Modified inverse exponentiated exponential Poisson distribution to Analyze air quality dataset of Kathmandu, Nepal

Introduction

In the field of statistics and probability, the development of new lifetime distributions is essential for modelling and analyzing lifetime data related to the longevity or duration of various phenomena. Lifetime distributions play a critical role in areas such as reliability analysis, survival studies, and actuarial science, among others. The exponential distribution (ED) offers a straightforward and elegant analytical solution for numerous challenges encountered in lifetime testing and reliability studies. However, the ED falls short when it comes to fitting certain real-world scenarios where hazard rates are not constant and exhibit monotonic patterns. In recent years, novel model classes have been developed to address this issue by modifying the traditional ED. Gupta and Kundu (1999) [14] presented a generalized ED (also known as exponentiated exponential distribution) that broadens its application to data with growing and decreasing failure rate functions dependent on the shape parameter. Many authors to generate new probability models have used compounding of the exponential distribution with various discrete distributions. Exponential-geometric model (Adamidis and Loukas, 1998) [1] was published combining an exponential & geometric distribution, demonstrating a decreasing failure rate. In a similar vein, Kus (2007) [20] introduced the exponential-Poisson (EP) distribution, defined by two parameters and exhibiting a decreasing failure rate. This distribution arises from the integration of an exponential distribution with a zero-truncated Poisson distribution. The cumulative distribution function (CDF) for EP distribution can be expressed as

Barreto-Souza and Cribari-Neto (2009) [4] improved the distribution provided by (Kus,2007) [20] by include a shape parameter. This modification increased the distribution's versatility in modeling real-world data. This extension, known as the Generalized Exponential Poisson (GEP) distribution, features failure rates that demonstrate decreasing, increasing, and upside-down bathtub patterns. The CDF of GEP distribution can be formulated by (2). 1 ( ; , , ) [ ( ; , )] 1 exp 1 ; 0, 0, 0, 0

In a similar fashion, Gupta & Kundu (2001) [15] created Exponentiated exponential family and Cancho et al. (2011) [7] introduced a new distribution family derived from the exponential distribution, which is characterized by its increasing failure rate function. This distribution is known as the Poisson exponential (PE) distribution. Bakouch et al. (2012) [4] introduced the Exponentiated Exponential Binomial (EEB) distribution, blending characteristics from exponentiated exponential and binomial distributions. This distribution is capable of accurately representing lifetime data characterized by increasing, decreasing, and upside-down bathtub-shaped failure rates. Mahmoudi and Sepahdar (2013) [23] introduced a novel distribution known as the Exponentiated Weibull-Poisson (EWP) distribution. This distribution is defined by four specific parameters and shows different patterns of failure rates: increasing, decreasing, bathtub-shaped, and unimodal. Ristic & Nadarajah (2014) [29] developed Exponentiated exponential Poisson (EEP) distribution characterized by three distinct parameters, exhibiting decreasing, increasing and upside-down bathtub shaped failure rates depending upon the shape parameter values. The unconditional CDF of EEP distribution can be formulated as (3)

Furthermore, the known lifetime compounded distributions created with exponential distribution and various discrete probability distributions by various writers are Pareto Poisson-Lindley Distribution (Asgharzadeh et al.,2013) [3] , Exponentiated Exponential-Geometric Distribution (Louzada et al.,2014) [22] , Extended Poisson exponential distribution (Fatima & Roohi, 2015) [11] , Poissonodd generalized exponential family of distributions (Muhammad,2018) [27] , Poisson Nadarajah-Haghighi Distribution (Mansoor et al.,2020). Likewise, Chaudhary and Kumar (2020) [8] developed Poisson NHE distribution with exhibiting rising, constant and a diverse range of monotone failure rates. The CDF of PNHE distribution can be formulated as

In the same way, life time probability distributions developed by various authors are New Exponential Extension Poisson Distribution (Joshi & Kumar, 2021) [18] , Poisson Inverse NHE Distribution (Chaudhary & Kumar,2020) [8] , Exponential Intervened Poisson and distribution (Jayakumar & Sankaran, 2021) [15] , Poisson Extended Exponential Distribution (Maya et al.,2022). Sousa-Ferreira et al. (2023) [32] introduced a new three-parameter lifetime distribution, named the Extended Chen-Poisson Lifetime Distribution. This distribution is created by combining the Chen distribution with the zero-truncated Poisson distribution, allowing it to accommodate various complex hazard shapes. Telee and Kumar (2023) [34] recommended new lifetime distribution known as Inverse Exponentiated Exponential Poisson (IEEP) distribution taking inverse of random variable X in EEP of (3). The CDF of IEEP distribution can be expressed as

