Complete Data & Analytics Mathematics Masterclass

📚 Topics Covered

• What is data science? The role of mathematics
• Types of data: numerical, categorical, ordinal
• Scales of measurement: nominal, ordinal, interval, ratio
• Population vs sample concepts
• Parameters vs statistics
• Data collection methods and sampling techniques
• Simple random sampling
• Stratified and cluster sampling
• Sampling bias and selection bias
• Data quality: missing data, outliers, errors

• Data cleaning principles
• Exploratory Data Analysis (EDA) philosophy

📚 Topics Covered

• Mean: arithmetic, geometric, harmonic
• Median and quartiles
• Mode and multimodal distributions
• Weighted averages and their applications
• Range and interquartile range
• Variance and standard deviation
• Coefficient of variation
• Mean absolute deviation
• Moments: skewness and kurtosis
• Box plots and five-number summary

• Z-scores and standardization
• Detecting outliers: IQR method, z-score method

📚 Topics Covered

• Principles of effective visualization
• Histograms and density plots
• Scatter plots and correlation patterns
• Bar charts and categorical data visualization
• Pie charts: when to use and avoid
• Heat maps and correlation matrices
• Time series plots and trends
• Parallel coordinates for high-dimensional data
• Q-Q plots and probability plots
• Interactive visualizations with Plotly

• Dashboard design principles
• Misleading visualizations and how to spot them

📚 Topics Covered

• Sample spaces and events
• Classical, frequentist, and subjective probability
• Probability axioms and properties
• Conditional probability and independence
• Bayes' theorem and applications
• Law of total probability
• Combinatorics for probability
• Permutations and combinations in data
• Birthday paradox and probability paradoxes
• Monte Carlo simulation basics

• Random number generation
• Probability in machine learning context

📚 Topics Covered

• Random variables: discrete vs continuous
• Probability mass functions
• Bernoulli and binomial distributions
• Geometric and negative binomial distributions
• Poisson distribution and rare events
• Hypergeometric distribution
• Multinomial distribution
• Expected value and variance calculations
• Moment generating functions
• Joint discrete distributions

• Marginal and conditional distributions
• Applications in A/B testing

📚 Topics Covered

• Probability density functions
• Uniform distribution
• Normal distribution: properties and importance
• Standard normal and z-tables
• Exponential distribution and memoryless property
• Gamma and beta distributions
• Chi-square distribution
• Student's t-distribution
• F-distribution
• Log-normal distribution

• Weibull distribution for reliability
• Distribution fitting and selection

📚 Topics Covered

• Vectors as data points
• Vector operations: addition, scalar multiplication
• Dot product and cosine similarity
• Vector norms: L1, L2, L-infinity
• Distance metrics: Euclidean, Manhattan, Minkowski
• Orthogonality and orthogonal projections
• Vector spaces and subspaces
• Linear independence in feature space
• Basis vectors and coordinate systems
• Change of basis for data transformation

• Gram-Schmidt for orthogonalization
• Applications to feature engineering

📚 Topics Covered

• Matrices as datasets and transformations
• Matrix multiplication as data transformation
• Special matrices: diagonal, symmetric, orthogonal
• Matrix transpose and properties
• Matrix rank and linear independence of features
• Invertible matrices and their meaning
• Determinants and volume interpretation
• Systems of linear equations in ML
• Gaussian elimination for solving systems
• LU decomposition

• Matrix norms and condition numbers
• Sparse matrices in big data

📚 Topics Covered

• Eigenvalues and eigenvectors intuition
• Characteristic equation
• Geometric interpretation of eigenvectors
• Diagonalization of matrices
• Spectral decomposition
• Positive definite matrices
• Covariance matrices in data
• Principal Component Analysis (PCA) theory
• PCA algorithm step-by-step
• Variance explained and scree plots

• PCA for dimensionality reduction
• Applications in data compression

📚 Topics Covered

• Sampling distribution concept
• Sampling distribution of the mean
• Standard error and its importance
• Central Limit Theorem (CLT)
• CLT applications and limitations
• Sample size determination
• Finite population correction
• Sampling distribution of proportions
• Sampling distribution of variances
• Chi-square distribution from normal samples

• t-distribution for small samples
• F-distribution for variance ratios

📚 Topics Covered

• Point estimates vs interval estimates
• Confidence level interpretation
• CI for population mean (known variance)
• CI for population mean (unknown variance)
• CI for population proportion
• CI for difference of means
• CI for paired differences
• CI for variance and standard deviation
• CI for ratio of variances
• Bootstrap confidence intervals

