### 1. Introduction

^{th}. India rules the list with 22 of the worst 30 cities globally [1]. Air pollution is an insidious destroyer of the country’s health and wealth. With every inhale, microscopic particles get deep into the lungs. Air pollution is the major cause of cancers, strokes, and heart disease [2]. In recent studies, it is found that pollution is stunting children’s growth and development [3, 4]. The total number of cases is likely to rise in the coming years if pollution is not regulated. Moreover, additional deaths as well as disabilities will be introduced [5]. Air pollution has reduced the average life expectancy to 1.8 years. $225 billion is the estimated financial cost in lost labor and a thousand times more cost in healthcare. Rising pollution and its severity have forced researchers and scientists to carry out various studies involving its health effects and future trends. Air pollution dynamics is a complex process due to randomness involved in air pollutants behavior. Adaptive Neuro-Fuzzy Inference System (ANFIS) is effective in dealing with non-linear real-time problems. ANFIS has been widely used in disaster management, rock engineering [6, 7] healthcare sector, finance, and many other real-time fields [8–10]. ANFIS deals with both regression and classification problems [11].

_{2.5}, oxides of nitrogen (NO, NO

_{2}, NO

_{x}) and sulphur dioxide are studied. The daily (24-h average) air pollutant concentrations are obtained from Central Pollution Control Board. PM

_{2.5}(

*μ*g/m

^{3}) concentrations are observed from March 2015 to June 2019 and NO (

*μ*g/m

^{3}), NO

_{x}(ppb), SO

_{2}(

*μ*g/m

^{3}), NO

_{2}(

*μ*g/m

^{3}) are observed from January 2010 to June 2019. The model developed is only dependent on the past values of the respective air pollutant. The study is independent of other variables. The authors have not encountered such work to the best of their knowledge. The article is organized into four sections explaining the problem undertaken and the data collected, methodology, results, and the last section concludes the work carried out.

### 2. Methodology

### 2.1. Study Area and Dataset

_{2.5}), oxides of nitrogen, and sulphur dioxide from January 2010–June 2019 was collected from CPCB for the current analysis. Fig. 1 depicts the areas monitored by CPCB in Delhi, India, and the study area-Shadipur.

### 2.2. Wavelet Transform

### 2.3. Adaptive-Neuro Fuzzy Inference System

#### 2.3.1. Particle swarm optimization

^{D}in PSO is based on position vector q and velocity vector w of the particle. For each iteration, velocities of all variables are modulated based on inertia weight (W), cognitive (d

_{1}), and social (d

_{2}) acceleration. For (n+1)

^{th}epoch, velocity and position are updated as:

##### (1)

$$\begin{array}{l}{w}_{iD}^{n+1}=W{w}_{iD}^{n}+{d}_{1}{s}_{1}^{n}({q}_{iD}^{n}-{y}_{iD}^{n})+{d}_{2}{s}_{2}^{n}({q}_{iD}^{n}-{y}_{iD}^{n})\hfill \\ {y}_{iD}^{n+1}={y}_{iD}^{n}+{w}_{iD}^{n+1}\hfill \end{array}\}$$*s*

_{1},

*s*

_{2}∈

*U*(0,1) and

*d*

_{1}+

*d*

_{2}≤ 4. Consider the particle’s best position as pbest. Compare the current position of the particle with that of pbest. If the current position is better than pbest, then pbest is the current position otherwise pbest is the best global position (gbest). The optimal values for tuning ANFIS parameters in this study are taken as W = 1, d

_{1}= 1, and d

_{2}= 2 using the trial and error process. The maximum iteration for all the algorithms is taken as 1000.

