Overview of gaussian copula

Gaussian copula is a mathematical construction, which defines the dependence relationship between the random variables, making the independent of their marginal distributional forms by simply linear transforming of multivariate normal distributions. Copula provides a strong model to express the relation between multivariate random variables because it creates clear distinction between dependence structure and distribution of individual variables. The methodology follows the copula theory whereby a multivariate distribution is conceived as an integration of the marginal distributions of the same with a copula. In the multivariate normal case, the Gaussian copula easily approximates the linear correlations and creates scope to allow the marginal distributions to differ with normal distribution [31].

Consider random variables , each with a marginal CDF . Gaussian copula transforms these variables to uniform random variables with respect to the range . The transformed variables retain the dependency patterns imbedded in the copula. We can mathematically articulate the joint CDF of random variables with marginal distributions , as follows:

(1)

The copula function with is the function that generalizes the dependence between the referred to variables. The theory of copula has its theoretical footing in the Sklars theorem. It postulates that any distribution of multivariate can be presented as a copula which is in observance jointed with the marginal distributions [32]. In particular:

(2)

where . This separation allows us to model the dependence structure, as shown in , independently of the marginals . For completely continuous distributions, the associated density function is defined as:

(3)

with is the marginal density, and is the copula density. The multivariate normal distribution forms the Gaussian copula. Let represent a vector of standard normal random variables characterised by a covariance matrix . The Gaussian copula exhibits the following characteristics:

(4)

Here, is the cumulative distribution function (CDF) of the multivariate normal distribution defined by the covariance matrix , and , where represents the inverse of the standard normal CDF, with for signifying the marginal uniform variables [33].

The dependency structure in the Gaussian copula is defined by the covariance matrix of a multivariate normal distribution. This structure allows it to capture and represent linear relationships among the variables. At the same time, the marginal distributions of the variables are flexible and do not need to follow a normal distribution, enabling diverse applications with customized marginal properties. However, they are limited in capturing tail dependencies, as the Gaussian structure inherently assumes symmetric dependency patterns. This restriction is in contrast to the -copula, and other copulas that have the ability of capturing greater tail dependencies. The Gaussian copula provides a density function for dependency modeling:

(5)

In which , are uniform random variables that are related to the original data through the marginal cumulative distributions, is the vector of inverse standard normal transformations, and is the covariance matrix with ones along the diagonal. is the standard normal density function, whereas signifies its cumulative distribution function (CDF). In this formulation there is a normalisation term to cover fixed marginal distributions [20]. The joint density is altered using the equation to make the marginal distributions fit a uniform distribution on the range [0,1]. Note that ; hence,

(6)

Then by putting (6) in density function (5) hence, we get. Focusing on dependency structure of the Gaussian copula, the term may be eliminated to arrive to:

(7)

(8)

The proportionality denotes the missing of the marginal density elements needed to uphold the regularity of the marginals. This is simplification, which points out to the ability of the copula to describe dependencies regardless of the specific marginal distributions. The joint density function of the Gaussian copula is expressed as follows:

(9)

Parameter estimation and marginal modeling

Gaussian copula models are often estimated using the maximum likelihood (ML) approach. In this strategy, both the marginal and copula parameters are combinedly computed. It means optimising the likelihood function of all parameters including the marginal distributions () and the covariance matrix (). But this method does not scale well as the space of the random variables in the model becomes dimensional, that is, growing exponentially with the number of random variables. The process of optimising the whole log likelihood requires large calculations especially in the cases of high dimensional data sets. The maximum likelihood two-stage inference of marginals (IFM) applied [34], usually deals with this difficulty. This method saves a lot of computational burden and at the same time estimates parameters correctly. The whole likelihood function for a Gaussian copula model can be articulated as follows:

(10)

In which the determinant of the covariance matrix is denoted as where, with , and is the marginal density of the i-th variable of the j-th data [35]. The associated log-likelihood function can be articulated as:

(11)

The IFM approach divides the estimating procedure into two phases: The marginal parameters () are initially computed for each variable individually. Subsequently, the copula parameters, particularly the covariance matrix (), are estimated while the marginal parameters remain unchanged. We first derive the marginal parameters by optimising the likelihood function for the marginal distributions. The marginal parameters are obtained by the marginal likelihood .

(12)

where . The computed marginal parameters serve as inputs for the copula likelihood function, facilitating the estimation of the copula parameters. The estimate of copula parameters is accomplished by optimising the likelihood function .

