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The growing complexity of the contemporary financial systems requires the emergence of sophisticated computational and statistical methods that are capable of managing uncertainty, lack of normality and structural variability of multivariate data. The TS charts defined by Hotelling are widely applicable but have been observed to be susceptible to asymmetrical distributions and outliers and are therefore inapplicable in a dynamic real-world example, such as cryptocurrency markets. We present a computationally efficient ambiguity-aware framework in this work, which generalizes the robust covariance estimation methods, which are MVE and MCD, into a neutrosophic logic-based framework. This adaptation also allows the proposed charts to model and react to the intrinsic data ambiguity and indeterminacy with improved robustness and additional multivariate process monitoring. The methodology is validated by a combination of simulation experiments and empirical research on high-frequency financial data of the Binance Exchange, with the focus on the BTCUSDT and ETHUSDT trading pairs. The evaluation of the performance is performed based on total and generalized variance measures that give a holistic picture of the sensitivity and adaptability of the method to noise in data and complexities arising in the presence of noise and complexity of data. The results demonstrate that the proposed approach is considerably superior to conventional TS charts and their robust variants, particularly in terms of detecting a small shift and trends of multivariate financial procedures. Thus, it is a contribution to the growing body of knowledge about applying computational statistics and data science to a scalable, uncertainty-sensitive system of high-dimensional process monitoring in volatile financial settings.

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1. Introduction

The strive for quality control over industrial processes has always been crucial to consistency in providing customer satisfaction and holding a competitive edge. Comparable monitoring schemes, including the Shewhart control charts, have also been applied as the basis for controlling deviations involving various single quality characteristics over a given period, Montgomery [1]. However, the current manufacturing and service processes are even more complex and therefore require the analysis of multiple variables, which needs better statistical tools. This is especially important in situations where several related quality variables must be controlled, using bivariate charts because the univariate charts do not adequately represent the dependency of the quality variables. Such developments have provided new opportunities for uncertainty modeling and developing robust monitoring models in other fields, particularly those that operate in dynamic and noisy environments.

Recent studies have continued to reveal that Statistical Process Control (SPC) methodologies, particularly control chart-based ones, can be extremely relevant in the field of finances and economics, where volatility, uncertainty, and dynamic dependencies are prominent. Initial foundational literature by Kovarik and Klimek [2] formalized the use of time-series control charts in financial process monitoring, pointing out the effectiveness in detecting deviations in non-stationary as well as autocorrelated conditions. This was further extended by Kovarik et al. [3], who incorporated control charts in a financial decision-making system to enhance real-time monitoring capabilities and support the decision process. Dumicic and Zmuk [4] used SPC charts for stock market trading and showed that the charts are useful in detecting regime changes and short-term deviations in trading patterns. Recently, under stochastic volatility, Bisiotis et al. [5] provided a review of SPC methods used in portfolio optimization, asset monitoring, and risk assessment. Yeganeh and Shongwe [6] developed a profile-tracking model that uses a hybrid of SPC and regression models to identify volatility shifts and systemic changes in financial time series. This paper presented a strong control chart designed specifically for nonlinear and heteroscedastic time series with Huber-support vector regression by Kim et al. [7], which is helpful in solving issues such as structural breaks and heavy-tailed errors. Simultaneously, Abu Bakar and Rosbi [8] demonstrated the ability of ARIMA models to predict cryptocurrency exchange rates, highlighting the potential of integrating classical time-series forecasting with modern control chart-based financial surveillance. All these papers help to highlight the increased applicability and flexibility of SPC models, especially robust and empirical versions, in dealing with uncertainty, heteroscedasticity, and volatility in modern financial markets.

The TS chart, developed by Hotelling in the middle of the twentieth century, represented a major advancement in quality control in the multivariate case by providing a scalar measure of shifts in mean vectors that simultaneously takes into account variability by covariance, Hotelling [9]. Its apparent effectiveness is overshadowed by two major weaknesses: owing to the reliance on the TS chart, the analysis heavily relies on classical statistical assumptions, including normality and non-outliers, which are often violated in real-life scenarios. In particular, estimates of vectors of mean values and covariance matrices are highly sensitive to outliers, which can stem from the errors or some exceptional events and outlying values and can produce rather weak control limits, Farcomeni and Greco [10].

To overcome such drawbacks, advanced statistical methods have come up as viable solutions. These methods range from the ones developed by Rousseuw and Leroy [11] to algorithms, which are specifically designed to minimize the effects of outliers, providing a better chance of correct monitoring of the process quality of the tested products, Ali et al. [12]. Researchers propose to improve the effectiveness of multivariate control charts, especially in identifying small shifts in a process, by incorporating stable estimators.

During the course of quality monitoring, data collected may be inaccurate or ambiguous for a number of reasons. As a result, the performance of conventional Hotelling TS charts does not give a loud statement. To address these problems, theorists have employed fuzzy logic, which many authors, including Wibawati et al. [13], have adopted. Subsequently, it emerged as a major component of dealing with uncertainty in databases. After them, there have been brought up many variations or a new technique of fuzzy logic, such as fuzzy attribute and variable control charts, to enhance their utility in the concerned fields, Ercan and Anagun [14]. However, while these approaches offer proper methods for handling the fuzziness of data, none of them capture the uncertainty of some of the data. This gap is filled through the measure of neutrosophic logic, which includes both the features of determinacy and indeterminacy; see Smarandache [15]. Meanwhile, the fuzzy system was initially formulated by Zadeh in the mid-nineteenth century, and it became a significant method incorporated in addressing the issues of ambiguity and uncertainty of data sets. In order to improve the applicability of fuzzy logic in different situations, different extensions of the original material, including the fuzzy attribute and variable control charts have been proposed in the later years, Fadaei and Pooya [16]. Nonetheless, these methods may be termed reasonable in handling fuzziness, and they do not effectively tackle the uncertain characteristic of some data.

Neutrosophic logic, an advanced generalization of fuzzy logic, bridges this gap by incorporating measures of both determinacy and indeterminacy, offering a comprehensive framework for analyzing uncertain observations. Smarandache [15]. In a wide number of contexts, neutrosophic control charts have been applied with some success tentatively in various settings, including to univariate control charts, whereas their use in multivariate is seen as being more limited. New innovations like the neutrosophic Hotelling TS chart expound the growth in sensitivity and capacity of dealing with uncertain data, especially on models with related variables.

Research published in the past few years regarding the neutrosophic extensions of standard statistical control charts—X-bar, S, and EWMA—is based on univariate processes, but the indeterminacy and uncertainty in measurements have been discussed by Ref. [17] and Ref. [18]. It is important to understand that these methodologies are useful but still need modification when used in multivariate processes where there is random observation. Along this line, Ref. [19] introduced Hotelling TS statistics in the framework of neutrosophic data, which broadens the general information structure of established statistics to cope with uncertainty. When this methodology was applied to chemical data, the newly constructed charts had better performance compared to Hotelling TS charts. On this basis, Wibawati et al. [20] have proposed the neutrosophic Hotelling TS control chart with better sensitivity to inherent indeterminacy of the production process and applied it to real-life glass production. These works can be regarded as important advancements in the utilization of neutrosophic logic for multivariate process monitoring, although several lines of research are also possible to extend its uses.

