Improving parameters estimation of a truncated Poisson regression model based on meta-heuristic optimization algorithms

The paper discusses computational and numerical challenges that are associated with the truncation of the information and which change the usual Poisson likelihood by the introduction of black kite optimization algorithm. Real data is used to demonstrate a significant improvement in healthcare and medical research. Truncated Poisson regression models (TPRM) are essential for analyzing count data where zero counts are unobserved, a common scenario in many real-world applications, such as healthcare and medical research. However, parameter estimation in such models often suffers from bias and inefficiency due to the complexity induced by truncation. This study proposes an improved parameter estimation approach for TPRM by leveraging meta-heuristic optimization algorithms. Specifically, we integrate state-of-the-art meta-heuristics, black kite optimization algorithm (BKA), to optimize the likelihood function and overcome the limitations of traditional iterative methods such as Newton-Raphson and quasi-Newton algorithms. Using extensive Monte Carlo simulations and real data application, we evaluate the performance of the proposed method under varying sample sizes and covariate structures. The results demonstrate that our meta-heuristic-based estimator significantly reduces mean squared error compared to conventional estimators, enhancing model reliability and predictive accuracy. The proposed approach offers a robust and efficient alternative for parameter estimation in truncated Poisson regression, with potential applications in epidemiology, ecology, and other fields dealing with truncated count data.

1 Introduction

Count data modeling is a statistical field of study directed at such variables as dependent variables that are identifications of a non-negative integer number of events or occurrences, i.e., the quantity of visits to a doctor, traffic incidents, or children within a family [1, 2]. Leading to the right skewed distribution, among other characteristics unlike continuous outcomes of count data are discrete that tend to have a distorted distribution centered on lower values with a high proportion of zeros [3].

Poisson regression, a classical modeling strategy of the count data, presupposes the idea of the counts being distributed according to Poisson distribution with its means (equidispersion). This model presents the expected count as an exponential of explanatory variables that makes its interpretation to be simple in the generalized linear model framework. Empirical data however, generally do not fit the equidispersion assumption, therefore assuming they are over dispersed, which is very common, or under dispersed, which is rare. Such cases are dealt with by substituting such models with newer models with more parameters such as the Negative Binomial regression that has an extra parameter to fit flexibility to variance independent of the mean. Other distributions intended to handle different dispersion are the Generalized Poisson Conway-Maxwell-Poisson (COM-Poisson) and similar distributions [4, 5].

The other difficulty which is commonly seen is the occurrence of excess zeros; that is more counts than are predicted by standard count models. At the cost of this, zero-inflated models and hurdle models have been designed. The two components have independent structure each component modeling the result on the binary zero-non zero and the count distribution of the positive values, and thus generates better fitting and inference in situations where zeros are produced by separate mechanisms [6, 7].

The truncated Poisson regression model is a statistical technique used to analyze a count-type of data that are truncated; such as dynamic observations are excluded in the sample when smaller or larger than some integer limits [8, 9]. Truncated Poisson regression contrasts with the usual Poisson regression model in that the counts take a truncated distribution due to the requirement that the counts exceed (left truncation) or are less than (right truncation) a given value [10]. This method would consider the fact that the realized counts do not fall into the grand scope of counts that might be observed hence avoid bias in parameter estimation and wrong inference that would have been made had truncation not been considered [11, 12].

Optimization means achieving the best result under certain conditions. Engineers must make many technical and managerial decisions throughout the design, construction, and upkeep of any engineering system. The main purpose of all these decisions is to either minimize the effort or to maximize the benefit. In any situation, the effort required or the amount of benefit can be defined as a function of certain decision variables. Optimization is then the process of choosing the conditions that result in the highest or lowest value of a function [13, 14].

There is no one way that can solve all optimization problems efficiently. Due to this, various optimization methods have been invented to handle many types of optimization problems. These methods are also called mathematical programming and are typically included in the field of operations research. Operations research uses scientific methods to solve problems related to decision making and to find the best solutions [13, 14]. In recent years, new optimization methods have gained popularity and are widely used to solve difficult engineering optimization problems. Such as genetic algorithms, simulated annealing, particle swarm optimization, ant colony optimization, black-winged kite optimization algorithm [13].

