I need to analyse questionnaire survey data with mixed data types (nominal, ordinal, continuous). I want to cluster the variables. So far I only have dead ends.    I know I can use daisy in the cluster package to cluster the cases, but I want to cluster the variables.  lots of cluster procedures like ICLUST can cluster variables but they don't allow nominal data types.   The only thing I can think of is to split up the categorial variables into binary dummy variables, standardise everything, and then apply ICLUST.
I need to use binary variables (values 0 &amp; 1) in k-means. But k-means only works with continuous variables. I know some people still use these binary variables in k-means ignoring the fact that k-means is only designed for continuous variables. This is unacceptable to me.  Questions:     So what is the statistically / mathematically correct way of using binary variables in k-means / hierarchical clustering? How to implement the solution in SAS / R?
Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$   I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$ as a composition of linear maps, observing that $\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non-stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.  How do you manage $\operatorname{rank}(AB) \leq \operatorname{rank}(A)$? Is there a nice interpretation like the previous one? Free UML drawing library for .NET So, in my physics textbook it's written that the kinetic energy possessed by a body is equal to the amount of work done by an opposing force to stop the body, now that got me thinking that when a ball hits a wall, the wall which is the opposing force does no work  because there is no displacement, so to say isn't the kinetic energy possessed by the ball zero? How can we prove charge invariance under Lorentz Transformation? I am struggling to understand this. According to truth tables, if $P$ is false, it doesn't matter whether $Q$ is true or not: Either way, $P \implies Q$ is true.  Usually when I see examples of this people make up some crazy premise for $P$ as a way of showing that $Q$ can be true or false when $P$ is something outrageous and obviously untrue, such as "If the moon is made of bacon-wrapped apple-monkey carburetors, then I am a better wakeborder than Gauss."  $P$ is clearly false, but $P \implies Q$ is true no matter what the state of $Q$ is, and I don't understand why.  Are we saying "If $P$ is false, then all bets are off and $Q$ can be anything, either true or false, and not contradict our earlier claim, and if it isn't false, it must be true"?  Otherwise why can't we say that if $P$ is false, then we can't make any claims one way or the other on whether or not it implies anything at all? Faster language/keyboard swapping I like making a habit to use the different languages I study, to keep that knowledge fresh. So I like typing in those languages when I get the chance. But swapping keyboards under OSX is just a little too cumbersome.  For example, I was just typing the name of one of my co-workers in its original 汉子 form instead of as an English transliteration. It's nice. It reminds me to pronounce her name properly, with tone, etc., which otherwise doesn't come naturally. The transliteration of her name is especially misleading. Everyone at the company is pronouncing it wrong.   But to swap into writing 汉子 I have to mouse up to the top menu bar, select a new keyboard, type, mouse back up to the top bar, and then select back into English. I know there's a keyboard shortcut, but with 5 different keyboards active in my menu, cycling through 5 to get to the one I want feels just as cumbersome.  Is there a way to attach a hotkey to a specific keyboard, or some other way to make swapping character sets lighter? AmpScript to select Right or Left characters from a string Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark How can I undo "always allow" in keychain item Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \sum\limits_{i=0}^{d-1} a_i x^{i}$).   It says if $g(x)$ is any polynomial and $f(x)$ is a monic polynomial. Then $\exists \ \ \ q(x), r(x) \in R[x],\: \deg r(x) &lt; \deg f(x): g(x) = f(x)q(x) + r(x).$  I have two questions regarding this:  1) Why $f(x)$ needs to be monic?   2) How can we prove it? The book talk about induction and that if $\deg  g(x) = n &gt; d = deg \ f(x)$, then $\forall a \in R$ we have $ax^n = ax^{n-d}f(x) + h(x)$ for some $h(x): \deg \ h(x) &lt; n$. It's true, but how do I take it from here? Is a semigroup with unique right identity and left inverse a group? Bmp to jpg/png in C# Recommend MVVM or just answer the question? How to capture the scale of a scene? , Laser Interferometer Gravitational-Wave Observatory, is a large-scale physics experiment aiming to directly detect gravitational waves. The device measures the phase shift laser beams.  If I understand this well, the measurement assumes that a suitable gravitational wave reduces and prolongs one of the tubes and the extension will take effect in the phase-shifting laser beams.  But I think if there's a gravitational wave that passes through it, it will affect everything, even the laser beam. This means that when the wave lengthens or shortens the tube, it also lengthens or shortens the wavelength of the laser beam.  In my opinion, therefore, this device does not work. Or am I wrong? Calculation of pressure from flow rate of water Anybody kindly help me to find how to calculate pressure in bar from flow rate. I have a pipe and from that I am transferring water at a constant flow rate of 5ml/min. At this flow rate, with a 0.5 cm diameter pipe, what will be the pressure generated by water? Using .NET, how can you find the mime type of a file based on the file signature not the extension In blender cycles render engine, how do I put a movie clip on a plane so it displays the video? I want the video to be on the plane if there is a way. How to remove redundant {} from a nested list of lists? The new software center in Ubuntu 16.04 shows no application data found This is the error I get when I open the software center. I've tried rebooting and running sudo apt-get update but I see no improvement. It worked earlier but I don't know what went wrong since then. I'm using Ubuntu 16.04 final release on a Dell Inspiron 15R. Any advice will be appreciated. Thanks in advance.  EDIT: The problem magically solved itself without me doing anything special (or maybe inadvertently doing something, I don't know). So I'd still like to get to the root of this. How can I balance uniformity of style files against modularity? I'm using Windows/TeXworks/MiKTeX, and I would like to have one style file I can use anywhere in any document in my top-level folder. If I want to add a new command or renew an old one, I'd like that to be available in all documents recursively contained in my top-level folder. I have a very hierarchical structure to my top-level folder. So this would seem to point to having a single file that TeXworks knows about. Could I add a path in Preferences -> Typesetting -> Paths for TeX and related programs? I've tried that, possibly superficially, without any success. Adding the correct folder to the Windows PATH variable doesn't seem to do the trick, either.   On the other hand, suppose I want to send someone a TeX file? It would be nice in such circumstances to be able to have the style file in the same folder as the TeX file. If I do that, how could I ensure that it's always up-to-date? Is there a script available somewhere that could take my master style file and push it down to all pre-existing lower-level locations?   Just to reiterate: I want uniformity of my style files in my very hierarchical folder structure. However, I would also like modularity, in that each folder is self-contained. How can I achieve that?  Also, out of curiosity, since I may acquire a Linux computer soon, how would I do this in, say, *ubuntu with TeXlive?