1.1. Research Gap

Outliers hold a big impact on central tendency and dispersion estimates, hence risking the efficiency of the conventional estimators. These estimators based on normal theory are highly sensitive to outliers. Efficient statistical methods have, however, been discovered as ways of coping with this challenge by providing statistical solutions that work in a manner similar to common statistical methods but in a way that remains immune to these outrageous values and marginal violations of model assumptions. Most classical estimation procedures either rely on assumptions like data error having infinitesimal normal distribution or apply the central limit theorem with an intention of obtaining normally distributed estimates, Ali and Saleh [21]. However, when the data includes some outlying values, then these sorts of estimators are much worse, as gauged by measures such as the influence function and the breakdown point; see Ref. [10]. So, robust statistics thus stand out as an important tool to enhance the analysis reliability under such worst-case scenarios.

Past studies on the TS charts have primarily focused on datasets that are imprecise, indefinite, and exhibiting uncertain or indeterminable control limit parameters in tandem with ambiguous interval statistics. Such datasets are described based on interval-based neutrosophic observations where all the values are presumed to lie within some predetermined intervals. Nevertheless, the review of literature revealed an absence of the use of TS charts, using robust covariance matrices like MCD and MVE, in neutrosophic environments, particularly when dealing with the data contaminated by outliers. Further, one has the impression that this area of statistics has not been fleshed out as much as it should yet remains a specific and unique area of statistics, and thus one might expect there to be far more written on the subject. The present study will attempt to make the first step towards filling this gap in the literature. As indeterminate data is more frequent than determinate data in real applications, there is a critical need to design more sophisticated neutrosophic statistical techniques to deal with such datasets.

1.2. Scope of the Study

In more detail, this work is intended to contribute to both theoretical and practical development of multivariate quality control by using statistical methods by: While other forms of multivariate control charts have been useful in monitoring multivariate processes, a number of them are traditional TS charts; the major drawbacks associated with them include the fact that their development requires normality (symmetry) and the fact that multivariate processes contain outliers, which are commonly seen in current and real environments. Some of the statistically significant techniques, like MCD and MVE matrices, can withstand outlying observations, and their application to neutrosophic atmospheres has been elaborated only marginally.

To address this gap and lessen the impact of data volatility and variability in a complex financial system, this research integrates a new neutrosophic model of synthesizing TS charts leveraging covariance matrices. Future adaptations of the method to multivariate frameworks, especially in the case of financial market data, where the interaction among variables and incidence of outliers are more prominent, will be made possible through present investigation, which further leads to the evolution of neutrosophic statistical techniques. The relevance of the suggested methodology is evaluated through real-world financial data, including high-frequency cryptocurrency information from the Binance Exchange, which is utilized to determine the applicability of the neutrosophic logic in multivariate process monitoring within financially hazardous circumstances. The ultimate objective of this effort is to assist in the integration of more resilient and versatile statistical monitoring systems embedded with more flexibility into dynamic financial shifts.

The structure of the article is organized into distinct sections that progressively and systematically discuss the proposed methodology and its applications. Section 2 contains quality control and adapted neutrosophic TS charts, where authors present the Hotelling TS charts concept and the problem of using them for further development. This paper reveals that neutrosophic logic can be used to overcome both problems with the help of determinacy, indeterminacy, and falsity for financial and manufacturing systems. Section 3 describes the method of developing the neutrosophic TS chart based on powerful instruments, that is, MVE and MCD. This section discusses the way the charts deal with the problem of outliers and make it more capable of noticing any changes in the process. Section 4 presents a numerical illustration with simulation and Binance Exchange financial data, which indicates that the proposed charts are better than conventional approaches in uncertainty and outlier management. Finally, Section 5 provides the recollections of the advantages reached in relation to the neutrosophic Hotelling TS charts, indicating more improved performance and proposing future research that may create these useful tools.

2. Quality Control and Adapted Neutrosophic TS Chart

A control chart usually tracks a single value at frequent equal intervals, for example, the machine usage time, the quality of the products, or the speed of delivery. But in most of the industrial and financial situations, total quality is a compound of several factors, and we need multivariate control charts. Of them, the Hotelling TS control chart is a relatively well-known method for monitoring multivariate variables with combined quality characteristics. It quantifies the importance of variation in a mean vector procedure, accounting for the dependence of the variables, consistent with process control. As we will see in sections below, while the Hotelling TS chart is advantageous, it has limitations, including a constant covariance matrix and normality, which may not be achievable in most cases.

In order to address these issues, one may consider the neutrosophic logic integration as a new solution because of paying attention to indeterminacy and uncertainty of the data. The neutrosophic TS charts, when compared to the multivariate control methods, incorporate determinacy, indeterminacy, and falsity measures, which makes them effective in monitoring processes with ambiguous data. Such an approach is especially significant in such a setting as the financial markets, where uncertainties are likely to occur. The relevance of TS neutrosophic charts to the real data of Binance Exchange Trading Bitcoin (BTCUSDT) and Ethereum (ETHUSDT) pairs of data validates the importance of prioritizing the use of neutrosophic TS charts by showing that they are more sensitive to minor fluctuations in the process and are able to remove the effect of responsible data noise. A significant breakthrough that improves the decision-making in the same direction in case of uncertain situations is the integration of traditional statistics and neutrosophic concepts in multivariate quality control. Thus, in the following paragraph, we defined some neutrosophic notations used to represent the Hotelling TS chart like in Willems et al. [22], Henning et al. [23] and Sedeeq et al. [24].

