Artificial Intelligence with PHP
  • Getting Started
    • Introduction
    • Audience
    • How to Read This Book
    • Glossary
    • Contributors
    • Resources
    • Changelog
  • Artificial Intelligence
    • Introduction
    • Overview of AI
      • History of AI
      • How Does AI Work?
      • Structure of AI
      • Will AI Take Over the World?
      • Types of AI
        • Limited Memory AI
        • Reactive AI
        • Theory of Mind AI
        • Self-Aware AI
    • AI Capabilities in PHP
      • Introduction to LLM Agents PHP SDK
      • Overview of AI Libraries in PHP
    • AI Agents
      • Introduction to AI Agents
      • Structure of AI Agent
      • Components of AI Agents
      • Types of AI Agents
      • AI Agent Architecture
      • AI Agent Environment
      • Application of Agents in AI
      • Challenges in AI Agent Development
      • Future of AI Agents
      • Turing Test in AI
      • LLM AI Agents
        • Introduction to LLM AI Agents
        • Implementation in PHP
          • Sales Analyst Agent
          • Site Status Checker Agent
    • Theoretical Foundations of AI
      • Introduction to Theoretical Foundations of AI
      • Problem Solving in AI
        • Introduction
        • Types of Search Algorithms
          • Comparison of Search Algorithms
          • Informed (Heuristic) Search
            • Global Search
              • Beam Search
              • Greedy Search
              • Iterative Deepening A* Search
              • A* Search
                • A* Graph Search
                • A* Graph vs A* Tree Search
                • A* Tree Search
            • Local Search
              • Hill Climbing Algorithm
                • Introduction
                • Best Practices and Optimization
                • Practical Applications
                • Implementation in PHP
              • Simulated Annealing Search
              • Local Beam Search
              • Genetic Algorithms
              • Tabu Search
          • Uninformed (Blind) Search
            • Global Search
              • Bidirectional Search (BDS)
              • Breadth-First Search (BFS)
              • Depth-First Search (DFS)
              • Iterative Deepening Depth-First Search (IDDFS)
              • Uniform Cost Search (UCS)
            • Local Search
              • Depth-Limited Search (DLS)
              • Random Walk Search (RWS)
          • Adversarial Search
          • Means-Ends Analysis
      • Knowledge & Uncertainty in AI
        • Knowledge-Based Agents
        • Knowledge Representation
          • Introduction
          • Approaches to KR in AI
          • The KR Cycle in AI
          • Types of Knowledge in AI
          • KR Techniques
            • Logical Representation
            • Semantic Network Representation
            • Frame Representation
            • Production Rules
        • Reasoning in AI
        • Uncertain Knowledge Representation
        • The Wumpus World
        • Applications and Challenges
      • Cybernetics and AI
      • Philosophical and Ethical Foundations of AI
    • Mathematics for AI
      • Computational Theory in AI
      • Logic and Reasoning
        • Classification of Logics
        • Formal Logic
          • Propositional Logic
            • Basics of Propositional Logic
            • Implementation in PHP
          • Predicate Logic
            • Basics of Predicate Logic
            • Implementation in PHP
          • Second-order and Higher-order Logic
          • Modal Logic
          • Temporal Logic
        • Informal Logic
        • Semi-formal Logic
      • Set Theory and Discrete Mathematics
      • Decision Making in AI
    • Key Application of AI
      • AI in Astronomy
      • AI in Agriculture
      • AI in Automotive Industry
      • AI in Data Security
      • AI in Dating
      • AI in E-commerce
      • AI in Education
      • AI in Entertainment
      • AI in Finance
      • AI in Gaming
      • AI in Healthcare
      • AI in Robotics
      • AI in Social Media
      • AI in Software Development
      • AI in Adult Entertainment
      • AI in Criminal Justice
      • AI in Criminal World
      • AI in Military Domain
      • AI in Terrorist Activities
      • AI in Transforming Our World
      • AI in Travel and Transport
    • Practice
  • Machine Learning
    • Introduction
    • Overview of ML
      • History of ML
        • Origins and Early Concepts
