this is the best description of using matrices to solve Linear Regression I have ever seen. With basic to intermediate knowledge of Linear Algebra Chasnow explains each step in clear detail and then rounds things of with a trivial example to explain all the math behind the powerful method of Linear regression. This man is both a Maths and communications genius. Greetings from Germany. Your students are incredibly lucky, you look like a professor who is determined to make sure all students understand the topics rather than one who wants students to marvel at their own math prowess.
Thanks a lot Prof Jeff for valuable lecture. @4:25 : Why AT A is invertible matrix - as per last lesson Invertible matrix which has inverse (AA-1=I) ,kindly clarify what make AT A invertible matrix? @5:21 : Projection Matrix (A(ATA)-1 AT - Is there any paper illustrate it as i think we didn't come to it last lectures .
at fisrt you start with 1) Ax=b and then you say 2) Ax = b(proj) you multiplied both sidesof (1) by A transposed, and then use (2). can't understand how you can use both in the same proof.
Professor,I still have one question you mean b-b_proj is orthogonal to Matrix A,wheteher do we know that at the beginning B is out of the column space of A, so b and b-b_proj is in the kernel of Matrix A,b-b_proj becomes just short or long
according to what you did , you found the slope and intercept for 3 points on a graph - how is this related to the least squares problem you never got around to the difference of ( y - y' ) where y' is the value lying on the least sqaures line. i hope this question makes sense
@@ProfJeffreyChasnov Professor,I realized I made a error when I want to describe the unequal relation between Ax and b. I should use Norm2(Ax)=Norm2(b) .May I know whether this expression is correct?
At the end with B0 = 1 and B1 = 1/2. If I plug in X = 1 then Y = 1 + 1/2 * 1 = 3/2 and not 1 like in the data. Same for other two datapoints. Am I missing something?
Great video... But there is a set of ounnecessary confusions: You name the vrctor of beta_i as x, matrix containing x_i as A, and vector from y as b. :/ Most people are lost at that point. :/
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this is the best description of using matrices to solve Linear Regression I have ever seen. With basic to intermediate knowledge of Linear Algebra Chasnow explains each step in clear detail and then rounds things of with a trivial example to explain all the math behind the powerful method of Linear regression. This man is both a Maths and communications genius. Greetings from Germany. Your students are incredibly lucky, you look like a professor who is determined to make sure all students understand the topics rather than one who wants students to marvel at their own math prowess.
Thank you Prof. Chasnov, we appreciate your good work.
Thanks a lot Prof Jeff for valuable lecture.
@4:25 : Why AT A is invertible matrix - as per last lesson Invertible matrix which has inverse (AA-1=I) ,kindly clarify what make AT A invertible matrix?
@5:21 : Projection Matrix (A(ATA)-1 AT - Is there any paper illustrate it as i think we didn't come to it last lectures .
thanks a lot mr. Chasnov ! really saved me hours of mental breakdown
Your explanation is superb!
Fantastic explaination!
Thank you so much Dear Prof. Chansov
Many thanks!!! You certainly have a huge talant for teaching as you know where the breakes in understanding sit to be given a particular attention!
at fisrt you start with
1) Ax=b
and then you say
2) Ax = b(proj)
you multiplied both sidesof (1) by A transposed, and then use (2). can't understand how you can use both in the same proof.
It should be like this A^TAx = A^T +A*T(b - bproj) however the b-bproj is just Null(AT) so that part just become zero
You made it look easy; thank yo so much, Professor Chasnov!
Professor,I still have one question you mean b-b_proj is orthogonal to Matrix A,wheteher do we know that at the beginning B is out of the column space of A, so b and b-b_proj is in the kernel of Matrix A,b-b_proj becomes just short or long
Amazing explanation
Good class professor
Thank you professor !!
Why would the columns of A typically be linearly independent in a least squares problem? (4:10)
Lot's of rows, very few columns.
At 10:08 why is b equal to y? I have seen the normal equation as A^T * A * x = A^T * y which im confused by because of the y in the equation
Because this lecture follows Lecture 26.
according to what you did , you found the slope and intercept for 3 points on a graph - how is this related to the least squares problem
you never got around to the difference of ( y - y' ) where y' is the value lying on the least sqaures line.
i hope this question makes sense
The solution minimizes the sum of the squares of the differences.
Super good.
Is this similar to finding a "regression plane"?
My understanding is that if you continue on with the Normal Equation and find the projection matrix you will solved the regression plane
You are awesome!
How does multiplying a vector in null space by transpose of that matrix get rid of the vector in the null space
Hello professor, thanks for your video. But I have one question if the Ax=b is overdetermined, why can we still use "=" instead of Ax>=b or Ax
With = there is no solution. What would be the meaning of >= or
@@ProfJeffreyChasnov thanks a lot, professor, I understand the meaning of the equation and the way to obtain solution.
@@ProfJeffreyChasnov Professor,I realized I made a error when I want to describe the unequal relation between Ax and b. I should use Norm2(Ax)=Norm2(b) .May I know whether this expression is correct?
Thank you.
Did you reflect the video or are you really writing backwards?
Of course, it's reflected ))
@@vmarchenkoff : )
Thanks a lot !
At the end with B0 = 1 and B1 = 1/2. If I plug in X = 1 then Y = 1 + 1/2 * 1 = 3/2 and not 1 like in the data. Same for other two datapoints. Am I missing something?
We try to do the line that gets as closer as possible to the data points
There is a wrong solution B_0 = 0 B_1 = 1 if you drow it, it is clear
superb
Isn't A(AtA)^(-1)At just the identity matrix?
Only if A is invertible. But here A is not even a square matrix.
Why do you use the normal equations to find x and not just directly use the x = (A'A)^-1 A' b equation you'd derived already?
It is computationally inefficient to compute inverses. Faster to do Gaussian elimination.
why is y = 1/2(x) better than simply y = x? Plotted it seems like y=x fits better
it minimizes the square of the errors. the errors being the difference between y value of the data and the y value of the line at each x value.
Great video... But there is a set of ounnecessary confusions:
You name the vrctor of beta_i as x, matrix containing x_i as A, and vector from y as b.
:/
Most people are lost at that point. :/
Thank you
thank you
Many thanks