I perform Principal Component Analysis using two variables that are standardized. This is done by applying a SVD on the correlation matrix of the concerned variates. However, the SVD gives me the same eigenvector (weights) irrespective of what the two variables are. It's always [.70710678, .70710678]. I find this strange. Of course, the eigenvalues differ.  My question is: How to interpret this?    PS. I wanted to conduct a total least squares regression on two variables. My statistical programme does not provide TLS, but TLS luckily equals Principal Component Analysis, as far as I know. Hence my question. The question is not about TLS directly, but why I get the same eigenvectors irrespective of which variables I use (as long as they are exactly 2).
I've been testing PCA via SVD to decompose a simple time series data matrix, $X$.  I have two signals $x_1(t)$ and $x_2(t)$ in a data matrix where $M$ rows represents each timepoint sample and each column represents $x_1$ and $x_2$.  The mean signal, $\hat{x}$, is defined as the mean along the row axis (average of $x_1$ and $x_2$ along each timepoint).  I normalize each column of $X$ by subtracting its mean and dividing by the standard deviation.  When I use [U S V] = svd(Xz) in matlab, regardless of how the variables are distributed (whether they are correlated or uncorrelated), one of the columns of the right singular matrix, V, always points in the same direction (to a multiplicative constant) as the mean vector $(1/2, 1/2)$.  But when add an additional third time series, this is never the case (where the mean vector is $(1/3, 1/3, 1/3)$.  Because I normalize the standard deviation for each vector, it does make sense that the direction of most variance given by PCA would be the diagonal 45 degree line.  But if both variables $x_1$ and $x_2$ are independent gaussians, couldn't the PCA direction be any direction since the distribution is radially symmetric?  MATLAB Code:  s = RandStream('mcg16807', 'Seed', 0); RandStream.setDefaultStream(s);  G = zeros(1000,2); G(:,1) = 40*randn(1000,1)-100; G(:,2) = tan(G(:,1)) + randn(1000,1); X = G - repmat(mean(G),[size(G,1), 1]); Xz = X./repmat(std(G),[size(G,1), 1]); [U S V] = svd(Xz); G_mean = mean(Xz,2); corrcoef(G_mean, U(:,1)) corrcoef(G_mean, U(:,2))
Can't find boot-repair package for the newest version of Ubuntu After upgrading Ubuntu 13.10 to 14.04, grub failed to load Ubuntu and threw the error symbol 'grub_term_highlight_color' not found. I followed the solution offered in the  but this failed for me, throwing the error failed to get canonical path of /cow.   Now, I'm trying to install Boot-Repair to fix this problem on a live pen-drive, using these commands.   sudo add-apt-repository ppa:yannubuntu/boot-repair sudo apt-get update sudo apt-get install -y boot-repair &amp;&amp; (boot-repair &amp;)   But after adding and updating the repositories, it shows ppa:yannubuntu/boot-repair 404 Not Found error.  W: Failed to fetch http://ppa.launchpad.net/yannubuntu/boot-repair/ubuntu/dists/trusty/main/binary-amd64/Packages  404  Not Found   Is this problem solvable or is there an alternative PPA for Boot-Repair? Open CSV File with Different Delimiter Dell Latitude E7270 I read here at  that holding Shift during boot can bring up the GRUB menu. However I try holding one or the other Shift keys, tapping, tapping then holding, nothing works. Am I missing something? How does one access the GRUB menu during boot time?   No dual boot here, just standard Lubuntu 14.04 install. How does the twitter bot work? I heard that there was a twitter bot which automatically posted questions and asked for answers for specific questions on every site. How do they work? Is there an algorithm for doing this or is it purely random? I know that it doesn't tweet closed questions, but that's it. In his lectures on String Theory (), Tong gives a proof of the Weyl anomaly, using equation $(4.36)$. It seems wrong to me.  Here he uses the OPE between the stress-energy tensors $T_{zz}T_{ww}$  obtained when trace vanishes, i.e. $T_{z \bar{z}} = 0$: this implies that they are holomorphic functions $T_{zz} = T_{zz}(z)$. But in this proof he starts from the fact that $T_{z \bar{z}} \neq 0$ (we want to proof this thing after all) and so $T_{zz}$ is not a holomorphic function anymore! In the OPE $(4.36)$ I should have also terms with $(\bar{z}- \bar{w})$.  I can't also understand why he uses in the rest of the proof only the singular term $(z-w)^{-4}$, neglecting the other ones $(z-w)^{-2}$, $(z-w)^{-1}$.  (The same proof is given in these lectures  on conformal field theory, equation $(6.9)$).  I'll be really thankful if someone could explain me this proof :) How to search a string in the list of answers added by me on Stack Overflow? I am able to view all the answers given by me and questions asked by me, but not able to search a string in them?  Currently I have no other choice than clicking each ques/ans link manually (very hectic) ? If low-cost index funds are considered the best investment, why are there so many high-cost, managed funds? The general consensus is that the single best indicator of a fund's future performance is the its expense ratio . If you look for more than 5 minutes online you discover the  approach to investing; the , etc. It looks like the less you pay for a fund (and therefore the less it's managed), the better it does in the long run.  So why is it that in peoples' 401(k)s there are so many high-cost actively managed funds? Furthermore, why are there so many high-cost actively managed funds in the first place? Shouldn't the market reflect the fact that actively-managed funds typically don't beat the index they benchmark to and and react by purging itself of these crappy funds? I'm starting to write a book on . Since I've been contributing on this tag for a long time, I had the opportunity to read &amp; answer several interesting threads on Stackoverflow related to this technology.  I would like to discuss some of the interesting cases published on Stackoverflow in my book, and I wonder if I could use the images and code shared on those threads in it.  