##### Semester I

- Single-variable analysis: Limits, continuity, sequences and series of real numbers, differentiation, maxima and minima, Riemann integration, improper integrals (like normal, exponential, gamma).
- Multi-variable analysis: Differentiation, convexity, gradient and Hessian of a multivariate function, Taylor's expansion, necessary and sufficient conditions for the existence of an extremal point, Newton's method, Lagrange multipliers, gradient and conjugate gradient methods.
- Sequences and series of functions.

**Textbook**

- Tom W. Apostol:
*Calculus: One-Variable Calculus with An Introduction to Linear Algebra*, Vol 1, Wiley, Second edition (2007). - Tom W. Apostol:
*Calculus: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability*, Vol 2, Wiley, Second edition (2007).

**Reference**

- P.G. Ciarlet:
*Introduction To Numerical Linear Algebra And Optimisation*, Cambridge University Press.

- Combinatorial probability, Independence of events, Conditional probabilities
- Random variables, densities, Expectation, Variance and moments, Standard univariate distributions, Independence of random variables, Moment Generating Functions
- Tchebychev's inequality and weak law of large numbers, Central Limit Theorem.
- Marginal Distribution, Conditional Distribution, Conditional expectation, Regression, Correlation, Bivariate normal distribution, Multivariate normal distribution
- Introduction to Statistics with examples of its use, Draw random samples, Descriptive statistics, Graphical statistics: Histogram, scatter diagram, Pie diagram, estimates sample moments, sample mean, sample standard deviation
- Sampling distributions based on normal populations - t, chi-square and F distributions
- Sufficient statistics. Point and Interval Estimation, Consistency, Minimum Variance Unbiased Estimator (statement only), method of moments estimators, maximum likelihood estimator, consistency and asymptotic normality of MLE's (statement only)
- Testing of Hypothesis: one sample and two sample tests based on t, chi-square and F distributions. - Error probabilities, statistical power of test, p-values, log-likelihood ratio test

**Textbook**

- Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye
*Probability and Statistics for Engineers and Scientists*, by Pearson, Ninth Edition (2013) - Sheldon Ross
*A First Course in Probability*, Pearson, Ninth Edition (2018) - Prabhanjan N. Tattar, Suresh Ramaiah, B. G. Manjunath,
*A Course in Statistics with R*; Wiley, (2018)

Introduction to basic programming principles using Python, including object-oriented design, big-oh notation, sorting and search algorithms, elementary data structures (lists, heaps, binary trees).

**Textbook**- Mark Pilgrim :
*Dive into Python*, available online. - T.H. Cormen, C.E. Leiserson, and R.L. Rivest : Introduction to algorithms, Prentice-Hall (2010)

- Some counting principles
- Basic logic
- Finite automata and regular languages

**Textbooks**

- Kenneth H Rosen,
*Discrete Mathematics and Its Applications*: McGrawHill (7th Edition) (2017)

**RDBAM and SQL**RDBMS overview, Queries on One table, joins, self joins, inner-join, outer-join, multiple relations between tables, set operations, agreegate operations, efficient queries, Structured Query Language, Commands in SQL, Datatypes in SQL,Data Manipulation and Data Processing with SQL

**Textbooks**

- Sumathi, S., Esakkirajan, S.
*Fundamentals of Relational Database Management Systems*Springer 2007

**Visualization**

- Graphical Practice: Graphical Excellence,Graphical Integrity
- Theory of Data Graphics: Data-Ink and Graphical Redesign, Data-Ink Maximization and Graphical Design,Multifunctioning Graphical Elements,Data Density and Small Multiples,Aesthetics and Technique in Data Graphical Design

**Textbooks**

- Edward R. Tufte
*The Visual Display of Quantitative Information*, (2001)

##### Semester II

- Jordan canonical form, other reductions to triangular and diagonal forms, projection matrices;
- Matrix norms, Rayleigh quotient, conditioning of a problem, floating point arithmetic, backward and forward stability of an algorithm;
- Direct and iterative methods for solving a linear system of equations: Gaussian elimination, LU factorization, Cholesky method, QR factorization, Householder's matrices, Jacobi's method, Gauss-Seidel method, successive over-relaxation methods (SOR);
- Eigenvalue-eigenvector methods: methods based on reduction to Hessenberg or tridiagonal forms (Arnoldi, Gram-Schmidt), power iteration, inverse iteration, QR iteration, Rayleigh quotient iteration, Jacobi's method, bisection method, divide-and-conquer, Krylov subspace methods, method of conjugate gradients;
- Singular value problems: Computing the SVD, elements of PCA;
- Least squares problems: normal equations, QR, SVD, solving rank-deficit least squares problems using SVD and QR.

