Introductory Mathematics for Artificial Intelligence

by Prof. Sang-Gu Lee

http://matrix.skku.ac.kr/intro-math4ai/

·                           01 Week   http://matrix.skku.ac.kr/intro-math4ai/W1/

·                          02 Week   http://matrix.skku.ac.kr/intro-math4ai/W2/

·                          03 Week   http://matrix.skku.ac.kr/intro-math4ai/W3/

·                          04 Week   http://matrix.skku.ac.kr/intro-math4ai/W4/

·                          05 Week   http://matrix.skku.ac.kr/intro-math4ai/W5/

·                          Week 6.  Matrix decompositions (LU, QR, SVD)   http://matrix.skku.ac.kr/intro-math4ai/W6/

·                          07 Week   http://matrix.skku.ac.kr/intro-math4ai/W7/

·                          08 Week   http://matrix.skku.ac.kr/intro-math4ai/W8/

·                          Week 9.  Gradient descent method    http://matrix.skku.ac.kr/intro-math4ai/W9/

·                          Week 10 http://matrix.skku.ac.kr/intro-math4AI/W10/

·                          Week 11 http://matrix.skku.ac.kr/intro-math4AI/W11/

·                          Week 12 PCA http://matrix.skku.ac.kr/intro-math4AI/W11/

·                          Week 13 ANN과 Backpropagation http://matrix.skku.ac.kr/intro-math4AI/W13/

·                          Week 14 MNIST Data Set and PCA http://matrix.skku.ac.kr/intro-math4AI/W14/

 

A summary  by DongNa (Chinese Student)

1) [Day 1] summary

2) [Day 1] Code-practice

                           3) [Day 2] Summary

                           4) [Day 2] Code-practice

                           5) [Day 3] Summary

                           6) [Day 3] Code-practice

                           7) [Day 4] Summary

                           8) [Day 4] Code-practice

9) [Day 5] Code-practice

10) [Day 5] Summary

11) [Day 6] Code-practice

12) [Day 6] Summary

 

 

1)      [Day 1] Summary

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(12) Least Square Solution

 

(13) The method of finding least square line

 

(14) Gram-Schmidt

 

(15) QR-decomposition

 

(16) 최소제곱문제를 QR 분해로 풀기

 

  [Day 1] Code-practice

1).Vector Projection

page43image54452352

Source: Linear Algebra

  Sang-Gu LEE with Jon-Lark KIM, In-jae KIM,Namyong LEE,

Ajit KUMAR,Phong VU,Victoria LANG,Jae Hwa LEE

Where p: the projection of y to x

      W: the component of y orthogonal to x

2).Distance Between a Point and a Plane

page45image54410064

Source: Linear Algebra

  Sang-Gu LEE with Jon-Lark KIM, In-jae KIM,Namyong LEE,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2)      [Day 2] Summary

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2)    Eigenvalue & Eigenvector

 

 

3)    Diagonalizable Matrix

 

4)    SVD

 

 

 

 

 

 

5)    Least square solution

 

6)    Pseudo-inverse

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3)      [Day 2] Code-practice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4)      [Day 3] Summary

 

 

 

 

 

 

 

 

 

 

 

 

1)    Differentiable

2)    Derivative

 

3)    Maximum & Minimum

4)    Integral

5)    Partial Derivative

6)    Chain Rule

 

7)    Gradient & Hessian

8)    Directional derivative

9)    Fermat’s theorem on critical points

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5)      [Day 3] Code-practice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

­­

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6)      [Day 4] Summary

2)    Taylor

·         1) Hessian Matrix

2)    Double Integration

3)  Jacobian Determinant

 

 

 

4)  Maximum & Minimum

 

5)  Quadratic form

 

 

 

 

 

 

 

 

 

 

 

 

 

6)  The relationship between vector & determinant

7)  Inverse Matrix

 

 

 

 

 

 

8)  The method of finding least square line

 

 

 

 

 

 

9)  Gram-Schmidt

 

 

10)        QR-decomposition

 

 

 

 

 

 

