{T}\) creates a higher dimension \(D\) vector, encoded to signify the data from the input series. The decoder NN framework proceeds the vector \(D\) as the input to attain the resultant series \(\left\{ {y_{1} ,y_{2} , \ldots ,y_{T} } \right\}\) across the loop directed. This computation, which involves the LSTM-seq2seq framework, is followed,
$$h_{t} = \psi \left( {x_{{t^{\prime}}} h_{t – 1} } \right)$$
(18)
$$D = \phi \left( {h_{{1^{\prime} }}, h_{t} } \right)$$
(19)
$$h_{t} = \theta \left( {y_{t – 1} ,h_{{t – 1^{\prime}}} D} \right)$$
(20)
While \({\grave{h}}_{t}\) and \({h}_{t}\) represent the HL in the decoder and encoder at the time step \(t,\) correspondingly \(\theta,\) \(\psi,\) and \(\phi\) denote non-linear function activation.
Assuming that \(y^{m \times 1}\) is the monitoring deformation data, and \(x^{m \times n}\) is the influencing factor sequence data, with \(m\) the instance counts and \(n\) the influencing factor counts, for example, time, temperature, water level, and more. The dual data sequences are first reordered into \(x^{{\left( {m/T} \right) \times n \times T}}\) and \(y^{{\left( {m/T} \right) \times 1 \times T}} ,\) respectively, by time steps \(T\). The previous output of the HL \({h}_{T}\) is then led to the decoder. Every time step of the output decoder is given below:
$$y_{t} = \sigma \left( {{\grave{h}}_{t} } \right)$$
(21)
They are associated with sequence to form the last fitting outcome, and the distortion \(y\) is lastly attained once the denormalization is over.
CAE classifier
The CAE effortlessly incorporates local convolution networks and conventional AEs, presenting a reconstruction feature to the convolutional method33. This feature maps the transformation from input to output called convolutional decoding. Applying the fundamental unsupervised greedy training characteristic of AEs, it is becoming possible to calculate the parameters for either encoder or decoder processes. Now, \(f\) signifies the convolutional encoder function, whereas \(f\) represents the decoder counterparts. The input contains feature maps \(x \in R^{{n \times l \times l}}\), both from the first layer or the previous opinion. This input includes \(n\) feature mapping, all spanning a region of l × l pixels. The convolution AE process includes \(m\) convolutional kernels, making \(m\) feature mapping within the output layer. When such feature mapping derives from the input layer, \(n\) describes the input channel counts. However, if they originated from previous layers, \(n\) represents the complete output feature mapping of that last layer. The dimensions of the convolution kernels stand at \(d\) × \(d\), guaranteeing \(d \le l.\)
The group of parameters \(\theta = \left\{ {W,{\grave{W}},b,{\grave{b}},} \right\}\) describes the learning basics of the convolution AE layer. During this, \(b\in{R}^{m}\) and \(W = \left\{ {w_{j} ,j = 1,2, \ldots ,m} \right\}\) be similar to the convolution encoding parameters. Now, every \(w_{j} \in R^{n \times l \times l}\) may additionally be characterized as a vector \(w_{j} \in R^{{nl^{2} }}\). However, \(W = \left\{ {w_{j} ,j = 1,2, \ldots ,m} \right\}\) and \({\grave{b}}\) denotes parameters for the convolution decoding. For this one, \({\grave{b}} \in R^{{nl^{2} }}\) and every \({\grave{w}}_{j} \in R^{{1 \times nl^{2} }}\).
Firstly, the input image experiences an encoder method. In this stage, size patches \(d\) × \(d\) pixels signified as \({x}_{i}\) while \(i = 1,2, \ldots ,p\), are removed from the input images. Then, for all patches, the weighting \({w}_{j}\) of the \({j}\)th convolutional kernel was applied to convolutional processes. These outcomes within the calculation of the values of the neuron \({0}_{ij}\) for \(j = 1,2, \ldots ,m\) denotes the output layer:
$$o_{ij} = f\left( {x_{i} } \right) = \sigma \left( {w_{j} \cdot x_{i} + b} \right)$$
(22)
While \(\sigma\) signifies a non-linear activation function, the ReLU activation function has been applied in this study.
$$Relu\left( x \right) = \left\{ {\begin{array}{*{20}l} x \hfill & {x \ge 0} \hfill \\ 0 \hfill & {x < 0} \hfill \\ \end{array} } \right.$$
(23)
After this, the \({o}_{ij}\) output from the convolutional decoder experiences encoder, where \({x}_{i}\) is reconstructed with \({o}_{ij}\) to yield \({\grave{x}}_{i}\)
$$x_{i} = f^{\prime}\left( {o_{ij} } \right) = \phi \left( {w_{i} \cdot o_{ij} + {\grave{b}}} \right)$$
(24)
Next, the convolutional encoder and decoder operations, \({\grave{x}}_{i}\), are produced for all samples. Deriving from the reconstruction process, \(P\) patches and all dimensions of \(d\) × \(d\) are gained. The cost function can be described as the MSE among the novel patches of the input images \({x}_{i}\) \(\left( {while \; i = 1,2, \ldots ,p} \right)\) and the reconstructed patches \({\grave{x}}_{i}\) \(\left( {while \; i = 1,2, \ldots ,p} \right)\). The particular procedure of cost function is offered in Eq. (25), whereas the reconstruction error can be specified in Eq. (26).