(13)

. The joint log-likelihood function is expressed as follows:

(14)

where signifies the marginal parameters and represents the covariance matrix that parameterises the copula. The optimisation process commences with the estimation of marginal parameters via . Upon determining these estimations are incorporated into the copula likelihood function . The copula parameters are subsequently determined by resolving . Collectively, these fulfil the system’s requirements:

(15)

In the Gaussian copula framework, marginal distributions may be estimated by either parametric or non-parametric methods, contingent upon the characteristics of the data. Kernel density estimation (KDE) is a prevalent non-parametric technique that estimates the probability density function of a random variable without presupposing any particular parametric structure. KDE operates by interpolating observable data points to generate a continuous representation of the underlying distribution. The KDE for a given set of data points is articulated as:

(16)

Let signify the total number of data points, indicate the kernel function, represent the -th observed data point, and imply the bandwidth parameter [36]. The choice of bandwidth influences the continuity of the density function estimation. Augmenting bandwidth yields more fluid and generalised estimates, whereas reducing bandwidth produces sharper and more reactive estimates that discern small intricacies in the data. KDE is advantageous when parametric assumptions regarding marginal distributions are inadequate or when the data exhibits intricate structures that parametric distributions cannot adequately represent, due to its inherent flexibility.

Symbolic feature construction with ECDF and DFDV

The ECDF serves as a statistical tool to characterise the distribution of pixel intensities in an image. For a certain image , let represent a random variable corresponding to the pixel values, where . The ECDF, denoted as , is computed using the following formula:

(17)

In this equation, denotes the total pixel count in the image, indicates the intensity of the -th pixel in the -th image, and is an indicator function that equals when the pixel intensity is less than or equal to , and otherwise [37].

The ECDF function represents the proportion of pixels in the image with intensity values less than or equal to a given intensity level . It provides a thorough summary of pixel intensities across the entire image. The ECDF value increases with rising , spanning from (signifying that no pixels have values less than or equal to ) to (showing that all pixels satisfy the criterion ). The ECDF elucidates the distribution of pixel intensities, enabling the examination and comparison of image attributes such as brightness and contrast.

Let denote the ECDF of the pixel values from the -th image, where . Upon calculating the ECDF for a collection of pictures, the DFDV consolidates the ECDF values at designated distinguishing points. The DFDV at a particular discriminating point is defined as the probability that the ECDF values at that point are less than or equal to a certain value . The DFDV function at point can be articulated as:

(18)

signifies the ECDF of the -th image, whereas denotes the aggregation of ECDFs from all images. The term denotes a certain pixel intensity level utilised as a differentiating criterion. The variable , which ranges from to , represents the cumulative probability at , namely the value of the ECDF assessed at point [25]. These distinguishing features encapsulate essential traits in the distribution of pixel intensities among various image categories. The discrepancies in the choice of these sites give different shape of features hence influence the discriminative ability of the features obtained. The creation of ECDF and the further conversion into the DFDV is a procedure described in [25] and provides a well-structured approach to represent the data of the image in symbolic form by using the cumulative distribution-based features.

The DFDV classifies different images according to the distribution of their pixels intensity. It finds important distribution patterns which distinguishes one type of photos over another by comparing DFDV at certain distinguishing points. The representation is more concezed symbolic data representation, unlike traditional pixel-based data. DFDV data is output data which is minimised and therefore computationally powerful whilst preserving the capability of portraying complex relationships existing between pixel intensities inside the visual image [38,39].

Cluster-based point selection distinction

The point of distinction is a particular value of a pixel which is used in the assessment of the ECDF at the point. The probability value , or the realisation of the ECDF at point is also represented as . DFDV function is given by and it is the probability that . This gives information on the detailed spread of pixel values in that area.

The K-means clustering algorithm detects the points of distinction. They assign the data , which are shown as , into cluster groups based on similarity [40,41]. It starts with the random choice of initial centroids, . We assign every piece of data to the cluster those with the nearest centroid, using the following measure of distance:

(19)

(20)

represents the cluster assigned to , whereas represents the centroid value of cluster for . Afterward, the recalibration of the centroids is performed by taking the mean of those values of the data points in every cluster [42]. is the number of data points of cluster . The redistribution of the data points and the switching of the centroid is repeated until the convergence of the clusters, which means the convergence of the algorithm [43].

