Deprived air quality in India is now considered an imminent risk to public health as well as the country’s ability to develop. The environment is significantly impacted by urban air pollution, especially in developing countries. Along with 10 other Indian cities that are even more impure, Kolkata, the third-largest city by population in India, is among the 25 cities with the highest pollution in the world. One of the Indian cities that requires involvement to ensure good air quality in the decades to come is Kolkata. The smog, manufacturing production, and other forms of pollution in the environment are affecting more places. One of the most undeveloped and dirty cities in the world is Kolkata, as stated [31] . Based on the research comparing data on air quality in four Indian cities, Kolkata has greater pollution levels compared to Mumbai and Chennai and is also close to Delhi [32] and listed in polluted cities [33] . The winter season is when air pollution in Kolkata is at its worst, being more severe than throughout other seasons [34] . On the contrary, during peak hours, the worst traffic intersections triple the city’s usual pollution levels [35] . The production of power, the transportation industry, soil and road dust, trash combustion, etc. are the main sources of pollution in India. The biggest contributor to air quality in Kolkata is transportation [36] [37] , where the prevalence of badly managed automobiles, the usage of gasoline, and inadequate regulations have made this industry the dominant source of pollutants in the air [38] [39] . Three thermal power plants are also operational close to Kolkata, in addition to a number of small businesses that also affect our environment [40] . According to the West Bengal Pollution Control Board’s research on the various causes of air pollution in Kolkata, automobiles are the main cause of air pollution (51.4%), followed by industry (24.5%) and dust particles (21.1%) [41] . Data on air pollution was collected hourly for the current study from the Central Pollution Control Board (CPCB) in India.

To assess information about the environment, numerous kinds of devices are accessible, such as respiratory detectors. The expected value of the sample position and taking into account classical analysis, patterns, and differences between neighboring stations may be used to identify particular data values that are not usual. The pattern analysis in R-programming [42] was an illustration of the expert structure of data and validation of an environmental parameter. The results were simply statistically examined for traditional interpretation. In order to extract conclusions, the suggested approach requires using a huge amount of data that already exists, with some incomplete findings. The amount of data that is saved in systems nowadays requires the use of machine-learning tools. The methodology of research discussed here is oriented towards the discovery of information in databases (Knowledge Discovery Database) (KDD) [43] . This provides a full data extraction procedure and yields an accessible technique for preparing data and examining the obtained results. In order to provide information and aid in conclusion creation, the KDD provides a clear and collective method of observing the design and model parameters that are practical for outlier identification, future prediction, and/or classification. We apply some steps to this research: Imputation for handling missing values, classical analysis, SPC, and FDA.

Incomplete datasets pose a challenge in data preprocessing, making machine learning algorithms ineffective for training models. Various data imputation approaches have been proposed to predict appropriate values using different algorithms. Accurate estimation of imputation methods is crucial for completing missing values, especially in air quality. The FCM method generates satisfactory estimation results for multiple-dimensional datasets, but clustering results are sensitive to membership degree and cluster centroids. Huang J. et al. [44] pointed out the fuzzy C means method for data imputation. The clustering methods are often used in data imputation when dealing with missing values. K-means is a centroid-based algorithm that partitions data into k clusters, while fuzzy K-means assigns membership degrees to each data point for soft clustering.

