Syllabus for
Master of Science (Statistics)
Academic Year  (2022)

Master of Science in Statistics at CHRIST (Deemed to be University) offers the students an amalgam of knowledge on theoretical and applied statistics on a broader spectrum. Further, it intends to impart awareness of the importance of the conceptual framework of statistics across diversified fields and provide practical training on statistical methods for carrying out data analysis using sophisticated programming languages and statistical softwares such as R, Python, SPSS, EXCEL, etc. The course curriculum has been designed in such a way to cater for the needs of stakeholders to get placements in industries and institutions on successful completion of the course and to provide those ample skills and opportunities to meet the challenges at the national level competitive examinations like CSIR NET in Mathematical Science, SET, Indian Statistical Service (ISS) etc.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: Identify the need and scope of Interdisciplinary research.

PO3: Enhance research culture and uphold scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO2: Demonstrate the execution of statistical experiments or investigations, analyse and interpret using appropriate statistical methods, including statistical software and report the findings of experiments or studies accurately.

PSO3: Demonstrate acquaintance with contemporary trends in industrial/research settings and innovate novel solutions to existing problems.

CIA - 50%

ESE - 50%

Probability is a measure of uncertainty and forms the foundation of statistical methods. This course makes students use measure-theoretic and analytical techniques for understanding probability concepts.  

Sets – functions - Sigma field – Measurable space – Sample space – Measure – Probability as a measure - Inverse function - Measurable functions – Random variable - Induced probability space - Distribution function of a random variable: definition and properties.

Expectation and moments: Definition and properties – Probability generating function - Moment generating functions - Moment inequalities: Markov’s, Chebychev’s, Holder, Jenson and basic inequalities - Characteristic function and properties (idea and statement only).

Random vectors – joint distribution function – joint moments - Conditional probabilities - Randon-Nikodym Theorem (Statement only) - Bayes’ theorem – conditional distributions – independence - Conditional expectation and its properties

Modes of convergence: Convergence in probability, in distribution, in rth mean, almost sure convergence and their inter-relationships - Convergence theorem for expectation 

Law of large numbers - Covergence of series of independent random variables - Weak law of large numbers (Kninchine’s and Kolmogorov’s) - Kolmogorov’s strong law of large numbers - Central limit theorems for i.i.d random variables: Lindberg-Levy and Liaponov’s CLT.

2.     Bhat, B.R, (2014), Modern Probability Theory, 4^th Ed., New Age International.

Probability distributions are used in many real-life phenomena. This course makes students understand different probability distributions and to model real-life problems using them.

Modified power series family and properties - Binomial - Negative binomial, Logarithmic series and Lagrangian distributions and their properties as special cases of the results from modified power series family - hypergeometric distribution and its properties. 

Unit II: Continuous Distributions Teaching Hours: 12

Pearsonian system of distributions - Beta, Gamma, Pareto and Normal as special cases of the Pearson family and their properties - Exponential family of distributions. 

Sampling distributions of the mean and variance from normal population - independence of mean and variance - chi-square, students t and F distribution and their non-central forms - Order statistics and their distributions.

Unit IV: Multivariate distributions Teaching Hours: 12

Bivariate Poisson, Multinomial distribution - Multivariate normal (definition only) - bivariate exponential distribution of Gumbel - Marshall and Olkin distribution - Dirichlet distribution.

Quadratic forms in normal variables: distribution and properties - Cochran’ theorem: applications.

2.Krishnamoorthy, K. (2016). Handbook of statistical distributions with applications. CRC Press.

2. Johnson N.L, Kotz S and Balakrishnan N (2017) Continuous univariate distributions I & II, John Wiley. 

3. Johnson N.L, Kotz S and Balakrishnan N (2000) Multivariate Distribution, 2nd Ed., John Wiley.

4.Arnold B.C, Balakrishnan N and Nagaraja H.N (2012) A first course in order statistics. 

5.Elderton, W. P., & Johnson, N. L. (2009). Systems of frequency curves, Cambridge University press.

This course is offered to make students understand the critical aspects of matrix theory and linear models used in different areas of statistics such as regression analysis, multivariate analysis, design of experiments and stochastic processes.

Matrix operations - Linear equations - row reduced and echelon form - Homogenous system of equations - Linear dependence

Vectors - Operations on vector space - subspace - nullspace and column space - Linearly independent sets - spanning set - bases - dimension - rank - change of basis.

Algebra of linear transformations - Matrix representations - rank nullity theorem - determinants - eigenvalues and eigenvectors - Cayley-Hamilton theorem - Jordan canonical forms - orthogonalisation process - orthonormal basis.

Reduction and classification of quadratic forms - Special matrices: symmetric matrices - positive definite matrices - idempotent and projection matrices - stochastic matrices - Gramian matrices - dispersion matrices

Fitting the model - ordinary least squares - estimability of parametric functions - Gauss – Markov theorem - applications: regression model - analysis of variance.

2. Lipschutz, S., & Lipson, M. L. (2018). Schaum's Outline of Linear Algebra. McGraw-Hill Education.

2. Rencher, A. C., & Schaalje, G. B. (2008). Linear models in statistics. John Wiley & Sons.

3. Khuri, A. I. (2003). Advanced calculus with applications in statistics. Hoboken, NJ: Wiley-Interscience.

4. Gentle, J. E. (2017) Matrix algebra- Theory, Computations and Applications in Statistics. Springer texts in statistics, Springer, New York.

5. Strang, G. (2006) Linear Algebra and its Applications: Thomson Brooks. Cole, Belmont, CA, USA.

CIA1                                                : 10

Mid Semester Examination (CIA II)     : 25

CIA III                                              : 10

Attendance                                        : 05

End Semester Exam                           : 50

Total                                                 : 100

This course aims to impart the concepts of survey sampling theory and the analysis of complex surveys, including methods of sample selection, estimation, sampling variance, standard error of estimation in a finite population, development of sampling theory for use in sample survey problems and sources of errors in surveys.

Sampling vs census, simple random sampling: with (SRS) and without replacement (SRSWOR) of units, estimators of mean, total and variance, determination of sample size, sampling for proportions, Stratified sampling scheme: estimation and allocation of sample size, comparison with simple random sampling schemes. 

Lab Exercises:

1. Drawing samples with SRSWR and SRSWOR and estimation of parameters

2. Estimation of parameters using a sample of proportions

3. Drawing stratified sample and estimation of parameters

Bias and mean square error, estimation of variance, confidence interval, comparison with mean per unit estimator, optimum property of ratio estimator, unbiased ratio type estimator, ratio estimator in stratified random sampling, Difference estimator and Regression estimator, comparison of regression estimator with mean per unit and ratio estimator, regression estimator in stratified random sampling.

Lab Exercises:

4. Estimation using ratio estimator

5. Estimation using regression estimator

6. Ratio estimator and regression estimator in stratified sampling

With and without replacement sampling schemes: PPS and PPSWR schemes, Selection of samples, estimators: ordered and unordered estimators. Π ps  sampling schemes.

Lab Exercises:

7. Exercise on the PPS scheme

8. Exercise on the PPSWR scheme

9. Exercise on Πps sampling scheme

Systematic sampling scheme: estimation of population mean and variance, comparison of systematic sampling with SRS and stratified random sampling, circular systematic sampling, Cluster sampling: estimation of population mean, estimation of efficiency by a cluster sample, variance function, determination of optimum cluster size, Multistage sampling: estimation population total with SRS sampling at both stages, multiphase sampling (outline only), quota sampling, network sampling; Adaptive sampling: introduction and estimators under adaptive sampling. Introduction to small area estimation.

