udpdates
This commit is contained in:
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</ul>
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@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Cancer Data again now with Decision Trees and other Methods',
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</ul>
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@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
|
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('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
|
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'___sec38'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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('Other measures in classification studies: Cancer Data again',
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</ul>
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@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
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('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
|
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('Cancer Data again now with Decision Trees and other Methods',
|
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'___sec38'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</ul>
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@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
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('Including more classes', 2, None, '___sec36'),
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('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
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'___sec38'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</ul>
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@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
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('Including more classes', 2, None, '___sec36'),
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('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
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|
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|
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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||||
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||||
</ul>
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||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
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('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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||||
|
||||
</ul>
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||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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('Extending to more predictors', 2, None, '___sec35'),
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||||
('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
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||||
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
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('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
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||||
('Cancer Data again now with Decision Trees and other Methods',
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|
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'___sec38'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
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||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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||||
('Extending to more predictors', 2, None, '___sec35'),
|
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('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
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('Cancer Data again now with Decision Trees and other Methods',
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|
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'___sec38'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
|
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('Other measures in classification studies: Cancer Data again',
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2,
|
||||
None,
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||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
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|
||||
('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
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|
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|
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
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|
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
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('Extending to more predictors', 2, None, '___sec35'),
|
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('Including more classes', 2, None, '___sec36'),
|
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('More classes', 2, None, '___sec37'),
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|
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|
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|
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
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||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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|
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('More classes', 2, None, '___sec37'),
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
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|
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|
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@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
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|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
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|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
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||||
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|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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||||
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|
||||
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||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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||||
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|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
@@ -227,7 +224,16 @@ f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any value from minus infinity to plus infinity. If we however let \( f(y\vert y) \) be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function \( f \), namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of \( y_i \) with a given \( x_i \).
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any
|
||||
value from minus infinity to plus infinity. If we however let
|
||||
\( f(y\vert y) \) be represented by the mean value, the above example
|
||||
shows us that we can constrain the function to take values between
|
||||
zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
|
||||
at our last curve we see also that it has an S-shaped form. This leads
|
||||
us to a very popular model for the function \( f \), namely the so-called
|
||||
Sigmoid function or logistic model. We will consider this function as
|
||||
representing the probability for finding a value of \( y_i \) with a given
|
||||
\( x_i \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
@@ -205,10 +202,11 @@ MathJax.Hub.Config({
|
||||
<h2 id="___sec28" class="anchor">The logistic function </h2>
|
||||
|
||||
<p>
|
||||
The perceptron is an example of a ``hard classification" model. We
|
||||
Another widely studied model, is the so-called
|
||||
perceptron model, which is an example of a ``hard classification" model. We
|
||||
will encounter this model when we discuss neural networks as
|
||||
well. Each datapoint is deterministically assigned to a category (i.e
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft"
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
|
||||
classifier that outputs the probability of a given category rather
|
||||
than a single value. For example, given \( x_i \), the classifier
|
||||
outputs the probability of being in a category \( k \). Logistic regression
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
@@ -207,13 +204,13 @@ MathJax.Hub.Config({
|
||||
<p>
|
||||
Till now we have mainly focused on two classes, the so-called binary
|
||||
system. Suppose we wish to extend to \( K \) classes. Let us for the sake
|
||||
of simplicity assume we have only two predictors. We have then
|
||||
following model
|
||||
of simplicity assume we have only two predictors. We have then following model
|
||||
|
||||
$$
|
||||
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
||||
$$
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
@@ -240,7 +237,7 @@ discussed in the material on <a href="https://compphysics.github.io/MachineLearn
|
||||
methods</a>.
|
||||
|
||||
<p>
|
||||
This will be discussed next week.
|
||||
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that <b>Scikit-learn</b> offers.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
@@ -202,7 +199,7 @@ MathJax.Hub.Config({
|
||||
<a name="part0039"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec38" class="anchor">Cancer Data again now with Decision Trees and other Methods </h2>
|
||||
<h2 id="___sec38" class="anchor">Wisconsin Cancer Data </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -102,10 +102,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -185,7 +182,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
|
||||
|
||||
</ul>
|
||||
|
||||
@@ -1435,7 +1435,16 @@ $$
|
||||
<p> <br>
|
||||
|
||||
<p>
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any value from minus infinity to plus infinity. If we however let \( f(y\vert y) \) be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function \( f \), namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of \( y_i \) with a given \( x_i \).
