practice.Rmd

---
title: "Feb 6 in-class practice"
author: "Yumou"
date: "2023-02-06"
output: html_document
---

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

## Question 1

Write a function to compute the Euclidean norm of a vector (square root of sum of squared elements in a vector).
Namely, for a vector $x = (x_1, x_2, \ldots, x_p)$, the Euclidean norm of $x$ is
$$|x|_2 = (x_1^2 + x_2^2 + \ldots + x_p^2)^{1/2}.$$
```{r}
EuclideanNorm = function(x){
  res = sqrt(sum(x^2))
  return(res)
}

EuclideanNorm(c(1 : 5))

EuclideanNorm(-10)
```


## Question 2

Write a loop function to calculate the euclidean norm of a function

```{r}
x = c(3:16)

EuclideanNorm(x)

d = 0
for (i in 1:length(x)) {
  d = d + x[i]^2
  cat("Iteration: ", i, ", x = ", x[i], "d = ", d)
}
sqrt(d)
```


## Question 3

Use loop to write he function for Euclidean Norm


```{r}
EuclideanNormLoop = function(x) {
  d = 0
  for (i in 1 : length(x)) {
    d = d + x[i]^2
  }
  
  return(sqrt(d))
}

EuclideanNormLoop(c(-10:10))
```

Write a function for finding variance of data

```{r}
Var = function(x) {
  d = 0
  v = sum(x)
  for (i in 1:length(x)) {
    d = d + (x[i] - v)^2
  }
  ret = 1/(length(x) - 1) * d
  return(ret)
}

Var(c(-10:10))

#R built-in
var(c(-10:10))
```

## Question 4

Write a function to calculate the smapel variances of a vector $x$

```{r}
MyVar = function(x) {
  n = length(x)
  xbar = sum(x) / n
  x.centered = x - xbar
  res = sum(x.centered^2) / (n - 1)
  return(res)
}

MyVar(c(-5:20))

var(c(-5:20))
```

## Question 5 (simulation example)

Suppose we have data from a Normal distribution with mean $\mu = 5$ and variance
$\sigma^2 = 1$. We Want to estimate $\thea = \mu^2$ (true value is 25)

Suppose $X_1, \ldots, X_n$ are the data. Wh have two estimators 

We have 2 estimators

1. $\hat{\theta}_1 = \bar{X}^2$ where $\bar{X} = \sum_{i = 1}^{n} X_i$

2. $\hat{\theta}_2 = \frac{1}{n(n-1)}\sum_{i \neq j} X_i X_j = \frac{1}{n(n-1)} \bigg\{ \bigg(\sum_{i=1}^{n} X_i \bigg)^2 - \sum{i = 1}^{n} X_i}$

```{r}
theta1 = function(x) {
  res = mean(x)^2
  return(res)
}

theta2 = function(x) {
    n = length(x)
    res = (sum(x)^2 - sum(x^2)) / (n * (n - 1))
    return(res)
}

# Simulaiton:
# SUppose the smaple size if 20 (n = 20)
# Simulation from normal distribution with mean 5 and variance 1
# SImulate 1000 times

diff1 = c()
diff2 = c()

for (rep in 1 : 1000) {
  z = rnorm(20, 5, 1)

  diff1[rep] = theta1(z) - 25
  diff2[rep] = theta2(z) - 25
}


hist(diff1)
hist(diff2)

MSE1 = sqrt(mean(diff1^2))
MSE2 = sqrt(mean(diff2^2))

MSE1
MSE2
```