The hazard rate function (HRF curve) of IEEP distribution demonstrates a unimodal pattern, characterized by both monotonically increasing and monotonically decreasing segments as well as it displays an upside-down bathtub-shaped hazard rate. The concept of developing a new distribution entail introducing a flexible distribution that can be applied in diverse areas including health, engineering, demography, survival analysis, and reliability analysis. Its flexibility and broad applicability make it a promising option for a wide range of statistical and scientific investigations. Therefore, the primary objective of this paper is to develop a novel lifetime distribution and perform data analysis on the PM 2.5 air quality dataset from Kathmandu, Nepal. This innovative initiative aims to understand the current air quality situation in Kathmandu, Nepal. Air pollution is a significant environmental concern in many urban areas worldwide, and Kathmandu, the capital city of Nepal, is no exception. Nestled in a valley surrounded by the majestic Himalayan Mountains, Kathmandu faces a unique set of challenges when it comes to air quality. The city's topography, rapid urbanization, and a variety of pollution sources contribute to elevated levels of air pollutants, negatively impacting the health and well-being of its residents. Kathmandu's air pollution results from a combination of natural and anthropogenic factors. The city experiences increased pollution during the winter months, primarily due to meteorological conditions, which trap pollutants close to the ground. Moreover, the rapid urbanization and industrialization of the city have led to increased emissions of pollutants from vehicles, industries, and construction activities. Inefficient public transportation systems and a lack of proper waste management contribute to this problem. The consequences of this pollution are far-reaching, affecting not only the health of Kathmandu's residents but also the environment, including its rich biodiversity and cultural heritage. In Kathmandu, Nepal, the winter and pre-monsoon seasons are characterized by an atmospheric phenomenon called the inversion effect. This phenomenon impedes the dispersal of pollutants in the absence of rainfall. Consequently, concentrations of PM 2.5 , PM 10 , and TSP (Total Suspended Particles) rise dramatically during these periods. In addition, pollution levels experience a notable increase from March to April owing to heightened occurrences of forest fires nationwide during this period. According to a study conducted by (Lamichhane et al. ,2023) [20] , out of 309 days of air quality monitoring, PM 2.5 levels exceeded the National Ambient Air Quality Standards on 153 days. Particulate matter (PM) are microscopic particles or droplets in the air that can be inhaled into the lungs and affect human health and the environment. These particles originate from a range of sources, including industrial operations, automobile emissions, and natural sources like dust and pollen. The size of particulate matter is a critical factor in its impact on health, as smaller particles can penetrate deeper into the respiratory system. PM, or particulate matter, is divided into various fractions according to size. Among these fractions, PM 10 (particles with a diameter of 10 micrometres or less) and PM 2.5 (particles with a diameter of 2.5 micrometres or less) are particularly worrisome due to their ability to potentially cause respiratory and cardiovascular issues, as documented by (WHO,2023). Shrestha (2021) [31] reported that the yearly average of PM 2.5 in Nepal dropped from 37.8µg/m 3 in 2019 to 35.8 µg/m 3 in 2020. However, in 2021 (unexpectedly and significantly), the average rose to 56.5 µg/m3 (through May). This increase was primarily caused by widespread forest fires that occurred throughout Nepal, peaking at the end of March. The simulated 24-hour average PM 2.5 concentrations range from 30 μg/m3 to 65 μg/m3. These simulated PM 2.5 values exceed the World Health Organization (WHO) standard of 25 μg/m3 for a 24-hour mean. The PM 2.5 levels in the air of Kathmandu are notably elevated, particularly during the dry winter month of December. According to Tuladhar et al. (2021), the urban air quality in Kathmandu is particularly harmful during this time period. A numerical scale of the Air Quality Index (AQI) lies between 0 and 500. As the AQI number goes up, air pollution gets worse, and the health risks also increase. For instance, an AQI of 50 or below means the air quality is excellent, whereas a reading over 300 indicates extremely hazardous. World Health Statistics report of 2022 says that most people worldwide breathe air that surpasses WHO guidelines for air quality. More than 20% of deaths from cardiovascular diseases are caused by the harmful effects of air pollution, totalling over 3.5 million deaths each year. The WHO reports an increasing number of people exposed to air pollution, especially in low and middle-income countries like Nepal, which strengthens the evidence linking air pollution to health problems in these areas. Chaudhary et al. (2024) [10] created innovative modal called a New Extended Kumaraswamy Exponential Distribution for conducting data analysis of air quality conditions of Kathmandu, Nepal. Banerjee et al. (2024) used statistical modelling to investigate the relationship between air pollution and climatic parameters in the city of Asansol, West Bengal, India. The following framework is employed to organize the different components of this study. In Section 2, we introduce the Modified Inverse Exponentiated Exponential Poisson (MIEEP) distribution, elucidating its mathematical and statistical characteristics. Section 3 explores estimation technique namely maximum likelihood (MLE). In Section 4, a dataset on air quality collected in Kathmandu, Nepal in 2021, specifically focusing on PM 2.5 values, is analyzed . Based on this dataset, we estimate model parameters and demonstrate several criteria for assessing the goodness of fit of the suggested model. In section 5, we conduct time series analysis of air quality data of P 2.5 in Ratnpark station of Kathmandu, Nepal. Section 6 has closing remarks. The R programming language is used to do numerical computations, contributing to a planned method that guides readers through a clear and well-organized framework. This approach enhances understanding and appreciation of the research methodology and results.

