# On the Estimation of Logistic Models with Banking Data Using Particle Swarm Optimization
###### Abstract
A bank is a financial institution with a main intermediary function, which is accepting deposits and providing loans. A bank can be classified as a commercial bank or a rural bank. The objects of this study are the data of assets of Indonesian commercial and rural banks. Commercial banks carry out their business activities conventionally or according to Sharia,in which there are also services in payment transmission services, while rural banks do not provide services in payment transmission. Commercial banks also have a wider range of financial products than rural banks, such as foreign exchange, insurance businesses, and equity participation. In terms of service areas, commercial banks can serve at the village, city, provincial, national, and international levels, and on the other hand, rural banks' service areas are limited to districts or cities. Therefore, the total assets of commercial banks are certainly far greater than rural banks; see Figure 1a,b. The monthly fluctuation of the assets can be seen in Figure 1c,d. The reason why we need to distinguish between commercial and rural bank data, without collecting them into a single data unit, is that we want to see whether the dynamics of growth in commercial banking assets (big banks) are different or the same as the dynamics of rural banking (small banks) asset growth. Whether the two data will fit into the same or different logistic models is our concern. In the application of a logistic model to modeling data, the problem of estimating the model's parameters is unavoidable. In the logistic model, it can be said that the parameters can be divided into two groups, the first group is the parameters with a small estimated value, and the second group is the carrying capacity parameter which may have a very large value. The gap between the two groups causes parameter estimation problems in a search area that is in the form of a very long rectangle. Such a search area will complicate classical numerical methods; for example, [PERSON]'s method with its gradient and Hessian which requires guessing the initial value. The large value of carrying capacity will interfere with the performance of the [PERSON] method. Meanwhile, a metaheuristic algorithm such as the spiral optimization (SpO) algorithm [24; 25], which is very powerful for an optimization problem in a square search area as it has an iteration update in the form of spiral rotation, will also experience weaknesses in estimating the parameters of the logistic model because the update in each iteration will create search points that exit the search area. Particle swarm optimization (PSO), an optimizer based on the bird or fish-flocking behavior in search of optimal food resources, is one of the metaheuristic methods that is unaffected by the size of the search area because the updating of search points in the PSO method can adapt to the size of the search area. Thus, the algorithm will be very suitable for estimating the parameters of logistic models. To avoid discovering search points outside the search area, the PSO method might include a condition that ensures all search points remain within the search area. In this study, numerous logistic models' parameters are estimated using PSO. In order to reduce the time required to complete iterations, we employ the Cauchy criterion in addition to the maximum number of iterations as a stop condition. Using the Cauchy criterion, we can automatically halt the algorithm if we suspect that it has attained convergence. Although there have been so many variants of PSO in the literature, including recent developments in [26; 27; 28; 29], this paper uses the standard PSO algorithm with a decreasing inertia weight [30; 31]. To check the algorithm's robustness, we apply the Taguchi method [32; 33] to determine the optimal parameters for PSO. Figure 1: Total assets of (**a**) commercial banks and (**b**) rural banks in Indonesia in the period January 2007 –January 2020. The monthly fluctuation of total assets of (**c**) commercial banks and (**d**) rural banks.