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Reading list

Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

What is Statistical Modeling? Definition, Types, Uses and More

V
Vidhi Last Updated : 12 Jul, 2024
7 min read

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.

Table of contents

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 Modeling?

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.

Also Read: All About Hypothesis Testing

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:

Statistical Modeling model specification

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.

model misspecification
Source: Author

Statistical Modeling vs Machine Learning

AspectMachine LearningStatistical Modeling
FocusAlgorithms that enable systems to learn patterns from data.Building mathematical models to explain relationships between variables.
GoalPrediction, classification, clustering, pattern recognition, etc.Inference, understanding relationships, hypothesis testing.
Data SizeHandles large and complex datasets with features selection.Can handle small to large datasets, but typically requires domain knowledge for feature selection.
FlexibilityAdaptable to various tasks and data types.Limited flexibility, often specific to a particular hypothesis.
ComplexityCan handle complex patterns and nonlinear relationships.Typically focuses on simpler models with interpretability.
AutomationEmphasizes automation and optimization of model performance.Requires manual feature engineering and model selection.
InterpretabilitySome models like decision trees are interpretable.Often provides more interpretable results, aiding in understanding relationships.
Training TimeLonger training times for complex models.Shorter training times for simpler models.
ExamplesNeural networks, Random Forests, Support Vector Machines.Linear regression, logistic regression, ANOVA.

Statistical Modelling vs Mathematical Modelling

AspectStatistical ModelingMathematical Modeling
FocusCaptures relationships and patterns in data.Represents real-world situations using equations.
Data UsageUtilizes empirical data to build models.Often uses theoretical or assumed data.
AssumptionsModels may rely on assumptions about data distribution.Relies on assumptions about relationships between variables.
GoalInference, hypothesis testing, understanding relationships.Solving complex problems through mathematical equations.
ApplicationsPredictive analytics, decision-making, hypothesis testing.Physical sciences, engineering, economic models.
Model ComplexityCan handle complex real-world patterns and noise.Can represent intricate systems and interactions.
InterpretabilityOften provides insights into data relationships.Focuses on understanding mathematical relationships.
VariablesIncorporates real data variables and interactions.Utilizes mathematical variables and constants.
ValidationInvolves testing against empirical data.Validates against theoretical results or experiments.
ExampleLinear regression, ANOVA.Differential equations, optimization models.

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.

Frequently Asked Questions

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.

 

V
Vidhi

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Responses From Readers

Clear

dhruv
dhruv

Thanks Maanvi. Interesting article. Just a suggestion: maybe you can think of adding a simple example of some of the cases, so newcomers to statistics find it even more easier to understand from a practical view point.

123

geraldine P. pacheco
geraldine P. pacheco

thank you for sharing this article it really help my study as a guide

123

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