The SARS-CoV-2 is the most recently identified member of the coronavirus family causing COVID-19 [1] . This respiratory disease started for Wuhan, China during late December, 2020 and spread to all the countries worldwide [2] . World health organization (WHO) declared this infectious disease as a public health emergency of international concern (PHEIC) on January 30, 2020 as it reached to more than 18 countries [3] and on Feb 11, 2020 named this "COVID-19" and later, declared this a pandemic on March 11, 2020 [4, 5] . After the ten months journey, this virus caused over 37 million positive cases and more than 1 million deaths worldwide till October 10, 2020 [6] . This disease is highly infectious and to control its spread, following three ways have been suggested as most desirable preventive measures: social distancing [7] , use of mask [8] and early identification of infected people.
Till now, there is no best-known medication or vaccine is available for the proper treatment of COVID-19; therefore to stop the infection the infected person must be quarantined to break the chain of infection. Real-time reverse transcription-polymerase chain reaction (RT-PCR) is the best known method of COVID-19 diagnosis. This came into effect after the rapid antibody tests showed unreliable results, because antibodies appear after 9-28 days of the infection and by this time, an infected person may spread the disease, if not isolated [9] . The RT-PCR test is a costly and time taking process, whereas inability to early-stage diagnosis and due to rapid growth of infections, medical experts are continuously trying to search some other means of diagnostic measures.
SARS-CoV-2 virus initially affects the respiratory system of the infected person and in later stages, it affects lungs that may cause severe pneumonia [10] . It is evident that the symptoms of COVID-19 are extremely heterogeneous, ranging from minimal symptoms to significant hypoxia with acute respiratory distress syndrome (ARDS) [11] . Like common flu, the common early symptoms of COVID-19 are fever, dry cough, nausea, diarrhea, muscle aches, vomiting, headache, loss of smell or taste, sore throat, etc. Statistics show that 40%-45% of COVID-19 patients remain asymptomatic, but the virus affects their respiratory system silently resulting in severe pneumonia. COVID-19 starts affecting the lungs as soon as it reaches through the nose or throat.
Aiming to the diagnosis at initial stage; specially for asymptomatic patients, chest imaging may play a key role because the study shows that COVID-19 pneumonia is different from normal pneumonia and tuberculosis [12] . Although for lung-related disease diagnosis, chest CT imaging has been proven more effective, the CXR is preferred because comparatively it is widely available, faster and cheaper. Since, SARS-CoV-2 attacks the epithelial cells that line respiratory tract, hence CXR image analysis can help diagnose pneumonia, inflammation, abscesses and enlarged lymph nodes. Identification of COVID-19 cases from other types of pneumonia and tuberculosis using CXR images is a crucial task and objective of this research is to develop a quicker way of COVID-19 diagnosis in presence of a minimum number of positive class samples.
Biomedical image segmentation and classification via machine learning algorithms have become admired areas of research to make the existing healthcare system more responsive and robust [13, 14, 15] . Concerning the same, the computing infrastructure advancements make it feasible to apply the deep learning approaches for complex biomedical image analysis tasks. It is evident from recent research works that the COVID-19 CXR images patients depicts certain abnormalities [12] . Deep learning architectures specially, convolution neural networks (CNN) are found to be very effective and efficient feature learning approaches in the field of biomedical image analysis [16] . For pneumonia detection a transfer learning based deep network approach pre-trained on ImageNet was proposed by Chouhan et al. [17] , whereas Wang et al. [18] proposed a customized VGG16 model to classify different types of pneumonia. Later, the importance of image augmentation with CNN was demonstrated by Ronneburger et al. [19] in presence of limited number of images. Afterwards, to diagnosis of pneumonia in chest X-rays, Rajpurkar et al. [20] proposed a 121-layered dense CNN, whereas Lakhani et al. [21] proposed a pneumonia detection model based on AlexNet and GoogLeNet along with image augmentation.