Let $x_{j k}^{N} \in [x_{j k}^{L}, x_{j k}^{U}]$ represent the neutrosophic observation for the k-th variable and j-th observation, where $x_{j k}^{L}$ is the least observation and $x_{j k}^{U}$ is the maximum observation, defining the indeterminacy interval. The neutrosophic data can be expressed as

$x_{j k}^{N} = x_{j k}^{L} + x_{j k}^{U} I_{N}, I_{N} \in [I_{L}, I_{U}],$

where

I_{N}

is the indeterminacy factor. The neutrosophic data matrix,

X^{N}

, can be written as

$X^{N} \in [[\begin{matrix} x_{11}^{L} & \dots & x_{1 p}^{L} \\ ⋮ & ⋱ & ⋮ \\ x_{n 1}^{L} & \dots & x_{n p}^{L} \end{matrix}], [\begin{matrix} x_{11}^{U} & \dots & x_{1 p}^{U} \\ ⋮ & ⋱ & ⋮ \\ x_{n 1}^{U} & \dots & x_{n p}^{U} \end{matrix}]] .$

The neutrosophic sample mean is given by

${\bar{x}}_{k}^{N} \in [\frac{1}{n^{L}} \sum_{j = 1}^{n^{L}} x_{j k}^{L}, \frac{1}{n^{U}} \sum_{j = 1}^{n^{U}} x_{j k}^{U}],$

which can be expressed as

${\bar{x}}_{k}^{N} = {\bar{x}}_{k}^{L} + {\bar{x}}_{k}^{U} I_{N}, I_{N} \in [I_{L}, I_{U}] .$

The sample variance under the neutrosophic environment is obtained as follows:

$s_{k}^{N} \in [\frac{1}{n^{L}} \sum_{j = 1}^{n^{L}} {(x_{j k}^{L} - {\bar{x}}_{k}^{L})}^{2}, \frac{1}{n^{U}} \sum_{j = 1}^{n^{U}} {(x_{j k}^{U} - {\bar{x}}_{k}^{U})}^{2}],$

or, equivalently,

$s_{k}^{N} = s_{k}^{L} + s_{k}^{U} I_{N}, I_{N} \in [I_{L}, I_{U}] .$

Within the neutrosophic framework, the covariance matrix is formulated as follows:

$S_{i k}^{N} \in [\frac{1}{n^{L}} \sum_{j = 1}^{n^{L}} (x_{j i}^{L} - {\bar{x}}_{i}^{L}) (x_{j k}^{L} - {\bar{x}}_{k}^{L}), \frac{1}{n^{U}} \sum_{j = 1}^{n^{U}} (x_{j i}^{U} - {\bar{x}}_{i}^{U}) (x_{j k}^{U} - {\bar{x}}_{k}^{U})] .$

Finally, the neutrosophic Hotelling $T_{N}^{2}$ statistic is defined as

$T_{N}^{2} = ({({\bar{x}}^{L} - μ_{0}^{L})}^{'} S_{L}^{- 1} ({\bar{x}}^{L} - μ_{0}^{L}), {({\bar{x}}^{U} - μ_{0}^{U})}^{'} S_{U}^{- 1} ({\bar{x}}^{U} - μ_{0}^{U})),$

where

T_{N}^{2} \in [T_{L}^{2}, T_{U}^{2}]

According to Henning et al. [23], the analytical procedure for applying the Hotelling TS chart consists of two sequential stages. So let us explain these steps in the neutrosophic framework. In phase I, the control boundaries for the neutrosophic Hotelling $T_{N}^{2}$ are given as

$U C L_{N} = \frac{p_{N} (m - 1) (n - 1)}{m n - m - p + 1} F_{α, p_{N}, m - p_{N} - 1}, L C L_{N} = 0,$

where

p_{N} \in [p_{L}, p_{U}]

denotes the no. of quality characteristics, m the sample number, n the sample points, and F represents Snedecor’s F distribution. Following Tracy et al. [25] and Bersims et al. [26] for phase II, the control limit boundaries of neutrosophic Hotelling

T_{N}^{2}

are given as

$U C L_{N} = \frac{p_{N} (m + 1) (n - 1)}{m n - m - p + 1} F_{α, p_{N}, m - p_{N} - 1}, L C L_{N} = 0 .$

The value of the LCL for both phases is set at zero. The other thing that has to be tried before constructing multivariate charts is to make some conformative checks; these are the normality and independence assumptions. However, outliers affect the normality assumption in the wrong way. In response to this, some neutrosophic robust TS charts are suggested here below to alleviate such difficulties in the next section.

3. Proposed Neutrosophic TS Charts

MVE and MCD are the two reliable statistical methods in estimating the central tendency and covariance matrix in large multivariate data sets. The MVE makes the sphere that engulfs as little as fifty percent of data, proving that our estimates are resistant to outrageous values. Meanwhile, MCD reduces the determinant of the covariance matrix, which is computed from a subset of data; thus, MCD provides a robust estimate of location and dispersion even if one half of the data contains extreme outliers. As with any techniques, both methods are markedly effective in dealing with the issues that the outliers and anomalies create and, thus, are recognized as fundamental to accomplishing multivariate analysis.

The employment of neutrosophic MVE and MCD in Hotelling TS charts contributes positively to the leads of quality control activities. These matrices increase the sensitivity and accuracy of the control charts owing to the fact that the central tendency and covariance structure of the probability density function are not influenced by outliers or uncertainty. The neutrosophic MVE and MCD matrices identified can be used in real-life scenarios, where the data is perturbable and uncertain, such as in manufacturing or an industrial process, to provide a higher monitoring and detection capability of a process change. By providing strong control limits, these matrices establish an initial creation of anomalies and better consistency in the process and increase the quality of the decision-making processes as well as the efficiency of the operations.

In this research, therefore, MVE and MCD are extended to develop credible TS charts that meet the requirements of uncertainty estimation and indeterminacy of decision-making in the neutrosophic environment. These methods incorporate deterministic and indeterminate aspects of observations, which give good covariance estimates to MVE and MCD. These methods incorporate both deterministic and indeterminate aspects of observations, producing robust covariance matrices for MVE and MCD. Hence, as compared to the classical matrices, which assume that all data values are precise and clear, these developed neutrosophic matrices work more efficiently where actual measurements cannot be depicted accurately. This increased operational robustness attenuates distortional impact stemming from outliers, ensures statistical efficiency and offers accurate point estimates even in conditions of data uncertainities.

Let $μ_{x . m v e}^{N} = (μ_{x_{1} . m v e}^{N}, μ_{x_{2} . m v e}^{N}, \dots, μ_{x_{p} . m v e}^{N})$ and $μ_{x . m c d}^{N} = (μ_{x_{1} . m c d}^{N}, μ_{x_{2} . m c d}^{N}, \dots, μ_{x_{p} . m c d}^{N})$ illustrate the robust mean vectors for MCD and MVE considering each component $i \in {1, 2, \dots, p}$ , consequently. Further, $μ_{x . m v e}^{N} \in [μ_{x . m v e}^{L}, μ_{x . m v e}^{U}]$ and $μ_{x . m c d}^{N} \in [μ_{x . m c d}^{L}, μ_{x . m c d}^{U}]$ . The robust covariance matrices, $SR (i)$ , are defined as

$SR (i) = (\begin{matrix} {SR}_{x_{1} x_{1}}^{(i)} & {SR}_{x_{1} x_{2}}^{(i)} & \dots & {SR}_{x_{1} x_{p}}^{(i)} \\ {SR}_{x_{2} x_{1}}^{(i)} & {SR}_{x_{2} x_{2}}^{(i)} & \dots & {SR}_{x_{2} x_{p}}^{(i)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {SR}_{x_{p} x_{1}}^{(i)} & {SR}_{x_{p} x_{2}}^{(i)} & \dots & {SR}_{x_{p} x_{p}}^{(i)} \end{matrix}),$

where the robust covariance matrix type is indicated by

i \in {1, 2}

as follows:

$SR (1)$ : Minimum Volume Ellipsoid (MVE);

$SR (2)$ : Minimum Covariance Determinant (MCD).