Lab Exercises:

9. Exercise on the systematic sampling scheme

10. Exercise on cluster sampling

11. Exercise on multi-stage sampling

12. Exercise on small area estimation

Sampling and non-sampling errors, the effect of unit nonresponse in the estimate, procedures for unit nonresponse

Lab Exercises:

13. Exercise on the sensitivity of efficiency due to sampling errors

14. Procedures for non-response

2. Singh, D. and Chaudharay, F.S. (2018) Theory and Analysis of Sample Survey Designs, New Age International.

2. Singh, S. (2003). Advanced Sampling: Theory and Practice. Kluwer.

3. Des Raj and Chandhok, P. (2013) Sampling Theory, McGraw Hill. 

4. Mukhopadhay, P (2009) Theory and methods of survey sampling, Second edition, PHI Learning Pvt Ltd., New Delhi.

5. Sampath, S. (2005) Sampling theory and methods, Alpha Science International Ltd., India.

6. Lumley, T. (2011). Complex surveys: a guide to analysis using R. John Wiley & Sons.

ESE - 50%

Python is a generic programming language that is extensively used in data science. This course equips students with programming skill in Python and associated statistical libraries and to apply in data analysis

  Installing Python - basic syntax - interactive shell - editing, saving  and  running  a script. The concept of data types -  variables -  assignments - mutable type - immutable types - arithmetic operators and expressions -  comments in the program - understanding error messages - Control statements - operators.

Practical Assignments:

1.  Lab exercise on data types

2.  Lab exercise on arithmetic operators and expressions 

3. Lab exercise on Control statements

Introduction to functions - inbuilt and user defined functions - functions with arguments and return values - formal vs actual arguments - named arguments - Recursive functions -  Lambda function - OOP Concepts -  classes - objects - attributes and methods - defining classes - inheritance -  polymorphism.

Practical Assignments:

4.  Lab exercise on inbuilt and user-defined functions

5.  Lab exercise on Recursive and Lambda function 

6. Lab exercise on OOP Concepts.

Introduction to Pandas - Pandas data series - Pandas data frames - data handling -  grouping - Descriptive statistical analysis and Graphical representation.

Practical Assignments:

7.  Lab exercise on Pandas data series, frame, handling and grouping

8.  Lab exercise on statistical analysis 

  Hypothesis testing - data modeling - linear regression models -  logistic regression model.

Practical Assignments:

9.  Lab exercise on Hypothesis testing

10.  Lab exercise on regression modelling

Line graph - Bar chart - Pie chart - Heat map - Histogram - Density plot - Cumulative frequencies - Error bars - Scatter plot - 3D plot.

Practical Assignments:

11.  Lab exercise on graphical and diagrammatic representation.

12.  Lab exercise on the density plot

13.  Lab exercise on scatter and 3D plot

To acquaint students with different methodologies in statistical research and to make them prepare scientific articles using LaTeX.

Objectives - Motivation - Utility - Concept of theory - empiricism - deductive and inductive theory - Characteristics of the scientific method - Understanding the language of research - Concept - Construct - Definition - Variable - Research Process  Problem Identification & Formulation - Research Question – Investigation Question - Logic & Importance

Principles of mathematical writing - LaTeX: installing packages and editor, preparing title page - mathematical expressions - tables - importing graphics - bibliography - writing a research paper - survey article - thesis writing - Beamer: preparing presentations

ESE - 50%

To provide a strong mathematical and conceptual foundation in the methods of parametric estimation and their properties.

Sufficiency - factorisation theorem - minimal sufficiency - exponential family and completeness - Ancillary statistics and Basu's theorem

UMVUE - Fisher Information and Cramer-Rao inequality - Chapman-Robbin’s and Bhattacharya bounds - Rao-Blackwell theorem - Lehman-Scheffe theorem - Unbiased estimation

Consistency - Weak and strong consistency - Marginal and joint consistent estimators - CAN estimators - equivariance - Pitman estimators

Methods of moments - Minimum chi square and its modification, Least square estimation, Maximum likelihood, Properties of maximum likelihood estimators, Cramer-Huzurbazar Theorem, Likelihood equation - multiple roots, Iterative methods, EM Algorithm.

Large sample confidence interval - shortest length confidence interval - Methods of finding confidence interval: Inversion of the test statistic, pivotal quantities, pivoting CDF- evaluation of confidence interval: size and coverage probability, loss function and test function optimality.

2.Silvey, S. D. (2017). Statistical inference. Routledge.

3.Trosset, M. W. (2009). An introduction to statistical inference and its applications with R. Chapman and Hall/CRC.

4.Dixit, U. J. (2016). Examples in parametric inference with R, Springer.

5.Lehmann, E. L., & Casella, G. (2006). Theory of point estimation, 2nd Ed. Springer.

6.Robert, C., & Casella, G. (2013). Monte Carlo statistical methods. Springer

ESE-50%

Regression models are mainly used in establishing a relationship among variables and predicting future values. It got applications in various domain such as finance, life science, management, psychology, etc. This course is designed to impart the knowledge of statistical model building using regression technique.

Linear Regression Model: Simple and multiple -  Least squares estimation -  Properties of the estimators - Maximum likelihood estimation -  Estimation with linear restrictions -  Hypothesis testing -  confidence intervals.

Practical Assignments:

1.     Build a simple linear model and interpret the data.

2.     Construct confidence interval for the simple linear model

3.     Build a multiple linear models and estimate its parameters.

4.     Construct confidence interval for multiple linear model

Residual analysis -  Departures from underlying assumptions -  Effect of outliers - Collinearity - Nonconstant variance and serial correlation -  Departures from normality -  Diagnostics and remedies. 

Practical Assignments:

5.     Carry out residual analysis and validate the model assumptions.

6.     Construct residual plots for checking outliers, leverage points and influential points.

7. Checking the assumption of homoscedasticity and its remedial measures 

8. Detecting multicollinearity and its remedial measures

selection of input variables and model selection -  Methods of obtaining the best fit - stepwise regression -  Forward selection and backward elimination

Practical Assignments:

9.     Selecting the best model using step wise regression.

10.     Selecting the best model using the forward and backward selection procedure.

Introduction to general non-linear regression - least-squares in non-linear case - estimating the parameters of a non-linear system - reparametrization of the model -  Non-linear growth models 

Practical Assignments:

11.Estimate parameters in non-linear models using the least square procedure

Linear absolute deviation regression - M estimators -  robust regression with rank residuals -  resampling procedures for regression models, methods and its properties (without proof) -  Jackknife techniques and least-squares approach based on M-estimators.

Practical Assignments:

12.     Illustrate resampling procedures in regression models.

13.     Build a regression model with robust regression procedures.

2. Draper, N. R., & Smith, H. (2014). Applied regression analysis. 3rd edition. John Wiley & Sons.

3. Montgomery, D. C., Peck, E. A., & Vining, G. G. (2021). Introduction to linear regression analysis. John Wiley & Sons.

2. Keith, T. Z. (2014). Multiple regression and beyond: An introduction to multiple regression and structural equation modelling. Routledge.

3. Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.

4. Fox, J., & Weisberg, S. (2018). An R companion to applied regression. Sage publications.

ESE - 50%

The programming skill in R helps students to perform statistical computations with ease. This course equips students with knowledge of R programming to develop statistical models for real-world problems

R and R studio - Variables - Functions - Vectors - Expressions and assignments - Logical expressions - Matrices - The workspace - R markdown.