Algorithm 1. K-means Clustering of ECDF points

Input: Set of ECDF values , for , number of clusters

Output: Cluster assignments , centroids

1: Initialize centroids randomly

2: Repeat:

  For each in data:

    Assign to cluster with nearest centroid

  For each cluster :

    Update centroid as mean of assigned points

3: Until centroids converge

In this study, the number of clusters was set to v = 10, corresponding to the ten digit classes (0–9) in the MNIST dataset. This design choice ensures that the clustering process aligns structurally with the class composition of the dataset. Euclidean distance was used as the similarity measure, and centroid updates were iterated until convergence.

This method clusters ECDF values to acquire the unique characteristics of pixel values distribution in each category of images that enable determination of the most suitable discriminative level to use in the classification. After grouping the data, the effectiveness of the obtained clusters can be evaluated using the Silhouette coefficient, which measures the effectiveness of the groups. Suppose that we have a dataset of points (where ), referred to the value of ECDF at point for the -th image.

The Silhouette coefficient for point is determined as follows:

(21)

(22)

(23)

Let mean to other data points be the average of all data points in the same cluster to . Conversely, denotes the mean distance from to points in the closest neighbouring cluster. To compute , assume cluster consists of n points that are part of the same cluster as , where d denotes the distance between and . To calculate , denote as the cluster including and as the nearest neighbouring cluster. Let and represent the amounts of points in clusters and , respectively. denotes the distance between point and , situated in distinct clusters [44]. The overall Silhouette coefficient , for the entire dataset is the mean of all individual coefficients.

(24)

The Silhouette coefficient ranges from to . A number close to indicates that a data point is well-aligned with its cluster and distinctly separates from others. A score of about indicates that the data point is closer to an alternative cluster, suggesting a potential clustering error [45]. The mean Silhouette coefficient for all clusters is calculated at various candidate points to determine the optimal distinguishing point that maximises the average Silhouette coefficient.

Distinguishing points are selected using a silhouette-based optimization strategy. First, ECDF are constructed for all images. A set of candidate threshold values is generated across the ECDF domain, and the corresponding ECDF values are computed. For each candidate threshold, K-means clustering (Algorithm 1) is performed with clusters, consistent with the ten MNIST digit classes, using Euclidean distance. The average silhouette coefficient is then calculated to assess clustering quality. The first distinguishing point is selected as the candidate maximizing , and the second is chosen as the next highest value excluding the first. In this study, the number of distinguishing points is set to , balancing clustering quality and model complexity within the copula framework. The overall procedure is illustrated in Fig 1.

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Fig 1. Flowchart for selecting two optimal distinguishing points based on average Silhouette coefficients.

The first distinguishing point is selected as the candidate maximizing the average silhouette coefficient , and the second is chosen as the next highest coefficient excluding the first selected point.

https://doi.org/10.1371/journal.pone.0346790.g001

Gaussian Copula with DFDV for image classification

This research employs a Gaussian copula methodology for image classification, utilising random variables characterised by the DFDV derived from the ECDF. The algorithm categorises photos into classes , with each image segmented into areas. For each partition, the points are utilised to assess pixel values, yielding random variables. These random variables encapsulate the fundamental distribution patterns of pixel intensities throughout the image. Implementing this technique necessitates several essential processes. Each of these procedures is detailed in the subsequent sections.

The distinguishing points are identified by the analysis of comprehensive data from classes. The ECDF is initially created for data points across all classes, and candidate distinguishing points are chosen. These points signify critical thresholds that may elucidate disparities in the pixel intensity distributions among various classes. The ECDF for a particular class , partition , and image is represented as , adhering to Equation (17). This function offers a cumulative depiction of pixel intensities for a specified area of the image. The DFDV, originating from the ECDF, encapsulates the distribution at designated locations , where . For partition ,and class , the DFDV is denoted as , as articulated in Equation (18). This transformation enables a concise symbolic representation of visual regions, hence enhancing subsequent analysis and classification.

The -means algorithm is employed to partition the data into clusters for each distinguishing point. The clustering procedure is directed by Equations (19) and (20), which guarantee the establishment of significant clusters. Equations (21–23) calculate the Silhouette coefficient to assess the quality of clustering at each prospective distinguishing point. This statistic assesses the extent to which each data point aligns with its assigned cluster compared to other clusters. We determine the optimal threshold as the point that maximises the average Silhouette coefficient. This ensures that the chosen distinguishing point embodies the characteristics that distinguish image classes.

