Properties of the multivariate Gaussian probability distribution
Пікірлер: 107
@alisalimy93874 жыл бұрын
Hard to find a good explanation of this problem, until i found this! Great job Alexander!!!
@dsilvavinicius7 жыл бұрын
Finally a good explanation of the geometry interpretation of two-dimensional Gaussian! Great job!
@rajanalexander49492 жыл бұрын
Great explanation -- especially the graphical interpretation and example. Thank you!
@user-sj2zu9rn9q4 жыл бұрын
Thanks for you. Alexander. The best one I have seen.
@MacMac07105 жыл бұрын
This is great because you explain notation as well as giving solid examples!
@blasttrash Жыл бұрын
at 6:30 at the bottom right there is a contour plot where its printed that (sigma_11)^2 > (sigma_22)^2 What exactly is sigma_11 in that diagram? Is it the distance from center point of the contour plot to first concentric circle? or is it distance from center to 2nd concentric circle? or is it distance from center to 3rd concentric circle? Or is it something else? Similarly what is sigma_22?
@prathamhullamballi837 Жыл бұрын
@@blasttrash When you look at the contour plot but only taking x axis, then the variance associated with distribution along x-axis is (sigma_11)^2. Similarly, for y-axis, it would be (sigma_22)^2. Look at how the 'spread' in the contour plot along x-axis is more than the same along y-axis? That is precisely what we mean by (sigma_11)^2 > (sigma_22)^2. Note that the circles are just contour plots and the distance from it to the centre doesn't necessarily mean it is sigma_11 or anything.
@amirkeramatian6537 жыл бұрын
Very helpful video with clear explanations. Thanks a lot!
@jiongwang764511 жыл бұрын
thank you very much, this is succinct and easy to understand, way better than many text books !!
@visheshsinha_3 жыл бұрын
Thank You so much , I was struggling to understand this , you made it really simple.
@christinhainan11 жыл бұрын
I find your KZfaq videos much more helpful to learn - compared to the class videos. Maybe because I suffer from short attention span.
@K4moo10 жыл бұрын
Thank you for sharing, very useful.
@renato56682 жыл бұрын
This is a great explanation, it helped a lot
@avijoychakma86785 жыл бұрын
Nice explanation. Thank you so much.
@karthiks323910 жыл бұрын
Really nice video.. Thanks a lot.. !
@amizan865310 жыл бұрын
that was extremely helpful, thanks for posting!
@technokicksyourass6 жыл бұрын
The summary at the end was the best part. I would have liked some more explanation on what the different shapes of the contour plot mean.
@omarebacc073 жыл бұрын
When covariance values in the covariance matrix (the non-diagonal values) are or tend to 1, means that the shapes of the contour will look like ellipses incline with aprox 45 degrees or follow a rect line(positive association between variables). In contrast, when the covariance values are equal to zero, means that the shape of the curves will be similir to a circle, i.e, there is no asociation between the variables (similar to figure in min 6:13).
@PravNJ4 жыл бұрын
Thank you. This was helpful!
@ProfessionalTycoons5 жыл бұрын
thank you for this post!
@nyctophilic17904 жыл бұрын
Thank you so much , awsome work
@tomt86917 жыл бұрын
This is fantastic! Thank you!
@ProfessionalTycoons5 жыл бұрын
clear explanation very good
@osamaa.h.altameemi559210 жыл бұрын
Very nice video thank you.
@chyldstudios2 жыл бұрын
Solid explanation.
@hcgaron6 жыл бұрын
is the vector x assumed to be a row vector? I ask only because we have x - mu which is a row vector inside the exponential. To subtract components, would we not assume that x is a row vector like mu?
@ZLYang11 ай бұрын
At 4:32, if x and μ are row vectors, [x-μ] should also be a row vector. Then how to multiply (Σ^(-1))* [x-μ]? Since the dimension of (Σ^(-1)) is 2*2, and the dimension of [x-μ] is 1*2.
@user-ob2pe2wx7u2 жыл бұрын
Ha, the approach of decomposing the covariance matrix would be a nice example of PCA!
