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_quarto.yml
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@ -57,9 +57,9 @@ website:
text: "📝 1 - Advanced Git and Contributing"
- href: "material/2_tue/git/tasks.qmd"
text: "🛠 1 - Git: Exercises"
- href: "material/2_tue/testing/slides.qmd"
- href: "material/2_tue/testing/slides.md"
text: "📝 2 - Testing"
- href: "material/2_tue/CI/missing.qmd"
- href: "material/2_tue/ci/slides.md"
text: "📝 3 - Continuous Integration"
- href: material/2_tue/codereview/slides.qmd
text: "📝 4 - Code Review"

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There are many good ones out there. One we can recommend is the [one from GitHub](https://education.github.com/git-cheat-sheet-education.pdf).

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Also [one from GitHub](https://github.github.io/actions-cheat-sheet/actions-cheat-sheet.pdf)

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---
type: slide
slideOptions:
transition: slide
width: 1400
height: 900
margin: 0.1
---
<style>
.reveal strong {
font-weight: bold;
color: orange;
}
.reveal p {
text-align: left;
}
.reveal section h1 {
color: orange;
}
.reveal section h2 {
color: orange;
}
.reveal code {
font-family: 'Ubuntu Mono';
color: orange;
}
.reveal section img {
background:none;
border:none;
box-shadow:none;
}
</style>
# Learning Goals
- Name and explain common workflows to automate in RSE.
- Explain the differences between the various continuous methodologies.
- Explain why automation is crucial in RSE.
- Write and understand basic automation scripts for GitHub Actions.
- s.t. we understand what `PkgTemplates` generates for us.
Material is taken and modified from the [SSE lecture](https://github.com/Simulation-Software-Engineering/Lecture-Material).
---
# 1. Workflow Automation
---
## Why Automation?
- Automatize tasks
- Run tests frequently, give feedback early etc.
- Ensure reproducible test environments
- Cannot forget automatized tasks
- Less burden to developer (and their workstation)
- Avoid manual errors
- Process often integrated in development workflow
- Example: Support by Git hooks or Git forges
---
## Typical Automation Tasks in RSE
- Check code formatting and quality
- Compile and test code for different platforms
- Generate coverage reports and visualization
- Build documentation and deploy it
- Build, package, and upload releases
---
## Continuous Methodologies (1/2)
- **Continuous Integration** (CI)
- Continuously integrate changes into "main" branch
- Avoids "merge hell"
- Relies on testing and checking code continuously
- Should be automatized
---
## Continuous Methodologies (2/2)
- **Continuous Delivery** (CD)
- Software is in a state that allows new release at any time
- Software package is built
- Actual release triggered manually
- **Continuous Deployment** (CD)
- Software is in a state that allows new release at any time
- Software package is built
- Actual release triggered automatically (continuously)
---
## Automation Services/Software
- [GitHub Actions](https://github.com/features/actions)
- [GitLab CI/CD](https://docs.gitlab.com/ee/ci/)
- [Circle CI](https://circleci.com/)
- [Travis CI](https://www.travis-ci.com/)
- [Jenkins](https://www.jenkins.io/)
- ...
---
# 2. GitHub Actions
---
## What is "GitHub Actions"?
> Automate, customize, and execute your software development workflows right in your repository with GitHub Actions.