This study consistently develops the copula model for each class, enhancing generality. The copula model for the -th class can be expressed as follows:

(25)

This formulation enables the copula function to encapsulate the dependency structure among random variables, while the marginal distributions manage the individual behaviour of each variable. Through this generalisation, the model is rendered applicable to a variety of image classes, thereby facilitating the consistent and systematic representation of the data. The random variables are defined as transformations that satisfy the equation , where the variables exhibit a uniform distribution over the interval . The correlation between these random variables is expressed as follows by a copula function:

(26)

Furthermore, the joint distribution function will be associated with the joint probability density function, which is defined as follows:

(27)

This study employed the Gaussian copula model to examine the interrelationship among random variables related to features, partitions, and distinguishing points within the image. The Gaussian copula function utilised in this work is articulated as follows:

(28)

The density function of the Gaussian copula is obtained from its distribution function as illustrated below:

(29)

With The joint density function of the Gaussian copula is expressed as follows:

(30)

We come up with the likelihood function in order to simultaneously involve both the copula and marginal distributions in establishing the parameters of the Gaussian copula model. This joint likelihood formulation consists of two elements: one for the Gaussian copula and another for the marginal components, as follows:

(31)

Denote the parameters correlated with the marginal distributions of the copula by the log-likelihood of the Gaussian component of the copula is . The second in the summation symbolizes the log-likelihood of the marginals distributions whereby; denotes the log-likelihood that is related to the marginal distributions of every territory and division.

Parameter estimation is performed by maximizing the joint log-likelihood function with respect to both and . To ensure computational efficiency and numerical stability, the optimization is conducted using the two-step Inference Functions for Margins (IFM) procedure described in Equations (12) and (13). In the first stage, the marginal parameters are estimated independently by maximizing the marginal likelihood function as given in Equation (12). Subsequently, the copula parameter is estimated by maximizing the copula log-likelihood in Equation (13), while keeping the marginal estimates fixed. This procedure enables coherent modeling of both the marginal distributions and their dependence structure within the Gaussian copula framework.

Classification procedure

The method used in this study is one of classification, but this is done using a probabilistic framework where the intention is to determine the likelihood of a particular test image being related to a particular class. Let and be the new input image and the -th class respectively. The conditional probability shows the assigned probability that an input falls in an appropriate class. It is computed by first calculating the joint probability of the input and the class and this is done by multiplying two factors namely: (1) probability of observing centering dot given the probability of class which is expressed as , and (2) the prior probability of class which has the probability expressed as . Probability is estimated using a Gaussian copula model trained on the data so far, but prior is calculated as training samples belonging to the -th class divided by the entire trained sample set [46].

After computing the conditional probabilities of each of the classes, the machine will assign the test image to the one with the greatest probability. This approach allows making classification judgements using the most dependable statistical records extracted out of the data and model. The confusion matrix evaluates the effectiveness of the classification model that counts and lists all correct and incorrect guesses of the model on each of the classes. The main performance indicator is the accuracy which refers to the number of successful predictions made to the total amount of forecasts. Moreover, precision rate, recall and F1-score are also obtained to have a more comprehensive evaluation of the success of this model. These measures are used together to test the competency of the model in distinguishing and disparity between classes of images to guarantee overall analysis of the accuracy of the model in prediction [47]. The overall structure of the proposed approach, including both training and testing phases, is presented in Fig 2.

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Fig 2. Block diagram of the proposed image classification method using symbolic ECDF and DFDV features.

Starting from data preprocessing and ECDF construction on partitioned images. Distinguishing points are selected based on clustering ECDF values and evaluating Silhouette coefficients (detailed in Fig 1). The DFDV and Gaussian copula parameter are estimated from training data. In testing, the model evaluates and computes the joint copula distribution for final classification.

https://doi.org/10.1371/journal.pone.0346790.g002

Fig 2 illustrates the overall classification framework. In the training phase, images are partitioned and their ECDFs are constructed. Optimal distinguishing points are selected based on the highest silhouette coefficients (see Fig 1). Using ECDF values at these points, the parameters of the DFDV , denoted by , are estimated for each class. The dependence structure among features is then modeled using a Gaussian copula with correlation parameter . In the testing phase, ECDF values at the selected distinguishing points are evaluated using the estimated parameters , and the joint copula density is computed using . Each test image is assigned to the class that maximizes the corresponding joint density value.