Recently, several deep-learning based COVID-19 detection techniques have been proposed using CXR and CT images, where Linda [22] introduced COVID-Net, a deep CNN based architecture for the detection of COVID-19 infection in CXR images. Afterwards, Wang et al. [23] achieved specificity, sensitivity and accuracy of 88.0%, 87.0 % and 89.5% respectively for COVID-19 identification using CT images. Recently, Punn and Agarwal proposed the random oversampling and weighted class loss function approach for unbiased fine-tuned learning (transfer learning) with various state-of-the-art deep learning approaches such as baseline ResNet, Inception-v3, Inception ResNet-v2, DenseNet169 and NASNetLarge. All above discussed deep learning approaches have used pneumonia CXR images from different datasets [24, 25, 26, 27] . The limited availability of COVID-19 positive samples may lead to biased outcome therefore, all the above discussed approaches follow oversampling to build the model. The oversampling of medical images is not always a reliable practice to increase the classification accuracy, hence this article proposes a one-class approach for COVID-19 diagnosis in the presence of a limited number of samples using pinball based OCSVM. One-class classification techniques, specially one-class support classifiers (OCSVCs) [28] are found more suitable for anomaly and novelty detection tasks [29] . In present research work, COVID-19 CXR images [24] , radiological society of North America (RSNA) images [27] and U.S. national library of medicine (USNLM) collected montgomery country -NLM(MC) [30] are utilized for experiments to justify the workability of proposed model and also used for comparison with existing deep learning approaches.
The identification of COVID-19 infection or pneumonia in CXR images is specifically considered as anomaly detection task which is specifically one-class classification problem, where the samples of only one-class are available and other class samples are totally absent or ill-defined. In the present research it is assumed that only COVID-19 infected CXR images are available, whereas other type of pneumonia and normal CXR images are either ill-defined or totally absent at the time of training. It is evident that OCSVCs (support vector data description (SVDD) [31] and OCSVM [28] ) are suitable in such circumstances where the negative class samples are totally absent [29, 32, 1] . For more precise results OCSVM has been extended under the supervision of pinball loss function [33] and the proposed model is named as PB-OCSVM, which possesses the following advantages:
• It is less sensitive to the noise.
• It's computational complexity is nearly equal to the conventional OCSVM, because pinball loss function does not add any computation overhead.
• PB-OCSVM reduces to the original OCSVM when pinball loss parameter tend to '1', and thus OCSVM could be considered as a particular case of the proposed PB-OCSVM.
• PB-OCSVM computes a hyperplane similar to original OCSVM.
In the rest of the manuscript, section 2 discusses the recent research articles and section 3 describes the utilized datasets. The proposed methodology is discussed in section 4 whereas the experiments and results are discussed in Section 5. Finally, the concluding remarks and future aspects are discussed in section 6.
CXR images are always preferred for disease diagnosis due to easy availability and easy handling of X-ray machines. As discussed in the preceding section, the conventional way of testing COVID-19 via means of RT-PCR technique is a time taking and error prone process, specially for asymptomatic patients. Hence, the CXR image analysis is claimed to be a more robust alternate way to diagnose COVID-19. Concerning the same, several studies (binary and multi-class classifications based deep learning models) have been reported using multiple CXR datasets [24, 25, 26, 27] for diagnosis of COVID-19 pneumonia. The objective of these studies is to develop a fully automatic diagnosis system, because the manual analysis is time taking process and unable to serve the purpose due to limited availability of radiology experts.
In the battle against the COVID-19 pandemic, several organizations and researchers are trying to apply machine learning and artificial intelligence (AI) approaches as quicker and alternate ways to identify COVID-19 infection on CXR images [12] . It is evident with deep insight into recent COVID-19 related publications that deep learning models are capable to diagnose and triage new infections, that supplant the conventional testing methods, thus helpful to reduce the growing burden on radiologists during this pandemic. It is also interesting that there is always a chance of infection while collecting samples for swab test, whereas CXR analysis is comparatively safer and easily manageable. Towards this objective, Kermany et al. [34] proposed a deep learning approach to detect pneumonia in CXR images with 92.80% accuracy, whereas seeking the same objective in addition to classical image augmentation Stephen et al. [35] proposed an efficient deep learning model with 4 convolution and 2 dense layers, and obtained 93.73% accuracy. Afterwards, Saraiva et al. [36] proposed a 7 convolution and 3 dense layers based childhood pneumonia diagnosis model and attained 95.30% accuracy. Later, Liang and Zheng [37] proposed a transfer learning based pediatric pneumonia diagnosis technique with 49 convolution and 2 dense layers with a deep residual network, and achieved 96.70% accuracy. Later, concerning the same, Wu et al. [38] proposed a pneumonia prediction model based on CNN and random forest, and achieved 97% accuracy for CXR images.
