Predictive Maintenance for Switch Machine Based on Digital Twins

As a unique device of railway networks, the normal operation of switch machines involves railway safe and efficient operation. Predictive maintenance becomes the focus of the switch machine. Aiming at the low accuracy of the prediction state and the difficulty in state visualization, the paper proposes a predictive maintenance model for switch machines based on Digital Twins (DT). It constructs a DT model for the switch machine, which contains a behavior model and a rule model. The behavior model is a high-fidelity visual model. The rule model is a high-precision prediction model, which is combined with long short-term memory (LSTM) and autoregressive Integrated Moving Average model (ARIMA). Experiment results show that the model can be more intuitive with higher prediction accuracy and better applicability. The proposed DT approach is potentially practical, providing a promising idea for switching machines in predictive maintenance.

Introduction
Today, the railway has developed rapidly and has become the most extensively used network, especially in China [1,2]. So, high security is required to keep trains operation safe [3]. The switch machine, which takes charge of pulling and pushing turnout blades, is significant to ensure railway safe and efficient operation. Any failure in turnout may lead to nasty accidents and even cause severe loss of life and property. Besides, statistics survey reports that the failure number of switch machines accounts for more than 40% of all railway signaling equipment [4]. However, the most used maintenance in railway on-sites depends on human experience [5], which tends to fail when facing complex problems. Additionally, maintenance personnel indeed needs a visual tool for observing the device for better maintenance. Hence, the intelligent maintenance model and the visualization for switch machines are essential [6].
Many scholars have researched switch machines in Prognostic and Health Management (PHM), including fault detection and diagnosis, prognosis, health prediction, etc. [7]. Compared to prognosis, fault diagnosis is relatively mature. However, diagnostics is, in essence, a classification problem. It can only classify existing fault phenomenon of switch machines and cannot handle the urgent problem since it belongs to corrective maintenance, lacking foresight ability. This paper focuses on the predictive maintenance for switch machines.
Predictive Maintenance (PM) has gradually become the promising solution in complex equipment Prognosis and Health Management (PHM) since it is more likely to effectively reduce operation and maintenance costs than breakdown maintenance and periodic maintenance [8]. PM evolves the maintenance pattern from the passive generation to active optimization selection [9] and improves the stability and reliability of the switch machine, as well as increases its service life. There are also some studies on the state prediction for the switch machine.
Aiming to forecast failure progression in railway turnout systems. Guclu presented an autoregressive moving average (ARMA) model to predict states in the slide chair of turnout system [10]. Liu proposed using polynomials to fit the relationship between temperature and gap offset data to achieve the purpose of predicting gap offset [11]. However, ARMA and polynomial fitting methods are difficult to build high-precision models based on nonlinear data. In other fields, some scholars have related research on prediction problems. Yang proposed support vector regression machine (SVR) to predict landslide displacement [12]. In terms of nonlinear fitting, SVR shows advantages. However, its prediction accuracy is relatively low. Hsu proposed the Grey forecasting model to predict the demand and sales in the global integrated circuit industry [13]. Grey prediction model has high prediction accuracy for short-term prediction, and its calculation is simple. Kong proposed a long short-term memory (LSTM) to solve the problem of volatility and uncertainly in electric load prediction [14]. Although the LSTM method has certain advantages for nonlinear data, it is relatively simple to fit more information. To solve this problem, Li proposed an optimally combined PSO-SVR-NGM model based on the entropy weight method to construct slope displacement prediction model [15]. The combination model can make full use of the information of various single prediction model and meet higher standards of prediction accuracy. The various prediction models are compared as shown in Table 1. The prediction accuracy is relatively low. Hsu Grey forecasting The prediction accuracy is relatively high for short term and calculation is simple.
It is not suit for long-term predictions. Kong LSTM It could capture nonlinear information and suit for long-term predictions.
It is relatively simple to fit more information.
Most of the research only analyzes the monitoring data, lacking comprehensive analysis for the real-time status of the equipment, which leads to one-sided and low real-time prediction results. Therefore, it is necessary to realize the interaction between the real operating status and virtual simulation by real-time monitoring data. It will be conducive to the comprehensive prediction of the equipment health status.
Digital Twins (DT) provide important theoretical basis and technical support for the connection and real-time interaction between virtual space and physical space [16]. DT is a digital model of the physical system that expresses all components and their states, and the model dynamically updates by monitoring the system state in real-time [17], which in other words, it would show the current state and predict its future state immediately and intuitively. Recently, the concept of Digital Twins (DT) has been proposed and gradually attracted widespread attention in smart manufacturing and complex system [18,19]. The most popular application is for PHM [20], especially in the aerospace field. Researchers have realized that the DT has the potential to optimize maintenance performance. The first application of DT was in 2011. AFRL proposed a DT conceptual model to predict the life of aircraft structure and assure its structural integrity [21]. Using the concept of dynamic Bayesian network, Li constructed wing health monitoring DT to predict the probability of crack growth and realized DT vision [22]. Luo studied a DT model and DT data approach to realize reliable PM of CNC machine tools [23]. Combining the machine learning model, Binderberger created a digital representation of humans, which could predict individual stress levels to monitor health [24]. Ding proposed a prediction method of shearer health status driven by the fusion of Digital Twins and deep learning [25].
Researchers have tried to construct the DT model in aircraft, industrial equipment, medical, and other fields using data-driven technology for PHM. Currently, there is no DT research applied to PHM in the railway field.
In order to solve the problems of poor accuracy and real-time of prediction results, DT can be introduced into PHM for switch machines. This paper proposed a PM approach for switch machines based on the DT, which improves the accuracy of the prediction results and fills research of the DT on the railway field for PHM. Modeling technologies are used to establish a high-guaranteed switch machine visualization model. Sensors obtain equipment parameters in real-time, and a large amount of historical data is processed and stored in the cloud storage files. To solve the problem of low prediction accuracy, a model combined LSTM and ARIMA by entropy weights is proposed, which comprises a real-time digital model that reflects and predicts the switch machine state. In addition, the prediction results are easy to be demonstrated in the visualization model, which helps the maintenance personnel to grasp the mechanism and more details.
The outline is organized as follows. Section 2 proposes the framework and methods of the DT of PM for switch machines. Then, Section 3 takes the switch machine gap as a case for verification and analyzes experimental results. Section 4 makes a summarization.

Overview
According to the five-dimension DT model proposed by Tao  (1) PE is a physical device of the switch machine, which provides parameters and data for DD. (2) VE is the core of the DT. It concludes visualization model and rule model, which is the key to realizing the switch machine's visualization and prediction. (3) DD contains equipment static data, environmental data and real-time operating data collected by the Internet of Things. DD takes the change of data processing and data cleaning. (4) CN transmits information through the data communication mechanism. (5) Ss can visually present the prediction results to the maintenance personnel and provide solutions to the problem.
Among them, VE is the most important part in DT. VE includes the behavior model and rule model.

Behavior Model Construction
Firstly, a three-dimensional model is constructed based on solid modeling according to the physical parameters of switch machine elements. Then, after format conversion, import these elements into Unity3D and complete assembly based on cooperation and constraints of mechanical components. Operating data and dynamic parameters make a virtual model update and drive its behavior simulation.
Secondly, a rule model should be constructed by data-driven approach. The concept of data-driven has been extensively researched in many areas. This method could mine knowledge from the large amount of historical data of the switch machine and learn the healthy/unhealthy states of the switch machine. The constructed rule model could examine the switch machine performance and predict the possibility of equipment degradation.
Finally, the calculation of the rule model can be shown in the visualization model to provide guidance to operators.

Rule Model Construction
This paper utilizes the combined prediction model of LSTM and ARIMA based on entropy weight. This model can capture both linear patterns and nonlinear patterns. It involves the LSTM model, ARIMA model, and entropy weight method theory. Firstly, two single prediction models are established. Then, a combination prediction model based on the entropy weighting theory is obtained using the statistical error information from single prediction models. The process of combination model construction is shown in Figure 2.

LSTM Model
LSTM is a particular type of RNN, which could process and analyze time series [27]. LSTM can learn long-term dependency information.
The LSTM model contains a memory unit and three gate controllers (forget gate, input gate, and output gate). The forget gate determines what information should be discarded, the input gate determines which new input information should be saved in the memory unit, the output gate determines what information should be output, and the memory unit is adopted to store information for use in the next stage.
The training algorithm of the LSTM network is a back-propagation algorithm. The main steps are as follows [28].
(1) Based on the forward calculation algorithm, calculate forget gate output f t , input gate output i t , output gate output o t , and the output of the LSTM layer h t at time t.
 is bias. σ and tanh are sigmoid and hyperbolic tangent activation functions, respectively, which play the role of gate. They describe the throughput of each part. They describe the throughput of each part.
(2) Reverse calculation of the error term of each LSTM cell. The error term can be represented by the mean square error (MSE).
whereŷ is the network prediction value, and y is the actual value. There are two directions of error term propagation. One is along the time, and another is to spread to the previous layer.
(3) According to the error term, calculate the gradient of each weight.
(4) Update the weights by the gradient optimization algorithm.

