daviddoji/summerschool_simtech_2023
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Another update

Browse Source
This commit is contained in:
Marco Oesting 2023-10-09 16:14:00 +02:00
parent 08951f610d
commit 39892ad1c1
1 changed files with 32 additions and 30 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

62
material/3_wed/regression/MultipleRegressionBasics.qmd
View File

@ -10,7 +10,7 @@ editor:
### Introductory Example: tree dataset from R
``` julia
``` julia
using Statistics
using Plots
using RDatasets
@ -25,7 +25,7 @@ scatter(trees.Volume, trees.Girth,
the *explanatory variable/covariate* `girth`? Can we predict the volume
of a tree given its girth?
``` julia
``` julia
scatter(trees.Girth, trees.Volume,
legend=false, xlabel="Girth", ylabel="Volume")
plot!(x -> -37 + 5*x)
@ -68,7 +68,7 @@ rather use Julia to solve the problem.
\[use Julia code (existing package) to perform linear regression for
`volume ~ girth`\]
``` julia
``` julia
lm(@formula(Volume ~ Girth), trees)
```
@ -183,7 +183,7 @@ the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the
output above.
``` julia
``` julia
r2(linmod1)
r2(linmod2)
@ -232,31 +232,33 @@ $$
For the models above, these are:
+--------------+---------------------+--------------------+
| Type of Data | Distribution Family | Link Function |
+==============+=====================+====================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+--------------+---------------------+--------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+--------------+---------------------+--------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | rac{x}{1-x}\right) |
| | | $$ |
+--------------+---------------------+--------------------+
+---------------+-------------------+------------------+
| Type of Data | Distribution | Link Function |
| | Family | |
+===============+===================+==================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+---------------+-------------------+------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+---------------+-------------------+------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | ra |
| | | c{x}{1-x}\right) |
| | | $$ |
+---------------+-------------------+------------------+
In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,
@ -274,7 +276,7 @@ $$
In the Gaussian case, the maximum likelihood estimator is identical to
the least squares estimator considered above.
``` julia
``` julia
using CSV
using HTTP