Practical Assignments:

1. Demonstrate variables and functions in R

2. Creating vectors and matrices and associated operations in R

3. Logical and arithmetic operations in R

Loops: if, for, while - Program flow - Basic debugging  - Good programming habits -  Input and outputs: Input from a file and output to a file

Practical Assignments:

4. Illustration of control structures: if, else, for

5. Illustration of control structures: while, repeat, break, next and ifelse

Functions - Optional arguments and default values - Vector-based programming using functions - Recursive programming - Debugging functions - Sophisticated data structures - Factors -Dataframes - Lists - The apply family.

Practical Assignments:

7. Creating user-defined functions and doing vector-based programming

8. Creating lists and data frames and associated operations

9. Demonstration of recursive functions, apply functions in R

Visualising data - Graphical summaries of data: Bar chart, Pie chart, Histogram, Box-plot, Stem and leaf plot, Frequency table - Plotting of probability distributions and sampling distributions - P-P plot - Q-Q Plot  - ggplot2 - lattice – 3D plots,  -  par -graphical augmentation.

Practical Assignments:

10. Visualization of univariate data

11. Visualization of numerical variables in R using ‘base R’, ‘ggplot2’ and ‘lattice 3D’ packages

12. Contingency tables and visualization of categorical variables using ‘base R’, ‘ggplot2’ and ‘lattice 3D’  packages

13. Construction  of probability plots and quantile plots in R

Simulating iid uniform samples - Congruential generators - Seeding - Simulating discrete random variables - Inversion method for continuous random variables -  Rejection method - generation of normal variates: Rejection with exponential envelope, Box-Muller algorithm.

Practical Assignments:

14.  Simulation of discrete variables in R

15. Simulation of continuous variables- inversion method, rejection method

 ESE 50%

This course provides a strong foundation for data science and application area related to it and caters for the underlying core concepts and emerging technologies in data science.

Introduction – Big Data and Data Science  – Data science Hype – Getting Past the Hype – The Current Landscape – Role of Data Scientist – Exploratory Data Analysis –  Data Science Process Overview – Defining goals – Retrieving data – Data preparation – Data exploration – Data modelling – Presentation. Problems in handling large data – General techniques for handling large data – Big Data and its importance, Four Vs, Drivers for Big data, Big data analytics, Big data applications, Algorithms using map-reduce, Matrix-Vector Multiplication by Map Reduce. Steps in big data – Distributing data storage and processing with Frameworks – Data science ethics – valuing different aspects of privacy – The five C’s of data.

 Practical Assignments:

1.      1. Lab exercise for feature engineering

2. Lab exercise for big data processing

Data Warehousing - Defining Feature – Data warehouses and data marts –Metadata in the data warehouse – Data design and Data preparation - Dimensional Modeling - Principles of dimensional modeling – The star schema – star schema keys – Advantages of the star schema – Updates to the dimension tables – The snowflake schema – Aggregate fact tables – Families Oo Stars – MDX queries – Reporting services.

Practical Assignments:

11. Lab Exercise on Analysis Services

12. Lab Exercise on Reporting Services

2. D J Patil, Hilary Mason, Mike Loukides, (2018), Ethics and Data Science, O’ Reilly.

3. LiorRokach and OdedMaimon, (2010), Data Mining and Knowledge Discovery Handbook. 

This course provides an understanding of various statistical methods in describing and analyzing biological data. Students will be equipped with an idea about the applications of statistical hypothesis testing,  related concepts and interpretation in biological data.

Presentation of data - graphical and numerical representations of data - Types of variables, measures of location - dispersion and correlation - inferential statistics - probability and distributions - Binomial, Poisson, Negative Binomial, Hyper geometric and normal distribution.

Practical Assignments:

1.Exercise on the representation of data

2.Exercise on reporting data by descriptive statistics

Parametric methods - one sample t-test -  independent sample t-test - paired sample t-test - one-way analysis of variance - two-way analysis of variance - analysis of covariance - repeated measures of analysis of variance - Pearson correlation coefficient - Non-parametric methods: Chi-square test of independence and goodness of fit - Mann Whitney U test - Wilcoxon signed-rank test - Kruskal Wallis test - Friedman’s test - Spearman’s correlation test. 

Practical Assignments:

3.Exercise on various parametric methods of analysis

4.Exercise on various non-parametric methods of analysis

Review of simple and multiple linear regression - introduction to generalized linear models - parameter estimation of generalized linear models - models with different link functions   - binary (logistic) regression - estimation and model fitting - Poisson regression for count data -  mixed effect models and hierarchical models with practical examples. 

Practical Assignments:

5.Exercise on simple linear and multiple linear regression

6.Exercise on logistic regression

7.Exercise on Poisson regression

Introduction to epidemiology, measures of epidemiology, observational study designs: case report, case series correlational studies, cross-sectional studies, retrospective and prospective studies, analytical epidemiological studies-case control study and cohort study, odds ratio, relative risk, the bias in epidemiological studies.

Practical Assignments:

8.Exercise on analysis of observational study data 

9.Exercise on analysis of cross-sectional study data

10.Exercise on analysis of case-control study data

11.Exercise on analysis of cohort study data

Introduction to demography, mortality and life tables, infant mortality rate, standardized death rates, life tables, fertility, crude and specific rates, migration-definition and concepts population growth, measurement of population growth-arithmetic, geometric and exponential, population projection and estimation, different methods of population projection,  logistic curve, urban population growth, components of urban population growth.

Practical Assignments:

12.Exercise on preparing life tables

13.Exercise on measures of population growth

2.David Moore S. and George McCabe P., (2017) Introduction to practice of statistics, 9th Edition, W. H. Freeman.

3.Sundar Rao and Richard J., (2012) Introduction to Biostatistics and research methods, PHI Learning Private limited, New Delhi.

2.Gordis Leon (2018), Epidemiology, 6th Edition, Elsevier, Philadelphia

3.Ram, F. and Pathak K. B., (2016): Techniques of Demographic Analysis, Himalaya Publishing house, Bombay.

4.Park K., (2019), Park's Text Book of Preventive and Social Medicine, Banarsidas Bhanot, Jabalpur.

ESE-50%

This course is designed to train the students to develop their modelling skills in mathematics through various methods of optimization. The course helps the students to understand the theory of optimization methods and algorithms developed for solving various types of optimization problems. 

Introduction to optimization – convex set and convex functions – simplex method: iterative nature of simplex method – additional simplex method: duality concept - dual simplex method – generalized simplex algorithm - revised simplex method: revised simplex algorithm – development of the optimality and feasibility conditions.   

Practical Assignments:

1. Formulate the LPP.

2. Solve the LPP using simplex method.

3. Solve the LPP using revised simplex method.                                                                                                                  

Branch and bound algorithm – cutting plane algorithm – transportation problem: north-west method, least-cost method, vogel’s approximation and method of multipliers – assignment problem: mathematical statement, Hungarian method, variations of assignment problems.

 Practical Assignments:

4. Solve integer LPP by cutting plane method.

5. Formulate and solve transportation problems.

6. Formulate and solve assignment problems.  

Introduction – unimodal function – one dimensional optimization: Fibonacci method – golden Section Method – quadratic interpolation methods - cubic interpolation methods – direct root method: newton method and quasi newton method – Multidimensional unconstrained optimization: univariate method – Hooks and Jeeves method – Fletcher – Reeves method - Newton’s method and quasi newton’s method.

Practical Assignments:

7. Solve a non LPP problem.

8. Solve an unconstrained optimization problem by a univariate method

Single variable optimization – multivariable optimization with no constraints: semi-definite case and saddle point – multivariable optimization with equality constraints: direct substitution – method of constrained variation – method of Lagrange multipliers - Kuhn-Tucker conditions - constraint qualification – convex programming problem.