The black kite optimization algorithm (BKA) is a meta-heuristic optimization algorithm based on the migration and hunting patterns of the black kite. The BKA brings together the Cauchy mutation technique and the Leader strategy to increase its ability to find global solutions and speed up convergence. This method manages achieves a balance both global and local knowledge to find good solutions [15]. It should be noted that not all problems can be solved by one algorithm. There is no meta-heuristic algorithm that is optimal in all optimization problems. A meta-heuristic algorithm can do very well with certain types of issues, but may not be as successful with others. The advancement of technology and the rise in problem complexity make it hard for some traditional algorithms to solve them [15].

The key shortcoming of parameter estimation in the truncated Poisson regression models can be viewed as the complex effect of the likelihood that is induced by truncation as it may necessitate the employment of the non-linear optimization tools and cause other problems with acquiring consistent and efficient estimators. Truncated Poisson models can be estimated using maximum likelihood in which the likelihood functions are complex with the normalizing constant being dependent upon the truncation limits. That increases the computational burden of parameter estimation and frequently requires non-linear optimization techniques iterative in nature, which have the potential to converge slowly or at local as opposed to global optima [16].

The weaknesses of quasi-Newton algorithms in parameter estimation of the truncated Poisson regression model mostly follow these issues because of the difficulty of the truncated likelihood function, which influence convergence and speed of computation. In the truncated Poisson measuring and computing log-likelihood, the log-likelihood contains a normalizing constant that relies on the truncation that results in the likelihood function to be non-linear and consequently more strenuous compared with that in the conventional Poisson. Quasi-Newton optimization methods do not compute the Hessian explicitly but can become sensitive to this complexity and thus slow to converge or to find global (as opposed to local) maxima. Quasi-Newton optimization can be sensitive to starting parameter values, and is iterative, whereas truncation or small sample size limits often render it insensitive to such start values. Such sensitivity may induce instability or convergence to irreliable parameter estimates [17, 18].

The paper therefore concerns the difficulty of causing the parameters of the truncated Poisson regressions to get the accurate estimation of parameters based on maximum likelihood approach algorithms. In particular, the paper discusses computational and numerical challenges that are associated with the truncation of the information and which change the usual Poisson likelihood by the introduction of black kite optimization algorithm. The complexity typically gets in the way of rapid convergence, arithmetical precariousness and can be biased when using traditional optimization schemes like Newton-Raphson, Gauss-Newton, or perhaps quasi-Newton procedures.

2 Poisson regression

The modeling of count data is based on Poisson regression. It was the earliest model to explicitly model counts and it remains at the foundation of the numerous kinds of count models that can be used by analysts, it is also called log-linear regression. The dependent variable in this model is Poisson distributed and the dependent variable is linked with the independent variables using a log link function which yields a linear equation [4, 6].

The Poisson regression model states that the y_i is a random variable distributed as Poisson with parameter λ_i, which depends on the regressors x_i. The main formula of the model is [19]:

$\begin{array}{ll}P\left(Y_i=y_i|x_i\right)=\frac{e^{-\lambda }{\lambda }^{y_i}}{y_i!},y_i=0,1,2\dots ,& (1)\end{array}$

The most popular formulation for λ_i is the loglinear model

$\begin{array}{ll}ln{\lambda }_i=x_i^T\beta & (2)\end{array}$

where $x_i^T\beta ={\beta }_0+{\beta }_1{x}_{i1}+\dots +{\beta }_n{x}_{in}$, $x_i^T$ represents the vector of independent variables and β is the regression coefficients. The expected number of events is given by

$\begin{array}{ll}E\left[y_i|x_i\right]=Var\left[y_i|x_i\right]={\lambda }_i=e^{x_i^T\beta }& (3)\end{array}$

The log likelihood function is

$\begin{array}{ll}lnL={\displaystyle \sum _{i=1}^n}\left[-{\lambda }_i+y_i{x}_i^T\beta -ln{y}_i!\right]& (4)\end{array}$

3 Truncated Poisson distribution

Let y a discrete random variable follow Poisson distribution with mean and variance (λ) then the probability mass function of Poisson distribution is [19, 20]:

$\begin{array}{ll}P\left(y_i;\lambda \right)=\frac{e^{-\lambda }{\lambda }^{y_i}}{y_i!},y_i=0,1,2,\dots ,\lambda >0& (5)\end{array}$

A sample is considered truncated if the observations are limited to a certain portion of the population distribution. The truncated distribution is a subset of the untruncated distribution, with truncation occurring either from the left side [left truncated (y > l)] at a point, l, from the right side [right truncated (y < k)] at a point, k, or from both sides within the interval [l, k]. The probability density function of the shortened random variable can be described as a condition a distribution, as demonstrated below [19, 21]:

