﻿ 大学院科目（過年度） | 東北大学大学院情報科学研究科 システム情報科学専攻 尾畑研究室－システム情報数理学II研究室－

## 平成29(2017)年度 Probability Models

### Objectives and Outline

Probability models are essential in mathematical analysis of random phenomena. In these lectures, we focus on Markov chains as basic models of random time evolution. Starting with fundamental concepts in probability theory (random variables, probability distributions, etc.), we study fundamentals on Markov chains (transition probability, recurrence, stationary distributions, etc.). Moreover, we overview random walks, birth-and-death processes and Poisson processes, and their wide applications. Background knowledge on elementary probability is required.

### Basic References

[1] 尾畑伸明「確率モデル要論」牧野書店, 2012.
[2] D. L. Minh: Applied Probability Models, Duxbury, 2001.
[3] Further references are found in the resume.

• Cover page
• Chapter 1. Probability spaces and random variables
• Chapter 2. Probability distributions
• Chapter 3. Independence and dependence
• Chapter 4. Limit theorems
• Chapter 5. Markov chains
• Chapter 6. Stationary distributions
• Chapter 7. Topics in Markov chains I: Recurrence
• Chapter 8. Topics in Markov chains II: Absorbing states
• Chapter 9. Galton-Watson branching processes
• Chapter 10. Poisson processes
• Chapter 11. Queueing theory

## Data Science Basic "A Rudimentary Knowledge of Multivariate Analysis"

This has been delivered every fall semester since 2015, as a part of DSP (Data Science Program) Tohoku University. The materials are available, see here

### Basic References

[1] P. G. Hoel: Introduction to Mathematical Statistics, Wiley
[2] A. J. Dobson and A. G. Barnett: An Introduction to Generalized Linear Models, 3rd Edition, CRC Press, 2008. [Japanese translation available for 2nd Edition]
[3] P. McCullaghand J. A. Nelder: generalized Linear Models, 2nd Edition, Chapman & Hall, 1989.

## 平成28(2016)年度 Probability Models

### Objectives and Outline

As an introduction to mathematical analysis of random phenomena we learn probability models, their construction and analysis. We start with fundamental concepts in probability theory (random variables, probability distributions, and so on). For the time evolution of random phenomena we study basic properties of random walks, Markov chains, Markov processes, and take a bird's-eye view of their wide applications. Background knowledge on elementary probability is required.

### Basic References

[1] 尾畑伸明「確率モデル要論」牧野書店, 2012.
[2] D. L. Minh: Applied Probability Models, Duxbury, 2001.
[3] Further references are found in the resume.

### Resume

• Cover page and Overview
• Chapter 1. Random variables and probability distributions
• Chapter 2. Independence and dependence
• Chapter 3. Markov chains
• Chapter 4. Stationary distributions
• Chapter 5. Topics in Markov chains
• Chapter 6. Topics in random walks
• Chapter 7. Galton-Watson branching processes
• Chapter 8. Poisson processes
• Chapter 9. Queueing theory
• Chaps. 1-9

## 2016 Fall: Data Science Basic "A Rudimentary Knowledge of Multivariate Analysis"

This is a part of DSP (Data Science Program) Tohoku University.

### Basic References

[1] P. G. Hoel: Introduction to Mathematical Statistics, Wiley
[2] A. J. Dobson and A. G. Barnett: An Introduction to Generalized Linear Models, 3rd Edition, CRC Press, 2008. [Japanese translation available for 2nd Edition]
[3] P. McCullaghand J. A. Nelder: generalized Linear Models, 2nd Edition, Chapman & Hall, 1989.

## 平成27(2015)年度 Probabilistic Models

### Objectives and Outline

Mathematical analysis is important for the understanding of random phenomenon apperaing in the various fields of natural, life and social sciences, and the probabilistic approach is essential. We start with the most fundamental concepts in probability theory and learn basic tools for probabilistic models. In particular, for the time evolution of random phenomenon we study basic properties of random walks, Markov chains, Markov processes, and take a bird's-eye view of their wide applications.

### Basic References

[1] 尾畑伸明「確率モデル要論」牧野書店, 2012.
[2] D. L. Minh: Applied Probability Models, Duxbury, 2001.
[3] Further references are found in the resume.