In the neutrosophic context, the robust covariance matrix ${SR}^{N} (i)$ is expressed as an interval-valued matrix

${SR}^{N} (i) \in [(\begin{matrix} {SR}_{x_{1} x_{1}}^{L (i)} & {SR}_{x_{1} x_{2}}^{L (i)} & \dots & {SR}_{x_{1} x_{p}}^{L (i)} \\ {SR}_{x_{2} x_{1}}^{L (i)} & {SR}_{x_{2} x_{2}}^{L (i)} & \dots & {SR}_{x_{2} x_{p}}^{L (i)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {SR}_{x_{p} x_{1}}^{L (i)} & {SR}_{x_{p} x_{2}}^{L (i)} & \dots & {SR}_{x_{p} x_{p}}^{L (i)} \end{matrix}), (\begin{matrix} {SR}_{x_{1} x_{1}}^{U (i)} & {SR}_{x_{1} x_{2}}^{U (i)} & \dots & {SR}_{x_{1} x_{p}}^{U (i)} \\ {SR}_{x_{2} x_{1}}^{U (i)} & {SR}_{x_{2} x_{2}}^{U (i)} & \dots & {SR}_{x_{2} x_{p}}^{U (i)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {SR}_{x_{p} x_{1}}^{U (i)} & {SR}_{x_{p} x_{2}}^{U (i)} & \dots & {SR}_{x_{p} x_{p}}^{U (i)} \end{matrix})] .$

Alternatively, the neutrosophic robust covariance matrix can be expressed as

${SR}^{N} (i) = {SR}^{L} (i) + {SR}^{U} (i) I_{N}, I_{N} \in [I_{L}, I_{U}],$

where

I_{N}

represents the indeterminacy factors, and

{SR}^{L} (i)

and

{SR}^{U} (i)

are the neutrosophic [lower, upper] bounds of the robust covariance matrix for type i. This formulation integrates two types of powerful covariance estimation techniques, namely MVE and MCD, into the neutrosophic environment to handle uncertainty and outliers in multivariate data sets. The statistics of the proposed robust neutrosophic TS charts are as follows:

$T_{N_{m v e}}^{2} = ({({\bar{x}}^{L} - μ_{x . m v e}^{L})}^{'} {({SR}^{L} (1))}^{- 1} ({\bar{x}}^{L} - μ_{x . m v e}^{L}), {({\bar{x}}^{U} - μ_{x . m v e}^{U})}^{'} {({SR}^{U} (1))}^{- 1} ({\bar{x}}^{U} - μ_{x . m v e}^{U})),$

$T_{N_{m c d}}^{2} = ({({\bar{x}}^{L} - μ_{x . m c d}^{L})}^{'} {({SR}^{L} (2))}^{- 1} ({\bar{x}}^{L} - μ_{x . m c d}^{L}), {({\bar{x}}^{U} - μ_{x . m c d}^{U})}^{'} {({SR}^{U} (2))}^{- 1} ({\bar{x}}^{U} - μ_{x . m c d}^{U})),$

where

T_{N_{m v e}}^{2} \in [T_{L_{m v e}}^{2}, T_{U_{m v e}}^{2}]

and

T_{N_{m c d}}^{2} \in [T_{L_{m c d}}^{2}, T_{U_{m c d}}^{2}]

. Note that

{\bar{x}}^{N} \in [{\bar{x}}^{L}, {\bar{x}}^{U}]

are the traditional average estimates.

In the next section, the performance of the proposed neutrosophic robust Hotelling TS charts $(T_{N_{m v e}}^{2}, T_{N_{m c d}}^{2})$ will be evaluated against the classical neutrosophic Hotelling TS chart $T_{N}^{2}$ . To compare the $T_{N}^{2}$ and $(T_{N_{m v e}}^{2}, T_{N_{m c d}}^{2})$ charts, total variance and generalized variance can serve as evaluation metrics. The chart that yields the lowest values for these variances is deemed the most effective (Ali et al., [27]).

4. Numerical Illustration

The use of neutrosophic robust TS charts is relatively new, and, to the knowledge of the authors of the paper, this subject has not yet been investigated in the literature. Consequently, a comparative analysis was carried out in order to quantify the total and the general variances in relation to the adapted and proposed charts. The next subsections are devoted to the findings and analyses.

4.1. Simulation Study

In the first experiment, for the purpose of generating neutrosophic data originally, normal data was initially generated in R randomly with the help of the following mean vector info $(5, 10)$ and covariance info $[\begin{matrix} 4 & 2 \\ 2 & 3 \end{matrix}]$ . The generated data are as follows: main sample size of $n = 5$ observers per subgroup, a total of $m = 10$ sub-groups, and two variables $p = 2$ , which means 50 observations. To apply neutrosophic characteristics, differences in percentage were defined as intervals of lower and upper ends of results. This was achieved by applying an uncertainty factor (20% of the standard deviation) to the generated values, which was added to and subtracted from the range definition. The outliers were then included in the dataset to model realistic situations in practice. Namely, for five random indices, the values of the first independent variable were increased by 10 units, and the second dependent variable was reduced by 10 units. This process generated extreme values to signify slight variation or inaccuracy that always characterizes sample data. These steps helped make sure that our dataset is not only going to contain normal types of data but also include issues with uncertainty and outliers that are often experienced in real-life problems.

A neutrosophic interval was established for each observation to simulate uncertainty and indeterminacy in a controlled environment. The upper and lower bounds of each variable were given as

$[I_{L}, I_{U}] = [x_{i} - δ σ, x_{i} + δ σ] .$

In this formulation $x_{i}$ , $σ$ , and $δ$ stand for the simulated observation, the standard deviation (SD) of the variate, and a scalar uncertainty constant. Here we employed the standard deviation of 20% of the variable as the indeterminacy interval of each observation, that is, 0.2.

This strategy captures a controlled level of uncertainty into the simulated data, representing measurement imprecision and environmental variability that could affect process monitoring. The resulting interval-valued data were then analyzed using classical, MVE, and Fast-MCD robust covariance estimators to evaluate the stability and sensitivity of the provided neutrosophic control framework. For further information related to indeterminacy, see [28].

The neutrosophic covariance and correlation matrices were calculated using the adapted and two proposed robust methods, as presented in Table 1 and Table 2. The results of the first simulation experiment, which included Mahalanobis distance values, are illustrated in Figure 1 and Figure 2, which are present to illustrate the neutrosophic lower and upper bounds, respectively. These figures show that when using the MVE and FMCD methods, five outliers were detected. The adapted and proposed control charts, configured after excluding out-of-control points in Phase 1, are depicted in Figure 3, Figure 4 and Figure 5.

