# 查看完整版本 : Numerical Linear Algebra 定 Stochastic Processes 黎緊要take 兩個其中一個sequence 黎讀。

22point 2019-9-19 04:00 AM

## Numerical Linear Algebra 定 Stochastic Processes 黎緊要take 兩個其中一個sequence 黎讀。

narius 2019-9-19 05:52 AM

Both are interesting, useful topics. Whether you should take both of them depends on your goals and aptitude. Sit in both at least, if you don't to do all the work.

Do you have the syllabus and learning objectives. Those will tell you a little more about how much of the class is conceptual and how much is application. Just two titles tell me little. Not everyone teaches these topics the same way.

I am teaching a class now with a bit of stochastic modeling but everything is embedded in a big application, which is very different than just doing the math. In fact ,my students program more than doing math, in this class.

hoyh 2019-9-19 11:40 AM

[quote]原帖由 [i]22point[/i] 於 2019-9-19 04:00 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=506960811&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Numerical Linear Algebra
[url=https://nptel.ac.in/courses/111107106/]https://nptel.ac.in/courses/111107106/[/url]

Stochastic Processes: Data Analysis and Computer Simulation

[quote]原帖由 [i]22point[/i] 於 2019-9-19 04:00 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=506960811&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

icantstop 2019-9-19 07:15 PM

[quote]原帖由 [i]22point[/i] 於 19-9-2019 04:00 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=506960811&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

innominate 2019-9-19 07:48 PM

Linear Algebra方便明白machine learning.過來人

[quote]原帖由 [i]innominate[/i] 於 2019-9-19 07:48 PM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=506999271&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]
Linear Algebra方便明白machine learning.過來人 [/quote]
Notes: Numeral Linear Algebra =/= Linear Algebra.

CMCJ 2019-9-21 12:29 AM

Numerical Linear Algebra - 所有 Math courses, 如果見到 Numerical 呢個字, 呢科大既係讀一d點樣用另類方法去解決 Math 問題, 而呢類方法主要係用向工程計算, 例如, 以前無計算機時, 如果想計 sin 53 度, log 15, 咁 Numerical Methods 就可以提供方法去解決呢類問題. 而 Numerical Linear Algebra 就係點樣用另類方法去解決關於 Linear Algebra 既問題, 例如點樣去解 system of equations, 兩三個 variables 就好昜用人手去解, 但 variables 去到十個八個或更多, 就一定要用 computer 去解, 所以近代既 Numerical Methods 發展主要係點樣用 algorithms 去解決 Math 問題.

Stochastic Processes 本身係屬於 Probability Theory, 在 Statistical Theory 上, 係無絕對 random, 所以點樣 produce 一個 random number 就已經係一個學問, Stochastic Processes 就係有一個 stochastic system 包含一 set 既 sample space, 而每個 sample 都係一個 random value / function, 而呢 set sample space 係會隨時間而改變. Stochastic Processes 所用到既 Pure Math 工具相當多, 例如, calculus, algebra, set theory, 甚至 topology, 所以 Stochastic Processes 係一門幾深既學問, 但應用範圍相當廣闊.

CMCJ 2019-9-21 12:40 AM

[quote]原帖由 [i]22point[/i] 於 2019-9-19 04:00 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=506960811&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 12:40 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507060538&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

numerical linear algebra呢D可以等到左grad school, confirm左真係搞algorithm, scientific computing, CSE果類果陣先再讀都唔遲.

CMCJ 2019-9-21 02:21 AM

[quote]原帖由 [i]竹劍[/i] 於 2019-9-21 01:15 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507061419&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

numerical linear algeb ... [/quote]

narius 2019-9-21 03:26 AM

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 12:29 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507060301&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Shaking my head. Almost all stats methods are about dimensionality reduction. Stat is much more related to linear algebra than you realized. Heck, I don't teach that at my own undergrad classes.

But still .. you must have seen the matrix formulation of regression, and the collinearity issues in regression is very much related to spans and sub-spaces in the data space. Heck, the whole Principal Component Analysis is basically a big linear algebra exercise.

Stat and LA is not separate like you imply.

narius 2019-9-21 03:32 AM

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 02:21 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507062614&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Well, everyone in the business should learn at least one, if not more platforms. (BTW, R or Python would be much better than matlab in terms of library support).