**Textbook**

- Lloyd N. Trefethen and David Bau, III:
*Numerical linear algebra,*SIAM (1997)

Association rules, frequent itemsets; Finding high-correlation with low-support; Classifiers -- Bayesian, Nearest Neighbour; Decision Trees; Clustering techniques; Vector space (TF-IDF) model; Stop words and stemming; Supervised learning : Bayesian Networks, Support Vector Machines; Semisupervised learning: Expectation maximization; Web search: HITS and PageRank;

**Textbook**- Jiawei Han, Micheline Kamber:
*Data mining: concepts and techniques*(2nd ed), Morgan Kaufman (2006). - Bing Liu:
*Web Data Mining: Exploring Hyperlinks, Contents and Usage Data,*Springer (2006). - Soumen Chakrabarti:
*Mining the Web: Discovering knowledge from hypertext data,*Elsevier (2003). - Christopher D Manning, Prabhakar Raghavan and Hinrich Schutze :
*An Introduction to Information Retrieval,*Cambridge University Press (2009)

A quick revision of sorting, searching, selection and Big Oh; Divide and Conquer; Dynamic Programming; Graphs, BFS, DFS, connectivity; Algorithms on Matrices; Combinatorial Optimization --- Linear Programming, Simplex, Duality, Primal Dual Algorithms (shortest paths, max flow, matching).

**Online Materials**- Introduction to distributed computing,
- Cloud computing overview
- Clocks in distributed systems
- Consistency and replication (shared memory)
- Distributed File Systems
- Dissect the Hadoop File system and its internals
- Spark as system for computing where Hadoop is not adequete.
- Distributed Data Stores (No SQL)

**Textbooks**

- Martin Van Steen and Andrew S Tanenbaums:
*Distributed Systems*3rd Edition (2017); Available Online

- Foundations, characterization, an overview of data processing pipelines in enterprise computing. limitations of the previous generation of data processing tools need for tools and algorithms to process big data. Elements of modern processing environments - like the lambda architecture.
- Computational tools for big data - map reduce; Hadoop; spark; Dask with Google Clod Platform
- Motivate the need for algorithms for big data - the need for sketching, bloom filters etc. Discussion of theory. Motivate and discuss basic machine learning techniques for learning from big data - PCA, logistic regression, linear regression. Discuss how convexity can be exploited to create tractable solutions for big data problems using stochastic gradient descent.
- Applications using big data: Recommender Systems and maybe similarity methods

**Online Materials**

##### Summer Internship (compulsory)

##### Semester III

- Introduction to Predictive Analytics, Case studies
- Least-Square Method, Overview of Supervised Learning,
**Regression**: Linear Models, Gauss-Markov Theorem, Multiple Linear Regression, Variable Selection, Bayesian Linear Regression, Ridge Regression, LASSO, Elastic Net, Principal Component Regression, Functional Regression, Gaussian Process Regression- Outlier detection, Influential point, Cooks distance, Model Selection via AIC and BIC
**Classification**: Linear Classifiers, Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis, Logistic Regression, CART, CHAID,

**Textbook**

- Hastie, Tibshirani, Friedman,
*The Elements of Statistical Learning, Data Mining, Inference and Prediction,*(2010) Second Edition, Springer Series in Statistics. Available Here

- Deep Learning Philosophy, Deep Neural Network, Pytorch, Keras
- Reinforcement Learning, LSTM
- Graphical Network Models
- Hidden Markov Models

**Textbooks**

- Ian Goodfellow, Yoshua Bengio and Aaron Courville:
*Deep Learning*MIT Press 2016 Available Here - Sutton and Barto
*Reinforcement Learning: An Introduction*MIT Press 2018 Available Here

Appropriate courses will be offered from the list of elective course. For more detail contact your faculty advisor.