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any
|
||||
value from minus infinity to plus infinity. If we however let
|
||||
\( f(y\vert y) \) be represented by the mean value, the above example
|
||||
shows us that we can constrain the function to take values between
|
||||
zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
|
||||
at our last curve we see also that it has an S-shaped form. This leads
|
||||
us to a very popular model for the function \( f \), namely the so-called
|
||||
Sigmoid function or logistic model. We will consider this function as
|
||||
representing the probability for finding a value of \( y_i \) with a given
|
||||
\( x_i \).
|
||||
</section>
|
||||
|
||||
|
||||
@@ -1443,10 +1452,11 @@ This expression implies however that \( f(y_i\vert x_i) \) could take any value
|
||||
<h2 id="___sec28">The logistic function </h2>
|
||||
|
||||
<p>
|
||||
The perceptron is an example of a ``hard classification" model. We
|
||||
Another widely studied model, is the so-called
|
||||
perceptron model, which is an example of a ``hard classification" model. We
|
||||
will encounter this model when we discuss neural networks as
|
||||
well. Each datapoint is deterministically assigned to a category (i.e
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft"
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
|
||||
classifier that outputs the probability of a given category rather
|
||||
than a single value. For example, given \( x_i \), the classifier
|
||||
outputs the probability of being in a category \( k \). Logistic regression
|
||||
@@ -1687,8 +1697,7 @@ $$
|
||||
<p>
|
||||
Till now we have mainly focused on two classes, the so-called binary
|
||||
system. Suppose we wish to extend to \( K \) classes. Let us for the sake
|
||||
of simplicity assume we have only two predictors. We have then
|
||||
following model
|
||||
of simplicity assume we have only two predictors. We have then following model
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
@@ -1696,6 +1705,7 @@ $$
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and
|
||||
<p> <br>
|
||||
$$
|
||||
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
||||
@@ -1758,12 +1768,12 @@ discussed in the material on <a href="https://compphysics.github.io/MachineLearn
|
||||
methods</a>.
|
||||
|
||||
<p>
|
||||
This will be discussed next week.
|
||||
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that <b>Scikit-learn</b> offers.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec38">Cancer Data again now with Decision Trees and other Methods </h2>
|
||||
<h2 id="___sec38">Wisconsin Cancer Data </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
|
||||
@@ -96,10 +96,7 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -1365,7 +1362,16 @@ f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any value from minus infinity to plus infinity. If we however let \( f(y\vert y) \) be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function \( f \), namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of \( y_i \) with a given \( x_i \).
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any
|
||||
value from minus infinity to plus infinity. If we however let
|
||||
\( f(y\vert y) \) be represented by the mean value, the above example
|
||||
shows us that we can constrain the function to take values between
|
||||
zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
|
||||
at our last curve we see also that it has an S-shaped form. This leads
|
||||
us to a very popular model for the function \( f \), namely the so-called
|
||||
Sigmoid function or logistic model. We will consider this function as
|
||||
representing the probability for finding a value of \( y_i \) with a given
|
||||
\( x_i \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1373,10 +1379,11 @@ This expression implies however that \( f(y_i\vert x_i) \) could take any value
|
||||
<h2 id="___sec28">The logistic function </h2>
|
||||
|
||||
<p>
|
||||
The perceptron is an example of a ``hard classification" model. We
|
||||
Another widely studied model, is the so-called
|
||||
perceptron model, which is an example of a ``hard classification" model. We
|
||||
will encounter this model when we discuss neural networks as
|
||||
well. Each datapoint is deterministically assigned to a category (i.e
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft"
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
|
||||
classifier that outputs the probability of a given category rather
|
||||
than a single value. For example, given \( x_i \), the classifier
|
||||
outputs the probability of being in a category \( k \). Logistic regression
|
||||
@@ -1590,13 +1597,13 @@ $$
|
||||
<p>
|
||||
Till now we have mainly focused on two classes, the so-called binary
|
||||
system. Suppose we wish to extend to \( K \) classes. Let us for the sake
|
||||
of simplicity assume we have only two predictors. We have then
|
||||
following model
|
||||
of simplicity assume we have only two predictors. We have then following model
|
||||
|
||||
$$
|
||||
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
||||
$$
|
||||
@@ -1651,12 +1658,12 @@ discussed in the material on <a href="https://compphysics.github.io/MachineLearn
|
||||
methods</a>.
|
||||
|
||||
<p>
|
||||
This will be discussed next week.
