Tutorial_1-R.Rmd

---
title: "Applied Statistics with R (STAT 215)"
subtitle: "Tutorial 1: The R Language"
author: 
  - "Ziling Ma"
  - "ziling.ma@kaust.edu.sa"
date: "September 29, 2024"
output:
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    highlight: vignette
    toc: true
    number_sections: true
---

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# Assigning Objects {#Ex1}
a. Create an object that stores the value $32 × \frac{41}{8}$.
b. Overwrite your object in (a) by itself divided by $2.33$. Print the result to the console.
c. Create a new object with the value $−8.2 × 10^{−3}$.
d. Print directly to the console the result of multiplying (b) by (c).

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# Vectors {#Ex2}
a. Create and store a sequence of values from 5 to −11 that progresses in steps of 0.3.
b. Overwrite the object from (a) using the same sequence with the order reversed.
c. Repeat the vector c(-1,3,-5,7,-9) twice, with each element repeated 10 times, and store the result. Display the result sorted from largest to smallest.
d. Extract the first and last elements of your vector from (a), storing them as a new object.

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# Matrix operations {#Ex3}
In `R`, use just one line of code to create the following mtrices and assign them to the variables `W`, and `X`, respectively.
\[
W=\begin{pmatrix}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9
\end{pmatrix};
X = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix};
\]
Also, compute $W\odot X$ and $W\cdot X$, such that $\odot$ represents the element-wise multiplication and $\cdot$ represents matrix multiplication.

Given a matrix Y, where
```{r}
 Y <- matrix(c(2,3,1,4),nrow = 2)
 Y
```
How can you compute the inverse matrix of Y?


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# Matrix and functions {#Ex4}
a. Create a matrix $A\in\mathbb R^{3\times3}$, such that
\[
a_{ij} = i + j,\quad\forall i,j.
\]
b. Now, we consider a NON-NUMERIC case. Create $B\in \mathbb R^{3\times3}$, where the element is obtained 
using the function below and the values in matrix A, such that
\[
b_{ij}=\begin{cases}
\text{TRUE}, & a_{ij}\text{ is even}, \\
\text{FALSE}, &\text{otherwise}.
\end{cases}
\]
c. Finally, create $C\in \mathbb R^{3\times3}$, such that
\[
c_{ij}=\begin{cases}
1, & a_{ij} \in \{1,2,3\}, \\
0, &\text{otherwise}.
\end{cases}
\]
d. Confirm the dimensions of the matrix from (a) are $2 × 3$ if you remove any one row.

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# Recursive functions 1 {#Ex5}
Write a function for computing $f(n)=n!,\ (n\in \mathbb N)$. Print the 10 first numbers from the sequence $f$.
Also, for $n=10$, compare your result with the `gamma()` function from `R` (recall that, for any positive integer $n$, $\Gamma(n) = (n-1)!$).


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# Graph function {#Ex6}
Use the functions `outer` y `persp` to graph the following functions\n\n")
$$f(x, y) = 1 - exp(-1 / (x^2 + y^2)), \text{for}\: x, y ∈ [-5, 5]$$
$$f(x, y) = sin(x)cos(y), \text{for}\: x, y ∈ [-\pi/2, \pi/2]$$
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# Linear regression {#Ex7}
Suppose that we model the height of male teenagers as
\[
\mathrm{height}_i = \beta_0 + \beta_1 \cdot \mathrm{age}_i + \epsilon_i,\ \text{s.t. } \epsilon_i \sim \mathcal N\left(0, \sigma^2\right).
\]
Now, for a random sample of size $n=20$, we want to estimate $\boldsymbol \beta = (\beta_0,\beta_1)^\intercal$ as
\[
\boldsymbol \beta = (X^\intercal X)^{-1} X^{\intercal} \boldsymbol y,
\]
where
\[
X = \begin{pmatrix}
1 & \textrm{age}_1 \\
\vdots & \vdots \\
1 & \textrm{age}_{20}
\end{pmatrix},\quad
\boldsymbol y = \begin{pmatrix}
\textrm{height}_1 \\
\vdots\\
\textrm{height}_{20}
\end{pmatrix}.
\]
Compute $\hat {\boldsymbol \beta}$ and plot $\hat {\boldsymbol y} =X\hat {\boldsymbol \beta}$. For this exercise, generate the data set using the following code:
```{r}
set.seed(999)
beta.0 <- 140
beta.1 <- 2
ages <- sample(x = 12:18, size = 20, replace = TRUE)
heights <- round(x = beta.0 + beta.1 * ages + rnorm(n = 10, mean = 0, sd = 2), 
                 digits = 2)
data <- data.frame(heights = heights, ages = ages)
```
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# `R Markdown` and `R Sweave`

* Very elegent and powerful tools to use with `Rstudio`.
* Work very well for homework, reports, presentations, etc.
* `R Markdown` is mainly for generating `.html` files though it can also generate `.pdf` files and many more (this file is generated by `R Markdown`).
* `R Sweave` is like $\LaTeX$ but with `R` chunks within. It generates standard `.pdf` files.



# A fun exercise if you would like to try (Recursive functions) 
Create 3 `N`-element row vectors `p`, `q`, and `r`:
\[
\begin{cases}
\mathtt{p}: &\text{prime number sequence},\\
\mathtt{q}: &\text{Fibonacci sequence},\\
\mathtt{r}: &r_n=\lfloor\sqrt{n\pi}\rfloor.
\end{cases}
\]
Do NOT compute the terms manually (write functions!).
Also, create matrices:
\[
A=\begin{pmatrix}
\mathtt p \\ \mathtt q \\ \mathtt r
\end{pmatrix},
B=\begin{pmatrix}
\mathtt p^\intercal & \mathtt q^\intercal & \mathtt r^\intercal
\end{pmatrix}
\]