It is necessary to note that having out-of-control points is also expressed when using charts, which shows the charts’ ability to identify outliers. As shown in Figures, plotted values fall inside the control limits in the used charts, such as in Figure 3, Figure 4 and Figure 5. Hence, it proves the chart’s reliability and validity in the Phase 2 context. Additionally, the results of the initial simulation experiment concerning the adapted and proposed charts are displayed in Table 3. The findings in Table 3 clearly show that the proposed robust charts outperform the adapted chart with respect to control limits, total and generalized variances, and that the second proposed chart yields the most favorable results.

Moreover, the experiment was conducted ten times more, i.e., 1000 runs for $p = 2, 3$ with $n = 5, 7, 10$ . In order to present a comparative analysis, the averages of the control limits, total and generalized variances were calculated, and the summarized results are represented in Table 4 and Table 5, respectively. These tables suggest that the variances for $p = 3$ are greater than those for the $p = 2$ for the corresponding values of $n = 5, 7, 10$ . Further, both of the variations in MCD are slightly higher than those of MVE but significantly lower than those of CLASSICAL. This means that MVE results in the best performance among the methods. It is also important to realize that the variations/fluctuations in methods are likely to reduce with the rise in the sample size across the board. Therefore, the adapted and the proposed charts give better results as sample size increases.

Discussion on Simulated Data Results

Deeper insights have been provided by expanding the discussion of the simulated results as suggested by the reviewer.

It is evident from the estimates that the application of neutrosophic intervals is an effective way to increase the robustness of the control charts by accounting for the uncertainty and imprecision of the data of the simulated processes. A comparison of Classical, MVE and Fast-MCD estimators revealed that the Classical method is quite vulnerable to outliers, whereas the robust estimators, especially MVE, remain stable in both total and generalized variance even in perturbed situations. This validates the fact that the proposed neutrosophic framework is able to model uncertainty without compromising on statistical efficiency.

The Figure 3, Figure 4 and Figure 5 further indicate that once the out-of-control points are eliminated, the process acquires a stable behavior where all the subgroup means fall within the defined control limits set. The fact that neutrosophic control limits are identical in all the repeated runs also illustrates that the detection capabilities of the proposed charts are similar. This demonstrates the flexibility of the neutrosophic method of modeling dynamic process behavior in the face of uncertainty.

Moreover, the theoretically anticipated pattern of decreasing variations with increased subgroup size is demonstrated by the results; findings support the notion that more data aggregation enhances the stability of the process. The better performance of MVE in different sample sizes implies that it will offer a better blend of robustness and sensitivity to actual changes in the processes. Overall, the findings of the simulation confirm that the given framework is effective in spotting abnormalities and remains reliable in the condition of indeterminacy.

4.2. Binance Exchange Data

In this section, we assessed the results of the suggested charts using real data collected on Binance Exchange to demonstrate their usefulness in financial settings. The neutrosophic robust $T^{2}$ control charts are used for one-minute interval historical trading data of two major cryptocurrencies, such as Bitcoin (BTCUSDT) and Ethereum (ETHUSDT), recorded between 02:05:00 and 12:26:00 on 15 December 2021. Emphasis is laid on the high and low price, which is a short-run volatility and directional movement of the market behavior. Trading pairs like BTCUSDT and ETHUSDT are traded versus the Tether (USDT), which is pegged to the US dollar, being a very liquid and responsive pair to the market events.

The purpose of the suggested charts is to identify structural shifts in the mean process of these interval-valued price data. These changes, when observed by the charts (e.g., subgroup points are above control limits), can be interpreted by the traders as possible entry or exit indicators. Bullish and bearish tones are indicated by upward and downward trends. The charts are early signals of regime changes by means of capturing these shifts in situations of uncertainty and noise. In this way, the methodology does not only help in price reconstruction but also in improving real-time monitoring and decision-making in the volatile financial environment.

Fluctuations in prices and other external influences introduce uncertainty, ambiguity, and noise into financial time series. Thus, in this research, high or low prices of BTCUSDT and ETHUSDT digital currencies are neutrosophically expressed to reflect this inherent indeterminacy. The data were obtained from the publicly available website https://kaggle.com, and ethical approval was not required.

Unlike the simulated environment where the indeterminacy element $I N \in [I_{L}, I_{U}]$ was reflected as a ratio of the subgroup SD, the Binance Exchange data are naturally represented by interval-valued data on a time-point basis. In the case of every minute-level record, the limits of the bounds $I_{L}$ and $I_{U}$ are determined by the minimum of the low prices and the maximum of the high prices between BTCUSDT and ETHUSDT,

$[I_{L}, I_{U}] = [min ({Low}^{BTC}, {Low}^{ETH}), max ({High}^{BTC}, {High}^{ETH})]$

The methodology is an attempt to capture the sensitivity of prices over time, and therefore, it is a representation of the actual data-driven indeterminacy rather than a parametric estimate. The bounds were later applied to shape the neutrosophic modelling of uncertainty and subgroup variation in our control chart technique.

The plots of the Mahalanobis distance of the Binance Exchange data are displayed in Figure 6 and Figure 7, which show how the Classical and Robust (MVE and MCD) estimators are behaving in real-world price movements. In the figures, outliers are represented by the color red to indicate their nonconformity with the central data distribution. Neutrosophic intervals, based on the actual high-low dispersion of BTC and ETH, are used to reflect the market dynamics and offer a significant contribution to the suggested architecture of the charts.

The control limits, total and generalized variances for all techniques used on the real data are summarized in Table 6. The findings demonstrate that the Classical approach is highly sensitive to outliers and market noise, and thus it is not as dependable in unstable financial circumstances. The MVE and MCD estimators, on the contrary side, show significantly lower total and generalized variance, which indicates greater stability and robustness in the representation of the real data structure. Specifically, MCD exhibits the least variance among all subgroup sizes, which demonstrates its exceptional performance.

Discussion on Binance Data Results

The Binance data analysis reveals the efficiency of the offered neutrosophic robust control charts in the real-life environment marked by uncertainty and nonlinearity. The suggested system is specially made to handle the natural indeterminacy of the real-time interval values of high and low prices. Despite simulated cases that employed artificial generation of indeterminacy, this case takes advantage of the real-market-based uncertainty.

The outcomes support the visual analysis of Figure 6 and Figure 7 control chart soundness: Although the Classical procedure demonstrates the unnecessary dispersion and classification of outliers, the robust strategies (particularly MCD) afford the increased control limits and appropriately detect the volatility-induced anomalies. This interval data capability and flexibility to fit fluctuation are essential in financial settings where there are price increases and decreases in brief periods of time.

