Principal Component Analysis (The Math) : Data Science Concepts
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- Опубліковано 22 вер 2019
- Let's explore the math behind principal component analysis!
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Very well presented. You are a great teacher. Hopefully you are going to cover the entire AI space.
That is the goal!
@@ritvikmath Oh wow!!! I'm so happy to see you're taking this on. I'm a huge fan and this is a real highlight for me. Thanks for all you do!!
@@ritvikmath i fully support that goal! I just started with data science bro. Loving your videos, you're a great teacher.
Finally, a video that explains the math behind PCA so clearly. Went through all the other videos and it helped a lot! Thank you!
Love your teaching style. Keep these videos coming!
I've been wrestling to get all intuitional and computational components for doing pca for a while, and seeing it all come together here helps tremendously! Great as always, 10/10 video :)
This is a very strong video. It requires proper study. I hope you do more of this great stuff. Thank You!
Such a well curated explanation of PCA, thanks so much!
Thanks infinitely for all your videos, you're literally the best at explaining these concept in a clear and excellent way in order to continue with what we have to study/ do! Huge respect man.
You're very welcome!
Byfar the most accessible description of pca...finally was able to clearly connect the covar matrix and the eigen values to variance maximization
Simple and straight to the point. aBsolutely welldone!
Beautifully explained! Thanks so much!
Thanks Ritvik. Excellent explanation of PCA. Good job, well done!
Like everyone else has mention, amazing clarity and style.
Thank you very much for the explanations - very very well done. Your references to the mathematical backround is key!
Cleared most of my doubts. Thanks a lot.
Really great video! Thanks for explaining this concept wonderfully!
Very appreciative of the explanation why we end up with using vectors corresponding to the biggest Eigenvalues. Thanks so much
You have made it clear. Thank you
Thank god I found your channel. I am studying masters degree in computer science in a prestigious university and cost me a lot of money but your channel is very useful to dig deeper and understand many things. Stay on the good work!
Thanks for sharing your knowledge. It's great to have people like you helping out!
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
Simply excellent!
wow! thank you!
I watched all the videos before watching this one, they really helps a lot!
I stopped at 01:33 and I am going to watch the other 5 videos. you are such a blessing mate.
Ahh such a clean explanation. I really appreciate! I will have practical statics for astrophysics exam soon, and I was having some problem with the theory part. All your videos were very helpful! I hope I am gonna get a good grade from the exam. :)
I had to go thru the prerequisite videos to clarify my concepts first, but after that this PCA explanation is amazing! I think you are equivalent to 10 college professors out there in terms of teaching skills. I hope you get that proportion money and the college professors feel ashamed and work harder to catchup to your standards. Again, amazing!
You have a gift for teaching.
Brilliant explanation of why eigen vector is the one from maximum optimisation, never saw such great explanation before. Wish your course is in Coursera. I do not think any text book explains the eigen value as Lagrangian Multiplier and eigen vector as maximising variance. Thanks so much.
Hi Ritvikmath, thank you for your super informative videos! I took all courses on this topic but I was wondering if you could expand it with factor analysis and correspondence analysis. It would be interesting to know how different methods work and relate to each other because it would provide a deeper perspective. Thanks
Phenomenal video, thank you for the hard work 👏
Just perfect !! Thank you :)
I'm loving your content, you're showing a part of math that is not usually shown. The part where you actually use it, where you make your choices and why are you choosing them. Like it's nice to understand the equations and why it gives you a 0 on the sweet spot, but it's also nice to remind that it not only works but it was build to work with that intention.
So in the end you still need to figure out how do you get your problem to fit in one of those, what can you choose in these big generic operations to fit it into your problem.
Thanks for the feedback! I do try to focus a lot more on the "why" questions rather than the "how" questions.
Everything was clearly understood from math side! Thank you for your link on Medium account!
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
Dear rivitmath,
Thank you so much sir for your clear explanation. Even being in my last year of college, I am still struggling with the basics of statistics. With your help, I have been striving exponentially in class and looking to graduate from college in this semester. Your videos have been so so so helpful and i wish you an amazing health to continue with your content. I wish you could have been my professor in college. Thank you for putting out the high quality contents. Words can't describe how much I appreciate you, sir. Thank you. You have changed my life.
Thanks for the kind words. Wishing you much success!
This is a great explanation, thanks a lot. It'll be great if you can also make a video showing a practical example with some data set, showing how you use the eigenvectors projection matrix to transform the initial data set.