From: [https://docs.github.com/en/actions](https://docs.github.com/en/actions)
---
## General Information
- Usage of GitHub's runners is [limited](https://docs.github.com/en/actions/learn-github-actions/usage-limits-billing-and-administration#usage-limits)
- Available for public repositories or accounts with subscription
- By default Actions run on GitHub's runners
- Linux, Windows, or MacOS
- Quickly evolving and significant improvements in recent years
---
## Components (1/2)
- [Workflow](https://docs.github.com/en/actions/using-workflows): Runs one or more jobs
- [Event](https://docs.github.com/en/actions/using-workflows/events-that-trigger-workflows): Triggers a workflow
- [Jobs](https://docs.github.com/en/actions/using-jobs): Set of steps (running on same runner)
- Steps executed consecutively and share data
- Jobs by default executed in parallel
- [Action](https://docs.github.com/en/actions/creating-actions): Application performing common, complex task (step) often used in workflows
- [Runner](https://docs.github.com/en/actions/learn-github-actions/understanding-github-actions#runners): Server that runs jobs
- [Artifacts](https://docs.github.com/en/actions/learn-github-actions/essential-features-of-github-actions#sharing-data-between-jobs): Files to be shared between jobs or to be kept after workflow finishes
---
## Components (2/2)
<img src="https://docs.github.com/assets/cb-25535/mw-1440/images/help/actions/overview-actions-simple.webp" width=95%; style="margin-left:auto; margin-right:auto; padding-top: 25px; padding-bottom: 25px; background: #eeeeee">
From [GitHub Actions tutorial](https://docs.github.com/en/actions)
---
## Setting up a Workflow
- Workflow file files stored `${REPO_ROOT}/.github/workflows`
- Configured via YAML file
```yaml
name: learn-github-actions
on: [push]
jobs:
check-bats-version:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v2
- uses: actions/setup-node@v2
with:
node-version: '14'
- run: npm install -g bats
- run: bats -v
```
---
## Actions
```yaml
- uses: actions/checkout@v3
- uses: actions/setup-node@v2
with:
node-version: '14'
```
- Integrated via `uses` directive
- Additional configuration via `with` (options depend on Action)
- Find actions in [marketplace](https://github.com/marketplace?type=actions)
- Write [own actions](https://docs.github.com/en/actions/creating-actions)
---
## Some Useful Julia Actions
- Find on [gitHub.com/julia-actions](https://github.com/julia-actions/)
```
- uses: julia-actions/setup-julia@v1
with:
version: '1.9'
```
- More:
- `cache`: caches `~/.julia/artifacts/*` and `~/.julia/packages/*` to reduce runtime of CI
- `julia-buildpkg`: build package
- `julia-runtest`: run tests
- `julia-format`: format code
---
## User-specified Commands
```yaml
- name: "Single line command"
run: echo "Single line command"
- name: "Multi line command"
run: |
echo "First line"
echo "Second line. Directory ${PWD}"
workdir: tmp/
shell: bash
```
---
## Events
- Single or multiple events
```yaml
on: [push, fork]
```
- Activities
```yaml
on:
issue:
types:
- opened
- labeled
```
- Filters
```yaml
on:
push:
branches:
- main
- 'releases/**'
```
---
## Artifacts
- Data sharing between jobs and data upload
- Uploading artifact
```yaml
- name: "Upload artifact"
uses: actions/upload-artifact@v2
with:
name: my-artifact
path: my_file.txt
retention-days: 5
```
- Downloading artifact
```yaml
- name: "Download a single artifact"
uses: actions/download-artifact@v2
with:
name: my-artifact
```
**Note**: Drop name to download all artifacts
---
## Test Actions Locally
- [act](https://github.com/nektos/act)
- Relies extensively on Docker
- User should be in `docker` group
- Run `act` from root of the repository
```text
act (runs all workflows)
act --job WORKFLOWNAME
```
- Environment is not 100% identical to GitHub's
- Workflows may fail locally, but work on GitHub
---
## Further Reading
- [What is Continuous Integration?](https://www.atlassian.com/continuous-delivery/continuous-integration)
- [GitHub Actions documentation](https://docs.github.com/en/actions)
- [GitHub Actions quickstart](https://docs.github.com/en/actions/quickstart)
---
# 3. Demo: Automation with GitHub Actions
---
## Setting up a Test Job
- Import [Julia test package repository](https://github.com/uekerman/JuliaTestPackage) (the same code we used for testing)
- Set up workflow file
```bash
mkdir -p .github/workflows
cd .github/workflows
vi format-check.yml
```
- Let's check whether our code is formatted correctly. Edit `format-check.yml` to have following content
```yaml
name: format-check
on: [push, pull_request]
jobs:
format:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v3
- uses: julia-actions/setup-julia@v1
with:
version: '1.9'
- name: Install JuliaFormatter and format
run: |
julia -e 'using Pkg; Pkg.add(PackageSpec(name="JuliaFormatter"))'
julia -e 'using JuliaFormatter; format(".", verbose=true)'
- name: Format check
run: |
julia -e '
out = Cmd(`git diff --name-only`) |> read |> String
if out == ""
exit(0)
else
@error "Some files have not been formatted"
write(stdout, out)
exit(1)
end'
```
- `runs-on` does **not** refer to a Docker container, but to a runner tag.