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Terms and Principles
      • Machine Learning Life Cycle
      • Problems and Challenges
    • ML Capabilities in PHP
      • Overview of ML Libraries in PHP
      • Configuring an Environment for PHP
        • Direct Installation
        • Using Docker
        • Additional Notes
      • Introduction to PHP-ML
      • Introduction to Rubix ML
    • Mathematics for ML
      • Linear Algebra
        • Scalars
          • Definition and Operations
          • Scalars with PHP
        • Vectors
          • Definition and Operations
          • Vectors in Machine Learning
          • Vectors with PHP
        • Matrices
          • Definition and Types
          • Matrix Operations
          • Determinant of a Matrix
          • Inverse Matrices
          • Cofactor Matrices
          • Adjugate Matrices
          • Matrices in Machine Learning
          • Matrices with PHP
        • Tensors
          • Definition of Tensors
          • Tensor Properties
            • Tensor Types
            • Tensor Dimension
            • Tensor Rank
            • Tensor Shape
          • Tensor Operations
          • Practical Applications
          • Tensors in Machine Learning
          • Tensors with PHP
        • Linear Transformations
          • Introduction
          • LT with PHP
          • LT Role in Neural Networks
        • Eigenvalues and Eigenvectors
        • Norms and Distances
        • Linear Algebra in Optimization
      • Calculus
      • Probability and Statistics
      • Information Theory
      • Optimization Techniques
      • Graph Theory and Networks
      • Discrete Mathematics and Combinatorics
      • Advanced Topics
    • Data Fundamentals
      • Data Types and Formats
        • Data Types
        • Structured Data Formats
        • Unstructured Data Formats
        • Implementation with PHP
      • General Data Processing
        • Introduction
        • Storage and Management
          • Data Security and Privacy
          • Data Serialization and Deserialization in PHP
          • Data Versioning and Management
          • Database Systems for AI
          • Efficient Data Storage Techniques
          • Optimizing Data Retrieval for AI Algorithms
          • Big Data Considerations
            • Introduction
            • Big Data Techniques in PHP
      • ML Data Processing
        • Introduction
        • Types of Data in ML
        • Stages of Data Processing
          • Data Acquisition
            • Data Collection
            • Ethical Considerations in Data Preparation
          • Data Cleaning
            • Data Cleaning Examples
            • Data Cleaning Types
            • Implementation with PHP
          • Data Transformation
            • Data Transformation Examples
            • Data Transformation Types
            • Implementation with PHP ?..
          • Data Integration
          • Data Reduction
          • Data Validation and Testing
            • Data Splitting and Sampling
          • Data Representation
            • Data Structures in PHP
            • Data Visualization Techniques
          • Typical Problems with Data
    • ML Algorithms
      • Classification of ML Algorithms
        • By Methods Used
        • By Learning Types
        • By Tasks Resolved
        • By Feature Types
        • By Model Depth
      • Supervised Learning
        • Regression
          • Linear Regression
            • Types of Linear Regression
            • Finding Best Fit Line
            • Gradient Descent
            • Assumptions of Linear Regression
            • Evaluation Metrics for Linear Regression
            • How It Works by Math
            • Implementation in PHP
              • Multiple Linear Regression
              • Simple Linear Regression
          • Polynomial Regression
            • Introduction
            • Implementation in PHP
          • Support Vector Regression
        • Classification
        • Recommendation Systems
          • Matrix Factorization
          • User-Based Collaborative Filtering
      • Unsupervised Learning
        • Clustering
        • Dimension Reduction
        • Search and Optimization
        • Recommendation Systems
          • Item-Based Collaborative Filtering
          • Popularity-Based Recommendations
      • Semi-Supervised Learning
        • Regression
        • Classification
        • Clustering
      • Reinforcement Learning
      • Distributed Learning
    • Integrating ML into Web