Case (1): left truncated Poisson at zero (y_i > 0)

The zero-truncated Poisson distribution (ZTP) was first introduced by David and Johnson [36], which is distributed as y ~ ZTP(λ) and it is one of the models of logarithmic linear regression. The probability mass function of the zero truncated Poisson (ZTP) distribution is [3, 22, 23]:

$\begin{array}{l}P\left(y_i|y_i>0\right)=\frac{P\left(y_i;\lambda \right)}{Pr\left[y_i>0\right]}\\ =\frac{\frac{e^{-\lambda }{\lambda }^{y_i}}{y_i!}}{1-Pr\left[y=0\right]}\\ =\frac{e^{-\lambda }{\lambda }^{y_i}}{\left(1-e^{-\lambda }\right)y_i!},y_i=1,2,3,\dots & (6)\end{array}$

Case (2): right truncated Poisson (y_i ≤ k)

The Poisson distribution becomes right-truncated when truncation occurs at (k) where (y_i ≤ k). The probability mass function of the right truncated Poisson distribution takes the following form [24, 25]:

$\begin{array}{l}P\left(y_i|y_i\le k\right)=\frac{P\left(y_i;\lambda \right)}{Pr\left[y_i\le k\right]}\\ =\frac{e^{-\lambda }{\lambda }^{y_i}}{y_i!\left({\displaystyle {\sum }_{z=0}^k}\frac{e^{-\lambda }{\lambda }^z}{z!}\right)}\\ =\frac{{\lambda }^{y_i}}{\left({\displaystyle {\sum }_{z=0}^k}\frac{{\lambda }^z}{z!}\right)y_i!},y_i=0,1,2,\dots ,k& (7)\end{array}$

Case (3): double truncated Poisson (l ≤ y_i ≤ k)

The Double truncated Poisson data result from merging left truncated and right truncated Poisson data types. The probability mass function of the double truncated Poisson distribution takes the following form [26].

$\begin{array}{l}P\left(y_i|l\le y_i\le k\right)=\frac{P\left(x_i;\lambda \right)}{Pr\left[l\le y_i\le k\right]}\\ =\frac{{\lambda }^{y_i}}{\left({\displaystyle {\sum }_{z=l}^k}\frac{{\lambda }^z}{z!}\right)y_i!},y_i=l,l+1,l+2,\dots ,k& (8)\end{array}$

The analysis of right truncation and double truncation for count data has received less scholarly focus compared to left truncation. One possible explanation is that Left-truncation occurs more frequently than right-truncation [26].

4 Truncated Poisson regression model

The truncated Poisson regression model is one of the models of logarithmic linear regression for the dependent variable (y_i) and is defined by the following formula [19, 21, 27, 28]:

$\begin{array}{ll}{y}_i=e^{x_i^T\beta +U_i}U~P\left(\lambda \right)& (9)\end{array}$

$where{x}_i^T\beta ={\beta }_0+{\beta }_1{x}_{i1}+\dots +{\beta }_n{x}_{in}$

$\begin{array}{l}{y}_i~P\left(\lambda \right)\end{array}$

The distribution parameter of the response variable (y_i) can be expressed as [29–31]:

$\begin{array}{r}{\lambda }_i=e^{x_i^T\beta }\\ ln{\lambda }_i=x_i^T\beta \end{array}$

In The truncated Poisson regression model the observations of (y_i, x_i) are obtained only for part of the population. The main goal of regression analysis involves parameter estimation to understand the relationship between dependent variable and independent variables. The maximum likelihood estimator serves to calculate parameter estimates for truncated Poisson regression models.

5 Maximum likelihood estimation (MLE)

The estimation of parameters represents a fundamental research topic that attracts mathematical statistics researchers, because new estimation methods require accurate parameter estimation and optimal estimator identification [32].

Case (1): zero truncated Poisson regression model

The maximum likelihood function for the zero truncated Poisson regression model derives from the conditional probability function of the zero truncated Poisson distribution shown in Equation 6.

