Maths for Data Science

Key skills required

Modeling a process (physical or informational) by probing the underlying dynamics
Constructing hypotheses
Rigorously estimating the quality of the data source
Quantifying the uncertainty around the data and predictions
Identifying the hidden pattern from the stream of information
Understanding the limitation of a model
Understanding mathematical proof and the abstract logic behind it

Logarithm, exponential, polynomial functions, rational numbers
Basic geometry and theorems, trigonometric identities
Real numbers and complex numbers (basic properties)
Series, Summation, inequalities
Graphing and plotting, Cartesian coordinates / polar coordinates, conic sections

Data summaries and descriptive statistics, central tendency, variance, covariance, correlation
Basic probability: basic idea, expectation, probability calculus, Bayes' theorem, conditional probability
Probability distribution functions: uniform, normal, binomial, chi-square, Student's t-distribution, central limit theorem
Sampling, measurement, error, random number generation
Hypothesis testing, A/B testing, confidence intervals, p-values
ANOVA, t-test
Linear regression, regularization
dimensionality reduction, principle component analysis [^1]

Basic properties of matrix and vectors: scalar multiplication, linear transformation, transpose, conjugate, rank, determinant
Inner and outer products, matrix multiplication rule and various algorithms, matrix inverse
Special matrices: square matrix, identity matrix, triangular matrix, idea about sparse and dense matrix, unit vectors, symmetric matrix, Hermitian, skew-Hermitian and unitary matrices
Matrix factorization concept/LU decomposition, Gaussian/Gauss-Jordan elimination, solving Ax=b linear system of equation
Vector space, basis, span, orthogonality, orthonormality, linear least square
Eigenvalues, eigenvectors, diagonalization, singular value decomposition

Functions of a single variable, limit, continuity, differentiability
Mean value theorems, indeterminate forms, L’Hospital’s rule
Maxima and minima
Product and chain rule
Taylor’s series, infinite series summation/integration concepts
Fundamental and mean value-theorems of integral calculus, evaluation of definite and improper integrals
Beta and gamma functions
Functions of multiple variables, limit, continuity, partial derivatives
Basics of ordinary and partial differential equations

Sets Basics, subsets, power sets
Counting functions, combinatorics, countability
Basic proof techniques: induction, proof by contradiction
Basics of inductive, deductive, and propositional logic
Basic data structures: stacks, queues, graphs, arrays, hash tables, trees
Graph properties: connected components, degree, maximum flow/minimum cut concepts, graph coloring
Recurrence relations and equations
Growth of functions and O(n) notation concept

Basics of optimization, how to formulate the problem
Maxima, minima, convex function, global solution
Linear programming, simplex algorithm
Integer programming
Constraint programming, knapsack problem
Randomized optimization techniques: hill climbing, simulated annealing, genetic algorithms