ARIMA Model
The Autoregressive Integrated Moving Average (ARIMA) model is one of the most typical and widely used linear statistical models [29]. In the ARIMA model, the current value is assumed to have a linear relationship with the historical value and random interference. The goal of ARIMA is to find a linear function to express this linear relationship and predict current value based on historical value.
ARIMA can be regarded as the ARMA model after differential processing. The ARIMA model can be shortened as ARI MA(p, d, q)(P, D, Q) s .
Here, p, d, and q, respectively, represent the order of the autoregression (AR) of the model, the number of nonseasonal differences, and the order of the moving average (MA) of the model. P, D, and Q, respectively, represent the order of the seasonal autoregressive, the number of seasonal differences, and the order of the seasonal moving average.
Building a model includes the following three steps, model identification, parameter estimation, and diagnostic checking [30].
(1) Firstly, choose the appropriate value of d to convert non-stationary series into stationary series. Then, test the stationarity of series according to the Augmented Dickey-Fuller (ADF). Analyze the autocorrelation coefficient function (ACF) and partial autocorrelation function (PACF) plots to determine the parameters of the seasonal part (P and Q), and the nonseasonal part (p and q).
(2) The best parameters combination can be estimated by Bayesian Information Criterion. ( 3) The estimated model should be checked whether it fits these data. If the prediction error is white noise, it demonstrates that the model extracts the information from the original. Meanwhile, the ACF of the prediction error will be very low.

Entropy Weight Method
Shannon introduced the concept of entropy into information theory to describe the amount of object's information in 1984 and named it information entropy. Information entropy is an objective measure of the degree of disorder of information. The idea of the entropy weight method is that the greater the amount of information in the indicator, the lower the entropy is, and the higher the weight is. Therefore, the entropy weight method has strong operability and objectivity. The processes of calculating weights are as follows.
Supposed there are m objects and n indexes. Let X be a known index matrix, where the element x ij represents the j-th index of the i-th evaluation object. X is the normalized matrix of X. benefit object [31].
The entropy p i of the i-th evaluation object could be calculated by where it is assumed that, when x ij = 0, x ij lnx ij = 0. The weight w i of the i-th object can be calculated by

Experiment
The PM system based on DT for switch machine can monitor and predict the state of the switch machine, such as the closed state of turnout, the loss of the automatic switch, the loss of the reducer, etc. This paper takes the prediction of turnout closed state as an example for research. This paper selects the ZD6 electric switch machine as an example object. Experiment data were sampled from September to November 2020 at a station, and the sampling interval is 2 h. There are total 648 time series points to be researched for verifying the validity of the approach. The change of the switch gap can be considered as a time series to study.

Principle of Switch Machine Gap
The task of the switch machine changes the switch rails position and locks it in their place. The turnout contact state can be expressed by switch machine gap value. As shown in Figure 3, the gap size (∆L1 or ∆L2) can reveal the relative displacement of the indication rod and indirectly measure the degree of contact of the switch rail and the stock rail, supervising the state of the road in the terminal position.
As shown in Figure 4a, under normal conditions, the switch rail would closely contact with the stock rail and the gap size would be normal. The corresponding circuits could complete disconnected and connected work. As shown in Figure 4b, under abnormal conditions, if the switch rail does not closely contact with the stock rail, the gap size will be too large or too small [32], even the indicator piece could not fall into the notch as shown in Figure 4c.
The switch machine gap value is related to the operating state of the switch machine. Whether the switch machine will keep working in a normal state can be judged by predicting the change of gap value. If the switch machine gap size exceeds the standard, maintenance personnel need to check operation on-site to ensure the normal operation of the switch machine.

The Construction of Behavior Model
According to the method mentioned in Section 2, the visualization model is constructed. Figure 5 shows the behavior simulation of the physical switch machine in normal and reverse directions. The virtual model mirrors and visualizes the real-time state of the switch machine in physical space and provides a decision-making solution for its PM.

Combination Prediction Model
The gap value data in DD is selected to verify the predictive performance of the combined prediction model. The data from November 3rd to November 10th data is used as a test set. All other data is used as a training set to construct a model. The first step is to train the LSTM model and ARIMA model. The LSTM prediction model involves a large number of parameters. The experimental platform is based on the Keras framework. Based on experience and multiple experiment comparisons, the LSTM model parameters that minimize the average prediction error are determined. The length of the segmentation window for the model is set as 30. There are three hidden layers, and the number of hidden layer neurons are 128, 32, and 64, respectively. In order to prevent over-fitting, a Dropout layer is added after each layer of the loop structure, and the value is set to 0.2. Epochs and batch size are set to 350 and 32, respectively. The batch size is 32, the loss function is the mean square error (MSE), the activation function is tanh function, and the optimizer chooses Nadam.
The most important process of ARIMA model construction is to determine the orders. It can be seen that the time series is non-stationary and seasonal fluctuations. Hence, we applied both nonseasonal (d = 1) difference and seasonal difference (D = 1) to eliminate non-stationarity. Analyze its ACF plot and PACF plot after first-order difference. Figure 6a,b, respectively, show the ACF plot and PACF plot of differential time series. For the seasonal part of the ARIMA model, there was a significant spike at lag 12 in the ACF plot (Q = 1), but there was no significant spike at lag 12 or 24 in the PACF plot (P = 0). For the nonseasonal part of the ARIMA model, in the first cycle, there were two significant spikes (lag 1 and lag 10) in the ACF plot and three significant spikes (lag 2, lag 10, and lag 11) in the PACF plot. After further fitting and bias information criteria for comparison, we finally chose (11, 1, 2)(0, 1, 1) 1 2 ARIMA model. The second step is to determine the weight of the model according to the information entropy of the models. The selection of evaluation indexes and evaluation objects is crucial. We select the absolute value of the error between the actual value and the predicted value of the two single prediction methods as the evaluation indexes (n = 2). Meanwhile, five days before a test day are selected as the evaluation objects (m = 60). The weights of the two models of the combination prediction model are obtained by calculation. The weights of the two single models are shown in Table 2.  Figure 7 shows the prediction results of the two single models and the combination model.

Results and Analysis
From the comparison, it can be seen that the prediction result of the combination prediction model is closer to the actual value than the prediction result of the single prediction models, and the prediction effect is better. There are three general indicators for evaluating the performance of time series prediction models: Root Mean Squared Error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The computing formulas for the RMSE is as follows [33]: Table 3 shows the RMSE of the two single method prediction models and the entropyweighted combination prediction model. It shows that RMSE, MAE, and MAPE of the combination model were less than those of the other two single models in the numerical values, which shows that the combination prediction model based on entropy weight has the better fitting ability and more robust stability than the single prediction method. The entropy weight method evaluates the amount of information contained in a single prediction method and then calculates and sets the weights of various single methods in different prediction days. Hence, the combination prediction model based on the entropy method utilizes the explicit and implicit information contained in each prediction models, avoids the concentrated influence of multiple factors on the prediction accuracy, balances the deviation of each single prediction model, and fully integrates information. Finally, the prediction accuracy can be improved, indicating the superiority and feasibility of the proposed method. The gap prediction results can be visualized in the virtual model. If there is an abnormality in the prediction result of the gap in the next day, the indicator rod in the model will show an exception to remind the maintenance personnel to arrange an inspection. Figure 8 is the diagram of the virtual model pre-warning.

Discussion
Overall, this paper establishes the DT model of switch machine. Taking the gap in the switch machine as an example, combining the current behavior simulation and the prediction result of the switch machine gap, we can estimate the future state of switch machine. On switch machine PM task, compared with other prediction methods, the approach proposed has good performance and strong applicability and makes switch machine maintenance proactive, reliable, and economical.

Conclusions
To reduce maintenance cost, improve the prediction accuracy, and provide a visualization tool in switch machines on PM tasks, we proposed a DT approach. Firstly, we built a high-fidelity model of the physical switch machine to map it, which realizes the real-time mapping between the virtual model and the physical entity, and visually displays the physical equipment's state. Then, the LSTM-ARIMA combination model was utilized as the inference algorithm to predict the state of the switch machine indirectly. The intelligent switch machine DT system realized the visual monitoring, as well as the state prediction. Meanwhile, it can provide technical support for its maintenance issues. Combining the visual model of the switch machine and the state prediction results, the maintenance personnel can reasonably arrange the maintenance plan. This approach can predict the switch machine state in advance to improve the reliability of the switch machine and avoid affecting the driving efficiency.
In future work, data from different sources of switch machines can be fed to the DT model. The DT framework can be applied to the PM for switch machines, and even other equipment.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:


T product Tensors Part I: Inequalities

The T product operation between two three order tensors was invented around 2011 and it arises from many applications, such as signal processing, image feature extraction, machine learning, computer vision, and the multiview clustering problem. Although there are many pioneer works about T product tensors, there are no works dedicated to inequalities associated with T product tensors. In this work, we first attempt to build inequalities at the following aspects: (1) trace function nondecreasing and convexity; (2) Golden Thompson inequality for T product tensors; (3) Jensen T product inequality; (4) Klein T product inequality. All these inequalities are related to generalize celebrated Lieb concavity theorem from matrices to T product tensors. This new version of Lieb concavity theorem under T product tensor will be used to determine the tail bound for the maximum eigenvalue induced by independent sums of random Hermitian T product, which is the key tool to derive various new tail bounds for random T product tensors. Besides, Qi et. al introduces a new concept, named eigentuple, about T product tensors and they apply this concept to study nonnegative (positive) definite properties of T product tensors. The final main contribution of this work is to develop the Courant Fischer Theorem with respect to eigentuples, and this theorem helps us to understand the relationship between the minimum eigentuple and the maximum eigentuple. The main content of this paper is Part I of a serious task about T product tensors. The Part II of this work will utilize these new inequalities and Courant Fischer Theorem under T product tensors to derive tail bounds of the extreme eigenvalue and the maximum eigentuple for sums of random T product tensors, e.g., T product tensor Chernoff and T product tensor Bernstein bounds.

