---
title: "Linear Models and Their Generalizations"
subtitle : "Code Chapter 1: Linear gaussian regression"
author: "Katia Meziani"
output:
#  pdf_document:
#    toc: yes
html_document:
code_folding: hide
css: ./style.css
df_print: paged
highlight: tango
number_sections: no
theme: flatly
toc: yes
toc_float:
collapsed: no

---

```{r  setup, include=FALSE}
knitr::opts_chunk$set(warning = FALSE, message = FALSE)
```









#  Motivations

<span style="color: blue;">
    <span class="underline">**The Ozone Example**</span> (With <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span> )
</span>

This dataset (Air Breizh, Summer 2001) contains 112 observations and 13 variables.

```{r}
library(rmarkdown)
ozone1 <- read.table("ozone1.txt", header = TRUE, sep = " ", dec = ".")
paged_table(ozone1)
```

Let's correct the incorrect declaration of variables:



```{r}
library(dplyr)
ozone1$Date <- as.Date(as.factor(ozone1$Date), format = "%Y%m%d")
ozone1$vent <- as.factor(ozone1$vent)
ozone1$pluie <- as.factor(ozone1$pluie)
ozone1 <- ozone1 %>% mutate(across(where(is.character), as.numeric))
ozone1 <- ozone1 %>% mutate(across(where(is.integer), as.numeric))
str(ozone1)
```







# 1. Linear regression model

##  Simple linear model




Let's plot this line (here with  <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span> ) is called the least squares line.

```{r,   out.width = "60%", fig.align = 'center', message = FALSE, warning = FALSE}
library(ggplot2)
library(plotly)
p1 <- ggplot(ozone1) +
  aes(x = T12, y = maxO3) +
  geom_point(col = 'red', size = 0.5) +
  geom_smooth(method = "lm", se = FALSE)
ggplotly(p1)
```


With  <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span> , we can compute $\widehat {\boldsymbol{\beta}}=(\widehat{\beta}_0, \widehat{\beta}_1)$

```{r}
library(kableExtra)
mod <- lm(maxO3 ~ T12, data = ozone1)
mod$coefficients %>% kbl(col.names = "Estimation")
```











# 2. Ordinary least squares estimator (OLSE)







##  Declaration of the model with Under <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>

Let's revisit the ozone example and, for this chapter, select only the numerical regressors with  <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>

```{r}
names(ozone1)
Ozone <- ozone1[, 2:12]
names(Ozone)
```



Under <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>   ,we can declare the linear model in two ways: either by naming the variables or by using a `.`.

```{r}
mod <- lm(maxO3 ~ T9 +
  T12 +
  T15 +
  Ne9 +
  Ne12 +
  Ne15 +
  Vx9 +
  Vx12 +
  Vx15 +
  maxO3v, data = Ozone)
mod <- lm(maxO3 ~ ., data = Ozone)
```

Now, let's display the estimated values $\widehat{\boldsymbol{\beta}}$ of $\boldsymbol{\beta}$.

```{r}
coefficients(mod)
```




# 3. Residuals analysis



```{r}
summary(mod$residuals)
```





We can display Residual standard error with <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>
```{r}
cat("Residual standard error:", summary(mod)$sigma, "on", summary(mod)$df[2], "degrees of freedom\n")
```











# 4. Inference in the Gaussian linear regression model


##  Confidence intervals/regions







<span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>  Display the confidence interval for $\beta_j$


```{r}
mod <- lm(maxO3 ~ ., data = Ozone)
cbind(mod$coefficient, confint(mod, level = 0.95))
```


<span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>  Display the confidence interval for
$\sigma^2$


```{r}

n <- length(Ozone$maxO3)
p <- length(coef(mod))
alpha <- 0.05

sigma2_hat <- summary(mod)$sigma^2

IC_sigma2 <- sigma2_hat * c((n - p) / qchisq(1 - alpha / 2, n - p),
                            (n - p) / qchisq(alpha / 2, n - p))

cat("estimation of sigma : ", sigma2_hat, "\n")
cat("Intervalle de confiance pour sigma^2 (95%) : [", IC_sigma2[1], ", ", IC_sigma2[2], "]\n")
```














# 5. Hypothesis testing

##  Test: $\mathscr{H}_0:\,c^\top{\beta}=a$




Declare our linear model and display with  <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>

```{r}
library(kableExtra)
mod <- lm(maxO3 ~ ., data = Ozone)
summary(mod) %>% coefficients() %>% kbl()
```




##  Global Fisher test





Under <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span> ,  the `summary` displays the Global Fisher test


```{r, eval=FALSE}
F-statistic: 32.67 on 10 and 101 DF,  p-value: < 2.2e-16
```



##  Fisher test for nested models




<span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>  Consider 2 nested models
$$
\left\{\begin{array}{l}
\text{mod}_1:\, \text{maxO3=T9+Ne9+Vx9+maxO3v}+\epsilon,\\
\text{maxO3=T9+T12+Ne9+Ne12+Vx9+Vx12+maxO3v}+\epsilon
\end{array}\right.
$$
and display the nested Fisher test
$$
\left\{\begin{array}{l}
\mathscr{H}_0: \, \text{mod}_1 \text{ is adequat}\\
\mathscr{H}_1: \, \text{mod}_2 \text{ is adequat}
\end{array}\right.
$$

```{r}
mod_1<-lm(maxO3~T9+Ne9+Vx9+maxO3v,data=Ozone)
mod_2<-lm(maxO3~T9+T12+Ne9+Ne12+Vx9+Vx12+maxO3v,data=Ozone)

anova(mod_1,mod_2)
```




# 6. $R^2$-coefficient




Let display these coefficient with <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span>

```{r}
cat("Multiple R-squared:", summary(mod)$r.squared, ", Adjusted R-squared:", summary(mod)$adj.r.squared, "\n")

```






# 7. Prediction


## 7.1. Fitted values




Note that with <span class="solution"><span style="color: blue;">**Ⓡ**</span> </span> , we can display the fitted values $\widehat Y_i$

```{r}
haty<-predict(mod,data=Ozone)
# or equivalently
#haty<-mod$fitted.values
y<-Ozone$maxO3
cbind.data.frame(y,haty)%>%rmarkdown::paged_table()
```