- Add, commit, push
- After the push, inspect "Action" panel on GitHub repository
- GitHub will schedule a run (yellow dot)
- Hooray. We have set up our first action.
- Failing test example:
- Edit settings on GitHub that one can only merge if all tests pass:
- Settings -> Branches -> Branch protection rule
- Choose `main` branch
- Enable "Require status checks to pass before merging". Optionally enable "Require branches to be up to date before merging"
- Choose status checks that need to pass: `test`
- Click on "Create" at bottom of page.
- Create a new branch `break-code`.
- Edit some file, violate the formatting, commit it and push it to the branch. Afterwards open a new PR and inspect the failing test. We are also not able to merge the changes as the "Merge" button should be inactive.
---
## act Demo
- `act` is for quick checks while developing workflows, not for developing the code
- Check available jobs (at root of repository)
```bash
act -l
```
- Run jobs for `push` event (default event)
```bash
act
```
- Run a specific job
```bash
act -j test
```
---
# 4. Exercise
Set up GitHub Actions for your statistics package. They should format your code and run the tests. To structure and parallelize things, you could use two separate jobs.

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}
</style>
## Learning Goals of the Git Lecture
# Learning Goals
- Refresh and organize students' existing knowledge on Git (learn how to learn more).
- Students can explain difference between merge and rebase and when to use what.

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@ -3,6 +3,7 @@
1. Work with any forge that you like and create a user account (we strongly recommend GitHub since we will need it later again).
2. Push your package `MyStatsPackage` to a remote repository.
3. Add a function `printOwner` to the package through a pull request. The function should print your (GitHub) user name (hard-coded).
4. Use the package from somebody else in the classroom and verify with `printOwner` that you use the correct package.
5. Fork this other package and contribute a function `printContributor` to it via a PR. Get a review and get it merged.
6. Add more functions to other packages of classmates that print funny things, but always ensure a linear history.
4. Start a new Julia environment and use your package through its url: `]add https://github.com/[username]/MyStatsPackage`.
5. Now use the package from somebody else in the classroom instead and verify with `printOwner` that you use the correct package.