11)         최소제곱문제를 QR 분해로 풀기

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7)      [Day 4] Code-practice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Day 5] Code-practice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Day 5] Summary

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Day 6] Code-practice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Day 6] Summary

 

3)    The properties of PMF & PDF

4)    Permutation & combination

 

 

18)Expectation & Variance & Standard Deviation

 

 

 

5)    Covariance Matrix

 

  

6) Gamma Distribution & Beta Distribution

7) Joint Density Function

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12)  Central Limit Theorem

 

13) Covariance & Correlation Coefficient

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14) Normal Distribution & Standard Normal Distribution

·        

 

 

 

 

 

 

 

 

15) Joint Probability Function & Marginal Probability Distribution

 

 

 

 

 

 

 

 

 

16) Expectation & Variance & Standard Deviation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

<Project Proposal and Some result>

Name / e-mail (이름과 이-메일): Dong Na

Team Project (Tentative, just idea now):

[First One] Economics key points +Mathematics AI key points

There is more detailed information in the picture as shown below.

[Second one] Blockchian(Bitcoin) & Prabability

I have already presented first topic last time, so I want to represent another topic which would relate to bitcoin & probability.

 

The total frame of the report

 

 

The aim of Economics is an effort to find optimal resource allocation under resource constraint. One of the classifications of economics is that static economics and dynamic economics. In terms of static economics, I think the most important thing is optimization. More specifically, I will apply two specific examples. The first example is how to find profits maximization. The method is to use Critical Value & Hessian Matrix from Basic Mathematics for AI. And the second example is the method of finding Utility Maximization by using Stationary Point & Bordered Hessian from Basic Mathematics for AI. Moreover, in terms of Dynamic Economics, Firstly, I will apply some economics definitions which can be calculated via using Differentiation & Integration from Basic Mathematics for AI; Secondly, I will take a specific case for solving the exhaustible resource problem for the optimal extraction path through using Differentiation & Integration from Basic Mathematics for AI.

 

 

 

Static Economic

 

First Example for Profit Maximization

 

By using: Critical value + Hessian Matrix

 

 

To find critical point of the given function by using sage-code

 

To determine the positive sign or negative sign of Hessian Matrix, then we can determine whether the maximum or minimum of the given function at the critical point

 

Thus, in my opinion, in terms of the first example for Profit Maximization, we can also calculate this problem by using sage-code. However, I am not good at using sage-code, so it is very hard for me to calculate the solution in this way. As we can see, It’s a very complicated way to find the solution without using sage-code. Therefore, my next goal is to try my best to study sage-code.

Static Economic

The Second Example for Utility Maximization

By using: Stationary Point + Bordered Hessian

 

 

 

 

 

 

 

Total Conclusion for Maximum & Minimum

 

 

 

 

Dynamic Economic

 

Some Economic definitions by using Differentiation + Integration

 

 

 

 

Dynamic Economic

 

One specific Example for solving the exhaustible resource problem for the optimal extraction path by using Differentiation + Integration

 

 

 

 

 

 

 

Conclusion, these problems of economics can all be easily and quickly solved by using sage-code. Thus, I think I should study sage-code as soon as possible. Sage-code is a very useful thing in many areas.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The reason why we firstly to study linear algebra is that all of things in this world can be described by using vectors. Moreover, linear algebra applies a best way to solve system of linear equations by using determinant & many types of matrix. Especially Hessian determinant & Jacobian determinant & inverse matrix & diagonal matrix & orthogonal matrix. Moreover, Hessian Determinant & Jacobian determinant can also be used in differential and probability. Moreover, we can use them to solve n-dimension of dataset. There are two main ways to reduce the higher dimension including SVD & PCA.

PCA is a dimension reduction method, which means to reduce the rank of the covariance matrix by using SVD. More specifically, the process of dimension reduction is that to delete some relative unimportant eigenvectors, then use the rest of eigenvectors to create a reduced space (a smaller size of matrix). Although this method has to lost some eigenvectors, the relative important eigenvector will be preserved in the end in order to make sure total information of the dataset is as same as possible. Moreover, PCA method is more efficient than the linear regression method due to the fact that there is minimum distance from each data to the linear function. (Each data is orthogonal to the linear function). Thus, if we use the reduced size of matrix to analysis and application of such an amount of dataset, we will efficiently deal with data.

