Here is a a comprehensive road-map for learning mathematics for working in the field of Data Science and Machine Learning. This road-map includes applications of each subject for each field and resources and tutorials for each topic.

# Basics of Mathematics

Start with the fundamentals of mathematics such as basic algebra, calculus, and statistics.

Algebra is used to manipulate data sets and solve data-related problems, such as time-series. Calculus is used in optimization functions and variables, while statistics is used for data analysis and modelling.

## Topics and Concepts

- Algebra
- Basics of Linear Algebra

- Calculus
- Single Variable Calculus
- Differentiation
- Integration
- Concepts of Extrema, Minima or Maxima
- Limits

- Statistics
- Basics of probability theory
- Basics of Inferential Statistics

- The concept of “Expected Value” in statistics and integration

## Learning Resources

- Basics of Linear Algebra
- Calculus
- Statistics
- Khan Academy – High school statistics: Probability, Probability distributions & expected value

# Linear and Nonlinear Algebra

Study matrices and their operations, linear transformations and nonlinear transformations, and eigenvectors. These are important concepts in Regression Models and performing predictions, or grouping tasks(classification and clustering).

## Applications

- Performing mathematical operations on multidimensional datasets
- Classification
- Regression Models
- Dimensionality Reduction
- High-Dimensional Data Processing(E.g. video, image, voice, …)
- Optimization Tasks(E.g. Gradient Descent)
- Classification
- Deep Learning
- Feature Extraction

## Topics and Concepts

- Vector Spaces (transformations and matrices)
- Mathematical Operations of Matrices
- Determinants
- Eigenvalues and Eigenvectors
- Matrix Decomposition

## Learning Resources

- Khan Academy – Algebra 2
- Khan Academy – College Algebra
- Khan Academy – Linear algebra
- Imperial College London – Mathematics for Machine Learning: Linear Algebra

# Calculus and Optimization

Learn differential and integral calculus, which are essential for understanding optimization algorithms and plays a great role in machine learning algorithms. Then continue with Multivariable Calculus which is an advanced course in calculus that includes topics such as gradients, optimization, and multivariate integration.

Finally, study optimization algorithms and techniques which plays an essential role in machine learning algorithms.

Note that familiarity with calculus functions to the point of knowing their behavior and changes is enough to work on most optimization algorithms.

## Applications

- Optimization
- Machine Learning & Deep Learning including training Artificial Neural Networks(ANN)
- Data Analysis
- Signal Processing

## Topics and Concepts

- Multivariable calculus
- Differential
- Integral Calculus
- Multiple Integral

- Partial Derivatives
- Higher Order Derivatives
- Hessian Derivatives
- Jacobian Derivatives

- Integral Transform, specially Fourier transform
- Vector-Values Functions
- Distribution
- Hessian Distributions
- Jacobian Distributions
- Laplacian Distributions
- Lagrangian Distributions

- Optimization techniques
- Convex Optimization
- Gradients and their properties, Including:
- Directional Gradients
- Exploding Gradient
- Stochastic Gradient

- Other techniques used in Hyperparameter Optimization of ANNs
- Bayesian Optimization
- Concept and algorithms behind Evolutionary Methods

## Learning Resources

- Khan Academy – Calculus 1
- Khan Academy – Calculus 2
- Khan Academy – Khan Academy – multivariable calculus
- Khan Academy – Differential equations
- MIT – Multivariable Calculus
- Imperial College London – Multivariate Calculus

# Probability & Statistics

Study probability and statistics, which are the foundations of machine learning and data science. Focus on concepts like random variables, expectation, variance, hypothesis testing, and Bayesian statistics.

## Applications

- Inference and Estimation
- Machine Learning
- Data Analysis
- Predictive Modelling
- Experimental Design and Analysis

## Topics and Concepts

- Probability theory
- Sets and measures
- Events and Rules of probability
- Probability Functions
- Expected value
- Random Variables
- Distributions & Densities
- Entropy
- Bias and Variance
- Hypothesis Testing

- Statistics
- Descriptive Statistics & it’s measures, Inferential Statistics
- Bayesian Statistics & Chain Rule
- Parameter Estimation
- Concepts of Fitting(OverFitting and Underfitting)

## Learning Resources

# Further Steps for Mastering Data Science and Machine Learning

## Computer Science Fundamentals

Gain a good understanding of computer science concepts such as algorithms, data structures, and programming. You will need this knowledge to implement machine learning algorithms.

## Programming Languages

Learn at least one programming language to practice and master data science and machine learning by doing real experiments and creating real-world applications. Python is the best programming language for this purpose, however those who enter this field from the field of mathematics and statistics often prefer R, While web developers often prefer JavaScript or Java.

## Data Modelling and Analysis Techniques

Once you have learned programming and theoretical knowledge, it’s time to put them to work. Learn more about statistical modelling techniques such as linear regression, logistic regression, decision trees, and neural networks by running real experiments.

## Advanced Machine Learning Techniques

Study advanced topics in machine learning, such as deep learning and reinforcement learning. These are techniques that are used in artificial intelligence to develop intelligent agents and models.

## Real-World Applications and Projects

Finally, work on real-world data science and artificial intelligence projects to apply the knowledge you have gained. Building predictive models, working with big data, and developing intelligent bots and virtual assistants are some examples of projects you can start with.

# References

Here are most popular reference books for Data Science & Machine Learning:

- Probability Theory: The Logic of Science by E. T. Jaynes
- The Matrix Calculus You Need For Deep Learning by Terence Parr, Jeremy Howard
- Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong
**Pattern Recognition and Machine**Learning by Christopher M. Bishop**The Elements of Statistical Learning**by Jerome H. Friedman, Robert Tibshirani, Trevor Hastie