6. Fork this other package and contribute a function `printContributor` to it via a PR. Get a review and get it merged.
7. Add more functions to other packages of classmates that print funny things, but always ensure a linear history.

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---
format: revealjs
type: slide
slideOptions:
transition: slide
width: 1400
height: 900
margin: 0.1
---
<style>
.reveal strong {
font-weight: bold;
color: orange;
}
.reveal p {
text-align: left;
}
.reveal section h1 {
color: orange;
}
.reveal section h2 {
color: orange;
}
.reveal code {
font-family: 'Ubuntu Mono';
color: orange;
}
.reveal section img {
background:none;
border:none;
box-shadow:none;
}
</style>
# Learning Goals
- Justify the effort of developing tests to some extent

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############################################################################
#### Execute code chunks separately in VSCODE by pressing 'Alt + Enter' ####
############################################################################
using Statistics
using Plots
using RDatasets
using GLM
##
trees = dataset("datasets", "trees")
scatter(trees.Girth, trees.Volume,
legend=false, xlabel="Girth", ylabel="Volume")
##
scatter(trees.Girth, trees.Volume,
legend=false, xlabel="Girth", ylabel="Volume")
plot!(x -> -37 + 5*x)
##
linmod1 = lm(@formula(Volume ~ Girth), trees)
##
linmod2 = lm(@formula(Volume ~ Girth + Height), trees)
##
r2(linmod1)
r2(linmod2)
linmod3 = lm(@formula(Volume ~ Girth + Height + Girth*Height), trees)
r2(linmod3)
##
using CSV
using HTTP
http_response = HTTP.get("https://vincentarelbundock.github.io/Rdatasets/csv/AER/SwissLabor.csv")
SwissLabor = DataFrame(CSV.File(http_response.body))
SwissLabor[!,"participation"] .= (SwissLabor.participation .== "yes")
##
model = glm(@formula(participation ~ age), SwissLabor, Binomial(), ProbitLink())

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---
editor:
markdown:
wrap: 72
---
# Multiple Regression Basics
## Motivation
### Introductory Example: tree dataset from R
\[figure of raw data\]
*Aim:* Find relationship between the *response variable* `volume` and
the *explanatory variable/covariate* `girth`? Can we predict the volume
of a tree given its girth?
\[figure including a straight line\]
First Guess: There is a linear relation!
## Simple Linear Regression
Main assumption: up to some error term, each measurement of the response
variable $y_i$ depends linearly on the corresponding value $x_i$ of the
covariate
$\leadsto$ **(Simple) Linear Model:**
$$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad i=1,...,n,$$
where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$ are independent
normally distributed errors with unknown variance $\sigma^2$.
*Task:* Find the straight line that fits best, i.e., find the *optimal*
estimators for $\beta_0$ and $\beta_1$.
*Typical choice*: Least squares estimator (= maximum likelihood
estimator for normal errors)
$$ (\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x} \beta_1\|^2 $$
where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$ is the
vector of covariates and $\mathbf{1}$ is a vector of ones.
Written in matrix style:
$$
(\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}) \left( \begin{array}{c} \beta_0\\ \beta_1\end{array}\right) \right\|^2
$$
Note: There is a closed-form expression for
$(\hat \beta_0, \hat \beta_1)$. We will not make use of it here, but
rather use Julia to solve the problem.
\[use Julia code (existing package) to perform linear regression for
`volume ~ girth`\]
*Interpretation of the Julia output:*
- column `estimate` : least square estimates for $\hat \beta_0$ and
$\hat \beta_1$
- column `Std. Error` : estimated standard deviation
$\hat s_{\hat \beta_i}$ of the estimator $\hat \beta_i$
- column `t value` : value of the $t$-statistics
$$ t_i = {\hat \beta_i \over \hat s_{\hat \beta_i}}, \quad i=0,1, $$
Under the hypothesis $\beta_i=0$, the test statistics $t_i$ would
follow a $t$-distribution.
- column `Pr(>|t|)`: $p$-values for the hyptheses $\beta_i=0$ for
$i=0,1$
**Task 1**: Generate a random set of covariates $\mathbf{x}$. Given
these covariates and true parameters $\beta_0$, $\beta_1$ and $\sigma^2$
(you can choose them)), simulate responses from a linear model and
estimate the coefficients $\beta_0$ and $\beta_1$. Play with different
choices of the parameters to see the effects on the parameter estimates
and the $p$-values.
## Multiple Regression Model
*Idea*: Generalize the simple linear regression model to multiple
covariates, w.g., predict `volume` using `girth` and \`height\`\`.
$\leadsto$ **Linear Model:**
$$y_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_p x_{ip} + \varepsilon_i, \qquad i=1,...,n,$$where
- $y_i$: $i$-th measurement of the response,
- $x_{i1}$: $i$ th value of first covariate,
- ...
- $x_{ip}$: $i$-th value of $p$-th covariate,
- $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$: independent normally
distributed errors with unknown variance $\sigma^2$.
*Task:* Find the *optimal* estimators for
$\mathbf{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)$.