• Prediction intervals vs confidence intervals
• Sample size for desired margin of error

📚 Topics Covered

• Null and alternative hypotheses
• Type I and Type II errors
• Significance level and p-values
• Test statistics and rejection regions
• One-sample t-test
• Two-sample t-test (equal and unequal variances)
• Paired t-test
• Z-test for proportions
• Chi-square test for variance
• F-test for equal variances

• One-tailed vs two-tailed tests
• Power of a test and effect size

📚 Topics Covered

• One-way ANOVA principles
• F-statistic and ANOVA table
• Assumptions of ANOVA
• Post-hoc tests: Tukey, Bonferroni, Scheffé
• Two-way ANOVA
• Interaction effects
• Repeated measures ANOVA
• MANOVA basics
• Kruskal-Wallis test (nonparametric)
• Multiple testing problem

• False Discovery Rate (FDR)
• Benjamini-Hochberg procedure

📚 Topics Covered

• Complete statistical analysis pipeline
• Integration of all Phase 1 concepts
• Report writing and presentation
• Reproducible research practices
• Statistical consulting simulation

📚 Topics Covered

• Linear relationship between variables
• Least squares estimation
• Geometric interpretation of least squares
• Normal equations derivation
• Properties of least squares estimators
• Gauss-Markov theorem
• R-squared and adjusted R-squared
• Residual analysis
• Assumptions of linear regression
• Diagnostic plots: Q-Q, residual plots

• Outliers and influential points
• Confidence and prediction intervals for regression

📚 Topics Covered

• Multiple regression model
• Matrix formulation of regression
• Partial regression coefficients
• Multicollinearity detection and handling
• Variance Inflation Factor (VIF)
• Feature selection methods
• Forward, backward, stepwise selection
• All subsets regression
• Adjusted R-squared and model selection
• AIC, BIC, Mallows' Cp

• Cross-validation for model selection
• Interaction terms and polynomial regression

📚 Topics Covered

• Overfitting and bias-variance tradeoff
• Ridge regression (L2 regularization)
• Ridge regression geometry
• LASSO regression (L1 regularization)
• LASSO for feature selection
• Elastic Net regression
• Choosing regularization parameters
• Cross-validation for lambda selection
• Bayesian interpretation of regularization
• Group LASSO

• Adaptive LASSO
• Applications in high-dimensional data

📚 Topics Covered

• Exponential family of distributions
• Link functions and canonical links
• Logistic regression for binary outcomes
• Maximum likelihood estimation for GLMs
• Newton-Raphson and IRLS algorithms
• Odds ratios and interpretation
• Poisson regression for count data
• Negative binomial regression
• Ordinal logistic regression
• Multinomial logistic regression

• Deviance and goodness of fit
• Quasi-likelihood methods

📚 Topics Covered

• Kernel regression
• Local polynomial regression (LOESS)
• Bandwidth selection
• Spline regression
• Smoothing splines
• Penalized splines
• Generalized Additive Models (GAMs)
• Regression trees basics
• K-nearest neighbors regression
• Gaussian Process regression introduction

• Quantile regression
• Robust regression methods

📚 Topics Covered

• Classification vs regression
• Bayes classifier and Bayes error
• Linear Discriminant Analysis (LDA)
• Quadratic Discriminant Analysis (QDA)
• Fisher's linear discriminant
• Naive Bayes classifier
• Gaussian Naive Bayes
• Multinomial and Bernoulli Naive Bayes
• K-Nearest Neighbors classification
• Distance metrics for KNN

• Decision boundaries visualization
• Class imbalance problems

📚 Topics Covered

• Decision trees for classification
• Entropy and information gain
• Gini impurity
• Tree pruning methods
• Cost complexity pruning
• Random Forests algorithm
• Out-of-bag error estimation
• Feature importance from trees
• Extremely Randomized Trees
• Gradient Boosting Machines (GBM)

• XGBoost mathematics
• LightGBM and CatBoost

📚 Topics Covered

• Maximum margin classifiers
• Hard margin SVM
• Soft margin SVM and slack variables
• Lagrangian formulation
• Dual problem and support vectors
• Kernel trick and Mercer's theorem
• Common kernels: RBF, polynomial, sigmoid
• Multi-class SVM strategies
• SVM for regression (SVR)
• Nu-SVM and One-class SVM

