Mathematical Modeling Series – Introduction

Mathematics is not just limited to a subject of any curriculum rather it can help to model real world situations. These models that are derived using Mathematical language always give concrete and full proof solutions to the problem. However, the real challenge is to transform the problem solving process (design) into mathematical language. My “Mathematical Modeling Series” is small and focused effort to transform real world Data Analysis related problems into Mathematical Models. As a part of this series I’ll try to cover all the Maths topics required for Data analysis. Upcoming posts will also cover the topics from programming, Data structure and computations where Mathematical implementation play a major role. To start, I’ll remain focused on “what” and “how” of the mathematical foundation to model the Data analysis problems. Before dwelling into specific topic of the mathematics it is must to know “why” we actually need “Mathematical model”?

What is mathematical model? When a real-world situation can be described in mathematical language, the description is a mathematical model. For example, the natural numbers constitute a mathematical model for situations in which counting is essential. Mathematical models are abstracted from real-world situations. The mathematical model may give results that allow us to predict what will happen in the real-world situation. If the predictions are inaccurate or the results of experimentation do not conform to the model, the model must be changed or discarded. Mathematical modeling is often an ongoing process. For example, finding a mathematical model that will provide an accurate prediction of population growth is not a simple task. Any population model that one might devise would need to be reshaped as further information is acquired.

So, which Mathematics topics are primarily focus on the Data analysis? the answer is simple Algebra and Calculus. And why? because your Data points are in some space (i.e. in some span or position) and there is some relation between these Data points. In this process the first step is to build the mathematical model to prove: yes there a relation [mathematical expression to encode the relation]. In the next step of modeling is go to an extent to prove how strong/weak the relation between these Data points [i.e. model optimization]. Here, the first step uses Algebra to model the relation and in the next step Calculus take care of optimization. However, in some scenarios even algebra field can also help in optimization. Quick example: Consider a milk box which manufacture wants to design in such a way that it can fill the milk properly.

Algebra Problem:
The sum of the height, width, and length of a box is 207 mm. If the height is three
times the width and the length is 7 mm more than the width, find the dimensions
of the box.

Calculus Problem:
A protein energy drink box is to hold (6.75 fl oz) of protein energy drink.
If the height of the box must be twice the width, what dimensions will minimize
the surface area of the box?

Hopefully this example clarified enough about Algebraic and Calculus way of looking the problem. In the next post in this series we’ll go deep into how to look at the Data from Mathematical modeling perspective.

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