Preprocessing data

This experiment preprocesses the handwritten numeric dataset MNIST to maximise on the feature extraction to increase the performance of the classification accuracy. The dataset is freely available at https://www.kaggle.com/datasets/hojjatk/mnist-dataset. All of the images are rescaled to a 20 × 20 grid, which is enough to memorise the basic outline of the numeric shapes and reduces excess information [48]. Each of the enlarged pictures is partitioned into equal sections to better reflect characteristics of the photos. This decomposition allows the study of localised properties and these are necessary in symbolic representations of data. The results of the previous studies [25] showed that the best modelling results were achieved after dividing the images into four horizontal parts as can be seen in Fig 3.

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Fig 3. Image partitioning into four horizontal segments.

Each digit image is divided into four horizontal parts to extract localized ECDF features.

https://doi.org/10.1371/journal.pone.0346790.g003

This finding highlights the need of determining an optimal size of partition to achieve effective representation of distribution-related features during image classification. The pixel values are normalised using the min-max scaling approach a process in which after normalisation, the values adopt a range of [0,1] to normalise the data hence making manipulation possible.

Outcomes of the distance points

The identifications of distinguishing points entail application of special pixel values that can best distinguish the features of classes. Clustering is executed for each of candidate points (with ), ECDF values are clustered using K-Means. The Silhouette coefficient is computed to evaluate clustering quality. This coefficient assesses the degree to which each data point corresponds with its respective cluster in relation to other clusters. We identify the two points with the highest Silhouette coefficient as the distinguishing point, as it effectively differentiates the characteristics of many classes. Selecting 10 candidate points ensures adequate coverage of pixel variations while maintaining computational efficiency. Limiting the final selection to two distinguishing points reduces model complexity and mitigates the risk of overfitting. Fig 4 depicts the methodology, including the methods for pinpointing these locations. We produced the ECDF for all images spanning the ten classes.

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Fig 4. Illustration of distinguishing point determination selection using K-means clustering and Silhouette coefficients.

The first step is constructing the ECDF for all data. ECDF values are evaluated at candidate thresholds, where in this case . At each candidate point, the ECDF values are clustered into groups using K-means clustering, and the clustering quality is assessed using the average Silhouette coefficient . the two thresholds with the highest values are selected as optimal distinguishing points..

https://doi.org/10.1371/journal.pone.0346790.g004

We select two distinguishing points for each partition. The resulting points may vary for each occurrence, depending on the value of the acquired Silhouette coefficient. The chosen distinguishing qualities are those that most efficiently differentiate the image characteristics among its categories. We construct the Gaussian copula model utilising the IFM approach and optimal distinguishing points. Table 1 presents the mean Silhouette coefficient values.

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Table 1. Silhouette coefficient values for 10 candidate distinguishing points. Each row corresponds to a specific image partition , and each column represents a candidate distinguishing point with . For each partition, the two candidate points with the highest Silhouette coefficient values (in bold) are selected as optimal distinguishing points.

https://doi.org/10.1371/journal.pone.0346790.t001

For each , the pixel thresholds associated with the two highest Silhouette coefficients are selected as distinguishing points. This guarantees that the chosen points yield optimal separability for clustering among classes. At partition 1, the maximum Silhouette value is recorded at with a Silhouette coefficient of 0.6897, indicating that this threshold yields the most distinct clustering for this scenario. The distinguishing points for each threshold are selected based on the two highest Silhouette values to provide excellent separability. In partition 1, the positions and demonstrate the highest silhouette values. Likewise, for partition 2, the chosen points are  = 0.03 and  = 0.5. Transitioning to partition 3, the best points are  = 0.3 and  = 0.5. Finally, for partition 4, the points and have the optimal clustering performance. These values serve as determining thresholds that distinctively separate the classes of images to be analyzed later on. The selected distinctive points represent pixel value which provides the most reliable feature of separation between image classes. These can provide points to use as the basis of the ensuing clustering or classification in the pipeline of the analysis.