      • Open-Source Projects
      • Introduction to EasyAI-PHP
    • Key Applications of ML
    • Practice
  • Neural Networks
    • Introduction
    • Overview of NN
      • History of NN
      • Basic Components of NN
        • Activation Functions
        • Connections and Weights
        • Inputs
        • Layers
        • Neurons
      • Problems and Challenges
      • How NN Works
    • NN Capabilities in PHP
    • Mathematics for NN
    • Types of NN
      • Classification of NN Types
      • Linear vs Non-Linear Problems in NN
      • Basic NN
        • Simple Perceptron
        • Implementation in PHP
          • Simple Perceptron with Libraries
          • Simple Perceptron with Pure PHP
      • NN with Hidden Layers
      • Deep Learning
      • Bayesian Neural Networks
      • Convolutional Neural Networks (CNN)
      • Recurrent Neural Networks (RNN)
    • Integrating NN into Web
    • Key Applications of NN
    • Practice
  • Natural Language Processing
    • Introduction
    • Overview of NLP
      • History of NLP
        • Ancient Times
        • Medieval Period
        • 15th-16th Century
        • 17th-18th Century
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • NLP and Text
      • Key Concepts in NLP
      • Common Challenges in NLP
      • Machine Learning Role in NLP
    • NLP Capabilities in PHP
      • Overview of NLP Libraries in PHP
      • Challenges in NLP with PHP
    • Mathematics for NLP
    • NLP Techniques
      • Basic Text Processing with PHP
      • NLP Workflow
      • Popular Tools and Frameworks for NLP
      • Techniques and Algorithms in NLP
        • Basic NLP Techniques
        • Advanced NLP Techniques
      • Advanced NLP Topics
    • Integrating NLP into Web
    • Key Applications of NLP
    • Practice
  • Computer Vision
    • Introduction
  • Overview of CV
    • History of CV
    • Common Use Cases
  • CV Capabilities in PHP
  • Mathematics for CV
  • CV Techniques
  • Integrating CV into Web
  • Key Applications of CV
  • Practice
  • Robotics
    • Introduction
  • Overview of Robotics
    • History and Evolution of Robotics
    • Core Components
      • Sensors (Perception)
      • Actuators (Action)
      • Controllers (Processing and Logic)
    • The Role of AI in Robotics
      • Object Detection and Recognition
      • Path Planning and Navigation
      • Decision Making and Learning
  • Robotics Capabilities in PHP
  • Mathematics for Robotics
  • Building Robotics
  • Integration Robotics into Web
  • Key Applications of Robotics
  • Practice
  • Expert Systems
    • Introduction
    • Overview of ES
      • History of ES
        • Origins and Early ES
        • Milestones in the Evolution of ES
        • Expert Systems in Modern AI
      • Core Components and Architecture
      • Challenges and Limitations
      • Future Trends
    • ES Capabilities in PHP
    • Mathematics for ES
    • Building ES
      • Knowledge Representation Approaches
      • Inference Mechanisms
      • Best Practices for Knowledge Base Design and Inference
    • Integration ES into Web
    • Key Applications of ES
    • Practice
  • Cognitive Computing
    • Introduction
    • Overview of CC
      • History of CC
      • Differences Between CC and AI
    • CC Compatibilities in PHP
    • Mathematics for CC
    • Building CC
      • Practical Implementation
    • Integration CC into Web
    • Key Applications of CC
    • Practice
  • AI Ethics and Safety
    • Introduction
    • Overview of AI Ethics
      • Core Principles of AI Ethics
      • Responsible AI Development
      • Looking Ahead: Ethical AI Governance
    • Building Ethics & Safety AI
      • Fairness, Bias, and Transparency
        • Bias in AI Models
        • Model Transparency and Explainability
        • Auditing, Testing, and Continuous Monitoring
      • Privacy and Security in AI
        • Data Privacy and Consent
        • Safety Mechanisms in AI Integration
        • Preventing and Handling AI Misuse
      • Ensuring AI Accountability
        • Ethical AI in Decision Making
        • Regulations & Compliance
        • AI Risk Assessment
    • Key Applications of AI Ethics
    • Practice
  • Epilog
    • Summing-up
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  • Assumptions of Simple Linear Regression
  • Assumptions of Multiple Linear Regression
  1. Machine Learning
  2. ML Algorithms
  3. Supervised Learning
  4. Regression
  5. Linear Regression

