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152
material/2_tue/testing/slides.md
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@ -34,11 +34,11 @@ slideOptions:
# Learning Goals
- Justify the effort of developing tests to some extent
- Get to know a few common terms of testing
- Work with the Julia unit testing package `Test.jl`
Material is taken and modified, on the one hand, from the [SSE lecture](https://github.com/Simulation-Software-Engineering/Lecture-Material), which builds partly on the [py-rse book](https://merely-useful.tech/py-rse), and, on the other hand, from the [Test.jl docs](https://docs.julialang.org/en/v1/stdlib/Test/).
Material is taken and modified from the [SSE lecture](https://github.com/Simulation-Software-Engineering/Lecture-Material), which builds partly on the [py-rse book](https://merely-useful.tech/py-rse), and from the [Test.jl docs](https://docs.julialang.org/en/v1/stdlib/Test/).
---
@ -48,30 +48,21 @@ Material is taken and modified, on the one hand, from the [SSE lecture](https://
## What is Testing?
- Smelling old milk before using it!
- A way to determine if a software is not producing reliable results and if so, what is the reason.
- Manual testing vs. automated testing.
- Smelling old milk before using it
- A way to determine if a software is not producing reliable results and if so, what is the reason
- Manual testing vs. automated testing
---
## Why Should you Test your Software?
- Improve software reliability and reproducibility.
- Make sure that changes (bugfixes, new features) do not affect other parts of software.
- Improve software reliability and reproducibility
- Make sure that changes (bugfixes, new features) do not affect other parts of software
- Generally all software is better off being tested regularly. Possible exceptions are very small codes with single users.
- Ensure that a released version of a software actually works.
---
## Nomenclature in Software Testing
- **Fixture**: preparatory set for testing.
- **Actual result**: what the code produces when given the fixture.
- **Expected result**: what the actual result is compared to.
- **Test coverage**: how much of the code do tests touch in one run.
---
## Some Ways to Test Software
- Assertions
@ -83,29 +74,27 @@ Material is taken and modified, on the one hand, from the [SSE lecture](https://
## Assertions
- Principle of *defensive programming*.
```julia
@assert condition "message"
```
- Principle of *defensive programming*
- Nothing happens when an assertion is true; throws error when false.
- Types of assertion statements:
- Precondition
- Postcondition
- Invariant
- A basic but powerful tool to test a software on-the-go.
- Assertion statement syntax in Python
```julia
@assert condition "message"
```
- A basic but powerful tool to test a software on-the-go
---
## Unit Testing
- Catching errors with assertions is good but preventing them is better!
- Catching errors with assertions is good but preventing them is better.
- A *unit* is a single function in one situation.
- A situation is one amongst many possible variations of input parameters.
- User creates the expected result manually.
- Fixture is the set of inputs used to generate an actual result.
- Actual result is compared to the expected result by `@test`.
- User creates the **expected result** manually.
- **Actual result** is compared to the expected result by `@test`.
---
@ -113,109 +102,44 @@ Material is taken and modified, on the one hand, from the [SSE lecture](https://
- Test whether several units work in conjunction.
- *Integrate* units and test them together in an *integration* test.
- Often more complicated than a unit test and has more test coverage.
- A fixture is used to generate an actual result.
- Actual result is compared to the expected result by `@test`.
- Often more complicated than a unit test and gives higher test coverage.
---
## Regression Testing
- Generating an expected result is not possible in some situations.
- Compare the current actual result with a previous actual result.
- Compare the *current* actual result with a *previous* actual result.
- No guarantee that the current actual result is correct.
- Risk of a bug being carried over indefinitely.
- Main purpose is to identify changes in the current state of the code with respect to a past state.
---
## Test Coverage
- Coverage is the amount of code a test touches in one run.
- Aim for high test coverage.
- There is a trade-off: high test coverage vs. effort in test development
---
## Comparing Floating-point Variables
- Very often quantities in math software are `float` / `double`.
- Such quantities cannot be compared to exact values, an approximation is necessary.
- Comparison of floating point variables needs to be done to a certain tolerance.