Moreover, I am very interested in deep neural network. Amazing! The algorithm of deep neural network is that in order to decrease the error between predictive value and correct value, which means to update the weight by using back propagation and gradient descent method.

 

Math4AI,  Professor: Sang-Gu LEE

(Summary by Dong-Na)

 

 

1)  [Final OK by SGLee] Finalized by 손재민,김미리[6주차] 결합밀도함수 및 주변밀도함수(정리와 예제 풀이 추가)

[My Comments]: Jacobian can be used in probability & One example

 

2)[Final OK by TA] Re-Finalized by 이범수[6주차] Finalized by 박건영, 이상구, 손재민 (연속확률분포에 대한 질문)

 

[My Comment] the reason why I chose this topic is that I was interested in the python code. Excellent logic & beautiful graph.  What a smart person!   

3)[Final OK by SGLee] Finalized by 홍정명,유민솔,박건우-[6주차]질문-이항분포를 정규근사할 수 있는 이유

[My Comments] When n is large enough, in order to easily calculate the binomial distribution, we can use the approximate solution of standard normal distribution. (pf & one example)

4)[Final OK by SGLee]Re-finalized ver.2 by류재헌,박정현,이상구 교수님,이지용,이혜연,권서영,박건영-PCA가 데이터 분석에서 갖는 의의

 

 

 

 

[My Comments] PCA is a dimension reduction method, which means to reduce the rank of the covariance matrix by using SVD. More specifically, the process of dimension reduction is that to delete some relative unimportant eigenvectors, then use the rest of eigenvectors to create a reduced space (a smaller size of matrix). Although this method has to lost some eigenvectors, the relative important eigenvector will be preserved in the end in order to make sure total information of the dataset is as same as possible. Moreover, PCA method is more efficient than the linear regression method due to the fact that there is minimum distance from each data to the linear function. (Each data is orthogonal to the linear function). Thus, if we use the reduced size of matrix to analysis and application of such an amount of dataset, we will efficiently deal with data.

[Source]https://cn.ourladylakes.org/978756-singular-value-decomposition-algorithm-KBVLJQ

 

[Source] https://www.cnblogs.com/pxzheng/p/12690150.html

 

5)[Final OK by TA]Finalized by 권서영,이상구 교수님(comment on last year’s PBL report/신뢰구간에 대한 고찰)

[My Comments]: An example of in terms of normal distribution

 

Source: https://www.simplypsychology.org/confidence-interval.html

 

6) [Final OK by SGLee] Finalized by 이범수 오혜준[6주차] 연속확률분포와 파이썬의 구현

 

 

 

 

[My Comments]: the python code is very interesting. I analyzed one of pictures of Normal Distribution PDF as shown below.

7) [Final OK by SGLee]Re-finalized by 류재헌,김은진,권서우,유민솔,이상구 교수님,이상원,박수연,권서영 – 중심극한정리의 질문과 실습

[My Comments]

 

 

Source: https://medium.com/analytics-vidhya/central-limit-theorem-and-machine-learning-part-1-af3b65dc9d32

 

 

8) [Final OK by SGLee][6주차]Normal Distribution 이론,실습([Final OK by SGLee]Finalized by 이지용,안은선:[6주차]지수분포,정규분포 요약 글 인용하셨습니다)

 

[My Comments]

9) [Final OK by SGLee][Final OK by TA][5주차]Re-Finalized by 정승민,박건영,정진웅,양지원,박정현[포아송분포 유도하기,실습,조건,코드추가]

[My Comments] One pf.

10) [Final OK by SGLee] Re-finalized by 류재헌,유민솔,이지용,이상구 교수님,정현목,김은진-결합분보 질문

[My Comments] Summary of how to find 결합확률분포

 

 

 

 

 

 

 

[열린문제 1] 다른 교재에서 찾은 몇 가지 다항함수의 개형을 그리시오.