Assumptions of Linear Regression

PreviousGradient DescentNextEvaluation Metrics for Linear Regression

Last updated 1 month ago

Assumptions of Simple Linear Regression

Assumptions of Linear Regression

Linear regression relies on several key assumptions for optimal performance:

  1. Linearity: The relationship between independent and dependent variables is linear.

  2. Independence: Observations are independent of each other.

  3. Homoscedasticity: The variance of residual errors is constant across all levels of the independent variables.

  4. Normality: Residuals (errors) are normally distributed.

  5. No Multicollinearity (in multiple linear regression): Independent variables should not be highly correlated with each other.

Violating these assumptions can lead to inaccurate predictions, so it’s essential to check them before finalizing a model.

Assumptions of Simple Linear Regression

Linear regression is a powerful tool for understanding and predicting the behavior of a variable, however, it needs to meet a few conditions in order to be accurate and dependable solutions.

  1. Linearity: The independent and dependent variables have a linear relationship with one another. This implies that changes in the dependent variable follow those in the independent variable(s) in a linear fashion. This means that there should be a straight line that can be drawn through the data points. If the relationship is not linear, then linear regression will not be an accurate model.\

  2. Independence: The observations in the dataset are independent of each other. This means that the value of the dependent variable for one observation does not depend on the value of the dependent variable for another observation. If the observations are not independent, then linear regression will not be an accurate model.

  3. Homoscedasticity: Across all levels of the independent variable(s), the variance of the errors is constant. This indicates that the amount of the independent variable(s) has no impact on the variance of the errors. If the variance of the residuals is not constant, then linear regression will not be an accurate model.\

    Homoscedasticity in Linear Regression\

  4. Normality: The residuals should be normally distributed. This means that the residuals should follow a bell-shaped curve. If the residuals are not normally distributed, then linear regression will not be an accurate model.

Assumptions of Multiple Linear Regression

For Multiple Linear Regression, all four of the assumptions from Simple Linear Regression apply. In addition to this, below are few more:

  1. No multicollinearity: There is no high correlation between the independent variables. This indicates that there is little or no correlation between the independent variables. Multicollinearity occurs when two or more independent variables are highly correlated with each other, which can make it difficult to determine the individual effect of each variable on the dependent variable. If there is multicollinearity, then multiple linear regression will not be an accurate model.

  2. Additivity: The model assumes that the effect of changes in a predictor variable on the response variable is consistent regardless of the values of the other variables. This assumption implies that there is no interaction between variables in their effects on the dependent variable.

  3. Feature Selection: In multiple linear regression, it is essential to carefully select the independent variables that will be included in the model. Including irrelevant or redundant variables may lead to overfitting and complicate the interpretation of the model.

  4. Overfitting: Overfitting occurs when the model fits the training data too closely, capturing noise or random fluctuations that do not represent the true underlying relationship between variables. This can lead to poor generalization performance on new, unseen data.

Multicollinearity

Multicollinearity is a statistical phenomenon that occurs when two or more independent variables in a multiple regression model are highly correlated, making it difficult to assess the individual effects of each variable on the dependent variable.

Detecting Multicollinearity includes two techniques:

  • Correlation Matrix: Examining the correlation matrix among the independent variables is a common way to detect multicollinearity. High correlations (close to 1 or -1) indicate potential multicollinearity.

  • VIF (Variance Inflation Factor): VIF is a measure that quantifies how much the variance of an estimated regression coefficient increases if your predictors are correlated. A high VIF (typically above 10) suggests multicollinearity.