• Kernel selection and tuning
• SMO algorithm basics

📚 Topics Covered

• K-means clustering algorithm
• K-means++ initialization
• Elbow method and silhouette analysis
• Hierarchical clustering
• Agglomerative vs divisive
• Linkage methods: single, complete, average
• Dendrograms and cutting trees
• DBSCAN algorithm
• Mean Shift clustering
• Gaussian Mixture Models (GMM)

• EM algorithm for GMMs
• Spectral clustering

📚 Topics Covered

• Confusion matrix and derived metrics
• Precision, recall, F1-score
• ROC curves and AUC
• Precision-recall curves
• Multi-class metrics
• Cross-validation strategies
• Stratified and time series CV
• Nested cross-validation
• Bootstrap validation
• Learning curves

• Validation curves
• Grid search and random search

📚 Topics Covered

• Convex sets and convex functions
• Convexity in machine learning
• Gradient descent algorithm
• Learning rate selection
• Momentum and Nesterov acceleration
• AdaGrad and RMSprop
• Adam and AdamW optimizers
• Second-order methods: Newton, L-BFGS
• Stochastic gradient descent (SGD)
• Mini-batch gradient descent

• Convergence analysis
• Constrained optimization for ML

📚 Topics Covered

• Perceptron algorithm
• Multi-layer perceptrons
• Universal approximation theorem
• Activation functions: sigmoid, tanh, ReLU
• Forward propagation mathematics
• Backpropagation derivation
• Chain rule in neural networks
• Gradient vanishing and exploding
• Weight initialization strategies
• Xavier and He initialization

• Batch normalization mathematics
• Dropout as regularization

📚 Topics Covered

• Convolution operation in 1D and 2D
• Padding and stride calculations
• Pooling operations: max, average
• Parameter sharing and translation invariance
• Receptive field calculations
• Backpropagation through convolutions
• Popular architectures: LeNet, AlexNet, VGG
• ResNet and skip connections
• Inception modules
• Depthwise separable convolutions

• Transfer learning mathematics
• Feature map visualization

📚 Topics Covered

• Vanilla RNN formulation
• Backpropagation through time (BPTT)
• Gradient clipping
• Long Short-Term Memory (LSTM)
• LSTM gates mathematics
• Gated Recurrent Units (GRU)
• Bidirectional RNNs
• Encoder-decoder architectures
• Attention mechanism basics
• Sequence-to-sequence models

• Applications to time series
• RNN regularization techniques

📚 Topics Covered

• End-to-end ML pipeline
• Model deployment considerations
• A/B testing for ML
• Model monitoring
• ML system design

📚 Topics Covered

• Bayesian vs Frequentist philosophy
• Prior distributions and their selection
• Likelihood functions
• Posterior distributions
• Conjugate priors
• Beta-Binomial model
• Normal-Normal model
• Gamma-Poisson model
• Credible intervals vs confidence intervals
• Bayesian point estimates

• MAP vs MLE estimates
• Empirical Bayes methods

📚 Topics Covered

• Monte Carlo integration review
• Importance sampling
• Markov chains for sampling
• Metropolis algorithm
• Metropolis-Hastings algorithm
• Gibbs sampling
• Convergence diagnostics
• Gelman-Rubin statistic
• Effective sample size
• Hamiltonian Monte Carlo

• NUTS sampler
• Variational inference basics

📚 Topics Covered

• Bayesian linear regression
• Predictive distributions
• Bayesian model averaging
• Spike and slab priors
• Horseshoe prior
• Bayesian logistic regression
• Probit regression
• Gaussian processes for regression
• Kernel selection in GPs
• Sparse Gaussian processes

• Bayesian neural networks
• Uncertainty quantification

📚 Topics Covered

• Hierarchical Bayesian models
• Random effects models
• Mixed effects models
• Nested data structures
• Shrinkage and pooling
• Dirichlet process mixtures
• Infinite mixture models
• Chinese Restaurant Process
• Stick-breaking construction
• Latent Dirichlet Allocation

• Topic modeling
• Hidden Markov Models

📚 Topics Covered

• Bayes factors
• Model evidence and marginal likelihood
• Savage-Dickey density ratio
• DIC and WAIC
• LOO cross-validation
• Posterior predictive checks
• Model comparison strategies
• Bayesian hypothesis testing
• Reversible jump MCMC
• Bayesian variable selection

• Bayesian model averaging
• Ensemble methods in Bayesian context

📚 Topics Covered

• Deep learning optimization landscape
• Loss surface geometry
• Mode connectivity
• Lottery ticket hypothesis
• Neural tangent kernels
• Double descent phenomenon
• Implicit regularization
• Neural architecture search
• AutoML for deep learning
• Meta-learning basics