I'm hoping somebody from StackExchange could clarify about the copyright status of the material shared on Stackoverflow questions/answers. Should you pick a baby up everytime it cries? My husband and i have differing opinions on this. I don't believe in the term 'spoilt'. I would be very happy if my daughter was 'spoilt' in the sense that she expects to be treated with the highest possible respect and love and affection in all her relationships.   Also it would not bother me if she think she is special and unique (that's exactly what I want her to think). Mother's face is a mirror for the baby - through the mother's reactions and attention, the child learns that she is a lovable and valuable human being.   I spend all my time with my baby when she is awake. I play games with her, I smile, and laugh, and I never ever let her cry. I rush to pick her up immediately, even if it's the middle of the night, I force myself to get up, no matter how tired I am (I admit it's hard sometimes, and I do let her cry for 10-20 seconds sometimes, but never more). How to prove $\sum_{k=1}^{n}F_k = F_{n+2}-1$ by induction when $F_n$ is the Fibonacci sequence Let $F_n$ be the Fibonacci sequence where  $F_0$ = 0  , $F_1$ = 1 and  $F_n$ = $F_{n-1}$ + $F_{n-2}$.                   I want to prove the following by induction. $$\sum_{k=1}^{n}F_k = F_{n+2}-1$$ Here is what I have so far.  Can anybody tell me if I am right?  Base case = n=1  $F_k$ = 1 = LHS  $F_{1+2}$ - 1 = 2 - 1 = 1 = RHS  Statement is true for n=1.  Assume statement is true for n=i.  $$\sum_{k=1}^{i}F_k = F_{i+2}-1$$  Prove statement is true for n=i+1.  $$\sum_{k=1}^{i+1}F_{k} = F_{((i+1)+2)}-1$$  $F_1$ + $F_2$ +...+ $F_i$ + $F_{i+1}$ = $F_{((i+1)+2)}-1$  $F_{i+2}$ - 1 + $F_{i+1}$ = $F_{((i+1)+2)}-1$  $F_{((i+1)+2)}-1$ = $F_{((i+1)+2)}-1$  Statement holds for n=i+1 In the decimal system: $...-3, -2, -1, 0, 1, 2, 3...$ In the binary system: $...-11, -10, -1, 0, 1, 10, 11...$ In the unary system: $-111, -11, -1, \text{ ??? }, 1, 11, 111...$   I was only able to find a related question but none with an answer so far: Where are my Minecraft saves located? I want to copy/modify/delete a Minecraft world. Where can I find my saves folder? New UI encourages askers to confirm or dispute duplicate votes , the banner shown to askers whose questions have attracted at least one duplicate close vote has a couple of new options:    If the author clicks the first button, they're shown a confirmation that clarifies the results of this action somewhat:    If they click "Ok", the question is instantly closed:    If, instead, they edit, they'll be offered this guidance:    ...and once an edit is submitted the choice will go away until another duplicate close-vote is cast.   In just the past day, 80 questions  network-wide have been closed in response to confirmation from the asker; edits by askers are up significantly as well.   We'll be monitoring the results to see if further adjustments are needed. A few open questions surrounding this change include:    Perhaps if the number of votes is below some threshold?   I'm posting this separately from  to allow room for discussion and . How to convert a column number (e.g. 127) into an Excel column (e.g. AA) How do I survive in the Nether? Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose   $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm f}\left(x\right) - {\rm f}\left(b\right)}{x - b} =0 $$  Show that there exists $\xi \in \left(a, b\right)$ such that $\displaystyle{% \left\vert\vphantom{\Large A}\,{\rm f}''\left(\xi\right)\right\vert \geq \frac{4\left\vert\vphantom{\Large A}% \,{\rm f}\left(a\right) - {\rm f}\left(b\right)\right\vert} {\left(b - a\right)^{2}}}$.  I don't know how to start. Any hints ?. Which option is grammatical?        There will be readings from Nikki Giovanni’s and Alice Walker’s writings.   There will be readings from Nikki Giovanni and Alice Walker's writings.      Saying it out loud the latter sounds right, but looking at it the former looks better. Ignore compile errors How can I set up a permanent compile loop that keeps trying even when there are errors? I'm using latexmk with a remote compile setup as discussed  and . However, if I save a document with a bug (either a legitimate bug, or I save it before closing a tag), then latexmk halts. Any suggests how I can have it pause, try again, and keep trying again, for ever and ever? Can I delete user directory after deleting user? The reason I ask is because I recently made the mistake of installing Oracle 12c on my sub-performance laptop. Noticing the excessive amount of resources this added up to on startup, I decided to uninstall it.  What I failed to see through, however, was the fact that it created a new Windows user account in order for the initial installation to work out. In my case, it was mandatory because I only had an administrator account and I didn't see the need to use the built-in account. So, after failing to successfully install 12c twice, I wound up creating not just one, but two Windows user accounts. This had to do with something in regards to how Oracle sets up the Oracle Home business and whatnot, but I have no idea what any of that detailed, so I just went ahead and did it.  After uninstalling what I could in regards to Oracle and the database, I decided the next thing to do would be to remove the created users.  Having Windows 10 Home installed, I couldn't use the local users &amp; groups view/tool to delete these from the GUI. SO, I did the next best thing and Googled a solution. In my case, I used the following command:  net user username /DELETE   When I executed the command, the console gave the response of having successfully deleted the account for both accounts specified.  To verify that this had worked, I went to the users directory located in %SYSTEMROOT%/Users/ and came to find that the user directories were still there.  What does this mean and if I should delete them, will anything bad happen?  Can I even delete these directories?  Regards,  --T.R.G