My question in full is as follows - &quot;What is meant by Principal Component Analysis. including in the explanation the terms: geometric information; covariance matrix; orthogonal transformation; Spectral Theorem My understanding of Principal Component Analysis is that it reduces a number of variables x1, x2... to a smaller set of principal components that store as much of the original information from the original variables in these newly created principal components. However I don't fully understand all of these other terms or how they relate to PCA.
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?
How do you run Ubuntu Server with a GUI? What does functools.wraps do? In a comment on this , someone said that they weren't sure what functools.wraps was doing. So, I'm asking this question so that there will be a record of it on StackOverflow for future reference: what does functools.wraps do, exactly? The words in the node of tree I want to plot this graph, but I don't know how to code the words in the nodes.     Here is my code:  \documentclass{article} \usepackage{amsmath} \usepackage{titletoc} \usepackage{tikz} \usetikzlibrary{trees,positioning,fit}  \begin{document}{\allowdisplaybreaks[4]}  \tikzstyle{bag} = [text width=2em, text centered] \tikzstyle{end} = []  \begin{center} \begin{tikzpicture}[sloped] \node (a) at ( 0,0) [bag] {$\ln r_0$}; \node (b1) at ( 3,1.5) [bag] {$\ln r_{10} + \sigma_1\sqrt{\Delta t_1}$}; \node (b2) at ( 3,-1.5) [bag] {$\ln r_{10}$}; \node (c1) at ( 6,3) [bag] {$\ln r_{20} + 2\sigma_2\sqrt{\Delta t_2}$}; \node (c2) at ( 6,1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c3) at ( 6,0) [bag] {$\ln r_{20} + \sigma_2\sqrt{\Delta t_2}$}; \node (c4) at ( 6,-1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c5) at ( 6,-3) [bag] {$\ln r_{20}$}; \draw [-&gt;] (a) to node [below] {} (b1); \draw [-&gt;] (a) to node [above] {} (b2); \draw [-&gt;] (b1) to node [below] {} (c1); \draw [-&gt;] (b1) to node [below] {} (c2); \draw [-&gt;] (b1) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c4); \draw [-&gt;] (b2) to node [below] {} (c5); \end{tikzpicture} \end{center}  \end{document}       Update code use  \draw [-&gt;] (a) to node [below] {} (b1.south east); \draw [-&gt;] (a) to node [above] {} (b2.north east); \draw [-&gt;] (b1.south east) to node [below] {} (c1.west); \draw [-&gt;] (b1.south east) to node [below] {} (c2.west); \draw [-&gt;] (b1.south east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c4.west); \draw [-&gt;] (b2.north east) to node [below] {} (c5.west); Prevent sent emails treated as junk mails using php mail function Canonical replacing Unity with gnome-shell in Ubuntu 18.04 what now? The default desktop environment in Ubuntu 18.04 is going to be gnome-shell instead of Unity 8.  Does this mean development of Unity is stopped? or just that we can expect it to be beta a bit longer?  source on  post by Mark Shuttleworth. phoronix also posted about it  and here about the CentOS 7 DVD ISO vs Everything ISO different What does "it" refer to in "it's raining"? 554 5.7.1 : Relay access denied centos postfix I have problem in send mail from postfix in centos  I have following  for sending mail but I am getting error.  As in the link I tried following commands  telnet localhost smtp  Trying ::1... Connected to localhost. Escape character is '^]'. 220 mydomain.com ESMTP Postfix ehlo localhost 250-mydomain.com 250-PIPELINING 250-SIZE 10240000 250-VRFY 250-ETRN 250-ENHANCEDSTATUSCODES 250-8BITMIME 250 DSN mail from:&lt;domain.com&gt; 250 2.1.0 Ok rcpt to:&lt;test@gmail.com&gt; 554 5.7.1 &lt;test@gmail.com&gt;: Relay access denied   Edit-1  In terminal this works  echo TEST | mail -v -s "Test mail" testing@gmail.com   my postconf -n shows belog information  alias_database = hash:/etc/aliases alias_maps = hash:/etc/aliases command_directory = /usr/sbin config_directory = /etc/postfix daemon_directory = /usr/libexec/postfix data_directory = /var/lib/postfix debug_peer_level = 2 home_mailbox = Maildir/ html_directory = no inet_interfaces = localhost inet_protocols = all mail_owner = postfix mailq_path = /usr/bin/mailq.postfix manpage_directory = /usr/share/man mydestination = $myhostname, localhost.$mydomain, localhost, $mydomain mydomain = dummy.com myhostname = dummy.com mynetworks = all mynetworks_style = host myorigin = $mydomain newaliases_path = /usr/bin/newaliases.postfix queue_directory = /var/spool/postfix readme_directory = /usr/share/doc/postfix-2.6.6/README_FILES sample_directory = /usr/share/doc/postfix-2.6.6/samples sendmail_path = /usr/sbin/sendmail.postfix setgid_group = postdrop unknown_local_recipient_reject_code = 550   What configuration I have to perform to send mails from my server. How to visualize the gradient as a one-form? I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I usually visualize gradients as vector fields while I visualize one-forms with level sets. How to visualize the gradient as a one-form? Is it ever acceptable to have a  in your C or C++ application?  