Traditional statistical analysis seeks to analyze the observed frequency distribution, which yields the absolute frequency of occurrence of each of the potential results of a discrete class [23] . If there are just a finite number of various outcomes (a discrete example), if the distribution function is utilized in the situation of an indefinitely frequent and randomly trustworthy calculation and each result is different, the outcome of relative frequency will not be very enlightening. This returns all values of the absolute frequency of occurrence that are less than x in this example [45] . By applying SPC to monitor the system, it is possible to identify the outliers. But in conditions where the points do not reach the defined limit, the analysis focuses on substantially low and high measurements. To study individual observation, the techniques can be used to study individual or average maps. You should split the dataset into logical subgroups [46] . The ability to cluster variance and simply identify variability in the presence of unique causes makes it crucial for the creation of rational groupings. For instance, when a measurement occurs again in the same way, it changes due to laboratory or analytical error, unless it is impractical to use rational subgroups. The method of gathering the data is the logical subgroup. The data collected shows that some of them display intrinsic variation, which is the common cause of variation; we can ignore this. The approach that makes it possible to identify special-cause variation, which may have an unfavorable impact on the subgroups when it is avoided, In addition, if the mechanism is excessively violated, the limit of the control chart that establishes the border to be defined is established based on variability within each subgroup. Therefore, only subgroups that replicate the common cause variance of the process should be gathered [25] . When the data have some missing observations, the data will have been imputed, and if normality has been established, then the data will be correctly structured. If we reject the null hypothesis, the data can be normalized in two different ways. The first is to utilize customized procedures for non-normal distributions to turn them into normal distributions or to modify data to normalize the data set [47] . A classical process analysis can be divided into two stages: the first, when a test is performed to remove normality and unusual measurement from the outcomes, and the second, when the pattern is examined and circumstances outside of control are experienced. The average, UCL, and LCL are specified at the first level. The average is defined precisely by the control model and signifies the objective point. Then, we set the confidence interval as the standard deviation of the process [48] . The Shewart control chart is the most commonly used for SPC due to its high success rate in spotting significant modifications in a process. The control chart is more accurately described as a monitoring system for graphical statistical processes. While the underlying distribution of processes is understood, it is usually built into a traditional control chart to monitor process parameters. Although the most current data is used in these figures, the slight or gradual improvement in the procedure is not proven. Different criteria have been devised by different writers to identify specific deviations [49] [50] and to add to the basic rules. Using these extra criteria [23] makes Shewart’s control charts more vigilant and leads to a significant capacity for detecting a non-random sample. Once the data is structured and normality test has done and the data is not normal then use Box-cox transformation [51] . The transformation is as follows:

X J ( λ ) = { X J ( λ ) − 1 λ ,     if   λ ≠ 0 log ( X J ) ,     if   λ = 0

where λ denotes the maximizes the profile likelihood of the data X J .

The average run length (ARL) is the most commonly used and simple mode to measure the capacity of a control chart with supplementary run rules. In the control charts, the run rule is used before the warn alarm indication when the process is not controlled. The important thing to do if this happens is to identify it as soon as possible. On the other hand, it would be reasonable to have a few false alarms when the mechanism is statistically under control. This term is defined specifically as an error of α (type I) and an error of β (type II). The technique of sensitivity is also defined and is highly linked to the number of outliers. It must be considered that the potential to identify out-of-control techniques is high; for this reason, there will be a lot of points that fall outside [52] .

FDA is a collection of methods for studying curves and functions to analyze data across time [53] . Begin by converting vector samples into functional samples. The beginning points, which come from the study’s generated discrete values, are used to create the curves. Smoothing is the process of transforming vector points into a continuous function over time. This data composition is valuable in the research of air pollution since it takes all of the values from the day as a single unit. As a result, a day with NO₂, SO₂, and O₃ values of varying variability may have an average identical to the other days, and the vectorial approach detects the outliers. These days would be identified as possible outliers by the functional analysis. For outlier detection in these types of investigations, functional techniques have always been shown to be superior.

Let x ( t f ) represent the initial observations, t f ∈ R signifies the time steps, and p represents the number of observations ( f = 1 , 2 , ⋯ , p ). The individual value of the function x ( t ) ∈ x ⊂ F , where F is a functional space, can be observed. The functional space F = s p a n ( ϕ 1 , ϕ 2 , ⋯ , ϕ p ) is used to estimate x(t), where ϕ g is the set of basis functions ( g = 1 , 2 , ⋯ , n b ) and p is the number of basis functions necessary to generate a functional sample. In statistics, there are various types of bases, but the Fourier basis is the most commonly employed. Furthermore, for periodic data like the ones we have in our study, the Fourier basis is the best option [25] .

min x ∈ F ∑ f = 1 p ( z f − x ( t f ) ) 2 + λ Γ ( x ) (1)

z f = x ( t f ) + ϵ f Where x is the observing point at t f , ϵ f is the random noise with zero mean, λ is the level of regulization and Γ is penalized, operator.

x ( t ) = ∑ g = 1 p c g ϕ g ( t ) (2)

where { c g } g = 1 p is the coefficient that multiplied the basis function. We can write the problem of smoothing as:

min c { ( z − ϕ c ) T ( z − ϕ c ) + λ c T R c } (3)

z = ( z 1 , ⋯ , z p ) T , the expansion of vector coefficient c = ( c 1 , ⋯ , c p ) T , a ( p , n b ) -matrix ϕ whose elements are ϕ f g = ϕ g ( t f ) ; and a ( p , n b ) -matrix R whose elements are:

R g l = 〈 D 2 ϕ g , D 2 ϕ l 〉 L 2 ( T ) = ∫ T D 2 ϕ g , D 2 ϕ l d t (4)