T-product Tensors
The T-product operation between two three order tensors was introduced by Kilmer and her collaborators in [2,3]. It has been shown as a powerful tool in many fields: signal processing [4,5], machine learning [6], computer vision [7,8], image processing [9], low-rank tensor approximation [10][11][12] etc. Due to wide applications of T-product, T-SVD and tubal ranks, Qi et. al [1]. extend eigentuple concept first defined from [13] to study properties for symmetry of T-product tensors and positive (nonnegative) semidefiniteness of T-product tensors by defining a T-quadratic form, whose variable is an m × p matrix, and whose value is a p-dimensional vector. They further show that a T-quadratic form is positive semidefinite (definite) if and only if the smallest eigentuple of the corresponding T-symmetric tensor is nonnegative (positive). Besides T-quadratic form, general functions for T-product tensors and their properties are also studied based on T-SVD, see [14][15][16][17]. However, none of these works discussed further issues about inequalities associated with T-product tensors. The first part of this work about T-product tensors is to develop several new T-product tensors inequalities, which are the main topics discussed by this paper. In the matrix setting, there are many useful applications about these similar inequalities under the traditional matrix product, e.g., quantum information processing [18]. The second part of this work is to apply these new inequalities about T-product tensors to tail bounds estimation of the maximum eigenvalue and the maximum eigentuple for sums of random T-product tensors, e.g., Chernoff and Bernstein bounds. We will introduce these new inequalities about T-product tensors obtained at this Part I work at the next subsection.

New Inequalities about T-product Tensors
In this work, we define trace, denoted by Tr, as the summation of f-diagonal entries of a given symmetric T-product tensor C ∈ C m×m×p and study properties of trace for T-product tensors. Our first main inequality about trace is following theorem: Theorem 1.1 (Monotonicity and Convexity of T-product Trace Function) Let f : R → R be a continuous function with non-decreasing / convex / strictly convex properties, then so is the mapping C → Tr (f (C)).
From trace definition, we will prove Golden-Thompson inequality for two Hermitian T-product tensors which will be utilized to prove T-product tensors martingale inequalities. A Hermitian T-product tensor is a tensor equal to its Hermiitan transpose, which is defined by Eq. (11). Theorem 1.2 (Golden-Thompson inequality for T-product Tensors) Given two Hermitian T-product tensors C, D ∈ C m×m×p , we have where ⋆ is the product operation between two T-product tensors defined by Eq. (15).
The next inequality we will show is Jensen's operator inequality (positive semidefinite relation between two T-product tensors). A tensor with T-positive definite (or T-positive semi-definite) will be abbreviated as TPD (or TPSD), see Section 2.3 for its definition. If we have a TPSD relation between two T-product tensors C and D represented as C D, then the difference T-product tensor (D − C) is a T-positive semi-definite tensor. Let I mmp ∈ C m×m×p be the identity tensor defined by Eq. (13).
where X i ∈ C m×m×p are bounded, Hermitian T-product tensors with all eigenvalues in the interval I and The immediate application of Theorem 1.1 is to prove Klein's inequality for T-product tensor.
In both situations, if f is strictly convex, equality holds if and only if C = D.
Previous theorems will help us to establish the following main theorem of this paper about Lieb's concavity for T-product tensors since tail bounds for sums of random T-product tensors will be derived based on such concavity relation.

Theorem 1.5 (Lieb's concavity theorem for T-product tensors) Let H be a Hermitian T-product tensor. Following map
is concave on the positive-definite cone.
We are ready to present the theorem for the tail bound of the maximum eigenvalue induced by independent sums of random Hermitian T-product tensors and this theorem will play a key role to establish various new tail bounds of the maximum eigenvalue generated by independent sums of random T-product tensors. Theorem 1.6 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigenvalue) Given a finite sequence of independent Hermitian T-product tensors {X i }, we have Similarly, we can generalize master tail bound for independent sum of random Hermitian T-product tensors with respect to eigenvalue from Theorem 1.6 to eigentuple version by the following theorem 1.7. We begin with operation defined in Proposition 2.1 from work [1]. Let a = (a 1 , a 2 , · · · , a p ) T ∈ C p , then operator circ to the vector a can be defined as and circ −1 (circ(a)) def = a. Suppose that a, b ∈ C p , we define where · is the standard matrix and vector multiplication. Then, we are ready to present the following theorem. Tail Bound for Independent Sum of Random T-product Tensors for Eigentuple)  Given a finite sequence of independent random Hermitian T-product

Theorem 1.7 (Master
where e tb ∈ C p is the exponential for the vector tb with respect to operation.
The last important theorem is the Courant-Fischer theorem for T-product tensors. This theorem will be used to figure out the relationship between the maximum eigentuple and the minimum eigentule of a T-product tensor.
where is the division (inverse operation) under .
All these inequalities and maximum/minimum eigentuples relation about T-product tensors will be utilized to derive a serious of new tail bounds for the extreme eigenvalue and eigentuple for sums of random T-product tensors. These new inequalities different from author Chang's previous works about bounds for sums of random tensors based on Einstein product [19,20].

Paper Organization
The rest of this paper is organized as follows. In Section 2, basic notions of T-product tensors are introduced. Lieb's concavity theorem under T-product will be studied in Section 3. General tail bounds for random Tproduct tensors are provided in Section 4. Courant-Fischer Theorem under T-product is given in Section 5. Finally, conclusion will be drawn in Section 6. Nomenclature: The sets of complex and real numbers are denoted by C and R, respectively. The symbol def = denotes mathematical definition.

T-product Tensors
In this section, we will review T-product operations briefly and discuss related properties in Sec. 2.1. The T-SVD decomposition of T-product tensors and T-Symmetric tensors will be presented in Sec. 2.2

What are T-product Tensors
For a third order tensor C ∈ C m×n×p , we define bcirc operation to the tensor C as: where C (1) , · · · , C (p) ∈ C m×n are frontal slices of tensor C. The inverse operation of bcirc is denoted as bcirc −1 with relation bcirc −1 (bcirc(C)) def = C. For a third order tensor C ∈ C m×m×p , we define Hermitian transpose of C, denoted by C H , as And a tensor D ∈ C m×m×p is called a Hermitian T-product tensor if D H = D. Similarly, we also define ranspose of C, denoted by C T , as And a tensor D ∈ C m×m×p is called a Symmetric T-product tensor if D T = D.
The identity tensor I mmp ∈ C m×m×p can be defined as: where I mp is the identity matrix in R mp×mp . A zero tensor, denoted as O mnp ∈ C m×n×p , is a tensor that all elements inside the tensor as 0.
In order to define the T-product operation, we need to define another kind of operation over a third order tensor. For a third order tensor C ∈ C m×n×p , we define unfold operation to the tensor C as: (2) . . .
where unfold(C) ∈ C mp×n , and the inverse operation of unfold is fold with the relation fold(unfold(C)) def = C. Given C ∈ C m×n×p and D ∈ C n×k×p , we define the T-product between C and D as where C ⋆ D ∈ C m×k×p .
where ≥ is the elementwise comparison between two vectors.
We define the T-product tensor trace for a tensor C = (c ijk ) ∈ C m×m×p , denoted by Tr(C), as following which is the summation of all entries in f-diagonal components. Then, we have following lemma about trace properties.

Lemma 1
For any tensors C, D ∈ C m×m×p , we have where c, d are two contants. And, the transpose operation will keep the same trace value, i.e., Finally, we have Proof: Eqs. (17) and (18) are true from trace definiton directly. From Eq. (15), the i-th frontal slice matrix of D ⋆ C is similarly, the i-th frontal slice matrix of C ⋆ D is Because the matrix trace of Eq. (20) and the matrix trace of Eq. (21) are same for each slice i due to linearity and invariant under cyclic permutations of matrix trace, we have Eq. (19) by summing over all frontal matrix slices.
Below, we will define the determinant of a T-product tensor C ∈ C m×m×p and its asscoiate properties. The determinant of a m × m × p tensor C is the m-linear alternating form defined as where V i ∈ C m×p is the i-th lateral matrix of the tensor C. Moreover, we require that det(I mmp ) = 1.
Given two tensors C, D ∈ C m×m×p , the determinant of C ⋆ D is det(C ⋆ D) = λdet(D) for some value λ.
If we set D as I mmp , we have Then, we have

T-SVD Decomposition
Given a tensor C ∈ C m×n×p , Theorem 4.1 in [2] proposed a T-singular value decomposition (T-SVD) for C as: where U ∈ C m×m×p and V ∈ C n×n×p are orthogonal tensors, and S ∈ C m×n×p is a f-diagonal tensor. We also have U T ⋆ U = I mmp and V T ⋆ V = I nnp . We define σ(C) be the spectrum of C, i.e., the set of s ∈ C, where s are nonzero entries from tensor S. We use · for the spectral norm, which is the largest singular value of a T-product tensor. Given any integer k and B ∈ C m×m×p , we define B k as where B k ∈ C m×m×p . Then, we have following corollary from T-SVD in Eq. (25).
Corollary 1 Suppose B ∈ C m×m×p is a Hermitian T-product tensor, and S −1 exists, where S is f-diagonal tensor obtained from the T-SVD of the tensor C. Then, we have Then, we can define the T-product tensor exponential function and the T-product tensor logarithm function under T-product as below with tensor power.
Definition 2 Given a tensor X ∈ C m×m×p , the tensor exponential of the tensor X is defined as where X 0 is defined as the identity tensor I mmp . Given a tensor Y, the tensor X is said to be a tensor logarithm of Y if e X = Y.
From T-SVD in Eq. (25), we can express a Hermitian T-product tensor C ∈ C m×m×p as where s iik are eigenvalues of the tensor C, and U i is obtained from the i-th lateral slice (matrix) of the tensor U with column vectors as u Note that we have U From Corollary 1 and Eq. (29), we can have following spectral mapping lemma.

