I applied PCA on an $n\times p$ matrix where $n$ means the number of samples and $p$ means the number of variables, and I am using the 1st principal component (PC) as the new representation of the entire dataset. I confirmed that the 1st PC has the highest variance among all PCs, as expected. But I don't know how to interpret the values in the 1st PC, while before applying PCA a negative or positive value had a meaning. I observed that row means of the original data has a $-0.99$ correlation with the 1st PC, meaning that if a sample had a positive value across all variables, then this sample had a highly negative value in the 1st PC, while a consistent negative value across variables in the original data corresponds to a highly positive sample in the 1st PC. This is extremely confusing. How can I interpret that behavior?  I also realized that all loadings for the 1st PC are negative. I think this might be related to what I observe. Any thoughts?
I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?  Consider this:  set.seed(999) prcomp(data.frame(1:10,rnorm(10)))$x              PC1        PC2  [1,] -4.508620 -0.2567655  [2,] -3.373772 -1.1369417  [3,] -2.679669  1.0903445  [4,] -1.615837  0.7108631  [5,] -0.548879  0.3093389  [6,]  0.481756  0.1639112  [7,]  1.656178 -0.9952875  [8,]  2.560345 -0.2490548  [9,]  3.508442  0.1874520 [10,]  4.520055  0.1761397  set.seed(999) princomp(data.frame(1:10,rnorm(10)))$scores          Comp.1     Comp.2  [1,]  4.508620  0.2567655  [2,]  3.373772  1.1369417  [3,]  2.679669 -1.0903445  [4,]  1.615837 -0.7108631  [5,]  0.548879 -0.3093389  [6,] -0.481756 -0.1639112  [7,] -1.656178  0.9952875  [8,] -2.560345  0.2490548  [9,] -3.508442 -0.1874520 [10,] -4.520055 -0.1761397   Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?    In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)?
Wine or virtual machine (VirtualBox) to run apps (eg. Photoshop) I am wondering if it will be better (stability, convenience and speed) to run apps like Photoshop with Wine or a virtual machine like VirtualBox? Where to put my code: plugin or functions.php? (Re?)Install Ubuntu on a Dell laptop that had Ubuntu preinstalled I have Dell Laptop that is 15 5567 7th Generation. So it was maybe pre-installed as Dell says and I can see in the product information that this laptop is Ubuntu built in and based not Windows.  So I want to get my Ubuntu OS again. How can I do that? Confused at + and ++ in TikZ I can't seem to figure out the difference between + and ++ relative coordinates in TikZ.   I understand ++. ++ specifies delta coordinates that are added to the last current point, the current point is then set to the new absolute current point (correct?):  \draw (0,0) -- ++(1,0) -- ++(0,1) means:  set Current Point to (0,0) add (1,0) to Current Point = (1,0) set Current Point to (1,0) add Current Point (1,0) to (0,1) = (1,1) set Current Point to (1,1)   I can't seem to figure out that + does, when I try it out it looks like the coordinates in the +(x,y) are absolute coordinates rather than delta coordinates, but the manual says otherwise We all know that $\sqrt[3]{3}\not\in \mathbb{Q}({\sqrt[3]{2}})$ intuitively. We even know that $\sqrt[3]{p}\not\in \mathbb{Q}(\sqrt[3]{q})$ for two distinct primes $p, q$. However, I don't know how to prove these things rigorously. In case of $\sqrt{p}\not\in \mathbb{Q}(\sqrt{q})$, we can just set assume $\sqrt{p}=a+b\sqrt{q}$ for some $a, b\in\mathbb{Q}$ and deduce a contradiction. However, in cubic case, expansion of $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{3}$ is very complicated and even I expand this I couldn't get any contradiction by this.  To be specific, it gives us 3 diophantine equations  $$a^{3}+2b^{3}+4c^{3}+12abc=3$$ $$a^{2}b+2b^{2}c+2c^{2}a=0$$ $$ab^{2}+2bc^{2}+ca^{2}=0$$  and I don't know how to check solvability of this kind of diophantine equations (over $\mathbb{Q}$). Even if there exists a way to check solvability, I cannot convince that it can be used to prove $\sqrt[3]{p}\not\in \mathbb{Q}(\sqrt[3]{q})$.   So I tried to use Galois theory, but actually I don't know where to go. I assumed $\mathbb{Q}(\sqrt[3]{3})=\mathbb{Q}(\sqrt[3]{2})$ and take a Galois closure, i.e. $K=\mathbb{Q}(\sqrt[3]{3}, w)=\mathbb{Q}(\sqrt[3]{2}, w)$ where $w^{3}=1$ then tried to use field norm maps. We can check that $\beta=\sqrt[3]{3}\cdot\sigma(\sqrt[3]{3})\cdot\sigma^{2}(\sqrt[3]{3})\in \mathbb{Q}$ when $\sigma\in Gal(K/\mathbb{Q}), \sigma(\sqrt[3]{2})=\sqrt[3]{2}, \sigma(w)=w^{2}$. But I cannot get any information from this.   Edit : It would be great if there is a simple algorithm to check whether $\alpha\in K$ or not for a given algebraic number $\alpha$ and a number field $K$. In Star Trek, does the original die in teleportation? I am just curious.  If I am teleported is the original me dead and the teleporter just makes another copy of me there[at destination]? From the third week ( when does it actually begin) This describes a certain natural process that takes 15 weeks  It describes what a certain component looks like at some point in time during a naturallly occurring process  What I don't get is if it means it appears this way starting in the third week or after ?  Here's the full sentence  "The malleus first appears as a cartilage model from the third week"  Thanks again for you support guys How to use my aliases in my crontab? Getting a black screen when installing or Live booting Ubuntu (any version) in my laptop I have an ACER Aspire laptop. Model number is E5-573G-56RG. I bought it only 15 days back. There is no DVD drive in it. I have tried various versions of Ubuntu- 14.04.2, 14.04.3, 15.10, 12.04.5. Tried disabling secure boot, fast startup. I current have Windows 10 Home and Windows 10 PRO (not-activated) installed with GPT partition in UEFI bios. I actually want only the Home edition and Ubuntu 14.04.3 (preferably because of LTS) or 15.10. In case more information is required, kindly ask me..  