book

Essential Math for Data Science

by Thomas Nield

May 2022

Intermediate to advanced

352 pages

9h 15m

English

O'Reilly Media, Inc.

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Conventions Used in This BookUsing Code ExamplesO’Reilly Online LearningHow to Contact UsAcknowledgments
Number TheoryOrder of OperationsVariablesFunctionsSummationsExponentsLogarithmsEuler’s Number and Natural LogarithmsEuler’s NumberNatural LogarithmsLimitsDerivativesPartial DerivativesThe Chain RuleIntegralsConclusionExercises
Understanding ProbabilityProbability Versus StatisticsProbability MathJoint ProbabilitiesUnion ProbabilitiesConditional Probability and Bayes’ TheoremJoint and Union Conditional ProbabilitiesBinomial DistributionBeta DistributionConclusionExercises
What Is Data?Descriptive Versus Inferential StatisticsPopulations, Samples, and BiasDescriptive StatisticsMean and Weighted MeanMedianModeVariance and Standard DeviationThe Normal DistributionThe Inverse CDFZ-ScoresInferential StatisticsThe Central Limit TheoremConfidence IntervalsUnderstanding P-ValuesHypothesis TestingThe T-Distribution: Dealing with Small SamplesBig Data Considerations and the Texas Sharpshooter FallacyConclusionExercises
What Is a Vector?Adding and Combining VectorsScaling VectorsSpan and Linear DependenceLinear TransformationsBasis VectorsMatrix Vector MultiplicationMatrix MultiplicationDeterminantsSpecial Types of MatricesSquare MatrixIdentity MatrixInverse MatrixDiagonal MatrixTriangular MatrixSparse MatrixSystems of Equations and Inverse MatricesEigenvectors and EigenvaluesConclusionExercises
A Basic Linear RegressionResiduals and Squared ErrorsFinding the Best Fit LineClosed Form EquationInverse Matrix TechniquesMatrix DecompositionGradient DescentOverfitting and VarianceStochastic Gradient DescentThe Correlation CoefficientStatistical SignificanceCoefficient of DeterminationStandard Error of the EstimatePrediction IntervalsTrain/Test SplitsMultiple Linear RegressionConclusionExercises
Understanding Logistic RegressionPerforming a Logistic RegressionLogistic FunctionFitting the Logistic CurveMultivariable Logistic RegressionUnderstanding the Log-OddsR-SquaredP-ValuesTrain/Test SplitsConfusion MatricesBayes’ Theorem and ClassificationReceiver Operator Characteristics/Area Under CurveClass ImbalanceConclusionExercises
When to Use Neural Networks and Deep LearningA Simple Neural NetworkActivation FunctionsForward PropagationBackpropagationCalculating the Weight and Bias DerivativesStochastic Gradient DescentUsing scikit-learnLimitations of Neural Networks and Deep LearningConclusionExercise
Redefining Data ScienceA Brief History of Data ScienceFinding Your EdgeSQL ProficiencyProgramming ProficiencyData VisualizationKnowing Your IndustryProductive LearningPractitioner Versus AdvisorWhat to Watch Out For in Data Science JobsRole DefinitionOrganizational Focus and Buy-InAdequate ResourcesReasonable ObjectivesCompeting with Existing SystemsA Role Is Not What You ExpectedDoes Your Dream Job Not Exist?Where Do I Go Now?Conclusion
Using LaTeX Rendering with SymPyBinomial Distribution from ScratchBeta Distribution from ScratchDeriving Bayes’ TheoremCDF and Inverse CDF from ScratchUse e to Predict Event Probability Over TimeHill Climbing and Linear RegressionHill Climbing and Logistic RegressionA Brief Intro to Linear ProgrammingMNIST Classifier Using scikit-learn

Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7

Content preview from Essential Math for Data Science

Chapter 3. Descriptive and Inferential Statistics

Statistics is the practice of collecting and analyzing data to discover findings that are useful or predict what causes those findings to happen. Probability often plays a large role in statistics, as we use data to estimate how likely an event is to happen.

It may not always get credit, but statistics is the heart of many data-driven innovations. Machine learning in itself is a statistical tool, searching for possible hypotheses to correlate relationships between different variables in data. However there are a lot of blind sides in statistics, even for professional statisticians. We can easily get caught up in what the data says that we forget to ask where the data comes from. These concerns become all the more important as big data, data mining, and machine learning all accelerate the automation of statistical algorithms. Therefore, it is important to have a solid foundation in statistics and hypothesis testing so you do not treat these automations as black boxes.

In this section we will cover the fundamentals of statistics and hypothesis testing. Starting with descriptive statistics, we will learn common ways to summarize data. After that, we will venture into inferential statistics, where we try to uncover attributes of a population based on a sample.

What Is Data?

It may seem odd to define “data,” something we all use and take for granted. But I think it needs to be done. Chances are if you asked any person what data is, ...