Lemma 2
For any continous function f : R → R and any Hermitian T-product tensor C, we have (31)

Positive Semidefinite T-product Tensors
Given a Hermitian T-product tensor C ∈ C m×m×p , and a tensor X ∈ C m×1×p obtained from treating the matrix X ∈ C m×p as a tensor with dimensions R m×1×p . We define following quadratic form with respect to the matrix X as and we say that a tensor C is T-positive definite (TPD) (or T-positive semi-definite (TPSD)) if F C (X) > 0 (or F C (X) ≥ 0 ) for any X ∈ C m×p , where 0 is a zero vector with size p.
We now define eigentuples and eigenmatrices of a Hermitian T-product tensor which will be used to characterize TPD or TPSD for a given tensor. For a matrix X ∈ C m×p = (x (1) , · · · , x (p) ) , we define unfolding opeartion with respect to the matrix X columns, denoted by cunfold(X), as (1) x (2) . . .
where cunfold(X) ∈ C mp . Then, suppose that C ∈ C m×m×p is a Hermitian T-product tensor, we define C ⋆ X as where C ⋆ X ∈ C m×p . We also define a new product operation between a vector d = (d 1 , d 2 , · · · , d p ) T and a matrix X ∈ C m×p , denoted by •, as where · is the standard matrix multiplication. Suppose that X ∈ C m×p and X = O, and d ∈ C p , if we have we call d as an eigentuple of C, and X as an eigenmatrix of C corresponding to the eigentuple d. From Theorem 4.1 in [1], a T-square tensor C ∈ C m×m×p with eigentuples arranged as f-diagonal tensor S according to the standard form provided by Definition 1, i.e., s 1 ≥ s 2 ≥ · · · ≥ s m . Then C is TPD (or TPSD) if and only if the smallest eigentuple s m > (or ≥ )0. We use C vec to represent the spectral norm of eigentuple of the tensor C, which is defined as

T-product Tensors Analysis
We will begin with monotonicity and convexity discussions of the trace function.

Lemma 3 For a given continous and non-decreasing function f : R → R, the associated trace function on a Hermitian T-product tensor C is given by
Then we have Proof: We first assume that the function f is differentiable, then the first derivative of f is greater or equal than zero (monotonicity). We further define a trace function g(t) def = Tr (f (D + t(C − D))). Then, we have where we apply Lemma 1 at the last equality, and the last inequality comes from the nonnegative of f ′ . By applying the standard continuity argument, we can relax the requirement that f is continuously differentiable to the requirement that f is continuous. The next lemma will be used to show the convexity of trace function on a Hermitian T-product tensor C.
where V i is an eigenmatrix of C and it's the only case if f is strictly convex.
Proof: From spectral representation by Eq. (29), we have where the only inequality comes from the convexity of the function f . Since for each i, k, we have for some i ′ , k ′ , and is 0 otherwise.
From Lemma 3 and Lemma 4, we have the following theorem about convexity and monotonicity of a trace function. We recall theorem 1.1.

Theorem 1.1 (Monotonicity and Convexity of T-product Trace Function)
Let f : R → R be a continuous function with non-decreasing / convex / strictly convex properties, then so is the mapping C → Tr (f (C)).
Proof: Given two C, D ∈ C m×m×p Hermitian T-product tensors, f as a convex function, and V

[k]
i for 1 ≤ i ≤ m and 0 ≤ k ≤ p − 1 be an orthonormal basis of C m×p consisting of eigenmatrices of C+D 2 . Then, from Lemma 4, we have where inequalities come from Lemma 4. This demonstrates that the map C → Tr (f (C)) is midpoint convex.
For the strict convexity of f and Tr f ( i . From Lemma 4, the equality will be true when V [k] i are eigenmatrices for both tensors C and D. Then, we have which indicates that C = D. An obvious continuity argument now shows that if f continuous as well as convex, C → Tr (f (C)) is convex, and strictly convex so if f is strictly convex. Therefore, this Theorem is proved from Lemma 3 and above arguments.
From T-SVD, we have following relation for Hermitian T-product tensor: Above Eq. (46) is named as transfer rule. We have defined tensor exponential under Definition 2, and the exponential of an Hermitian T-product tensor is always TPD due to the spectral mapping Lemma 2. From transfer rule Eq. (46), the tensor exponential satisfies following relations for a Hermitian T-product tensor C ∈ C m×m×p that we will use at later theory development: and From Theorem 1.1 and the monotonicity of the exponential function, we have Below, we want to prove the monotonicity and the concavity of the logarithm function. We will begin definitions about T-product tensor monotonicity and convexity first and present several lemmas used to establish the monotonicity and the concavity of the logarithm function. Given two Hermitian T-product tensors C, D ∈ C m×m×p , a function f : R → R is said to be T-product tensor monotone if the following relation holds: A function f is said as a T-product tensor convex function if we have: where 0 ≤ t ≤ 1. Also, a function f is said as a T-product tensor concave function if −f is a T-product tensor convex function. The following derivation about the monotonicity and the concavity of the logarithm function is extended from matrices according to works in [21] and [22] to T-product tensors.
Proof: Since eigenvalues are roots of the characteristic polynomial, it is enough to show that det(λI mmp − C ⋆ D) = det(λI mmp − D ⋆ C). We first assume that C has inverse, then from Eq. (24), we have This shows that σ(C ⋆ D) = σ(D ⋆ C).
If C is not invertible, we choose a sequence {ǫ n } in C\σ(C) with ǫ n → 0, with property that all new tensors C n def = C − ǫ n I mmp are invertibale for each n. Then, Lemma 6 For every tensor C ∈ C m×m×p and every function f on σ(C H ⋆ C), we have Proof: Since σ(C H ⋆ C) = σ(C ⋆ C H ) from Lemma 5 and C ⋆ C H ⋆ C n = C ⋆ C H n ⋆ C, for n ∈ N, this lemma is hold for f is a polynomial. If the function f is an arbitary function on σ(C H ⋆ C) = [s 1 , · · · s n ], we define the Lagrance interpolation polynomial as where we have p(s i ) = f (s i ) for 1 ≤ i ≤ n. Then, we also have and this Lemma is proved. Following Lemma is adopted from Corollary 12 from [22].
is T-product tensor monotone function; 2.
x/f (x) is T-product tensor monotone function; is T-product tensor concave function;

1/f (x) is T-product tensor convex function.
Proof: The proof is based in Corollary 12 from [22]. But those facts about using Theorem 2.5.2 and Theorem 2.5.3 from [21] should be modified from matrices settings to T-product tensors settings. With help from Lemma 6 and transfer rules provided by Eq. (46), the proof about Theorem 2.5.2 and Theorem 2.5.3 from [21] for T-product tensors is straightforward by replacing matrix multiplication opertions to T-product operations.
We are ready to prove that the logarithmic function is T-product tensor monotone and concave function on (0, ∞).
In general, it is not practical to always working with Hermitian T-product tensor, we will apply dilations techqnique to expand any Ttensor into a Hermitian T-product tensor. For any tensor C ∈ C m×n×p , a dilation for the tensor C, denoted as D(C), will be where D(C) ∈ C (m+n)×(m+n)×p and we have (D(C)) H = D(C) (Hermitian T-product tensor after dilation). Also, we have Since Eq. (59) is zero trace, the largest eigenvalue of D(C) will be the same with the largest singular of C.
Since the expectation of a random T-product tensor can be considered as a convex combination, expectation preserves the TPSD order as: Also from Lemma 7, we know that the quadratic function f (x) = x 2 is T-product tensor convex, thus, we have We will present one more theorem in this section about Golden-Thompson inequality for T-product tensors. We recall theorem 1.2 Theorem 1.2 (Golden-Thompson inequality for T-product Tensors) Given two Hermitian T-product tensors C, D ∈ C m×m×p , we have where ⋆ is the product operation between two T-product tensors defined by Eq. (15).
Proof: From T-SVD decomposition and Eq. (29), we can express the tensor C as where λ are eigenvalues and P λ are corresponding projectors (T-product tensors) which are mutually orthgonal. Given X O, we define following mapping with respect to the tensor C as : Then, we have following properties about mapping P C (X ) 1. P C (X ) commutes with C; 2. Tr (P C (X ) ⋆ C) = Tr (X ⋆ C); The third property of P C (X ) is true due to the following relation: where Let he equality = 1 comes from the fact that the trace is multiplicative under the Kronecker product. The inequality ≤ 2 follows from inequality from the third property of P A ⊗m 2 (A ⊗m 1 ), the monotone of log and Tr exp ( ) functions, and the number of eigenvalues of A ⊗n 2 growing polynomially with n due to the fact that the number of distinct eigenvalues of A ⊗n 2 is bounded by the number of different types of sequences of mp symbols of length n, see Lemma II.1 in [23]. The equality = 3 utilizes the commutativity property for tensors P A ⊗m 2 (A ⊗m 1 ) and A ⊗m 2 based on the first property. Finally, the equality = 4 applies trace properties from the second property of the mapping P A ⊗m 2 (A ⊗m 1 ). If n → ∞, the result of this theorem is established.

Lieb's Concavity Under T-product
In this section, we will extend several trace inequalities to T-product tensors: Jensen's T-product tensor inequality in Section 3.1 and Klein's T-product tensor inequality in Section 3.2. These new T-product tensor inequalities will play important roles in establishing a new version of Lieb's concavity theorem under Tproduct in Section 3.3.

Jensen's T-product Inequality
In this subsection, we will derive Jensen's T-product tensor inequality in Theorem 1.3. We begin with a lemma which will be used in later proof in Thereom 1.3. Given two natural numbers m, n, we define a T-product tensor θ ∈ C m×m×p as exp(2π √ −1/n)×I mmp . Then, we can have tensor D ∈ C mn×mn×p obtained by where diag θ, θ 2 , · · · , θ n−1 , I mmp will be a matrix with entries as T-product tensors, and the diagonal part of this matrix is compsoded by tensors (θ, θ 2 , · · · , θ n−1 , I mmp ). Let D ∈ C mn×mn×p be another matrix of T-product tensor, i.e., entries d i,j as T-product tensors. We define a new operation * between two T-product tensors, A, B with dimensions belong to C mn×mn×p as: where both a i,k and c k,j are T-product tensors. Therefore, given any tensor C ∈ C mn×mn×p , the i, j-th entry (a T-product tensor) of C * D becoms exp(2π √ −1j/n) × c i,j , where c i,j ∈ C m×m×p is a T-product tensor.