Formulation of Modified Inverse Exponentiated Exponential Poisson distribution

In order to create a more versatile probability model, we've incorporated an additional shape parameter, denoted as θ, into (5) of the Cumulative Distribution Function (CDF) of the Inverse Exponentiated Exponential Poisson (IEEP) distribution as described by (Telee & Kumar, 2023) [34] . This enhanced probability model is referred to as the Modified Inverse Exponentiated Exponential Poisson distribution with three parameters (α, β, λ, θ) > 0, represented as MIEEP. Consequently, its CDF is expressed by (6). ;,,,) 1 1 exp 1 ; 0, 0, 0, 0, 0

The associated PDF of equation ( 6) is included as

Survival function (S(x))

The equation (8) represents the survival function of the proposed model MIEEP.

Hazard rate function (h(x))

The hazard rate represents the rate at which failures occur instantly, as defined by equation (9).

Reversed hazard rate function

The formula (10) defines the model's reversed hazard rate, h rev (x).

Cumulative hazard rate Expression (11) is the function for cumulative hazard rate H(x) =ln ( )

Quantile function

Quantile function for the MIEEP is defined in (12).

Asymptotic behavior

The asymptotic behavior of the density function may be tested by demonstrating that 0 lim ( ) lim ( )

If the model fits the asymptotic conditions, the modal value will be clearly specified. Considering limits at the endpoints,

, suggesting that the mode of the model will exist.

Skewness

Al-Saiary et al. (2019) [2] gave quartile-based of skewness as

Kurtosis

The Octiles coefficient of Kurtosis, as defined by (Moors, 1988), can be computed using the following formula:

Methods for Estimation of model constants

Method of Maximum likelihood estimation (MLE) approach for estimating the parameters. This estimation method depends on maximizing the log likelihood function of the model. Let's say we have a random sample of 'n' observations from MIEEP, represented by

x .The log likelihood function is formulated as follows:

( , , , | ) log( ) log(1 ) 1 exp 1 1

To find the first and second order partial derivatives of the log-likelihood function, differentiate equation ( 13) with respect to α, β, λ, and θ. By setting the first-order derivatives to zero and solving for the parameters of the proposed model, we can estimate their values. However, in cases where obtaining a solution for the first-order partial derivatives is not feasible, one can employ computer programming to solve the system of nonlinear equations.

Real Data Analysis

To assess the model's adaptability, we [20] is legal standards for air quality as Nepalese government. These guidelines provide specific upper limits of some parameters (table1) [20] . Status of Air Quality in Nepal, Annual Report, 2021. Department of Environment, Kathmandu, Nepal. 2 reveals that the mean P2.5 value at Ratnapark Station is higher than the daily median value. Figure 3 4 presents the histogram alongside the fitted PDF using the MLE estimation approach, in addition to the TTT plot of the suggested model. 2.475328 0.051073 We also created P-P and Q-Q plots for the suggested model to assess its validity, which are shown in figure 6. We compared five previously published probability models: the Lindley Inverse Weibull (LIW) distributions (Joshi & Kumar, 2020) [16] , the Weibull Extension (WE) distribution (Tang et al., 2003) [33] , the Generalized Weibull Extension (GWE) distribution (Sarhan & Apaloo, 2013) [30] , the Generalized Exponential Extension (GEE) distribution (Lemonte, 2013) [21] and the Half Logistic Nadarajah Haghighi (HLNHE) distribution (Joshi & Kumar, 2020) [16] .

