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MIT License
Copyright (c) 2023 s-ccs
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

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@ -42,7 +42,7 @@ website:
- href: material/1_mon/rse/rse_basics_slides.qmd - href: material/1_mon/rse/rse_basics_slides.qmd
text: "📊 1 - RSE" text: "📊 1 - RSE"
- href: "material/1_mon/why_julia/page.qmd" - href: "material/1_mon/why_julia/page.qmd"
text: "📊 2 - Why Julia" text: "📊 2 - Why Julia"
- href: "material/1_mon/firststeps/firststeps_handout.qmd" - href: "material/1_mon/firststeps/firststeps_handout.qmd"
text: "📝 3 - First Steps: Handout" text: "📝 3 - First Steps: Handout"
- href: "material/1_mon/firststeps/tasks.qmd" - href: "material/1_mon/firststeps/tasks.qmd"
@ -69,11 +69,11 @@ website:
contents: contents:
- href: material/3_wed/docs/handout.qmd - href: material/3_wed/docs/handout.qmd
text: "📝 1 - Docs: Handout" text: "📝 1 - Docs: Handout"
- href: material/3_wed/docs/tasks.qmd" - href: material/3_wed/docs/tasks.qmd
text: "🛠 1 - Docs: Exercises" text: "🛠 1 - Docs: Exercises"
- href: material/3_wed/vis/handout.qmd - href: material/3_wed/vis/handout.qmd
text: "📝 2 - Visualizations: Handout" text: "📝 2 - Visualizations: Handout"
- href: material/3_wed/vis/tasks.qmd" - href: material/3_wed/vis/tasks.qmd
text: "🛠 2 - Visualizations: Exercises" text: "🛠 2 - Visualizations: Exercises"
- href: material/3_wed/linalg/slides.qmd - href: material/3_wed/linalg/slides.qmd
text: "📝 3 - LinearAlgebra" text: "📝 3 - LinearAlgebra"
@ -84,7 +84,7 @@ website:
contents: contents:
- href: material/4_thu/sim/slides.qmd - href: material/4_thu/sim/slides.qmd
text: "📝 1 - Simulation" text: "📝 1 - Simulation"
- href: material/4_thu/stats/missing.jl - href: material/4_thu/bootstrap/Bootstrap.qmd
text: "📝 2 - Bootstrapping" text: "📝 2 - Bootstrapping"
- href: material/4_thu/parallel/slides.qmd - href: material/4_thu/parallel/slides.qmd
text: "📝 3 - Parallelization" text: "📝 3 - Parallelization"

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@ -40,6 +40,6 @@ And that's it! You should have a nice progress bar now
1. Implement a type `StatResult` with fields for `x`, `n`, `std` and `tvalue` 1. Implement a type `StatResult` with fields for `x`, `n`, `std` and `tvalue`
2. Implement an outer constructor that can run `StatResult(2:10)` and return the full type including the calculated t-values. 2. Implement an outer constructor that can run `StatResult(2:10)` and return the full type including the calculated t-values.
3. Implement a function `length` for `StatResult` to multiple-dispatch on 3. Implement a function length for StatResult (using multiple-dispatch) which returns the `.n` field. Overload "Base.length"
4. **Optional:** If you have time, optimize the functions, so that mean, sum, length, std etc. is not calculated multiple times - you might want to rewrite your type. Note: This is a bit tricky :) 4. **Optional:** If you have time, optimize the functions, so that mean, sum, length, std etc. is not calculated multiple times - you might want to rewrite your type. Note: This is a bit tricky :)

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---
---
::: callout ::: callout
[Link to Slides](slides.qmd) [Link to Slides](slides.qmd)
::: :::
@ -7,6 +11,7 @@
Solve [task 1](tasks.qmd#1) Solve [task 1](tasks.qmd#1)
---- ----
# Documenter.jl # Documenter.jl
### File-structure overview ### File-structure overview
@ -44,7 +49,9 @@ makedocs(
``` ```
### How to generate ### How to generate
`julia --project=docs/ docs/make.jl` or `]activate docs/; include("make.jl")` or use `LiveServer.jl` - `julia --project=docs/ docs/make.jl`, or
- `]activate docs/; include("make.jl")`, or
- `LiveServer.jl` + `deploydocs()`
### How to write ### How to write

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@ -10,17 +10,31 @@
``` ```
docs/ docs/
├── src/ ├── src/
├── src/mydocs.jl ├── src/index.md
└── make.jl └── make.jl
``` ```
### add some docs ### add some docs
2. with mydocs containing 2. with mydocs containing
````{verbatim} ````
Show docstring of a single function `func`
```@docs ```@docs
func(x) func
``` ```
Provide documentation for all doc_string-equiped functions
```@autodocs
Modules = [MyDocsPackage]
```
Execute a code-block, but hide it's output
```@example MyScope
x = [1,2,3]
nothing #hide
```
```` ````
and and

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--- ---
--- ---
## Slides ## Slides
The slides are available [in pptx format here](Julia_Matrices_Optimization_JuMP_Stuttgart2023.pptx). Note that there are a few extra slides in case you are motivated to learn more! The slides are available [in pptx format here](Julia_Matrices_Optimization_JuMP_Stuttgart2023.pptx). Note that there are a few extra slides in case you are motivated to learn more!
