First of all, I would like to note that I have read similar topics in CrossValidated but I am not fully satisfied.   I have a dataset which consists of an $N\times M$ binary matrix. 1 means that an action is performed and 0 that it is not.   I apply PCA to the dataset and surprisingly get very good results, especially when I reduce it to only two dimensions. I am looking for the intuition behind performing PCA on such a dataset (i.e. where each attribute contains categorical data; you can give whatever example you think is more understandable) and whether a more appropriate technique can be applied. I am working with MATLAB and I need the data in a clustering friendly form.
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?
Is linear algebra more “fully understood” than other maths disciplines? In , it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood, and hence a natural target to reduce pretty much anything to.   Now, it's evident enough that such a linearisation, if possible, tends to make hard things easier. Find a Hilbert space hidden in your domain, obtain an orthonormal basis, and bam, any point/state can be described as a mere sequence of numbers, any mapping boils down to a matrix; we have some good theorems for existence of inverses / eigenvectors/exponential objects / etc..  So, LA sure is convenient.  OTOH, it seems unlikely that any nontrivial mathematical system could ever be said to be thoroughly understood. Can't we always find new questions within any such framework that haven't been answered yet? I'm not firm enough with G&ouml;del's incompleteness theorems to judge whether they are relevant here. The first incompleteness theorem says that discrete disciplines like number theory can't be both complete and consistent. Surely this is all the more true for e.g. topology.  Is LA for some reason exempt from such arguments, or does it for some other reason deserve to be called the best understood branch of mathematics? I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is:  If $k&lt;n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has $n-$dimensional Lebesgue measure zero.  I've seen some proofs that use Sard's lemma but I'm not really familiar with that subject and I've never seen a proof of said lemma so I'd appreciate a proof that doesn't use it if possible.  Thanks in advance! jQuery animate() Deleting my own question Uninstalling Ubuntu from dual boot / changing bootloader Intuitive interface for Composing Boolean Logic? I'm interested to know how people have, or would construct an interface which simplifies constructing logical boolean conditions? (Yes, it's a database query builder) In my mind a bad interface has them typing lots of AND, NOT IN, OR keywords etc. Or else has a million combo boxes.  I'm wondering if anyone has a good example I could glean ideas from? Or some suggestions?  At the moment I'm considering a split screen with a grid for composing the query and a canvas to display a visual representation of the sets of conditions and how they overlap etc. Like a Venn Diagram. This still gives me a problem with the usability/intuitiveness of the controls on the grid.  EDIT:  I'm very interested in ideas which simplify it for less technical users. However, ideas for a version of the UI for advanced users are very helpful as well. I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is $$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\in\mathbb{N}$$. why is there no non-degenerate 2-forms on 4-sphere? Sphere rolling up a ramp In this question, I did energy conservation, with the states being the initial one and the one where it is at the top , and I got $$ .5 I w^2 + .5 mv^2 = mgh$$ However, in  the answer key they have done, $$ .5 mv^2 + .5 Iw^2 = .5 Iw^2 + mgh$$ Now why would the k.e be turned to p.e but not the rolling energy?? How do I reset a lost password (using recovery mode requires me to type the password)? I'm sure there must be a simple reason for this but I can't seem to be able to import a shp file or a feature class from another file or sde geodatabase into a Feature Dataset without getting an error.  The feature classes come in fine if I don't don't try to import them into a Feature Dataset.  The feature dataset definitely has the same projection as the feature classes I'm trying to import so I don't think that's the problem.  The Error I get is ERROR 000210 and just says Failed to convert and cannot create output and Failed to execute (CopyFeatures).    Any ideas what I may be doing wrong here.  Seems like something so simple.  I'm trying to use the Feature Datasets to better organize my data which I think is there purpose.   Update:  So as you can see below, Jason helped me find the answer.   As suggested by Matt below, I added a new Idea to the ArcGIS Ideas site.  Here is the link to that, so please promote this up if you agree a better error message is needed here: Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group. Prove that $(\mathbb{Z}_n , +)$, the integers  $\pmod{n}$ under addition, is a group.  To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition is closed-- rather, we show these three items):  $(a)$ Associative Law  $(b)$ Existence of Identity  $(c)$ Existence of Inverse  This is what I have been working with so far:     $(a)$ Associative Law:   First, assume that $n$ is a composite number and that $a,b \in \mathbb{Z}$ s.t. $a=nx$ and $b=ny$ for some integers, $x$ and $y$. Then $a+nx+nx=n(x+y)$.     