$\begin{array}{l}L\left(\beta |y\right)={\Pi }_{i=1}^n\frac{e^{-\lambda }{\lambda }^{y_i}}{\left(1-e^{-\lambda }\right)y_i!}\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_{i}ln\left(\lambda \right)-\lambda -ln\left({1-e}^{-\lambda }\right)-ln\left(y_i!\right)\right]\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_i\beta -e^{x_i^T\beta }-ln\left({1-e}^{-x_i^T\beta }\right)-ln\left(y_i!\right)\right]& (10)\end{array}$

The zero truncated Poisson maximum likelihood estimators require the derivative of Equation 10 with respect to β to obtain their values as:

$\begin{array}{ll}\frac{\partial lnL}{\partial \beta }={\displaystyle \sum _{i=1}^n}\left[y_i-\frac{e^{x_i^T\beta }}{\left(1-e^{-e^{x_i^T\beta }}\right)}\right]x_i=0& (11)\end{array}$

Case (2): right truncated Poisson regression model

The maximum likelihood function for the right truncated Poisson regression model derives from the conditional probability function of the right truncated Poisson distribution shown in Equation 7.

$\begin{array}{l}L\left(\beta |y\right)={\Pi }_{i=1}^n\frac{{\lambda }^{y_i}}{\left({\displaystyle {\sum }_{z=0}^k}\frac{{\lambda }^z}{z!}\right)y_i!}\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_{i}ln\left(\lambda \right)-ln\left(y_i!\right)-ln\left({\displaystyle {\sum }_{z=0}^k}\frac{{\lambda }^z}{z!}\right)\right]\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_i\beta -ln\left(y_i!\right)-ln\left({\displaystyle {\sum }_{z=0}^k}\frac{{\left(x_i^T\beta \right)}^z}{z!}\right)\right]& (12)\end{array}$

We derive maximum likelihood right truncated Poisson regression estimators by taking the first derivative of β and setting it to zero according to the following:

$\begin{array}{ll}\frac{\partial lnL}{\partial \beta }={\displaystyle \sum _{i=1}^n\left[y_i-\frac{{\displaystyle {\sum }_{z=0}^k\frac{x_{i}z{(x_i^T\beta )}^z}{z!}}}{{\displaystyle {\sum }_{z=0}^k\frac{{(x_i^T\beta )}^z}{z!}}}\right]}=0& (13)\end{array}$

Case (3): double truncated Poisson regression model

The maximum likelihood function for the double truncated Poisson regression model derives from the Equation 8.

$\begin{array}{l}L\left(\beta |y\right)={\Pi }_{i=1}^n\frac{{\lambda }^{y_i}}{\left({\displaystyle {\sum }_{z=l}^k}\frac{{\lambda }^z}{z!}\right)y_i!}\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_{i}ln\left(\lambda \right)-ln\left(y_i!\right)-ln\left({\displaystyle \sum _{z=l}^k}\frac{{\lambda }^z}{z!}\right)\right]\\ lnL\left(\beta |y\right)={\displaystyle \sum _{i=1}^n}\left[y_i\beta -ln\left(y_i!\right)-ln\left({\displaystyle \sum _{z=l}^k}\frac{{\left(x_i^T\beta \right)}^z}{z!}\right)\right]& (14)\end{array}$

The maximum likelihood double truncated Poisson regression estimators can be determined differentiating Equation 14 with respect to β, giving

$\begin{array}{ll}\frac{\partial lnL}{\partial \beta }={\displaystyle \sum _{i=1}^n\left[y_i-\frac{{\displaystyle {\sum }_{z=l}^k\frac{x_{i}z{(x_i^T\beta )}^z}{z!}}}{{\displaystyle {\sum }_{z=l}^k\frac{{(x_i^T\beta )}^z}{z!}}}\right]}=0& (15)\end{array}$

Equations 11, 13, 15 contain non-linear relationships between parameters which require iterative methods including Newton Raphson or Fisher scoring. In this research, we employ the traditional optimization method [Quasi newton method (BFGS)] alongside meta heuristic algorithms to estimate parameters of truncated Poisson regression model in three different cases, as well as improving a new algorithm to enhance the solution process.

6 Broyden–Fletcher–Goldfarb– Shanno Method (BFGS)

The BFGS method is one of the quasi-Newton algorithms for unconstrained optimization problem, for finding a point x^* ∈ Rⁿ let:

$\begin{array}{ll}minf\left(x\right)x*\in R^n& (16)\end{array}$

Where the objective function f:Rⁿ → R is a twice continuously differentiable objective function. The Broyden–Fletcher–Goldfarb–Shanno Method (BFGS) performs an iterative process that follows this procedure [13, 33]:

1. Start with x₁, an initial point and [H₁], a positive definite symmetric matrix n × n. where [H₁] is the identity matrix [I]. Set i = 1 the iteration number.