@ -8,4 +9,22 @@ The slides are available [in pptx format here](Julia_Matrices_Optimization_JuMP_
## Exercise ## Exercise
The exercise is rendered [as html here](Julia_Matrices_Optimization_JuMP_Stuttgart2023.ipynb) but can also be downloaded {{< downloadthis Julia_Matrices_Optimization_JuMP_Stuttgart2023.ipynb label="Download as ipynb" >}} The exercise is rendered [as html here](Julia_Matrices_Optimization_JuMP_Stuttgart2023.ipynb) but can also be downloaded {{< downloadthis Julia_Matrices_Optimization_JuMP_Stuttgart2023.ipynb label="Download as ipynb" >}}
Before starting the exercise, please download the files [Manifest.toml](Manifest.toml) and [Project.toml](Project.toml), put the files in the same folder that \*.ipynb file is located.
Start Julia in that directory and run the following commands:
```{julia}
using Pkg; Pkg.activate("."); Pkg.instantiate()
```
Now, you are ready to open the notebook in VSCode or alternatively in the same Julia console that you used for installation. You can run the following commands:
```{julia}
using IJulia; notebook(dir=".")
```

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@ -44,7 +44,7 @@ $$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad i=1,...,n,$$
where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$ are independent where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$ are independent
normally distributed errors with unknown variance $\sigma^2$. normally distributed errors with unknown variance $\sigma^2$.
*Task:* Find the straight line that fits best, i.e., find the *optimal* *Aim:* Find the straight line that fits best, i.e., find the *optimal*
estimators for $\beta_0$ and $\beta_1$. estimators for $\beta_0$ and $\beta_1$.
*Typical choice*: Least squares estimator (= maximum likelihood *Typical choice*: Least squares estimator (= maximum likelihood
@ -152,7 +152,7 @@ Defining the *design matrix*
$$ \mathbf{X} = \left( \begin{array}{cccc} $$ \mathbf{X} = \left( \begin{array}{cccc}
1 & x_{11} & \ldots & x_{1p} \\ 1 & x_{11} & \ldots & x_{1p} \\
\vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\
1 & x_{11} & \ldots & x_{1p} 1 & x_{n1} & \ldots & x_{np}
\end{array}\right) \qquad \end{array}\right) \qquad
(\text{size } n \times (p+1)), $$ (\text{size } n \times (p+1)), $$
@ -177,7 +177,7 @@ regression model, but we provide explicit formulas now:
- estimated standard errors: - 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} \hat s_{\beta_i} = \sqrt{([\mathbf{X}^\top \mathbf{X}]^{-1})_{ii} \frac 1 {n-p-1} \|\mathbf{y} - \mathbf{X} \beta\|^2}
$$ $$
- $t$-statistics: - $t$-statistics:
@ -187,21 +187,21 @@ regression model, but we provide explicit formulas now:
- $p$-values: - $p$-values:
$$ $$
p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p} p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p-1}
$$ $$
::: {.callout-caution collapse="false"} ::: {.callout-caution collapse="false"}
## Task 2 ## Task 2
1. Implement functions that estimate the $\beta$-parameters, 1. Implement functions that estimate the $\beta$-parameters, the
the corresponding standard errors and the $t$-statistics. corresponding standard errors and the $t$-statistics.