### Resume

• Chaps. 0-11
• Cover page and overview
• Chapter 1. Random variables and probability distributions
• Chapter 2. Independence and dependence
• Chapter 3. Limit theorems
• Chapter 4. Random walks
• Chapter 5. Topics in one-dimensional random walks
• Chapter 6. Markov chains (small errors are fixed Nov. 13)
• Chapter 7. Topics in Markov Chains
• Chapter 8. Poisson Processes
• Chapter 9. Queueing Theory
• Chapter 10. Brownian Motion -- An Intuitive Introduction
• Chapter 11. Galton-Watson Branching Processes (This is additional.)

## 2015 Fall: Data Science Basic "A Rudimentary Knowledge of Multivariate Analysis"

This is a part of DSP (Data Science Program) Tohoku University.

### Basic References

[1] P. G. Hoel: Introduction to Mathematical Statistics, Wiley
[2] A. J. Dobson and A. G. Barnett: An Introduction to Generalized Linear Models, 3rd Edition, CRC Press, 2008. [Japanese translation available for 2nd Edition]
[3] P. McCullaghand J. A. Nelder: generalized Linear Models, 2nd Edition, Chapman & Hall, 1989.

## 平成26(2014)年度 Probabilistic Models

### Objectives and Outline

Mathematical analysis is important for the understanding of random phenomenon apperaing in the various fields of natural, life and social sciences, and the probabilistic approach is essential. We start with the most fundamental concepts in probability theory and learn basic tools for probabilistic models. In particular, for the time evolution of random phenomenon we study basic properties of random walks, Markov chains, Markov processes, and take a bird's-eye view of their wide applications.

### References

Further references are found in the resume.

### Resume (English)

• Chapter 0. Cover page and overview
• Chapter 1. Random variables and probability distributions
• Chapter 2. Bernoulli trials
• Chapter 3. Random walks
• Chapter 4. Markov chains
• Chapter 5. Poisson processes
• Chapter 6. Queueing theory
• Chapter 7. Galton-Watson branching processes
• Chapter 8. Brownian motion

## 平成25(2013)年度 確率モデル論

### 資料

• Chapter 0. Overview
• Chapter 1. Random variables and probability distributions
• Chapter 2. Bernoulli trials
• Chapter 3. Law of large numbers and central limit theorem
• Chapter 4. Random walks
• Chapter 5. Markov chains
• Chapter 6. Poisson processes
• Chapter 7. Galton-Watson branching processes

## 平成25(2013)年度 量子確率論とその応用

• 1. 量子確率論の基礎概念
• 2. 量子分解
• 3. 代数的グラフ理論
• 4. スペクトル・グラフ理論
• 5. 大きなグラフの漸近的スペクトル解析

## 平成24(2012)年度

### 資料

• Chapter 0. Cover page (Japanese)
• Chapter 1. Random variables and probability distributions
• Chapter 2. Bernoulli trials
• Chapter 3. Law of large numbers and central limit theorem
• Chapter 4. Random walks
• Chapter 5. Markov chains
• Chapter 6. Poisson processes

## 平成23(2011)年度

### 資料

• 第0章 カバーページ
• 第1章 序論
• 第2章 確率変数と確率分布
• 第3章 ベルヌイ試行列
• 第4章 大数の法則と中心極限定理
• 第5章 ランダム・ウォーク
• 第6章 マルコフ連鎖
• 第7章 ポアソン過程第8章ブラウン運動
• 第9章ゴルトン・ワトソン分枝過程

## 2010 Spring

Spectral Analysis of Large Networks: Quantum Probabilistic Approach and Applications
Chungbuk National University, Korea

• 1. Graphs and Matrices
• 2. Spectra of Graphs
• 4. Quantum Probability5. Stieltjes Transform and Continue Fraction
• 6. Kesten Distributions
• 7. Catalan Paths and Applications
• 8. Graph Products and Independence
• 9. Quantum Central Limit Theorems
• 10. Deformed Vacuum States and Q-Matrices

## 2008 Summer

Quantum Probability and Applications to Complex Networks University of Wroclaw, Poland

#### Part 1 (57 pages)

• 1. Introduction
• 2. Graphs and adjacency matrices
• 3. Spectra of graphs