@spyhunter00662 жыл бұрын
I'd like to know how you call your x value for univariate caseü or x value set for multivariate case in your Gaussian distribuitons? Do you name them as "data set" or " variable set"? Also, what makes the mean value size same as the x data size? Thanks in advance. Should we think that we create one mean average for every added x data point in our data set? That's why we average them when we find the best estimated value in the end.
@andrew-kd4jk11 жыл бұрын
very good tutorial
@nates33612 жыл бұрын
Excellent explanation
@parshantjuneja48112 жыл бұрын
Thanks dude! I get it now! Well almost ;)
@spyhunter00662 жыл бұрын
In the formula at the minute 2.11, when you find the inverse of a Sigma matrix in the exp(...) , do you use unit matrix method, any coding , or some other method? Cheers.
@emirlanaliiarbekov87292 жыл бұрын
clearly explained!
@spyhunter00662 жыл бұрын
Could you explain more about the sum of the vectors in your notations for the maximum likelihood estimates at the minute 1.45? As far as I have noticed, there has been only one data set, namely one x vector. Thus, what actually are you summing up with j indices? Cheers.
@abdoelrahmanbashir40964 жыл бұрын
thank you teacher :)
@kaushik9007 жыл бұрын
At 11:02, you mean Xb=X*sqrt(EIGEN VALUE MATRIX) right?
@100uo10 жыл бұрын
awesome, thank you man!
@utsavdahiya37295 жыл бұрын
Thank youuuuuuuuuu♥️♥️♥️♥️♥️♥️♥️
@shivampadmani_iisc5 ай бұрын
Thank you so much so much sooooo much
@samarths7 жыл бұрын
thanks a lot
@alaraayhan77623 жыл бұрын
thank you !!
@CSEfreak10 жыл бұрын
AMazing thank you
@elumixor4 жыл бұрын
I think there is an error in the maximum likelihood formula in the order of vector multiplication. The way you have it makes the operation a dot product, not the outer product.
@georgestamatelis78123 жыл бұрын
thank you
@spyhunter00662 жыл бұрын
should we get x vector also as a row vector with length d just like nü (mean) vector at the minute of 1.44!
@RonnyMandal757 жыл бұрын
Haha, why would someone vote this down? This is great!
@boyangchen55445 жыл бұрын
exactly the best I can find
@chrischoir35944 жыл бұрын
They voted it down because hey are probably democrats and they don't like truth and facts
@llleiea4 жыл бұрын
Ronny Mandal maybe bc there are some small mistakes
@fupopanda4 жыл бұрын
He does have mistakes and really bad inconsistencies throughout the slides. Not enough to dislike though, but enough to not be surprised of the dislikes.
@LegeFles3 жыл бұрын
@@chrischoir3594 I thought the republicans don't like truth and facts
@ayasalama79656 жыл бұрын
in 12:45 shouldn't the expression on top of the graph be XD rather than XC ? great video !
@laurent__90325 жыл бұрын
Love your videos! Isn't there a small mistake where you place your transpose ? Should'nt it be $\Delta^2=(x-\mu)^T\Sigma(x-\mu)$ instead ?
@martynasvenckus4232 жыл бұрын
At 5:32, Alexander says "The scaling of the sigmas is accomplished by creating a diagonal covariance matrix". Could you explain what does "scaling of the sigmas" mean? Where are they being scaled? Thanks
@timvandewauw10452 жыл бұрын
When calculating the joint distribution p(x1)p(x2) for vector x_underlined = [x1 x2], he vectorizes (x1-mu2) and (x1-mu2) to the vector form (x_underlined-mu_underlined). I believe what he means by scaling of the sigmas, is a similar transformation from two seperate, scalar sigmas to a matrix, in this case the covariance matrix Sigma.
@GundoganFatih3 жыл бұрын
6:28 why do we create a diagonal cov. matrix. Let X be a feature set of two features (mx2), shouldn't sigma be cov(X)?
@hayekpower54643 жыл бұрын
Why does x is a row vector instead of column vector?