2. Test your functions with the `tree' data set and try to reproduce the 2. Test your functions with the \`tree' data set and try to reproduce
output above. the output above.
::: :::
Which model is the best? For linear models, one often uses the $R^2$ characteristic. Which model is the best? For linear models, one often uses the $R^2$
Roughly speaking, it gives the percentage (between 0 and 1) of the variance that can be explained by the linear model. characteristic. Roughly speaking, it gives the percentage (between 0
and 1) of the variance that can be explained by the linear model.
``` julia ``` julia
r2(linmod1) r2(linmod1)
@ -212,8 +212,11 @@ linmod3 = lm(@formula(Volume ~ Girth + Height + Girth*Height), trees)
r2(linmod3) r2(linmod3)
``` ```
::: {.callout-note} ::: callout-note
The more covariates you add the more variance can be explained by the linear model - $R^2$ increases. In order to balance goodness-of-fit of a model and its complexity, information criteria such as `aic` are considered. The more covariates you add the more variance can be explained by the
linear model - $R^2$ increases. In order to balance goodness-of-fit of a
model and its complexity, information criteria such as `aic` are
considered.
::: :::
## Generalized Linear Models ## Generalized Linear Models
@ -256,30 +259,30 @@ $$
For the models above, these are: For the models above, these are:
+----------------+------------------+--------------------------+ +----------------+-----------------+-------------------------+
| Type of Data | Distribution | Link Function | | Type of Data | Distribution | Link Function |
| | Family | | | | Family | |
+================+==================+==========================+ +================+=================+=========================+
| continuous | Normal | identity: | | continuous | Normal | identity: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g(x)=x | | | | g(x)=x |
| | | $$ | | | | $$ |
+----------------+------------------+--------------------------+ +----------------+-----------------+-------------------------+
| count | Poisson | log: | | count | Poisson | log: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g(x) = \log(x) | | | | g(x) = \log(x) |
| | | $$ | | | | $$ |
+----------------+------------------+--------------------------+ +----------------+-----------------+-------------------------+
| binary | Bernoulli | logit: | | binary | Bernoulli | logit: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g(x) = \log\left( | | | | g(x) = \log\left( |
| | | \frac{x}{1-x} | | | | \frac{x}{1-x} |
| | | \right) | | | | \right) |
| | | $$ | | | | $$ |
+----------------+------------------+--------------------------+ +----------------+-----------------+-------------------------+
In general, the parameter vector $\beta$ is estimated via maximizing the In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e., likelihood, i.e.,
@ -311,10 +314,33 @@ model = glm(@formula(participation ~ age^2),
``` ```
::: {.callout-caution collapse="false"} ::: {.callout-caution collapse="false"}
## Task 3: ## Task 3:
1. Reproduce the results of our data analysis of the `tree` data set 1. Reproduce the results of our data analysis of the `tree` data set
using a generalized linear model with normal distribution family. using a generalized linear model with normal distribution family.
2. Generate $n=20$ random covariates $\mathbf{x}$ and Poisson-distributed counting data with parameters $\beta_0 + \beta_1 x_i$. Re-estimate the parameters by a generalized linear model. 2. Generate $n=20$ random covariates $\mathbf{x}$ and
Poisson-distributed counting data with parameters
$\beta_0 + \beta_1 x_i$. Re-estimate the parameters by a generalized
linear model.
::: :::
## Outlook: Linear Mixed Models
In the linear regression models so far, we assumed that the response
variable $\mathbf{y}$ depends on the design matrix of covariates
$\mathbf{X}$ - which are assumed to be given/fixed - multiplied by the
so-called *fixed effects* coefficients $\mathbf{X}\beta$ and independent
errors $\varepsilon$. However, in many situations, there are also random
effects on several components of the response variable. These can be
included in the model by adding another design matrix $\mathbf{Z}$
multiplied by a random vector $u$, the so-called *random effects*
coefficients, that are assumed to be jointly normally distributed with
mean vector $0$ and variance-covariance matrix $\Sigma$ (typically *not*
a diagonal matrix). In matrix notation, we have the following form:
$$
\mathbf{y} = \mathbf{X} \beta + \mathbf{Z}u + \varepsilon
$$
Maximizing the likelihood, we can estimate $\beta$ and optimally
predict the random vector $u$.

