Below is an outline of research topics for each area of specialization within the Center for Mathematics and Society.
A. Actuarial Science specialization
As described in the 2023 Curriculum Book of the Undergraduate Mathematics Study Program, the Actuarial Science specialization focuses on the use of mathematics, probability and statistics, as well as economics and finance in the measurement and management of quantitative risk. Research conducted at the Center for Mathematics and Society in the Actuarial Science specialization can be grouped into the following topics.
1. Loss models
Before calculating premiums and reserves, modeling of losses consisting of claim counts and claim sizes is needed to determine patterns and parameters that can be used for further analysis. Loss modeling, which is usually understood as the construction of distributions for claim counts and claim sizes, is a part of actuarial science that focuses on investigating and constructing distributions using mathematical approaches. In addition, parameter estimation processes using various approaches are also often discussed here.
2. Statistical methods
Claim severity is a random variable, especially in general insurance data. The main problem in modeling this data is the large gap between claim amounts. In the center, claim severity modeling is carried out using quantitative methods, i.e., by fitting several heavy-tailed distributions to simulated claim severity data. To determine which distribution is suitable for use, tests are performed using Quantile-Quantile plots along with the Akaike Information Criterion (AIC) for each model. One of the research results conducted showed that a mixture of Lognormal, Inverse Gaussian, and Gamma distributions is a good model for large claim severity data. Using these models, the posterior probability for each model can be calculated using Bayes’ Theorem, as well as the pure premium for the excess layer.
B. Data Analytics Specialization
As described in the 2023 Curriculum Book of the Undergraduate Mathematics Study Program, the Data Analytics specialization focuses on data modeling, visualization, engineering, and its applications. Research conducted at the Center for Mathematics and Society in the Data Analytics specialization can be grouped into the following topics.
1. Non-Parametric Statistics
Kernel methods are part of nonparametric statistics usually used as alternatives to various empirical statistical methods by performing smoothing using a function called a kernel. Various classical statistical methods such as pdf estimation with histograms, empirical distribution functions, and goodness-of-fit tests have had their performance and convergence levels improved by kernel methods. However, smoothing with kernel functions raises a problem, commonly known as the boundary bias problem, when the data support is not the entire real line. This issue is a major obstacle to the use of kernel methods in actuarial science because almost all actuarial data are non-negative. Research conducted in the center in this field aims to introduce how bijective transformations can eliminate the boundary bias problem from various naive kernel methods and improve their performance when applied to data defined on a general interval.
2. Bayesian Statistics
Unlike classical statistical approaches that negate subjective bias in an analysis and base conclusions only on objective data, Bayesian statistics always takes into account the preferences, experiences, and personal knowledge of the researcher in decision-making. There are three important components in every Bayesian analysis, namely the prior distribution which represents subjective perspective, the model distribution which represents objective information in Bayesian Statistics, and the posterior distribution which is the result of “mixing” the two previous distributions. With the posterior distribution, Bayesian analysis and decision-making can be carried out.
3. Biostatistics
Bandung, one of the largest cities in Indonesia, has a serious problem with Dengue Fever (DF) which is transmitted by Aedes mosquitoes. In the center, the severity of DF in the city of Bandung is studied statistically. The quantile value of the severity level is estimated with quantile regression, using homoskedastic and heteroskedastic varying coefficient models, where the estimator coefficients are assumed to depend on time and are approximated using P-splines. In addition, the distribution of DF severity across districts in the city of Bandung is assessed quantitatively using relative risk, by applying the spatial model of Besag, York, and Mollie (BYM). The optimal model, determined using the Deviance Information Criterion (DIC), can be used to identify districts and times when the risk of DF is highest.
4. Supervised and Unsupervised Learning
The development of the digital world now produces a lot of data (big data) that can be explored to understand data/user behavior so that a system can work more efficiently and effectively. Existing machine learning methods can make a system work in real time and automatically. Broadly speaking, machine learning is divided into two types: supervised learning and unsupervised learning. A simple example of the two types is in animal image classification. The target data in supervised learning has labels (chicken, cat, ant, etc.), while in unsupervised learning there are no labels (classification results: four-legged animals, two-legged animals, etc.). The learning methods studied in the center’s research include (but are not limited to) regression, decision trees, SVM, nearest neighbors, neural networks, and reinforcement learning. Furthermore, machine learning topics studied include (but are not limited to) stock price prediction, sentiment analysis, disease classification, image classification, scheduling, shopping habit analysis, fraud detection, solving differential equations, and determining hydrodynamic parameters.