Practical Assignments:

9. Solve a single variable optimization problem.

10. Solve multivariable optimization problems with equality constraints.

11. Solve a  convex optimization problem.

Unconstrained minimization problem – solution of an unconstrained geometric programming problem using arithmetic-geometric inequality method – primal dual relationship - constrained minimization - dynamic programming: Dynamic programming algorithm – solution of linear programming problem by dynamic programming.

 Practical Assignments:

12. Formulate and solve a dynamic programming problem.

13. Solve LPP through dynamic programming problems.

14. Solve a  geometric programming problem.

ESE-50%

This course is designed to provide the strong conceptual foundations of testing of hypothesis, procedures of testing hypothesis, likelihood ratio tests, sequential tests and non-parametric tests

Robust estimation: The influence curve and empirical influence curve - M-estimation: Median, Trimmed and winsorized mean - Influence curve for M-estimators -  Limiting distribution of M-estimators - Resampling methods: Quenouille’s Jackknife estimation, parametric and non-parametric bootstrap methods.

Basic concepts in statistical hypotheses testing - Simple and composite hypothesis - Critical regions - Type-I and Type-II error - Significance level - p-value and power of a test - Randomised and non-randomised tests - Neyman-Pearson lemma and its applications - Generalization of NP lemma - Construction of tests using NP lemma - Most powerful test - Uniformly most powerful test - Monotone Likelihood Ratio (MLR) property - Testing in one-parameter exponential families - Unbiased and invariant tests - Locally most powerful tests.

One-sided uniformly most powerful tests - Unbiased and Uniformly Most Powerful Unbiased tests for different two-sided hypothesis - Extension of these results to Pitman family when only upper or lower end depends on the parameters - UMP test from α-similar tests and α-similar tests with Neyman structure

Likelihood ratio test (LRT) - asymptotic properties - LRT for the parameters of binomial and normal distributions - Generalized likelihood ratio tests - Chi-Square tests - t-tests - F-tests - Need for sequential tests - Sequential Probability Ratio Test (SPRT) - Wald’s fundamental identity - OC and ASN functions - Applications to Binomial, Poisson and Normal distributions.

Non-parametric tests: Sign test - Chi-square tests - Kolmogorov-Smirnov one sample and two samples tests - Median test - Wilcoxon Signed Rank test - Mann- Whitney U-test - Test for Randomness - Runs up and runs down test - Wald–Wolfowitz run test for equality of distributions - Kruskal–Wallis one-way analysis of variance - Friedman’s two-way analysis of variance - Power and asymptotic relative efficiency

Statistics, John Wiley and Sons.

2.  Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 2/e, John Wiley, NewYork.

2nd edition. Mc-Millan, New York.

4. Kale, B. K. and Muralidharan, K.  (2015). Parametric Inference: An Introduction. Alpha Science Int. Ltd.
5. Mukhopadhyay, P.(2015): Mathematical Statistics, Books and Allied (P) Ltd., Kolkata.
6. Gibbons J.K. (1971).  Non-Parametric Statistical Inference, McGraw Hill.

The exposure provided to the multivariate data structure, multinomial and multivariate normal distribution, estimation and testing of parameters, various data reduction methods would help the students in having a better understanding of research data, its presentation and analysis. This course helps to understand multivariate data analysis methods and their applications in various research areas.

Basic concepts on multivariate variables - Multivariate normal distribution - Marginal and conditional distribution - Concept of random vector - Its expectation and Variance - Covariance matrix. Marginal and joint distributions - Conditional distributions and Independence of random vectors - Multinomial distribution - Characteristic functions in higher dimensions - Multiple regressions and multiple correlations - Partial regression and Partial correlation (illustrative examples).

Multivariate analysis of variance (MANOVA) and Covariance (MANCOVA) of one and two-way classified data with their interactions - Univariate and Multivariate Two-Way Fixed-effects Model with Interaction.

Wishart distribution (definition, properties) - Construction of tests - Union - Intersection and likelihood ratio principles - Inference on mean vector - Hotelling's T2- Comparing Mean Vectors from Two Populations - Bartlett’s Test. 

Concepts of discriminant analysis - Computation of linear discriminant function (LDF) - Classification between k multivariate normal populations based on LDF - Fisher’s Method for discriminating two or several populations - Evaluating Classification Functions - Probabilities of misclassification and their estimation - Mahalanobis D2.

Factor analysis: - Orthogonal factor model - Factor loadings - Estimation of factor loadings - Factor scores and Its applications.

Cluster Analysis: - Distances and similarity measures - Hierarchical clustering methods - K- Means method. 

Wiley. New York.

2. Johnson, R.A. and Wichern, D.W. (2018). Applied Multivariate Statistical Analysis.

6th edn. Prentice- Hall. London.

Mathematical Statistics. 2nd edn. John Wiley & Sons. New York.

2. Srivastava, M.S. and Khatri, C.G. (1979). An Introduction to Multivariate Statistics.

North Holland.

3. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley. New

York.

CIA-50%

ESE-50%

This course considers statistical techniques to evaluate processes occurring through time. It introduces students to time series methods and the applications of these methods to different types of data in various fields. Time series modelling techniques including AR,MA,ARMA,ARIMA,SARIMA and their multivariate extensions will be considered with reference to their use in forecasting. The objective of this course is to equip students with various forecasting techniques and to familiarize themselves with modern statistical methods for analyzing time series data.

Stochastic Process - Time series as a discrete parameter stochastic process - Auto – Covariance - Autocorrelation and their properties - Exploratory time series analysis- graphical analysis - classical decomposition model - concepts of trend, seasonality and cycle - Estimation of trend and seasonal components-Elimination of trend and seasonality - Method of differencing - Moving average smoothing - Method of seasonal differencing

Practical Assignments:

1.Graphical representation of time series, plots of ACF and PACF and their interpretation

2. Examples of  trend, seasonal and cyclical time series and estimation of trend and seasonal  components

3. Exercise on Moving average smoothing to eliminate trend and illustration on the method of differencing to eliminate trend and seasonality.

4.Exercise on least-square fitting to estimate and eliminate the trend component.

Stationary time series models - Concepts of weak and strong stationarity - General linear Process - Auto-Regressive(AR), Moving Average(MA) and Auto-Regressive Moving Average (ARMA) processes – their properties - conditions for stationarity and invertibility -model identification based on ACF and PACF- Maximum likelihood estimation - Yule Walker Estimation - Order selection ( AIC and BIC ) - Residual Analysis - Box Jenkins methodology to the identification of stationary time series models

Practical Assignments:

5.Exercise on fitting AR, MA and ARMA models.

6.Model identification using ACF and PACF,  model selection using AIC and BIC.

7. Residual analysis and diagnosis check for AR, MA and ARMA models.

Concept of non-stationarity – Unit root - Spurious trends and regressions-unit root tests : Dickey-Fuller (DF) test - Augmented Dickey-Fuller(ADF) test – Auto-Regressive Integrated Moving Average(ARIMA(p,d,q)) models - Difference equation form of ARIMA- Random shock form of ARIMA – Seasonal Unit root test - Seasonal ARIMA models

Practical Assignments: 

8. Exercise on the identification of non-stationary series from various plots.

9. Exercise on testing non-stationarity using ADF test, Exercise on fitting ARIMA models.

10. Residual analysis and diagnosis check for ARIMA model.

11. Excercise on seasonal unit root test and SARIMA

In sample and out of sample forecast - Simple exponential and moving average smoothing - Holt Exponential Smoothing - Winter exponential smoothing - Forecasting trend and seasonality in Box Jenkins model:- Method of minimum mean squared error(MMSE) forecast - their properties - forecast error

Practical Assignments:

12.Exercise on Simple exponential smoothing and Holt Exponential Smoothing

13.Exercise on Winters exponential smoothing.

