---
title: "R Markdown demo"
author: "DP"
date: "September 4, 2016"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
## R Markdown

This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see <http://rmarkdown.rstudio.com>.

When you click the **Knit** button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:

\bigskip
\hrule\smallskip
``my cutoff''
\smallskip
\hrule

\newcommand{\bone}{\mathbbm1}
\def\Rlang{{\bfseries R}}
\setlength{\parindent}{2em}

Everything above the line (``my cutoff'') was created automatically when I created a new file by selecting
\qquad
File > New File > R Markdown
\qquad
in RStudio. I was prompted to provide a title and author.

The file originally contained more about R Markdown, below  my cutoff line.
I replaced that material by some Linear Models stuff then clicked on ``Knit PDF''
in the toolbar.

```{r}
set.seed(10)   # for reproducibility
mydata <- data.frame(y=rnorm(10),
	x1=1:10,x2= 11:20, x3= 0.5*(1:10)-3*(11:20))
out <- lm(y ~ ., data=mydata)
summary(out) 
```

Now let's try to figure out what \Rlang\ has done.
First  determine which matrix \Rlang\ fed to the $qr()$ function:
```{r}
M <- model.matrix(out) # should have out$qr equal to qr(M)
round(cbind(M,mydata)[1:4,],3) # for comparison
```
\noindent
As expected,  \Rlang\ prepended a column of $1$'s to the predictors in $mydata$.
You might want to compare $out\$qr$ with $qr(M)$.

Now extract the matrices for the QR decomposition of the model matrix:
```{r}
Q <- qr.Q(out$qr)
R <- qr.R(out$qr)
# What would you expect 
# round(cbind( Q %*% R,M),3) 
#to show?
round(Q[1:4,],2) # Why are there four columns?
round(R,3) # Why is it 4 by 4 ?
```
\noindent
Notice that R[3:4,] is all zeros. That means that only the first two columns of $Q$
are being used to span the model space; the model matrix has rank $2$.
Let me split both matrices in the way described in the QR.pdf handout:
```{r}
Q1 <- Q[,1:2]
R1 <- R[1:2,1:2]
R2 <- R[1:2,3:4]
# as a check look at
# round(cbind( M, Q1 %*% R1, Q1 %*% R2),3)
```

According to QR.pdf, the fitted vector $\widehat y$  should equal $Q_1Q_1^T y$:
```{r}
# round(out$fitted.values- Q1 %*% t(Q1) %*% mydata$y,4) # all zero?
```
\noindent
The matrix $Q_1Q_1^T$ projects ten-dimensional Euclidean space orthogonally onto the two-dimensional subspace spanned by a column of $1$'s and $x_1,x_2,x_3$.

If you read through QR.pdf you should see how to calculate other parts the summary using only~$y$ and~$out\$qr$.  Homework~1 essentially asks you to add some more calculations to this handout.