PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Probability and Statistics → Statistical Inference ↓


Regression Analysis


Regression analysis is a statistical method used for modeling and analyzing the relationship between a dependent variable and one or more independent variables. It is a fundamental tool in statistical inference, widely used to predict the value of a dependent variable based on the values of the independent variables. This method also helps us understand the strength and nature of the relationship between variables.

Introduction to regression analysis

At its core, regression analysis involves finding the best-fit line or curve that describes the data points in your dataset. This relationship is typically expressed as an equation where the coefficients represent the strength of the effect of each independent variable on the dependent variable.

There are different types of regression analysis depending on the type of data and the relationship we suspect. The most common types include:

  • linear regression
  • multiple linear regression
  • Polynomial regression
  • logistic regression

Linear regression

Let's start with linear regression, which is the simplest form of regression. In linear regression, we attempt to model the relationship between two variables by fitting a linear equation to the observed data. One variable is considered the explanatory variable (independent), and the other is considered the dependent variable.

Simple linear regression

Simple linear regression represents the relationship between a dependent variable y and an independent variable x by the following equation:

y = β₀ + β₁x + ε
  • y is the dependent variable we are trying to predict.
  • β₀ is the intercept of the line with the y-axis.
  • β₁ is the slope of the line.
  • ε is the error term, which represents the variability in y not explained by the model.

Example of simple linear regression

Suppose we are investigating the relationship between temperature and the number of ice creams sold. Here is a scatter plot showing this relationship:


    
    
    
    
    
    
    
    
    
    temperature
    ice cream sold

Each point on the graph represents one day. Our goal is to find a line that best fits all of these points, suggesting that as the temperature increases, more ice cream is sold. The fitting line is estimated using the least squares method, which minimizes the sum of squared differences between the observed values and the values predicted by the line.

Multiple linear regression

When a single independent variable is not enough to accurately predict the dependent variable, we use multiple linear regression. It involves more than one independent variable (x₁, x₂, ..., xn) to predict the dependent variable y. The equation looks like this:

y = β₀ + β₁x₁ + β₂x₂ + ... + βnxn + ε

Example of multiple linear regression

Consider predicting the price of a home based on the number of bedrooms, the size of the home in square feet, and the neighborhood quality index. The model might look something like this:

price = β₀ + β₁ * bedrooms + β₂ * size + β₃ * neighborhood + ε

Each β coefficient estimates the change in house price associated with a one unit change in the predictor variable, holding all other predictors constant.

Polynomial regression

Polynomial regression is an extension of linear regression that is used when the relationship between the independent variable x and the dependent variable y is curvilinear. The polynomial regression equation is expressed as:

y = β₀ + β₁x + β₂x² + ... + βnxⁿ + ε

Example of polynomial regression

An example of polynomial regression could be modeling plant growth over time, where the growth rate speeds up and then slows down as the plant matures.


    
    
    
    
    
    
    
    
    
    Time
    Plant growth

Logistic regression

Logistic regression is used to model the probability of a binary outcome based on one or more predictor variables. Unlike linear regression, in logistic regression, the outcome variable is categorical, meaning it is a binary outcome where the data can only fall into one of two categories.

The formula used in logistic regression is the logistic function:

p = 1 / (1 + e^-(β₀ + β₁x₁ + β₂x₂ + ... + βnxn))

Example of logistic regression

A practical example involves whether a customer will buy a product (1) or not (0) based on factors such as income and age.

Assumptions in regression analysis

For regression analysis to be valid, certain assumptions must hold:

  • Linearity: The relationship between the independent and dependent variables must be linear.
  • Independence: The residuals (errors) must be independent.
  • Homoskedasticity: The residuals should have constant variance at all levels of the independent variable.
  • Normality: The residuals should be normally distributed.

Conclusion

In conclusion, regression analysis is a powerful tool for understanding the relationships between variables. It is essential for making predictions and providing insights based on data. While linear regression is the simplest form of regression analysis, understanding the broader context of polynomial, and logistic regression provides a comprehensive toolkit for tackling a wide range of statistical estimation problems.

Applying regression analysis within the framework of these assumptions leads to more accurate and reliable predictive models, helping researchers and professionals to make informed decisions based on empirical data.


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