14.Exercise on forecasting using ARIMA models.

15.Exercise on forecasting using seasonal ARIMA models.

 Introduction to Cross covariance and correlation matrix – Vector Autoregressive Models – Vector Moving Average Models – Vector Autoregressive Moving Average Models – their properties – Concept of Cointegration - Johansen Cointegration test

Practical Assignments:

16. Exercise on fitting  of VAR, VMA, VARMA models

17.Illustration of cointegration procedure

2. Chatfield, C., & Xing, H. (2019). The analysis of time series: an introduction with R. CRC Press.

2. Brockwell, P. J., & Davis, R. A. (2016). Introduction to time series and forecasting. springer.

ESE - 50%

Java programming is essential tool for data science and statistical computation. This course enables students to create programs in Java which can be used for statistical analysis

Installing java Development kit - Java JVM, JRE and JDK - Working with the Terminal –First Program - Data Types: integer, float, char, boolean – Variables and constants -  - Operators - Input and Output - Expressions & Blocks - Comment

Practical Assignments:

1. Lab exercise on data types

2. Lab exercise on various operators in Java

3. Lab exercise on expressions & blocks 

Conditional Statements - Truth Tables - For Loops - While Loops – Functions – Declarations – Overloading - Return Values                                      

Practical Assignments:

4. Lab exercise on conditional statements

5. Lab exercise on loops

6. Lab exercise on functions

Primitive data structure  and Non-primitive data structure  - Array Processing - Multi-dimensional Arrays - String Methods - String Manipulation – Regular Expressions - Operations in Dictionaries

Practical Assignments:

7. Lab exercise on arrays

8. Lab exercise on strings and regular expressions

9. Lab exercise on dictionaries

Objects - Encapsulation – Classes – Inheritance – Inheritance Hierarchies - Polymorphisim – Abstraction – Protected Classes – Exceptions –Assertions –Loggin – Generic Programming

Practical Assignments:

10. Lab exercise on classes and objects

11. Lab exercise on inheritance property

12. Lab exercise exception handling and assertions

Recursive Functions - Recurrence Relation - Base Case Analysis - Motivation Behind Recursion

Practical Assignments:

13. Lab exercise on recursive functions

14. Lab exercise on base case analysis

2.     Schildt, H., & Coward, D. (2014). Java: the complete reference. New York: McGraw-Hill Education.

Machine learning has a wide array of applications that belongs to different fields, such as biomedical research, reliability of large structures, space research, digital marketing, etc. This course will equip students with a wide variety of models and algorithms for machine learning and prepare students for research or industry application of machine learning techniques. 

Statistical learning: definition-prediction accuracy and model interpretability-supervised and unsupervised learning-assessing model accuracy- important problems in data mining: classification, regression, clustering, ranking, density estimation- Concepts: training and testing, cross-validation, overfitting, bias/variance tradeoff, regularized learning equation- simple and multiple linear regression algorithms

 Practical Assignments:

1.     Lab exercise on data preparation and using simple linear regression

2.     Lab exercise on model assessment simple linear regression

3.     Lab exercise on data preparation with multiple linear regression

Logistic model- training and testing the model-linear discriminant analysis-quadratic discriminant analysis- Use of Bayes’ theorem-k- nearest neighbours - Naive Bayes’- Adaboost

 Practical Assignments:

1.     Lab exercise on the logistic model

2.     Lab exercise on discriminant analysis

3.     Lab exercise on  Naïve Bayes’ and k-NN classifiers

4.     Lab exercise on  Adaboost

Optimal model-shrinkage methods: ridge and lasso regression-Dimension reduction methods: principal component (PC) regression and partial least square (PLS) regression: Non-linear models: regression splines-polynomial – Generalized additive models

 Practical Assignments:

1.     Lab exercise on ridge regression

2.     Lab exercise on Lasso regression

3.     Lab exercise on PC regression

4.     Lab exercise on PLS regression

Decision tree-regression trees - bagging - random forests - boosting - classification trees-boosting-tree vs linear models

 Practical Assignments:

1.     Lab exercise on decision trees

2.     Lab exercise on regression trees

3.     Lab exercise on random forests

4.     Lab exercise on classification trees

Maximal classifier-support vector classifiers-support - rank boost (ranking algorithm) - hierarchical Bayesian modelling for density - resampling techniques-bootstrap- clustering algorithms: K-means algorithm. 

Practical Assignments:

1.     Lab exercise on SVM classifier

2.     Lab exercise on rank boost algorithm

3.     Lab exercise on kernel density estimation

4.     Lab exercise on k-means clustering

This course provides an understanding of various statistical methods in describing and analyzing biological data. Students will be equipped with an idea about the applications of statistical hypothesis testing,  related concepts and interpretation in biological data.

Presentation of data - graphical and numerical representations of data - Types of variables, measures of location - dispersion and correlation - inferential statistics - probability and distributions - Binomial, Poisson, Negative Binomial, Hyper geometric and normal distribution.

Practical Assignments:

1. Exercise on the representation of data
2. Exercise on reporting data by descriptive statistics

Parametric methods - one sample t-test -  independent sample t-test - paired sample t-test - one-way analysis of variance - two-way analysis of variance - analysis of covariance - repeated measures of analysis of variance - Pearson correlation coefficient - Non-parametric methods: Chi-square test of independence and goodness of fit - Mann Whitney U test - Wilcoxon signed-rank test - Kruskal Wallis test - Friedman’s test - Spearman’s correlation test.

Practical Assignments:

3. Exercise on various parametric methods of analysis

4. Exercise on various non-parametric methods of analysis

Review of simple and multiple linear regression - introduction to generalized linear models - parameter estimation of generalized linear models - models with different link functions   - binary (logistic) regression - estimation and model fitting - Poisson regression for count data -  mixed effect models and hierarchical models with practical examples.

Practical Assignments:

5.Exercise on simple linear and multiple linear regression

6. Exercise on logistic regression

7. Exercise on Poisson regression

Introduction to epidemiology, measures of epidemiology, observational study designs: case report, case series correlational studies, cross-sectional studies, retrospective and prospective studies, analytical epidemiological studies-case control study and cohort study, odds ratio, relative risk, the bias in epidemiological studies.

Practical Assignments:

8. Exercise on analysis of observational study data
9. Exercise on analysis of cross-sectional study data
10. Exercise on analysis of case-control study data
11. Exercise on analysis of cohort study data

Introduction to demography, mortality and life tables, infant mortality rate, standardized death rates, life tables, fertility, crude and specific rates, migration-definition and concepts population growth, measurement of population growth-arithmetic, geometric and exponential, population projection and estimation, different methods of population projection,  logistic curve, urban population growth, components of urban population growth.

Practical Assignments:

12. Exercise on preparing life tables
13. Exercise on measures of population growth

This course will provide knowledge in different probability models in the reliability evaluation of the system and its components. Reliability engineering is applied in the industry to reduce failures, ensure effective maintenance and optimize repair time.

Reliability of a system - failure rate - mean, variance and percentile residual life: identities connecting them - notions of ageing - IFR, IFRA, NBU, NBUE, DMRL, HNBUE, NBUC, etc. and their mutual implications - TTT transforms and characterization of ageing classes.

Practical Assignments:

1. Exercise on failure rate function and mrl function
2. Exercise on comparison ageing classes

Non-monotonic failure rates and mean residual life functions - study of lifetime models: exponential, Weibull, lognormal, generalized Pareto, gamma with reference to basic

concepts and ageing characteristics - bathtub and upside-down bathtub failure rate

distributions

Practical Assignments:

      3. Exercise on exponential lifetime model

      4. Exercise on Weibull lifetime model

      5. Exercise on bathtub shaped lifetime model

Reliability systems with dependents components: Parallel and series systems, k out of n

Systems - ageing properties with dependent and independents components -  concepts and

measures of dependence on reliability - RCSI, LCSD, PF2, WPQD.