Results for ECDF and DFDV

After identifying distinguishing features on the basis of the greatest Silhouette coefficients, there is the need to calculate the partitions and classes ECDF, respectively. Each pixel intensity distribution of a class in each partition produces an ECDF, so there are four ECDFs per a class, as there are four partitions. For each division, two distinguishing points are chosen, resulting in eight distinguishing points for each class. These ECDFs illustrate the cumulative probability of pixel intensities, reflecting variations in image attributes across distinct partitions. Figs 5 and 6 illustrates the ECDFs for each class and partition, offering a visual depiction of the pixel distributions that underpin the development of the Gaussian copula model.

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Fig 5. ECDF plots of four partitions for digit classes .

Each subplot shows the ECDF with rows corresponding to partitions and columns to digit classes. Red and blue vertical dashed lines denote the first and second selected distinguishing points, respectively.

https://doi.org/10.1371/journal.pone.0346790.g005

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Fig 6. ECDF plots of four partitions for digit classes .

Each subplot shows the ECDF with rows corresponding to partitions and columns to digit classes. Red and blue vertical dashed lines denote the first and second selected distinguishing points, respectively.

https://doi.org/10.1371/journal.pone.0346790.g006

ECDF plots represent full distribution of pixel brightnesses in the separate image divisions (rows) belonging to different classes of images (columns). The visualisation comprises two segments: the upper segment presents the ECDF results for classes 0–4, whereas the lower segment exhibits the ECDF results for classes 5–9. Every row is a representation of an image partition and every column is a representation of image classes numbered 0–9. The distribution patterns in the graphs underscore discrepancies among partitions and classes. In certain classes, pixel distributions are dispersed throughout the intensity spectrum, signifying variability in pixel attributes within those segments. Conversely, other classes have more concentrated distributions within particular intensity ranges, indicating a predominance of pixels with uniform attributes. This phenomenon is seen in specific partitions when the ECDF curves ascend sharply over a limited range, indicating that a significant proportion of pixels have the same intensity value.

The disparities in distribution patterns among classes and partitions indicate that each class exhibits distinct traits as evidenced by its pixel distributions. The divisions segment the image, facilitating a more nuanced study of pixel changes. The red and blue vertical lines in the plots serve as essential markers for comprehending the data distribution and constructing the Gaussian copula model.

The development of DFDV relies on the collection of ECDFs at the specified distinguishing points indicated in the plots. For each image class, a total of 8 DFDVs are produced at the distinguishing points for each partition. The DFDVs will be used as random variables in Gaussian model of copula. Illustration of the DFDV results is depicted in Fig 7.

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Fig 7. DFDV results for ten image classes.

Each subplot corresponds to a digit class , showing the estimated DFDV curves for each of the 8 features formed from 4 partitions and 2 selected distinguishing points. The variations in curve shapes across digit classes reflect how distributional differences are captured and used for classification.

https://doi.org/10.1371/journal.pone.0346790.g007

According to the DFDV plot shown above, it is observed that the DFDV distribution pattern is various in each image class ( to ). The curves of DFDV of each of the classes differ in their shape and in the points of distribution with higher concentrations attesting to the differences of their type. The DFDV distribution is more cluster in some classes, e.g., and while it is more dispersed with flatter peaks in other classes, e.g., and . This trend shows that the DFDV values differ among the image class and partition as a difference between the classes in the pixel intensity characteristic. The divergence in DFDV distribution contributes as an influential variable in comprehending the disparity in pixel features of each partition and forms the input in the Gaussian copula prototyping to better simulate the associations between partitions. These findings also reveal the suitability of DFDV in identifying the slight changes in distributions of pixel intensities.

The graphic of DFDV presented above shows the discrete patterns of DFDV distributions applying to each class of images ( through to ). Each of the classes represents a certain level of variance in DFDVs shape and in the locations of the peaks, resulting in the individual profile of classes. In some of the classes, like variation in shape and distribution peak points, it is possible to present the variations in the characteristics of classes. The distribution of the DFDV in certain classes, e.g., and is clustered where the peaks are pronounced whereas in certain other classes, e.g., and , the distribution is more spread with the peaks being low. This trend indicates that the DFDV values change depending on the image class and the partition by depicting variations in the intensity of pixels between the classes. The match inequalities in DFDV patterns are an important indicator to understand the deviation in pixel features in thousands of classes and is used as an input in Gaussian copula model to better describe the relationship between partitions. These findings also depict the sensitivity of DFDV to subtle variations in pixel intensity distributions.

