I was going through a document of Western Michigan University to understand the limitations of K-means clustering algorithms. Below is the link:  On slide no 33 its mentioned that K-means has problems when clusters are of different  Sizes Densities Non globular shapes  Since we explore our data and try to figure out the different groups that are present in our data through the k-means clustering algorithm, how would we know that the size of the clusters is different beforehand? We can visualize it if we have two-dimensional data but how it can be done if the data is n-dimensional? Is there any way to examine the data before proceeding to apply k-means. Also, the explanation for the limitation is: if we have different sizes of clusters, k-means will not give the desirable clusters as it tries to partition the clusters equally. But I don't think its always the case. I had applied k-means on the following dataset with k-means++ initialization  It gave me clusters with highly uneven distribution of 4346, 23, 3. I think I am missing some prerequisite steps before proceeding. Please help me clear my doubts. Thanks.
K-means is a widely used method in cluster analysis. In my understanding, this method does NOT require ANY assumptions, i.e., give me a dataset and a pre-specified number of clusters, k, and I just apply this algorithm which minimizes the sum of squared errors (SSE), the within cluster squared error.  So k-means is essentially an optimization problem.  I read some material about the drawbacks of k-means.  Most of them say that:   k-means assumes the variance of the distribution of each attribute (variable) is spherical;  all variables have the same variance; the prior probability for all k clusters is the same, i.e., each cluster has roughly equal number of observations;   If any one of these 3 assumptions are violated, then k-means will fail.  I could not understand the logic behind this statement. I think the k-means method makes essentially no assumptions, it just minimizes the SSE, so I cannot see the link between minimizing the SSE and those 3 "assumptions".
Ubuntu Wifi drivers not working after getting update Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain? Let $R$ be a Euclidean domain, i.e., a ring with a norm $N : R \rightarrow \mathbb N$ such that for any $a,b\in R$ with $b\not=0$, we may write $a = qb + r$ for some $q,r \in R$ with $N(r) &lt; N(b)$. Let $I$ be a prime ideal of $R$, so $R/I$ is a domain. Must $R/I$ in fact be a Euclidean domain?   I know that this is true if we replace "Euclidean domain" with "PID", and false if we replace it with "UFD", so there is no clear intuition to be gained from similar concepts. My first attempts at a proof was to borrow a construction from analysis and define a norm on $R/I$ by  $N(a + I) = \inf\{N(a+i) : i\in I\}$  but this didn't really get me anywhere.   So, this fact true? If so, how can I prove it? If not, whats a counterexample? Let $a&gt;0$, show that for $x&gt;0$, $1&lt;f(x)&lt;2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$  I could take the derivative, find the maximum of the function and conclude. But that involves heady algebra and you have to solve a high degree polynomial. Is there an easier way simply using inequalities? Thanks, much appreciated Whey I try to run the program Layout Editor (available on ) on Ubuntu 14.04.1 64-bit LTS I get the following output in the terminal:  $ layout layout: /usr/lib/x86_64-linux-gnu/libstdc++.so.6: version `GLIBCXX_3.4.20' not found (required by layout)   I tried the command strings on /usr/lib/x86_64-linux-gnu/libstdc++.so.6 as following and here is the input and the output:  $ strings /usr/lib/x86_64-linux-gnu/libstdc++.so.6 | grep GLIBCXX GLIBCXX_3.4 GLIBCXX_3.4.1 GLIBCXX_3.4.2 GLIBCXX_3.4.3 GLIBCXX_3.4.4 GLIBCXX_3.4.5 GLIBCXX_3.4.6 GLIBCXX_3.4.7 GLIBCXX_3.4.8 GLIBCXX_3.4.9 GLIBCXX_3.4.10 GLIBCXX_3.4.11 GLIBCXX_3.4.12 GLIBCXX_3.4.13 GLIBCXX_3.4.14 GLIBCXX_3.4.15 GLIBCXX_3.4.16 GLIBCXX_3.4.17 GLIBCXX_3.4.18 GLIBCXX_3.4.19 GLIBCXX_DEBUG_MESSAGE_LENGTH   How can I resolve this issue I somehow guess that it is related with GCC but cannot figure out how to fix the problem. Furthermore, I have tried a PPA as suggested on  and there is a similar question for an older version of Ubuntu at . However, after I ran sudo update I get numerous warnings and the list I obtained with the strings command did not have new entries after installing gcc 4.9, so I purged the PPA() with ppa-purge. Any help, suggestions on how to proceed are greatly appreciated. Integral of $\sin x \cos x$ using two methods differs by a constant? $$ \int \sin \theta\cos \theta~d \theta= \int \frac {1} 2  \sin 2\theta~ d \theta=-\frac {1} 4  \cos 2\theta$$ But, if I let  $$ u=\sin \theta , \text{ then }du=\cos \theta~d\theta $$ Then $$ \int \sin \theta\cos \theta~d \theta= \int u ~ du =\frac { u^2 } 2 =\frac {1} 2 \sin^2 \theta $$ Since $$ \sin^2 \theta =\frac {1} 2 - \frac {1} 2 \cos 2\theta$$ The above can be written as $$ \int \sin \theta\cos \theta~d \theta= \frac {1} 2 \sin^2 \theta =\frac {1} 2 \left( \frac {1} 2 - \frac {1} 2 \cos 2\theta \right)=\frac {1} 4-\frac {1} 4 \cos 2\theta   $$ Why are the two results differ by the constant $1/4$? Thank you. Apply function conditionally Avoiding numerical cancellation question for $\sin x -\sin y$ for $x \approx y$ When trying to avoid cancellation, one tries to reformulate the equation in order to avoid subtraction between almost equal terms.   