• Few-shot learning
• Self-supervised learning

📚 Topics Covered

• Self-attention mechanism
• Multi-head attention
• Positional encoding
• Transformer architecture
• BERT and GPT architectures
• Vision Transformers (ViT)
• Cross-attention mechanisms
• Efficient attention variants
• Linear attention
• Sparse transformers

• Transformer training dynamics
• Large language model mathematics

📚 Topics Covered

• Variational Autoencoders (VAE)
• ELBO derivation
• Reparameterization trick
• Beta-VAE and disentanglement
• Generative Adversarial Networks (GANs)
• Wasserstein GAN
• GAN training dynamics
• Mode collapse solutions
• Normalizing flows
• Diffusion models mathematics

• Score matching
• Energy-based models

📚 Topics Covered

• Graph representation learning
• Message passing neural networks
• Graph Convolutional Networks (GCN)
• GraphSAGE algorithm
• Graph Attention Networks (GAT)
• Spectral graph convolutions
• Graph pooling methods
• Link prediction
• Node classification
• Graph classification

• Knowledge graph embeddings
• Geometric deep learning

📚 Topics Covered

• Markov Decision Processes
• Bellman equations
• Value iteration
• Policy iteration
• Q-learning algorithm
• Deep Q-Networks (DQN)
• Policy gradient methods
• REINFORCE algorithm
• Actor-Critic methods
• Proximal Policy Optimization (PPO)

• Soft Actor-Critic (SAC)
• Multi-armed bandits

📚 Topics Covered

• Time series components: trend, seasonality, noise
• Stationarity and unit root tests
• Autocorrelation and partial autocorrelation
• ARIMA models
• Box-Jenkins methodology
• Seasonal ARIMA (SARIMA)
• State space models
• Kalman filtering
• Exponential smoothing methods
• Holt-Winters method

• STL decomposition
• Time series cross-validation

📚 Topics Covered

• Vector Autoregression (VAR)
• Cointegration and error correction
• GARCH models for volatility
• Long memory models (ARFIMA)
• Regime switching models
• Dynamic Factor Models
• Spectral analysis
• Wavelet analysis
• Deep learning for time series
• LSTM for forecasting

• Temporal Convolutional Networks
• Prophet algorithm

📚 Topics Covered

• Correlation vs causation
• Potential outcomes framework
• Average Treatment Effect (ATE)
• Randomized experiments
• Selection bias and confounding
• Propensity score matching
• Inverse probability weighting
• Doubly robust estimation
• Instrumental variables
• Regression discontinuity

• Difference-in-differences
• Synthetic control methods

📚 Topics Covered

• Path analysis
• Confirmatory factor analysis
• Structural equation models
• Latent variable models
• Measurement models
• Model identification
• Maximum likelihood for SEM
• Fit indices and model evaluation
• Multi-group SEM
• Growth curve models

• Mediation and moderation
• Directed Acyclic Graphs (DAGs)

📚 Topics Covered

• Advanced analytics pipeline
• Deep learning deployment
• Causal analysis project
• Research paper writing
• Industry presentation

📚 Topics Covered

• Big data characteristics: 5 V's
• Distributed computing principles
• CAP theorem and data consistency
• MapReduce paradigm
• Hadoop ecosystem overview
• HDFS architecture
• Apache Spark fundamentals
• RDDs and DataFrames
• Spark SQL and optimization
• Spark MLlib algorithms

• Streaming data concepts
• Lambda and Kappa architectures

📚 Topics Covered

• Distributed gradient descent
• Parameter server architecture
• Data parallelism vs model parallelism
• Federated learning
• Online learning algorithms
• Stochastic gradient methods at scale
• Approximate algorithms for big data
• Random sampling techniques
• Sketching algorithms
• MinHash and LSH

• Bloom filters
• Count-Min sketch

📚 Topics Covered

• Stream processing concepts
• Apache Kafka architecture
• Apache Flink fundamentals
• Window functions in streaming
• Watermarks and late data
• Exactly-once processing
• Complex event processing
• Real-time aggregations
• Streaming ML predictions
• Online learning in streams

• Anomaly detection in streams
• Time series databases

📚 Topics Covered

• NoSQL database types
• Document stores: MongoDB analytics
• Column stores: Cassandra, HBase
• Graph databases for analytics
• Time series databases: InfluxDB, TimescaleDB
• NewSQL systems
• Data modeling for NoSQL
• CAP theorem implications
• Consistency models
• Data warehousing concepts

