INDIAN STATISTICAL INSTITUTE

(Delhi Centre)

7 S. J. S. Sansanwal Marg, New Delhi 110016

Linear Algebra and Linear Models

M.Stat. Ist Year (NB-Stream) & MS(QE) IInd Year

Academic Year 2015 - 2016: Semester I

Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: Room # 208 on the first floor of the Faculty Block.
Instructor's Office Hours: Tuesday and Friday 1:15 PM - 2:15 PM

Teaching Assistant (TA): Gursharn Kaur
Email: gursharn (dot) kaur12r (at) isid (dot) ac (dot) in
TA's Office Hours: Wednesday 2:00 PM - 3:00 PM in the Library.

Class Time: Tuesday and Friday 11:15 AM - 01:15 PM (4 hours per week).

Lecture Hall: Room # 24 Academic Block (First Floor).

Course Duration (including examinations): July 27 - November 20, 2015
(Total of 17 Weeks = 7 Weeks of Classes + 1 Midterm Examination Week + 7 Weeks of Classes + 2 Final Examination Weeks).

Midterm Examination:
Date: Monday, September 14, 2015 (During September 14 - 18, 2015)
Duration: 10:00 AM - 12:00 noon (two hours)
Venue: Conference Hall, First Floor, Administrative Block

Final Examination:
Date: Tuesday, November 10, 2015
Duration: 10:00 AM - 01:30 PM (three and half hours)
Venue: Conference Hall, First Floor, Administrative Block

Course Outline:

• Linear Algebra (for approximately 4 weeks):
  • Review of Vector Space: Subspaces, linear dependence and independence, basis, dimension, sum and intersection of subspaces, inner product and norm, geometric interpretation, Gram-Schmidt orthogonalization, orthogonal projection, projection on a subspace.
  • Review of Matrices: Rank, trace, elementary operations, canonical reductions, Kronecker product, orthogonal matrices, symmetric matrices, inverse, sweep-out method, operations with partitioned matrices, determinants.
  • Linear equations, homogeneous and inhomogeneous systems, solution space, consistency and general solution, characteristic roots and vectors, Cayley-Hamilton theorem, canonical reduction of symmetric matrices, spectral decomposition, singular values and singular value decomposition.
  • Quadratic forms, definiteness, classification and transformations.
  • Generalized inverse of a matrix (done through assignments).
• Linear Models (approximately 10 weeks):
  • Linear statistical models/Gauss-Markov Models.
  • Least square estimation, Estimable linear functions, Normal equations, Best Linear Unbiased Estimates (BLUEs).
  • Gauss-Markov Theorem.
  • Variance of BLUEs.
  • Normality assumption of error. Maximum likelihood estimation vs Least square estimation/MLE vs BLUE. MVUE vs BLUE.
  • Estimation of error variance. Degrees of freedom. Fundamental Theorems of Least Square. Testing of linear hypotheses.
  • Fisher-Cochran theorem, distribution of quadratic forms.
  • Various different linear models and ANOVA.
    • One way classification.
    • Two way classification with and without interactions. Equal and unequal number of observations per cell.
    • Tukey's One Degree of Freedom test.
    • Nested classification model.
    • Three way classification, various different types of interactions.
    • Multiple regression.
  • Multiple comparisons.
  • Analysis of Covariance (ANCOVA).
  • Random and Mixed Effect Models.

References:

• Linear Algebra:
  • Linear Statistical Inference by C. R. Rao.
  • Linear Algebra and Linear Models, Elements of Sample Surveys and Design of Experiments by R. B. Bapat.
  • Linear Algebra by A. R. Rao and P. Bhimsankaram
• Linear Models:
  • Plane Answers to Complex Questions by R. Christensen.
  • Linear Statistical Inference by C. R. Rao.
  • Linear Models, An Integrated Approach by D. Sengupta and S. R. Jammalamadaka.
  • A Course in Linear Models by M. Kshirsagar.

Grading Policy:

• Assignments: 10% of the total credit.
• Quizzes: 15% of the total credit.
• Midterm Exam: 25% of the total credit.
• Final Exam: 50% of the total credit.

Assignment Policies:

• There will be a total of 14 sets of homework assignments each carrying a total of 10 points. The average of the 10 best assignment scores will be taken for the final grading.
• The assignments will be given in class on every Friday, starting from July 31, 2015. Each assignment will be due in class on the Friday of the following week. For example the first assignment is due on Friday, August 07, 2015.
• Each assignment will be based on the course materials which will be covered in the class in the week of the assignment.
• Late submission of an assignment will NOT be accepted. If you can not submit an assignment on time, don't worry about it and try to do well in the others. It will not count in your final grade since you have four extra assignment anyway.
• Graded assignments will be returned in the class an week after their submission. For example, the assignment which is due on August 07, 2015 (Friday) will be returned after grading on August 14, 2015 (Friday).
• Click here for downloading the assignments.

Quiz Policies:

• There will be four or more quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover materials done in the lectures given in the weeks prior to it.
• Each quiz will be of 15 points and will be of 30 minutes duration.
• Final grade for the quizzes will be (Best Score Before Midterm + Best Score Between Midterm and Final)/2.
• There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry, try doing well in the others.
• All quizzes will be part of the final grading.
• All quizzes will be closed note and closed book examinations.

Exam Policies:

• The Midterm and the Final Examinations will be open notes examinations. That means, students will be allowed to bring his/her own hand written notes, study materials, list of theorems etc. But no printed or photocopied materials will be allowed.
• Any unfair means used by any students in the examinations will be dealt with the strictest possible measures, as per the Institute rules. In particular, if any student is found to be using any kind of unfair practice during any of the examinations (including the quizzes) then he/she will be awarded ZERO in that examination.

Regrading Policy:

• Regrading of homeworks or exams will only be undertaken in cases where, you believe there has been a genuine error or misunderstanding. Please note that our primary aim in grading is consistency, so that all students are treated the same; for this reason, we will not adjust the score of one student on an issue of partial credit, unless the score allocated clearly deviates from the grading policy we adopted for that problem.
• If you wish to request a regrading of a homework or exam, you must return it to the instructor with a written note on a separate piece of paper explaining the problem.
• The entire assignment or the exam may be regraded, so be sure to check the solutions to ensure that your overall score will go up after regrading.
• All such requests must be received within one week from the date on which the homework or exam was made available for return.

Last modified November 08, 2015.