The problem can be solved with

c = ( ϕ ′ ϕ + λ R ) − 1 ϕ z (5)

The functional data allows us to determine whether or not different time intervals, such as days, weeks, or months, are higher than the mean feature and how far they differ. It also enables the removal of outliers that aren’t real but are caused by system failure. The notion of depth allows you to sort a collection of data in Euclidian space by how close it is to the sample core. In multivariate analysis, the concept of depth emerged and was generated to calculate a point centrality among a cloud of them. This idea started to be incorporated into practical data analysis over the course of the year. In this region, the centrality of a certain curve xi is defined by depth, and the center of the sample is the mean curve. The two depth measurements, Fraiman-Muniz depth (FMD) and H-model depth (HMD) [25] , are most usual in the sense of functional data.

Through the estimation of depths, it is also possible to classify outliers with a practical approach. In this case, it will take into account elements that have different behavioral designs than the rest. Instead of summarizing the curve observations into a single point, such as the average, the definition of depth makes it possible to deal with observations identified at a given interval in curve types. The depth technique is used for the identification of outliers and significance: there will be a low depth of an element that is distant from the sample. Thus, practical outliers are the curves with the least depth.

Firstly, the F n , t ( x e ( t ) ) is the cumulative empirical distribution function of the values of the curves { x e ( t ) } , ( e = 1 , 2 , ⋯ , n ) in a certain time t ∈ [ a , b ] it is contemplated. It can be defined as:

F n , t ( x e ( t ) ) = 1 n ∑ g = 1 n I ( x g ( t ) ≤ x e ( t ) ) (6)

where I(.) is an indicator function, next, the FMD for curve x_i is calculated as:

FMD n ( x e ( t ) ) = ∫ a b D n ( x e ( t ) ) d t (7)

where t ∈ [ a , b ] . The functional mode in HMD, on the other hand, is the element or curve that is most densely surrounded by the other curves in the dataset. HMD is written as:

HMD n ( x e , h ) = ∑ g = 1 n K ( ‖ x e − x g ‖ h ) (8)

In a functional space, with a kernel function K : R + → R + , a bandwidth parameter h and ‖   ⋅   ‖ as the norm. In a vast majority of cases, it is norm L₂, expressed as:

‖ x e ( t ) − x f ( t ) ‖ = ( ∫ ( x e ( t ) − x f ( t ) ) 2 d t ) 1 / 2 (9)

There are also a number of parameters for the kernel functions K(∙). The truncated Gaussian kernel is a popular one, can be expressed as:

K ( t ) = 2 2 π exp ( − t 2 2 ) ,     t > 0 (10)

A functional sample set may have elements that, although not containing error, exhibit characteristics that are distinct from the rest of the set. Instead of only comparing the mean values over the measurement time interval, the depth measurements mentioned above allow sets of observations over time fitted to curves to be compared in order to find outliers in functional samples. An outlier in a functional sample will therefore have significantly less depth because depth and outlier are opposite terms. In order to find functional outliers, the deepest curves are sought after. The value of bandwidth h was chosen as the 15th percentile of the empirical distribution using the HMD to create the outlier selection criterion { ‖ x e ( t ) − x f ( t ) ‖ 2 , e , f = 1 , 2 , ⋯ , n } . The cut-off C was chosen so that around 1% of accurate observations were incorrectly classified as outliers (type I error) [54] :

P r ( HMD n ( x e ( t ) ) ) < c = 0.01 ,     e = 1 , 2 , ⋯ , n . (11)

Unfortunately, the distribution of the selected functional depth is unknown, necessitating an estimate of C. For the purposes of this study, we selected a method based on bootstrapping [55] [56] [57] [58] the curves of the original set with a probability proportional to depth out of the several approaches to estimate this value. As an overview, the bootstrapping strategy is as stated:

1) By using sampling with replacement, a new sample is taken from the previous sample (each element is replaced after extraction so it can be chosen again). In addition, order 10 has been chosen for resampling.

2) The populational parameter of interest is estimated using this new sample as a basis to generate a statistic.

3) Repeat the steps overhead a significant number of times.

4) Finally, Determine the empirical statistical distribution.

