Principal Component Analysis (The Math) : Data Science Concepts

Let's explore the math behind principal component analysis!
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[@kaeruuuu_]

Necessary videos: 1. kzbin.info/www/bejne/jmibpX94jph1g80 (Vector Projections) 2. kzbin.info/www/bejne/nZ3EmoNoZ5d9jaM (Eigenvalues & Eigenvectors) 3. kzbin.info/www/bejne/bKC9hWpoYtOhr6s (LaGrange Multipliers) 4. kzbin.info/www/bejne/m2iWYWZpn7-Heas (Derivative of a Matrix) 5. kzbin.info/www/bejne/Z2aVpYaPqc6EmNk (Covariance Matrix)

[@loveena419]

Finally, a video that explains the math behind PCA so clearly. Went through all the other videos and it helped a lot! Thank you!

[@qaarloshilaal2778]

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.

[@ritvikmath]

You're very welcome!

[@pigtowndanzee]

Love your teaching style. Keep these videos coming!

[@TamNguyen-qi8di]

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.

[@ritvikmath]

Thanks for the kind words. Wishing you much success!

[@joachimguth6226]

Very well presented. You are a great teacher. Hopefully you are going to cover the entire AI space.

[@ritvikmath]

That is the goal!

[@warrenbaker4124]

@@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!!

[@Moiez101]

@@ritvikmath i fully support that goal! I just started with data science bro. Loving your videos, you're a great teacher.

[@clxdyy.luveditxs]

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!

[@silverstone-z2d]

You're a brilliant teacher! Thanks for sharing the Lagrange multipliers trick especially to derive the solution to the optimization problem. Most of the resources I referred to before this entirely skip this part and only state the solution to be the eigenvector corresponding to the max eigenvalue of the covariance matrix

[@resoluation345]

The best series to explain the maths behind PCA

[@bhajman123]

Byfar the most accessible description of pca...finally was able to clearly connect the covar matrix and the eigen values to variance maximization

[@amaramar4969]

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!

[@vinceb8041]

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 :)

[@paulbrown5839]

This is a very strong video. It requires proper study. I hope you do more of this great stuff. Thank You!

[@alphar85]

I stopped at 01:33 and I am going to watch the other 5 videos. you are such a blessing mate.

[@shivamkak7981]

Such a well curated explanation of PCA, thanks so much!

[@jaivratsingh9966]

Simply excellent!

[@worldpulse365]

Simple and straight to the point. aBsolutely welldone!

[@133839297]

You have a gift for teaching.

[@thinkingAutomata]

Thanks Ritvik. Excellent explanation of PCA. Good job, well done!

[@BleachWizz]

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.

[@ritvikmath]

Thanks for the feedback! I do try to focus a lot more on the "why" questions rather than the "how" questions.

[@_arkadij]

Very appreciative of the explanation why we end up with using vectors corresponding to the biggest Eigenvalues. Thanks so much

[@Chill_Magma]

Straight to the point and thorough you deserve to be subscribed from my 3 accounts

[@robertbillette4671]

Like everyone else has mention, amazing clarity and style.

[@mashakozlovtseva4378]

Everything was clearly understood from math side! Thank you for your link on Medium account!

[@zilezile4942]

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/

[@riteshsaha6881]

This is super helpful. Way better than my professor's explanation

[@DeRocks1607]

You are great teacher.. ultimately I understood

[@volsurf1274]

Concise, clear and superbly explained. Thanks!

[@ritvikmath]

Glad it was helpful!

[@christinejiang6386]

wow! thank you! I watched all the videos before watching this one, they really helps a lot!

[@vinceb8041]

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.

[@nahidakhter8646]

Beautifully explained! Thanks so much!

[@ShubhamYadav-ut9ho]

Amazing explanation as always

[@543phi]

Thanks for this video! As a Data Science student, your lecture helped to clarify a lot....I appreciate your teaching style.

[@zilezile4942]

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/

[@cll2598]

Epic explanation

[@MaxDavidsonArgentina]

Thanks for sharing your knowledge. It's great to have people like you helping out!

[@zilezile4942]

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/

[@arun_kanthali]

Great Explanation.. Thank-you 👍

[@subhabhadra619]

Awesomely represented..

[@berkoec]

Such a well-explained video - keep up the great work!

[@ritvikmath]

Thanks a ton!

[@sidddddddddddddd]

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

[@sandeepc2833]

Cleared most of my doubts. Thanks a lot.

[@ahmadawad4782]

Watched many videos about linear algebra and PCA. You're the one who made it clear for me. Thanks!

[@zilezile4942]

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/

[@jhonportella5618]

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

[@martinw.9786]

Thank you very much for the explanations - very very well done. Your references to the mathematical backround is key!

[@paulntalo1425]

You have made it clear. Thank you

[@aravindsaraswatula2561]

Awesome video

[@nuamaaniqbal6373]

cant thank u enough!! u r truly the boss!

[@kakabudi]

Really great video! Thanks for explaining this concept wonderfully!

[@Chill_Magma]

Seeing your videos increases my confidence on math stuff :DDD

[@cameronbaird5658]

Phenomenal video, thank you for the hard work 👏

[@yarenlerler67]

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. :)

[@simranjoharle4220]

Your videos are extremely helpful! Thank you!