Your videos help a lot man.. Thank you 👍
Straight to the point and thorough you deserve to be subscribed from my 3 accounts
cant thank u enough!! u r truly the boss!
What you've called the closed form of the covariance matrix is actually the biased estimator of the covariance matrix \Sigma. And if you divide by (N-1) instead of (N), you get the unbiased estimator of \Sigma. Awesone video! Thanks :D
12:20 Quick note on why going down the list of eigenvalues is legit, the covariance matrix is a symmetric matrix, and it can be shown that if such a matrix has more than one eigenvalues that are not the same, the corresponding eigenvectors will be orthogonal.
Necessary videos:
1. ua-cam.com/video/X78tLBY3BMk/v-deo.html (Vector Projections)
2. ua-cam.com/video/glaiP222JWA/v-deo.html (Eigenvalues & Eigenvectors)
3. ua-cam.com/video/6oZT72-nnyI/v-deo.html (LaGrange Multipliers)
4. ua-cam.com/video/e73033jZTCI/v-deo.html (Derivative of a Matrix)
5. ua-cam.com/video/152tSYtiQbw/v-deo.html (Covariance Matrix)
Amazing explanation as always
Watched many videos about linear algebra and PCA. You're the one who made it clear for me. Thanks!
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
The best series to explain the maths behind PCA
Great Explanation.. Thank-you 👍
Seeing your videos increases my confidence on math stuff :DDD
Thank you very much !! really helpful
Concise, clear and superbly explained. Thanks!
Glad it was helpful!
Great, great video I really appreciate your effort and good methodology to teach. I have a question on the projection math. on your projection video you obtained P=XUU but here you used P=U*XU. Maybe this is a silly question but I would really appreciate if you can tell me why this equivalence is possible. Many thanks
Your videos are extremely helpful! Thank you!
Glad you like them!
Awesomely represented..
Thanks for this video! As a Data Science student, your lecture helped to clarify a lot....I appreciate your teaching style.
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
Thanks for the amazing video! can anyone please explain why the projection is u1T . Xi * u?
In the projection video it is ( Xi . u ) u. Are they equivalent?
Hi Ritvik- Can you do a video on factor analysis. That would be huge! Thanks buddy!
thank you man appreciate it
Thanks Ritvik, I went through multiple resources to figure out this exact questions " why does eigen vectors and eigen values of a covariance matrix represent the direction and strength of the biggest increase in variance" . Thanks your video clarifies it beautifully.
One question still though, I understand the equation we use to maximise but why do we need the constraint(uT u =1)?
Such a well-explained video - keep up the great work!
Thanks a ton!
Clearly explained, helped me greatly in understanding the basis of PCA.
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
This video is super great! I was wondering why Covariance matrix is used to compute PCA, but this video made my doubts clear!!
Glad it was helpful!
Very well presented - well done!😊😊
Great video, it would be nice if you could show the big picture through the SVD decomposition :)
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
this video is amazing
Very Intuitive, Great Job Ritvik!
Good morning
If you have difficulty understanding the statistical models and programming them with the R software; You have difficulty understanding where the main components come from when you do principal component analysis; You need to discover the statistics for functional data in particular the analysis in functional principal components; you have no idea how to model by functional linear model ... You like clear and detailed explanations. Click on this link amikour.wordpress.com/nos-formations/
Hello, thanks for this video and also for the others, well done! On this video I have a doubt to ask. Where can I submit the question in order to not mess comments here?
Just finished the LA section in the Deep Learning book and I can tell this is going to help supplement and fill in this gaps of understanding. Good vid.
I hope so!
Really appreciate this! Any good book suggestion for PCA mathematical Framework in greater depth? Maybe another video (hard maths of pca)?
you rock, thank you
Excellent presentation and delivery … wish you all the success!
Thank you! You too!
Thanks a lot.
great explanation. Really appreciate it. thanks
Glad it was helpful!
@ritvikmath 5:02 I don't understand where is this formula of projection (proj(xi)=ut xi u) coming from. The projection video does not say that. What the projection video exactly says is that the proj(xi) = (xi dot u)*u. No transpose there! Where did you get that transpose from? And the dot product is missing ?
Another question, at 5:50 why do you take only the magnitude of the vector?
Do you have a video about instrumental variables? Because in general seems to be just regular manipulation, but in a more complex way.