In $\sin (x) - \sin (y), x \approx y$ the suggested solution is to reformulate it to $$2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$  But I don't understand how it is any better? The subtraction between $x, y$ remains. Is it because $\lvert\,\sin(x)-\sin(y)\,\rvert\leq \lvert\,x-y\,\rvert$, so cancellation is less likely to happen between $x,y$ than the sines? Delete one of the OS via Ubuntu I'm currently running on dual boot (win 7 and ubuntu). I'd like to know how to delete my Windows 7 OS using ubuntu This is problem 3.18 from Patrick Morandi's Field and Galois Theory.     Give an example of fields $k \subseteq K \subseteq L$ and $l \subseteq L$ for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed in $K$, and $lK = L$, but $l$ is not algebraically closed in $L$.   This question has been asked twice, namely  and , though neither had any answers, and only some inconclusive comments. Normally I wouldn't ask the question again, but after struggling with it for weeks at this point, I emailed Morandi, and he gave me some hints.  He told me that he remembers finding an example, but that he isn't sure. He told me that it would need to be over characteristic $p$ and that the extensions (at least some of them I suppose) could not be separable. Moreover, he told me that $K = k(x)(a)$, where $a$ is a root of an irreducible polynomial $f(x,y) \in k(x,y)$ (I'm not 100% what the root of a two variable polynomial is). Finally, he said that $l$ is an algebraic extension of $k$, where $f$ becomes reducible over $l$. Still however, I can't come up with anything. I was hoping that with these extra hints, someone would be able to figure it out.  He did point out that there is a chance that there is an error in the question, but I haven't managed to prove that such an example cannot exist. How do I disable auto-maximizing of newly launched windows in GNOME? 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); Please add TeX rendering in titles on the iOS app "Error: Invalid signature" when configuring dual boot Ubuntu 14.04 and Windows 10 I have Sanguinare Vampiris. I don't really want to become a vampire. How can I cure myself? Since Ubuntu 11.10 use a new kernel, it's very difficult to have a decent support for virtualization.   VirtualBox doesn't support guest additions for ubuntu 11.10, so I can't copy to and from my ubuntu desktop and windows, which I absolutely require, plus FreeBSD seems not to be able to use DHCP without guest additions.  Virt-manager instead gives an error on launch:  Unable to open a connection to the libvirt management daemon.  Libvirt URI is: qemu:///system  Verify that:  - The 'libvirt-bin' package is installed  - The 'libvirtd' daemon has been started  - You are member of the 'libvirtd' group   unable to connect to '/var/run/libvirt/libvirt-sock', libvirtd may need to be started: Permission denied  Traceback (most recent call last):   File "/usr/share/virt-manager/virtManager/connection.py", line 1146, in _open_thread     self.vmm = self._try_open()   File "/usr/share/virt-manager/virtManager/connection.py", line 1130, in _try_open     flags)   File "/usr/lib/python2.7/dist-packages/libvirt.py", line 102, in openAuth     if ret is None:raise libvirtError('virConnectOpenAuth() failed') libvirtError: unable to connect to '/var/run/libvirt/libvirt-sock', libvirtd may need to be started: Permission denied   The problem is solved by running virt-manager as root, but I don't like that. How do I change permissions to run Virt-Manager as user?  Is there a way to install guest additions on Ubuntu 11.10? Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$ How to create a database directory structure for MySQL? What info does Google backup? Does anyone know what exactly does Google backup (and restore) when you enable that option?  Also, are we able to see this list somewhere? (Maybe hidden n-levels deep in Google Account information?)  Reason I ask is I've had the unfortunate experience of my SD Card dying on me, put in a temporary SD Card and allowed Google to do the auto-restore.  Some things that were restored:   Saved wireless networks Background Some apps - strangely enough only a subset of my apps were restored (some with data, some without). All previous apps were installed via Android Market.   Then finally got my new SD Card, put that in and the following were restored   Saved wireless networks Some apps - again, same problem... only a subset (different subset, even fewer apps). This time no data seemed to have been restored either.   I think it would be very helpful to be able to view/manage a list of what is backed up, and thus would be restored (but I have yet to find this list... if it even exists). Should student or supervisor be corresponding author for publications based on student research? Papers published from an academic project (MSc or PhD) usually have two authors; the first author is the student who mainly conducted the research, and the second author is the professor who supervised the projects.  The corresponding author is the one who take the responsibility of a paper, and thus, some believe that students are not yet prepared to take this responsibility.  Ideally, who should be corresponding author for papers published by MSc or PhD projects?  I understand that it mainly depends on personal agreements and preferences, but I want to know which case is more reasonable from academic logic? When creating the id attributes for HTML elements, what rules are there for the value?