Probability Theory

Taught By Prof. Balakrishnan Ramakrishnan (Ex Professor & Head, ISI-NEC)

• Elementary concepts of Probability
• Conditional Probability, Independence, Bayes Theorem
• Random variable, probability distribution and properties; probability mass/density function, cumulative distribution function, expectation, variance, mean square error, moments.
• Bernoulli and Binomial, Poisson, Geometric and Negative Binomial, Hy- pergeometric, Uniform, Normal Exponential, Gamma, Beta distribu- tions.
• Chebyshev’s inequality, weak law of large numbers (statement and con- cept), central limit theorem (statement and concept).
• Distribution of a function of a random variable.
• Bivariate distribution for discrete and continuous random variables; joint, marginal and conditional distributions, moments, covariance, correlation coefficient. Bivariate Normal Distribution
• Independent random variables and their sums. Transformation of two random variables.
• Sampling distributions under the assumption of normality: chi-square, t, F.
• Reference Book: A First Course in Probability - Ross, S.

Statistical Methods I (Online + Offline Classes)

Taught by Dr. Souvik Roy (Ex Associate Professor, ISI-Kolkata, Now Prof.& Head, ISI-NEC) & Two Teaching Assistants

• Introduction to data, statistical problems and related data analysis. Con- cept of population, sample and statistical inference through examples.
• Summarization of univariate data; graphical methods, measures of loca- tion, spread, skewness and kurtosis; outliers and robust measures, sample moments. Empirical cumulative distribution function.
• Statistical computations: data summary and graphical display of data, basic statistics. Plotting empirical cumulative distribution function.
• Data simulations from discrete and continuous probability distributions: Bernoulli and Binomial, Poisson, Geometric and Negative Binomial, Hypergeometric, Uniform, Normal, Exponential, Chi-Square, Gamma, Beta.
• Analysis of discrete and continuous data: fitting some standard discrete and continuous probability distributions. Goodness of fit: Pearson’s Chi- square test, Kolmogorov-Smirnov test. Graphical methods of verifying the fit: Q-Q and P-P Plots. Shapiro-Wilks test for Normality.
• Introduction to resampling (bootstrap) and cross-validation techniques.
• Introducing the analysis of variance; one way analysis, F test, Kruskal- Wallis nonparametric test; two way analysis. Designs of experiment: principles of designing an experiment. Introduction to CRD, RBD, Bal- anced and Unbalanced Block Designs, cross over designs with applica- tions in industrial and clinical trials.
• Reference Book: Introduction to Probability and Statistics for Engineers and Scien- tists - Ross, S.

Statistical Inference (Online + Offline Classes)

Taught by Dr. Sudheesh Kumar Kattumannil (Associate Professor, ISI-Chennai)

• Brief introduction to random variable and probability distributions. Ran- dom sample and the concept of statistical Inference with examples.
• Point estimation: estimator and estimate. Desirable properties of an esti- mator: unbiasedness, smaller variance and mean squared error. Method of moments and maximum likelihood estimation. Asymptotic behaviour of MLE (statement on consistency and asymptotic normality).
• Interval estimation- Confidence interval and its basic properties, Con- struction of confidence interval for parameter of Uniform distribution, exponential distribution, mean of the normal distribution with known and unknown variance.
• Hypotheses and the concept of hypotheses testing. Null and alternative, simple and composite hypothesis, significance level, size, p-value and power. Introduction to likelihood ratio tests with examples.
• One sample problem: Test for randomness - run test. Test for mean under the assumption of normality with known and unknown variance. Nonparametric tests for median: signed test, Wilcoxon’s signed rank test. One sample test for proportion.
• Comparison of two samples- Two independent samples: graphical pro- cedures, K-S test. Comparing mean under the assumption of normality (two sample t test). Nonparametric test for medians - Mann-Whitney- Wilcoxon. Two sample test for proportion. Two dependent samples: paired t test under the assumption of normality, Nonparametric tests for two dependent samples.
• Reference Book: Introduction to Probability and Statistics for Engineers and Scien- tists - Ross, S.