• Data lakes vs data warehouses
• Apache Iceberg and Delta Lake

📚 Topics Covered

• Cloud computing for analytics
• AWS analytics services
• Google Cloud Platform ML tools
• Azure Machine Learning
• Serverless analytics
• Auto-scaling for ML workloads
• Cost optimization strategies
• Multi-cloud strategies
• Data governance in cloud
• Security and compliance

• MLOps in cloud
• Edge analytics

📚 Topics Covered

• Principles of experimental design
• Randomization, replication, blocking
• Completely randomized designs
• Randomized block designs
• Latin square designs
• Factorial designs
• 2^k factorial designs
• Fractional factorial designs
• Response surface methodology
• Central composite designs

• Optimal design theory
• Sample size calculations

📚 Topics Covered

• A/B testing fundamentals
• Statistical power in A/B tests
• Multiple testing corrections
• Sequential testing
• Bandits vs A/B tests
• Multi-armed bandits
• Thompson sampling
• Contextual bandits
• Variance reduction techniques
• CUPED method

• Stratification in experiments
• Network effects in testing

📚 Topics Covered

• Natural experiments
• Interrupted time series
• Regression discontinuity design
• Fuzzy RDD
• Difference-in-differences advanced
• Triple differences
• Synthetic control advanced
• Matching methods review
• Coarsened exact matching
• Genetic matching

• Sensitivity analysis
• Bounds for causal effects

📚 Topics Covered

• Censoring and truncation
• Survival and hazard functions
• Kaplan-Meier estimator
• Nelson-Aalen estimator
• Log-rank test
• Cox proportional hazards model
• Stratified Cox models
• Time-varying covariates
• Parametric survival models
• Accelerated failure time models

• Competing risks
• Recurrent events analysis

📚 Topics Covered

• Repeated measures data
• Mixed effects models review
• Generalized Estimating Equations (GEE)
• Growth curve models
• Latent growth models
• Cross-lagged panel models
• Dynamic panel data models
• Fixed effects vs random effects
• Hausman test
• Missing data in longitudinal studies

• Multiple imputation
• Attrition and selection bias

📚 Topics Covered

• Spatial data types
• Spatial autocorrelation
• Moran's I and Geary's C
• Spatial regression models
• Kriging and interpolation
• Point pattern analysis
• Spatial clustering
• Geographically weighted regression
• Space-time models
• Disease mapping

• Environmental statistics
• GIS integration

📚 Topics Covered

• Text preprocessing mathematics
• TF-IDF and BM25
• Word embeddings: Word2Vec, GloVe
• Document embeddings
• Topic modeling: LDA, NMF
• Named entity recognition
• Part-of-speech tagging
• Dependency parsing
• Sentiment analysis methods
• Text classification algorithms

• Sequence labeling with CRFs
• Transformer-based NLP

📚 Topics Covered

• Image processing fundamentals
• Convolution and filtering
• Edge detection algorithms
• Feature extraction: SIFT, SURF, HOG
• Image segmentation methods
• Object detection: R-CNN family
• YOLO architecture
• Semantic segmentation
• Instance segmentation
• Face recognition mathematics

• Optical flow
• 3D vision basics

📚 Topics Covered

• Collaborative filtering
• User-based vs item-based CF
• Matrix factorization methods
• Singular value decomposition
• Non-negative matrix factorization
• Alternating least squares
• Content-based filtering
• Hybrid recommendation systems
• Deep learning for recommendations
• Sequential recommendations

• Multi-stakeholder recommendations
• Evaluation metrics for RecSys

📚 Topics Covered

• ML system design patterns
• Feature engineering pipelines
• Feature stores
• Model versioning
• Experiment tracking
• Model registry
• CI/CD for ML
• Model serving architectures
• Batch vs online inference
• Model monitoring and drift detection

• A/B testing for ML models
• Rollback strategies
• Edge deployment
• Model compression techniques
• Quantization and pruning

📚 Topics Covered

• Bias in data and algorithms
• Fairness metrics
• Demographic parity
• Equalized odds
• Calibration
• Fair ML algorithms
• Model interpretability methods
• LIME and SHAP
• Counterfactual explanations
• Privacy-preserving ML

• Differential privacy
• Federated learning for privacy
• AI governance
• Regulatory compliance

📚 Topics Covered

• Reading research papers effectively
• Reproducing research results
• Writing technical reports
• Creating data science portfolio
• Interview preparation for DS roles
• Case study methodology
• Technical presentation skills
• Open source contribution
• Kaggle competition strategies
• Networking in data science

• Continuous learning strategies
• Career paths in data science