# On the Estimation of Logistic Models with Banking Data Using Particle Swarm Optimization
###### Abstract  
This paper presents numerical works on estimating some logistic models using particle swarm optimization (PSO). The considered models are the [PERSON] model, Pearl and Reed generalization model, [PERSON] model, [PERSON] model, [PERSON] model, hyper-Gompertz model, [PERSON] model, Turner et al. model, and [PERSON] model. We employ data on commercial and rural banking assets in Indonesia due to their tendency to correspond with logistic growth. Most banking asset forecasting uses statistical methods concentrating solely on short-term data forecasting. In banking asset forecasting, deterministic models are seldom employed, despite their capacity to predict data behavior for an extended time. Consequently, this paper employs logistic model forecasting. To improve the speed of the algorithm execution, we use the Cauchy criterion as one of the stopping criteria. For choosing the best model out of the nine models, we analyze several considerations such as the mean absolute percentage error, the root mean squared error, and the value of the carrying capacity in determining which models can be unselected. Consequently, we obtain the best-fitted model for each commercial and rural bank. We evaluate the performance of PSO against another metaheuristic algorithm known as spiral optimization for benchmarking purposes. We assess the robustness of the algorithm employing the Taguchi method. Ultimately, we present a novel logistic model which is a generalization of the existence model. We evaluate its parameters and compare the result with the best-obtained model. banking data; logistic growth model; parameter estimation; particle swarm optimization; Taguchi method +
Footnote †: journal: _Algorithms_  
0000-0001-9500-0001-9500][PERSON]. [PERSON]  
1 Department of Mathematics, Faculty of Science and Mathematics, Universitas Diponegoro,  
Semarang 50275, Indonesia1  
multiplication of micro-organisms that must be proportional to the current number and the lack of foods [4]. [PERSON] and [PERSON] derived the logistic function for representing the U.S. population growth [1]. Two years later, [PERSON] and [PERSON] made a generalization by taking the growth rate parameter as a function of time [2]. Years later, for empirical purposes, many researchers modified the logistic model directly or indirectly from the previous researchers. For instance, [PERSON] studied the growth of length and weight of guppy fish (_Lebistes reticulatus_) [14] and the connections between metabolism and the growth rate of living organisms [15], [PERSON] extended the von Bertalanffy model to become more flexible and tested the plant data [16], [PERSON] proposed a modification in the Verhulst model for modeling the population or organism size [17], [PERSON] et al. presented the generalization called generic growth model [18], and then [PERSON] generalized the generic growth model in [19]. It also needs to be noted that there is a Gompertz function [20] that belongs to the family of logistic functions. In [18], [PERSON] et al. also discussed the hyper-Gompertz function. In this paper, we use the logistic model to model banking assets growth. Many banking asset forecasting methods employ statistical methodologies that focus exclusively on short-term data predictions, such as [21; 22; 23]. Deterministic models are rarely utilized in banking asset forecasting, despite their ability to project data behavior over prolonged periods. Therefore, this work utilizes logistic model prediction. We use Indonesian banking data from the period January 2007-January 2020. The rationale for using the logistic model in analyzing the growth of banking assets is as follows. For a healthy bank, the amount of assets has a tendency to always go up exponentially, although, in some parts, it also fluctuates. However, for several reasons of the bank's internal and external limitations, there should be an upper threshold number for banking asset growth which can be seen as the carrying capacity in the logistic model.