[ Hint : plot ( -x^9 -5*x^4 -130*x^3 -53*x^2 +542*x +3 , (x, -10, 10)) ]

 

 

 

[열린문제 2] DRW00003aec3706그래프의 개형을 그리시오.

Plot(x*sin(1/x))

 

[열린문제 3] 앞에서 배운 함수들 중 학습한 함수의 합성함수를 만들고, 그래프를 그리시오.

 

 

 

[열린문제 4] 다음 학생들의 문제풀이를 참고하여 다양한 방정식의 (근사)해를 구하시오.

 

 

 

http://matrix.skku.ac.kr/math4ai/PBL-Record/ http://matrix.skku.ac.kr/2020-Math4AI-PBL/

http://matrix.skku.ac.kr/KOFAC/ 에서 실습하시오

[열린문제 5] 행렬에 대해 학습한 다른 교재의 행렬 연산을 시행해 보시오.

http://matrix.skku.ac.kr/KOFAC/ 에서 실습하시오

 

 

[열린문제 6] 인터넷이나 다른 교재에서 5차 (이상) 행렬을 찾아서, 전치행렬과 역행렬이 존재하는지를 확인하고, 존재하면 찾아보시오.

 

[열린문제 7] 거리 척도를 사용하여 유사도를 계산할 수 있는 데이터의 종류에는 어떤 것이 있는지 생각해보시오.

 

 

 

 

[열린문제 8] 위의 거리 척도로 유사도를 판단하기가 용이하지 않은 데이터의 경우에 유사도를 판단하는데 사용이 가능한 다른 척도는 무엇이 있을지 생각해보시오. (Hint: 방향이 같은 데이터/벡터들의 경우)

 

 

 

[열린문제 9] 두 개의 7차원 벡터(데이터) 사이의 거리(distance)를 직접 구하시오.

(Hint: (거리를 활용한) 데이터의 유사도 <실습실> 활용)

             http://matrix.skku.ac.kr/math4AI-tools/distance_similarity/

 

 

[열린문제 10] 어떤 데이터들이 코사인 유사도를 사용하여 분석 가능할지 생각해보시오.

[열린문제 11] 두 개의 5차원 데이터(벡터) 사이의 내적과 사잇각 DRW00003aec3708를 구하시오. (Hint: (사잇각을 활용한) 데이터의 유사도 <실습실> 활용)

http://matrix.skku.ac.kr/math4AI-tools/cosine_similarity

 

 

 

[열린문제 12] 다른 교재의 선형 연립방적식의 해를 위의 명령어로 구하시오.

(Hint: http://matrix.skku.ac.kr/2018-album/LA-Sec-3-5-lab.html 실습 활용)

 

 

[열린문제 13] 주어진 선형연립방정식이 유일해를 갖는지, 무수히 많은 해를 갖는지, 해가 존재하지 않는지를 판단하는 것은 첨가행렬 DRW00003aec370a의 RREE를 구하여 이것만 자세히 보면 바로 판단이 가능한 이유를 설명하시오.

 

1)    Type 1

2)    Tyope 2

 

3)    Type 3

 

 

열린문제 14] 앞서 구한 방법으로 DRW00003aec370c평면의 6개의 점에 (best fit 하는) 3차의 최소제곱곡선 DRW00003aec370e을 구할 수 있음에 대하여 토론하시오.

[열린문제 15] 예제4와 같은 방법으로 다른 교재에서 찾은 선형연립방정식의 최소제곱해를 구하시오. (Hint: http://matrix.skku.ac.kr/2018-album/LS-QR-decom.html 활용)

 

[열린문제 16] 예제 6과 같은 방법으로 다른 교재에서 찾은 행렬의 특잇값 분해(SVD)를 구하시오.  

 

 

[과제 2]

[열린문제 1] 다른 교재에서 찾은 (연속) 미분가능한 함수의 3계 도함수(3rd derivative)를 구하시오.

 

 

 

 

[열린문제 2] 주어진 구간에서 두 번 미분가능한 함수를 골라서 그 함수의 극댓값, 극솟값 및 그 구간에서의 최댓값, 최솟값을 찾아보시오.