$$J_{CAE} \left( \theta \right) = \frac{1}{p}\mathop \sum \limits_{i = 1}^{p} L\left[ {x_{i,} {\grave{x}}_{i} } \right]$$
(25)
$$L_{CAE} \left[ {x_{i,} {\grave{x}}_{i} } \right] = \left\| {x_{i} – {\grave{x}}_{i} } \right\|^{2} = \left\| {x_{i} – \phi \left( {\sigma \left( {x_{i} } \right)} \right)} \right\|^{2}$$
(26)
Applying SGD, the errors and weights are refined iteratively, resulting in the optimizer of the convolution AE layer. After training, this enhanced parameter yields the feature mapping, which is forwarded to the following layers.
During this study, the CAE method was accurately calculated using numerous layers, all providing particular functions for decoding and encoding the input data. This method begins using the input layer, which obtains the scalogram, which is then passed over consecutive convolution layers. This layer gradually decreases the image dimensionalities, separating the main features. Ensuing the encoder method, this method changes to the decoder stage. This reconstructed output is essential for classifying and identifying different error states in the pumps.
ICOA-based parameter selection
Eventually, the hyperparameter tuning method is implemented by ICOA to enhance the classification outcomes of ensemble models34. This model was chosen because of its ability to optimize hyperparameters in complex ML models. The ICOA technique improves the standard COA approach by incorporating adaptive mechanisms for improved exploration and exploitation during the search process. Unlike grid or random search, ICOA dynamically navigates the search space, mitigating computation time while achieving more accurate parameter tuning. Its robustness in averting local optima ensures optimal configurations for ensemble models, enhancing accuracy and stability. This method is particularly advantageous for high-dimensional parameter spaces, where conventional techniques often face difficulty with efficiency and precision.
COA is a new metaheuristic intellectual optimizer model. Coati’s behaviours stimulate it, specifically their techniques of hunting and attacking iguanas while avoiding predators, to tackle optimization issues.
Initialization
The below-mentioned formulation signifies the original members of the coati population:
$$x_{i,j} = lb_{j} + r \cdot \left( {ub_{j} – lb_{j} } \right),\quad i = 1,2, \ldots ,N,\;\;j = 1,2, \ldots ,m$$
(27)
Here, \({x}_{i,j}\) signifies the \(j\)th dimensional location of \(i\)th coati, \(r\) denotes a randomly generated actual number within an interval of \(\left[ {0,1} \right],\) \({ob}_{j}\) and \(l{b}_{j}\) indicate the upper and lower bounds in the dimension \(j\), respectively; \(N\) specifies an amount of coati population, \(m\) represents several sizes.
Hunt and attack tactics
Coatis will search iguanas by climbing trees, and the below-given formulation will signify the coati’s location in the tree:
$$\begin{aligned} x_{i,j}^{P1} & = x_{i,j} + r \cdot \left( {Iguana_{j} – I \cdot x_{i,j} } \right), \\ & \quad for\;i = 1,2, \ldots ,\left\lfloor \frac{N}{2} \right\rfloor \wedge j = 1,2, \ldots ,m \\ \end{aligned}$$
(28)
The subsequent formulations express the coati’s location on the ground and then the iguana’s arrival:
$$\begin{aligned} & Iguana_{j}^{G} = lb_{j} + r \cdot \left( {ub_{j} – lb_{j} } \right),\quad j = 1,2, \ldots ,m, \\ & x_{i,j}^{P1} = \left\{ {\begin{array}{*{20}l} {x_{i,j} + r \cdot \left( {Iguana_{j}^{G} – I \cdot x_{ij} } \right),} \hfill & {\quad F_{{Iguana^{G} }} < F_{i} ,} \hfill \\ {x_{i,j} + r \cdot \left( {x_{i,j} – Iguana_{j}^{G} } \right),} \hfill & {\quad else,} \hfill \\ \end{array} } \right. \\ & \quad \quad \quad for\;i = \left\lfloor \frac{N}{2} \right\rfloor + 1,\left\lfloor \frac{N}{2} \right\rfloor + 2, \ldots ,N \wedge j = 1,2, \ldots ,m \\ \end{aligned}$$
(29)
If the coati’s novel site improves an objective value of a function, then it is accepted; or else, it stays unmoved. This process is stated to as the greedy law and is expressed below:
$$x_{i,j}^{P1} = \left\{ {\begin{array}{*{20}l} {x_{i,j}^{P1} ,} \hfill & {\quad F_{i}^{P1} < F_{j}, } \hfill \\ {x_{i,j}, } \hfill & {\quad else.} \hfill \\ \end{array} } \right.$$
(30)
While, \({x}_{i,j}^{P1}\) signifies the new site of \(i\)th coati in dimension \(j\), \(r\) refers to a stochastic actual numeral in the range of \(\left[ {0,1} \right],\) \(Iguan{a}_{j}\) represents the dimensional \(j\) location of iguana, signifying the location of an optimum member of the population, \(I\) is a number which is selected at random in the group of {1, 2}. \(Iguan{a}_{j}^{G}\) indicates the \(j\)th dimensional location of arbitrarily generated iguana under the tree, \({F}_{i}\) represents the main function of \(ith\) coati value, and \({F}_{Iguan{a}^{G}}\) specifies the main value of function of iguana on the base.