Additionally, the stability of MCD on various time slices and the low variance values indicate that it offers a stable platform on which real-time monitoring can be carried out. The suggested method can be applied to multi-asset portfolios or multi-dimensional crypto markets in the future, where neutrosophic uncertainty modelling can be used to provide a high performance benefit over crisp or fuzzy control schemes.

It is significant to note that the suggested neutrosophic robust $T^{2}$ control charts have a different objective in contrast to conventional time-series models, like multi-GARCH, DCC-GARCH, or Bayesian change point detection. Although these models mainly concentrate on forecasting volatility and modeling dynamic correlation, the neutrosophic model pays attention to monitoring processes in real time and detecting structure changes in the face of uncertainty and indeterminate data. The suggested charts are complementary, as they offer quick insights into real-time information about process stability and abnormal market behavior without depending on distributional or stationarity assumptions by incorporating interval-valued information and implementing robust covariance estimation.

4.3. Limitations of the Study

Despite the positive outcomes of the suggested neutrosophic robust control charts, the following study limitations must be noted:

Although both the MVE and Fast-MCD estimators are resistant to anomalies, they can become computationally intensive in high-dimensional models or with larger sub-groups.

The neutrosophic indeterminacy bounds were defined using a fixed uncertainty factor, which was 20 percent of the standard deviation in the simulation study. Although this method offers a workable analytical system, the results may differ based on the context or application settings.

5. Conclusions

The proposed neutrosophic Hotelling TS charts solve several severe issues of multivariate quality control by applying neutrosophic logic to deal with uncertainty and indeterminacy as well as real data issues such as outliers and asymmetries. These charts improve upon conventional methods by the use of neutrosophic parameters as well as taking into consideration the correlation structures and therefore are more effective and sensitive approaches used in monitoring processes with dependencies. Supported by simulation and Binance Exchange financial data, the optimal charts outlined in this paper indeed offer detection of process changes while providing stability, which is beneficial for highly fluid environments, such as cryptocurrency markets. This paper recognizes the importance of improved modeling of statistical analysis within modern complex and uncertain data space and offers directions to future improvements of the quality control methods that involve uncertainty in order to be relevant and practical within the changing environment of the modern industries and systems. These neutrosophic approaches provided in this investigation can be further developed in future research following the work of Arslan et al. [29] with the inspirations of Alomair and Shahzad [30] and Arslan et al. [31]. The next step towards future growth is to incorporate the suggested ambiguity-conscious monitoring system with more sophisticated volatility modeling techniques, especially those that consider the asymmetric behavior and periodicity of financial markets [32,33], to make the proposed framework more pertinent to real-time financial surveillance and predictions. Also, the framework may be expanded by including the data of other significant exchanges like Coinbase and Kraken that would enable a more comprehensive demonstration of generalizability across financial platforms. Additionally, advanced preprocessing of statistical data balancing could be supported by the inclusion of trading volume and volatility measures. Lastly, a comparison of the neutrosophic approach with other uncertainty modeling strategies, such as fuzzy logic, could provide further details on the relative advantages and disadvantages of the neutrosophic approach to monitoring complex financial processes.

Author Contributions

Conceptualization, A.F.H., A.O.A. and I.A.; methodology, A.F.H. and N.A.; validation, A.O.A.; formal analysis, A.F.H. and I.A.; resources, N.A.; data curation, A.F.H. and I.A.; writing—original draft preparation, A.F.H., A.O.A., I.A. and N.A.; writing—review and editing, A.F.H., A.O.A., I.A. and N.A.; visualization, A.O.A.; supervision, I.A.; project administration, A.F.H.; funding acquisition, A.F.H. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

All the relevant data information is available within the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1 Mahalanobis distance for the neutrosophic ${Classical}^{L}$ , ${MVE}^{L}$ and ${MCD}^{L}$ methods.

Figure 2 Mahalanobis distance for the neutrosophic ${Classical}^{U}$ , ${MVE}^{U}$ and ${MCD}^{U}$ methods.

Figure 3 Neutrosophic adapted TS control charts.

Figure 4 Neutrosophic proposed MVE TS control charts.

Figure 5 Neutrosophic proposed MCD TS control charts.

Figure 6 Binance exchange, 1 min intervals, 2021 dataset, Mahalanobis distance for the neutrosophic ${Classical}^{L}$ , ${MVE}^{L}$ and ${MCD}^{L}$ methods.

Figure 7 Binance exchange, 1 min intervals, 2021 dataset, Mahalanobis distance for the neutrosophic ${Classical}^{U}$ , ${MVE}^{U}$ and ${MCD}^{U}$ methods.

Table 1

Neutrosophic covariance matrices.

{Classical}^{N} \in [(\begin{matrix} 12.10943 & - 7.84883 \\ - 7.84883 & 12.99963 \end{matrix}), (\begin{matrix} 12.170187 & - 7.859315 \\ - 7.859315 & 12.917903 \end{matrix})]

{MVE}^{N} \in [(\begin{matrix} 5.300765 & - 2.586071 \\ - 2.586071 & 3.161343 \end{matrix}), (\begin{matrix} 5.283302 & - 2.582179 \\ - 2.582179 & 3.171023 \end{matrix})]

{MCD}^{N} \in [(\begin{matrix} 5.738000 & - 3.214612 \\ - 3.214612 & 3.552192 \end{matrix}), (\begin{matrix} 5.716884 & - 3.210456 \\ - 3.210456 & 3.564996 \end{matrix})]

Table 2

Neutrosophic correlation matrices.

{Classical}^{N} \in [(\begin{matrix} 1.0000000 & - 0.6255723 \\ - 0.6255723 & 1.0000000 \end{matrix}), (\begin{matrix} 1.000000 & - 0.626816 \\ - 0.626816 & 1.000000 \end{matrix})]

{MVE}^{N} \in [(\begin{matrix} 1.0000000 & - 0.6255723 \\ - 0.6255723 & 1.0000000 \end{matrix}), (\begin{matrix} 1.000000 & - 0.626816 \\ - 0.626816 & 1.000000 \end{matrix})]

{MCD}^{N} \in [(\begin{matrix} 1.0000000 & - 0.6255723 \\ - 0.6255723 & 1.0000000 \end{matrix}), (\begin{matrix} 1.000000 & - 0.626816 \\ - 0.626816 & 1.000000 \end{matrix})]

Table 3

Estimated neutrosophic variances and control limits for the first experiment.

Methods	General Variance	Total Variance	LCL	UCL
${CLASSICAL}^{N}$	[95.814, 95.44447]	[25.10906, 25.08809]	[0, 0]	[5.978024, 5.978024]
${MVE}^{N}$	[10.06977, 10.08582]	[8.462109, 8.454324]	[0, 0]	[5.453636, 5.453636]
${MCD}^{N}$	[10.04875, 10.07364]	[9.290193, 9.281881]	[0, 0]	[5.631472, 5.631472]

Table 4

Average estimated neutrosophic variances and control limits for thousand experiments for $p = 2$ .