#### 2.3.2. Genetic algorithm

#### 2.3.3. Ant colony optimization

##### (2)

$$P(s,v)=\{\begin{array}{ll}\underset{v\in J(s)}{\text{argmax}}\left\{{[\eta (s,v)]}^{\alpha}{[\eta (s,v)]}^{\beta}\right\},\hfill & if\hspace{0.17em}r\le {r}_{0}\hfill \\ R\hfill & \mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a},otherwise\hfill \end{array}$$*η*(

*s*,

*v*) represents pheromone (desirability of (s,v)) on edge (s,v). r is the parameter that governs the relative value of desired, r

_{0}is initialized with 0 ≤ r

_{0}≤ 1, r belongs to rand([0,1]). r

_{0}= 0.5 for carrying out the present study. J(r) is the set of edges available at point r of decision. S is a random variable chosen according to the probability distribution given below:

##### (3)

$$P(s,r)=\{\begin{array}{ll}\frac{{[\eta (s,v)]}^{\alpha}{[\eta (s,v)]}^{\beta}}{{\displaystyle \sum _{v\in J(s)}}{[\eta (s,v)]}^{\alpha}{[\eta (s,v)]}^{\beta}},\hfill & if\hspace{0.17em}r\in J(s)\hfill \\ 0\hfill & ,otherwise\hfill \end{array}$$*η*(

*s*,

*v*) ← (1−

*ρ*)

*η*(

*s*,

*v*)+

*ρη*

_{0}, 0<

*ρ*<1

*ρ*is the pheromone evaporation coefficient representing. The global modified amount when all ants arrived at the destination is given by

*δ*is the global evaporation coefficient parameter, and the last term is the increase in desirability [18].

#### 2.3.4. Firefly algorithm

_{0}and v(0)=V

_{0}, and α is the light absorption coefficient. d is defined as [20, 21]:

_{j}and y

_{k}are fireflies positions j and k. Firefly is captivated by yet brightness-based firefly. The movement of the firefly is given by

*ηɛ*

*is the random movement in case of absence of brightness,*

_{j}*η*varies between 0 and 1 and the second term in the equation is the captivation factor with coefficient as

*β*

_{0}. FFA is used to tune the premise parameter of ANFIS with optimal values of coefficients as α=1 and β

_{0}=2, using trial and error process.

#### 2.3.5. Differential evolution

#### 2.3.6. Proposed algorithm

x(t) at time t is considered where x(t) represents daily (24 h) concentration of air pollutants on day t.

x(t) is decomposed using the wavelet transform as mentioned in section 2.3.1 using high and low filters. x(t) is the sum of a

_{n}(approximation at level n) and d_{1},d_{2},. . . .,d_{n}(details at level n). Daubechies wavelet(db5) is considered in the present study due to its property to extract fluctuations nicely.Instead of x(t), the smoothed decomposed series are used for further analysis. Let y(t) denote the approximation at level 5 i.e. y(t) = a

_{5}(t).The autocorrelation function (ACF) for y(t) is computed as described by Mohammad et al. [24] to find out the dependence of y(t) on lag values. Let the optimized lag is τ. The input set is past τ values of the series and the output value is y(τ+1).

The series is divided into training (70%) and testing (30%) datasets.

The parameters as described in section 2.3 are obtained. Using the algorithm described in sections 2.3.1–5, the premise and consequent parameters are trained.

The parameters are trained to obtain the optimized error. The simulations are carried using Matlab R2019a software.

Steps (4)–(7) are carried out for d

_{1},d_{2},. . . .,d_{n}. The final output is obtained adding the trained values of approximation and details. The test dataset is simulated using the model. Further effectiveness of the method is verified using trained and tested datasets.

### 2.4. Evaluation Criterion

_{O}(t) and y

_{p}(t) w.r.t time t) for training and testing datasets are considered. The determinism coefficient (R

^{2}) defines the relation between the aforementioned pairs. The parameters for determining errors are based on various values, and they primarily combine absolute and relative errors. Here are some of the error measurements:

##### (9)

$$\begin{array}{l}{R}^{2}=1-\frac{{\displaystyle \sum _{t=1}^{m}}{({y}_{o}(t)-{y}_{P}(t))}^{2}}{{\displaystyle \sum _{t=1}^{m}}{({y}_{o}(t)-\frac{1}{m}{\displaystyle \sum _{t=1}^{m}}{y}_{o}(t))}^{2}}\hfill \\ Mean\hspace{0.17em}Absolute\hspace{0.17em}Error,MAE=\frac{{\displaystyle \sum _{t=1}^{m}}|{y}_{o}(t)-{y}_{P}(t)|}{m}\hfill \\ Mean\hspace{0.17em}Absolute\hspace{0.17em}Percentage\hspace{0.17em}Error,MAPE=\frac{{\displaystyle \sum _{t=1}^{m}}\left|\frac{{y}_{o}(t)-{y}_{P}(t)}{{y}_{o}(t)}\right|}{m}\times 100\hfill \\ Root\hspace{0.17em}Mean\hspace{0.17em}Square\hspace{0.17em}Error,RMSE=\sqrt{\frac{1}{m}\sum _{t=1}^{m}{({y}_{o}(t)-{y}_{P}(t))}^{2}}\hfill \end{array}\}$$##### (10)

