Effect of interchanging response and predictor in linear regression I'm trying to run linear regression models using the R data &quot;Auto.csv&quot;, which has the data for miles per gallon and the displacement. Now I'm trying to run two different linear regression models fit &lt;- lm(mpg ~ displacement)  For this model, I get the result (intercept)     displacement 35.12            -0.06  Then, I interchange the response and predictor fit &lt;- lm(displacement ~ mpg)  For this model, I get the result (intercept)     displacement 447.50          -10.79  For both these models, I'm getting the same results for p-value, r^2 values. Which of these models if better and what metric do we use to ascertain that? Any explanation would really be helpful
What is the difference between linear regression on y with x and x with y? The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). This suggests that doing a linear regression of y given x or x given y should be the same, but I don't think that's the case.   Can someone shed light on when the relationship is not symmetric, and how that relates to the Pearson correlation coefficient (which I always think of as summarizing the best fit line)?
For usability reasons, for example in , the 'viewed' counter should show as  ...instead of... . It's much harder to decipher that there are a million views on the question just by looking at it.  Some people in the comments pointed out internationalization concerns with it. I do think that since there might be a problem with it, that spaces would probably be a better idea. Why does flag marking as helpful/declined not always correlate with moderator action? I flagged a post and (choose one)   my flag was marked as helpful, yet nothing was done my flag was marked as declined, yet the post was deleted/edited/closed   Why? Don't heavier objects actually fall faster because they exert their own gravity? The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space. When are "if" and "whether" equivalent? Proving a direct product of groups is a group I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.    Let $n \geq 1$ be an integer. Suppose $\{G_1,...,G_n\}$ is a finite set of groups. Define the set $G = G_1 \times G_2 \times ... \times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i \in G_i$. Show that the operation $\bullet$: $G \times G \rightarrow G$ given by $(g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.    Proof of associativity: $((g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...g'_n)) \bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) \bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) \bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) \bullet ((g'_1,g'_2,...,g'_n) \bullet (g''_1,g''_2,...,g''_n))$  \  Proof of the existence of an identity For each $G_i \in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) \bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.  \  Proof of the existence of an inverse: Since each $G_i \in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) \bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$. Setting nautilus-open-terminal to launch Terminator rather than gnome-terminal CW complex such that action induces action of group ring on cellular chain complex. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given by the isomorphism of $\pi$ with $\text{Aut}(\overline{X})$. Let $A$ be an abelian group and let $\mathbb{Z}[\pi]$ act trivially on $A$, $a \cdot \sigma = a$ for $\sigma \in \pi$ and $a \in A$.  Now, assume that $X$ is a CW complex. How do I see that $\overline{X}$ is a CW complex such that the action of $\pi$ on $\overline{X}$ induces an action of the group ring $\mathbb{Z}[\pi]$ on the cellular chain complex $C_*(\overline{X})$ such that each $C_q(\overline{X})$ is a free $\mathbb{Z}[\pi]$-module and$$C_*(X; A) \cong A \otimes_{\mathbb{Z}[\pi]} C_*(\overline{X})?$$ Clone hard drive to another, smaller hard drive Is there a good tool/program that allows me to clone a system drive to another, smaller hard drive?  I've bought a new 64GB SSD drive and would like to move my current Windows 7 Professional 64-bit installation to that new drive. My current drive is a 200GB SATA drive with one partition only. Of course there's enough free space to shrink my current partition to less than 64GB. Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$. I have Sanguinare Vampiris. I don't really want to become a vampire. How can I cure myself? Take most recent (or concatenate) edit comments when editing during grace period Recently, I edited  to update a link to a better resource. I clicked "Save" without entering a description of the edit (edit #1).  Then I realised that the question was asked by a new user who may not be able to easily identify the change that was made. So I edited it again, this time only to add a detailed edit description (edit #2).  Since edit #2 was made during the grace period (within 5 minutes of edit #1), the system has merged them into a single edit, but only the edit comment from edit #1 was recorded in the revision history, whereas edit #2's comment was simply discarded/ignored.  