To be fair, even with all the libraries, most platforms only give you the very standard stuff. Any "real" modeling work needs to be developed on your own. But computational considerations depend on the problem. I have small size data set that i will code with a "dumb" algorithm because it is more flexible to change and that saves me time in developing variations.

I also have models with hard coded parameters (i almost never do that) and clever algorithm taking advantage of that because of the huge computational requirement.

It is not a 0 and 1 approach. Most take a while before they can learn do navigate these issues.

CMCJ 2019-9-21 05:44 AM

[quote]原帖由 [i]narius[/i] 於 2019-9-21 03:26 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507063387&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Shaking my head. Almost all stats methods are about dimensionality reduction. Stat is much more related to linear algebra than you realized. Heck, I don't teach that at my own undergrad classes.
... [/quote]

Linear regression including collinearity is just a part of Statistics. Nonlinear regression is already beyond the scope of Linear Algebra but both linear and nonlinear regressions are just a part of Statistics. However, Other substantial parts of Statistics also need other mathematical tools such as multivariable calculus, probability theory and differential equations etc. I would say yes, linear (multiple or single) regression is much related (I think "related" is not appropriate because Linear Algebra is just a tool for solving linear regression, no one says that a car is related to a screwdriver) to Linear Algebra but Statistics. I agree that Matrix and Determinant are the substantial part of Linear Algebra but linear transformation and solving system of equations are just an introduction to Linear Algebra. Matrix Theory, Topological vector spaces and Multilinear algebra and tensors are beyond your imagination.

I didn't say Statistics is not much related to Linear Algebra. I just said ML is closer to Statistics than Linear Algebra. If someone thinks ML is more related to Statistics and Statistics is more related to Linear Algebra, then ML is more related to Linear Algebra. By the concept of Math, that is transitive. I don't wanna argue that.

CMCJ 2019-9-21 05:53 AM

[quote]原帖由 [i]narius[/i] 於 2019-9-21 03:32 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507063446&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Well, everyone in the business should learn at least one, if not more platforms. (BTW, R or Python would be much better than matlab in terms of library support).

To be fair, even with all the  ... [/quote]

Mathlab is different from R and Python. Mathlab has CAS (Computer Algebraic system) and it can perform symbolic computation. Can R and Python do the same thing (I don't really know)? I would not compare them directly.

narius 2019-9-21 07:13 AM

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 05:53 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507064496&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Mathlab is different from R and Python. Mathlab has CAS (Computer Algebraic system) and it can perform symbolic computation. Can R and Python do the same thing (I don't really know)? I would not c ... [/quote]

No. But in general, you use the best tool for the task. For data work, people use R & Python. For symbolic work, we use mathematica. BTW, R is faster, though Python has better libraries in certain areas. The two platform also talks to one another.

Matlab (btw, no "h" .. it is matlab .. not mathlab .. mat stands for matrix) is certainly less popular than the other two, in terms of data work, particularly in the silicon valley.

narius 2019-9-21 07:22 AM

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 05:44 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507064444&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Linear regression including collinearity is just a part of Statistics. Nonlinear regression is already beyond the scope of Linear Algebra but both linear and nonlinear regressions are just a part  ... [/quote]

shake head x 2 .. all these math topics are related. You know that the way you solve high order linear differential equations are based on linear algebra, right? Calculus (and you don't even have to go to the vector kind) is also intimately embedded (if you do not like the word relate) to linear algebra. Taylor series expansion is a good example. It decomposes a function into polynomial basis in the hilbert space. Fourier transform decomposes functions into another set of basis. That is all linear algebra.

You really should not look at all these math concepts in isolation. In addition, tensor is really not that big of a deal. Even undergrad can learn tensorflow (the popular tensor software library that is the foundation for deep learning network). Tensors are nothing more than matrices in more than 2 dimensions. Now if you go into tensor calculus, that can get a bit more challenging (not unlike vector calculus) but in general, you do not need that for the common ML/AI applications.

CMCJ 2019-9-21 08:48 AM

[quote]原帖由 [i]narius[/i] 於 2019-9-21 07:22 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507065369&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

shake head x 2 .. all these math topics are related. You know that the way you solve high order linear differential equations are based on linear algebra, right? Calculus (and you don't even have  ... [/quote]

I never heard that Taylor Series, and Fourier Series, Analysis and Transform are Linear Algebra. I never learned them from any linear algebra course. I learned them from Calculus and I saw them on some Numerical Methods books too. No, linear differential equations are not based on linear algebra, especially higher order DE. Yes, we can put a matrix or a vector as x variables in a Taylor Series but it does not mean Taylor Series is Linear Algebra. If so, ok, Linear Algebra is everything, no need to argue that. Yes, Fourier Transform involves Linear Transformation but it does not mean Fourier Transform is Linear Algebra. Fourier Transform should be more than this. As your logic, Series is also Linear Algebra too, right?