Table 5 displays the parameters of the models under consideration, which were computed using MLE. Table 5 also shows the SE for each of the distributions analyzed. Table 6 shows the information criteria values for MIEEP, as well as the models under consideration. Because the information criterion values for the suggested model are lower than those for the examined model, this suggests that the data fits the suggested model better than the examined models. We also exhibited the histogram and fitted the density curve for the MIEEP and other models under consideration. Figure 7 shows the model's ecdf as well as the fitted distribution functions for the other models investigated.

Time series Analysis of air pollution data of PM2.5

Pollution seems to be time series data. It generally depends on the seasons. In this section of the study, we have analyzed the PM 2.5 data using time series analysis and make some forecasting also. We have divided the whole data into four quarters. Five-year average of PM 2.5 value is maximum in first quarter equal to 66.164 and minimum equal to 14.072 in fourth quarters as seen in Figure 8. During the five years, 51.224 is maximum value in 2020 whereas 34.322 is minimum value in 2019 (Figure 8).

As data is time series data, so data may have trend, seasonal variation, cyclic variations or, random variations. To study the different variations and trend, we have plotted the graph of observed values in five years shown in left side of figure 9 and observed versus fitted curves in right side. We can observe from the left time series of PM 2.5 that there appears to be seasonal fluctuation every year. That is, there is a peak every first quarter, and a trough every fourth quarter. An additive model could be employed to interpret this time series because the seasonal fluctuations seem consistent in size over time, independent of the series' level, and the random fluctuations also appear relatively stable over time. Since data seems to be seasonal, it may contain seasonal, trend and irregular component. We have decomposed these components. The "decompose ()" function in R can be used to estimate the trend and seasonal components. The tables 9,10 and 11 contains estimated values for the seasonal, trend, and irregular components respectively. The estimated seasonal factors from first to fourth quarter are same across every year. The highest seasonal factor is 22.99534 for quarter 4 and smallest seasonal factor is -30.09482 for third quarter indicating that there seems to be peak in PM 2.5 in fourth quarter and lowest in third quarter. We also examined whether the forecast errors exhibit consistent variance over time and follow a normal distribution with a nonzero mean. To assess this, we plotted the forecast errors and a histogram on the right side of Figure 11. The analysis indicates that the predicted errors are most likely normally distributed with a mean of zero. Furthermore, the Ljung-Box test suggests minimal evidence of non-zero autocorrelation in the forecast errors over the sample period. This reinforces the observation that the forecast error distribution approximates normality with a mean of zero. Therefore, these findings imply that the simple exponential smoothing method offers a suitable predictive model for the PM 2.5 data spanning from 2017 to 2021.

Conclusion

In this study, a novel and versatile probability distribution known as the Modified Inverse Exponentiated Exponential Poisson distribution has been introduced. We thoroughly examined various statistical properties of this innovative model. To evaluate the appropriateness of our model, we analyze a real dataset that includes P 2.5 air quality measurements obtained from the Ratnapark station in Kathmandu, Nepal, over the course of 2021.We performed data analysis to estimate parameters, validate models, and compare models for P 2.5 data. However, concentrations of P 2.5 were at their lowest during the monsoon season and peaked during the winter season, indicating seasonal fluctuations. We used a variety of diagnostic methods to validate our model, including P-P and Q-Q graphs. In addition, we evaluated metrics such as log-likelihood values, the Corrected Akaike information criterion, Hannan-Quinn information criteria, Akaike information criterion and Bayesian information criterion. To assess the model's suitability and accuracy, we performed Anderson-Darling, Cramer-Von Mises, and Kolmogorov-Smirnov tests. We have also conducted the time series analysis of air pollution data of P 2.5 during years 2017-2021 in Ratnapark of Kathmandu, Nepal. Based on our findings, the air quality in Kathmandu, Nepal was consistently poor, except during the COVID-19 pandemic. Additionally, we determined that the simple exponential smoothing method effectively predicts PM 2.5 levels from 2017 to 2021.

Acknowledgement

We have also forecasted for future times not in our time series data. Original data is from first quarter 2017 to fourth quarter 2021, all together 20 quarters. We have forecasted for first quarter 2022 to fourth quarter 2030 (36 more quarters). Forecasted values are tabulated below in table12.

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