I have read many articles about PLS, but I could not understand the mathematical description yet. I know that it is quite similar to principal component regression (PCR), except that it takes into account the direction of the response variable.  Could you please provide for me a simple mathematical explanation e.g the formula and coefficient estimates form?
Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA?  I have looked at many sources online and have not found anything that had the right combination of rigor and accessibility.    I have looked into The Elements of Statistical Learning, which was suggested in a comment on a question asked on Cross Validated, , but I don't think that this reference does the topic justice (it's too brief to do so, and doesn't provide much theory on the subject).  From what I've read, PLS exploits linear combinations of the predictor variables, $z_i=X \varphi_i$ that maximize the covariance $ y^Tz_i $ subject to the constraints $\|\varphi_i\|=1$ and $z_i^Tz_j=0$ if $i \neq j$, where the $\varphi_i$ are chosen iteratively, in the order in which they maximize the covariance.  But even after all I've read, I'm still uncertain whether that is true, and if so, how the method is executed.
SQL-Like Inner Join on IDs in Two Layers in QGIS Using QGIS v2.6.0-Brighton.  This question is not about editing layers geographically in anyway. It is merely trying to find data from two layers as you would in an SQL Query and append an extra ID to the second layer.  I have one polyline layer each identified with two different IDs. There is a point layer also with one of these IDs, I wish to find this corresponding ID and append them as a column to the point layer.  How do I do this without finding each row individually and typing it in manually (i.e. larger scale)? I would prefer to use SQL like syntax but if there is a GUI as well that would be ideal. Happy to install official plugins. How can I get Google to stop thinking I'm a Bot? So, I'm getting the occasional CAPTCHA message, which is irritating but fine, but occasionally Google just blocks my network wholesale. I've submitted automated requests to unblock it, but I have no idea who I can email directly to clarify the issue.  I've run all manner of anti-virus, parsed my router's logs, combed through Fiddler to see if any errant packets were slipping by. Nothing. The only Google searches that happen are made by me through the web-browser.  Who or what do I have to do to get Google to stop falsely flagging my network as a spam-bot? Explicit bijection between $[0,1)$ and $(0,1)$ Proving that $[0,1)$ and $(0,1)$ have the same cardinality (without assuming any previous knowledge) can be done easily using Cantor-Bernstein theorem.  However I'm wondering if someone can build an explicit bijection between these sets.  It's easy to build a bijection between $(0,1)$ and $\mathbb R$, so a bijection from $[0,1)$ to $\mathbb R$ will also fit the bill. The word that always gets me mixed up : JUST I had just arrived at the hotel, checked in and gotten into my room. Just as I was about to unpack, I noticed that one of my bags was missing. At first I thought I had left it back at the airport, but it turned out that I had left it on the taxi.  Here, what's the meaning and role of "just"?  Why do you guys input the word "just" here?  What exactly does the word do grammatically?   Could you help me? The projection of an ellipse is still an ellipse While studying the problem of determining the orbit of a binary star system, my astronomy notes say: "the projection of an ellipse onto a plane is still an ellipse (except the special case in which it's a segment)". How could I prove this?  EDIT: As suggested i'm adding what I know about ellipses. I know the basics of ellipses in the plane, the equation of an ellipse in a plane in cartesia coordinates, the equation of an ellipse in polar coordinates, the fact that ellipses can be obtained by the intersection of planes with a cone. In my geometry course we defined ellipses as the the points of the plane such that the ratio between the distance from a point and a line is constant and less than 1. I also know that the sum of the distances of the points of the ellipse from the foci is constant. I am using PHP with Apache on Linux, with Sendmail. I use the PHP  function. The email is sent, but the envelope has the Apache_user@localhostname in MAIL FROM (example nobody@conniptin.internal) and some remote mail servers reject this because the domain doesn't exist (obviously). Using mail, can I force it to change the envelope MAIL FROM?  EDIT: If I add a header in the fourth field of the mail() function, that changes the From field in the headers of the body of the message, and DOES NOT change the envelope MAIL FROM.  I can force it by spawning sendmail with sendmail -t -odb -oi -frealname@realhost and piping the email contents to it. Is this a better approach?  Is there a better, simpler, more PHP appropriate way of doing this?  