How principal component analysis handled high multicolinearity in data set?
In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman?
Does the speed of light vary in non-inertial frames? How do I install, update and remove GNOME Shell Extensions? I am running Ubuntu GNOME and would like to know how exactly I can install, uninstall, upgrade, deactivate, and activate GNOME Shell Extensions. Why is negative times negative = positive? Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.  I went ahead and gave them a proof by contradiction like so:  Assume $(-x) \cdot (-y) = -xy$  Then divide both sides by $(-x)$ and you get $(-y) = y$  Since we have a contradiction, then our first assumption must be incorrect.  I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.  Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one? PuTTY inserts random characters during a session I recently started renting space on a remote server so that I could work on a project. I found that a relatively painless way to access it on a windows machine is through PuTTY. However, there is one thing that has always irked me when using it: for seemingly no reason random characters are sometimes inserted at the cursor.  Most of the time it is just a single tilde, but rarely it spits out what looks like some escape sequence ([[^8 or the like). It will only occur when I am focused on the window, whether I am typing or 20 feet away from the keyboard. If left for long enough, it will spit tildes at random intervals (average is about 1 minute).  Finally, this behavior seems to be inconsistant when running programs such as nano or the mysql interface: in nano, instead of inserting tildes, it will set marks (ctrl-^); in mysql, lines will become un-editable.  My question is this: Has anyone else experienced this sort of behavior in PuTTY? And if so, what can be done to prevent/correct this behavior? How should one maintain and update a MiKTeX installation? How and why do you use an image histogram? I realize that an image histogram is a graphical display of an images tonal distribution (i.e. horizontal darks to lights, vertical pixel distribution), but how does one really use it and why? I mean, can't you determine everything you need just by looking at the image? Approximating $\arctan x$ for large $|x|$ Conditional Linq Queries We're working on a Log Viewer. The use will have the option to filter by user, severity, etc. In the Sql days I'd add to the query string, but I want to do it with Linq. How can I conditionally add where-clauses? Dimensioning of a technical drawing in TikZ Why does the "Unanswered Questions" tab show questions that have answers? The following time-frequency correspondence illustration is in .    Also a gif animation version (if the animation doesn't show, please open the following image in a new window):    This picture has so much explanatory power, and I would like to replicate it in TikZ for future use.  Here is what I came up with:  \documentclass{minimal} \usepackage{tikz}  \begin{document} \begin{tikzpicture} [x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,blue!90] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,red!90] (0,0,0) -- (0,6.5,0) node[below] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \x in {0.5,1.5,2.5,3.5,4.5,5.5}{   \draw[blue!50] (\x,0,0)   \foreach \y in {0,0.02,...,6.28}{     -- ({\x},{\y},{sin(\x*\y*(157))/sqrt(2*\x)})   };   \draw[blue!90, thick] (\x,6.5,0) -- (\x,6.5,1/\x); }  \end{tikzpicture} \end{document}   This is the output so far.    I have two questions:   How to produce that red superposed sine wave of all the blue sine waves? I don't know if there is a sum function or I have to use loop yet again? How to make the camera projection in TikZ more similar to the perspective in that wikipedia illustration?   Any suggestion and tweaking of the parameters I used in the sample drawing are welcome as well.  Thanks in advance!    Update 1:  Here is a new version using tikz, more readable than the first one. Yet the superposition of the sine waves are done manually...I still don't know how to use foreach to produce a sum.  \begin{tikzpicture}[x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,black!70] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,black!70] (0,0,0) -- (0,6.5,0) node[below right] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \y in {0.5,1.5,...,5.5}{ \draw [cyan!50, domain=0:2*pi,samples=200,smooth]   plot (\y,\x, {sin(4*\y*\x r)/\y }); \draw[blue, ultra thick] (\y,6.5,0) -- (\y,6.5,1/\y); } \draw [red, thick, domain=0:2*pi,samples=200,smooth]  plot (0,\x, {sin(4*0.5*\x r)/0.5 + sin(4*1.5*\x r)/1.5 + sin(4*2.5*\x r)/2.5 + sin(4*3.5*\x r)/3.5 + sin(4*4.5*\x r)/4.5 + sin(4*5.5*\x r)/5.5} ); \end{tikzpicture}   The result is as follows: Why is this function continuous, unlike the Dirichlet function? How do the Chitauri "ships" propel themselves? In the 2012 The Avengers film, the Chitauri army arrives through the portal along with these big robot alien whale Chitauri dudes. Here's one chasing Iron Man:  These guys basically swim through the air like... giant robot alien whales. I don't see any means of propulsion. Other, more humanoid Chitauri soldiers fly around on things that seem to be mechanical &quot;speeders&quot; of sorts. What are these guys? Are they ships, creatures, or both? Is there an explanation anywhere in the Avengers universe for how they defy gravity and swim through the air? Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align} Novel about a dragon lord who has to wear gloves at all times How to add flavouring ingredients to steamed or boiled veggies? I bought a new SSD with 750 GB and made two partitions: 600 GB and 150 GB. The smaller one was thought to be a Bootcamp partition one day, but to date it was just another unused partition. When I needed more space, I decided to merge these two partitions. The partition was created with Mavericks, the merge was done with Yosemite.  I opened Disk Utility and the "minus" button. Disk Utility told me both partitions would be merged then. However it miserably failed. My main hard disc suddenly had errors which couldn't be repaired and even two of my other external hard discs could not be mounted any longer.  I eventually restored the hard disc from Time Machine, which worked well.  My second partition is now listed as free:    Unfortunately I can't resize the 600 GB partition nor can I create a new partition using the free space.  Diskutil reveils the following:  ~  diskutil list /dev/disk0    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *750.2 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:          Apple_CoreStorage                         599.6 GB   disk0s2    3:                 Apple_Boot Recovery HD             650.0 MB   disk0s3 /dev/disk1    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:                  Apple_HFS Macintosh HD           *599.3 GB   disk1                                  Logical Volume on disk0s2                                  XXXXXX-XXXX-XXXX-XXXX-XXXXXXXXXXXX                                  Unencrypted   Also diskutil brought up this:  ~  diskutil cs list CoreStorage logical volume groups (1 found) | +-- Logical Volume Group 9D2BA4BA-xxxx-xxxx-xxxx-xxxxxxxxxxxx     =========================================================     Name:         Macintosh HD     Status:       Online     Size:         599640592384 B (599.6 GB)     Free Space:   6111232 B (6.1 MB)     |     +-&lt; Physical Volume 29A9A02B-xxxx-xxxx-xxxx-xxxxxxxxxxxx     |   ----------------------------------------------------     |   Index:    0     |   Disk:     disk0s2     |   Status:   Online     |   Size:     599640592384 B (599.6 GB)     |     +-&gt; Logical Volume Family 2FA7892F-xxxx-xxxx-xxxx-xxxxxxxxxxxx         ----------------------------------------------------------         Encryption Status:       Unlocked         Encryption Type:         None         Conversion Status:       NoConversion         Conversion Direction:    -none-         Has Encrypted Extents:   No         Fully Secure:            No         Passphrase Required:     No         |         +-&gt; Logical Volume 7BF42B7B-xxxx-xxxx-xxxx-xxxxxxxxxxxx             ---------------------------------------------------             Disk:                  disk1             Status:                Online             Size (Total):          599282155520 B (599.3 GB)             Conversion Progress:   -none-             Revertible:            No             LV Name:               Macintosh HD             Volume Name:           Macintosh HD             Content Hint:          Apple_HFS   I have tried this:    fsck -fy (The volume Macintosh HD appears to be OK.)   but it wouldn't help. It said I couldn't recover because it wasn't created in a way it would support (sorry for missing the exact message).  I then have learned that Apple_CoreStorage might indicate my disk is encrypted, which might explain why I can't resize it. It doesn't explain why I can't create a new partition to me.  What are my options to make my hard disk available in a single partition with size 750 GB? I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. Run script as root at login (no sudoer user, shell session) Much like /etc/profile and ~/.profile but ran by root instead of the user is doing login. /etc/rc.local runs after boot but I need running the script before login. The user is not a sudoer one.   Thanks! I'm having difficulty with exercise 1.43 of Lang's Algebra. The question states     Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroup that is isomorphic to $G/H$.   Thinking about this for a bit, the only reasonable approach I could think of was to construct some surjective homomorphism $\phi\colon G\to K$ for $K\leq G$, and $\ker\phi=H$, and then just use the isomorphism theorems to get the result.  After a while of trying, I've failed to come up with a good map, since $H$ seems so arbitrary. I'm curious, how can one construct the desired homomorphism? This is just the approach I thought of, if there's a better one, I wouldn't mind seeing that either/instead. Thank you.