2. Determine the gradient ∇f_i at point x₁, then set:

$\begin{array}{ll}{S}_i=-\left[H_1\right]\nabla f_i& (17)\end{array}$

3. Determine the optimal step length ${\lambda }_i^{*}$ moving along direction S_i, then set:

$\begin{array}{ll}{x}_{i+1}=x_i+{\lambda }_i^{*}{S}_i& (18)\end{array}$

4. Check if the new point x_i+1 represents an optimal solution. If x_i+1 is optimal, stop. Otherwise, go to step 5.

5. Update the matrix [H₁] as:

$\begin{array}{l}\left[H_{i+1}\right]=\left[H_i\right]+\left(1+\frac{g_i^T\left[H_i\right]g_i}{d_i^T{g}_i}\right)\\ \times \frac{d_i{d}_i^T}{d_i^T{g}_i}-\frac{d_i{g}_i^T\left[H_i\right]}{d_i^T{g}_i}-\frac{\left[H_i\right]g_i{d}_i^T}{d_i^T{g}_i}& (19)\end{array}$

Where

$\begin{array}{ll}{g}_i=\nabla f\left(x_{i+1}\right)-\nabla f\left(x_i\right)=\nabla f_{i+1}-\nabla f_i& (20)\end{array}$

$\begin{array}{ll}{d}_i=x_{i+1}-x_i& (21)\end{array}$

Set i = i + 1 the new iteration number, and go to step 2.

7 Black winged kites algorithm (BKA)

In (2024) Wang et al. [15] introduced the black-winged kite optimization algorithm (BKA) which represents a groundbreaking meta-heuristic algorithm. The black-winged kite uses its survival techniques as the basis for its optimization algorithm. This bird utilizes excellent hovering capabilities in addition to its unexpected hunting proficiency. The black-winged kite feeds on insects together with birds and reptiles and small mammals. A model was developed through analysis of black-winged kite movement patterns and hunting abilities [15, 34, 35].

7.1 Initialization

The first step in BKA requires generating random solutions to establish the population as depicted in Algorithm 1. Each Black-winged kite receives its position through a uniform distribution:

$\begin{array}{ll}{X}_i=B{K}_{lb}+rand\left(B{K}_{ub}-B{K}_{lb}\right)& (22)\end{array}$

Where BK_lb, BK_ub represent the lower and upper bounds of i^th black kites, respectively, and rand ∈ [0, 1] is a random number [15, 34, 35].

Algorithm 1

Algorithm 1. Black winged kite algorithm.

7.2 Attacking

The black-winged kite waits quietly before dropping to attack its prey after matching its wings and tail to the wind speed. The black-winged kite underwent two different assault scenarios throughout the global exploration phase of the BKA. The kite maintains its hovering position in the air while it readjusts its position to reach the target at its optimal attack angle. The kite maintains its position in the air while scanning for targets before striking down the most vulnerable one it detects. The attack behavior model uses the following mathematical expression:

$\begin{array}{ll}{x}_{t+1}^{i,j}=\{\begin{array}{ll}{x}_t^{i,j}+n\left(1+sin\left(r\right)\right)\times x_t^{i,j}& p<r\\ x_t^{i,j}+n\times \left(2r-1\right)\times x_t^{i,j}& else\end{array}& (23)\end{array}$

$\begin{array}{ll}n=0.05\times e^{-2*{\left(\frac{t}{T}\right)}^2}& (24)\end{array}$

Where $x_{t+1}^{i,j},x_t^{i,j}$ is the position of the i^th Black-winged kites in the j^th dimension at iteration steps (t) and (t + 1)^th, respectively. r ∈ [0, 1] is a random number and p = 0.9 is a constant value. T represents the total number of iterations and t is the current iteration [15, 34, 35].