Seeking to identify COVID-19 infection using CXR images, Narin et al. [39] proposed a deep CNN based on ResNet50, InceptionV3 and Inception-ResNetV2 pretrained transfer models. Later, Apostolopoulos et al. [40] applied state-of-the-art CNN architectures for CXR image classification based on transfer learning approaches to deal with various abnormalities present in the samples. For experiments, two different datasets have been used where images belong to three different categories: COVID-19, viral/bacterial pneumonia and normal cases. This article validated the suitability of the deep learning approach via following performance parameters: accuracy, precision, sensitivity and specificity.
In the same race of COVID-19 diagnosis, Khalifa et al. [41] offered generative adversarial network (GAN) based approach for pneumonia detection using CXR images while ensuring the robustness against the over-fitting by generating more images. The dataset used for experiments contains 5863 chest x-ray samples of two classes: pneumonia and normal, in addition the GoogLeNet, Squeeznet, Resnet18 and AlexNet deep learning architectures have been experimented along with GAN. This is evident from this article that the Resnet18 along with GAN outperformed among other models. Further, Sethy et al. [42] introduced deep feature based support vector machine (SVM) model for COVID-19 infection classification of CXR images. It is found that features extracted by ResNet50 outperformed with SVM compared to other models. Later, workability of ResNet-50 architecture was proved by Bukhari et al. [43] under three categories of CXR images: non-COVID pneumonia, normal and COVID-19 pneumonia. The experimental results show that this approach efficiently diagnose COVID-19 infection from the other types of pneumonia.
As an alternate way of COVID-19 infection detection, a customized ResNet-50 CNN based deep learning architecture named 'COVIDResNet' has been proposed [44] , where for robust, faster and reliable training, conventional ResNet-50 is used along with different training techniques. For experiments the input images are progressively resized to 128 × 128 × 3, 224 × 224 × 3 and 229 × 229 × 3 pixels and the network is fine-tuned at each stage with automatic learning rate selection, whereas the proposed model attained higher accuracy with minimum computational complexity. Later, Zhang et al. [45] proposed a novel deep anomaly detection method for quicker and early diagnosis. To validate the performance of the proposed approach CXR images of two different categories have been considered: COVID-19 positive cases and other pneumonia. To eliminate the data imbalance problem in the collected samples, authors proposed a CXR based COVID-19 screening model through anomaly detection task [46] . Most recently, Punn and Agarwal [12] proposed a transfer learning based deep learning architecture for COVID-19 infection detection, whereas to handle the class imbalance problem random oversampling and weighted class approaches have been used. The experimental results show that the proposed approach outperformed other state-ofthe-art approaches.
From the above discussion, it is clear that all the proposed solutions for COVID-19 infection detection reported till date work based on different deep learning architectures. It is known that every deep learning model is hun-gry for data and consumes massive computing resources as well as time. Hence a faster way of diagnosis process is needed due to the rapid growth of COVID-19 cases and limited availability of healthcare experts (specially doctors and pathologists). Meanwhile, due to limited availability of COVID-19 CXR images, class imbalance appears the most critical issue that has been addressed by all recently proposed approaches. Therefore, a more robust solution is desired, capable of working in the presence of limited number of positive class samples whereas the other class samples are not present at all. Naturally, the diagnosis of COVID-19 cases can be formulated as an anomaly or novelty detection problem and with this notion following are the contributions of the present research:
• Formulated the COVID-19 diagnosis as a one-class classification problem.
• Only COVID-19 CXR images are considered as training samples.
• All training and other class samples are treated as test samples.
• Performance comparison with recent deep learning approaches is done with following parameters: accuracy, precision, specificity and sensitivity.