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## Backends ## Backends
Four backends: Four backends:
1. `CairoMakie` - SVG 1. `CairoMakie` - SVG
2. `GLMakie` - 2D/3D/fast interactivity 2. `GLMakie` - 2D/3D/fast interactivity
3. `WGLMakie` - Same as GLMakie, but in browser 3. `WGLMakie` - Same as GLMakie, but in browser

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@ -56,11 +56,11 @@ Note that there is no `cummean` function, but clever element-wise division in co
::: {.callout-tip collapse="true"} ::: {.callout-tip collapse="true"}
## click to show solution ## click to show solution
`cumsum(x) ./ 1:length(x)` `cumsum(x) ./ (1:length(x))`
::: :::
## 3. Plotting! ## 3. Plotting!
Now for your first plot. Use a `scatter` plot to visualize the cummulative mean output, if you do not generate a `Figure()` + `ax = f[1,1] = Axis(f)` manually, you can get it back by the scatter call. `f,ax,s = scatter()`. This is helpful as we later want to extend the `Axis` and `Figure` with other plot elements Now for your first plot. Use a `scatter` plot^[after a `using CairoMakie`] to visualize the cummulative mean output, if you do not generate a `Figure()` + `ax = f[1,1] = Axis(f)` manually, you can get it back by the scatter call. `f,ax,s = scatter()`. This is helpful as we later want to extend the `Axis` and `Figure` with other plot elements
Use `hlines!` to add a horizontal line at your "true" value Use `hlines!` to add a horizontal line at your "true" value
@ -70,7 +70,7 @@ Let's simulate 1000x datasets, each with a different seed, and take the mean ov
::: {.callout-tip collapse="true"} ::: {.callout-tip collapse="true"}
## click to show tip ## click to show tip
An easy way to call a function many times is to broadcast it on an array e.g. `1:1000` - you could also use `map` to do it, but I don't think it is as clear :) An easy way to call a function many times is to broadcast it on an array created e.g. via `1:1000` - you could also use `map` to do it, but I don't think it is as clear :)
::: :::

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# Resampling-Based Statistics
## Motivation: Fitting Models to Data
Situation considered yesterday: We have data and want to fit a model with certain parameters (e.g., a linear model) -- we estimate the parameter.
*Notation:*
- data: $\mathbf{x} = (x_1, \ldots, x_n)$
- model with unknown parameter $\theta$
- estimate $\widehat \theta(\mathbf{x})$
### Working example: Fitting a normal distribution
- data: $\mathbf{x} = (x_1, \ldots, x_n)$
- model: $\mathcal{N}(\theta, \sigma^2)$, i.e., a normal distribution with unknown mean $\theta$ (that we want to estimate) and variance $\sigma^2$ (that we are less interested in)
- use empirical mean as estimator: $\widehat \theta(\mathbf{x}) = \overline{x} = \frac 1 n \sum_{i=1}^n x_i$
``` julia
using Distributions
using Statistics
using StatsPlots
d = Normal(0.0, 1.0)
n = 100
x = rand(d, n)
θ = mean(x)
```
*Problem:* Estimator [never]{.underline} gives the [exact]{.underline} result -- if you have random data, also the estimate is random.
*Aim:* Find the distribution (or at least the variance) of the estimator $\widehat \theta$ in order to get standard errors, confidence intervals, etc.
In some easy examples, you can calculate the distribution of $\widehat \theta$ theoretically. *Example:* If $x_i$ is $\mathcal{N}(\theta,\sigma^2)$ distributed, then the distribution of $\widehat \theta(\mathbf{x})$ is $\mathcal{N}(\theta, \sigma^2/n)$. Strategy: Estimate $\sigma^2$, e.g. via the sample variance $$ \widehat \sigma^2 = \frac 1 {n-1} \sum_{i=1}^n (x_i - \overline{x})^2 $$ and take the standard error, confidence intervals, etc. of the corresponding normal distribution.