*Our choice again:* Least squares estimator (= maximum likelihood
estimator for normal errors)
$$
\hat \beta = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x}_1 \beta_1 - \ldots - \mathbf{x}_p \beta_p\|^2
$$
where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$\_j is the
vector of the $j$ th covariate and $\mathbf{1}$ is a vector of ones.
Written in matrix style:
$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}_1,\ldots,\mathbf{x}_p) \left( \begin{array}{c} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_p\end{array} \right) \right\|^2
$$
Defining the *design matrix*
$$ \mathbf{X} = \left( \begin{array}{cccc}
1 & x_{11} & \ldots & x_{1p} \\
\vdots & \vdots & \ddots & \vdots \\
1 & x_{11} & \ldots & x_{1p}
\end{array}\right) \qquad
(\text{size } n \times (p+1)), $$
we get the short form
$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{X} \mathbf{\beta} \|^2 = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
$$
\[use Julia code (existing package) to perform linear regression for
`volume ~ girth + height`\]
The interpretation of the Julia output is similar to the simple linear
regression model, but we provide explicit formulas now:
- parameter estimates:
$$
(\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
$$
- estimated standard errors:
$$
\hat s_{\beta_i} = \sqrt{([\mathbf{X}^\top \mathbf{X}]^{-1})_{ii} \frac 1 {n-p} \|\mathbf{y} - \mathbf{X} \beta\|^2}
$$
- $t$-statistics:
$$ t_i = \frac{\hat \beta_i}{\hat s_{\hat \beta_i}}, \qquad i=0,\ldots,p. $$
- $p$-values:
$$
p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p}
$$
**Task 2**: Implement functions that estimate the $\beta$-parameters,
the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the
output above.
## Generalized Linear Models
Classical linear model
$$
\mathbf{y} = \mathbf{X} \beta + \varepsilon
$$
implies that
$$ \mathbf{y} \mid \mathbf{X} \sim \mathcal{N}(\mathbf{X} \mathbf{\beta}, \sigma^2\mathbf{I}).$$
In particular, the conditional expectation satisfies
$\mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \mathbf{X} \beta$.
However, the assumption of a normal distribution is unrealistic for
non-continuous data. Popular alternatives include:
- for counting data: $$
\mathbf{y} \mid \mathbf{X} \sim \mathrm{Poisson}(\exp(\mathbf{X}\mathbf{\beta})) \qquad \leadsto \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \exp(\mathbf{X} \beta)
$$
Here, the components are considered to be independent and the
exponential function is applied componentwise.
- for binary data: $$
\mathbf{y} \mid \mathbf{X} \sim \mathrm{Bernoulli}\left( \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})} \right) \qquad \leadsto \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})}
$$
Again, the components are considered to be independent and all the
operations are applied componentwise.
All these models are defined by the choice of a family of distributions
and a function $g$ (the so-called *link function*) such that
$$
\mathbb{E}(\mathbf{y} \mid \mathbf{X}) = g^{-1}(\mathbf{X} \beta).
$$
For the models above, these are:
+----------------------+---------------------+----------------------+
| Type of Data | Distribution Family | Link Function |
+======================+=====================+======================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+----------------------+---------------------+----------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+----------------------+---------------------+----------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \frac{x}{1-x}\right) |
| | | $$ |
+----------------------+---------------------+----------------------+
In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,
$$
\hat \beta = \mathrm{argmax} \prod_{i=1}^n f(y_i \mid \mathbf{X}_{\cdot i}),
$$
which is equivalent to the maximization of the log-likelihood, i.e.,
$$
\hat \beta = \mathrm{argmax} \sum_{i=1}^n \log f(y_i \mid \mathbf{X}_{\cdot i}),
$$
In the Gaussian case, the maximum likelihood estimator is identical to
the least squares estimator considered above.
\[\[ Example in Julia: maybe `SwissLabor` \]\]
**Task 3:** Reproduce the results of our data analysis of the `tree`
data set using a generalized linear model with normal distribution
family.