Results for Gaussian Copula Model

After obtaining the values of the DFDV at the characteristics points, the next stage involves analysis of distributional properties of the DFDV. KDE method is applied as a non-parametric way of accurately and smooth drawing probability density function of the DFDV to help understand the distributional properties. KDE enables the evaluation and comparison of the DFDV distribution for each image class without necessitating any prior assumptions on the distribution’s structure. The estimation of the KDE parameter primarily involves determining the bandwidth, which regulates the smoothness of the computed density. Equation (16) estimates the parameter, while the Kolmogorov-Smirnov (KS) test assesses the derived bandwidth values for goodness-of-fit. The KS test quantitatively evaluates the degree of correspondence between the DFDV data and the distribution generated by KDE. Table 2 encapsulates the expected bandwidth parameters and the outcomes of the KS test.

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Table 2. Estimated Bandwidth Parameters and Results of the Kolmogorov-Smirnov Test. The table presents the bandwidth values (outside the parentheses) alongside the associated (inside the parentheses) for classes 0–9 over four partitions.

https://doi.org/10.1371/journal.pone.0346790.t002

This analysis assesses the bandwidth of KDE and the from the KS test over different partitions and thresholds ( and ). The minimal bandwidth values () signify narrow kernel smoothing. signify a strong correspondence between the ECDF and KDE distributions. The copula model employs a Gaussian copula approach to define the joint distribution of eight interrelated random variables, denoted as . The copula density function is intended to represent both the marginal behaviours of the variables and their dependence structure. The model is defined as:

(31)

Let and denotes the covariance matrix illustrating the interrelationships among the eight random variables. signifies the marginal cumulative distribution function of each variable, whereas symbolises the inverse of the ordinary normal cumulative distribution function. The terms represent the marginal density functions of the respective variables.

The parameters of the Gaussian copula model are determined by formulating a likelihood function based on the copula density function. This procedure incorporates the dependence structure represented by the correlation matrix and the marginal distributions . Parameter estimation is conducted via the IFM, facilitating a distinct and efficient delineation of the marginal components and dependencies, hence yielding optimal parameter estimations. The estimated parameters are displayed in Table 3.

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Table 3. Results of the estimated covariance matrix of the Gaussian Copula Model for digit classes . This table presents the estimated covariance matrices for the Gaussian copula model across all ten digits classes. Each matrix captures the dependence structure among the eight DFDV features derived from ECDF values at selected distinguishing points. Variations in the correlation strengths across classes reflect class-specific inter-feature dependencies used in the final classification stage.

https://doi.org/10.1371/journal.pone.0346790.t003

We performed clasfsification utilising the Bayesian technique, grounded in the predicted parameter values of the obtained covariance matrices. Following this approach, the data are grouped into the right categories by using the information regarding the interdependence of various variables which people have identified in the covariance matrices. Performance measures on the classification results were then carried out to estimate the aptitude of the model to categorize the categories accurately. The metrics used in the analysis will consist of accuracy, precision, recall and F1-score which will be presented in Table 4. A comparison is also made with the previous approach [25] that employed fixed distinguishing points.

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Table 4. Performance comparison between fixed and optimally selected distinguishing points. The table presents average and maximum classification performance metrics (accuracy, precision, recall, and F1-score) using fixed distinguishing points versus optimally selected points. The proposed method improves average accuracy and recall, indicating better generalization, while maintaining competitive maximum performance levels.

https://doi.org/10.1371/journal.pone.0346790.t004

As shown in Table 4, the proposed method, which employs optimally selected distinguishing points based on the highest Silhouette coefficients, achieves an average accuracy of 68.27% and recall of 66.70%, outperforming the previous method that used fixed distinguishing points (62.22% and 62.46%, respectively). This improvement indicates better generalization and enhanced ability to distinguish class-specific distributional features across partitions. Although the previous approach achieves slightly higher maximum values in accuracy (96.92%) and precision (95.30%), the proposed method maintains competitive maximum performance (up to 95.40% recall), while demonstrating more consistent results overall. These findings affirm the robustness of the model and highlight the effectiveness of using optimized symbolic features in enhancing the classification of handwritten digits. To provide a more detailed view of the classification performance, the confusion matrix for the proposed method is presented in Fig 8.

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Fig 8. Confusion matrix for the proposed method using optimally selected distinguishing points.

Cells show the number of predictions for each actual versus predicted digit. Darker colors indicate higher counts, highlighting the most frequent misclassifications.

https://doi.org/10.1371/journal.pone.0346790.g008