Introduction to Theory of Statistical Diagnosis
Lecturer:
Boris S.Darkhovsky, professor of "Mathematical Methods of Systems Analysis" ISA RAS Faculty of Computer Sciences, E-mail: bdarkhovskiy@hse.ru
doctor of physical & mathematical sciences,principal researcher of Institute for Systems
Analysis of Russian Academy of Sciences
1
Course Objective
The course will introduce the basic concepts of Statistical Diagnosis. This field of mathematical
statistics deals with the following problems:
a) detecting changes in probabilistic characteristics of random processes (fields) in off-line
regime
b) detecting changes in probabilistics characteristics of random processes (fields) in on-line
regime.
These problems arise in many applications and are known in the literature as “change-point
detection problems”. The goal of the course is to present to students the main ideas of nonparametric statistical diagnosis.
2
The position of the course in the structure of the educational program
Standard university courses of probability theory and functional analysis
Course duration: approx. 2 weeks: 16 hours.
Academic control forms are оne home assignment, one test.
2.1
Prerequisites of the course:



3
№
1.
Limit theorems of probability theory
Weak convergence of probabilistic measures
Space of continuos functions
Topic-Wise Curricula Plan
Topic name
Part 1 (16 hrs.)
Preliminary information
Mixing conditions
Weak convergence
Inequalities for maximum of sums of random
Course
hours, total
Audience hours
Practical
Lectures
studies
2
0
variables
Functionals of the maximum type
2.
Main ideas of the non-parametric approach to the
problems of statistical diagnosis
2
0
3.
Off-line change-points detection in random
sequences
Single and multiple abrupt changes in mean
Single and multiple abrupt changes in coefficients
of linear functional regression
Asymptotic analysis change-point estimation
2
2
4.
On-line change-points detection in random
sequences
Quality characteristics of sequential methods
Prior lower bounds of the quality criterion of the
detection
Non-parametric detection method
2
2
5.
Epsilon-complexity of continuous function and
model-free methodology of change-point detection
for time-series of arbitrary nature.
2
2
10
6
Part 1, totally:
16
3.References:
1.
2.
Billingsley P. Convergency of probability measures, Wiley, 1968.
B.E.Brodsky and B.S. Darkhovsky. Non-Parametric Statistical Diagnosis. Problems and
Methods. Kluwer, 2000.
3. Albert N. Shiryaev. Quickest Detection Problems: Fifty Years Later, Sequential Analysis,
v.29, 2010, 345-385.