I see that in PCA the first principal component maximizes the variances amongst all the points within the data set. What exactly does this mean, what does it show and what does every other principal component thereafter tell me?  I read these really nice simple explanation on PCA that helped me understand PCA as a whole:     To decide how many eigenvalues/eigenvectors to keep, you should   consider your reason for doing PCA in the first place. Are you doing   it for reducing storage requirements, to reduce dimensionality for a   classification algorithm, or for some other reason? If you don't have   any strict constraints, I recommend plotting the cumulative sum of   eigenvalues (assuming they are in descending order). If you divide   each value by the total sum of eigenvalues prior to plotting, then   your plot will show the fraction of total variance retained vs. number   of eigenvalues. The plot will then provide a good indication of when   you hit the point of diminishing returns (i.e., little variance is   gained by retaining additional eigenvalues).   However, it's still unclear to me as to really what each PC means and what PC2 now represents and PC3, etc.
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?
Help with completing the proof: Any unbounded sequence is either infinitely large or has a convergent subsequential limit Given $\{x_n\}$ is an unbounded sequence, prove:   $$ \lim_{n\to\infty}x_n = \infty\ \  \text{or}\ \ \lim_{n_k \to \infty} = A $$   In other words any unbounded sequence either infinitely large or has a convergent subsequence.     Everything before udpate section is wrong.  Part 1. Proving $|x_n| &gt; M \implies \lim_{n\to\infty} x_n = \infty$  $\Box$ Start with the definition of an unbounded sequence: $$ \forall M &gt; 0 \in \Bbb R \ \exists N \in \Bbb N: \forall n &gt; N\implies |x_n| &gt; M $$  Suppose that: $$ \lim_{n\to\infty}x_n \ne \infty $$ That means: $$ \forall n\in\Bbb N\ \exists C &gt;0 \in\Bbb R : |x_n| &lt; C $$  Now taking $M^\prime = \max\{M, C\}$ we have that: $$ \begin{cases} \begin{align} \exists N \in \Bbb N : &amp;\forall n&gt;N \implies &amp;|x_n| &gt; M^\prime \\ &amp;\forall n \in \Bbb N: &amp;|x_n| &lt; M^\prime \end{align} \end{cases} $$  So we have arrived to a contradiction which means the assumption is wrong and: $$ \lim_{n\to\infty}x_n = \infty $$ $\Box$     Part 2. It's not clear to me how to prove that if unboundedness does not imply infinity limit then it must imply the existence of a convergent subsequence.   Two questions is my mind:   Is part 1 correct?  How do I proceed with the second part?     I'm using the following definition. $$ \lim_{n\to\infty}x_n = +\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &gt; \epsilon \\ \lim_{n\to\infty}x_n = -\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &lt; -\epsilon \\ $$ A sequence is considered infinitely large when: $$ \lim_{n\to\infty}x_n = \infty\ \ \text{when}\ \ \lim_{n\to\infty}x_n = + \infty \ \ \text{or} \ \ \lim_{n\to\infty}x_n = - \infty $$    Update.   Looks like i've messed up a lot of things. I will try once again. As shown in comments and answers the statement holds only in case: $$ \lim_{n\to\infty} |x_n| = \infty \iff \forall \epsilon &gt; 0 \exists N \in \Bbb N: \forall n &gt; N \implies |x_n| &gt; \epsilon $$  Consider the following: $$ \lnot P = \lnot\left(\lim_{n\to\infty}|x_n| = \infty \right) \iff \exists \epsilon &gt; 0\ \forall N_1 \in \Bbb N : \exists n &gt; N_1 \land |x_n| &lt; \epsilon $$  But on the other hand it is given that $x_n$ is unbounded: $$ Q = \forall M &gt; 0\ \exists N_2 \in \Bbb N : |x_{N_2}| &gt; M $$  Construct a negative expression for boundedness: $$ \lnot P = \exists M &gt; 0\ \forall N_2 \in \Bbb N : |x_{N_2}| &lt; M $$  If $S = \lnot P \implies \lnot Q$ then $S = P \lor \lnot Q$  Let $\epsilon = M$, choose $N = \max\{N_1, N_2\}$ then both statements are true and: $$ \exists \epsilon &gt; 0\ \forall N = \max\{N_1, N_2\}: \exists n &gt; N \land |x_n| &lt; \epsilon $$  Now either $P$ is true, which would mean $\lim |x_n| = \infty$, or $\lnot Q$ is true which would mean the sequence is bounded and hence contains a convergent subsequence. Converting