```julia
@test 1 ≈ 0.999999 rtol=1e-5
```
- Get `≈` by Latex `\approx` + TAB
---
## Test-driven Development (TDD)
- Principle is to write a test and then write a code to fulfill the test.
- Advantages:
- In the end user ends up with a test alongside the code.
- Eliminates confirmation bias of the user.
- Writing tests gives clarity on what the code is supposed to do.
- Disadvantage: known to not improve productivity.
---
## Checking-driven Development (CDD)
- Developer performs spot checks; sanity checks at intermediate stages
- Math software often has heuristics which are easy to determine.
- Keep performing same checks at different stages of development to ensure the code works.
---
## Verifying a Test
- Test written as part of a bug-fix:
- Reproduce the bug in the test by ensuring that the test fails.
- Fix the bug.
- Rerun the test to ensure that it passes.
- Test written to increase code coverage:
- Make sure that the first iteration of the test passes.
- Try introducing a small fixable bug in the code to verify if the test fails.
---
# 2. Unit Testing in Julia with Test.jl
---
## Setup of Tests.jl
## Setup of Test.jl
- Standard library to write and manage tests, `using Test`
- Standardized folder structure:
```
├── Manifest.toml
├── Project.toml
├── src/
└── test
├── Manifest.toml
├── Project.toml
├── runtests.jl
└── setup.jl
```
```
├── Manifest.toml
├── Project.toml
├── src/
└── test
├── Manifest.toml
├── Project.toml
├── runtests.jl
└── setup.jl
```
- Singular `test` vs plural `runtests.jl`
- `setup.jl` for all `using XYZ` statements, included in `runtests.jl`
- Additional packages either in `[extra] section` of `./Project.toml` or in a new `./test/Project.toml` environment
- Additional packages in `[extra] section` of `./Project.toml` or in new `./test/Project.toml`
- In case of the latter: Do not add the package itself to the `./test/Project.toml`
---
## Run Tests
Various options:
- Directly call `runtests.jl` TODO?
- From Pkg-Manager `]test` when root project is activated
- Run: `]test` when root project is activated
---
@ -234,11 +158,11 @@ Various options:
- `@testset`: Structure tests
```julia
julia> @testset "trigonometric identities" begin
θ = 2/3*π
@test sin(-θ) ≈ -sin(θ)
@test cos(-θ) ≈ cos(θ)
end;
@testset "trigonometric identities" begin
θ = 2/3*π
@test sin(-θ) ≈ -sin(θ)
@test cos(-θ) ≈ cos(θ)
end;
```
- `@testset for ... end`: Test in loop
@ -251,6 +175,8 @@ Various options:
- [HiRSE-Summer of Testing Part 2b: "Testing with Julia" by Nils Niggemann](https://www.youtube.com/watch?v=gSMKNbZOpZU)
- [Official documentation of Test.jl](https://docs.julialang.org/en/v1/stdlib/Test/)
---
# 3. Test.jl Demo
We use [`MyTestPackage`](https://github.com/s-ccs/summerschool_simtech_2023/tree/main/material/2_tue/testing/MyTestPackage), which looks as follows:
@ -274,7 +200,7 @@ We use [`MyTestPackage`](https://github.com/s-ccs/summerschool_simtech_2023/tree
- Look at `MyTestPackage.jl` and `find.jl`: We have two functions `find_max` and `find_mean`, which calculate the maximum and mean of all elements of a `::AbstractVector`.
- Assertions were added to check for `NaN` values
- Look at `runtests.jl`:
- TODO: Why do we need `using MyTestPackage`?
- Why do we need `using MyTestPackage`?
- We include dependencies via `setup.jl`: `Test` and `StableRNG`.
- Testset "find"
- Look at `find.jl`

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material/3_wed/regression/Code_Snippets.jl
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@ -1,52 +1,48 @@
############################################################################
#### 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")
##
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())

586
material/3_wed/regression/MultipleRegressionBasics.qmd
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@ -1,292 +1,294 @@
---
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.
---
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 | Link Function |
| | Family | |
+===============+===================+==================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+---------------+-------------------+------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+---------------+-------------------+------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | ra |
| | | c{x}{1-x}\right) |
| | | $$ |
+---------------+-------------------+------------------+
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.