I'd suggest that either the most recent (non-default) edit comment is taken or that the comments are concatenated to provide a more useful edit history Hacking linear regression Let's say I perform linear regression on some data that produces the following $R^2$: $\text{RSS} = 1966815.13$ $\text{TSS} = 2145213.91$ $R^2 =  0.083$ Now let's say I bucket (take the average of successive equal size windows) the data and perform the exact same linear regression. This time the $R^2$ is: $\text{RSS} = 1187.56$ $\text{TSS} = 4758.32$ $R^2 =  0.75$ What have I just done here? I clearly haven't improved the explanatory power of the model over the original data, yet I've hacked the $R^2$ by averaging. What's the bias / variance argument of why this occurs? A graphical representation:  Edit: I've reduced the total variance of the data, but the bias remains unaffected. When (if ever) is this a valid statistical approach? Using the same variable as a fixed and random effect in Mixed Effect Models How to mount a VirtualBox shared folder at startup? sudo error, is mode 0777, should be 0440 pm-hibernate freezes and/or fails to resume on Dell XPS 13, 14.04, 3.19 kernel JavaScript RegExp objects I try to write a simple Markdown parser in JavaScript. Therefore I want to check for the [link content][link id] syntax. I use the following code:  data = data.replace( /\[(.*?)\][ ]*\[([0-9]+)\]/g, '&lt;a href="$2"&gt;$1&lt;/a&gt;' );   It works well, but now I want to do this with a RegExp object. So I set up the following bit of code:  var r = new RegExp( '\[(.*?)\][ ]*\[([0-9]+)\]', 'g' ); data = data.replace( r, '&lt;a href="$2"&gt;$1&lt;/a&gt;' );   But it doesn't work. It even says that my regular expression (which works since the first example does a good job) is invalid:     unmatched ) in regular expression   I think it must have to do with some RegExp-object peculiarities I am not aware of. What am I doing wrong and how can the problem be solved? Square roots -- positive and negative It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? I added a new field on Opportunity table to keep a count of records, as well as track the updates made on that record.  This is needed for an external system we use. Opportinity table already has around 5000 records now. I would like to update the new field that I added with some sequence order initially. Say each record to have a unique sequence number like 1,2,3 etc(can't use the Id field for unique). The field "Identifier__c" was added for this purpose with data type number and unique contraint. For that I created a temporary class and trigger. But it is throwing the error.     Review all error messages below to correct your data. Apex trigger OpportunitySAPIdentifierHandler caused an unexpected exception,   contact your administrator: OpportunitySAPIdentifierHandler: execution   of BeforeUpdate caused by: System.DmlException: Update failed. First   exception on row 0 with id 006E00000038FPSIA2; first error:   CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY, OpportunitySAPIdentifierHandler:   execution of BeforeUpdate caused by: System.DmlException: Update   failed. First exception on row 0 with id 006E00000038FPTIA2; first   error: CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY,   OpportunitySAPIdentifierHandler: execution of BeforeUpdate caused by:   System.DmlException: Update failed. First exception on row 0 with id   006E00000038FPSIA2; first error: SELF_REFERENCE_FROM_TRIGGER, Object   (id = 006E00000038FPS) is currently in trigger   OpportunitySAPIdentifierHandler, therefore it cannot recursively   update itself: [] Class.AutoNumber.OpportunityAutoNumber: line 76,   column 1 Trigger.OpportunitySAPIdentifierHandler: line 4, column 1: []   Class.AutoNumber.OpportunityAutoNumber: line 76, column 1   Trigger.OpportunitySAPIdentifierHandler: line 4, column 1: []:   Class.AutoNumber.OpportunityAutoNumber: line 76, column 1   Below is the trigger and class used.  trigger Opportuniwith sIdentifierHandler on Opportunity (before insert,before update) {          If(Trigger.isInsert || Trigger.isUpdate){             AutoNumber.OpportunityAutoNumber(Trigger.new);         }   }   The class is   public with sharing class AutoNumber {       public static void OpportunityAutoNumber (List&lt;Opportunity&gt; Opp) {          if(Trigger.isUpdate || Trigger.isInsert) {              for(Opportunity p: Opp) {                  List&lt;Opportunity&gt; oppList = new List&lt;Opportunity&gt;();                 oppList = [SELECT Id, createdDate, Identifier__c FROM Opportunity where Identifier__c = null and Id != :p.Id limit 100 FOR update];                 for (Opportunity oppL: oppList) {                      if(oppL.Identifier__c == null) {                          List&lt;Opportunity&gt; OpportunityList = new List&lt;Opportunity&gt;([select Identifier__c from Opportunity where Identifier__c != null order by Identifier__c desc limit 1]);                          if (!OpportunityList.isEmpty()) {                               Decimal maxval = OpportunityList.get(0).Identifier__c;                             oppL.Identifier__c = maxval + 1;                             update oppL;                         }                      }                     // integer currentCounter = Integer.valueOf(opp.Identifier__c);                     // oppr.Identifier__c = currentCounter + 1;                     // update oppr;                 }             }         }        }  } Limiting floats to two decimal points I want a to be rounded to 13.95.  &gt;&gt;&gt; a 13.949999999999999 &gt;&gt;&gt; round(a, 2) 13.949999999999999   The  function does not work the way I expected.