In present research, CXR images of COVID-19 [24] , radiological society of North America (RSNA) [27] and U.S. national library of medicine (USNLM) collected Montgomery country -NLM(MC) [30] datasets are considered for experiments. COVID-19 image dataset contains CXR images of following pneumonia classes: COVID-19, MERS, SARS and ARDS from multiple public domain resources available without infringing patient's confidentiality ( Fig. 1 (b) ). The dataset includes the statistics up to October 01, 2020 consisting of five categories of pneumonia such as SARSr-CoV-1 (SARS), SARSr-CoV-2 (COVID-19), Pneumocystis spp., Streptococcus spp. and ARDS with metadata: patient ID, offset, sex, age, finding, survival, view, modality, etc. For experiments other samples from COVID-19 infected images are considered as another class named as 'other pneumonia'. Another dataset used in this research has been introduced under RSNA pneumonia diagnose competition is a collection of 30,000 images chosen from the NIH CXR14 dataset [27] . Out of these samples, 15,000 images are pneumonia infected and out of remaining 15,000 samples, 7,500 had no findings whereas other 7,500 had different symptoms from pneumonia. Medical experts and radiologists annotated these images. A sample image is shown in Fig. 1(c) . Initially, this dataset was published with 25,684 training and 1,000 test images and later, the 1000 test images were merged to the existing training set to from 26,684 training images whereas 3,000 new images were introduced as test set. For robust testing and comprehensive coverage of the comparative analysis, NLM(MC) [30] dataset is also utilized that consists of 138 chest posterioranterior x-rays samples of tuberculosis and normal cases. A sample image is represented in Fig. 1(d) . Table 1 presents the class summary details of the fused dataset resulting from the above discussed datasets, which is utilized for training and testing of the proposed approach. The fused dataset is composed of 1214 posteroanterior chest x-ray samples with classes labeled as COVID-19 (108), other pneumonia (515), tuberculosis (58) and normal (533).
From preceding sections, it is inferred that deep learning approaches are playing a decisive role in advanced biomedical image analysis tasks. For the recent pandemic, several deep learning based models have been proposed and following key issues have been identified with these approaches:
• These models need massive computation power to train the neural network.
• Fails to work if only single class samples are available for training. This phenomenon is the motivation of this research.
• May give biased outcomes if the dataset suffers with class imbalance problem.
The above issues have been marked as the research gap and this research proposes a one-class based approach to solve these issues. OCSVM (a variant of OCSVCs) and the proposed PB-OCSVM are used as a one-class classifiers, capable of working with only target class (also known as class of interest (CoI)) samples. One-class classification problem is different from conventional binary and multiclass classification tasks, where only target class is well defined and the other class samples are either totally absent or poorly sampled. In present research, only COVID-19 samples are used as training samples and the performance evaluation is done in two phases: considering all COVID-19 samples as test samples and considering other class samples as test samples. The objective of this research is to maximize the learning ability in presence of limited number of samples while ensuring minimum false-rejection and false-acceptance. Fig. 2 describes the proposed approach where it is assumed that only the target class samples are available. Initially, data preprocessing is performed to remove the noise from the images and then preprocessed images are fed to PB-OCSVM for training. After training, test samples (inclass samples and other class samples) are tested. 
Data pre-processing is an important step in the presence of heterogeneous samples. In this article, COVID-19 infected posteroanterior CXR images of COVID-19 dataset are used to train the PB-OCSVM, whereas the other class samples are used as test samples. To validate the performance of the proposed approach the other pneumonia samples from COVID-19 dataset, RSNA [27] and NLM(MC) [30] datasets are used as test samples. The RSNA and NLM(MC) datasets consist of samples of pneumonia and tuberculosis respectively along with normal cases. It is evident that the CXR images in the aggregated dataset suffers with undesired artifacts such as varying resolutions, pixel level noise, bright texts and unnecessary symbols, hence pre-processing is desired for further analysis. To overcome from the textual and symbolic noise, in-painting is performed with the image mask generated using binary thresholding [47] (Eq. 1), whereas image resizing is performed for a fixed size resolution of 331 × 331 × 3.
where for input image i(x, y), the mask is designed with minimum and maximum threshold values defined as T h min and T h max respectively. Even after filtering out undesired information, there is always a possibility of uncertainty at the deep pixel level representation [48] . To preserve the original pixel value distribution, the adaptive total variation method [49] is opted to eliminate such uncertainties. Let, I is a grayscale image and γ is a bounded set over R 2 , such that γ ⊂ R 2 , denoising image D that closely matches to observed image x = (x 1 , x 2 ) γ -pixels, given as:
where γ(x) = 1 1+p mod Gσ * ∆D , G σ is the Gaussian kernel, σ is variance , p > 0 is contrast parameter and * is convolution operator. The complete data pre-processing operation is illustrated in Fig. 3 , where the histogram of resulting distributed pixels during every step of pre-processing is also shown. It is evident that the opted pre-processing approach is capable to eliminate the irregular intensities while preserving the original nature of the pixel distribution. 