Lemma 9
Given any tensor C ∈ C mn×mn×p and the tensor D defined by Eq. 67, we have where D k is the self-product of the tensor D by * operation k-times.
Proof: By direct computation with * , we have following: where this summation is zero for i = j, otherwise, it is c i,i . We are ready to prove theorem 1.3. Eq. (51). we have the following TPSD relation for each natural number n:

Theorem 1.3 (Jensen's T-product Inequality) For a continuous T-product tensor convex function f defined on an interval I. The definition of T-product tensor convex is given by
where X i ∈ C m×m×p are bounded, Hermitian T-product tensors with all eigenvalues in the interval I and Proof: Let us define a unitary tensor U = (u i,j ) ∈ C mn×mn×p for 1 ≤ i, j ≤ n as u i,j = C i , D = diag θ, θ 2 , · · · , θ n−1 , I mmp defined by Eq. (67), and define the tensor X ∈ C mn×mn×p as diag (X 1 , · · · , X n ). From Lemma 9, we have where the inequality comes from that the function f is a T-product tensor convex function.

Klein's T-product Inequality
The immediate application of Theorem 1.1 is to prove Klein's inequality for T-product tensor. We recall theorem 1.4.

Theorem 1.4 (Klein's T-product Inequality) For all C, D Hermitian T-product tensors and a differen-
tiable convex function f : R → R or for all C, D Hermitian T-product tensors and a differentiable convex function f : (0, ∞) → R, we have

In both situations, if f is strictly convex, equality holds if and only if C = D.
Proof: We define function F (t) as where t ∈ (0, 1). From Theorem 1.1, F (t) is a convex function. Then, we have then we obatin Klein's ineqaulity under T-product by rearrangement and substitution with Eq. (72).

Lieb's Concavity Theorem Under T-product
In this section, we will extend Lieb's concavity theorem to T-product tensors and we begin with the definition about the relative entropy between two T-product tensors.

Definition 3 Given two TPD tensors A ∈ C m×m×p and tensor B ∈ C m×m×p . The relative entropy between two T-product tensors A and B is defined as
We apply perspective function concept for T-product tensor convex and introduce the following lemma about the convexity of a T-product tensor convex function [24].
Lemma 10 Given f as a convex function, two commuting tensors X , Y ∈ C m×m×p , i.e., X ⋆ Y = Y ⋆ X , and the existence of the Y −1 , then the following map h is jointly convex in the sense that, given t

then we have
where ≤ 1 is based on the condition provided by Eq. (78) and Theorem 1.3.
Following lemma is given to establish the joint convexity property of relative entropy for T-product tensors.

Lemma 11 (Joint Convexity of Relative Entropy for T-product Tensors) The relative entropy function of two TPD tensors is a jointly convex function. That is
where t ∈ [0, 1] and all the following four tensors A 1 , B 1 , A 2 and B 2 , are TPD tensors.
Proof: From the definition 3, we wish to show the joint convexity of the function D(A B) with respect to the tensors A, B ∈ C m×m×p . Let us define tensor operators F(X ) def = A ⋆ X and G(X ) def = X ⋆ B for the variable tensor X ∈ C m×m×p . Then, we have F(X ) and G(X ) commuting on the inner product operation F(X ), G(X ) defined as: is jointly convex with respect to tensors A and B.
Lieb's concavity theorem is recalled below by theorem 1.5.

Theorem 1.5 (Lieb's concavity theorem for T-product tensors) Let H be a Hermitian T-product tensor. Following map
is concave on the positive-definite cone.
Proof: From Klein's inequality obtain from Theorem 1.4, the convexity of map t → t log t (which is strictly concave for t > 0) and Hermitian T-tensors X , Y, we have If we replace Y by e H+log A , we then have where D(X A) is the quantum relative entropy between two tensor operators. For real number t ∈ [0, 1] and two positive-definite tensors A 1 , A 2 , we have TrX H − D(X tA 1 ) + TrX where the first and last equalities are obtained based on the variational formula provided by Eq. (84), and the inequality is due to the joint convexity property of the relative entropy from Leamm 11. Based on Lieb's concavity theorem for T-product tensors, we have the following corollary.

Corollary 2 Let A be a fixed Hermitian T-product tensor, and let X be a random Hermitian T-product tensor, then we have
Proof 2 Define the random tensor Y = e X , we have where the inequality is based on Lieb's concavity theorem for T-product tensors obtained by Theorem 1.5 and Jensen's T-product tensor inequality by Theorem 1.3.

T-product Tensor Moments and Cumulants
Since the expectation of a random T-product tensor can be treated as convex combination, expectation will preserve the semidefinite order as From Jensen's T-product tensor inequality by Theorem 1.3, we also have Suppose a random Hermitian T-product tensor X having tensor moments of all orders, i.e., E(X n ) existing for all n, we can define the tensor moment-generating function, denoted as M X (t), and the tensor cumulant-generating function, denoted as K X (t), for the tensor X as where t ∈ R. Both the tensor moment-generating function and the tensor cumulant-generating function can be expressed as power series expansions: where ψ n is called tensor cumulant. The tensor cumulant ψ n can be expressed as a polynomial in terms of tensor moments up to the order n, for example, the first cumulant is the mean and the second cumulant is the variance: Finally, in this work, we also assume that all random variables are sufficiently regular for us to compute their expectations, interchange limits, etc.

Tail Bounds By Concatenation of Lieb's Concavity
The goal of this section is to develop several important tools which will be applied intensively in the proof of probability inequalities for the extreme eigentule (or eigenvalue) of a sum of independent random T-product tensors. The first tool is the Laplace transform method for T-product tensors discussed in Section 4.1, and the second tool is the tail bound for independent sums of random Hermitian T-product tensors presented by Section 4.2.

Laplace Transform Method for T-product Tensors
We extend the Laplace transform bound from matrices to T-product tensors based on [25]. Following lemma is given to establish the Laplace transform bound for the maximum eigenvalue of a T-product tensor.
Lemma 12 (Laplace Transform Method for T-product Tensors: Eigenvalue Version) Let X be a random Hermitian T-product tensor. For θ ∈ R, we have Proof 3 Given a fix value t, we have The first equality uses the homogeneity of the maximum eigenvalue map, the second equality comes from the monotonicity of the scalar exponential function, and the last relation is Markov's inequality. Because we have where the first equality used the spectral mapping theorem from Lemma 2, and the inequality holds because the exponential of an Hermitian T-product tensor is TPD and the maximum eigenvalue of a TPD tensor is dominated by the trace from Eq. (16). From Eqs (94) and (95), this lemma is established.
The Lemma 12 helps us to control the tail probabilities for the maximum eigenvalue of a random Hermitian T-product tensor by utilizing a bound for the trace of the tensor moment-generating function introduced in Section 3.4.
Since T-product tensors also have notions about eigentuples, we then extend Lemma 12 from eigenvalues version to eigentuples version. We begin with the derivation of Markov's inequality for random vectors.
Lemma 13 (Markov's inequality for Random Vector) If X ∈ R p is a nonnegative random vector and a > 0, then the probability that X is at least a = [a i ] can be bounded as: Proof: Because we have therefore, we have the desired inequality shown by Eq. (96).
We are ready to present following lemma about Laplace transform method for T-product tensors with eigentuples.

Lemma 14 (Laplace Transform Method for T-product Tensors: Eigentuple Version)
Let X ∈ C m×m×p be a random T-positive definite (TPD) tensor and an all one vector 1 p = [1, 1, · · · , 1] T ∈ C p . Suppose we have where t > 0 1 . Then, for b ∈ R p , we obtain where d max is the maximum eigentuple of the TPD tensor X .
Proof: Given a fix value t, we have The first equality uses the homogeneity of the maximum eigenvalue map, the second equality comes from the monotonicity of the exponential function with operation defined in Proposition 2.1 from work [1], and the last relation is Markov's inequality for random vector obtained from Lemma 13 since both E e dmax(tX ) and e tb are vectors with p entries. Then, we have where the first inequality comes from the relation that d max (tX ) ≤ λ max (tX )1 p , and the second inequality holds because e λmax(tX ) = λ max (e tX ) and the relation 1 p λ p max (e tX ) + 1 − 1 p ≤ Tr(e tX ). From Eqs (100) and (101), this lemma is established.

Tail Bounds for Independent Sums of Random T-product Tensors
This section will present the tail bound for the sum of independent random T-product tensors and several corollaries according to this tail bound for independent sums. We begin with the subadditivity lemma of tensor cumulant-generating functions.

Lemma 15 Given a finite sequence of independent random Hermitian T-product tensors
Proof: We begin with the following definition for the tensor cumulant-generating function for X i as: Then, we define the Hermitian T-product tensor H k as By applying Eq. (104) to Theorem 1.5 repeatedly for k = 1, 2, · · · , n, we have .
where the equality = 1 is based on the law of total expectation by defining E i as the conditional expectation given X 1 , · · · , X i . We are ready to present the theorem for the tail bound of independent sums of random Hermitian Tproduct tensors with respect to the maximum eigenvalue. We recell theorem 1.6 Theorem 1.6 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigenvalue) Given a finite sequence of independent Hermitian T-product tensors {X i }, we have Proof: By substituting the Lemma 15 into the Laplace transform bound provided by the Lemma 12, this theorem is established.
Several useful corollaries will be provided based on Theorem 1.6.