EDIT: The bottom line is I should have RTM. Thanks for the answers folks, the fifth parameter works and all is well. Where can I install ubuntu if I have 4 primary partitions? I want to triple boot Windows 7, Backtrack, and Ubuntu. Windows and Backtrack are already installed. The problem is that both of them take 4 partitions: 3 for Windows, and 1 for Backtrack. When I try to make another partition, it says I can't have over 4. How can I install Ubuntu with this problem? Note: The 3 Windows partitions are PQSERVICE, SYSTEM RESERVED, and eMachines. I don't really know if I can delete one. I need to upload my signature in a PDF document, but I do not know how to edit the background of this image.   I wish to remove the bluish-tint background, and make it pure white.   Can anyone help me with this, please? SemanticImport vs Import differ on same CSV file? This is one thing that keeps bugging me, and maybe there's a direct answer. Grammatically, which one is more correct of these two?  Does it make a difference?  I tried not to do that. I tried to not do that. Now the cubs are growing, fed on mom's milk, but mom herself hasn't eaten in five months, and is getting thinner by the day.   What does in mean here, and how would the meaning of that piece change if for were used? Project point onto line in Latitude/Longitude Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D.  Diagram: How to convert from one grading scheme to another? Why don't electromagnetic waves require a medium? How do I disable negative scaling? By default blender allows you to scale past 0 - how do I prevent this?  i.e. it's common in other 3D modelling software to resize a set of vertices against an axis to align them - however this isn't possible when negative scaling is enabled. Number of pairs of strings satisfying the given condition We are provided with a string A (may contain repeated characters as well ). One needs to make all possible permutations of A , and find out how many pairs of strings chosen from these permutations are not similar.  Two strings are similar if we can make both of them exactly same by carrying out at most 1 swap in each of them .  Example :- if we are provided with string abcd  , then for permutation abcd , the permutations which are not similar to it are bcda , bdac , cadb , cdba , dabc , dcab . Similar is the case for other permuations as well. So overall number of desired pairs are 24*6=144 .  I tried solving it for each possible pair by approach mentioned here     But as the permutations can be large enough , I need a better approach of solving this. Front suspension or full suspension mountain bike How to change the shadow color in Blender Cycles? I want to change the color of the shadow casting by a cube to green. While it's very easy to do it in Blender Internal (you select the lamp, go to the Lamp data header and change the shadow's color in the Shadow panel) in Cycles there're no such options. How to achieve it in Cycles? How to determine best cutoff point and its confidence interval using ROC curve in R? I have the data of a test that could be used to distinguish normal and tumor cells. According to ROC curve it looks good for this purpose (area under curve is 0.9):    My questions are:    How to determine cutoff point for this test and its confidence interval where readings should be judged as ambiguous? What is the best way to visualize this (using ggplot2)?   Graph is rendered using ROCR and ggplot2 packages:  #install.packages("ggplot2","ROCR","verification") #if not installed yet library("ggplot2") library("ROCR") library("verification") d &lt;-read.csv2("data.csv", sep=";") pred &lt;- with(d,prediction(x,test)) perf &lt;- performance(pred,"tpr", "fpr") auc &lt;-performance(pred, measure = "auc")@y.values[[1]] rd &lt;- data.frame(x=perf@x.values[[1]],y=perf@y.values[[1]]) p &lt;- ggplot(rd,aes(x=x,y=y)) + geom_path(size=1) p &lt;- p + geom_segment(aes(x=0,y=0,xend=1,yend=1),colour="black",linetype= 2) p &lt;- p + geom_text(aes(x=1, y= 0, hjust=1, vjust=0, label=paste(sep = "", "AUC = ",round(auc,3) )),colour="black",size=4) p &lt;- p + scale_x_continuous(name= "False positive rate") p &lt;- p + scale_y_continuous(name= "True positive rate") p &lt;- p + opts(             axis.text.x = theme_text(size = 10),             axis.text.y = theme_text(size = 10),             axis.title.x = theme_text(size = 12,face = "italic"),             axis.title.y = theme_text(size = 12,face = "italic",angle=90),             legend.position = "none",             legend.title = theme_blank(),             panel.background = theme_blank(),             panel.grid.minor = theme_blank(),              panel.grid.major = theme_line(colour='grey'),             plot.background = theme_blank()             ) p   data.csv contains the following data:  x;group;order;test 56;Tumor;1;1 55;Tumor;1;1 52;Tumor;1;1 60;Tumor;1;1 54;Tumor;1;1 43;Tumor;1;1 52;Tumor;1;1 57;Tumor;1;1 50;Tumor;1;1 34;Tumor;1;1 24;Normal;2;0 34;Normal;2;0 22;Normal;2;0 32;Normal;2;0 25;Normal;2;0 23;Normal;2;0 23;Normal;2;0 19;Normal;2;0 56;Normal;2;0 44;Normal;2;0 Detecting and acting on keyboard direction keys in Java