Calculus in Data Science and its Uses

What is Mathematics in Data Science?

In data science, calculus is also known as analysis in the branch of mathematics. Calculus  uses data science to study the rate of change of quantities, length, area, and volume of objects. It is divided into two different methods: differential and integral calculus.

Differential Calculus – divide something into small pieces to find how it changes.

Integral Calculus – integrates the small pieces to find how much there is.

Behind every data science model is an optimisation algorithm that relies heavily on calculus. Let’s learn the Gradient Descent Approximation (GDA) and how it can be used to build a simple linear regression estimator.

Gradient Descent Approximation

A gradient approximation measures how much the output of a function changes if you change the inputs to a little bit. For example, let’s take the maximum and minimum of a function in one- dimension, using derivatives with simple quadratic functions. The formula is:

$\hat{Y} = B_0 + B_1(X)$.

Also, the constant number ($\alpha$) is a small positive constant called the learning rate. Ensure the following points:

• when $X_n > X_{min}$, $f'(X_n) > 0$: it ensures that $X_{n+1} < X_n$. Thus, we are taking steps in the left direction to get to the minimum.
• when $X_n < X_{min}$, $f'(X_n) < 0$: it ensures that $X_{n+1} > X_n$. Thus, we are taking steps in the right direction to get to $X_{min}$.

Enroll yourself in the data science course to practice and implement this algorithm, the function of several variables can be minimised. In higher dimensions, the function of several variables can be reduced.

A standard approach to solving this type of problem is to define an error function (also known as a cost function). It helps to measure how good an algorithm is. The function will take in a pair $(\hat{Y}, Y)$ and return an error value based on how well it fits the data. For Linear Regression, a common cost function is the Mean Squared Error (MSE), which for $n$ data points is given by:

$$MSE = \frac{1}{n} \sum_{i=1}^{n} (Y_i – \hat{Y}_i)^2$$

It squares distances between each point’s predicted value ($\hat{Y}_i$) and the actual value ($Y_i$).

It’s conventional to square this distance to ensure it is a positive point and to form an error function differentiable.

The learning rate ($\alpha$) variable holds how large of a step we take downhill during each iteration. If we take too big of a step, we may skip the minimum. However, taking small steps will require many iterations to arrive at the minimum.

Data science is complex once you don’t know about the basic concept and nothing is better than an online data science program.

Conclusion

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