I tried installing through Wubi, when booting if I choose Ubuntu it automatically restarts and goes into Windows 10 Home! (I didn't have Windows 10 PRO at that time)  NONE OF THE SOLUTIONS I REFERRED HELPED ME.  I have spent 5 days (literally at least 20 hours reading all of them and trying out all kinds of editing in the grub command line, like removing quiet splash, adding and replacing with nomodeset, no splash, acpi_osi=off, acpi_osi=Linux, acpi_backlight=vendor,  acpi_backlight=legacy, noveau.modeset=0, i915.i915_enable_rc6=1 and such things! But I was not able to succeed installing which is why I doubt if there is some more thing to be done or know exactly what has to be done. Is it necessary to dispose System.Timers.Timer if you use one in your application? I am using System.Timers.Timer class in one of the classes in my application. I know that Timer class has Dispose method inherited from the parent Component class that implements IDisposable interface. Instances of the class below are created many times during my application lifecycle; each of them has an instance of Timer class that generates Elapsed events continuously during the class's lifecycle. Should I implement IDisposable interface in the class that uses Timer class to dispose the timer object? (I have seen code that doesn't do this at all). I am afraid that some unmanaged resources will not be freed if I use the class below like this:  SomeClass someClass = new SomeClass(); someClass.DoSomething(); someClass = null;   The class:  using System.Timers;  public class SomeClass {     private Timer m_timer;      public SomeClass()     {                    m_timer = new Timer();         m_timer.Interval = 1000;         m_timer.Elapsed += new ElapsedEventHandler(m_timer_Elapsed);         m_timer.AutoReset = false;         m_timer.Start();                            }      public void DoSomething()     {      }      private void m_timer_Elapsed(object sender, ElapsedEventArgs e)     {         try         {             //Do some task         }         catch (Exception ex)         {             //Ignore         }         finally         {             if (m_timer != null)             {                 //Restart the timer                 m_timer.Enabled = true;             }         }     } } I am currently writing a paper with several multiple regression analyses. While visualizing univariate linear regression is easy via scatter plots, I was wondering whether there is any good way to visualize multiple linear regressions?   I am currently just plotting scatter plots like dependent variable vs. 1st independent variable, then vs. 2nd independent variable, etc. I would really appreciate any suggestions. Find directories and files with permissions other than 775 / 664 After you get the Reindeer Stables, you can add decorations through a text form to increase production speed. I've added a few decorations myself, but I was wondering what all of them are. I know that some entries seem to be "synonyms". For example, if you enter "wreath" and then "garland", you'll only add the "a Christmas wreath" and trying to put up the garland will return "We already have a Christmas wreath".  These are the decorations I've already added along with discovered synonyms.   Nutcracker (Nutcracker) a Christmas tree (tree) baubles (bauble) glass ornaments (ornament) a star (star) Christmas lights (lights) a Christmas wreath (wreath, garland) mistletoe (mistletoe) a Christmas stocking (stocking) candy canes (candy cane) a chimney decoration (chimney, fireplace) a Santa decoration (santa) an elf decoration (elf) a reindeer decoration (reindeer) a nativity scene (stable) toys (toy) wrapping paper (wrap) candles (candles) tinsel (tinsel) a star (star) an angle (angle) holly (holly) paper chains (chains) Christmas crackers (cracker) Christmas cards (card) a figure of Mary (mary) a figure of Joseph (joseph) a figure of Jesus (jesus) wise men figures (wise men) a donkey decoration (donkey) a sleigh decoration (sleigh) a snowman (snowman) snow decoration (snow) Santa hats (hat) bells (bell) ribbons (ribbon) a turkey decoration (turkey) ? is_active_sidebar() Always Returns False How should a user's death be handled? What happened to Startups SE? I really enjoyed that site. I've only posted on it once, but I enjoyed seeing other people's posts and appreciated reading about entrepreneurial dilemmas. Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$ Error in batch file to copy folder: 'Invalid number of parameters' I am making a batch file in Windows XP to copy my folders I need to another folder on my PC. I am getting an error.  I get the error "Invalid number of parameters".  xcopy /s/z D:\Documents and Settings\%username%\Desktop C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\Favorites C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\Start Menu C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\My Documents C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\PrintHood C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\NetHood C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\Templates C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\Searches C:\SomeRandomFolder\ xcopy /s/z D:\Documents and Settings\%username%\Local Settings\Application Data\Microsoft\Outlook C:\SomeRandomFolder\ pause   Then I changed the batch and get this error: "File not found - Desktop"   xcopy /s/z D:\...\%username%\Desktop C:\SomeRandomFolder\ pause   How do I fix these errors? Sort ArrayList of custom Objects by property Interpretation of ROC curve I have a ROC curve like the figure below:   What can be inferred given this kind of curve in general?  What are the differences of states 1, 2 and 3 in the figure?  What causes the diagram to jump straight from state 1 to state 3? Most of the ROC curves on the internet look like the figure below: