Machine Learning Mathematics

Machine Learning is the field of study that gives computers the capability to learn without being explicitly programmed. Math is the core concept in machine learning which is used to express the idea within the machine learning model.

Mathematics for Machine Learning

In this tutorial, we will look at different mathematics concepts and will learn about these modules from basic to advance with the help particular algorithm.

Linear Algebra and Matrix	Regression
Geometry	Dimensionality Reduction
Vector Calculus	Vector Models
Probability Distribution	Miscellaneous

Linear Algebra and Matrix

Linear Algebra is an algebra extension to an undefined number of dimensions. Linear Algebra concerns the focus on linear equation systems. 

• Vectors and Matrices
  • Matrix Introduction
  • Matrix Addition
    • Matrix Addition using NumPy Arrays
  • Matrix Multiplication
    • Matrix Multiplication using Python
  • Matrix Manipulation using NumPy Arrays
  • Inverse of a Matrix
    • Evaluating Inverse using NumPy Arrays
  • Transpose of a Matrix
    • Evaluating Transpose using NumPy Arrays
  • Properties of Matrix
  • Determinant
  • Trace
• System of Linear Equations
  • System of Linear Equation
  • Solving Linear Equations using Gaussian Elimination 
  • LU Decomposition of Linear Equation
  • Matrix Inversion
• Matrix Factorization
  • Gram-Schmidt Process
  • QR Decomposition
  • Cholesky Decomposition
  • Singular Value Decomposition
  • Matrix Factorization
  • Diagonalization
  • Eigenvalues and Eigenvectors
  • Eigenspace
• Vector Spaces
  • Vector Operations
  • Vector Spaces and SubSpaces
  • Basis and Dimension
• Row Echelon Form
• Linear Mappings
• Least Square and Curve Fitting
• Affine Spaces

Statistics

Statistics is the collection of data, tabulation, and interpretation of numerical data, and it is applied mathematics concerned with data collection analysis, interpretation, and presentation.

• Mean, Standard Deviation, and Variance
  • Calculating Mean, Standard Deviation, and Variance using Numpy Arrays
• Sample Error and True Error
• Bias Vs Variance and Its Trade-Off
• Hypothesis Testing
  • T-test
  • Paired T-test
  • p-value
  • F-Test
  • z-test
• Confidence Intervals
• Correlation and Covariance
• Correlation Coefficient
• Covariance Matrix
• Normal Probability Plot
• Q-Q Plot
• Residuals Leverage Plot
• Robust Correlations
• Hypothesis Testing
  • Null and Alternative Hypothesis
  • Type 1 and Type 2 Errors
  • p-value interaction
  • Parametric Hypothesis Testing
    • T-test
    • Paired Samples t-test
    • ANOVA Test
  • Non-Parametric Hypothesis Testing
    • Mann-Whitney U test
    • Wilcoxon signed-rank test
    • Kruskal-Wallis test
    • Friedman test
• Theory of Estimation
  • Difference between Estimators and Estimation
  • Methods of Estimation
    • Method of Moments
    • Bayesian Estimation
    • Least Square Estimation
    • Maximum Likelihood Estimation
  • Likelihood Function and Log-Likelihood Function
  • Properties of Estimation
    • Unbiasedness
    • Consistency
    • Sufficiency
    • Completeness
    • Robustness
• Confidence Intervals

Geometry

Geometry is the branch of mathematics that deals with the forms, angles, measurements, and proportions of ordinary objects.

• Vector Norms
• Inner, Outer, Cross Products
• Distance Between Two Points
• Distance Measures
  • Euclidean Distance
  • Manhattan Distance
  • Minkowski Distance
  • Chebysev Distance
• Similarity Measures
  • Cosine Similarity
  • Jaccard Similarity
  • Pearson Correlation Coefficient
  • Kendall Rank Correlation Measure
  • Pearson Product-Moment Correlations
  • Spearman’s Rank Correlation Measure
• Orthogonality and Orthogonal Projections
  • Orthogonality and Orthonormal Vectors
  • Orthogonal Projections
  • Rotations
• Geometric Algorithms
  • Nearest Neighbor Search
  • Voronoi diagrams
  • Delaunay Triangulation
  • Geometric intersection and Proximity queries
• Constraints and Splines
• Box-Cox Transformations
  • Box-Cox Transformation using Python
• Fourier transformation
  • Properties of Fourier Transform
• Inverse Fast Fourier Transformation

Calculus

Calculus is a subset of mathematics concerned with the study of continuous transition. Calculus is also known as infinitesimal calculus or “infinite calculus.” The analysis of continuous change of functions is known as classical calculus

• Differentiation
  • Implicit Differentiation
  • Inverse Trigonometric Functions Differentiation
  • Logarithmic Differentiation
  • Partial Differentiation
  • Advanced Differentiation
• Mathematical Intuition Behind Gradients and their usage
  • Implementation of Gradients using Python
  • Optimization Techniques using Gradient Descent
• Higher-Order Derivatives
• Multivariate Taylor Series
• Application of Derivation
  • Application of Derivative – Maxima and Minima
  • Absolute Minima and Maxima
  • Constrained Optimization
  • Unconstrained Optimization
  • Constrained Optimization – Lagrange Multipliers
  • Newton’s Method
• Uni-variate Optimization
• Multivariate Optimization
• Convex Optimization
• Lagrange’s Interpolation
• Area Under Curve

Probability and Distributions

Probability and distributions are statistical functions that describe all the possible values.

• Probability
• Chance and Probability
• Addition Rule for Probability
• Law of total probability
• Bayes’ Theorem
• Discrete Probability Distributions
  • Discrete Uniform Distribution
  • Bernoulli Distribution
  • Binomial Distribution
  • Poisson Distribution
• Continuous Probability Distributions
  • Continuous Uniform Distribution
  • Exponential Distribution
  • Normal Distribution
  • Beta Distribution
    • Beta Distribution of First Kind
    • Beta Distribution of Second Kind
  • Gamma Distribution
• Sampling Distributions
  • Chi-Square Distribution
  • F – Distribution
  • t – Distribution
• Central Limit Theorem
  • Implementation of Central Limit Theorem
• Law of Large Numbers
• Change of Variables/Inverse Transformation

Regression

Regression is a statistical process for estimating the relationships between the dependent variables or criterion variables 

• Parameter Estimation
• Bayesian Linear Regression
• Quantile Linear Regression
• Normal Equation in Linear Regression
• Maximum Likelihood as Orthogonal Projection

Dimensionality Reduction

Dimensionality reduction is a technique to reduce the number of input variables in training data.

• Introduction to Dimensionality Reduction
• Projection Perspective in Machine Learning
• Eigenvector Computation and Low-Rank Approximations
• Mathematical Intuition Behind PCA
  • PCA implementation in Python
• Latent Variable Perspective
• Mathematical Intuition Behind LDA
  • Implementation of Linear Discriminant Analysis (LDA)
• Mathematical Intuition Behind GDA
  • Implementation of Generalized Discriminant Analysis (GDA)
• Mathematical Intuition Behind t-SNE Algorithm
  • Implementation of the t-SNE Algorithm