``` julia
σ = std(x)
est_d = Normal(θ, σ/sqrt(n))
plot(est_d, legend=false)
ci_bounds = quantile(est_d, [0.025,0.975])
vline!(ci_bounds)
```
*Problem:* In more complex examples, we cannot calculate the distribution.
## The 'ideal' solution: Generate *new* data
In theory, one would ideally do the following:
1. Generate new independent data $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(B)}$ (each sample of size $n$)
2. Apply the estimator separately to each sample $\leadsto$ $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$
3. Use the empirical distribution $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$ as a proxy to the theoretical one.
``` julia
B = 1000
est_vector_new = zeros(B)
for i in 1:B
x_new = rand(d, n)
est_vector_new[i] = mean(x_new)
end
histogram(est_vector_new, legend=false)
ci_bounds_new = quantile(est_vector_new, [0.025, 0.975])
vline!(ci_bounds_new)
```
::: callout-attention
*But:* In most real world situation, we can not generate new data if the distribution is unknown. We have to work with the data we have ...
:::
## The practical solution: Resampling / Bootstrap
*Idea:* Use samples $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(B)}$ that are not completely new, but obtained from [resampling]{.underline} the original data $\mathbf{x}$.
*Question:* How can one obtained another sample of the same size $n$? $\leadsto$ (re-)sampling with replacement
The overall procedure is as follows:
1. Generate $B$ samples $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(B)}$ of size $n$ by independently resampling from $\mathbf{x}$ with replacement.
2. Apply the estimator separately to each sample $\leadsto$ $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$
3. Use the empirical distribution $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$ as a proxy to the theoretical one.
``` julia
est_vector_bs = zeros(B)
for i in 1:B
x_bs = rand(x, n)
est_vector_bs[i] = mean(x_bs)
end
histogram(est_vector_bs, legend=false)
ci_bounds_bs = quantile(est_vector_bs, [0.025, 0.975])
vline!(ci_bounds_bs)
```
If the sample $\mathbf{x} = (x_1,\ldots,x_n)$ consists of independent and identically distributed data, the resampling procedure often provides a code proxy to the true (unknown) distribution of the estimator.
::: callout-note
The above resampling procedure is called bootstrap (from ''To pull oneself up by one's bootstraps.'') as only data are used that are already available.
:::
::: {.callout-caution collapse="false"}
## Task 1
1. Reconsider the `tree` data set and the simple linear regression model `Volume ~ Girth`. Calculate a 95% confidence interval for $\beta_1$ via bootstrap and compare to the Julia output of the linear model.
2. Use bootstrap to estimate the standard error for the predicted volume of a tree with `Girth=10` the output above.
:::
*Problem:* If the data are *not* independent, the above (i.i.d.) bootstrap samples would have a misspecified dependence structure and therefore lead to a bad uncertainty estimate. For some situations, there are specific modifications of the bootstrap procedure (e.g. block bootstrap for time series), but they tend to work well only if dependence is sufficiently weak.
## Parametric Bootstrap
There are situations where it is hardly possible to construct reasonable confidence intervals or estimate the standard error. But one could at least get a [rough guess]{.underline} of the uncertainty by the following thought experiment:
Assume that the estimated parameter value $\theta^*$ would be equal to the true one. How uncertain would an estimate be in that case?
The answer is given by the following procedure, called *parametric bootstrap*:
1. Generate independent data $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(B)}$ (each sample of size $n$) from the model with parameter $\theta^*$
2. Apply the estimator separately to each sample $\leadsto$ $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$
3. Use the empirical distribution $\widehat \theta(\mathbf{x}^{(1)}), \ldots, \widehat \theta(\mathbf{x}^{(B)})$ as a proxy to the theoretical one.
::: {.callout-caution collapse="false"}
## Task 2
1. Consider the following function that generate $n$ correlated samples that are uniformly distributed on $[\mu-0-5,\mu+0.5]$.
``` julia
myrand = function(mu, n)
rho = 0.9
res = zeros(n)
res[1] = rand(1)[1]
if n > 1
for i in 2:n
res[i] = rho*res[i-1] .+ (1-rho)*rand(1)[1]
end
end
res .= mu - 0.5 .+ res
return(res)
end
```
The additional parameter `rho` (between 0 and 1) controls the strength of dependence with 0 meaning independence and 1 meaning full dependence.