 

 

 

 

 

[열린문제 3] 함수 DRW00003aec3710의 최솟값을 구하시오. 단 DRW00003aec3712, DRW00003aec3714, DRW00003aec3716으로 한다.

I modified this solution.

 

 

 

 

The new solution shown as below.

 

 

 

[열린문제 4] 위의 그림과 같이 다양한 미분가능 함수인 경우, 일단 그래프의 개형을 그리고, 눈으로 확인되는 극솟값을 포함하는 작은 구간들을 정한다. 그 후, 각각의 구간에서 시작점을 잡아 경사하강법을 적용한다. 결과값(output)에 대하여 토론하시오.

 

* 양방향 그래프 그리기 (Grapher) : 

  http://matrix.skku.ac.kr/cal-lab/sage-grapher.html 

 

* picewise 함수 : 

  http://matrix.skku.ac.kr/cal-book/part1/CS-Sec-2-2-Sol.html 

* 양방향 그래프 그리기 (Grapher) : 

  http://matrix.skku.ac.kr/cal-lab/sage-grapher.html  

* 매개변수 함수 (Parametric Equation) Grapher : 

  http://matrix.skku.ac.kr/cal-lab/sage-grapher-para.html  

* 극좌표 함수 (Sage-Calculus-Polar Equation) Grapher  

  http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html  

* 음함수 (Implicit Function) Grapher  

  http://matrix.skku.ac.kr/cal-lab/sage-grapher-imp.html  

* 실습실: http://matrix.skku.ac.kr/KOFAC/ 

 

 

 

 

 

 

[ Midterm PBL 보고서 ]

보고서 양식 다운로드: http://matrix.skku.ac.kr/PBL-Form/PBL.hwp

(English Version MS Word file PBL report (Form and Sample) :

Download: http://matrix.skku.ac.kr/PBL-Form/PBL-Report-Form-English.docx  )

 

[ 과제 1]

[열린문제 1] 베이즈 정리(Bayes’ theorem)가 적용되는 예의 하나인 조건부 확률에 관한 몬티홀(Monty Hall) 문제에 대하여 토론하시오. https://destrudo.tistory.com/5

Monty Hall ­

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

[Source : https://en.wikipedia.org/wiki/Monty_Hall_problem]

My answer is I will switch my choice, which means I will choose door 2. The reason is that before the host open the door 3, the probability of winning the car at

door 2 is 1/3, after the host open the door 3 and he knows there is no car in it, the

probability of winning the car at door 2 will increase at 2/3.

In terms of an intuitive idea is there is no car in door 3, thus the probability of

Winning the car at door 1 & door 2 both are 1/2. Therefore, there is the same

probability of winning the car whether switch or not. However, I think this intuition

is not correct.

    Next, I will give more detailed explanation combination with Bayes’s theorem.

To put it simply, Bayes’s theorem is based on conditional probability. It describes

some certain information and conditions are known under some situations, then to

calculate the probability, which means to update the probability by using information.

 

For instance, there are two tennis players A and B, and you know nothing about

them. Now, the question is that do you think who will win in the next Olympic games.

Absolutely, the answer is 1/2, 1/2 respectively. However, if I tell you A has already

continuously won B ten times recently, what is your answer? You may update the

probability of winning, and you should choose A.

 

Then answer the question of Monty Hall by using Bayes’ theorem. When the host

open the door 3, how should the winning probability be updated at door 1 and door2 ?  

The probability of winning the car at door 1 is 1/3, which means you choose the

correct one at the first time. Thus, the probability of winning the car at door 2 &

door 3 is 2/3. Since no matter which door you choose, the host will always find a

door which contained a goat to open it. Therefore, when the host open the door 3,

the probability of winning car at door 1 does not to update. More specifically, no

matter which door be opened by host, there is no useful information about whether

the door 1 has a car. However, door 3 has no car is a very useful information for

whether door 2 has a car. The reason is that after host opened the door 3, the

probability of winning a car at door 3 would decrease to zero. According to the

probability of winning a car at door 3 & door 2 is still 2/3, thus, we can get the

probability of door 2 has a car is 2/3.  (2/3-0=2/3)  So, I think I should switch my

choose from door 1 to door 2.