Methods	General Variance	Total Variance	LCL	UCL
$n = 5$
${CLASSICAL}^{N}$	[107.111392, 107.098724]	[25.359650, 25.359184]	[0, 0]	[5.976781, 5.976789]
${MVE}^{N}$	[12.151025, 12.180094]	[8.987679, 8.991027]	[0, 0]	[5.941751, 5.941823]
${MCD}^{N}$	[12.838855, 12.844238]	[9.217895, 9.205715]	[0, 0]	[5.935058, 5.935248]
$n = 7$
${CLASSICAL}^{N}$	[81.159672, 81.163674]	[20.482124, 20.483049]	[0, 0]	[5.760528, 5.760590]
${MVE}^{N}$	[11.183228, 11.161603]	[8.549376, 8.560232]	[0, 0]	[5.562929, 5.562657]
${MCD}^{N}$	[11.583293, 11.583965]	[8.645173, 8.648573]	[0, 0]	[5.533587, 5.534255]
$n = 10$
${CLASSICAL}^{N}$	[81.159672, 60.139426]	[20.482124, 16.583932]	[0, 0]	[5.618259, 5.618165]
${MVE}^{N}$	[11.183228, 10.258034]	[8.549376, 8.151636]	[0, 0]	[5.349814, 5.349813]
${MCD}^{N}$	[11.583293, 10.416979]	[8.645173, 8.183327]	[0, 0]	[5.316084, 5.314856]

Table 5

Average estimated neutrosophic variances and control limits for thousand experiments for $p = 3$ .

Methods	General Variance	Total Variance	LCL	UCL
$n = 5$
${CLASSICAL}^{N}$	[2556.105421, 2556.183924]	[42.100900, 42.100044]	[0, 0]	[8.514900, 8.117900]
${MVE}^{N}$	[1118.895725, 1121.924917]	[38.902512, 38.808156]	[0, 0]	[8.371053, 7.974053]
${MCD}^{N}$	[1234.780571, 1240.172181]	[39.969485, 39.967560]	[0, 0]	[8.404675, 8.007675]
$n = 7$
${CLASSICAL}^{N}$	[1545.080684, 1545.583963]	[35.802500, 35.805237]	[0, 0]	[8.474901, 8.077901]
${MVE}^{N}$	[672.259702, 670.025247]	[32.126835, 32.055045]	[0, 0]	[8.371049, 7.974049]
${MCD}^{N}$	[714.877133, 32.055045]	[32.623379, 32.677841]	[0, 0]	[8.404599, 8.007599]
$n = 10$
${CLASSICAL}^{N}$	[989.575800, 987.280931]	[30.889645, 30.873818]	[0, 0]	[8.474881, 8.076881]
${MVE}^{N}$	[422.503455, 423.456425]	[27.402689, 27.427499]	[0, 0]	[8.371037, 7.973037]
${MCD}^{N}$	[440.562856, 440.364927]	[27.658751, 27.688465]	[0, 0]	[8.404595, 8.006595]

Table 6

Estimated neutrosophic variances and control limits for Binance Exchange data.

Methods	General Variance	Total Variance	LCL	UCL
$p = 2 n = 4$
${CLASSICAL}^{N}$	[1,155,100, 963,645.7]	[20,033.48, 21,890.21]	[0, 0]	[5.936273, 5.975135]
${MVE}^{N}$	[305,335.1, 213,185.1]	[3405.115, 3967.122]	[0, 0]	[5.698028, 5.106025]
${MCD}^{N}$	[231,737.6, 252,710.5]	[3982.413, 4146.657]	[0, 0]	[5.850653, 5.850653]
$p = 2 n = 5$
${CLASSICAL}^{N}$	[5,755,853,000, 5,783,421,000]	[487,147,424, 488,362,823]	[0, 0]	[5.862421, 5.862421]
${MVE}^{N}$	[2,310,641, 1,724,584]	[4533.86, 5005.636]	[0, 0]	[5.587021, 5.617023]
${MCD}^{N}$	[2,853,570, 1,791,341]	[5666.694, 5785.527]	[0, 0]	[5.370852, 5.490204]
$p = 2 n = 10$
${CLASSICAL}^{N}$	[1,512,340,000,000, 864,360,382,472]	[990,530,924, 992,820,816]	[0, 0]	[5.855444, 5.855443]
${MVE}^{N}$	[297,467,155,398, 134,380,269,263]	[1,675,552,920, 1,649,051,731]	[0, 0]	[5.497021, 5.606023]
${MCD}^{N}$	[250,202.9, 168,152.3]	[2377.548, 2481.075]	[0, 0]	[5.388108, 5.388106]

References

1. Montgomery, D.C. Introduction to Statistical Quality Control; John Wiley and Sons: Hoboken, NJ, USA, 2007.

2. Kovarik, M.; Klimek, P. The usage of time series control charts for financial process analysis. J. Compet.; 2012; 4, pp. 29-45. [DOI: https://dx.doi.org/10.7441/joc.2012.03.03]

3. Kovarik, M.; Sarga, L.; Klimek, P. Usage of control charts for time series analysis in financial management. J. Bus. Econ. Manag.; 2015; 16, pp. 138-158. [DOI: https://dx.doi.org/10.3846/16111699.2012.732106]

4. Dumicic, K.; Zmuk, B. Statistical control charts: Performances of short term stock trading in croatia. Bus. Syst. Res. Int. J. Soc. Adv. Innov. Res. Econ.; 2015; 6, pp. 22-35.

5. Bisiotis, K.; Psarakis, S.; Yannacopoulos, A.N. Control charts in financial applications: An overview. Qual. Reliab. Eng. Int.; 2022; 38, pp. 1441-1462. [DOI: https://dx.doi.org/10.1002/qre.2962]

6. Yeganeh, A.; Shongwe, S.C. A novel application of statistical process control charts in financial market surveillance with the idea of profile monitoring. PLoS ONE; 2023; 18, e0288627. [DOI: https://dx.doi.org/10.1371/journal.pone.0288627]

7. Kim, C.K.; Yoon, M.H.; Lee, S. Robust control chart for nonlinear conditionally heteroscedastic time series based on Huber support vector regression. PLoS ONE; 2024; 19, e0299120. [DOI: https://dx.doi.org/10.1371/journal.pone.0299120]

8. Abu Bakar, N.; Rosbi, S. Autoregressive integrated moving average (ARIMA) model for forecasting cryptocurrency exchange rate in high volatility environment: A new insight of bitcoin transaction. Int. J. Adv. Eng. Res. Sci.; 2017; 4, pp. 130-137. [DOI: https://dx.doi.org/10.22161/ijaers.4.11.20]

9. Hotelling, H. Multivariate quality control-illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis; McGraw Hill: New York, NY, USA, 1947.