---
editor:
markdown:
wrap: 72
---
# Multiple Regression Basics
## Motivation
### Introductory Example: tree dataset from R
```{julia}
using Statistics
using Plots
using RDatasets
trees = dataset("datasets", "trees")
scatter(trees.Volume, trees.Girth,
legend=false, xlabel="Girth", ylabel="Volume")
```
*Aim:* Find relationship between the *response variable* `volume` and
the *explanatory variable/covariate* `girth`? Can we predict the volume
of a tree given its girth?
```{julia}
scatter(trees.Girth, trees.Volume,
legend=false, xlabel="Girth", ylabel="Volume")
plot!(x -> -37 + 5*x)
```
First Guess: There is a linear relation!
## Simple Linear Regression
Main assumption: up to some error term, each measurement of the response
variable $y_i$ depends linearly on the corresponding value $x_i$ of the
covariate
$\leadsto$ **(Simple) Linear Model:**
$$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad i=1,...,n,$$
where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$ are independent
normally distributed errors with unknown variance $\sigma^2$.
*Task:* Find the straight line that fits best, i.e., find the *optimal*
estimators for $\beta_0$ and $\beta_1$.
*Typical choice*: Least squares estimator (= maximum likelihood
estimator for normal errors)
$$ (\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x} \beta_1\|^2 $$
where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$ is the
vector of covariates and $\mathbf{1}$ is a vector of ones.
Written in matrix style:
$$
(\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}) \left( \begin{array}{c} \beta_0\\ \beta_1\end{array}\right) \right\|^2
$$
Note: There is a closed-form expression for
$(\hat \beta_0, \hat \beta_1)$. We will not make use of it here, but
rather use Julia to solve the problem.
\[use Julia code (existing package) to perform linear regression for
`volume ~ girth`\]
```{julia}
lm(@formula(Volume ~ Girth), trees)
```
*Interpretation of the Julia output:*
- column `estimate` : least square estimates for $\hat \beta_0$ and
$\hat \beta_1$
- column `Std. Error` : estimated standard deviation
$\hat s_{\hat \beta_i}$ of the estimator $\hat \beta_i$
- column `t value` : value of the $t$-statistics
$$ t_i = {\hat \beta_i \over \hat s_{\hat \beta_i}}, \quad i=0,1, $$
Under the hypothesis $\beta_i=0$, the test statistics $t_i$ would
follow a $t$-distribution.
- column `Pr(>|t|)`: $p$-values for the hyptheses $\beta_i=0$ for
$i=0,1$
**Task 1**: Generate a random set of covariates $\mathbf{x}$. Given
these covariates and true parameters $\beta_0$, $\beta_1$ and $\sigma^2$
(you can choose them)), simulate responses from a linear model and
estimate the coefficients $\beta_0$ and $\beta_1$. Play with different
choices of the parameters to see the effects on the parameter estimates
and the $p$-values.
## Multiple Regression Model
*Idea*: Generalize the simple linear regression model to multiple
covariates, w.g., predict `volume` using `girth` and \`height\`\`.
$\leadsto$ **Linear Model:**
$$y_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_p x_{ip} + \varepsilon_i, \qquad i=1,...,n,$$where
- $y_i$: $i$-th measurement of the response,
- $x_{i1}$: $i$ th value of first covariate,
- ...
- $x_{ip}$: $i$-th value of $p$-th covariate,
- $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$: independent normally
distributed errors with unknown variance $\sigma^2$.
*Task:* Find the *optimal* estimators for
$\mathbf{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)$.
*Our choice again:* Least squares estimator (= maximum likelihood
estimator for normal errors)
$$
\hat \beta = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x}_1 \beta_1 - \ldots - \mathbf{x}_p \beta_p\|^2
$$
where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$\_j is the
vector of the $j$ th covariate and $\mathbf{1}$ is a vector of ones.