So, this is where I am stuck for the Associative Law. I'm not sure if I am headed down the correct path to show that $\mathbb{Z}_n$ is associative.      $(b)$ Existence of Identity:   Generally, when showing the existence of an identity we can use this information for addition :  $a + 0 = 0 + a =a$.    I am unsure how to insert the modular portion of the definition into the existence of this identity. I realize that for addition, we generally look at $0$ (rather than $1$ for multiplication).     $(c)$ Existence of Inverse: I need to show that for each element $a \in \mathbb{Z}_n$ that $a^{-1}=-a$. Again, I do not know how to insert the modular portion of definition into this aspect of the proof.   My question is how to incorporate this modular information into the proof that this is a group. We have not covered the integers modulo $n$, binary operations, nor groups in my intro to proofs course. I am trying to read my text to figure out how to prove this, but have a difficult time reading and interpreting math speak. We are given with  $$S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0  \}$$ Now observe that $-\sqrt n\leq x\leq\sqrt n $ and similarily, $-\sqrt n\leq y\leq\sqrt n $. This means that $-\left \lfloor{\sqrt n}\right \rfloor\leq x\leq \left \lfloor{\sqrt n}\right \rfloor$ and  $-\left \lfloor{\sqrt n}\right \rfloor\leq y\leq \left \lfloor{\sqrt n}\right \rfloor$. Through this result one can easily write code and figure out the cardinality of set $S_i$ but I was wondering whether an expression can be found that would allow one to directly compute $|S_i|$. I am running QGIS 2.18.14 (QGIS code revision: ff83b9a479) on a windows 7 machine, it comes with GRASS 7.2.2 (Code Revision r71496), but it does not come with r.in.lidar. I have noticed QGIS 2.18.16.1 has the same problem and it shares the same GRASS 7.2.2 (Code Revision r71496).  I know there were some issues with it and later solved as I read , I also certailny used it short ago in 2017, so I wonder which version should I go for before starting the always risky attempt of a new installation. Why was Dumbledore worried by the cut in Harry's arm? Making yogurt at home is a fairly straight-forward process.  Heat milk to 180F (82C), cool that milk to at least 110F (43C), add live cultures, usually in the form of yogurt with said cultures, incubate, then chill.   There are several variables in the process, some of which are: The fat content of the milk (I typically use whole), how long to hold the milk during the initial heat step, how long it takes to cool the milk, the temperature of the incubation step, and the length of time for incubation, to name a few.  While length of time for incubation, and temperature at this stage, clearly contribute to sourness, I am trying to nail down the variable that contributes to a creamy texture.  I'm not referring to breaking or clumping, as described in .  My yogurt comes out smooth.    My initial hypothesis is that holding the milk for a length of time at the initial 180F(82C) (rather than immediately removing from heat and/or chilling) has an impact on final texture. Any thoughts or experience that might inform my hypothesis? The direction of static friction? In my book, the following are described to be the steps that one must follow if he wishes to find the direction of static friction force acting on an object:  &quot;(1) Draw the F.B.D(free body diagram) of the object in question with respect to the other object on which it is kept. (2) Include the pseudo force also, if the contact surface is accelerating. (3) Decide the direction of the resultant force and resolve this force into two components; one along the surface of contact and the other along the normal to this surface. (4) The direction of static friction is opposite to the component of the result force along the contact surface.&quot;  Firstly, are these steps correct? Will I always obtain the correct answer by following these steps? Secondly, consider the following problem:  In questions like these, when I try to find the resultant force and then assign the direction to static friction, I find that my answer is incorrect! Are there some other steps to follow when I know that the object is definitely accelerating in one direction or another? Edit 1: I approached the above question by doing exactly what my book suggests; first, I identified all the forces acting on the smaller block, being the normal force applied by the block A and its weight mg downward. Adding these two forces, I get another force that actually acts on the body like so:  I wonder if I'm not supposed to take either of the contact forces into consideration when drawing the F.B.D to figure out the direction of static friction acting on the body. Also, are there other steps for finding the direction of kinetic friction? Edit 2: From Judge's answer, I have been able to figure out my mistake, it being that I completely missed step (3) and didn't resolve the resultant force into components! Sheesh. Sorry! Much thanks in advance! :) Regards. How do I create a WiFi hotspot sharing wireless internet connection (single adapter)? Why do airplanes experience negligible Coriolis force while bullets experience the Coriolis force in long range shooting, even though the mass of airplane is much bigger than a bullet? Occasionally, I've come across a webpage that tries to pop open a new window (for user input, or something important), but the popup blocker prevents this from happening.  What methods can the calling window use to make sure the new window launched properly?