string of 1s and 0s into binary value I'm trying to convert an incoming sting of 1s and 0s from stdin into their respective binary values (where a string such as "11110111" would be converted to 0xF7). This seems pretty trivial but I don't want to reinvent the wheel so I'm wondering if there's anything in the C/C++ standard libs that can already perform such an operation? Bluetooth headset never gives me sound options invalid EFI file path on windows 8 Cannot change backlight brightness Ubuntu 14.04 I have a 64-bit machine that has Windows 7 32-bit and Ubuntu 32-bit installed as a dual boot setup. Can I run 64-bit software on top of these 32-bit OSes, or are they a "bottleneck" limiting the true capability of the underlying hardware? Reduce file size without losing quality I would like to save files in Photoshop but keep them as small as possible.   I use Save for Web (and Devices) and save as a JPEG. Here are the settings I used:     I really do not want to lower the quality settings because I use text and sometimes I see artifacts if the setting is too low.   This produces file sizes of several hundred kilobytes depending on size of image. While this isn't a huge issue these images are used for the web and smaller is better. What are techniques without reducing quality to reduce the file size?   Update:  To add some more information here:   Many of the images I use and create get sent via email. This severally limits what I can do with placing of text using HTML and CSS over the image.   Also I use a variety of fonts. Some I could use for the web using CSS's @font-face but you can run into large file sizes and in some cases you cannot upload the font to the web legally. I could use a service like  but again many fonts I use are not available with Typekit.   Here is an example image: How to delay a command block I have created a short command block stack, one block teleports the player and the other plays block.piston.extend. My problem is that it teleports the player before the player can hear the sound. I would like to know if there is a way to delay the command from inside that command block. I have looked at other solutions but they all involve some complex scoreboard timer thing. If you could give me a solution that doesn't use that system or maybe you could explain it better.  This is the command I need to delay:  /tp @p -586.634 64 -430.697   If anyone could help I would really appreciate it, so thanks in advance! Find $\lim\limits_{x \to \infty} \frac{x}{(\log{x})^n}$ Find $\displaystyle \lim_{x \to \infty} \dfrac{x}{(\log{x})^n}$.   My book says that $\displaystyle \lim_{x \to \infty} \dfrac{(\log{x})^n}{x} = \lim_{y \to \infty} \dfrac{y^n}{e^y} = 0$, but I don't see how that is true. I have data with a column containing a number value, and I want to divide the value of each feature by the total sum of all the values in the column (to get a percentage).  So I would like to know if there is some simple syntax in the QGIS field calculator to get the sum of a column, preferably without resorting to for loops and what not.    I have a shapefile where feature A has the value X in column B. How can I summerize the values for all features in column B in order to divide value X to get a percentage for each feature? Cardinality of the set of all (real) discontinuous functions This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6.     (Show that) The cardinality of the set of all discontinuous functions is $2^{2^{\aleph_0}}$. [Hint: Using exercise 2.5, show that $|\mathbb{R}^\mathbb{R}-C|=2^{2^{\aleph_0}}$ whenever $|C|\leq 2^{\aleph_0}$.]   Exercise 2.5, as referred to in the hint, provides the following result:     For $n&gt;0$, $n \cdot 2^{2^{\aleph_0}} = \aleph_0 \cdot 2^{2^{\aleph_0}} = 2^{\aleph_0} \cdot 2^{2^{\aleph_0}} = 2^{2^{\aleph_0}} \cdot 2^{2^{\aleph_0}} = (2^{2^{\aleph_0}})^n = (2^{2^{\aleph_0}})^{\aleph_0} = (2^{2^{\aleph_0}})^{2^{\aleph_0}}=2^{2^{\aleph_0}}$.   