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Morten Hjorth-Jensen
2022-09-05 12:23:56 +02:00
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@@ -118,6 +118,10 @@ div.toc p,a {
2,
None,
'regression-analysis-and-resampling-methods'),
('Description of two-dimensional function',
3,
None,
'description-of-two-dimensional-function'),
('Part a): Paper and pencil part (also as weekly exercise for '
'week 36)',
3,
@@ -204,10 +208,34 @@ MathJax.Hub.Config({
<p>The main aim of this project is to study in more detail various
regression methods, including the Ordinary Least Squares (OLS) method,
In addition to the scientific part, in this course we want also to give you an experience in writing scientific reports.
The format for the delivery of your answers is namely that of a scientific report. At for example <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a> we detail how to write a report. Furthermore, at <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ReportExample/" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ReportExample/</tt></a> you can find examples of previous reports. How to write reports will also be discussed during lectures and at the various lab sessions.
In addition to the scientific part, in this course we want also to
give you an experience in writing scientific reports. The format for
the delivery of your answers is namely that of a scientific report. At
for example
<a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a>
we detail how to write a report. Furthermore, at
<a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ReportExample/" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ReportExample/</tt></a>
you can find examples of previous reports. How to write reports will
also be discussed during lectures and at the various lab sessions.
</p>
<p><b>A small recommendation when developing the codes here</b>. Instead of
jumping on to the two-dimensional function described below, we
recommend to do the code development and testing with a simpler
one-dimensional function, similar to those discussed in the exercises
of week 35. A simple test, as discussed during the lectures the first
two weeks is to set the design matrix equal to the identity
matrix. Then our model should give a mean square error which is exactly equal to zero.
When you are sure that your codes function well, you can then replace
the one-dimensional test function with the two-dimensional <b>Franke</b> function
discussed here.
</p>
<p>The Franke function serves as a stepping stone towards the analysis of
real topographic data. The latter is the last part of this project.
</p>
<h3 id="description-of-two-dimensional-function">Description of two-dimensional function </h3>
<p>We will first study how to fit polynomials to a specific
two-dimensional function called <a href="http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf" target="_blank">Franke's
function</a>. This
@@ -310,9 +338,9 @@ plt<span style="color: #666666">.</span>show()
</div>
<h3 id="part-a-paper-and-pencil-part-also-as-weekly-exercise-for-week-36">Part a): Paper and pencil part (also as weekly exercise for week 36) </h3>
<p>This part can be included in your theory description of the report.</p>
<p>This part should be included in your theory description of the report.</p>
<p>This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of <a href="https://www.springer.com/gp/book/9780387848570" target="_blank">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>).</p>
<p>This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of <a href="https://www.springer.com/gp/book/9780387848570" target="_blank">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>).</p>
<p>The assumption we have made is
that there exists a continuous function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim N(0, \sigma^2) \)