Practical Two.qmd

---
title: "Practical Two"
author: "Katinka Wilkinson"
date: "07/02/2025"
format: html
---

## Question 1

## Part 1: Generating Simulated Data
```{r}
library(ggplot2)
set.seed(1)
x <- 1:100
y <- sin(x/10) + rnorm(100, 0, 0.2^2)
head(y)
```
```{r}
ggplot(mapping = aes(x = x, y = y)) + 
  geom_point() + 
  labs(title = "Raw data")
```

## Part 2: Implementing the LOWESS Algorithm

```{r}
customLowess <- function(x, y, f){
# A function that implements the LOWESS algorithm and returns smoothed values.
# 
# Args:
#   x (numeric vector) - the x values in the data set
#   y (numeric vector) - the y values in the data set
#   f (numeric) - the spac of the smoothing where 0<f<1
# 
# Return:
#   y (numeric vector) - the smoothed y values
  
  # Span
  k <- floor(f*length(x)) # computing the number of neighbouring points used for smoothing

  smoothened_ys <- numeric(length(x))
  
  for (i in 1:length(x)) {
    # Computing weights
    current_x <- x[i]
    distances <- abs(x - current_x)
    all_values <- cbind(x, distances, y)
    sorted_values <- all_values[order(all_values[,2]),]
    k_neighbours <- sorted_values[1:k,]
    d_max <- k_neighbours[k,2]
    weights <- (1-(k_neighbours[,2]/d_max)^3)^3
    
    # Weighted Regression
    W = diag(weights, ncol=k)
    x_neigh <- cbind(1,k_neighbours[,1]) # here, x is a design matrix, meaning that the first column is simply a coulmn on 1s
    y_neigh <- k_neighbours[,3]
    betas <- solve(t(x_neigh) %*% W %*% x_neigh) %*% t(x_neigh) %*% W %*% y_neigh
    smoothened_ys[i] = betas[1,1] + betas[2,1]*current_x
  }
  return(smoothened_ys)
}

smooth_ys <- customLowess(x,y,0.2)

ggplot(mapping = aes(x = x, y = smooth_ys)) +
  geom_point() +
  labs(title = "Data smoothened using custom LOWESS function", x = "x", y = "y")
```
## Question 3

```{r}
lowess_output <- lowess(x, y, 0.2)

ggplot(mapping = aes(x = x, y = lowess_output$y)) + 
  geom_point() +
  labs(title = "Data smoothened using built-in lowess function",x = "x", y = "y")
```