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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

Understanding the Log-normal Distribution

N
NISHANT TIWARI Last Updated : 18 Jun, 2024
4 min read

Introduction

The log-normal distribution is a fascinating statistical concept commonly used to model data that exhibit right-skewed behavior. This distribution has wide-ranging applications in various fields, such as biology, finance, and engineering. In this article, we will delve into the log-normal distribution, its key parameters, and how to interpret them, as well as explore some practical examples to enhance understanding.

Overview

  • A log-normal distribution models data where the natural logarithm of a variable follows a normal distribution, showing positive skewness.
  • Understand the shape (ฯƒ), scale (m or eฮผ), and location (ฮผ) parameters to interpret and apply the log-normal distribution.
  • The log-normal distribution is connected to the normal distribution; if X is log-normal, ln(X) is usually distributed, and vice versa.
  • Estimate parameters ฮผ ฯƒ from data using techniques like Maximum Likelihood Estimation, which involves log transformation and calculating the mean and standard deviation.
  • The log-normal distribution is widely used in biology, finance, reliability engineering, and environmental science to model right-skewed data such as growth rates, stock prices, and time to failure.

Table of contents

What is a Log-normal Distribution?

A log-normal distribution describes the probability distribution of a random variable when its logarithm follows a normal distribution. In simpler terms, if the natural logarithm of a variable X follows a normal distribution, then X follows a log-normal distribution. This distribution remains continuous and features positive skewness, meaning it has a long right tail.

Key Parameters

There are primarily three parameters as follows:

  1. Shape Parameter (ฯƒ): This parameter affects the general shape of the distribution. It is also the standard deviation of the log-transformed variable.
  2. Scale Parameter (m or eฮผ): This parameter stretches or shrinks the distribution’s graph. In this distribution, the scale parameter is often called the median.
  3. Location Parameter (ฮผ): This parameter determines where on the x-axis the graph is located. It is the mean of the log-transformed variable.

These parameters are critical in understanding how this distribution behaves and how it can be applied to real-world data.

Also Read: What is Normal Distribution : An Ultimate Guide

Probability Density Function

The probability density function (PDF) of a log-normal distribution is given by:

Probability Density Function | Log-normal Distribution

where x>0, ฮผ is the mean of the variable’s logarithm, and ฯƒ is the standard deviation of the variable’s logarithm. This formula shows that the log-normal distribution is defined for positive values only, as the logarithm is not defined for non-positive values.

Relationship with the Normal Distribution

One of the most interesting aspects of its relationship with the normal distribution. If X follows a log-normal distribution, Y = ln(X) follows a normal distribution. Conversely, if Y follows a normal distribution, X = eY follows a log-normal distribution. This relationship allows us to use well-established methods for normal distributions to analyze log-normal data by transforming the data using logarithms.

Calculating Parameters from Data

We often use methods such as Maximum Likelihood Estimation (MLE) to estimate the parameters of this form of distribution from data. Hereโ€™s a simplified approach to estimate ฮผ and ฯƒ:

  1. Log-transform the data: Take the natural logarithm of all data points.
  2. Calculate the log-transformed data’s sample mean and standard deviation: These statistics will be the estimates for ฮผ and ฯƒ.

For example, consider a dataset of log-normally distributed incomes. By taking the natural logarithm of each income, we can compute the mean and standard deviation of these log-transformed values to estimate ฮผ and ฯƒ.

Practical Applications

This distribution is widely used in various fields due to its ability to model skewed data. Here are some examples:

  1. Biology: In biological studies, organisms’ growth rates often follow a log-normal distribution because growth rates are multiplicative rather than additive.
  2. Finance: Stock prices are commonly modeled using log-normal distributions because the percentage change in prices is normally distributed.
  3. Reliability Engineering: The time to failure of certain products can be modeled using a log-normal distribution, especially when the failure process is multiplicative.
  4. Environmental Science: The distribution of particle sizes in aerosols or the amount of rainfall in a given period.

Example Calculation

Letโ€™s consider a practical example to calculate the parameters of a log-normal distribution. Assume we have the following income data (in thousands): 20, 22, 25, 27, 30.

  • Log-transform the data:
Example of Log-normal Distribution
  • Calculate the sample mean ฮผ:
Example of Log-normal Distribution
  • Calculate the sample standard deviation (ฯƒ):
Example of Log-normal Distribution

Thus, the estimated parameters for the log-normal distribution are ฮผ approx 3.2005 and ฯƒ approx 0.1504.

Interpreting the Parameters

  • ฮผ: This is the mean of the log-transformed data. In our example, a ฮผ of 3.2005 indicates that the average of the natural logarithms of the incomes is around this value.
  • ฯƒ: This is the standard deviation of the log-transformed data. A ฯƒ of 0.1504 suggests that the log-transformed incomes are relatively close to the mean on a logarithmic scale.

Conclusion

The log-normal distribution is a powerful tool for modeling right-skewed data. We can effectively analyze and interpret data in various fields by understanding its key parameters and their relationship with the normal distribution. Whether dealing with financial data, biological growth rates, or reliability metrics, it offers a robust framework for understanding and predicting behavior.

Frequently Asked Questions

Q1. How do you interpret a lognormal distribution?

A. A lognormal distribution describes a variable whose logarithm is normally distributed, meaning the original variable is positively skewed and multiplicative factors cause its variation.

Q2. What is the log of a normal distribution curve?

A. The log of a normal distribution curve converts a lognormal distribution into a normal distribution, meaning if ๐‘‹, is lognormally distributed, ln(๐‘‹), is normally distributed.

Q3. Why is the log-normal distribution important?

A. The log-normal distribution is important because it models many natural phenomena and financial variables where values are positively skewed, and it helps in understanding and predicting multiplicative processes.

Log-normal Distribution
N
NISHANT TIWARI

Seasoned AI enthusiast with a deep passion for the ever-evolving world of artificial intelligence. With a sharp eye for detail and a knack for translating complex concepts into accessible language, we are at the forefront of AI updates for you. Having covered AI breakthroughs, new LLM model launches, and expert opinions, we deliver insightful and engaging content that keeps readers informed and intrigued. With a finger on the pulse of AI research and innovation, we bring a fresh perspective to the dynamic field, allowing readers to stay up-to-date on the latest developments.

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