7.3 Migration

Bird migration occurs as an intricate behavior because both climate conditions and food availability serve as influential environmental elements. Bird migration exists as an adaptation to seasonal changes through which numerous birds move from northern regions to southern areas for improved living conditions and resources. Migration teams follow leaders who need excellent navigation abilities to achieve success. Our hypothesis relies on bird migration principles which state that if the fitness value of the current population is lower than that of the random population, the leader will give up leadership and join the migratory population, indicating that it is not suitable to lead the population forward. On the other hand, if the fitness value of the current population is higher than that of the random population, then the population will be guided to its destination. The approach enables automatic selection of superior leaders to achieve migration success. The mathematical model describes for the migration patterns of black-winged kites as follows:

$\begin{array}{ll}{x}_{t+1}^{i,j}=\{\begin{array}{cc}{x}_t^{i,j}+C\left(0,1\right)\times \left(x_t^{i,j}-L_t^j\right)& F_i<F_{ri}\\ x_t^{i,j}+C\left(0,1\right)\times \left(L_t^j-m\times x_t^{i,j}\right)& else\end{array}& (25)\end{array}$

$\begin{array}{ll}m=2\times sin\left(r+\pi /2\right)& (26)\end{array}$

The parameter (m) is used to scale the current position of the kite in the update term, and scales the step size toward the leader perturbed by Cauchy mutation.

Where $L_t^j$ is the leading scorer of the black-winged kites in the j^th dimension of the t^th iteration so far. F_i is the fitness value of the current position obtained by any black-winged kite in the j^th dimension of the t^th iteration. F_ri is the fitness value of the random position obtained from any black kites in the j^th dimension of the t^th iteration, and C(0, 1) is the Cauchy mutation. The probability density function of the Cauchy distribution is [15, 34, 35]:

$\begin{array}{ll}f\left(x,\delta ,\mu \right)=\frac{1}{\pi }\frac{\delta }{{\delta }^2+{\left(x-\mu \right)}^2}-\infty <x<\infty & (27)\end{array}$

When δ = 0, μ = 1, then the standard form of the Cauchy distribution becomes the following:

$\begin{array}{ll}f\left(x,\delta ,\mu \right)=\frac{1}{\pi }\frac{1}{x^2+1}-\infty <x<\infty & (28)\end{array}$

8 Proposed algorithm

In this algorithm, the classical optimization algorithm, BFGS, was combined with the black-winged kite optimization algorithm (BFGS-BKA). where the randomness and speed of the black-winged kite optimization algorithm are used to find the optimal step length ${\lambda }_i^{*}$ in each iteration, while using parameter values before truncated as initial values for BFGS algorithm. The basic steps of this algorithm can be describes as:

Step (1): start with x₁, an initial point and [H₁], a positive definite symmetric matrix n × n. where [H₁] is the identity matrix [I]. Set i = 1 the iteration number.

Step (2): determine the gradient ∇f_i at point x₁, then set:

S_i = −[H₁]∇f_i

Step (3): find the optimal step length ${\lambda }_i^{*}$ by black-winged kite optimization algorithm:

1. Randomly generate the initial population of black-winged kite between λ_lb and λ_ub as λ₁, λ₂, ........, λ_N. Evaluate the fitness value of each black-winged kite as (λ₁), f(λ₂), ......, f (λ_N). Set t = 1 the iteration number.

2. BKA chooses the individual with the best fitness value to become the leader λ_L in the initial population.

3. Find f_best and λ_L, the black-winged kite algorithm initiates its global exploration and search during its attack behavior according the following equation:

$\begin{array}{l}{\lambda }_{t+1}^{i,j}=\{\begin{array}{cc}{\lambda }_t^{i,j}+n\left(1+sin\left(r\right)\right)\times {\lambda }_t^{i,j}& p<r\\ {\lambda }_t^{i,j}+n\times \left(2r-1\right)\times {\lambda }_t^{i,j}& else\end{array}\end{array}$

4. In Bird migration, will be the position is update based on the fitness value of the leader as:

$\begin{array}{l}{\lambda }_{t+1}^{i,j}=\{\begin{array}{cc}{\lambda }_t^{i,j}+C\left(0,1\right)\times \left({\lambda }_t^{i,j}-L_t^j\right)& F_i<F_{ri}\\ {\lambda }_t^{i,j}+C\left(0,1\right)\times \left(L_t^j-m\times {\lambda }_t^{i,j}\right)& else\end{array}\end{array}$

5. Test the convergence of the current solution. If the convergence criterion is not satisfied, go to step (3) and update the iteration number as t = t + 1, until convergence occurs and the optimal value of λ is determined.