 

In my opinion, the most important thing is host knows whether the doors have a car,

there is information asymmetry between host and me.

 

 

 

 

 

[열린문제 2] 다른 교재에서 찾은 연속확률변수 DRW00003aec3718의 기댓값과 분산, 표준편차를 구하시오.

 

[열린문제 3] 다른 교재에서 찾은 데이터의 공분산행렬을 구하시오.

 

 

 

 

 

 

[열린문제 4] 최소제곱직선과 회귀분석에서 사용하는 선형회귀 사이의 같은 점과 다른 점에 대하여 이해한 바를 토론하시오.

Source: http://matrix.skku.ac.kr/math4ai/part1/

        <<Linear Algebra and its Applications>> Written by David C.Lay,Steven R.Lay, Judi Mcdonald

For example, linear function y = a+b*x

In a linear regression, the coefficients a and b were used to determine in the

direction of minimizing the error.

In terms of PCA, a and b determine the minimum distance from each point to

The linear function y = a +b*x

In conclusion, PCA method is more efficient than the linear regression method due to

the fact that there is minimum distance from each data to the linear function. (Each

data is orthogonal to the linear function).

Source: http://matrix.skku.ac.kr/2020-AI-translation/

 

 

 [열린문제 1] 주성분분석에서 특잇값 분해(SVD)가 어떻게 사용되는지 설명해 보시오.

(1)  The first way:

Source: http://matrix.skku.ac.kr/2020-AI-translation/

(2) The second way:

PCA is a dimension reduction method, which means to reduce the rank of the covariance matrix by using SVD. More specifically, the process of dimension reduction is that to delete some relative unimportant eigenvectors, then use the rest of eigenvectors to create a reduced space (a smaller size of matrix). Although this method has to lost some eigenvectors, the relative important eigenvector will be preserved in the end in order to make sure total information of the dataset is as same as possible. Moreover, PCA method is more efficient than the linear regression method due to the fact that there is minimum distance from each data to the linear function. (Each data is orthogonal to the linear function). Thus, if we use the reduced size of matrix to analysis and application of such an amount of dataset, we will efficiently deal with data.

Source: http://matrix.skku.ac.kr/2020-AI-translation/

(3) Compare about these two ways

[Source] https://www.cnblogs.com/pxzheng/p/12690150.html

[열린문제 2] 본인 전공에서 찾은 데이터행렬을 가지고, 위의 주성분분석(PCA)의 알고리즘을 그 행렬에 적용해보고 토론하시오,

[열린문제 3] 공분산 행렬에 대한 주성분분석(PCA)을 하여, 차원을 축소하는 과정에 대하여 이해한 바를 토론하시오.

차원축소(dimension reduction)is a process of reducing the high-dimensional dataset to a low-dimensional dataset. The reason is that a low-dimensional dataset is relatively easy to visualize and compute the dataset. Moreover, the new dataset which is a lower dimension what contained the eigenvectors are linear independence, namely principal components.

            PCn: called the nth principal component, which represents to maintain the distribution of the original data as much as possible. In order to get the principal component, we should to find new axis.

PCA is one of the dimension reduction methods. The dataset of old coordinate by using PCA, which means rotation change of coordinate system to project data in this new coordinate system. PC1 represents the first coordinate axis of new coordinate system, PC2 represents the second coordinate axis of new coordinate system and so on. The square of the coordinate value of the data on each axis represents the variance of the corresponding variable, the goal of PCA is to find the variable with the largest variance. The reason is that the larger the variance represents the greater the degree of data dispersion. Which means contained the lager the information.

 

Then, the more detailed information have already shown on the solution of [열린문제1 of 기말고사 과제 2]

[참고자료] 담당교수의 PCA 추가설명 https://youtu.be/ukIttphmM_4

 

[열린문제 4] 최소제곱직선과 회귀분석에서 사용하는 선형회귀 사이의 같은 점과 다른 점에 대하여 이해한 바를 토론하시오.