What if you allocate some memory and use it until the very last line of code in your application (for example, a global object's destructor)? As long as the memory consumption doesn't grow over time, is it OK to trust the OS to free your memory for you when your application terminates (on Windows, Mac, and Linux)? Would you even consider this a real memory leak if the memory was being used continuously until it was freed by the OS.  What if a third party library forced this situation on you? Would refuse to use that third party library no matter how great it otherwise might be?  I only see one practical disadvantage, and that is that these benign leaks will show up with memory leak detection tools as false positives. Is there a way to preserve a weapon in Magicka within a saved multiplayer game Time taken to hit a pattern of heads and tails in a series of coin-tosses Inspired by Peter Donnelly's talk at , in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two patterns 'hth' and 'htt', it calculates how long it takes (i.e. how many coin tosses) on average before you hit one of these patterns.  coin &lt;- c('h','t')  hit &lt;- function(seq) {     miss &lt;- TRUE     fail &lt;- 3     trp  &lt;- sample(coin,3,replace=T)     while (miss) {         if (all(seq == trp)) {             miss &lt;- FALSE         }         else {             trp &lt;- c(trp[2],trp[3],sample(coin,1,T))             fail &lt;- fail + 1         }     }     return(fail) }  n &lt;- 5000 trials &lt;- data.frame("hth"=rep(NA,n),"htt"=rep(NA,n))  hth &lt;- c('h','t','h') htt &lt;- c('h','t','t')  set.seed(4321) for (i in 1:n) {     trials[i,] &lt;- c(hit(hth),hit(htt))     } summary(trials)   The summary statistics are as follows,        hth             htt          Min.   : 3.00   Min.   : 3.000    1st Qu.: 4.00   1st Qu.: 5.000    Median : 8.00   Median : 7.000    Mean   :10.08   Mean   : 8.014    3rd Qu.:13.00   3rd Qu.:10.000    Max.   :70.00   Max.   :42.000    In the talk it is explained that the average number of coin tosses would be different for the two patterns; as can be seen from my simulation. Despite watching the talk a few times I'm still not quite getting why this would be the case. I understand that 'hth' overlaps itself and intuitively I would think that you would hit 'hth' sooner than 'htt', but this is not the case. I would really appreciate it if someone could explain this to me. Does a run scored on a tag out on a forced runner? The situation:   Runners at third and first, ball is hit to the second basemen, 2nd basemen goes to tag the runner out. Runner stops, runner on third scores and then 2nd basement tags the runner, does not step on base.Does the run count? What's the 1 inch square knob sticking out of the wall? How to use GROUP BY to concatenate strings in SQL Server? Integrating $\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$ How to make sure that Thinkpad discharges external slice battery first? The thing is that when an external slice battery pack is connected, the laptop first discharges the internal one. This is a problem because then it is impossible to switch between multiple external battery packs without powering off the laptop. It is also not possible to just disconnect the external battery pack when you e.g. need to insert it into a docking station.  Thus, my question: how to change the discharge order under Linux?  For my use cases following order would make much more sense: when an external slice battery is connected, first discharge it completely before using power from the internal battery.  Background: For some Thinkpad models (e.g. X220) there are slice battery packs () available that you can connect to the bottom of the laptop (similar to a docking station).  They seem to be supported well on Linux - e.g. the Gnome Shell battery applet includes both batteries (internal and external) when connected - including displaying the current capacity for each. data & synchronization - manually sync mail. calender and contacts Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ For all $a, m, n \in \mathbb{Z}^+$,   $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ How to divide LaTeX table cells vertically? I want my table to have separated cells. Basically double lines but I can't figure how to have horizontal lines disconnected. I tried \hline and \cline but those connect too.  This is best I can do:    This is how I want my table to look:   \begin{tabular}{|c||c||c||c|}   \hline   \multicolumn{2}{|c||}{First Cell}&amp;   Second Cell&amp;   Third Cell\\   \hline   \hline   \begin{tabular}{@{}l@{}} Data\\ Data\\ Data\\   \end{tabular}&amp;   \begin{tabular}{@{}l@{}}     Data\\     Data\\     Data\\   \end{tabular}&amp;   \begin{tabular}{@{}l@{}}     Data\\     Data\\     Data\\   \end{tabular}&amp;   \begin{tabular}{@{}l@{}}     Data\\     Data\\     Data\\   \end{tabular}\\    \hline  \end{tabular}