Step (4): find new point:

$\begin{array}{l}{x}_{i+1}=x_i+{\lambda }_i^{*}{S}_i\end{array}$

Step (5): check if the new point x_i+1 represents an optimal solution. If x_i+1 is optimal, stop. Otherwise, go to step 6.

Step (6): update the matrix [H₁] as:

$\begin{array}{l}\left[H_{i+1}\right]=\left[H_i\right]+\left(1+\frac{g_i^T\left[H_i\right]g_i}{d_i^T{g}_i}\right)\frac{d_i{d}_i^T}{d_i^T{g}_i}-\frac{d_i{g}_i^T\left[H_i\right]}{d_i^T{g}_i}-\frac{\left[H_i\right]g_i{d}_i^T}{d_i^T{g}_i}\end{array}$

Where

$\begin{array}{l}{g}_i=\nabla f\left(x_{i+1}\right)-\nabla f\left(x_i\right)=\nabla f_{i+1}-\nabla f_i\\ d_i=x_{i+1}-x_i\end{array}$

Step (7): set i = i + 1 the new iteration number, and go to step 2.

9 Real data

This section contains real dataset to demonstrate the empirical importance of the algorithm (BFGS-BKA), where the real dataset is for the zero truncated Poisson regression model study. We use the (Arizona MedPar database, 1991). The dataset contains 1,495 observations the response variable, Lose, represents length of hospital stay. The explanatory variables for this model include an indicator of White (Patient identifies themselves as Caucasian, binary), Hmo (Patient belongs to a Health Maintenance Organization, binary), Type2 (Urgent admission, binary) and Type3 (Elective admission, binary).

As illustrated by Table 1, across all four methods, the parameter estimates are remarkably consistent, indicating stable estimation despite the choice of optimization algorithm. The numerical differences between the methods on each parameter are very small showing that all methods converge to nearly identical solutions. The MSE measures the average squared difference between observed and predicted values, serving as an indicator of model fit quality and estimator accuracy. Newton Raphson, Davidon–Fletcher–Powell (DFP) method, and BFGS yield the same MSE value of 0.0225, whereas the BFGS-BKA method achieves a slightly lower MSE of 0.0211. This implies that integrating the BKA algorithm with BFGS potentially enhances estimation accuracy or predictive performance. This improvement indicates that BFGS-BKA may better navigate the complex truncated likelihood surface to find a more optimal parameter set or avoid local optima.

Table 1

Table 1. Results zero-truncated Poisson regression model for the medpar data.

10 Simulation study

This section presented a simulation study to demonstrate the empirical importance of the algorithm (BFGS-BKA) to evaluate the accuracy of MLEs of parameters estimation of a truncated Poisson regression model of three cases left truncated Poisson regression model (LTPRM), right truncated Poisson regression model (RTPRM) and double truncated Poisson regression model (DTPRM). We consider dimension P = 3, 7, 12 and the sample sizes n = 25, 50, 100, 200.

10.1 Simulation results of the left truncated Poisson regression model

This section represented results for fit the left truncated Poisson regression model, with responses truncated at L = 0. Where independent variables are randomly generated according to a normal distribution and the data were simulated with true values beta = [0.5; −0.3; 0.2], beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2] and beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2; 0.4; −0.25; 0.15; 0.05; −0.05]. Tables 2–4 show the simulation results for LTPRM.

Table 2

Table 2. Simulation results for estimating parameters of a left truncated Poisson regression model at zero (y > 0; P = 3).

Table 3

Table 3. Simulation results for estimating parameters of a left truncated Poisson regression model at zero (y > 0; P = 7).

Table 4

Table 4. Simulation results for estimating parameters of a left truncated Poisson regression model at zero (y > 0; P = 12).

10.2 Simulation results of the right truncated Poisson regression model

This section represented results for fit the right truncated Poisson regression model, with responses truncated at U = 5. Where independent variables are randomly generated according to a normal distribution using matlab randn (n,p), also the data were simulated with true values beta = [0.5; −0.3; 0.2], beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2] and beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2; 0.4; −0.25; 0.15; 0.05; −0.05]. Tables 5–7 show the simulation results for (RTPRM).

Table 5

Table 5. Simulation results for estimating parameters of a right truncated Poisson regression model (P = 3).

Table 6

Table 6. Simulation results for estimating parameters of a right truncated Poisson regression model (P = 7).

Table 7

Table 7. Simulation results for estimating parameters of a right truncated Poisson regression model (P = 12).

