# Essential Math for Data Science

Mathematics is the bedrock of any contemporary discipline of science.

Almost all the techniques of modern data science, including machine learning, have a deep mathematical underpinning.

It goes without saying that you will absolutely need all the other pearls of knowledge—programming ability, some amount of business acumen, and your unique analytical and inquisitive mindset—about the data to function as a top data scientist.

But it always pays to know the machinery under the hood, rather than just being the person behind the wheel with no knowledge about the car.

Therefore, a solid understanding of the mathematical machinery behind the cool algorithms will give you an edge among your peers.

The knowledge of this essential math is particularly important for newcomers arriving to data science from other professions: hardware engineering, retail, the chemical process industry, medicine and health care, business management, etc.

Although such fields may require experience with spreadsheets, numerical calculations, and projections, the math skills required in data science can be significantly different.

Consider a web developer or business analyst.

They may be dealing with a lot of data and information on a daily basis, but there may not be an emphasis on rigorous modeling of that data.

Often, the emphasis is on using the data for an immediate need and moving on, rather than on deep scientific exploration.

Data science, on the other hand, should always be about the science (not the data).

Following that thread, certain tools and techniques become indispensable.

Most are the hallmarks of sound scientific process:Modeling a process (physical or informational) by probing the underlying dynamicsConstructing hypothesesRigorously estimating the quality of the data sourceQuantifying the uncertainty around the data and predictionsIdentifying the hidden pattern from the stream of informationUnderstanding the limitation of a modelUnderstanding a mathematical proof and the abstract logic behind itData science, by its very nature, is not tied to a particular subject area and may deal with phenomena as diverse as cancer diagnoses and social behavior analysis.

This produces the possibility of a dizzying array of n-dimensional mathematical objects, statistical distributions, optimization objective functions, etc.

Here are my suggestions for the topics to study to be at the top of the game in data science.

Functions, Variables, Equations, and GraphsThis area of math covers the basics, from the equation of a line to binomial theorem and its properties:Logarithm, exponential, polynomial functions, rational numbersBasic geometry and theorems, trigonometric identitiesReal and complex numbers, basic propertiesSeries, sums, inequalitiesGraphing and plotting, Cartesian and polar coordinates, conic sectionsWhere You Might Use ItIf you want to understand how a search runs faster on a million-item database after you’ve sorted it, you will come across the concept of “binary search.

” To understand the dynamics of it, you need to understand logarithms and recurrence equations.

Or, if you want to analyze a time series, you may come across concepts like “periodic functions” and “exponential decay.

”Where You Can Learn ItCoursera: Data science math skillsedX: Introduction to algebraKhan Academy: Algebra IStatisticsCredit: geralt/Pixabay/CC0The importance of having a solid grasp over essential concepts of statistics and probability cannot be overstated.

Many practitioners in the field actually consider classical (non-neural network) machine learning to be nothing but statistical learning.

The subject is vast, and focused planning is critical to cover the most essential concepts:Data summaries and descriptive statistics, central tendency, variance, covariance, correlationBasic probability: basic idea, expectation, probability calculus, Bayes’ theorem, conditional probabilityProbability distribution functions: uniform, normal, binomial, chi-square, student’s t-distribution, central limit theoremSampling, measurement, error, random number generationHypothesis testing, A/B testing, confidence intervals, p-valuesANOVA, t-testLinear regression, regularizationWhere You Might Use ItIn interviews.

If you can show you’ve mastered these concepts, you will impress the other side of the table fast.

And you will use them nearly every day as a data scientist.

Where You Can Learn ItCoursera: Statistics with R specializationCoursera: Business statistics and analysis specializationedX: Statistics and probability in data science using PythonLinear AlgebraThis is an essential branch of mathematics for understanding how machine-learning algorithms work on a stream of data to create insight.

Everything from friend suggestions on Facebook, to song recommendations on Spotify, to transferring your selfie to a Salvador Dali-style portrait using deep transfer learning involves matrices and matrix algebra.

Here are the essential topics to learn:Basic properties of matrix and vectors: scalar multiplication, linear transformation, transpose, conjugate, rank, determinantInner and outer products, matrix multiplication rule and various algorithms, matrix inverseSpecial matrices: square matrix, identity matrix, triangular matrix, idea about sparse and dense matrix, unit vectors, symmetric matrix, Hermitian, skew-Hermitian and unitary matricesMatrix factorization concept/LU decomposition, Gaussian/Gauss-Jordan elimination, solving Ax=b linear system of equationVector space, basis, span, orthogonality, orthonormality, linear least squareEigenvalues, eigenvectors, diagonalization, singular value decompositionWhere You Might Use ItIf you have used the dimensionality reduction technique principal component analysis, then you have likely used the singular value decomposition to achieve a compact dimension representation of your data set with fewer parameters.

All neural network algorithms use linear algebra techniques to represent and process the network structures and learning operations.

Where You Can Learn ItedX: Linear algebra: foundations to frontiersCoursera: Mathematics for machine learning: linear algebraCalculusWhether you loved or hated it in college, calculus pops up in numerous places in data science and machine learning.

It lurks behind the simple-looking analytical solution of an ordinary least squares problem in linear regression or embedded in every back-propagation your neural network makes to learn a new pattern.

Here are the topics to learn:Functions of single variable, limit, continuity, differentiabilityMean value theorems, indeterminate forms, L’Hospital’s ruleMaxima and minimaProduct and chain ruleTaylor’s series, infinite series summation/integration conceptsFundamental and mean value-theorems of integral calculus, evaluation of definite and improper integralsBeta and gamma functionsFunctions of multiple variables, limit, continuity, partial derivativesBasics of ordinary and partial differential equationsWhere You Might Use ItEver wondered how exactly a logistic regression algorithm is implemented?.There is a high chance it uses a method called “gradient descent” to find the minimum loss function.

To understand how this works, you need to use concepts from calculus: gradient, derivatives, limits, and chain rule.

Where You Can Learn ItedX: Pre-university calculusKhan Academy: Calculus ICoursera: Mathematics for machine learning: multivariable calculusDiscrete MathThis area is not discussed as often in data science, but all modern data science is done with the help of computational systems, and discrete math is at the heart of such systems.

A refresher in discrete math will include concepts critical to daily use of algorithms and data structures in analytics project:Sets, subsets, power setsCounting functions, combinatorics, countabilityBasic proof techniques: induction, proof by contradictionBasics of inductive, deductive, and propositional logicBasic data structures: stacks, queues, graphs, arrays, hash tables, treesGraph properties: connected components, degree, maximum flow/minimum cut concepts, graph coloringRecurrence relations and equationsGrowth of functions and O(n) notation conceptWhere You Might Use ItIn any social network analysis, you need to know properties of graph and fast algorithm to search and traverse the network.

In any choice of algorithm, you need to understand the time and space complexity—i.

e.

, how the running time and space requirement grows with input data size, by using O(n) (Big-Oh) notation.

Where You Can Learn ItCoursera: Introduction to discrete mathematics for computer science specializationCoursera: Introduction to mathematical thinkingUdemy: Master discrete mathematics: sets, math logic, and moreOptimization and Operation Research TopicsThese topics are most relevant in specialized fields like theoretical computer science, control theory, or operation research.

But a basic understanding of these powerful techniques can also be fruitful in the practice of machine learning.

Virtually every machine-learning algorithm aims to minimize some kind of estimation error subject to various constraints—which is an optimization problem.

Here are the topics to learn:Basics of optimization, how to formulate the problemMaxima, minima, convex function, global solutionLinear programming, simplex algorithmInteger programmingConstraint programming, knapsack problemRandomized optimization techniques: hill climbing, simulated annealing, genetic algorithmsWhere You Might Use ItSimple linear regression problems using least-square loss function often have an exact analytical solution, but logistic regression problems don’t.

To understand the reason, you need to be familiar with the concept of “convexity” in optimization.

This line of investigation will also illuminate why we must remain satisfied with “approximate” solutions in most machine-learning problems.

Where You Can Learn ItedX: Optimization methods in business analyticsCoursera: Discrete optimizationedX: Deterministic optimizationSome Parting WordsPlease don’t feel overwhelmed.

Though there are a lot of things to learn, there are excellent resources online.

After a refresher on these topics (which you probably studied as an undergrad) and learning new concepts, you will be empowered to hear the hidden music in your daily data analysis and machine-learning projects.

And that’s a big leap toward becoming an amazing data scientist.

Further ResourcesKDnuggets: 15 mathematics MOOCs for data scienceEliteDataScience: How to learn math for data science, the self-starter wayData Science Weekly: How much math and stats do I need on my data science resume?Analytics Vidhya: 19 MOOCs on mathematics and statistics for data science and machine learningY Combinator: Learning math for machine learningThis story was also featured by KDnuggets, thanks to Matthew Mayo.