XGBoost (Extreme Gradient Boosting) is one of the superior tree boosting algorithms and a popular choice among machine learning competition participants for regression and classification problems. Boosting is a “combination” technique, where the error of the old model is corrected by sequentially adding new models, until no further improvements can be obtained. In gradient boosting, the gradient descent algorithm is used to minimize the loss function when adding new models. XGBoost is an optimization of standard gradient boosting. The XGBoost algorithm is known to be scalable, portable, and accurate, and excels in processing speed and performance. In research at the center, the XGBoost algorithm is applied to study various types of industrial and actuarial data. Machine learning methods are applied to scheduling problems and research on ocean waves.
5. Digital Image Processing
The condition of ocean waves, both in coastal areas and in the open sea, is very important to know. One of the measurement techniques is remote sensing using radar. Radar can generally capture ocean waves over a wide range, up to a 2 km radius. The radar capture is in the form of a digital image which is then processed to obtain data in the form of estimates of wave parameter values (period, wavelength, and spectrum) and seabed depth. In coastal areas, this data is needed for port development planning and monitoring of natural conditions such as erosion or sedimentation. In the open sea, this data is needed to predict incoming waves for ships, to help captains navigate. In addition to radar, images can also be obtained from drones. Unlike radar images that rely on electromagnetic signals, drone images are produced from light reflections so that objects appear as seen by the eye. In research at the center, image processing is carried out by applying mathematical concepts such as continuous and discrete Fourier transforms, dispersive wave equations, optimization, interpolation, polar and Cartesian coordinates, and radar data sampling.
C. Optimization dan Dynamics Specialization
As described in the 2023 Curriculum Book of the Undergraduate Mathematics Study Program, the Optimization and Dynamics specialization focuses on modeling to solve problems in industry, health, and finance. Research conducted at the Center for Mathematics and Society in the Optimization and Dynamics specialization can be grouped into the following topics.
1. Time-Evolution of Financial Quantities
In Indonesia, IFRS 17 (International Reporting Financial Standards), which has been in effect since January 1, 2023, is translated into Financial Accounting Standard Statement 74 (PSAK 74) and must be implemented by January 1, 2025 at the latest. One of the consequences is the need for scientific methods —that can be scientifically justified— to determine interest rate assumptions based on the IGSYC (Indonesia Government Securities Yield Curve). For this purpose, interest rate movements need to be modeled mathematically using time series models along with other stochastic models. Research conducted at the center aims to compare the performance of the geometric Brownian motion model, the Vasicek model, and the CIR (Cox-Ingersoll-Ross) model for interest rate movements.
2. Valuation of Financial Instruments
Financial instruments (derivatives) are products whose values are determined by the price of more fundamental variables or underlying assets (such as stocks, stock indices, commodities, and so on). One example of a financial instrument is an option. Research at the center in this field focuses on employee stock options (ESOs), which are call options granted by a company to a group of its employees to purchase shares of the company. Employee stock options in Indonesia (granted by several companies such as Bank BRI, Krakatau Steel, etc.) have characteristics that make them quite complex in terms of mathematical modeling. In classical mathematical models for employee stock options, a single psychological barrier is used, and valuation is carried out using the binomial method. Research at the center in this field aims to contribute by developing mathematical models for employee stock options, such as modifications of the Hull-White model with a moving psychological barrier and bino-trinomial methods, as well as the application of these models, for example for valuing employee stock options in Indonesia.
3. Portfolio Optimization
A portfolio, once formed, needs to be monitored to ensure it provides optimal returns for the investor. To achieve this, rebalancing must be carried out so that the portfolio remains optimal. In research at the center, the portfolio will be rebalanced when there is a trend reversal in a stock index. To determine this trend reversal, logistic regression is used with several technical indicators as predictors. At this stage, the risk-aversion coefficient is adjusted so that the investor can rebalance the portfolio to obtain optimal results. The portfolio model used is the one that maximizes the rate of return while minimizing risk as measured by the semi-absolute deviation. The proposed framework in this research will be applied to portfolio construction consisting of LQ45 stocks.
4. Inventory Models
In managing a goods company, a good understanding of the dynamics of the inventory system for the goods sold is needed, so that appropriate strategies can be determined to minimize total expenditure or maximize total revenue of the company. One way to understand the dynamics of the inventory system is with the help of mathematical models. Researchers at the center in this field construct and analyze mathematically —both exactly and numerically— deterministic and probabilistic mathematical models for inventory systems that take into account various factors.
