平成29(2017)年度 Probability Models

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

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.

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
For further reading
[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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

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
For further reading
[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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

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
For further reading
[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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

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

尾畑伸明「確率モデル要論」牧野書店, 2012.
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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

授業科目の目的・概要

理工系科学・生命系科学をはじめ人文社会系科学に至るまで、ランダム現象の数理解析はますます重要になってきている。

本講義では、そのために必要不可欠となる確率論の基礎概念からはじめ、確率モデルの構成と解析手法を学ぶ。
特に、ランダム現象の時間発展を記述する確率過程として、ランダムウォーク・マルコフ連鎖・マルコフ過程の典型例をとりあげて、その幅広い応用を概観する。

参考書

尾畑伸明「確率モデル要論」牧野書店, 2012.

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

名城大学集中講義「無限次元解析特論」量子確率論とその応用 (2013.9.30-10.4)

  • 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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

授業科目の目的・概要

自然科学・生命科学をはじめ人文社会科学に至るまで、ランダム現象の数理解析はますます重要になってきている。
本講義では、そのために必要不可欠となる確率論の基礎概念からはじめ、 確率モデルの構成と解析手法を学ぶ。
特に、ランダム現象の時間発展を記述する確率過程として、ランダムウォーク・マルコフ連鎖・マルコフ過程の典型例をとりあげて、その幅広い応用を概観する。

参考書

尾畑伸明「確率モデル要論」牧野書店, 2012.

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

確率モデル論 (情報科学研究科・国際高等研究教育院) 応用解析学 (工学研究科)共通

授業科目の目的・概要

自然科学・生命科学をはじめ人文社会科学に至るまで、ノイズ・ゆらぎ・乱雑さ・不確定さから逃れられない現象には枚挙にいとまがなく、そのようなランダム現象の数理解析はますます重要になってきている。
本講義では、確率論の基本的な考え方になじみながら、確率モデルの構成と解析手法を学ぶ。
特に、時間発展を含むランダム現象を記述する確率過程としてマルコフ連鎖の基本的事項を学び、その幅広い応用を概観する。

参考書

尾畑伸明「確率モデル要論」牧野書店, 2012.

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
  • 3. Adjacency Algebras
  • 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
  • 4. Adjacency algebras
  • 5. Spectral distributions of distance-regular-graphs
  • 6. Adjacency matrices as algebraic random variables
  • 7. Hamming graphs

Part 2 (63 pages)

  • 8. Homogeneous trees
  • 9. Deformed vacuum states and free Poisson distributions
  • 10. Stieltjes transform and continued fraction
  • 11 Growing regular graphs
  • 12 Graph products
  • 13 Random graphs
  • 14 Quantum walks

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