Vectors & Matrices (Module-Based) - Before Mid-Term

Taught by Dr. Mridu Prabal Goswami (Assistant Professor, ISI-NEC)

• Introduction to Vectors and matrices.
• Vectors: Definition and examples, vector spaces and subspaces, basis of a vector space, linear dependence/independence
• Matrices: Definition and examples, matrix as a linear transformation, elementary matrices and elementary matrix operations, basic matrix op- erations including those of partitioned matrices, rank, nullity, trace, de- terminant and inverse of a matrix, idempotent matrix and its properties. Solutions of system of equations
• Spectral theory: eigenvalues and eigenvectors of matrices, decomposition of matrices, quadratic forms and definiteness of a matrix (with applica- tions in Statistics)
• Reference Book: Linear Algebra by Kenneth Hoffman

Regression Methods (Module-Based) - After Mid-Term

Taught by Dr. Kushal Banik Chowdhury (Assistant Professor, ISI-NEC)

• Introduction to Classical Linear Regression Model
• OLS method of estimation; fitted values, prediction of the response vari- able, tests of hypotheses.
• Residuals. Validation of assumptions using graphical techniques. Re- gression Diagnostics
• Use of dummy variables in regression
• Variable selection, multicollinearity. Model selection using AIC and BIC criteria.
• Concepts of robust and nonparametric regression
• Reference Book: Basic Econometrics by Damodar N Gujarati

Programming in R (Module-Based) - After Mid-Term

Taught by Dr. Sanjit Maitra (Assistant Professor, ISI-NEC)

• Introduction to packages- R: overview of packages, data han- dling, input-output operations. Basic programming: data types, arrays, loops etc.; functions and graphics.
• Improvement of the initial solution using methods of bisection, sort and search, Regula Falsi and Newton-Raphson.
• Fixed point iterative schemes, significant digits, round-off errors, finite computational processes and computational errors. Order of convergence and degree of precision.
• Matrix computations - basic operations, finding determinant, inverse, eigen roots and eigen vectors of a matrix, matrix decomposition, solving system of equations.
• Computational aspects of constrained optimization
• Unconstrained optimization: Newton, Quasi-Newton method.
• Experimentation designs: Obtaining global optimal solutions from local optimum solutions using iterative experimentation like Response Surface Methodology.
• Reference Book: An Introduction to R by CRAN R Studio

Programming in Python (Module-Based) - Before Mid-Term

Taught by Dr. Koyel Mandal (Ex Reseach Assistant, ISI-NEC, Now SERB NPDF, ISI-K)

• Introduction to packages- Python: overview of packages, data han- dling, input-output operations. Basic programming: data types, arrays, loops etc.; functions and graphics.
• Fixed point iterative schemes, significant digits, round-off errors, finite computational processes and computational errors. Order of convergence and degree of precision.
• Matrix computations - basic operations, finding determinant, inverse, eigen roots and eigen vectors of a matrix, matrix decomposition, solving system of equations.
• Reference Book: Dive into Python - Pilgrim, M.

Statistical Methods II (Online + Offline Classes)

Taught by Dr. Souvik Roy (Ex Associate Professor, ISI-Kolkata, Now Pro.& Head, ISI-NEC) & Two Teaching Assistants

• Multivariate Data Exploration/visualization, multivariate data handling, random vector, mean and variance-covariance matrix, introduction to multinomial and multivariate normal distributions.
• Applied Multivariate techniques: Principal components analysis and Factor Analysis.
• Introduction to discrete time Markov chains, finite and countable state space. Introduction to Markov chain Monte Carlo (MCMC) methods and applications of MCMC method in statistics. Introduction to HMM algorithm.
• Handling missing data; various methods of imputations including hot deck algorithm (MICE in R), EM algorithm
• Advanced regression techniques: Ridge, Principal component regression, LASSO and Spline smoothing
• Multiple Hypotheses testing.
• Introduction to MANOVA and Hotelling’s t2 .
• Reference Book:

Statistical Machine Learning

Taught by Dr. Sanjit Maitra (Assistant Professor, ISI-NEC)

• Introduction to bootstrap based machine learning. Assessment and model selection: confusion matrix and various criteria of evaluation, training and testing error rates.
• Pattern Recognition and Classification techniques
• Unsupervised learning: clustering procedures: hierarchical and non-hierarchical, k-means; association rules, ROCs.
• Supervised learning: Linear and quadratic discriminant analysis; Bayesian classifier, nearest neighbour classifier, Entropy based classifier.
• Tree based classification methods: predictive modeling using decision trees (CART), random forests.
• Support vector machine. Introduction to boosting and adaptive boosting algorithm.
• Introduction to Natural Language Processing (NLP), information re- trieval and text analysis: stop words, TF-IDF measure, vector space models.
• Introduction to neural networks, Convolutional NN, Deep NN.
• Reference Book: The Elements of Statistical Learning With Applications in R - James, G., Witten, D., Hastie, T. and Tibshirani, R.

Statistical Modeling (Online + Offline Classes)

Taught by Dr. Partha Sarathi Mukherjee (Associate Professor, ISI-Kolkata) & One Teaching Assistant

• Logistic regression; odds ratio, concordance-discordance measures, Lo- gistic Regression as a classsifier. Probit Regression. Introduction to Multilogit models.
• Modeling count data: Poisson Regression, Poisson models for zero in- flated data.
• Introduction and visualizing categorical data. Measures of association. Loglinear Models, Models for nominal and ordinal response.
• Survival Data Modeling: Time-to-event data and survival probabilities, notion of censoring, survival curve and other ways of representing sur- vival distribution, Kaplan-Meier and Nelson-Aalen estimates, log-rank test, Cox’s proportional hazard model. Parametric survival models for Exponential, Gamma, Wiebull distributions.
• Bayesian Inference and Modeling: Prior and posterior distributions, Bayesian models, Bayesian regression, Hierarchical Bayes models
• Introduction to mixture models
• Reference Book: An Introduction to Categorical Data Analysis (2nd edition & 2rd edition)- Agresti, A.

Time Series (Module-Based) - Before Mid-Term

Taught by Dr. Kushal Banik Chowdhury (Assistant Professor, ISI-NEC)

• Exploratory analysis and graphical display; trend, seasonal and cyclical components. Decomposition of time series into components, Smooth- ing.
• Stationary Time Series: Brief Introduction to AR, MA and ARMA mod- els; Box-Jenkins correlogram analysis, ACF and PACF, introduction to periodogram, choice of AR and MA orders.
• Non-Stationary Time Series: introduction to ARIMA model; determin- istic and stochastic trends; introduction to ARCH and GARCH mod- els.
• Forecasting: basic tools, using exponential smoothing and Box-Jenkins method. Residual analysis.
• Reference Book: Introduction to Time Series Analysis and Forecasting - Montgomery, D.C., Jennings, C.L., Kulachi, M.

Statistical Finance (Module-Based) - After Mid-Term

Taught by Dr. Mridu Prabal Goswami (Assistant Professor, ISI-NEC)

• Introduction to stock prices, returns and log-returns. Distribution of returns, Assessing Normality using skewness, kurtosis and q-q plots.
• Market return and risk free rate. Capital Asset Pricing Model (CAPM). Estimating beta and testing for CAPM.
• Options. Arbitrage and risk-neutral measure. European and American options. Option pricing using Binomial model: 1 and 2 step. Black- Scholes model (statement only), interpretation of drift and volatility.
• Value at risk and expected shortfall. Quantile estimation. Estimation of tail-index.
• Markowitz Portfolio Theory. Resampling for assessing estimation of Ef- ficient Portfolio.
• Reference Book: Statistics and Finance - Ruppert, David

Project

Our Project Guide was Dr. Mridu Prabal Goswami (Assistant Professor, ISI-NEC)