Would it be valid to interpret it instead as the row space, given that the row space defines the independent variables and pre-determines in which space the observations can move. Row vectors seem to make more sense as all observation are restricted to the # of independent variables present.
no, u hat is the distance between the orthogonal projection of y over the space (in this example represented as mu hat on a plane) spanned (obtained through linear combinations) by the columns or vectors (column space) and the actual y. there is a unique linear combination obtained through the Ols method that minimizes the value of u hat. the parameters of such "optimal" combination are the betas of the regression. u hat can be also interpreted as the portion of y that cannot be expressed through a linear combination of the columns (i.e. the part of y that does not lie on the column space). why is u hat perpendicular to the row space? due to the fact that u hat lies in the so called "left null space", which as stated in the fundamental theorem of linear algebra is orthogonal to the column space.
Hi, thanks for great explanation. I've been looking for something like this for ages ;) I guess I don't get one thing. Isn't it so that span of 3 independent vectors in 3D space cover entire space (as we can get any point by using linear combination) instead of just a plane?
what bounds all x into 1 plane? . if y1 is a step explained by [1 x11 x12] then it means there are 3 steps needed to explain y1 thats all, no one is saying that those all are in one plane. kindly explani
Agreed; there's also another error: the vector in the column space should be XB (in the graph now it's u) and the difference vector should be y-XB = u (instead of y-u as in the video 4:54)
Sometimes when you're studying advanced stuff, you get confused about the very basics. Thank you for this quick and easy explanation!!
Among all the intuitions we can possibly find, this is exactly the one I was looking for, thx a lot Ben!!!
Hey thanks for the video. Why at the end do you say u hat equals XBeta hat?
I thought it is y hat = XBeta hat.. thanks
Thank you very much. Clarified this subject very well :)
Would it be valid to interpret it instead as the row space, given that the row space defines the independent variables and pre-determines in which space the observations can move. Row vectors seem to make more sense as all observation are restricted to the # of independent variables present.
you are a life saver, thank you so much!!
I think u hat, in this case, doesn't indicate the residual, it only means the orthogonal projection of y onto col(X)... is that right?
no, u hat is the distance between the orthogonal projection of y over the space (in this example represented as mu hat on a plane) spanned (obtained through linear combinations) by the columns or vectors (column space) and the actual y. there is a unique linear combination obtained through the Ols method that minimizes the value of u hat. the parameters of such "optimal" combination are the betas of the regression. u hat can be also interpreted as the portion of y that cannot be expressed through a linear combination of the columns (i.e. the part of y that does not lie on the column space). why is u hat perpendicular to the row space? due to the fact that u hat lies in the so called "left null space", which as stated in the fundamental theorem of linear algebra is orthogonal to the column space.
Is it possible for y to be in the column space of X?
Hi, thanks for great explanation. I've been looking for something like this for ages ;) I guess I don't get one thing. Isn't it so that span of 3 independent vectors in 3D space cover entire space (as we can get any point by using linear combination) instead of just a plane?
2:59 The space described in the video is 4D, albeit drawn on a 2D graphic tablet. So 3 independent vectors in 4 dimensional space do span a "plane".
@@kottelkannim4919 yeah! Thats a hyperplane (sounds much fancier than it is)
@@kottelkannim4919 n
Amazing!!
that was excellent
what bounds all x into 1 plane? . if y1 is a step explained by [1 x11 x12] then it means there are 3 steps needed to explain y1 thats all, no one is saying that those all are in one plane. kindly explani
Thank you!!!!
The end of the video is not correct. When you write u_hat = XBeta_hat, thats not correct. It should be u_hat = y_hat - X Beta_hat
Agreed; there's also another error: the vector in the column space should be XB (in the graph now it's u) and the difference vector should be y-XB = u (instead of y-u as in the video 4:54)