##### Semester III

##### Elective Courses

- Difference Equations, Lag Operators, pth-Order Difference Equations
- White Noise, Expectations, Stationarity, and Ergodicity
- Linear Models, Autoregressive Processes , Moving Average Processes, Stationary ARMA Process
*Forecasting*: Principles of Forecasting, Forecasts with Gaussian Processes, Wold's Decomposition and the Box-Jenkins Modeling Philosophy- Parameter Estimation, Maximum Likelihood Estimation
- Vector Autoregressions; Bivariate Granger Causality Tests
- Understanding Kalman Filter
- Models of Nonstationary Time Series: Why Linear Time Trends and Unit Roots? Random Walk Models, Brownian Motion, Geometric Brownian Motion
- Cointegration, Canonical Correlation
- Time Series Models of Heteroskedasticity: ARCH and GARCH model

**Textbook**

- James D. Hamilton,
*Time Series Analysis*by Princeton University Press, NEW JERSEY, 2012 - Robert H. Shumway, David S. Stoffer
*Time Series Analysis and Its Applications: With R Examples*(Springer Texts in Statistics) 2017 - Richard J. Meinhold; Nozer D. Singpurwalla
*Understanding the Kalman Filter***The American Statistician**Vol. 37, No. 2. (May, 1983), pp. 123-127

- Mathematical optimization, LPP
- Convex sets and convex functions, convex optimization and applications to approximation
- Fitting and statistical estimation
- unconstrained optimization methods (descent methods, Newton's method)
- constrained optimization (interior point methods)

**Textbook**

- Boyd and Vandenberghe,
*Convex Optimization*, Cambridge University Press 2008, Available Here

**Topology part**: simplicial complexes, persistent homology, Betti numbers and bar codes,**Statistics part**: dimensionality reduction, clustering, data reconstruction and inference techniques.- A substantial part of the course will be devoted to hands on work like using various TDA packages, working through papers that describe applications of TDA to various data sets.

**Textbook**

- Robert Ghrist,
*Elementary Applied Topology*Available Here

- Introduction
- Objective vs Subjective Definition of Probability
- Axiomatic Definition of Probability
- Bayes Theorem
- Applications of Bayes Theorem
- Decision Theoretic framework and major concepts of Bayesian Analysis
- Likelihood, Prior and posterior
- Loss function
- Bayes Rule
- Conjugate priors and other priors
- Sensitivity Analysis
- Posterior Convergence
- One-parameter Bayesian models
- Poisson Model for Count data
- Binomial Model for Count data
- Multi-parameter Bayesian models
- Univariate Gaussian Model
- Multivariate Gaussian Model
- Covariance Matrix with Wishart Distribution
- Bayesian solution for high-dimensional problem in Covariance matrix for Portfolio Risk Analysis
- Multinomial-Dirichlet Allocation Models for Topic Model
- Bayesian Machine Learning
- Hierarchical Bayesian Model
- Regression with Ridge prior, LASSO prior
- Classification with Bayesian Logistic Regression
- Discriminant Analysis
- Bayesian Computation with stan
- Estimation of Posterior Mode with Optimization
- Estimation of Posterior Mean and other summary with Monte Carlo Simulation
- Accept-Rejection Sampling
- Importance Sampling
- Markov Chain and Monte Carlo
- Metropolis-Hastings
- Hamiltonial Monte Carlo
- Gaussian Process Regression
- Introduction
- Gaussian Process Regression for Big Data
- Bayesian Optimization

**Textbook**

- John Kruschke:
*Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan*(2014), Academic Press - Carl Edward Rasmussen and Christopher K. I. Williams:
*Gaussian Processes for Machine Learning*, MIT Press (2006) Available Online - Sourish Das, Sasanka Roy, Rajiv Sambasivan :
*Fast Gaussian Process Regression for Big Data*,**Big Data Research**, Volume 14, December 2018, Pages 12-26: Preprint Available Here; Python Implementation - Sourish Das, Aritra Halder, Dipak K. Dey
*Regularizing Portfolio Risk Analysis: A Bayesian Approach*,**Methodology and Computing in Applied Probability**September 2017, Volume 19, Issue 3, pp 865-889, Preprint Available Online

- Multi-armed Bandits
- Markov Decision Processes
- Dynamic Programming
- Monte Carlo Methods
- Temporal Difference Learning
- Model Based RL

**Textbook**

- Sutton, R. S and Barto, A. G.
*Reinforcement Learning: An Introduction*, MIT Press 2018 Available Here

This course covers material useful for an understanding of both theoretical and empirical finance. It is not intended as a comprehensive survey of economics. The approach is analytical (as befits a graduate math course) and stresses understanding of concepts. Topics from both micro and macro economics are covered.