@user-ru9rm3rc7u15 күн бұрын
Thanks for wonderful explanation Do you share slides?
@dc69404 жыл бұрын
So, when features are independent, finding P(x1) and P(x2) individually and then multiplying is same as finding using multivariate gaussian distribution 6:13 ? Is my understanding correct?
@junlinguo773 жыл бұрын
yes
@heyptech17266 жыл бұрын
nice
@d-rex70432 жыл бұрын
This should be mandatory viewing, before being assaulted with the symbolic derivations!
@snesh933 жыл бұрын
From 4:12 to 6:24 where is an explanation on the Independent Gaussian models, I have a basic doubt on the Sigma calculation. I am finding hard to understand that sigma needs to be a diagonal matrix of (sigma_1*sigma_1 , sigma_2*sigma_2), shouldnt it be a matrix of the form [[sigma_1*sigma_1, sigma_1*sigma_2], [sigma_2*sigma_1, sigma_2*sigma_2]] ? Can anyone explain that to me ?
@AlexanderIhler2 жыл бұрын
The covariance matrix of a zero man Gaussian has entries sig_ij = E[xi xj]. So if xi and xj are independent, this is zero except along the diagonal. I think you’re describing a rank 1 matrix? Which is different from independence in probability.
@thomasbloomfield40707 жыл бұрын
At 11:00 isn't that the eigenvalue matrix, not the eigenvector matrix? Thanks for the great video!
@pr7497 жыл бұрын
Yes, it is the singular value matrix. (square root of eigenvalue matrix)
@lemyul4 жыл бұрын
thanks alexa
@spyhunter00662 жыл бұрын
At the minute of 1.34, the maximum likelihood estimates formula has 1 over N coefficient. On the other hand, at the minute of 3.13, there is 1 over m coefficients. We know that N and m is the total number of values in the sums, but what is the reason you used different notations as N and m. Is it just to seperate univariate and multivariate cases while they keep their definitions (or meaning)? Also, the j values in the lower and upper limits of sum sembols are not so clear in this notation. Should we write j=1 to j=m or N for instance?
@farajlagum9 жыл бұрын
Thumb up!
@muratakjol14373 жыл бұрын
Summary: 13:02
@user-bz8nm6eb6g4 жыл бұрын
wow
@spyhunter00662 жыл бұрын
One more question about the example at the minute of 4.24, you said independent x1 and x2 variables. Independendent of what??? As far as I see, you can have 2 univariate formula like in this example, but when you combine them to see the combined likelihood, you have to have a mean vector in size of 2 and Sigma matrix iin size of 2x2. That's always the case, right? The size of the mean vector and the Sigma matrix look like defined by the number of combination of x values. Is that right? I saw another example somewhere else, you can have L(μ=28 ,σ=2 | x1=32 and x2=34) for instance to find the combined likelihood at x1=32 and x2=34, and he uses only one mean and sigma for both. REF:kzfaq.info/get/bejne/etRmlZyXqK-5oIE.html&ab_channel=StatQuestwithJoshStarmer
@samfriedman50316 ай бұрын
4:07 MLE for sigma-hat should be X by X-transpose (outer product) not X-transpose by X (inner product)
@quangle57013 жыл бұрын
Can anyone explain how to vectorize the formula at 5:16? Thanks
@livershotrawmooseliver249810 жыл бұрын
What is meant by compressing a 2D Gaussian function in 3D?
@AlexanderIhler10 жыл бұрын
Sorry; where is that? Most likely I simply meant that, to draw a 2D Gaussian distribution requires a 3D drawing -- 2 variables x1,x2, plus the probability p(x1,x2). It's inconvenient to try to render 3D functions, so we usually plot contours in 2D instead (x1 and x2), with the contours indicating the lines of equal probability, p(x1,x2)=constant.
@livershotrawmooseliver249810 жыл бұрын
Is it possible to compress a 2D Gaussian function?