The only way I could answer this question was by making use of two theorems stated later  in the same textbook. First, it is easy to prove that $|\mathbb{R}^\mathbb{R}-C|$ is an infinite set, since if it was finite we would have \begin{equation}2^{2^{\aleph_0}}=|\mathbb{R}^\mathbb{R}|=|\mathbb{R}^\mathbb{R}-C|+|C| \leq n+2^{\aleph_0}=2^{\aleph_0}&lt; 2^{2^{\aleph_0}},\end{equation}which is a contradiction. Now I make use of a theorem requiring the axiom of choice: For every infinite set $S$ there exists a unique aleph $\aleph_\alpha$ such that $|S|=\aleph_\alpha$. So I let $|\mathbb{R}^\mathbb{R}-C|=\aleph_\alpha$ for some ordinal $\alpha$. Now assume $\aleph_\alpha &lt; 2^{2^{\aleph_0}}$. Here I make use of another theorem, which i am not sure I am allowed to do: For every $\alpha$ and $\beta$ such that $\alpha \leq \beta$, we have $\aleph_\alpha+\aleph_\beta=\aleph_\beta$. So I basically then say either $|\mathbb{R}^\mathbb{R}-C|+|C|=\aleph_\alpha$ or $|\mathbb{R}^\mathbb{R}-C|+|C|=2^{\aleph_0}$, depending on whether $\aleph_\alpha \leq 2^{\aleph_0}$ or vice-versa. The assumption I am making is that $2^{\aleph_0}=\aleph_1$, which as I understand it is basically equivalent to the continuum hypothesis, as we are saying that the least uncountable number is $\aleph_1$ (but by the previous theorem requiring the axiom of choice this seems reasonable?). So anyway assuming I can do that, we again have a contradiction since then we are saying $|\mathbb{R}^\mathbb{R}|=2^{\aleph_0}&lt;2^{2^{\aleph_0}}$ or $|\mathbb{R}^\mathbb{R}|=\aleph_\alpha &lt; 2^{2^{\aleph_0}}$.  My main problem with my proof is that I am not making use of the result given in exercise 2.5 as suggested in the hint. Can anyone help with a proof making use of this result (stated in the second block quote above). How to use Alt+Shift to switch keyboard layouts? I recently upgraded to Ubuntu 13.10, with the new keyboard handling system. The little "switch keyboard" option does not let me set it to Alt+Shift. How can I get this keyboard binding, as I have absolutely no interest in using the Mac-esque Super+Space combo? What is the best way to store fresh herbs like parsley or cilantro?  I find I have about the same luck with them whether they are left in a plastic bag or I put them in a cup of water. At what age is the Force detectable in children? How do you join on the same table, twice, in mysql? Ratios in Regression, aka Questions on Kronmal Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading up on this, leading eventually to Kronmal 1993.  I want to make sure that I'm correctly interpreting his suggestions on how to model these.   For a model with a ratio with the same denominator on both the dependent and independent side: $ Z^{-1}Y = Z^{-1}1_n\beta_0 + Z^{-1}X\beta_X + \beta_Z + Z^{-1}\epsilon $   Regress dependent ratio on the (inverse) denominator variable in addition to the other ratios Weight by the (inverse) denominator variable  For a model with dependent variable as a ratio: $ Y = \beta_0 + \beta_XX + Z1_n\alpha_0 + ZX\alpha_X + Z^{-1}\epsilon $   Regress numerator by original variables, denominator, and denominator times original variables [what about categorical variables?] Weight by (inverse) denominator  For model with only independent variable ratios: $ Y = \beta_0 + X\beta_X + Z^{-1}1_n\beta_{Z^{-1}} + W\beta_W  + Z^{-1}W\beta_{Z^{-1}W} + \epsilon $   Include numerator and (inverse) denominator as main effects, ratio as interaction term.    Are my interpretations here correct? I'm hoping someone can give me a nudge in the right direction...     Let $F$ be a finite field, and let $f(x)$ be a nonconstant polynomial whose derivative is the zero polynomial.  Prove that $f$ cannot be irreducible over $F$.   I've got that every root of $f$ is a multiple root and that for $F=\mathbb{F}_{p^r}$, the exponent of every term of $f$ is a multiple of $p$. True displacement in EEVEE? I have a Blender 2.8 scene with animated true displacement. It is created using material nodes and works fine in Cycles. However, EEVEE doesn't displace my meshes truly, only normals are changed. Is it possible and how to solve the problem now or in the future? Is there something similar to sprintf() in C#?  I would for instance like to convert an integer to a 2-byte byte-array.  Something like:  int number = 17; byte[] s = sprintf("%2c", number); Short story - "One man's meat is another man's poison"