Written in matrix style:
$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}_1,\ldots,\mathbf{x}_p) \left( \begin{array}{c} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_p\end{array} \right) \right\|^2
$$
Defining the *design matrix*
$$ \mathbf{X} = \left( \begin{array}{cccc}
1 & x_{11} & \ldots & x_{1p} \\
\vdots & \vdots & \ddots & \vdots \\
1 & x_{11} & \ldots & x_{1p}
\end{array}\right) \qquad
(\text{size } n \times (p+1)), $$
we get the short form
$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{X} \mathbf{\beta} \|^2 = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
$$
\[use Julia code (existing package) to perform linear regression for
`volume ~ girth + height`\]
The interpretation of the Julia output is similar to the simple linear
regression model, but we provide explicit formulas now:
- parameter estimates:
$$
(\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
$$
- estimated standard errors:
$$
\hat s_{\beta_i} = \sqrt{([\mathbf{X}^\top \mathbf{X}]^{-1})_{ii} \frac 1 {n-p} \|\mathbf{y} - \mathbf{X} \beta\|^2}
$$
- $t$-statistics:
$$ t_i = \frac{\hat \beta_i}{\hat s_{\hat \beta_i}}, \qquad i=0,\ldots,p. $$
- $p$-values:
$$
p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p}
$$
**Task 2**: Implement functions that estimate the $\beta$-parameters,
the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the
output above.
```{julia}
r2(linmod1)
r2(linmod2)
linmod3 = lm(@formula(Volume ~ Girth + Height + Girth*Height), trees)
r2(linmod3)
```
## Generalized Linear Models
Classical linear model
$$
\mathbf{y} = \mathbf{X} \beta + \varepsilon
$$
implies that
$$ \mathbf{y} \mid \mathbf{X} \sim \mathcal{N}(\mathbf{X} \mathbf{\beta}, \sigma^2\mathbf{I}).$$
In particular, the conditional expectation satisfies
$\mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \mathbf{X} \beta$.
However, the assumption of a normal distribution is unrealistic for
non-continuous data. Popular alternatives include:
- for counting data: $$
\mathbf{y} \mid \mathbf{X} \sim \mathrm{Poisson}(\exp(\mathbf{X}\mathbf{\beta})) \qquad \leadsto \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \exp(\mathbf{X} \beta)
$$
Here, the components are considered to be independent and the
exponential function is applied componentwise.
- for binary data: $$
\mathbf{y} \mid \mathbf{X} \sim \mathrm{Bernoulli}\left( \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})} \right) \qquad \leadsto \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})}
$$
Again, the components are considered to be independent and all the
operations are applied componentwise.
All these models are defined by the choice of a family of distributions
and a function $g$ (the so-called *link function*) such that
$$
\mathbb{E}(\mathbf{y} \mid \mathbf{X}) = g^{-1}(\mathbf{X} \beta).
$$
For the models above, these are:
+--------------+---------------------+--------------------+
| Type of Data | Distribution Family | Link Function |
+==============+=====================+====================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+--------------+---------------------+--------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+--------------+---------------------+--------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | rac{x}{1-x}\right) |
| | | $$ |
+--------------+---------------------+--------------------+
In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,
$$
\hat \beta = \mathrm{argmax} \prod_{i=1}^n f(y_i \mid \mathbf{X}_{\cdot i}),
$$
which is equivalent to the maximization of the log-likelihood, i.e.,
$$
\hat \beta = \mathrm{argmax} \sum_{i=1}^n \log f(y_i \mid \mathbf{X}_{\cdot i}),
$$
In the Gaussian case, the maximum likelihood estimator is identical to
the least squares estimator considered above.
```{julia}
using CSV
using HTTP
http_response = HTTP.get("https://vincentarelbundock.github.io/Rdatasets/csv/AER/SwissLabor.csv")
SwissLabor = DataFrame(CSV.File(http_response.body))
SwissLabor[!,"participation"] .= (SwissLabor.participation .== "yes")
model = glm(@formula(participation ~ age^2),
SwissLabor, Binomial(), ProbitLink())
```
**Task 3:** Reproduce the results of our data analysis of the `tree`
data set using a generalized linear model with normal distribution
family.