BruceCampbell_ST503_ch8-test2.Rmd

---
title: "NCSU ST 503 Discussion 8"
subtitle: "Probem  8.7 Faraway, Julian J. Linear Models with R CRC Press."
author: "Bruce Campbell"
fontsize: 12pt
output: pdf_document
---

---
```{r setup, include=FALSE,echo=FALSE}
rm(list = ls())
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(dev = 'pdf')
knitr::opts_chunk$set(cache=TRUE)
knitr::opts_chunk$set(tidy=TRUE)
knitr::opts_chunk$set(prompt=FALSE)
knitr::opts_chunk$set(fig.height=5)
knitr::opts_chunk$set(fig.width=7)
knitr::opts_chunk$set(warning=FALSE)
knitr::opts_chunk$set(message=FALSE)
knitr::opts_knit$set(root.dir = ".")
library(latex2exp)   
library(pander)
library(ggplot2)
library(GGally)
```

# 8.8 Gammaray analysis
The gammaray dataset shows the x-ray decay light curve of a gamma ray burst. Build a model to predict the flux as a function time that uses appropriate weights.

First we plot the data 

```{r}
rm(list = ls())
require(nlme)
data(gammaray, package="faraway")
plot(gammaray$time,gammaray$flux, main="time versus flux")
plot(gammaray$time,gammaray$error, main="time versus measurement error")

```


Now we transform the data to be more amenable to linear modelling.  We choose $(flux)^{\frac{1}{8}} \sim log(time)$ as the basis for our modelling efforts. 

We fit a simple linear model here so we have something to compare to when we perform weighted regression;

```{r}
lm.fit <- lm((flux)^.125 ~log(time), data = gammaray)
summary(lm.fit)
plot(log(gammaray$time),(gammaray$flux)^.125, main = TeX("$(flux)^{\\frac{1}{8}} \\sim log(time)$"))
abline(coef(lm.fit),lty=5)

plot(residuals(lm.fit) ~ log(time), gammaray, main ="Residuals versus log(time) for simple linear model")
```

```{r}
wlm.fit <- gls((flux)^.125 ~log(time), data=gammaray, weights = ~(error)^.125)
summary(wlm.fit)

plot(log(gammaray$time),(gammaray$flux)^.125, main = TeX("$(flux)^{\\frac{1}{8}} \\sim log(time)$"))
abline(coef(wlm.fit),lty=5)

plot(residuals(wlm.fit) ~ log(time), gammaray, main ="Residuals versus log(time) for weighted regression")
```