Practical Assignments:

6. Exercise on reliability evaluation of series system

7. Exercise on reliability evaluation of a parallel system

8. Exercise on reliability evaluation of k out of n system

9. Exercise on reliability evaluation of dependent component system

Reliability estimation using MLE: exponential, Weibull and gamma distributions based on censored and non-censored samples - UMVU estimation of reliability function - Bayesian reliability estimation of exponential and Weibull models

Practical Assignments:

10. Exercise on ML estimation under non-censored samples.

11. Exercise on ML estimation under censored samples.

12. Exercise on  Bayesian estimation of reliability.

Life testing: basics – modelling lifetime – Accelerated Life Time (ALT) models- cumulative exposure models (CEM) - exponential CEM – stress-strength reliability – exponential stress-strength model.

Practical Assignments:

13. Exercise on basic life testing procedure.

14. Exercise on exponential CEM model.

15. Exercise on stress-strength reliability.

2.     Bain, L. (2017). Statistical analysis of reliability and life-testing models: theory and methods. Routledge.

 Java programming is essential tool for data science and statistical computation. This course enables students to create programs in Java which can be used for statistical analysis.

Installing java Development kit - Java JVM, JRE and JDK - Working with the Terminal –First Program - Data Types: integer, float, char, boolean – Variables and constants -  - Operators - Input and Output - Expressions & Blocks - Comment

Practical Assignments:

1. Lab exercise on data types

2. Lab exercise on various operators in Java

3. Lab exercise on expressions & blocks

Conditional Statements - Truth Tables - For Loops - While Loops – Functions – Declarations – Overloading - Return Values                                       

Practical Assignments:

4. Lab exercise on conditional statements

5. Lab exercise on loops

6. Lab exercise on functions

Primitive data structure  and Non-primitive data structure  - Array Processing - Multi-dimensional Arrays - String Methods - String Manipulation – Regular Expressions - Operations in Dictionaries

Practical Assignments:

7. Lab exercise on arrays

10. Lab exercise on strings and regular expressions

11. Lab exercise on dictionaries

Objects - Encapsulation – Classes – Inheritance – Inheritance Hierarchies - Polymorphisim – Abstraction – Protected Classes – Exceptions –Assertions –Loggin – Generic Programming

Practical Assignments:

13. Lab exercise on classes and objects

14. Lab exercise on inheritance property

15. Lab exercise exception handling and assertions

Recursive Functions - Recurrence Relation - Base Case Analysis - Motivation Behind Recursion

Practical Assignments:

17. Lab exercise on recursive functions

18. Lab exercise on base case analysis 

2.     Schildt, H., & Coward, D. (2014). Java: the complete reference. New York: McGraw-Hill Education.

This course will provide the basic theory and computing tools to perform nonparametric tests, including the Sign test, Wilcoxon signed-rank test, Median test etc. Kruskal-Wallis for one-way and multiple comparisons, linear rank test for location and scale parameters and measure of association in bivariate populations are other nonparametric tests covered in this course. The aim of the course is the in-depth presentation and analysis of the most common methods and techniques of non-parametric statistics such as sign test, rank test, run test, median test etc.

The quantile function - the empirical distribution function - statistical properties of order statistics- confidence interval for a population quantile -hypothesis testing for a population quantile -the sign test and confidence interval for the median - rank-order statistics -treatment of ties in rank tests-  Wilcoxon signed-rank test and confidence interval

PracticalAssignments:

1. Exercise on confidence interval estimation and hypothesis test for a population Quantile
2. Exercise on sign test and confidence interval for the median
3. Exercise on rank-order statistics -treatment of ties in rank tests
4. Exercise on Wilcoxon signed-rank test and confidence interval

Wald-Wolfowitz runs test - Kolmogorov-Smirnov two-sample test - median test - the control median test - the Mann-Whitney U test                                                                                    

Practical Assignments:

 5.Exercise on Wald-Wolfowitz runs test

6.Exercise on Kolmogorov-Smirnov two-sample test.

7.Exercise on Median test and control median test.

8.Exercise on Mann-Whitney U test.

Definition of linear rank statistics - Wilcoxon rank-sum test - mood test - Freund-Ansari-Bradley-David-Barton tests - Siegel-Tukey test

Practical Assignments:

9.Exercise on Wilcoxon rank-sum test and  mood test

10.Exercise on Freund-Ansari-Bradley-David-Barton tests.

11. Exercise on Siegel-Tukey test

Extension of the median test - the extension of the control median test - the Kruskal-Wallis one-way ANOVA test and multiple comparisons - tests against ordered alternatives - comparisons with a control -  Chi-Square test for k proportions

Practical Assignments:

12.Exercise on the extension of the median test and control median test.

13.Exercise on Kruskal-Wallis one-way ANOVA test.

14.Exercise on chi-square test for k  proportions

Introduction: definition of measures of association in a bivariate population - Kendall’s Tau coefficient - Spearman’s coefficient of rank correlation - relations between R and T; E(R), t, and r 

Practical Assignments:

15.Exercise on Kendall’s Tau coefficient.

16.Exercise on Spearman’s coefficient of rank correlation

1. Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods (Vol. 751). John Wiley & Sons.

2. Lewis, N. D. C. (2013). 100 Statistical Tests in R. Heather Hills Press.

ESE 50%

Game theory and the decision is a branch of Mathematics and Statistics that enables to study of the strategic interactions amongst rational decision-makers. Traditionally, game-theoretic tools have been applied to solve problems in Economics, Business, Political Science, Biology, Sociology, Computer Science, Logic, and Ethics. In recent years, applications of game theory have been successfully extended to several areas of engineered / networked system such as wireline and wireless communications, static and dynamic spectrum auction, social and economic networks.

Theory of games - zero-sum game - minimax - maxmin -  dominance strategy - the value of the game - Basic elements of game and Decision -  Comparison of the two theories - Decision function and Risk function; Randomization and optimal decision rules - Form of Bayes rules for estimation.                                                                                                         

Practical Assignments:

1. Two-person zero-sum game
2. Game with minimax and maximin strategies
3. Game with dominance rule and value of the game.

Admissibility and completeness - Fundamental theorems of Game and Decision theories - Admissibility of Bayes rules - Existence of Bayes decision rules - Existence of minimal complete class - Essential completeness of the class of non-randomized decision rules - Minimax theorem -  The complete class theorem -  Methods for finding minimax rules.                   

Practical Assignments:

1. Admissibility of Bayes rules
2. Examining the completeness
3. Problems on the non-randomized decision rule

Sufficient Statistics and essentially complete class of rules based on Sufficient Statistics - Complete Sufficient Statistics - Continuity of the risk function.         

Practical Assignments:

1. Examining the complete sufficient statistic
2. Rules-based on sufficient statistic

Invariant decision problems and rules - Admissible and minimax invariant rules -  Minimax estimates of location parameter - Minimax estimates for the parameters of normal distribution.                                                                                                        

Practical Assignments:

1. Minimax estimates of Location Parameter
2. Minimax estimates for the normal parameters.
3. Invariant minimax rules

Monotone Multiple decision problems - Bayes rules in multiple decision problems -  Slippage problems.

Practical Assignments:

1. Bayes rules in multiple decision problems
2. Slippage problems

    .

1.  Robert Winkler (Jan 15, 2003) An Introduction to Bayesian Inference and Decision, Second  Edition.
2. T.S. Ferguson (1967), Mathematical Statistics – A Decision Theoretic, Approach,  Academic Press.