Corollary 3 Given a finite sequence of independent Hermitian random tensors
and a sequence of non-random Hermitian T-product tensors {A i } with following condition: Then, for all θ ∈ R, we have Proof: From the condition provided by Eq. (106) and Theorem 1.6, we have where the second inequality holds since we bound the trace of a TPD T-product tensor by the dimension size m × p multiplied by the maximum eigenvalue; the last equality is based on the spectral mapping theorem since the function f is nonnegative.

Corollary 4 Given a finite sequence of independent Hermitian random tensors
Proof: From T-tensor logarithm concavity property provided by Lemma 8, we have and from the trace exponential monotone property provided by Lemma 3, we have where the last inequality holds since we bound the trace of a positive-definite tensor by the dimension size m × p multiplied by the maximum eigenvalue and apply spectral mapping theorem twice. Similarly, we can generalize master tail bound for independent sum of random Hermitian T-product tensors for eigenvalue version from Theorem 1.6 to master tail bound for independent sum of random Hermitian T-product tensors for eigentuple version by the following Theorem 1.7.

Theorem 1.7 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigentuple) Given a finite sequence of independent random Hermitian T-product tensors
where e tb ∈ C p is the exponential for the vector tb with respect to operation.
Proof: By substituting the Lemma 15 into the Laplace transform bound provided by the Lemma 14, this theorem is established.
Some useful corollaries will be provided based on Theorem 1.7.

Corollary 5 Given a finite sequence of independent random Hermitian T-product tensors
Then, for all b ∈ R p and n i=1 tX i satisfing Eq. (98), we have Proof: From the condition provided by Eq. (112) and Theorem 1.7, we have where the second inequality holds since we bound the trace of a TPD T-tensor by the eigenvalue size with m × p multiplied by the maximum eigenvalue; the last equality is based on the spectral mapping theorem since the function f is nonnegative. This corollary is proved. Ee tX i ,

Corollary 6 Given a finite sequence of independent random Hermitian T-product tensors
and from the trace exponential monotone property provided by Lemma 3, we also have where the last inequality holds since we bound the trace of a TPD tensor by the eigenvalue size with m × p multiplied by the maximum eigenvalue; and spectral mapping theorem for log and exp functions.

Courant-Fischer Theorem under T-product Tensors and Minimum Eigenvalue/Eigentuple
In this section, Courant-Fischer theorem for T-product tensors will be proved and this theorem will be used to show the relationship between the maximum eigentuple and the minimum eigentule of TPD T-product tensors. Let us recall theorem 1.8. j } ∈ C m×p be orthnomal matrices for 1 ≤ j ≤ m and 0 ≤ l ≤ p − 1, S k be the space spanned by {U j } for k ≤ j ≤ m and 0 ≤ l ≤ p − 1. Then, we have where is the division (inverse operation) under .
Proof: We will just prove the first characterization of d k . The other can be proved similarly. First, we wish to show that d k is achievable. As S k is the space spanned by {U j } for 1 ≤ j ≤ k and 0 ≤ l ≤ p − 1. For every X ∈ S k , we can express X as where e = (1, 0, · · · , 0) T ∈ C p . To verify that this is the maximum eigentuple, as T k is the space spanned by {U j } for k ≤ j ≤ m and 0 ≤ l ≤ p − 1, for any S k with dimension k × p the intersection of S k with T k is non-empty. Then, we also have Any such X can be expressed as Therefore, all subspace of S k with dimension k × p, we have This theorem is proved since d k is achievable and is the maximum eigentuple. By applying Theorem 1.8, we have following relations: d min (X ) = −d max (−X ) and λ min (X ) = −λ max (−X ) (124)

Conclusion
In this Part I work, we try to establish following inequalities about T-product tensors: (1) trace function nondecreasing/convexity; (2) Golden-Thompson inequality for T-product tensors; (3) Jensen's T-product inequality; (4) Klein's T-product inequality. All these inequalities are used to generalize celebrated Lieb's concavity theorem from matrices to T-product tensors. Then, this new version of Lieb's concavity theorem under T-product tensor is utilized to build master tail bounds for the maximum eigenvalue and the maximum eigentuple induced by independent sums of random Hermitian T-product. In order to find the relationship between the minimum eigentuple and the maximum eigentuple, we also extended the Courant-Fischer Theorem from matrices to T-product tensors. How these new inequalities and Courant-Fischer Theorem under T-product are used to derive new tail bounds of the extreme eigenvalue and eigentuple for sums of random T-product tensors is the main goal of our Part II paper.


A Bi-fold Approach to Detect and Classify COVID-19 X-Ray Images and Symptom Auditor

In this paper, we propose an ensemble-based transfer learning method to predict the X-ray image of a COVID-19 affected person. We have used a weighted Euclidean distance average as the parameter to ensemble the transfer learning model viz. ResNet50, VGG16, VGG19, Xception, and InceptionV3. Image augmentations have been carried out using generative adversarial network modelling. We took 784 training images, and 278 test images to validate our model accuracy, and the accuracy of our proposed model was around 98.67% for the training data set and 95.52% for the test data set. Along with that, we also propose a genetic algorithm optimized classification algorithm, to analyze the symptoms of COVID-19 for low, medium, and high-risk patients. The accuracy for the optimized set overshadowed the accuracy of un-optimized classification, and the optimized accuracy is as high as 88.96% for the optimized model. The novelty of this paper lies in the bi-sided model of the paper, i.e., we propose two major models, and one is the genetic algorithm optimized model to analyze the symptoms for a patient of varied risk and the other is to classify the X-ray image using an ensemble-based transfer learning model.

Introduction
The arrival of COVID-19 shock the world with its arrival, first detected at Wuhan, Hubei province of China in the month of December. It was first noted as unknown pneumonia clustering in the region and later it spread rapidly and took the shape of a deadly pandemic. Initially, it has been assumed that the transmission is among the animals but soon it was proved that human transmission is also occurring, and from then social distancing became our new normal. The virus's nature is similar to previous attacks of SARS-CoV, and MERS through SARS-CoV2's spreading rate is higher than the former ones but the mortality rate is lower than those. The typical symptoms which are seen in the patients affected by the virus are fever, cough, fatigue, sore through, muscle pain, shortness of breath, etc.
Rapid testing has been suggested across all nations to cut the spread of the virus. The major aim is to separate the infected population part from the susceptible population part. The testing kit is the need of the hour to carry on the tests and one of the most common tests to detect the presence of nCoV is the reverse transcription-polymerase chain reaction (RT-PCR) test. Though the RT-PCR test is extremely costly in nature and major laboratories do not have access to these kits to carry on the test; thus, the speed of rapid testing is declining at a brisk pace. Whereas, on the other hand, the availability of X-ray is widely available across India, and thus, it can be a potential solution for rapid testing if it could be merged with our proposed model.
In this paper, we present an ensemble-based transfer learning model to easily classify the X-ray image into 2 major classes' COVID-19 positive or, COVID-19 negative. The patches start to appear from 0 to 2 days from the onset of the disease, and heavy patches are being created with the passing time. We have proposed a Euclidean average weights method to initialize the weights of the pre-trained models viz. ResNet50, VGG16, VGG19, Xception, and InceptionV3 while making the ensemble model classify the X-ray images with more accuracy. Initially, all the models are given the same importance and all the models are being assigned as (1,1, 1,1, 1) in the vector space for the ensemble. Minimizing the false-negative count is one of the major aims while designing the classifier, and the ensembled proposed model has done that pretty well is can be shown using the confusion matrix. The pre-trained models are chosen that have a lower number of parameters to make our model computationally fast and economical keeping the accuracy factor in mind.
The images which have been passed through the classifier have been heavily pre-processed to provide the best image quality to maximize the prediction accuracy. At first, all the images were scaled to (224 × 224) size as those pretrained models are trained in such dimensions. There was a lack of training data and the data set consisted of around 784 images; thus, we have data augmented the image using DCGAN architecture. The new synthetic images which have been created by the DCGAN architecture have been subsequently passed through as training images, thus diminish the class-imbalance problem for the classifier. The preprocessing followed by noise removal from the image using Gaussian filter model, then we have used shadow removal and Image Enhancement using Canny edge detection and histogram equalization. The lung portion is being segmented to analyze the match with higher sensitivity.
Along with that we have also presented a symptom analysis model. In this section the symptoms of the patients are taken into account viz. fever, tiredness, dry cough, sore throat, no symptoms, muscle pain, age, nasal congestion, running nose, and gender to classify them according to low, medium, and high severity. In this section, the classifier is being optimized using the Genetic Algorithm to get the fittest offspring from a series of a generation. The optimized model showed better results than the former one. We took 10 generations for the iteration and with that 20 offspring per generation has been taken into account for measuring the fitness factor. The proposed model is can be a groundbreaking result as we can easily predict the severity of the patients with the help of symptoms shown by the patients across the day.
The novelty of the paper lies in bi-fold modeling. On one hand, we have proposed a novel ensemble-based transfer learning method, where the proposed architecture surpasses the accuracy metrics with respect to the traditional models. In addition, on the other hand, we have proposed another classification model which can model the severity of a patient from the onset of the symptoms. Along with that, the classifier model has been optimized using the genetic algorithm to tune the accuracy of the model. The paper is structured as "Literature Review" contains the literature review of the work, followed by ensemblebased proposed model in "Proposed Approach of Ensemble-Based Transfer Learning Model Using Euclidian Weighted Average" and Symptom classification analysis using genetic algorithm are proposed in "Symptom Analysis Using Hybrid Approach of Genetic Algorithm and Classifying Algorithm" and finally concluding remark and future scope of study in "Concluding Remark and Future Scope of Study"