[@ritvikmath]

Glad you like them!

[@proxyme3628]

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.

[@erfanbayat3974]

this video is amazing

[@wandering-byte]

Just what I was looking for

[@Rockyzach88]

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.

[@ritvikmath]

I hope so!

[@pratik.patil87]

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)?

[@ajanasoufiane3903]

Great video, it would be nice if you could show the big picture through the SVD decomposition :)

[@zilezile4942]

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/

[@shashanksundi5669]

Just perfect !! Thank you :)

[@bilalbayrakdar7100]

bro you are the best, thanks for you effort

[@akrylic_]

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

[@ritvikmath]

Very true, thanks for filling in the missing step!

[@zechengchang3444]

Can you explain more? How does (AB)^T =(B^T)(A^T) have anything to do with u^T*(xi-xbar)? Thanks.

[@ItahangLimbu]

@@zechengchang3444 (A^T B)^T = B^T * A

[@Tankwell-cq5ky]

Very well presented - well done!😊😊

[@knp4356]

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.

[@ritvikmath]

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 :)

[@fahimfaisal4660]

Excellent

[@nandhinin799]

Clearly explained, helped me greatly in understanding the basis of PCA.

[@zilezile4942]

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/

[@かいじゅ-y3j]

This video is super great! I was wondering why Covariance matrix is used to compute PCA, but this video made my doubts clear!!

[@ritvikmath]

Glad it was helpful!

[@mwave3388]

I'm preparing for a job interview. Thanks, the best PCA video I found.

[@mmarva3597]

Thank you very much !! really helpful

[@Sriram-kj6kl]

Your videos help a lot man.. Thank you 👍

[@rajathjain314]

Very Intuitive, Great Job Ritvik!

[@zilezile4942]

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/

[@Cybrean1]

Excellent presentation and delivery … wish you all the success!

[@ritvikmath]

Thank you! You too!

[@georgegkenios486]

Amazing work mate!

[@ritvikmath]

Thanks a lot!

[@seetaramdantu3190]

excellent...well explained

[@ritvikmath]

Glad it was helpful!

[@herberthubert6828]

you rock, thank you

[@gc6327]

Hi Ritvik- Can you do a video on factor analysis. That would be huge! Thanks buddy!

[@ОлегЗалесский-й1б]

great explanation. Really appreciate it. thanks

[@ritvikmath]

Glad it was helpful!

[@kisholoymukherjee]

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

[@ArpitAnand-yd7tr]

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

[@ernestanonde3218]

great video

[@mainakmukherjee3444]

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.

[@suvikarhu4627]

@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?

[@ItahangLimbu]

he wrote (x.u)u and dot product is just x^T*u so he write (x^T * u) u here `u` is a vector and (x^T * u) is a constant or say scaling factor

[@ItahangLimbu]

Second question: he is calculating the variance in the projecting say i project vector v1 and v2 to a vector `v' we are calculating the amount variance in the projection and the projected amount is (x^T * u) the constant scaling factor so he only used (x^T * u)

[@muhammadghazy9941]

thank you man appreciate it

[@MohamedMostafa-kg6gk]

Thank you for this great explanation .

[@ritvikmath]

You are welcome!

[@GeoffryGifari]

Hmmm i noticed that if two categories are strongly correlated, the plot will look close to a straight line. Going to multidimensional space, that "line" looks like the vector u1 in the video, on which the data are projected. Does that mean PCA will perform better the more correlated two (or more) categories are?

[@santiagolicea3814]

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.

[@deplo]

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

[@thirumurthym7980]

@ 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.

[@quark37]

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?

[@yurongluo447]

Your video is helpful for us. Can you create one video to explain Independent Component Analysis in detail? Thanks.

[@rabiizahir2885]

Thanks a lot.

[@brofessorsbooks3352]

Good!

[@zilezile4942]

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/

[@Markks100]

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?

[@XXZSaikou]

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?

[@AshishKGor]

Thanks sir.

[@poornanagasai262]

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?

[@PR-ud4fp]

Thanks 😊

[@brianogrady37]

I wish you specified what values represented the Principal Conponents earlier on. But great video regardless.

[@alejandropalaciosgarcia2767]

Bro, you are awsome

[@404nohandlefound]

Could you please explain how this links to SVD

[@DarkShadow-tm2dk]

I HOPE U WILL REPLY 🛑🛑🛑 Aren't we suppose to standardize data before applying PCA and if we do standardize data then mean = 0 At 8:12 the second part of the equation will get cancelled right? So the equation changes

[@darshansolanki5535]

Best video!!

[@zilezile4942]

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/

[@iOSGamingDynasties]

Great video. However, I am not quite understanding why projection can be written as u_1^T x_i u. What's the relationship between u and u_1? They have to have the same direction right? Also for the projection formula, it seems that the projection formula should equal to (u_1^T X_i) / ||u_1||^2 * u_1 but it seems that there has to be an extra 1/||u_1|| in the projection formula.

[@ahmad3823]

Amazing

[@ritvikmath]

Thank you! Cheers!