- Theory of consumer choice. Utility theory and preferences. Demand, revealed preferences, comparative statics.
- Extension of basic choice models to include time and uncertainty.
- Markets. Perfect competition, Monopoly, Monopolistic competition (Dixit-Stiglitz). Walrasian equilibrium.
- Risk aversion, risk sharing. Contingent claims.
- Game theory. Introduction to cooperative and non-cooperative games.
- Introduction to externalities and market failures.
- Adverse selection, moral hazard, principal-agent contracts.
- Introduction to auctions.
- Macroeconomics: Aggregate consumption, aggregate investment, money and financial markets, introduction to components of national income accounts.
- IS-LM in a closed economy.
- IS-LM analysis in an open economy

**Textbook**- Varian, Hal R
*Intermediate Microeconomics : A Modern Approach*Ninth Edition, 2014 - Olivier Blanchard:
*Macroeconomics*, 6th ed. 2017 - Rudiger Dornbusch, Stanley Fischer, and Richard Startz:
*Macroeconomics*, 11th Edition, 2017

- Time value of Money, Introduction to Primary Securites, Bonds and Equity, Risk free rate of interest, Financial Returns, Net Return, Log Return, Compounding, Annuities
- Discounting, Zero Coupon Bond and Regular Bond, Fundamentals of Bond Valuation, Spot Rate Curve, Yield Curve, Clean and Dirty Price of Bond, Term Structure, Pricing Yield Curve with Nelson-Siegel Model, Simulate Bond Prices
- Portfolio Theory, Geometric Brownian Motion, No-Arbitrage, Efficient Market Hypothesis, Efficient Frontier, CAPM, Asset pricing models Hands on practical with R

**Textbook**

- Sheldon M. Ross
*An Elementary Introduction to Mathematical Finance*(2011) Third Edition - Steven Shreve
*Stochastic Calculus for Finance I: The Binomial Asset Pricing Model*, Springer

- Financial Data and Their Properties
- Linear Models for Financial Time Series and Case Studies
- Asset Volatility and Volatility Models
- Applications of Volatility Models
- High Frequency Financial Data
- Value at Risk

**Textbooks**

- Ruey S. Tsay
*An Introduction to Analysis of Financial Data with R*Wiley, 2013

- Multivariate Gaussian Distribution; The Distribution of Linear Combinations of Normally Distributed Variates; Conditional Distributions and Multiple Correlation Coefficient
- Estimation of the Mean Vector and the Covariance Matrix
- The sampling distribution of sample mean vector
- The Distribution of the Sample Covarirance Matrix: Wishart Distribution
- Principal Component Analysis
- Patterns of Dependence: Graphical Models
- Copula: Define Multivariate Distribution using Copula Functions
- Hands-on with Copula package in R

**Textbooks:**

- T.W.Anderson
*An Introduction to Multivariate Statistical Analysis,*Third Edition, Wiler 2009 - Johnson/Wichern
*Applied Multivariate Statistical Analysis*Sixth Edition, Pearson 2015

- Natural Language Basics
- Understanding Text: Text Tokenization, Text Normalization, Understanding Text Syntax and Structure
- Text Classification
- Text Summarization: Topic Modeling
- Text Similarity and Clustering
- Semantic and Sentiment Analysis

**Textbooks**

- Bengfort Benjamin, Bilbro Rebecca and Ojeda Tony,
*Applied Text Analysis with Python*, O'Reilly, 2018 - Sarkar Dipanjan,
*Text Analytics with Python*, apress, 2016

We partner with industry where a student will work on a problem defined by the industry and develop a solution. This course is subject to availability of faculty from industry.