---
title: "Lab #1:  Practice in R"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
set.seed(1)
```

The purpose of this lab is to get you comfortable working in R and trying a few things from the first lecture.

## Simulating data:

We will simulate something like the weight vs. age data from a cubic function and add some noise and then apply a few methods.

1. Write a function in R to input the value  (age) and output, “noise-free” weight: 
$f(x) = -0.0004x^3 + 0.025x^2 + 2x + 50$
```{r}
```

2. Next we will simulate some values for $x$. Use the sample function to generate a random sample of integers between 40 and 80 of size 1000 (you will need the replace=TRUE option to make sure the numbers can be resampled otherwise you will run out of values quickly). Don’t forget that you can type `?sample` to get help as to how to use the function.
```{r}
```

3. Now generate a vector of the output of your function to generate the noise-free weights for each of the 1000 individuals.
```{r}
```

4. Use `rnorm` to generate some normally distributed noise with mean 0.0 and sd of 10.0 for each of the  values.
```{r}
```

5. Add the noise-free weights to the noise to get your simulated outcome data.
```{r}
```

6. Put these together in a dataframe with columns for age, weight (the observed outcome with noise), trueFx (the noise-free weight value), and noise.
```{r}
```

7. Generate a plot of weight against age.
```{r}
```

## Applying smoothing kernels:
1. Fit a nearest neighbors curve with ksmooth using a bandwidth of 10 and the `box` kernel.
```{r}
```

2. Fit another curve this time using the `normal` kernel.
```{r}
```

3. Plot the data with the two fitted curves and compare them. Hint: look at the object you have generated with ksmooth (i.e. type `name_of_the_object` or `print(name_of_the_object)`. Also, try `names(name_of_the_object)`. Use the command `lines` to plot the curves.
```{r}
```

4. Fit another 2 curves with the "normal" kernel using bandwidths of size 5, 10, and 20. How do they compare?
```{r}
```


## Fitting linear models:
1. Fit a linear regression to the data using the `lm` command.
```{r}
```

2. Run `summary(your_linear_model_name)` to get an idea of the fit
```{r}
```

3. Now fit quadratic and cubic models: you will need the `I` function to set quadratic and cubic terms in the regression, e.g. `I(x^2)`
```{r}
```

4. Add the fitted models to the plots. For the linear model, use the `abline` command. For the polynomial fits, you can use the `curve` command (you will need the option `add = TRUE`).
```{r}
```

5. Which curve do you prefer? How would you perform a hypothesis test for this?
```{r}
```

## Practice installing an R package
6. R packages are easy to install using the `install.packages` command. Try installing the `e1071` package. We'll use this package later in the course for fitting support vector machines.
```{r}
```