Also, do you have videos applying this concepts? Could be using R or Python. That would be very nice.
Excellent
Fun video. Thank-you. And thanks for all the pre-req videos.
Question: I've seen other videos that describe PCA vectors as orthogonal, but using eigenvectors they would not necessarily be orthogonal, right? What is the correct way to think about the orthogonality of PCA vectors? Thanks.
*
I think I answered my own question. The eigenvectors in question are of the covariance matrix of the related variables. This matrix is symmetrical so the eigenvectors will be orthogonal. Correct?
There's a property of transposes around 6:45 that you could have mentioned, and I got tripped up for a second. The reason why you can write u^T*(xi-xbar) as (xi-xbar) ^T*u is because
(AB)^T =(B^T)(A^T)
It's a cool trick, but not obvious
Very true, thanks for filling in the missing step!
Can you explain more? How does (AB)^T =(B^T)(A^T) have anything to do with u^T*(xi-xbar)? Thanks.
nicely explained! but I noticed you didn't mention the need to standardize the original data for PCA. Is standardization a little trick to make things faster or is it needed in the underlying math?
@ 4.54 - you are referring about projection video - on how you arrive projections formula. There is no such mention of U transpose in that projections video.
Thank you for this great explanation .
You are welcome!
Hi ritvik, thanks for the video. Can you please tell me how the vector projection formula is being used to calculate the projection of xi on u here? The formulae in the two videos seem to be quite different. Would really appreciate if you could help understand the underlying math
That's just a dot product between the potential u1 and Xi. It gives the magnitude of the projection in the direction of the unit vector u
Hey Ritvik, It would be great if you can generate some problems for viewers to solve. Watching is great but if you can supplement with actual problems then it would drive the points into viewers head. You can then further post solutions on your medium site. Hopefully at least 4-5 problems per each video. I've watched many videos on DS subjects but something in your teaching method is making it simpler to understand. Thanks.
I honestly really appreciate that you're trying to help me be more effective at what I do. I think it's a great idea and I'll look into it. Thanks :)
great video
Since principal component analysis is used to reduce the dimensions, thus lessen the curse of dimensionality, can you calculate the maximum amount of dimensions you need for a given dataset to find patterns?
Thanks for such amazing videos.
Have 1 question:
In the projection video you derived projection as X. V / ||V|| * u here you took started with u1T Xi u. What is the difference?
Will be helpful if you can point me to some resources !
I don't understand why the projected form of Xi on U1 is U1^TXiU. From your lecture on vector projections, P=(X.U)U, so why the change?
Amazing work mate!
Thanks a lot!
Your video is helpful for us. Can you create one video to explain Independent Component Analysis in detail? Thanks.
We find the equation of the variance of the vector, on which we are going to project the data, and then tried maximizing it, because, the vector, for which the variance will be highest (max eigen value), is gonna retain most of the information of the data, after dimensionality reduction.
Thanks sir.
Thank you:)
Shouldn't you also constrain u_2 to be orthogonal of u_1 if you want 2 dimensions? Such that each dimensions principal component will be orthogonal to each other. Or is that just an SVD and PCA relationship thing?
Thanks 😊
Is there any way to find the projected data points using pythagorean theorem
nice video. Very useful to me. You are also mentioning about link to couple of external resources @13.18 , could you please share?. thanks.
It's really a great explanation and one question I got is from the video of vector projection it is clear that the vector onto which we wanna project has the value is (u.x)u where (u.x) is the magnitude and u being the unit vector. Here comes my question in this present video(math behind pca) you used (u^T .x)u as the vector magnitude of the vector which is projected on to. What is the difference in using u and u^t(u transpose)? Can you please answer me?
I'm preparing for a job interview. Thanks, the best PCA video I found.
Bro, you are awsome
Could you please explain how this links to SVD
Hi Ritvik- Great video! In the first part of the video, you impose the constraint that u is a unit vector. So to arrive at the maximization problem, we have already imposed this constraint. Does this introduce any logical problems?
Unit vector is just because we want the direction that maximises the variance there is no problem infact is simplifies the maths
excellent...well explained
Glad it was helpful!
I think the links to those videos are missing in the notes section.
and after you figred out the eigenvector that corresponds to the biggest eigenvalue, I'm assuming you project the data points on it and plot it?
Can you explain how to apply PCA to non-square matrix cases using SVD?