I didn't isolate Math Concepts. Mathematics has algebraic and geometry meaning. I can derive algebraic meaning from geometry meaning and versa. They cannot isolate from each other or the system will crash down. That is the characteristic of Math. I just would not enlarge one small part to draw a whole picture or we don't need to divide Mathematics into many different areas. Nice, Tensor Calculus is a bit more challenging for you. Yes, Tensor Calculus can be an undergrad course. No problem, different people have different abilities. You do it your way; I do it my way.

CMCJ 2019-9-21 09:07 AM

[quote]原帖由 [i]narius[/i] 於 2019-9-21 07:13 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507065221&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

No. But in general, you use the best tool for the task. For data work, people use R & Python. For symbolic work, we use mathematica. BTW, R is faster, though Python has better libraries in certain ... [/quote]

I don't know what you're talking about. I didn't recommend OP to use a mathematical tool. I said "如果你會用 Mathlab 或者其他既計算工具". At that moment I wrote this message, I just remembered the tool called 'Matlab'. I didn't google what tools are available, which one is better or even if the name is correct. The point is not the tool, ok? The point is whether OP "決定用自己所學到既 Numerical Linear Algebra 去解決 Linear Algebra 既問題" or not. You talked about R, Python, Mathematica and Silicon Valley, they all are irrelevant to this topic and our discussion.

narius 2019-9-21 09:09 PM

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 08:48 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507067281&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

I never heard that Taylor Series, and Fourier Series, Analysis and Transform are Linear Algebra. I never learned them from any linear algebra course. I learned them from Calculus and I saw them on ... [/quote]

Apparently you did not learn it well enough, or deep enough. Look up Hilbert space and Banach space (essentially the space made up of functions .. it has infinite number of dimensions). A linear differential equation is basically a restriction on the general space and you can solve it by finding its basis. That is why you can turn a high order linear differential equation into a root finding exercise of a polynomial equation.

[url]https://math.stackexchange.com/questions/424955/on-the-vector-spaces-of-taylor-series-and-fourier-series[/url]

Heck, have you heard of eigen-functions? That is the basic foundations of quantum mechanics. You know how the Schrodinger's equation (basically a partial differential equation in 3-space & time) is analyzed, right? The perfect example of how a functional space is decomposed into its basis. The eigenvalues are the energy levels. I know for sure that they teach this even in undergrad courses (since my kid just had it) and you don't need to go to grad school to get it.

[quote]原帖由 [i]CMCJ[/i] 於 2019-9-21 02:21 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=507062614&ptid=28538459][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

[[i] 本帖最後由 竹劍 於 2019-9-22 05:25 AM 編輯 [/i]]

CMCJ 2019-9-22 12:48 AM

Linear Algebra 係討論點、線、面、空間的數學，有subspace的概念，同有vector、matric的應用。

Stochastic Process 係討論過程的輸出都係隨機變數，X1、X2...Xn 或 Xt，有鞅 Martingale: E(Xt|Xt-1) 係等於 Xt-1同semi等概念，係唸得明明Ito Calculus同build Black Scholes formula等的基本概率模型。

Matlab program 則可寫 monte-carlo computing

:smile_30:

[[i] 本帖最後由 台風中心的救星 於 2019-9-24 05:46 PM 編輯 [/i]]

[[i] 本帖最後由 台風中心的救星 於 2019-9-24 06:11 PM 編輯 [/i]]

22point 2020-1-1 06:00 AM

Stochastic Processes I
Random vectors, multivariate densities, covariance matrix, multivariate normal distribution. Random walk, Poisson process. Other topics if time permits.

Stochastic Processes II
Markov chains in discrete and continuous time, random walk, recurrent events. If time permits, topics chosen from stationary normal processes, branching processes, queuing theory.

Introduction to Numerical Analysis: Linear Algebra
Analysis of numerical methods for linear algebraic systems and least squares problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue and singular value computations. Knowledge of programming recommended

Introduction to Numerical Analysis: Approximation and Nonlinear Equations
Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Interpolation. Approximation of functions. Knowledge of programming recommended.