Escape from predators
Once a coati challenges a predator, the below-mentioned formulations can signify the random position of the coati’s escape:
$$\begin{aligned} & lb_{j}^{local} = \frac{{lb_{i} }}{t},ub_{j}^{local} = \frac{{ub_{i} }}{t},\quad where\;t = 1,2, \ldots ,T. \\ & x_{i,j}^{P2} = x_{i,j} + \left( {1 – 2r} \right) \cdot \left( {lb_{j}^{local} + r \cdot \left( {ub_{j}^{local} – lb_{j}^{local} } \right)} \right), \\ & \quad \quad \quad i = 1,2, \ldots ,N,\;j = 1,2, \ldots ,m, \\ \end{aligned}$$
(31)
\(l{b}_{j}^{local}\) and \(u{b}_{j}^{local}\) signify the local and upper bounds of the \(j\)th dimension, respectively; \(t\) denotes the present number of iterations, while \(T\) specifies the maximum iteration count. \({x}_{i,j}^{P2}\) refers to the novel location of \(i\)th coati in \(j\)th dimension throughout the 2nd phase.
The initialized population quality is essential in the meta-heuristic technique, considerably manipulating both the convergence speed and the accuracy of the last solution. The traditional COA utilizes a randomly generated model for initialization, which leads to a non-uniform spread of solution individuals. Here, the population initialization procedure is enhanced by applying the refractive opposite learning tactic to supplement the model’s performance by enlarging its range of searches.
The refractive index \(n\) was determined from the regular relationship.
$$n = \frac{sin\alpha }{{cos\beta }} = \frac{{d\left( {\left( {lb + ub} \right)/2 – x} \right)}}{{d\left( {x – \left( {lb + ub} \right)/2} \right)}}$$
(32)
Assume that \(k=l/{l}\), comprised in the abovementioned formulation and prolonged to the multi-dimensional space, yields the refractive opposite solution \({x}_{i,j}\):
$$x_{i,j} = \frac{{lb_{j} + ub_{j} }}{2} + \frac{{lb_{j} + ub_{j} }}{2k} – \frac{{x_{i,j} }}{k}$$
(33)
In the COA expansion stage, the coati adapts its site throughout the search procedure as per the present individual optimal, resulting in an early convergence to a local optimal solution by creating its efficiency in global exploration. This study provides a Levy flight method to improve the location upgrade procedure and tackle this limitation.
This technique yields randomized step distances and widens the exploration area, possibly improving the range of the coati population. The improved model for upgrading locations of coati is given below:
$$\sigma = \left( {\frac{{\Gamma \left( {1 + \beta } \right)sin\left( {\frac{\pi \beta }{2}} \right)}}{{\Gamma \left( {\frac{1 + \beta }{2}} \right)\beta \cdot 2^{{\frac{\beta – 1}{2}}} }}} \right)$$
(34)
$$Levy\left( \beta \right) = 0.01 \cdot u \cdot \frac{{r_{5} }}{{v^{{\frac{1}{\beta }}} }}$$
(35)
\(\varGamma\) denotes the usual function of Gamma, \(\beta\) refers to an arbitrarily produced variable within the interval of \(\left[ {0,2} \right],\) \({r}_{5}\) signifies the generated variable at random within the range \(\left[ {0,1} \right]\), \(u\) and \(v\) obey the usual distributions \(u \sim N\left( {0,\sigma_{2} } \right)\) and \(v \sim N\left( {0,1} \right)\), correspondingly. The mathematical formulation is expressed below:
$$x_{i,j}^{P2} = Levy\left( \beta \right) \cdot x_{i,j} + \left( {1 – 2r} \right).$$
$$\left( {lb_{j}^{local} + r \cdot \left( {ub_{j}^{local} – lb_{j}^{local} } \right)} \right)$$
(36)
The Levy flight approach has been presented to the COA growth stage to improve its global exploration capability and alleviate early convergence. By making longer-distance jumps in the solution space, Levy flight permits a more varied population distribution by enhancing the model’s capability to run away from local goals and efficiently discover the global optimal.
Fitness selection is one of the great factors inducing the outcome of the ICOA approach. The range of the hyperparameter model comprises the solution-encoded system for estimating the efficiency of the candidate solution. Currently, the ICOA approach studies accuracy as the foremost standard for planning FF.
$$Fitness = max\left( P \right)$$
(37)
$$P = \frac{TP}{{TP + FP}}$$
(38)
Where \(TP\) signifies the positive value of true, and \(FP\) represents the positive value of false.
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