Write functions that estimate the standard deviation of the estimated mean via (a) generating new samples from the true unknown distribution, (b) i.i.d. bootstrap, (c) parametric bootstrap 2. Use the functions for different values of `rho` and compare the results.
:::

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using Distributions
using Statistics
using StatsPlots
d = Normal(0.0, 1.0)
n = 100
x = rand(d, n)
θ = mean(x)
#---
σ = std(x)
est_d = Normal(θ, σ/sqrt(n))
plot(est_d, legend=false)
ci_bounds = quantile(est_d, [0.025,0.975])
vline!(ci_bounds)
#---
B = 1000
est_vector = zeros(B)
for i in 1:B
x_new = rand(d, n)
est_vector[i] = mean(x_new)
end
histogram(est_vector, legend=false)
ci_bounds_new = quantile(est_vector, [0.025, 0.975])
vline!(ci_bounds_new)
#---
est_vector_bs = zeros(B)
for i in 1:B
x_bs = rand(x, n)
est_vector_bs[i] = mean(x_bs)
end
histogram(est_vector_bs, legend=false)
ci_bounds_bs = quantile(est_vector_bs, [0.025, 0.975])
vline!(ci_bounds_bs)
#---
# the following function generates a sample of (strongly)
# correlated normal data with mean μ and variance 1
myrandn = function (mu=0.0, n=1)
rho = 0.9
res = zeros(n)
res[1] = randn(1)[1]
if n > 1
for i in 2:n
res[i] = rho*res[i-1] .+ sqrt(1-rho^2)*randn(1)[1]
end
end
res .= mu .+ res
return(res)
end
## Assume we obtained the following data
n = 20
true_mu = 1.0
y = myrandn(true_mu, n)
est_mu = mean(y)
## theoretical standard error
B = 1000
est_vector = zeros(B)
for i in 1:B
y_new = myrandn(true_mu, n)
est_vector[i] = mean(y_new)
end
std(est_vector)
## classical i.i.d. bootstrap
est_vector_bs = zeros(B)
for i in 1:B
y_bs = rand(y, n)
est_vector_bs[i] = mean(y_bs)
end
std(est_vector_bs)
## parametric bootstrap
est_vector_pbs = zeros(B)
for i in 1:B
y_pbs = myrandn(est_mu, n)
est_vector_pbs[i] = mean(y_pbs)
end
std(est_vector_pbs)
# ---
myrand = function(mu, n)
rho = 0.9
res = zeros(n)
res[1] = rand(1)[1]
if n > 1
for i in 2:n
res[i] = rho*res[i-1] .+ (1-rho)*rand(1)[1]
end
end
res .= mu - 0.5 .+ res
return(res)
end

47
social.qmd
View File

@ -4,22 +4,29 @@ title: Social Program
## Overview ## Overview
+------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| What | When | Where | | What | When | Where |
| ---------------------------------------------------------------------------- | ---------------------------------- | --------------------------------------------------------------------------------------------- | +==============================================================================+==================================+==================================================================================================+
| Get-together | Sunday (8th October) 7 pm | At a restaurant called [Metzgerei](https://maps.app.goo.gl/BchdVZWqegxgtHXi8) | | Get-together | Sunday (8th October) 7 pm | At a restaurant called [Metzgerei](https://maps.app.goo.gl/BchdVZWqegxgtHXi8) |
| BBQ | Monday (9th October) | | +------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| Hike to [Bärenschlössle](https://maps.app.goo.gl/xKQCuboiAR9YQksC7) + dinner | Tuesday (10th October) 6:10pm | We start at the entrance of the [SimTech building](https://maps.app.goo.gl/M8VgfDRkFiZk8wri8) | | Meet & Greet | Monday (9th October) | |
| Get-together/Games night | Tuesday (10th October) ~9:15 pm | At a student pub called [Unithekle](https://maps.app.goo.gl/NHdeJvjHiEPfvRh59) | +------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| Conference dinner | Wednesday (11th October) | | | Hike to [Bärenschlössle](https://maps.app.goo.gl/xKQCuboiAR9YQksC7) + dinner | Tuesday (10th October) 6:10pm | We start at the entrance of the [SimTech building](https://maps.app.goo.gl/M8VgfDRkFiZk8wri8) |