@bingbingsun630411 ай бұрын
学习
@torTHer683 жыл бұрын
ale beka xd
@ilyaskapenko80894 жыл бұрын
at kzfaq.info/get/bejne/m86fa9t5mKuanXk.html Why Delta^2 = (x-mu) * Σ^-1 * (x-mu)^T, not Delta^2 = (x-mu)^T * Σ^-1 * (x-mu)?
@austikan5 жыл бұрын
this guy sounds like Archer.
@thedailyepochs3383 жыл бұрын
Lanaaaaaaa!!!!!!
@Tokaexified5 жыл бұрын
I fell asleep watching this video with both hands under my head…when I woke up both of them had fell seep asleep and wouldn't wake up in a while..
@amitcraul6 жыл бұрын
at 9:24 Σ= UΛU^-1 instead of Transpose
@AlexanderIhler6 жыл бұрын
U is a unitary matrix, so they're the same
@ProfessionalTycoons5 жыл бұрын
Orthogonal matrix inverse == transpose
@harshitk112 жыл бұрын
x needs to be a column vector instead of row vector.
@spyhunter00662 жыл бұрын
At 5.23, you should have said (x-mu) transpose.
@AlexanderIhler2 жыл бұрын
These slides have a number of transposition notation errors, due to my having migrated from column to row notation that year. Unfortunately KZfaq does not allow updating videos, so the errors remain. It should be clear in context, since i say “outer product” for the few non inner products.
@spyhunter00662 жыл бұрын
@@AlexanderIhler NO worries, we spot them.
@OrhaninAnnesi7 жыл бұрын
please stop using probability density and probability interchangeably. The formula for a normal distribution never gives a probability, but a probability density, which can be greater than 1.
@umbhutta4 жыл бұрын
wow 1.5K supporter and just 40 haters :P
@danny-bw8tu6 жыл бұрын
it is not 2 dimension, it is 3 dimension
@spyhunter00662 жыл бұрын
Can you tell me the diffference between bivariate and multivariate case ? Can you also mention about when the parameters are dependent where we add extra dependence coefficient parameter? There is a sample video to refer for you give a better idea: kzfaq.info/get/bejne/e86dY9CU0cDXZWg.html
@AlexanderIhler2 жыл бұрын
Bivariate = 2 variables; multivariate = more than one variable. So bivariate is a special case, in which the mean is two-dimensional and the covariance is 2x2. Above 2 dimensions it is hard to visualize, so I usually just draw 2D distributions; but the mathematics is exactly the same.
@spyhunter00662 жыл бұрын
@@AlexanderIhler Your initial case of 1D Gaussian with only one x value is indeed a bivariate case with one x value with two parameters,the mean and the sigma value, right? Also, bivariate case can be called the simplest case of multivariate occasion, right? If we have a data set x and a multiple variable of mean and sigmas, we have to use your MULTIVARIATE CASE with a vector of x values and mean values with a covariance matrix for the sigma values, shouldn't we? Thanks for the help in advance.
@AlexanderIhler2 жыл бұрын
No, those are the parameters; if “x” (the random variable) is scalar, it is univariate, although the distribution may have any number of parameters. So, if x is bivariate, x=[x1,x2], the mean will have 2 entries and the covariance 4 (3 free parameters, since it is symmetric), so the distribution has 5 parameters total.
@spyhunter00662 жыл бұрын
@@AlexanderIhler x is your data point, right! If it is only one scalar value, the case is called univariate case, but if it is a vector of scalar values of two, it is called bivariate by definition. That's it. For bivariate and multivariate case where the data x variable is a vector of size d, the mean is also a vector of the same size of x vector. Thus, the covariance matrix by definition the square matrix has to have d by d matrix if x and mean has d dimension as you said . I assume you said 5 parameters in total, because symmetric terms are equal in covariance matrix, so 4-1=3 parameters coming from that Sigma matrix with size d x d .
@fupopanda4 жыл бұрын
Too many mistakes in the slides. But otherwise good explanation.
@joschk83316 жыл бұрын
the video is great but your audio sucks. buy an adequate microphone
@jfrohlich5 жыл бұрын
I can understand everything he's saying just fine.