3. M.H. DeGroot (1976), Optimal Statistical Decisions, McGraw Hill.

ESE - 50%

This will equip the student to apply statistical methods they have studied in various courses and present their work through research articles.

1. Apply various statistical methods in solving a real-life problem.

2. Comparison with the existing models or results.

3. Writing research article 

4. Presentation of the article

ESE 50%

Operations research helps in solving problems in different environments that need decisions. The module includes topics that include: linear programming, integer programming, nonlinear programming, simple queueing models and inventory models. The purpose of the course is to provide students with the concepts and tools to help them understand operations research and mathematical modelling methods. 

General Linear programming problem - Formulation -  Solution through graphical, Simplex, Big-M and Two phase methods - Revised Simplex method - Big-M and Two phase Revised Simplex methods - Duality - Primal-dual relationships - Dual Simplex method.

Gomory’s All-Integer Cutting-Plane Method - Construction of Gomory’s Constraint - Gomory’s Mixed-Integer Cutting-Plane Method -  Construction of Additional Constraint for Mixed-Integer Programming Problem - Branch and Bound Method.

General nonlinear programming problem - Constrained optimization with equality constraints - Necessary conditions for a generalized NLPP (without proof) - Sufficient conditions for a general NLPP with one constraint (without proof) - Sufficient conditions for a general problem with m(<n) constraints (without proof) - Constrained optimization with inequality constraints - Kuhn-Tucker conditions for general NLPP with m(<n) constraints (without proof)

Basics of queuing model - Probability distribution in a queueing system - Distribution of arrivals (Pure birth model) - Distribution of departures (Pure death model) -  Poisson queuing model:  (M/M/1) : (GD/∞/∞) - (M/M/1) : (N/FCFS/∞) - (M/M/c) : (∞/FCFS/∞) - (M/M/c) : (N/FCFS/∞).

Deterministic inventory Models - Economic Order Quantity(EOQ) models - Classic EOQ models - Problems with no shortages - The fundamental EOQ Problems: EOQ problems with several production runs of unequal length - Problems with price breaks - One price break - More than one price break - Probabilistic inventory models - Single Period Problem without set-up cost - I.

2. Taha, H. A. (2013). Operations research: an introduction. Pearson Education India.

3. Shortle, J. F., Thompson, J. M., Gross, D., & Harris, C. M. (2018). Fundamentals of queueing theory (Vol. 399). John Wiley & Sons.

4. Sharma, J. K. (2016). Operations research: theory and applications. Trinity Press, an imprint of Laxmi Publications Pvt. Limited.

This course will provide students with a mathematical background of various basic designs involving one-way and two-way elimination of heterogeneity and characterization properties. To prepare the students in deriving the expressions for analysis of experimental data and selection of appropriate designs in planning a scientific experimentation

Basic principles of design of experiments - Randomization - Replication and Local control - Uniformity trials - Size and Shape of plots and blocks - Elements of linear estimation - Analysis of variance - Completely Randomized Design (CRD) - Randomized Complete Block Design (RCBD) and Latin Square Design (LSD) -  Missing plot techniques.

Analysis of covariance - Ancillary/Concomitant variable and study variable - Linear model for ANCOVA - Adjustment of treatment sum of squares in ANCOVA - One - Way and two-way classification with a single concomitant variable in CRD and RCBD designs.

Factorial experiments - Simple experiment (single factor) vs Factorial experiments - Mixed and Fixed factor experiments - Treatment combination in a factorial experiment - Simple effect - Main effect and Interaction effect in a factorial experiment - Yates method of computing factorial effects totals - Complete and partial confounding in symmetrical factorial experiments (22, 23, 33, 2nand 3n series) - Gain in the factorial experiments.

Split - Plot, Split - Split plot and Strip - Plot (Split Block) design - Situation for the usage of the design - Layout and analysis of the designs - Difference in the error components in the designs - Selection of factor for allocation in plots (main/sub) - Combined experiments -  Cross - Over designs.

Balanced Incomplete Block (BIB) designs - General properties and Analysis with and without recovery of information - Construction of BIB designs - Parameter relationship - Intra and inter-block Analysis - Partially Balanced Incomplete Block Design (PBIBD) - Youden square designs - Lattice designs.

2.Cochran, W.G. and Cox, G.M. (1992). Experimental Designs. John Wiley. 

3.Dean, A.M. and Voss, D. (1999). Design and Analysis of Experiments. Springer.

4.Das, M.N. and Giri, N.C. (1986). Design and Analysis of Experiments. New Age.

5.Lawson, J. (2015). Design and Analysis of Experiments with R. CRC Press

6.Dey, A. (1986). Theory of Block Designs. Wiley Eastern Ltd.

ESE-50%

This course provides an introduction to the application of statistical tools in the industrial environment to study, analyze and control the quality of products

Meaning and scope of statistical quality control - Causes of quality variation - Control charts for variables and attributes - Rational subgroups - Construction and operation of, σ, R, np, p, c and u charts - Operating characteristic curves of control charts. Process capability analysis using histogram, probability plotting and control chart - Process capability ratios and their interpretations.

Specification limits and tolerance limits - Modified control charts - Basic principles and design of cumulative - sum control charts – Concept of V-mask procedure – Tabular CUSUM charts - Construction of Moving range - moving-average and geometric moving-average control charts.

Acceptance sampling: Sampling inspection by attributes – single, double and multiple sampling plans – Rectifying Inspection - Measures of performance: OC, ASN, ATI and AOQ functions - Concepts of AQL, LTPD and IQL - Dodge – Romig and MIL-STD-105D tables

Sampling inspection by variables - known and unknown sigma variables sampling plan - Merits and limitations of variables sampling plan - single, double and multiple sampling plans - Derivation of OC curve – determination of plan parameters.

Continuous Sampling Plans (CSP): CSP-1- CSP-2 - CSP-3 - Skip-Lot Sampling Plans (SkSP): SkSP-1 - SkSP-2  with SSP as reference plan - Chain Sampling Plans (ChSP - 1) with SSP as reference plan - Tighten-Normal-Tighted (TNT) sampling plan with SSP as reference plan– Decision Lines.

2. Schilling, E. G., and Nuebauer, D.V. (2009). Acceptance Sampling in Quality Control, Second Edition, CRC Press, New York

3. Duncan, A. J. (2003.). Quality Control and Industrial Statistics, Irwin-Illinois, US.

The objective of this course is to provide fundamental knowledge of neural networks and deep learning. This course gives a brief idea of the basics of neural networks, shallow and deep neural networks and other methods to build various research projects. 

Fundamental concepts of Artificial Neural Networks (ANN) - Biological neural networks - Comparison between biological neuron and artificial neuron - Evolution of neural networks - Scope and limitations of ANN - Basic models of ANN - Learning methods - Activation functions - Important terminologies of ANN: Weights - Bias - Threshold - Learning Rate - Momentum factor - Vigilance parameters.

Practical  Assignments:

1. Exercise on construction of Artificial Neural Networks.
2. Exercise on training and testing the data

Concept of supervised learning algorithms - Perceptron networks - Adaptive linear neuron (Adaline) - Multiple adaptive linear neuron - Back-Propagation network: Learning factors - Initial weights - Learning rate ɑ - Momentum factor - Generalization - Training and testing of the data.

Practical  Assignments: 

1. Exercise on multiple adaptive linear neurons 
2. Exercise on Back-propagation networks

Concept of unsupervised learning algorithms - Fixed weight competitive net: Maxnet - Mexican Hat net - Hamming networks - Kohonen self-organizing feature maps - Learning vector quantization.