| Bowling night | Thursday (12th October) 7:45 pm | At the [bowling centre in Möhringen](https://maps.app.goo.gl/yi4YfoqFXtz5uEhi6) | +------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| Get-together/Games night | Tuesday (10th October) \~9:15 pm | At a student pub called [Unithekle](https://maps.app.goo.gl/NHdeJvjHiEPfvRh59) |
+------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| Conference dinner\ | Wednesday (11th October) 7:00pm | Either at the entrance of the SimTech building (6:20pm) or directly at Sophies Brauhaus (7:00pm) |
| at [Sophies Brauhaus](https://sophies-brauhaus.de/en/) | | |
+------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
| Bowling night | Thursday (12th October) 7:45 pm | At the [bowling centre in Möhringen](https://maps.app.goo.gl/yi4YfoqFXtz5uEhi6) |
+------------------------------------------------------------------------------+----------------------------------+--------------------------------------------------------------------------------------------------+
## Sunday ## Sunday
Get-together at 7pm at a restaurant called [Metzgerei](https://maps.app.goo.gl/BchdVZWqegxgtHXi8) (vegetarian and vegan options available) Get-together at 7pm at a restaurant called [Metzgerei](https://maps.app.goo.gl/BchdVZWqegxgtHXi8) (vegetarian and vegan options available)
- Closest U-Bahn station(s): Schwab-/Bebelstraße (U2, U29) - Closest U-Bahn station(s): Schwab-/Bebelstraße (U2, U29)
- Closest S-Bahn station(s): Schwabstraße (S1, S2, S3, S4, S5, S6, S60) - Closest S-Bahn station(s): Schwabstraße (S1, S2, S3, S4, S5, S6, S60)
## Monday ## Monday
@ -41,15 +48,15 @@ Conference dinner (and Bene's birthday!)
## Thursday ## Thursday
On Thursday we will meet slightly before 20:00 at [bowling alley Möhringen](https://www.bowling-moehringen.de/) for a friendly tournament between groups. On Thursday we will meet slightly before 20:00 at [bowling alley Möhringen](https://www.bowling-moehringen.de/) for a friendly tournament between groups.
[Bowling centre in Möhringen :world_map:](https://maps.app.goo.gl/yi4YfoqFXtz5uEhi6) (8pm-10pm) [Bowling centre in Möhringen :world_map:](https://maps.app.goo.gl/yi4YfoqFXtz5uEhi6) (8pm-10pm)
- We want to start bowling at 8pm. Please be there about 10 minutes earlier to rent bowling shoes.
- Closest U-Bahn station(s): SSB-Zentrum (U3, U8, U12), Vaihinger Straße (U3, U5, U6, U8, U12)
- Closest bus stop(s): Wallgraben (N1)
- Closest S-Bahn station(s): Vaihingen (S1, S2, S3)
- 25 min walk
- We recommend taking the S-Bahn from University to Vaihingen and then change to U-Bahn
Beforehand you'll have time to get some food; we prepared [a map](https://www.google.com/maps/d/u/1/edit?mid=1vJQ4JtYGulsYW2M9HvyN73yKr6ESNlk&usp=sharing) with some recommendations on the way to the bowling alley, if you head there straight from campus. - We want to start bowling at 8pm. Please be there about 10 minutes earlier to rent bowling shoes.
- Closest U-Bahn station(s): SSB-Zentrum (U3, U8, U12), Vaihinger Straße (U3, U5, U6, U8, U12)
- Closest bus stop(s): Wallgraben (N1)
- Closest S-Bahn station(s): Vaihingen (S1, S2, S3)
- 25 min walk
- We recommend taking the S-Bahn from University to Vaihingen and then change to U-Bahn
Beforehand you'll have time to get some food; we prepared [a map](https://www.google.com/maps/d/u/1/edit?mid=1vJQ4JtYGulsYW2M9HvyN73yKr6ESNlk&usp=sharing) with some recommendations on the way to the bowling alley, if you head there straight from campus.