Literature Review
A significant amount of work is being carried out in this field to help the frontline workers to some extent. Radiologists discovered major findings like Kong et al. [1] observed opaque inferior airspaces in the CT images of the lung. Yoon Kong et al. [2] observed opacity in the left lower lung region in nodal structure through CT images. Vascular dilation and several patches have been seen by Zhao et al. [3] with some irregular opacities in the lung region. Li and Xia [4] observed GGO and patches across the lungs of the COVID-19 affected patients, along with that they reported signs of air bronchogram. Along with that vascular expansion is also seen in CT images which became a common factor in the patients and most of them showing similar traits. Zhu et al. [5] reported that approximately 33% of chest CT scans have rounded patches. In a work, a convolutional neural network (CNN) model has been developed to classify the CT images in respective classes of COVID-19 positive or negative [6]. Another model has also been created to classify images and it worked with an accuracy of around 89% [7]. Recent works have also been done to classify images using CNN architectures. Hemdan et al. [8] proposed a CNN architecture consisting of 8 convolution layers. Wang and Wong [9] proposed another CNN model which yielded an accuracy of around 92.54% in classifying 3 classes of pneumonia viz. none, non-COVID patches, and COVID patches. Ioannis et al. [2] proposed a model with 224 images and they got an accuracy of around 93.48% in their test data set. Sethy and Behera [101] proposed a method to classify images using the SVM algorithm. Apart from these, there are several deep learning works, where they proposed the model using various transfer learning methods [11][12][13][14][15][16][17].
Feature engineering is an important aspect in the field of machine learning and artificial intelligence. Selecting the subset from the original set to increase the performance is the key factor and the genetic algorithm helps to select the best attributes with high accuracy [18]. In a genetic algorithm, each chromosome is considered a solution. The algorithm operates on population and selects the best possible genes out of it. John et al. [19] used the medical field in the medical field for optimizing chemotherapy and how an artificial immune system can be implemented is illustrated. There may be many features that may not be highly correlated to the target and so the feature selection is set not fixed and here feature selection using this algorithm plays an important role and get preference over sequential forward and backward selection. H. Kim et al. [20] used for the SN Computer Science non-linear optimization problem. It will highly improve the model [21,22], and the removal of redundant features also helps to enhance the accuracy classification model. Hussein et al. in their work illustrated feature weighting and selection and used it in character recognition and pattern recognition [23]. Yang et al. [24] used it for feasibility testing of this algorithm in a neural network to improve the accuracy. Saidi et al. used big data and he stated that if the data is too large. For big data, it is very difficult to use this algorithm and thus parallel selection using this algorithm along with the map-reduced tools was used to overcome the problem. Santosh [25] proposed artificial intelligence-based solutions for the prediction of the different outbreaks of the epidemic across the world. It will help to identify COVID-19 outbreaks as well as forecast their nature of spread. Bhapkar et al. [26] proposed mathematical modeling to forecast the severity of the pandemic situation by predictive modeling of death tolls. As both the recovery and mortality rate change over the period, thus the authors used progressive recovery rates and progressive mortality rate for the predictive modeling. Dey et al. [27] showed a comparative study of the human mind under a prolonged lockdown period and how people reacted to such time window. Results showed under lockdown people paved their way for their passion in their house (results obtained after analyzing collected tweets), moreover, with the upliftment of the lockdown people slowly faded from their passion to their profession. Analyzing the emotions from the tweets showed most were neutral and a good share went for the worrying category. Mukherjee et al. [28] proposed a CNN architecture, where the network can be trained on both CT scan images and CXR images giving much more training data to learn the model. Their tailored DNN yielded 96.28% accuracy and a mere false negative percentage. Santosh et al. [29] proposed a dynamic model which can include complex parameters into consideration while predicting the epidemic outcome by not just only relying on SIER modeling, where many parameters are overlooked. The author also focused on the data-driven model which can auto tune the parameter value with the recent parametric values automatically. Das et al. [30] in this paper, a custom CNN network namely truncated inception net is proposed to segregate COVID-19 positive CXRs from other non-COVID cases. The proposed architecture achieved an accuracy of 99.96% in classifying COVID-19 positive cases. Mukherjee et al.
[31] proposed a light-weighted CNN architecture to classify CXR samples with COVID positive cases. The proposed architecture has been designed with much fewer parametric values as compared to other models. The proposed architecture achieved an accuracy of 99.69%.
All the works in the CNN classifier model have been concentrated by both using several convolution layers and building the model from the scratch or they have used independent transfer learning models to classify, where they tested on few models and compared the accuracies and chose the best. Most of the work is flawed due to lack of input images and class imbalance for the convolution layer; thus, in our work, we overcame this problem by introducing DCGAN Architecture as a data augmentation tool. Along with that the models shown a tendency of being overfitting and only 1 transfer learning method is not just enough for reliable predictability in such a sensitive case, where the major aim is to minimize the false negative count. The world is going through a pandemic situation and as the new stains of COVID-19 is still not fully interpretable and so search space is very difficult to understand and so here genetic algorithm is an effective way to compute the conditions based on different symptoms. Hereby boosting and grid search method the best parameter is identified, and thus, hyperparameter of the classifier model is tuned. By applying GA operators on conditions the highly correlated factors here, considered as genes are identified and then the classifier model is developed to give the best accuracy and to reduce the number of false negatives cases. If the false negatives are not regulated then it may lead to the worst condition.

Proposed Approach of Ensemble-Based Transfer Learning Model Using Euclidian Weighted Average
In the current situation of pandemic radiological lung imaging of patients plays a key role as the initial screening test. The X-ray predictability can be treated as a preliminary test for COVID-19 patients and only those patients will undergo RT-PCR test if and only if they are being confirmed by X-ray prediction. The patches start developing from day 2 and get significant on day 7, as shown in Fig. 1a, b.

Data Set and Data Pre-processing
In this study, we have considered data sets taken from Open Source authentic organizations. The images are being shared as open-source from various radiologists. One of the sources is Cohen JP, and the entire data set is being compiled from few other sources also. Italian Society of Medical and Interventional Radiology (SIRM) COVID-19 DATABASE, Novel Corona Virus 2019 Data set developed by Joseph Paul Cohen, images extracted from different publications [32]. In this study, we have taken 784 training images and 278 validation images and to knockout the class imbalance problem we have augmented images using DCGAN architecture. All the training images are being scaled to (224 × 224) size as those pre-trained models are trained in such dimensions [33,34].

Noise Removal
The CT images came with some minimal noise and to increase the model accuracy we have tried to minimize the noise. This is one of the steps of image quality being enhanced by applying noise removal methodology. We have used the average filtering method to reduce the noise. The transformation is shown in Fig. 2.

Shadow Removal
The images often contain some shadows due to the availability and position of illumination. In this step, we have tried to segregate the shadow component from the original images (Fig. 3). In this step, we have used Canny Edge detection to remove the shadows. [13,35]

Image Enhancement
In this pre-processing step, we aim to adjust and tune the color feature of the image. Here we have used an adaptive histogram equalizer to enhance the picture and contrast quality of the image by processing the histogram curve (Fig. 4).

Data Augmentation Using DCGAN Architecture
In the current work, the situation is extremely dynamic and changing every day and the data set is pretty small for accurate prediction of CT image class. Thus, we have augmented data to minimize the class imbalance problem and minimize the overfitting of models.
GAN works on the zero-sum principle and has 2 blocks to generate the synthetic image namely generator and discriminator.
• Generator: the generator or the Generative neural network is mainly responsible for creating a synthetic image to get undetected. It generates the image without training the features of the image of the input data set, i.e., without learning the semantics of the input image data. The loss equation for the generator function is given in the following equation:   to classify that the given sample is from the same data distribution or not (Fig. 5). The major goal of a discriminator network is to detect the fake content in the set. It is a classifier network that classifies whether the image is real or not. The loss equation for the discriminator function is given in the following equation: The combined loss function for the entire model is shown in Eq. 5: The images formed using DCGAN architecture are shown in Table 1.

Proposed Algorithm
The pre-trained state-of-the-art CNN classifier model is being tested at first with classifying the images into 2 major classes namely, COVID and non-COVID. The pre-trained models were used namely, ResNet50, VGG16, VGG19, Xception, and InceptionV3. Thus, a combined model always gives an upper hand in terms of accuracy and prediction. Thus, in this section, we have proposed a weighted Euclidian average method (WEAM) for higher model accuracy.
At first, all the models are being used independently, and afterward, they have been combined as an ensemble model. It is expected that the ensembled model will provide a more robust prediction. We have chosen the ResNet50 model as it is one of the contemporary models that are being used with fewer parameters with high accuracy. This model in particular is easy to train and converges fast. Other architectures which are chosen here give us higher accuracy with the considerable low amount of parameters, thus making it faster. For the InceptionV3 model, there are 11 inception layers present in the architecture.
The salient feature of our proposed model is to make a weighted average network, where the providing equation is based on weighted Euclidean average method. Let say our   model is working better than the other models i.e., it has a lower validation error in comparison to the other models which is being there. This implies that it assigned weights better to the classes better than the other models [36].
Assuming the accuracy for a kth model is being q 1 . Therefore, the validation accuracy error is 100 − q 1 . Thus, we define a new weighted factor: All the models are assigned (1, 1, 1, 1, 1) value initially, signifying all the models hold equal weightage and importance: Therefore, weight for the ith network is given as 2 )]

,
Now, let us assume that the output probabilities for class 1 and class 0 are in the form of x 0 , x 1 . Now, similarly, the predictive output probability for all the 5 networks be, x 01 , x 11 , x 02 , x 12 , x 03 , x 13 , x 04 , x 14 , x 05 , x 15 for the models 1, 2, 3, 4, 5, respectively. Now, let the weight for the models are, respectively, for models be w 1 , w 2 , w 3 , w 4 , w 5 calculated from Eq. 10. Therefore, average weight A is calculated using Eq. 11 [37]:

Results and Discussion
All the models have been trained for 50 epochs on early stopping callbacks being applied so that the epoch could be exited before the model goes overfitting (patience = 8 epochs). We have used Adam optimizer for a faster rate of convergence. Parameters used = 0.0001, 1 = 0.9, 2 = 0.889, ∈= 1 × 10 −8 . The training curves for the model are shown in Figs. 6, 7, 8, 9, 10, and 11, respectively [37,38].