10. Farcomeni, A.; Greco, L. Robust Methods for Data Reduction; CRC Press: Boca Raton, FL, USA, 2016.

11. Rousseeuw, P.J.; Leroy, A.M. Robust Regression and Outlier Detection; John Wiley and Sons: Hoboken, NJ, USA, 2003.

12. Ali, T.H.; Hayawi, H.A.A.M.; Botani, D.S.I. Estimation of the bandwidth parameter in Nadaraya-Watson kernel non-parametric regression based on universal threshold level. Commun. -Stat.-Simul. Comput.; 2023; 52, pp. 1476-1489. [DOI: https://dx.doi.org/10.1080/03610918.2021.1884719]

13. Wibawati,; Mashuri, M.; Purhadi,; Irhamah,. A fuzzy bivariate poisson control chart. Symmetry; 2020; 12, 573. [DOI: https://dx.doi.org/10.3390/sym12040573]

14. Ercan-Teksen, H.; Anagun, A.S. Intuitionistic fuzzy c-control charts using defuzzification and likelihood methods. J. Intell. Fuzzy Syst.; 2020; 39, pp. 6465-6473. [DOI: https://dx.doi.org/10.3233/JIFS-189110]

15. Smarandache, F. Introduction to Neutrosophic Statistics; Sitech and Education Publisher: Craiova, Romania, Romania-Educational Publisher: Columbus, OH, USA, 2014; 123.

16. Fadaei, S.; Pooya, A. Fuzzy U control chart based on fuzzy rules and evaluating its performance using fuzzy OC curve. TQM J.; 2018; 30, pp. 232-247. [DOI: https://dx.doi.org/10.1108/TQM-10-2017-0118]

17. Aslam, M.; Al-Marshadi, A.H.; Khan, N. A new X-bar control chart for using neutrosophic exponentially weighted moving average. Mathematics; 2019; 7, 957. [DOI: https://dx.doi.org/10.3390/math7100957]

18. Khan, Z.; Gulistan, M.; Chammam, W.; Kadry, S.; Nam, Y. A new dispersion control chart for handling the neutrosophic data. IEEE Access; 2020; 8, pp. 96006-96015. [DOI: https://dx.doi.org/10.1109/ACCESS.2020.2995998]

19. Aslam, M.; Arif, O.H. Multivariate analysis under indeterminacy: An application to chemical content data. J. Anal. Methods Chem.; 2020; 2020, 1406028. [DOI: https://dx.doi.org/10.1155/2020/1406028]

20. Wibawati, W.; Ahsan, M.; Khusna, H.; Qori’atunnadyah, M.; Udiatami, W.M. Multivariate control chart based on neutrosophic hotelling T2 statistics and its application. JTAM (J. Teor. Dan Apl. Mat.); 2022; 6, pp. 85-92. [DOI: https://dx.doi.org/10.31764/jtam.v6i1.5442]

21. Ali, T.H.; Saleh, D.M. Proposed hybrid method for wavelet shrinkage with robust multiple linear regression model: With simulation study. Qalaai Zanist J.; 2022; 7, pp. 920-937.

22. Willems, G.; Pison, G.; Rousseeuw, P.J.; Van Aelst, S. A robust Hotelling test. Metrika; 2002; 55, pp. 125-138. [DOI: https://dx.doi.org/10.1007/s001840200192]

23. Henning, E.; da Maia, M.T.; Walter, O.M.F.C.; Konrath, A.C.; da Cunha Alves, C. Application of hotelling’s T² control chart for a machining process of the inside diameter of a steel cylinder. Rev. Gestão da Produção Operações e Sist.; 2014; 15, 155. [DOI: https://dx.doi.org/10.15675/gepros.v34i2.1015]

24. Sedeeq, B.S.; Muhammad, Z.A.; Ali, I.M.; Ali, T.H. Construction Robust-Chart and Compare it with Hotelling’s T2-Chart. Zanco J. Hum. Sci.; 2024; 28, pp. 140-157.

25. Tracy, N.D.; Young, J.C.; Mason, R.L. Multivariate control charts for individual observations. J. Qual. Technol.; 1992; 24, pp. 88-95. [DOI: https://dx.doi.org/10.1080/00224065.1992.12015232]

26. Bersimis, S.; Psarakis, S.; Panaretos, J. Multivariate statistical process control charts: An overview. Qual. Reliab. Eng. Int.; 2007; 23, pp. 517-543. [DOI: https://dx.doi.org/10.1002/qre.829]

27. Ali, T.H.; Sedeeq, B.S.; Saleh, D.M.; Rahim, A.G. Robust multivariate quality control charts for enhanced variability monitoring. Qual. Reliab. Eng. Int.; 2024; 40, pp. 1369-1381. [DOI: https://dx.doi.org/10.1002/qre.3472]

28. Balegh, M.; Ghezal, A. Dynamical analysis of a system of fuzzy difference equations with power terms. Int. J. Dyn. Control; 2025; 13, pp. 1-14. [DOI: https://dx.doi.org/10.1007/s40435-025-01876-3]

29. Arslan, M.; Anwar, S.; Gunaime, N.M.; Shahab, S.; Lone, S.A.; Rasheed, Z. An improved charting scheme to monitor the process mean using two supplementary variables. Symmetry; 2023; 15, 482. [DOI: https://dx.doi.org/10.3390/sym15020482]

30. Alomair, A.M.; Shahzad, U. Neutrosophic mean estimation of sensitive and non-sensitive variables with robust hartley–ross-type estimators. Axioms; 2023; 12, 578. [DOI: https://dx.doi.org/10.3390/axioms12060578]

31. Arslan, M.; Shahzad, U.; Yeganeh, A.; Zhu, H.; Malela-Majika, J.C.; Ahmad, S. A Robust L-Comoments Covariance Matrix-Based Hotelling’s T² Control Chart for Monitoring High-Dimensional Non-Normal Multivariate Data in the Presence of Outliers. Qual. Reliab. Eng. Int.; 2025; 41, pp. 3308-3317. [DOI: https://dx.doi.org/10.1002/qre.70025]

32. Alzeley, O.; Ghezal, A. On an asymmetric multivariate stochastic difference volatility: Structure and estimation. Aims Math.; 2024; 9, pp. 18528-18552. [DOI: https://dx.doi.org/10.3934/math.2024902]

33. Ghezal, A.; Alzeley, O. Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility. Aims Math.; 2024; 9, pp. 11805-11832. [DOI: https://dx.doi.org/10.3934/math.2024578]

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Ambiguity-Informed Modifications to Multivariate Process Analysis Using Binance Market Data

Content area

Abstract

Full text

1. Introduction

1.1. Research Gap

1.2. Scope of the Study

2. Quality Control and Adapted Neutrosophic TS Chart

3. Proposed Neutrosophic TS Charts

4. Numerical Illustration

4.1. Simulation Study

4.2. Binance Exchange Data

4.3. Limitations of the Study

5. Conclusions