For example, linear function y = a+b*x

In a linear regression, the coefficients a and b were used to determine in the

direction of minimizing the error.

In terms of PCA, a and b determine the minimum distance from each point to

The linear function y = a +b*x

In conclusion, PCA method is more efficient than the linear regression method due to

the fact that there is minimum distance from each data to the linear function. (Each

data is orthogonal to the linear function).

Source: http://matrix.skku.ac.kr/2020-AI-translation/

 

[열린문제 5] 인공신경망과 오차 역전파법(back propagation)에 대하여 아는 대로 서술하시오.

[참고자료] 담당교수의 인공신경망(은닉망의 수학) 설명https://youtu.be/d4WercT_OnU ?

The algorithm of deep neural network is that in order to decrease the error between predictive value and correct value, which means to update the weight by using back propagation and gradient descent method.

 

 

 

 

 

 

 

 

 

Source: http://matrix.skku.ac.kr/2020-AI-translation/

 

 

 

 

[열린문제 6] MNIST 숫자 인식, 얼굴인식, 음성인식, 자연어생성 기술이 현재 어느 정도 활용되고 있는지 알아보시오,

 

 

Source: https://en.wikipedia.org/wiki/List_of_datasets_for_machine-learning_research

 

 

 

 

http://matrix.skku.ac.kr/KOFAC2/

[열린문제 7] 인공지능에 활용되는 수학에 대하여 알아보고 토론하시오.

Source: https://en.wikipedia.org/wiki/List_of_datasets_for_machine-learning_research

 

 

 

 

Natural Language Processing is one of the applications of AI. It can be solved by using Naive Bayes Algorithm. The reason is that Speech recognition is something like according to received signal sequence to conjecture the signal sequence actually emitted by the speaker. Moreover, the signal sequence contained the sentences that the speaker spoke and the meanings what he or her would to express. Therefore, the problems of speech recognition can be transformed into the problems of communication recognition which can be further simplified mathematics.  Moreover, this mathematic problem can be solved by using Bayes Algorithm.

Source: https://www.koreascience.or.kr/article/JAKO201907752706277.page

 

 [Final PBL 보고서 -개인별 포트폴리오]

◆ Final comment

[From Day 1 to Day 4] I am very interested in robot because I like watching science fiction film since I was a little girl. So, this is the reason why I chosen this course. However, at the beginning of this course, I have no idea on how to learn this course, after I saw many excellent students who uploaded the file contained summary & code-practice for every Day, I tried to go through the same way. Thus, I uploaded summary & code-practice from Day 1 to Day 4, but I did not participate in finalized projects. This is all my fault and I missed many good opportunities for discussing together and studying from each other. Therefore, I’m going to actively participate in finalized projects. (Up to now, I still don’t know the process of finalized projects). I will try my best to practice writing code, though I am not good at in this way.

[From Day 5 to Day 6] My participation in Q&A was not activated, I should make more questions and discussions in Q&A. Moreover, I had not worked hard enough, I should make more of an effort to study this course. I am very grateful to professor for his help including the translated websites and many Youtube videoes both are in English. Professor helped me a lot. Thank you for your help!!! I really appreciate and I will work hard on this course. 

[7 Day] PCA & ANN is very important component of AI. Although I still feel confused about PCA and it is a very tough period for me, this part is very magical. It is a great honor to be your student!!! I gained a lot from this course! Thanks for your encouragement and lots of help! During this semester, I had so many times want to give up, your encouragements gave me courage to move on.

Memorable course & Professor & classmates constitute my unforgettable and meaningful summer in 2021. Best wishes for all of you!

(Added part)Blockchain & AI

Blockchain technology is generated base on the invention of Bitcoin 1.0. An upgraded version of bitcoin called Ethereum or Bitcoin 2.0. The Ethereum are the second largest cryptocurrency and generate a very useful thing called smart contract by upgrading the script of Bitcoin. Information can flow freely on the internet nowadays, but payment cannot. However, we can use blockchain technology to make it happen in the future.