10.3 Simulation results of the double truncated Poisson regression model

This section represented results for fit the double truncated Poisson regression model, with responses truncated at L = 0 and U = 5. Where Independent variables are randomly generated according to a normal distribution using matlab randn (n,p), also the data were simulated with true values beta = [0.5; −0.3; 0.2], beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2] and beta = [0.5; −0.3; 0.2; 0.1; −0.1; 0.3; −0.2; 0.4; −0.25; 0.15; 0.05; −0.05]. Tables 8–10 show the simulation results for (DTPRM).

Table 8

Table 8. Simulation results for estimating parameters of a double truncated Poisson regression model (P = 3).

Table 9

Table 9. Simulation results for estimating parameters of a double truncated Poisson regression model (P = 7).

Table 10

Table 10. Simulation results for estimating parameters of a double truncated Poisson regression model (P = 12).

From all the Tables above, we can concluded that across all sample sizes, parameter estimates from all four methods are highly similar, demonstrating consistent convergence to approximately the same values. This consistency suggests each optimization method is capable of locating reliable parameter estimates even in smaller samples. As sample size increases, the parameter estimates stabilize and vary less across methods. This reflects the expected property of maximum likelihood estimators: increased sample size yields more precise and stable estimates. As sample size increases (from 25 to 200), the estimated MSE decreases monotonically for all methods, demonstrating improved estimation precision with larger data consistent with statistical theory. BFGS-BKA consistently achieves the lowest MSE at each sample size, indicating superior estimation accuracy. Classical methods such as Newton-Raphson, DFP, and BFGS provide similar MSE values, slightly higher than BFGS-BKA. The performance gap, though sometimes small in absolute terms, illustrates the advantage of using the Black Kite Optimization (BKA) to enhance the classical BFGS method, improving optimization over the complex truncated likelihood surface. The similarity of parameter estimates across methods confirms robustness of the numerical algorithms when applied to truncated Poisson regression models.

Integrating the Black Kite Optimization algorithm with BFGS consistently improves the optimization process, yielding more accurate and stable estimates. This suggests that metaheuristic approaches like BKA help avoid local optima and improve convergence speed when dealing with truncated likelihood functions. For small sample sizes where numerical instability and local optima are more problematic, BFGS-BKA provides meaningful improvements. For larger datasets, while all methods perform well, BFGS-BKA still maintains a measurable edge in accuracy.

Figures 1–3 show the performance of Newton's, DFP, BFGS, and BFGS-BKA algorithms in terms of time that represents the average computation time (in seconds) used by the all algorithms in the case of successful runs. We can see how the proposed algorithm is better than the others because of the Cauchy mutation that causes better exploration and the leader strategy that causes faster convergence leading to a balance between global and local search.

Figure 1

Figure 1. Computational time of left truncated Poisson regression model at zero (y > 0) (a) P = 3, (b) P = 7 and (c) P = 12.

Figure 2

Figure 2. Computational time of right truncated Poisson regression model (a) P = 3, (b) P = 7 and (c) P = 12.

Figure 3

Figure 3. Computational time of double truncated Poisson regression model (a) P = 3, (b) P = 7 and (c) P = 12.

11 Conclusions

This paper addressed the critical challenge of accurately estimating parameters in truncated Poisson regression models, where standard maximum likelihood estimation is complicated by a truncated likelihood function that includes a non-trivial normalizing constant. The paper proposed the use of the BKO algorithm, a metaheuristic inspired by the hunting and migratory behavior of black kites, which aims to enhance exploration and exploitation capabilities when searching the parameter space. By leveraging BKO, the study seeks to improve the stability, convergence speed, and accuracy of parameter estimates in truncated Poisson models. Simulation studies and empirical analyses included in the paper demonstrate the superior performance of this approach compared to classical optimization methods, highlighting its potential as a robust and efficient solution for parameter estimation in truncated count data settings.

Data availability statement

The data analyzed in this study is subject to the following licenses/restrictions: the data will be under request. Requests to access these datasets should be directed to Zakariya Algamal, emFrYXJpeWEuYWxnYW1hbEB1b21vc3VsLmVkdS5pcQ==.

Author contributions

GB: Conceptualization, Formal analysis, Software, Writing – original draft, Writing – review & editing. SW: Conceptualization, Formal analysis, Methodology, Resources, Writing – original draft, Writing – review & editing. ZA: Supervision, Validation, Writing – review & editing.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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