lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

changes to project 1 due to typos

Browse Source
This commit is contained in:
mhjensen
2019-08-29 04:34:07 +02:00
parent 488b7269cc
commit 08e6aa2893
16 changed files with 2267 additions and 54 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/Projects/2019/Project1/html/Project1.html
+8 -8
View File
@@ -49,7 +49,10 @@ div { text-align: justify; text-justify: inter-word; }
3,
None,
'___sec1'),
('Part b) Resampling techniques', 3, None, '___sec2'),
('Part b) Resampling techniques, adding more complexity',
3,
None,
'___sec2'),
('Part c): Bias-variance tradeoff', 3, None, '___sec3'),
('Part d): Ridge Regression on the Franke function with '
'resampling',
@@ -248,7 +251,7 @@ $$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
<h3 id="___sec2">Part b) Resampling techniques </h3>
<h3 id="___sec2">Part b) Resampling techniques, adding more complexity </h3>
<p>
Perform a resampling of the data where you split the data in training
@@ -266,7 +269,7 @@ approximately \( 2/3 \) to \( 4/5 \) of the data as training data.
<p>
Implement the \( k \)-fold cross-validation algorithm (write your own
code) and and evaluate again the MSE and the \( R^2 \) functions resulting
code) and evaluate again the MSE and the \( R^2 \) functions resulting
from the test data. You can compare your own code with that from
<b>Scikit-Learn</b> if needed.
@@ -306,16 +309,13 @@ The parameters \( \boldsymbol{\beta} \) are in turn found by optimizing the mean
squared error via the so-called cost function
$$
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{\
y}-\boldsymbol{\tilde{y}})^2\right].
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
$$
<p>
Show that you can rewrite this as
$$
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\bm\
{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])\
^2+\sigma^2.
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
$$
<p>