Blockchain is a database of distributed ledger which can successfully store data by using the format of block. More specifically, by using storage hash link-table with the features including tamper-resistant and traceability. If you want to modify any data, the other data which connect with this data will change represented in hash value will also change. Moreover, the run rules of blockchain and full data information both are open transparent. According to tamper-resistant & traceability & transparency of this database, blockchain can be used to promote the development of AI in data collection and data processing. To put it simply, blockchain focuses on maintaining accurate records, certification and implementation. However, in terms of AI, which focuses on making decisions, evaluating and understanding certain patterns and data sets. Moreover, AI has mature module recourses and algorithm resources. Thus, if we apply AI to blockchain, blockchain will become more energy-saving and, safer and more efficient, and the smart contract can also get better by using the optimize the construction of public blockchain, private blockchain and alliance blockchain. On the contrary, in terms of AI, blockchain can solve the allocation of the overall Ai system, and enable free flow of data, algorithm and model resources by using the features of blockchain including distribution and decentralization. The reason why I think these two technologies should combinate with each other is that they all have their own absolute advantage. More specifically, AI mainly trains the centralized intelligence of closed data platform, blockchain mainly promotes distributed applications in an opening data enframement. If we combinate them properly to get two relative advantages, we will get profit & utility maximization.  

 

 

◆ Final comment

 

[From Day 1 to Day 4] I am very interested in robot because I like watching science fiction film since I was a little girl. So, this is the reason why I chosen this course. However, at the beginning of this course, I have no idea on how to learn this course, after I saw many excellent students who uploaded the file contained summary & code-practice for every Day, I tried to go through the same way. Thus, I uploaded summary & code-practice from Day 1 to Day 4, but I did not participate in finalized projects. This is all my fault and I missed many good opportunities for discussing together and studying from each other. Therefore, I’m going to actively participate in finalized projects. (Up to now, I still don’t know the process of finalized projects). I will try my best to practice writing code, though I am not good at in this way.

[From Day 5 to Day 6] My participation in Q&A was not activated, I should make more questions and discussions in Q&A. Moreover, I had not worked hard enough, I should make more of an effort to study this course. I am very grateful to professor for his help including the translated websites and many Youtube videoes both are in English. Professor helped me a lot. Thank you for your help!!! I really appreciate and I will work hard on this course.   

 

 

 

 

 

§ Final PBLhttp://matrix.skku.ac.kr/2020-Math4AI-Final-pbl2/

          http://matrix.skku.ac.kr/2020-math4ai-final-pbl/

§ Midterm PBL  http://matrix.skku.ac.kr/2020-Mid-PBL-2/

          http://matrix.skku.ac.kr/2020-Mid-PBL-1/

§  

http://matrix.skku.ac.kr/intro-math4ai/PIC9443.png

Cyber Lab  in English

§  01 Week   http://matrix.skku.ac.kr/intro-math4ai/W1/

§ 02 Week   http://matrix.skku.ac.kr/intro-math4ai/W2/

§ 03 Week   http://matrix.skku.ac.kr/intro-math4ai/W3/

§ 04 Week   http://matrix.skku.ac.kr/intro-math4ai/W4/

§ 05 Week   http://matrix.skku.ac.kr/intro-math4ai/W5/

§ Week 6.  Matrix decompositions (LU, QR, SVD)   http://matrix.skku.ac.kr/intro-math4ai/W6/

§ 07 Week   http://matrix.skku.ac.kr/intro-math4ai/W7/

§ 08 Week   http://matrix.skku.ac.kr/intro-math4ai/W8/

§ Week 9.  Gradient descent method    http://matrix.skku.ac.kr/intro-math4ai/W9/

§ Week 10 http://matrix.skku.ac.kr/intro-math4AI/W10/

§ Week 11 http://matrix.skku.ac.kr/intro-math4AI/W11/

§ Week 12 PCA http://matrix.skku.ac.kr/intro-math4AI/W11/

§ Week 13 ANN과 Backpropagation http://matrix.skku.ac.kr/intro-math4AI/W13/

§ Week 14 MNIST Data Set and PCA http://matrix.skku.ac.kr/intro-math4AI/W14/