Introduction to Mathematical Modeling
There is a great deal of interesting work happening in data sciences, especially in the realms of Machine Learning (ML) and Deep Learning (DL), and they are popular for good reason. However, the more tried and tested old-timer, mathematical modeling, is not much talked about. Mathematical modeling methods are no less relevant and are complementary to ML. To create successful data products that solve real business problems, we must often deploy the whole breadth of available mathematical tools, far beyond ML.
A model is a simplified representation of a real system and captures the essence of the system. A mathematical model uses variables, operators, functions, equations, and equalities. Under the hood of mathematical models, there are first-principle models based on physical laws, stochastic models based on distributions, averages, and empirical models based on patterns or historical data. Based on the particular type of modeling, qualitative or quantitative recommendations can be made for the system under consideration. A mathematical model facilitates design and prototyping and substantiates decisions. To formulate a mathematical model, one needs the input and output, the constants and variables, the domain and boundary, or initial conditions and constraints. The solution can be analytic or numerical; in either case, it determines the typical behavior and critical parameters of the system, trends, dependency, and operating regimes. Systems can be deterministic, wherein we know the cause-effect relationship, or they may be stochastic, involving probability distributions.
A few mature tools in mathematical modeling are in the following areas:
- Mathematical optimization
- Signal processing
- Control theory
We will explore these mathematical modeling approaches in the following sections. A narrow focus on ML misses out on many relevant features of pure mathematical optimization in many use cases. Successful solutions across disparate domains blend the new world of ML with classical mathematical modeling techniques. For example, one can combine state-space modeling methods with ML to infer unobserved parameters of systems in a parameter estimation problem.