Evaluation Metrics
The evaluation of the proposed model has been done using the standard evaluation metrics available namely, classification accuracy, sensitivity, and F1 score. They are being calculated using Eqs. 12, 13, 14, respectively [39,40]: Here, TP represents true positive, TN as true negative, FN as false negative, FP as false positive. All the values are being calculated from the confusion matrix of our proposed algorithm.
The comparative results and metrics are shown in Table 2.

Symptom Analysis Using Hybrid Approach of Genetic Algorithm and Classifying Algorithm
In this section, we analyze the symptoms of nCoV affected patients using a hybrid model of genetic algorithm and classifier algorithms. The aim is to find out the level of severity  The input features which has been incorporated in this study are fever, tiredness, dry cough, sore throat, no symptoms, muscle pain, age, nasal congestion, running nose, and gender. At first, we have analyzed and measured the accuracy of the classifier model applying to the data set and then we have optimized our model using the genetic algorithm to get the optimized data set, and then it was passed through the classifier model and the accuracy of the latter one overshadowed the former one.

Data Set Description
Data set is COVID-19 Symptoms Checker, where different symptoms. It is open-source data. It has 316,800 and 27 columns. The above-mentioned data set consists of anonymous data from an infected person reported positive and admitted to the hospital. Some of the important features in the data set are 'breathing problem', 'fever', 'dry cough', 'sore throat', 'running nose', 'asthma', 'chronic lung disease', 'headache', 'tiredness' etc. This data set will predict whether anyone is having a COVID-19 or not depending on symptoms. Other features include Experience of symptoms like nasal congestion, runny nose, and severity, etc.
In the data set COVID-19 exposure: the exposures of symptoms are considered. For visualization and understanding the risk, pre-processing the raw data having columns like date of the test, date of the visit have been reduced to exposure period, diagnose period, days elapsed from showing symptoms, days are taken for visiting the hospital are considered. It can be illustrated in Fig. 12.

Bayesian Modeling and Feature Visualization for GA Feature Selection
In the last decades, the implementations of the Bayesian hierarchical model (BHM) gained a lot of ground in Artificial Intelligence. Applying this prediction of uncertainty can be done and it is very helpful in the creation of a framework for risk analysis in the healthcare section. Health is the major wealth of humans. It is an organized representation of probabilistic relationships between input and output. It gives conditional independencies. We have modeled for severity or condition or risks of the patient. We have used the severity class as our target variable to assist the much serious patient in the first treatment. To assess the relationship among the variables with the target variable we plot the Bayesian graph, which is acyclic in nature (Fig. 13).

Classifying Algorithm
One of the key formulation and category of it is the classification. In classification main target is to categorize the data into different classes based on features: The model is trained on training sample to say I, where I = x 1 c 1 , x 2 c 2 , … where x is the input vector and c is the specified class (Fig. 14). The key features ( f i ) that influence categorization is part of input x, where ∈ x is known as the best features and it is selected based on some scores [41].
So for building a good classifier understanding the redundant features and extraction and selection of the best is a very important part of it. After the removal of redundant or irrelevant features, the accuracy of the model increases and the computation time decreases. Therefore, basically, it is a method, where a subset: select/vectors X x 1 , x 2 , … .x n where n is the total length of the input vector to optimize the objective function. Now, there are mainly divided into three categories of feature selection filters, wrappers, and embedded selectors.
where0 < i ≤ m where m < n(nis total no of input features),

Support Vector Machine (SVM)
Support vector machine (SVM) is a supervised machine learning algorithm that is applicable for both regressions rather than logistic regression and classification. It makes a non-probabilistic binary discriminative linear classifier. It creates hyperplanes in infinite dimensions and output or prediction is done based on the optimal hyperplane. Kernel function (fx, y) is defined. In higher dimension hyperplane considered as a bunch of observable points whose scalar product is constant. Therefore, for higher dimension, k(x, y) modifies to k x i , y and so for the hyperplane, we can say that A higher value of k is preferable. In addition, the functional margin has to be maintained. If there are n points then w.x − b = 0 is the equation of hyperplane, where w is the normal vector and b w is the offset of the hyperplane. [42]

Random Forest
The Random Forest model works on the count of impurity and the split occurs in the direction, where the impurity is least. The count of impurity is measured by 2 factors namely Gini index and entropy. Entropy is described as the amount of information needed to correctly describe a sample. A homogenous sample will give o entropy, while heterogeneous will yield 1. It is calculated as represented in Eq. 8. Similarly, the Gini index is measured using inequality in the sample. Gini index value 0 denotes that the sample is

Naïve Bayes
Naïve Bayes algorithm gives us the probability of a point x which can belong to a specified class C . The probability is calculated based on conditional probability model P(x i |C K ) . The classification involves assigning the value to a class; thus, the proportion can be found using Eq. 19 [43]: Thus, the class can be found by mathematical notation given in Eq. 20:

Genetic Algorithm
One key factor in optimizing the function and evaluation of the fitness/accuracy of the model and genetic algorithm is an evolutionary, heuristic, domain-independent algorithm widely used for feature selection and optimization. The algorithm is inspired by Darwin's theory of genetics. It is wellsuited for multi-featured optimization.
At first, it initializes the randomly generated population. The commencement of the algorithm is from the generation of a population and selection of optimal individuals by measuring objective function and evaluate the fitness function, which takes into account about it fitness in the environment and iterates the process, till the convergence condition is established. In other words, individuals represent one of the solutions to the problem and a set of chromosomes from the population, and the best solution is evaluated using fitness function and the best fitted is considered.

Initialization and Chromosome Encoding
The first step is to define the population and the algorithm changes populations of chromosomes (Fig. 15). They are the portrayal of the solution to the problem in string format and a locus or a specified position is known as gene and the alphabet at that place is allele. In GA, we prefer encoding in {0,1}.

Fitness Function and Selection of Features
Assessment of the nature of the chromosome is done and according to criteria it is checked that whether it can be considered as a solution or not. The key selection building block in this algorithm is such that it can act as a guide to the evolution of the Chromosomes and so recombination is also done and having. Higher the rank/score/value more is the chance of getting selected. Some of the methods are Roulette Wheel, Random Stochastic Selection, and Truncation Selection, etc.

Recombination
The 2 major parts of recombination are crossover and mutation. Crossover is the exchange of genes, where the chromosomes are of 2 parents and mutation refers to all possible combinations (swapping and other) of the allele. After this results are fitted to the successor population (Fig. 16).

Evolution
The GA algorithm iterates until and unless the stopping criteria are being reached. After recombination, a new generation is being created and it also undergoes a similar process to evolve (Table 3). A widely used evolutionary technique is used called replacement-with-elitism. Here there is an almost complete replacement of the wide population in the successor population; this model ensures that the highest fitness does not get lost in the next generation [44].
The features tabulated in Table 4 give us the features which contribute most to the COVID-19 positive cases. Fig. 15 Showing key components [22]

Proposed Algorithm
Along with the feature selection, another important paradigm to increase the model performance is to optimize the hypermeters of the algorithms, and for that, we have used Grid Search Algorithm. Grid search takes n equally spaced points considering each interval [a i , b i ] including a i and b i . Therefore, the total of n m possible grid points is possible and it is calculated. Later, points (pair) are considered, the maximum of these values is chosen and the best with the highest value is returned. To overcome the overfitting ensemble of the Grid search, stratified cross validation is implemented, where the division of k-folds is done. Let lower bounds a = (a 1 , a 2 ,…, a m ) and a vector of upper bounds b = (b 1 , b 2 , …, b m ) for each component of ν, where ν = (ν 1 , ν 2 , …ν m ) where target is to maximise p value.
The parametric values which are fed into the Grid Search model to get the optimized value are tabulated in Table 5. Table 6 gives us the results of our model, where the optimized values of hyperparameters are given along with a comparative result of models in where one has been Page 12 of 14 SN Computer Science optimized after using the genetic algorithm as feature selection and the other is where the classifier model acted without any feature selection.

Concluding Remark and Future Scope of Study
In this study, we have considered a bi-folded study of classifying the images using a proposed weighted transfer learning methodology. The Euclidean Weighted Average is used as the selection equation in the proposed model. As a result, our proposed model gave a high accuracy of 98.67%. Our    proposed algorithm outperformed the traditional models in both terms of validation accuracy and sensitivity. The class imbalance problem has been handled using adding more images and it has data augmented using DCGAN architecture. All the images are pre-processed by removing shadows, noise, etc. for higher classifying accuracy.
In the next section, we have proposed a dual-stage classifier to classify patients into the risk category of low, medium, and high based on the symptoms they are showing. To optimize the result we have used genetic algorithm and the classifier algorithm which are being used are SVM, Naïve Bayes, and Random Forest Classifier. The optimized model has a much higher accuracy which is being expected as 88.96%. Based on this model we can assess and analyze the risk associated with a patient with nCoV symptoms.
The work can be extended using introducing a voice classifier model by which the patient can be classified by the sound of his or her cough. Moreover, other nature inspired algorithms could be incorporated like PSO, Artificial Ant Colony, etc. [46,47].