5. Models in Ecology and Epidemiology
Various natural phenomena involving quantities that change over time can be modeled in the form of dynamic systems. Two important examples of such phenomena are predator-prey interactions in an ecosystem and the spread of infectious diseases in a population. Research at the center in this field aims to construct, analyze from a dynamic systems perspective, numerically simulate, and apply to real case studies various mathematical models for predator-prey interactions and for the spread of COVID-19 and other infectious diseases.
6. Iteration of Arithmetical Maps
Almost all standard theories about the dynamical behavior of iterated maps are developed using calculus, so they can only be applied to maps defined on a manifold. For maps defined on non-manifold sets, the theory of their dynamical behavior must be constructed from scratch. The aim of research at the center in this field is to build and develop a theory of the dynamic behavior of such maps, for example the mean-median map that can be viewed as a map on the set of all finite sequences of real numbers or as a map on the set of all finite multisets of piecewise affine continuous functions, the generalized sum of remainders map, the prime power map, the greedy map, and maps that involve spatial discretization operators such as the floor function.
D. Pure and Educational Mathematics Specialization
As described in the 2023 Curriculum Book of the Undergraduate Mathematics Study Program, the Pure and Educational Mathematics specialization focuses on equipping future academics with strengthened pure mathematics concepts and basic aspects of teaching. Research conducted at the Center for Mathematics and Society in the Pure and Educational Mathematics specialization can be grouped into the following topics.
1. Max-plus Algebra
If the usual operations of addition and multiplication are replaced with the maximum and addition operations, a new mathematical system called max-plus algebra is obtained. In max-plus algebra, for example, the sum and product of 2 and 3 are respectively max{2, 3} = 3 and 2 + 3 = 5. In research at the center, max-plus algebra is applied to real-life problems such as scheduling the departure of transportation vehicles and scheduling production stages in industry.
2. Methods for Solving Differential Equations
Differential equations that model complex natural phenomena are generally not easily solved analytically. Therefore, the behavior of the solutions of such equations is more often studied using numerical methods, such as Euler’s method, Runge-Kutta method, and finite difference methods. In addition to numerical methods, there is also a collection of other methods that can be used, called semi-analytic methods, such as the variational iteration method, differential transform method, homotopy analysis method, and Adomian decomposition method. At the center, research is conducted applying these methods to various differential equations.
3. Morrey Spaces
Morrey spaces are a generalization of Lebesgue spaces Lp, which are spaces of functions defined using a generalization of the ppp-norm known in finite-dimensional vector spaces. At the center, research on Morrey spaces focuses on the properties satisfied by functional operators such as fractional integral operators and maximal operators defined on Morrey spaces or on a variant of Morrey spaces called Morrey-Adams spaces.
4. Herz Space
Like Morrey spaces, Herz spaces are a generalization of Lebesgue spaces Lp. Generally, Morrey spaces consider the ratio of the integral of a function over a ball to a power of the size of the ball. In Herz spaces, the integral of the function is considered over an annulus centered at zero. At the center, research on Herz spaces also focuses on the properties of functional operators such as fractional integral operators and maximal operators. In addition, the properties of the Hardy operator and the Hilbert operator are also studied.
5. Development of Assessment Methods
Since learning activities shifted from offline to online due to the COVID-19 pandemic, various challenges have arisen for educators in maintaining learning quality, one of which is in assessment. As a result, research developing objective assessment methods has become increasingly relevant. At the center, research is conducted to develop both offline and online assessment methods that minimize dishonesty.
6. Development of Enrichment Materials
Mathematics learning activities, especially at the university level, are always open to development not only pedagogically but also in terms of content. At the center, research is conducted to develop interesting materials that can be added as enrichment or as project materials in certain basic undergraduate mathematics courses.
7. Statistical Literacy
Statistical literacy is one of the key skills that a person must have in an information-rich world, and formal education in schools is expected to play a role in this. Statistical literacy is described as the ability to interpret, critically evaluate, and communicate statistical information and messages. Statistical literacy is believed to be the result of interconnected joint activities among components: literacy, statistical and mathematical skills, contextual understanding, and critical thinking. Kahneman discusses why we find it difficult to think statistically, and why it is easier to think associatively, with analogies, or cause-and-effect reasoning. He explains that statistics require us to think about many things at once, and the limitations of human reason —often too confident in what it knows but unable to measure its own ignorance and uncertainty— is one of the causes. Research at the center in this field examines data on statistical literacy abilities and develops statistical literacy assessment instruments and both qualitative and quantitative methods that can be used.