This section gives brief description of OCSVM, variants of OCSVM with different loss functions and Pinball-SVM. This discussion will help to understand the formulation of proposed PB-OCSVM.
One-class classification (OCC) techniques are majorly used for anomaly/novelty detection; specially when the negative class is ill-defined or totally absent. The objective of OCC is to maximize the learning ability using only the target class samples. For unbiased operation of binary or multi-class classifiers, presence of two well-defined classes is necessary; but if the test sample belongs to unknown class, the classifier may exhibit biased behaviour. One-class classification techniques are proven as robust solutions to handle this problem and majorly applicable for concept learning and outliers/novelty detection [50] .
Tax et al. [31] proposed a novel OCC model termed as SVDD, where the target class samples are enclosed by a hypersphere, and the boundary points are called support vectors. If a data sample falls outside of the hypersphere, SVDD treats it as outlier and rejects (Fig. 4) . The objective function of SVDD is defined as:
(3) subject to:
x i − a 2 ≤ R 2 + ξ i , where ξ i ≥ 0 ∀ i where radius of the hypersphere is R, x i and a are outlier and center of hypersphere respectively, the parameter C controls the trade-off between the volume and the errors, and slack variable ξ penalizes the outlier. With the Lagrange multipliers α i ≥ 0, γ i ≥ 0 where i ∈ {1, 2, . . . , N }, the objective is to minimize R for reducing the volume of the hypersphere to cover all CoI samples with slack variable penalty for outliers. By putting partial derivatives to zero and substituting the constraints into Eq. 3, following is obtained: If the description value of a test sample x i is greater than C then it is treated as an outlier. Kernels can be used to reformulate the SVDD as follows:
The SVDD outcome can be computed as follows:
The output of Eq. 6 is positive for samples inside the boundary and negative for an outliers. Later, Schlökopf et al. [28] proposed one-class support vector machine (OCSVM) as an alternate OCC approach to SVDD, where a hyperplane separates the target class samples from outliers as shown in Fig. 5 . In OCSVM, the samples of CoI are separated by a hyperplane with the maximal margin from the origin, whereas the negative class samples reside in the subspace of the origin. The OCSVM is defined as following quadratic equation (Eq. 7): max w,ξ,ρ
where in feature space the data sample x i is represented by φ and the outlier is penalized by slack variable ξ i . υ (0,1] decides the lower bound on the number of support vectors and upper bound on the fraction of outliers.
The dual optimization problem of Eq. 7 is defined as follows:
T and α i is the Lagrange multiplier, whereas the weight-vector w can be expressed as:
ρ is the margin parameter and computed by any x i whose corresponding Lagrange multiplier satisfies 0 < α i < 1
The decision function with kernel expansion can be written as:
Label of any test sample x can be decided with the following:ŷ = sign(f (x)) (12) where sign(.) is sign function. It is observed that both SVDD and OCSVM perform same with Gaussian kernel and origin plays a key role because all outliers reside there. Norm of the centre of SVDD is equal to margin of a hyperplane of OCSVM in unit norm feature space is [51] as depicted in Fig. 6(a) . Thus, reformulation of SVDD can be done by a hyperplane as follows:
where a represents the the centre the hypersphere. w svdd and ρ svdd are the normal vector and the bias respectively of SVDD hyperplane and defined as below:
In feature space, the virtual hyperplane passes through the origin and the sample margin is defined by its distance from the image of the data x as shown in Fig. 6(b) . The SVDD's sample margin is defined as below:
where y(x) is the image of data x in feature space and the sample margin of OCSVM is defined as follows: Figure 6 : Representation of SVDD and OCSVM in unit norm space [35] .
Because data samples exist on the surface of a unit hypersphere, the maximum and minimum value of sample margin is defined as '1' and '0' respectively as follows:
Also, sample margin of unbounded support vectors x U SV ( 0 <α x U SV < 1 υN ) are the same as the margin of hyperplane, therefore:
In feature space, the distribution of samples is represented by the sample margiin. Fig. 6(c) shows the distribution of sample margin of training data and hyperplane of OCSVM in feature space. It is also observed that for RBF kernel both SVDD and OCSVM perform equally, hence in the present research, OCSVM is chosen for enhancement with the help of pinball loss function.