Practical  Assignments:

1. Exercise on Maxnet and Mexican Hat net.
2. Exercise on Hamming networks
3. Exercise on Kohonen self-organizing feature maps
4. Exercise on Learning vector quantization

Introduction - Components of Convolution Neural Networks (CNN) architecture: Padding - Strides - Rectified linear unit layer - Exponential linear unit - Pooling - Fully connected layers  - Local response normalization - Hierarchical feature engineering -  Training CNN using Backpropagation through convolutions - Case studies: AlexNet - GoogLeNet.

Practical  Assignments:

1. Exercise on building CNN with the rectified linear unit and exponential linear unit
2. Exercise on Hierarchical feature engineering
3. Exercise on training CNN using Backpropagation through convolutions
4. Exercise on feature prediction using AlexNet and GoogLeNet. 

Stateless algorithms: Naive algorithms - Upper bounding methods - Simple reinforcement learning for Tic-Tac-Toe - Straw-Man algorithms - Bootstrapping for value function learning - One policy versus off policy methods: SARSA - Policy gradient methods: Finite difference method - Likelihood ratio method - Monte Carlo tree search.

Practical  Assignments:

1. Exercise on Naive and upper bounding algorithms for data classification.
2. Exercise on Bootstrapping for value function learning
3. Exercise on SARSA method
4. Exercise on Finite difference and likelihood ratio methods
5. Exercise on Monte Carlo tree search method. 

This course has been conceptualized in order to understand the fundamental and applied concepts of spatial statistics that describe the diverse set of methods to model and analyze the various types of Spatial data. 

Spatial data - Types of spatial data- Geostatistical data, Lattice data, Point pattern data with examples  - Visualizing spatial data: Traditional plots, lattice plots and interactive plots – Exploratory spatial data analysis - Intrinsic stationarity, Square-Root-Differences Cloud -  The Pocket plot – Decomposing the data into large and small scale variation -  Analysis of residuals – Variogram of residuals.

Practical Assignments:

1.     Exercise on the visualization of spatial data using traditional plots,

2.     Exercise on the visualization of spatial data using lattice and interactive plots

3.     Exercise on exploratory data analysis

Stationary Processes: Variogram, Covariogram and Correlogram - Estimation of variogram: Comparison of the variogram and covariogram estimation, exact distribution theory of the variogram - Robust estimation of variogram –  Spectral representations: Valid covariograms and variograms - Variogram model fitting: Criteria for fitting a variogram model, properties of variogram-parameter estimators, Cross-validating the fitted variogram.

Practical Assignments:

4.     Exercise on exploratory variogram analysis

5.     Exercise on variogram

6.     Exercise on variogram modelling

7.     Exercise on residual variogram modelling 

Scale of variation -  Ordinary Kriging: Effect of variogram parameters on Kriging,  Lognormal and Trans-Gaussian Kriging, Cokriging – Robust Kriging – Universal Kriging : Estimation of variogram for Universal Kriging – Median-Polish Kriging: Gridded and non-gridded data, Median Polishing spatial data, Bias in Median-Based  covariogram estimators – Applications of Geostatistics.

Practical Assignments:

8.     Exercise on Ordinary Kriging

9.     Exercise on Robust Kriging

10.     Exercise on Universal Kriging

Lattices – Spatial data analysis, Trend removal -  Conditionally and simultaneously specified spatial gaussian models – Markov random fields – Conditionally specified spatial models for discrete and continuous data – Parameter estimation for Lattice models using gaussian maximum likelihood estimation– Properties of estimators – Statistical image analysis and remote sensing.

Practical Assignments:

11.     Exercise on the estimation of parameters of lattice models

12.     Exercise on spatial autocorrelation

13.     Exercise on the fitting of lattice models 

Spatial point patterns data analysis: Complete spatial randomness, regularity and clustering – Kernel estimators of intensity function – Distance methods: Nearest-Neighbor methods – Statistical spatial analysis of point processes:  Stationary and Isotropic point processes – Palm distribution – Models and model fitting: Inhomogeneous Poisson, Cox and Poisson cluster process

Practical Assignments:

14.     Exercise on plotting spatial point patterns under the boundary

15.     Exercise on distance methods

16.    Exercise on Kernel smoothing

17.     Exercise on Inhomogeneous Poisson process

This course has been designed to train the students in handling different types of Big data sets and provide knowledge about the methods of handling these types of data sets.  

Concepts of Data Analytics: Descriptive, Diagnostic, Predictive, Prescriptive analytics -Big Data characteristics: Volume, Velocity, Variety, Veracity of data - Types of data: Structured, Unstructured, Semi-Structured, Metadata - Big data sources: Human-Human communication, Human-Machine Communication, Machine-Machine Communication - Data Ownership - Data Privacy.

Practical Assignments:

1. Setting up infrastructure and Automation environment

2. Case study for identifying Data Characteristics

Standard Big data architecture - Big data application - Hadoop framework - HDFS Design goal - Master-Slave architecture - Block System - Read-write Process for data - Installing HDFS - Executing in HDFS: Reading and writing Local files and Data streams into HDFS - Types of files in HDFS - Strengths and alternatives of HDFS - Concept of YARN.

Practical Assignments:

3.  Exercise on Installing HDFS

4.  Exercise on Reading and Writing Local files into HDFS

5.  Exercise on Reading and Writing Data streams into HDFS

Introduction to MapReduce - Sample MapReduce application: Wordcount - MapReduce Data types and Formats - Writing MapReduce Programming - Testing MapReduce Programs - MapReduce Job Execution - Shuffle and Sort - Managing Failures - Progress and Status Updates.

Practical Assignments:

6. Exercise on MapReduce applications

7. Exercise on writing and testing MapReduce Programs

8. Exercise on Shuffle and Sort

9. Exercise on Managing Failures

Stream processing Models and Tools - Apache Spark - Spark Architecture: Resilient Distributed Datasets, Directed Acyclic Graph - Spark Ecosystem - Spark for Big Data Processing: MLlib, Spark GraphX, SparkR, SparkSQL, Spark Streaming - Spark versus Hadoop

Practical Assignments:

10. Exercise on installing Spark

11. Exercise on Directed Acyclic Graph

12. Exercise on Spark using MLlib, Spark GraphX

13. Exercise on Spark using SparkR, Spark Streaming

Hive Architecture - Components - Data Definition - Partitioning - Data Manipulation - Joins, Views and Indexes - Hive Execution - Pig Architecture - Pig Latin Data Model - Latin Operators - Loading Data - Diagnostic Operators - Group Operators - Pig Joins - Row Level Operators - Pig Built-in function - User-defined functions - Pig Scripts.

Practical Assignments:

14. Exercise on Hive Architecture

15. Exercise on Pig Architecture

ESE - 50%

This course has been conceptualized in order to understand the high dimensional data and the statistical techniques such as principal component analysis, multidimensional scaling, independent component analysis and projection pursuit that are used to analyze the most challenging multidimensional real life problems.

Introduction to high dimensional data: high dimensional data in various fields with examples - need of high dimensionality - Curse and blessings of Dimensionality - Visualization of multidimensional data - parallel coordinate plots - multivariate random vectors - Multivariate normal distribution and estimation of parameters using likelihood estimation.                                                                           

Practical Assignments:

1.   Exercise on visualization of high dimensional data

      2.  Exercise on multivariate normal distribution

Principal component analysis - Principal components and dimension reduction - Visualization of principal components - Scree, Score, Projection plots and estimates of the density of the scores - Properties of principle components - Uses and interpretation of principal components - Standardized and high dimensional data - Sparse principal component analysis with LASSO and elastic nets - Consistency of principal components as the dimension grows.

Practical Assignments: