Vidhi — Updated On October 9th, 2023

## Introduction

Have you ever wondered how businesses predict market trends or scientists forecast climate changes? Welcome to the world of statistical modeling, where data transforms into knowledge. In this article, we’ll explore the fascinating realm of statistical modeling. What exactly is it? How does it work? What are its real-world applications? Whether you’re new to the concept or seeking deeper insights, join us on a journey to uncover the principles and significance of statistical modeling in deciphering the mysteries hidden within data.

This article was published as a part of the Data Science Blogathon.

Modeling is an art, as well as a science and, is directed toward finding a good approximating model … as the basis for statistical inference

Burnham & Anderson

## What is a Statistical Model?

A statistical model is a type of mathematical model that comprises of the assumptions undertaken to describe the data generation process.

Let us focus on the two highlighted terms above:

1. Type of mathematical model? Statistical model is non-deterministic unlike other mathematical models where variables have specific values. Variables in statistical models are stochastic i.e. they have probability distributions.
2. Assumptions? But how do those assumptions help us understand the properties or characteristics of the true data? Simply put, these assumptions make it easy to calculate the probability of an event.

Quoting an example to better understand the role of statistical assumptions in data modeling:

• Assumption 1: Assuming that we have 2 fair dice, and each face has equal probability to show up i.e. 1/6. Now, we can calculate the probability of two dice showing up 5 as 1/6*1/6. As we can calculate the probability of every event, it constitutes a statistical model.
• Assumption 2: The dice are weighted and all we know is that probability of face 5 is 1/8 which makes it easy to calculate the probability of both dice to show 5 as 1/8*1/8. But we do not know the probability of other faces, so we cannot calculate the probability of every event. Hence this assumption does not constitute statistical model.

## Why Do We Need Statistical Modeling?

The statistical model plays a fundamental role in carrying out statistical inference which helps in making propositions about the unknown properties and characteristics of the population as below:

#### Estimation

It is the central idea behind Machine Learning i.e. finding out the number which can estimate the parameters of distribution.

Note that the estimator is a random variable in itself, whereas an estimate is a single number which gives us an idea of the distribution of the data generation process. For example, the mean and sigma of Gaussian distribution

#### Confidence Interval

It gives an error bar around the single estimate number i.e. a range of values to signify the confidence in the estimate arrived on the basis of a number of samples. For example, estimate A is calculated from 100 samples and has a wider confidence interval, whereas estimate B is calculated from 10000 samples and thus has a narrower confidence interval

#### Hypothesis Testing

It is a statement of finding statistical evidence. Let’s further understand the need to perform statistical modeling with the help of an example below:

Objective is to understand the underlying distribution to calculate the probability that a randomly selected researcher would have written, let’s say, 3 research papers .

We have a discrete random variable with 8 (9-1) parameters to learn i.e., probability of 0,1,2.. research papers. As the number of parameters to be estimated increase, so is the need to have those many observations, but this is not the purpose of data modeling.

So, we can reduce the number of unknowns from 8 parameters to only 1 parameter lambda, simply by assuming that the data is following Poisson distribution.

Our assumption that the data follows Poisson distribution might be a simplification as compared to the real data generation process, but it is a good approximation.

## Types of Modeling Assumptions

Now that we understand the significance of statistical modeling, let’s understand the types of modeling assumptions:

• Parametric: It assumes a finite set of parameters which capture everything about the data. If we know the parameter θ which very well embodies the data generation process, then predictions (x) are independent of the observed data (D).
• Non-parametric: It assumes that no finite set of parameters can define the data distribution. The complexity of the model is unbounded and grows with the amount of data.
• Semi-parametric: It’s a hybrid model whose assumptions lies between parametric and non-parametric approaches. It consists of two components – structural (parametric) and random variation (non-parametric). Cox proportional hazard model is a popular example of semi-parametric assumptions.

## Definition of a Statistical Model: (S,P)

S: Assume that we have a collection of N i.i.d copies such as X1, X2, X3…Xn through a statistical experiment (it is the process of generating or collecting data). All these random variables are measurable over some sample space which is denoted by S

P: It is the set of probability distributions on S that contains the distribution which is an approximate representation of our actual distribution.

Let’s internalize the concept of sample space before understanding how a statistical model for these distributions could be represented:

• Bernoulli : {0,1}
• Gaussian : (-∞, +∞)

So now we have seen a few examples of sample space of some of the distribution’s family, now let’s see how a statistical model is defined:

• Bernoulli : ({0,1},(Ber(p))p∈(0,1))
• Gaussian: ((-∞, +∞),(N(𝜇,0.3))𝜇∈R)

## Specified and Misspecified Models

Model specification consists of selecting an appropriate functional form for the model. For example, given “personal income” (y) together with “years of schooling” (s) and “on-the-job experience” (x), we might specify a functional relationship y=f(s,x)} as follows:

## Model Misspecification

Has it ever happened with you that the model is converging properly on simulated data, but the moment real data comes, its robustness degrades, and it is no more converging? Well, this could typically happen if the model you developed does not match the data which is generally known as Model Misspecification. It could be because the class of distribution assumed for modeling does not contain the unknown probability distribution p from where the sample is drawn i.e. the true data generation process.

## When to Use Statistical Modelling in Data Science?

Statistical modeling in data science is invaluable in various contexts:

1. Exploratory Data Analysis: At the outset of a project, statistical models help identify trends, outliers, and relationships within the dataset, setting the stage for further analysis.
2. Hypothesis Testing: When you have a research question or hypothesis, statistical models facilitate rigorous testing, confirming or refuting assumptions.
3. Feature Selection: Statistical modeling aids in choosing relevant features for predictive models, enhancing model accuracy and interpretability.
4. Regression Analysis: When exploring relationships between variables, regression models reveal how one variable influences another, enabling predictions and insights.
5. Classification: Statistical models assist in classifying data into distinct categories, essential for tasks like sentiment analysis or disease diagnosis.
6. Anomaly Detection: Statistical models uncover unusual patterns, anomalies, or outliers in data, crucial for fraud detection or quality control.
7. Time Series Forecasting: For data with a temporal component, statistical models forecast future values, aiding in inventory management and financial predictions.
8. Segmentation Analysis: Models divide data into clusters based on similarities, enhancing customer segmentation and personalized marketing.
9. A/B Testing: Statistical modeling validates the effectiveness of changes or interventions by comparing control and experimental groups.
10. Predictive Modeling: In machine learning, statistical models predict outcomes based on historical data, essential for business forecasts and decision support.

## Conclusion

Statistical modeling is indispensable and assumptions shape our models’ quality. As you venture into data-driven decision-making, remember that a strong foundation in statistical modeling can guide you through the intricacies of real-world data. The insights gained from this journey will enhance your analytical prowess and empower your ability to unravel the patterns and difficulties hidden within complex datasets. As you embark on this path, consider taking the bold step toward mastering statistical modeling through the Blackbelt program. Equip yourself with the knowledge and skills needed to wield data as a strategic asset and harness the potential to drive innovation and informed choices across diverse domains.

Q1. What is statistical modeling with an example?

A. Statistical modeling is a process of using data to create mathematical representations of real-world phenomena. For instance, predicting housing prices based on factors like location, size, and features is a statistical model.

Q2. What is statistical modeling used for?

A. Statistical modeling helps to analyze data, make predictions, and understand relationships between variables. It aids decision-making in various fields, from finance to healthcare.

Q3. What is statistical modeling in Python?

A. Statistical modeling in Python involves using libraries like StatsModels or scikit-learn to build models. It enables data scientists to perform regression, hypothesis testing, and other analyses.

Q4. How do you write a statistical model?

A. Write a statistical model, define variables, choose an appropriate model type (e.g., linear regression), fit the model to your data, interpret results, and assess model accuracy using metrics like R-squared.  