For every machine learning tasks such as classification, regression, etc., the loss functions plays vital role. Consider the training set S = {x 1 , x 2 , . . . x n } , where x i ∈ R n are inputs and target class y is defined as y ∈ {1}. Let, f : R n − → R is a mapping from x i ∈ R n to y ∈ {1}. During training and testing the predicted output value of samples is supervised by the associated loss functions.
• OCSVM with hinge loss function:
Like other machine learning tasks the hinge loss for OCSVM [52] is defined as follows:
This unbounded loss function is capable to deal with the shortest distance of the target class samples, but sensitive to the noise. The hinge loss OCSVM can be written as:
where z i = wφ(x i ). The constraints defined in Eq. 21 are integrated into H loss (z i ). The samples satisfying z i = w.φ(x i ) ≥ ρ, are positioned above the hyperplane and no penalty is added, hence H loss (z i ) = 0, whereas samples satisfying z i = wφ(x i ) < ρ are located on the other side of the hyperplane and some penalty should be inflicted, therefore H loss (z i ) = ρ − z i > 0, which says that if the samples move far away from the hyperplane the penalty increases.
• OCSVM with ramp loss function:
As described above, the hinge loss function is unbounded and sensitive to the noise, the ramp loss function was introduced to OCSVM to make the model more robust [52] . The ramp loss function is described as follows:
where 0 < l < 1. When z i > lρ, the ramp loss is same to the hinge loss; whereas for z i ≤ lρ, the ramp loss is a constant that is an advantage over hinge loss which increases as z i decreases. This loss function is capable to deal with the shortest distance of the target class samples and less sensitive to the noise. The ramp loss OCSVM can written as:
where z i = wφ(x i ). Due to non-convexity of ramp loss the OCSVM problem is no longer a convex optimization. This problem is solved by the phenomenon that the R loss is actually the difference between two hinge loss functions and the modified formulation of the ramp loss OCSVM consists of the convex part and the concave part. This new problem is solved by "Concave-Convex Procedure" (CCP) procedure [53] .
• Pin ball SVM: The pinball loss function is given as follows, which can be regarded as a generalized l 1 loss.
where range of τ is [0, 1] [33] . For quantile regression [54, 55] , the pinball loss P τ has been successfully applied.
Huang et al. [33] derived quantile classification using the idea of support vector machine (SVM). It has been shown that pinball and hinge loss SVM have similar consistency property and computational complexity, moreover the pinball loss SVM is less sensitive to noise.
Huang et al. [33] introduced pinball loss into the SVM. Consider the data points to be classified denoted by the set T . To solve the classification task, w ∈ R n and b ∈ R must be identified such that:
where φ(.) is a nonlinear mapping from the input space to a new feature space. The optimal hyperplane w T φ(x) + b = 0, lies exactly between the supporting parallel hyperplanes given by:
and separates the binary class samples from each other with a margin of 1 w on each side. Data points residing on the supporting hyperplanes (Eq. 26) are termed as support vectors. The classifier is obtained by maximizing the margin. If the data points of two different classes are not linearly separable in feature space then to separate all the data points correctly, pinball loss P τ (x, y, f (x)) is introduced to allow the existence of data points that violate the constraints y i (w T φ(x i )+b) ≥ 1. Finally, the nonlinear Pin-SVM is formulated as follows:
After employing the pinball loss function in Eq. 27, following quadratic programming problem (QPP) come into existence for Pin-SVM:
s.t.
where ξ = (ξ 1 , ξ 2 , ..., ξ N ) T is a slack variable and c > 0 is a penalty parameter. The parameter c determines the weight between the two terms w 2 and t i=1 ξ i . Pin-SVM has an advantage of noise insensitivity. Meanwhile, Pin-SVM has similar time complexity to that of standard SVM. Note that the second constraint of Eq. 29 becomes ξ ≥ 0 when τ = 0, and thus Pin-SVM reduces to the hinge loss SVM.