$${PM}_{2.5}\hspace{0.17em}Air\hspace{0.17em}Quality\hspace{0.17em}Subindex=\{\begin{array}{l}\frac{X*50}{30},0<X\le 30\hfill \\ 50+\frac{(X-30)*50}{30},30<X\le 60\hfill \\ 100+\frac{(X-60)*100}{30},60<X\le 90\hfill \\ 200+\frac{(X-90)*100}{30},90<X\le 120\hfill \\ 300+\frac{(X-120)*100}{130},120<X\le 250\hfill \\ 400+\frac{(X-250)*50}{130},X>250\hfill \end{array}$$_{2.5}concentration in

*μ*g/m

^{3}.

##### (11)

$${SO}_{2}\hspace{0.17em}Air\hspace{0.17em}Quality\hspace{0.17em}Subindex=\{\begin{array}{l}\frac{X*50}{40},0<X\le 40\hfill \\ 50+\frac{(X-40)*50}{40},40<X\le 80\hfill \\ 100+\frac{(X-80)*100}{300},80<X\le 380\hfill \\ 200+\frac{(X-380)*100}{420},380<X\le 800\hfill \\ 300+\frac{(X-800)*100}{800},800<X\le 1600\hfill \\ 400+\frac{(X-1600)*50}{800},X>1600\hfill \end{array}$$_{2}concentration in

*μ*g/m

^{3}.

##### (12)

$${NO}_{x}\hspace{0.17em}Air\hspace{0.17em}Quality\hspace{0.17em}Subindex=\{\begin{array}{l}\frac{X*50}{40},0<X\le 40\hfill \\ 50+\frac{(X-40)*50}{40},40<X\le 80\hfill \\ 100+\frac{(X-80)*100}{100},80<X\le 180\hfill \\ 200+\frac{(X-180)*100}{100},180<X\le 280\hfill \\ 300+\frac{(X-280)*100}{120},280<X\le 400\hfill \\ 400+\frac{(X-400)*100}{120},X>400\hfill \end{array}$$_{x}concentration in ppb(parts per billion).

### 3. Results and Discussions

_{2}, PM

_{2.5}, and SO

_{2}which play a significant role in determining the air quality. Some of the important work is shown in table 1. The problem was previously dealt with using ANFIS modeling but the large error between the predicted and observed value was observed (refer to Table 1). The drawback found in classic ANFIS modeling was due to the gradient descent (GD) method. It is observed that in classic ANFIS modeling, the GD algorithm gets trapped in local minima. The tuning of premise parameters was inadequate leading to inadequate predictions. Moreover, too many parameters are also one of the reasons for erroneous prediction. The present work is an attempt to minimize the parameters at the same time improving the air quality prediction. In the current scenario, oxides of nitrogen, particulate matter, and sulphur dioxide are prominent pollutants. Though the air quality index is dependent on NO

_{X}, PM

_{2.5}, and SO

_{2}, NO and NO

_{2}are also predicted in the present study as both have a significant correlation with NO

_{X}. The air quality index is dependent on the prominent air quality sub-index corresponding to the pollutant. The proposed hybrid models were validated for the Shadipur area of India. The hybrid model combines the decomposition filter, fuzzy inference system, algorithm for optimizing parameter, and neuronal networks.

_{2}, NO

_{x}, and PM

_{2.5}proposed Wavelet Transform-Adaptive Neuro-Fuzzy Inference System-Particle Swarm Optimization (WANFIS-PSO) gives better results compared to other models. The genetic algorithm is very much close to PSO. The next step for the best model was to check the computation time.