The Second and third phase of NO₂, SO₂ and O₃ analysis includes the X ¯ charts with hourly rational groups for 298 days, as well as functional data analysis of hourly data with depth. During learning stage, non-normality measurements were observed, requiring Box-Cox transformation to verify normality assumption using monthly concentration averages. The Box-cox transformation is applied to the data set. The X ¯ Charts shows the annual cycle for working days. It is observed that there are 248 outliers for NO₂, 231 outliers for SO₂, and 136 outliers for O₃ as shown in Figures 1-3. The lowest concentration of NO₂ values takes place March to September and highest concentration in between January, February and October to December. For SO₂ the lowest concentration takes place January to March and November, December and the highest concentration in between April to October. And for O₃ the lowest concentration in between January, February and September to November and the highest concentration in between March to August and December. The results obtained with functional depth method are shown in Figures 4-9. There are 76, 63, and 22 outliers for NO₂, SO₂, and O₃ concentration and are shown in Figure 5, Figure 7, and Figure 9.

In this case the variable NO₂, SO₂ and O₃ analysis includes the X ¯ charts with hourly rational groups for 67 days, as well as functional data analysis of hourly data with depth. The X ¯ Charts shows the annual cycle for working days. There are 53 outliers for NO₂, 52 outliers for SO₂, and 30 outliers for O₃ as shown in Figures 10-12. The results obtained with functional depth method are shown in Figures 13-18. There are 76, 63, and 22 outliers for NO₂, SO₂, and O₃ concentration and are shown in Figure 14, Figure 16, and Figure 18.

It was discovered during the phase of learning that the data sets identifying days as the rational subgroups of the chart for working and nonworking days are not under control due to non-normality. Despite the fact that after applying Box-cox transformation the chart was not under control and has a high degree of variation, it should be noted that after outlier detection is complete, the study is regarded as having accomplished its aim, and no back transformation is conducted. Shewart graphs for the control stage revealed the considerable variability of the pollutant’s average. Figures 1-3 and Figures 10-12 shows control charts for working and non-working days of NO₂, SO₂, and O₃ recorded in Kolkata station. The red dot in the graph indicates a process that is not statistically controlled, whereas the black dots show a process that is statistically controlled. On the contrary, the visual interpretation of these graphs indicates that there is similar behavior in both working and nonworking (NO₂, SO₂, and O₃) cases, resulting in outcomes above the control limits.

In the functional methodology technique, the initial step is to generate a sample curve based on discrete measurements taken each hour. The graph shows 298 functions for working and 67 functions for non-working derived from 24-hour data. If the data has been translated into functional form, i.e., the curves with 24 points in a day, each of which takes into consideration the correlation between the NO₂, SO₂, and O₃ readings and may be examined for outliers, the data can be analyzed. When the depths are taken into account, the functional analysis results let us discover days with aberrant functional points, even if there are no outliers. Despite the fact that the daily limit values were exceeded, the NO₂, O₃, and SO₂ absorption may demonstrate aberrant behavior throughout the course of a day. The functional technique, on the other hand, detects any variation from normal daily NO₂, O₃, and SO₂ emission behavior without depending on any distribution limits. This can be seen in Figure 5, Figure 9, Figure 11, Figure 14, Figure 16, Figure 18, which shows the functional outliers found in both cases for NO₂, SO₂, and O₃. Outliers are indicated by black lines in Figure 5, Figure 9, Figure 11, Figure 14, Figure 16, Figure 18.

The traditional method detects a large number of outliers, even identifying outliers in more than half of the data in certain circumstances. This problem happens regardless of the station or compound review. As seen in the previous figure of functional outlier detection, the number of outliers found by functional analysis is significantly smaller in both working and nonworking cases. It should also be highlighted that the outliers revealed by functional analysis are among those identified by the traditional method. The non-normality of the data is analyzed and may be the cause of this finding. When the data are characterized by their normality or when the sets under study have a known statistical distribution, the classical approaches are very robust. The effectiveness of this approach, however, declines when it is used to analyze sets without these characteristics. Since the traditional methods examine the data as punctual observations, the detection of outliers in this situation is complicated. When studying atmospheric pollution, the actual pollution caused by a particular substance is dependent on a sequence of measurements that exceed the limits over a certain amount of time rather than a single measurement. The functional analysis, on the other hand, treats the data as a temporal series rather than as punctual observations, which is truly an appropriate solution to this case. Instead of an individual measurement, the airborne pollution caused by a certain compound is supplied by a group of observations that exceed the stated limitations over a specific time interval. This methodology explains why the classical analysis identifies a higher proportion of outliers than the functional analysis. Additionally, this methodology eliminates potential measurement errors, which are anomalous punctual measurements that are classified as outliers by conventional analysis. As a result, the results are more trustworthy when the instrumental error is removed in the first stage.