From above discussion it is clear that the hinge loss is sensitive to the noise. To overcome from this problem and to improve the performance of OCSVM, pinball loss function [33] is fused to standard OCSVM to obtain pinball OCSVM (PB-OCSVM). This enhanced version deals with quantile distance [54] that reduces the sensitivity to the noise. After introducing pinball loss function, the conventional OCSVM can be rewritten as follows. max w,ξ,ρ
where P τ is pinball loss as shown in Eq. 24 with z i = w T φ(x i ) which helps to the slope on the target class. Substituting the pinball loss into Eq. 30, following QPP is obtained:
Now after introducing a kernel based formulation to the PB-OCSVM, the Lagrangian with α i ≥ 0, β i ≥ 0 of Eq. 31 is as follows:
According to:
. . , N the dual problem of Eq. 32 is obtained as follows:
With kernel expansion the decision function can be defined as follows:
Finally, the test instance x can be labelled as follows:
where sign(.) is sign function. It is well known that the computational complexity of OCSVM is O(N 3 Tc ) where N Tc is the number of the only target class samples, whereas the computational complexity of PB-OCSVM is found same as OCSVM i.e. O(N Tc 3 ).
Therefore, the complexity of the PB-OCSVM is equal to conventional OCSVM, hence pinball loss function does not increase the computation time of PB-OCSVM.
COVID-19 positive samples of COVID-19 dataset [24] are used as training samples for OCSVM and the proposed PB-OCSVM whereas other samples of this dataset along with samples of datasets RSNA [27] and U.S. national library of medicine (USNLM) collected Montgomery country -NLM(MC) [30] are used for testing. Once training is done with suitable parameters, OCSVM and PB-OCSVM are used for testing in two scenarios: test with training samples and other samples of all three datasets. For performance evaluation four benchmark parameters have been used to show effectiveness of the proposed model: accuracy, precision, sensitivity, specificity and confidential interval (CI) with 95% and 98% confidence (Eq. 37). Comparison of performance of OCSVM and PB-OCSVM is done with recently proposed deep learning models as shown in Table 2 . It is known that all deep learning models need huge amounts of data and at the same time all classes are desired to be well defined. Hence, it is necessary to oversampling of COVID-19 samples to any deep learning model. It is observed that all recent deep learning architectures [56, 57, 58, 40, 59] used oversampling to deal with the data imbalance problem and after massive computational efforts were able to give significant performance.
T N + T P T N + F N + T P + F P P recision = T P T P + F P Specif icity = T N T N + F P Sensitivity = T P T P + F N Conf idence − Interval = z * ( accuracy * (1 − accuracy)/N (37) where z is standard deviation (1.96 for 95% confidence and 2.33 for 98% confidence) and N = 1214 is number of samples.
There are total four classes in fused dataset: COVID-19, normal, Tuberculosis and other pneumonia. Some existing models have been proposed considering all four classes whereas some articles considered Tuberculosis and other pneumonia classes as a single class that gives three classes: COVID-19, normal and other pneumonia. Few papers considered normal, Tuberculosis and other pneumonia as non COVID class that gives two classes COVID and non COVID. For existing deep learning architectures, experiments have been performed in three categories with consideration of 4, 3 and 2 classes. For exhaustive comparative analysis, all cases have also been considered with OCSVM and PB-OCSVM. It is observed that both the models gave the consistent performance because of their working principle. Whereas it is identified that deep learning models perform better in case of binary class classification approach. It is evident from the experiments that the OCSVM and PB-OCSVM are more stable and perform better compared to exisiting deep learning approaches without oversampling and efficiently work in limited availability of COVID-19 images. In addition the proposed model takes negligible computational power for training due to its simple formulation.
The objective of this research is to ensure a quicker way of early detection of COVID-19 with minimum computational cost. All existing deep learning based approaches for COVID-19 diagnosis work as binary or multi-class classifiers, and suffer with class imbalance problem therefore need enormous multiclass data samples for unbiased operation. To resolve these issues, it is evident that all these deep learning based approaches are using collection of multiple datasets and prefer oversampling strategies to ensure smooth functioning. The COVID-19 is a newly known deadly pandemic and precise characteristics are still unknown. Hence, oversampling cannot be considered as the reliable suite to achieve higher diagnosis accuracy.
In this research, a novel variant of OCSVM is proposed with pinball loss function to diagnose COVID-19 in presence of limited number of CRX images. Considering the available COVID-19 positive CXR images, OCSVM and proposed PB-OCSVM are trained and tested on in-class samples and other class samples of fused dataset. The performance of one-class approach is compared with existing deep learning models and it is evident that both OCSVM and PB-OCSVM outperformed. Meanwhile, it is also evident that the proposed model ensures negligible falsepositive and false-negative rates compared to OCSVM and attains significantly better accuracy, precision, specificity and sensitivity. It is believed that the same work can also be extended to other target class mining tasks.