|
||||
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that <b>Scikit-learn</b> offers.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec38">Cancer Data again now with Decision Trees and other Methods </h2>
|
||||
<h2 id="___sec38">Wisconsin Cancer Data </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
|
||||
@@ -101,10 +101,7 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Extending to more predictors', 2, None, '___sec35'),
|
||||
('Including more classes', 2, None, '___sec36'),
|
||||
('More classes', 2, None, '___sec37'),
|
||||
('Cancer Data again now with Decision Trees and other Methods',
|
||||
2,
|
||||
None,
|
||||
'___sec38'),
|
||||
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
||||
('Other measures in classification studies: Cancer Data again',
|
||||
2,
|
||||
None,
|
||||
@@ -1370,7 +1367,16 @@ f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any value from minus infinity to plus infinity. If we however let \( f(y\vert y) \) be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function \( f \), namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of \( y_i \) with a given \( x_i \).
|
||||
This expression implies however that \( f(y_i\vert x_i) \) could take any
|
||||
value from minus infinity to plus infinity. If we however let
|
||||
\( f(y\vert y) \) be represented by the mean value, the above example
|
||||
shows us that we can constrain the function to take values between
|
||||
zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
|
||||
at our last curve we see also that it has an S-shaped form. This leads
|
||||
us to a very popular model for the function \( f \), namely the so-called
|
||||
Sigmoid function or logistic model. We will consider this function as
|
||||
representing the probability for finding a value of \( y_i \) with a given
|
||||
\( x_i \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1378,10 +1384,11 @@ This expression implies however that \( f(y_i\vert x_i) \) could take any value
|
||||
<h2 id="___sec28">The logistic function </h2>
|
||||
|
||||
<p>
|
||||
The perceptron is an example of a ``hard classification" model. We
|
||||
Another widely studied model, is the so-called
|
||||
perceptron model, which is an example of a ``hard classification" model. We
|
||||
will encounter this model when we discuss neural networks as
|
||||
well. Each datapoint is deterministically assigned to a category (i.e
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft"
|
||||
\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
|
||||
classifier that outputs the probability of a given category rather
|
||||
than a single value. For example, given \( x_i \), the classifier
|
||||
outputs the probability of being in a category \( k \). Logistic regression
|
||||
@@ -1595,13 +1602,13 @@ $$
|
||||
<p>
|
||||
Till now we have mainly focused on two classes, the so-called binary
|
||||
system. Suppose we wish to extend to \( K \) classes. Let us for the sake
|
||||
of simplicity assume we have only two predictors. We have then
|
||||
following model
|
||||
of simplicity assume we have only two predictors. We have then following model
|
||||
|
||||
$$
|
||||
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
||||
$$
|
||||
@@ -1656,12 +1663,12 @@ discussed in the material on <a href="https://compphysics.github.io/MachineLearn
|
||||
methods</a>.
|
||||
|
||||
<p>
|
||||
This will be discussed next week.
|
||||
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that <b>Scikit-learn</b> offers.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec38">Cancer Data again now with Decision Trees and other Methods </h2>
|
||||
<h2 id="___sec38">Wisconsin Cancer Data </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
|
||||
Binary file not shown.
@@ -1691,14 +1691,24 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This expression implies however that $f(y_i\\vert x_i)$ could take any value from minus infinity to plus infinity. If we however let $f(y\\vert y)$ be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have $0 \\le f(y_i\\vert x_i) \\le 1$. Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function $f$, namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of $y_i$ with a given $x_i$. \n",
|
||||
"This expression implies however that $f(y_i\\vert x_i)$ could take any\n",
|
||||
"value from minus infinity to plus infinity. If we however let\n",
|
||||
"$f(y\\vert y)$ be represented by the mean value, the above example\n",
|
||||
"shows us that we can constrain the function to take values between\n",
|
||||
"zero and one, that is we have $0 \\le f(y_i\\vert x_i) \\le 1$. Looking\n",
|
||||
"at our last curve we see also that it has an S-shaped form. This leads\n",
|
||||
"us to a very popular model for the function $f$, namely the so-called\n",
|
||||
"Sigmoid function or logistic model. We will consider this function as\n",
|
||||
"representing the probability for finding a value of $y_i$ with a given\n",
|
||||
"$x_i$.\n",
|
||||
"\n",
|
||||
"## The logistic function\n",
|
||||
"\n",
|
||||
"The perceptron is an example of a ``hard classification\" model. We\n",
|
||||
"Another widely studied model, is the so-called \n",
|
||||
"perceptron model, which is an example of a ``hard classification\" model. We\n",
|
||||
"will encounter this model when we discuss neural networks as\n",
|
||||
"well. Each datapoint is deterministically assigned to a category (i.e\n",
|
||||
"$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n",
|
||||
"$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a \"soft\"\n",
|
||||
"classifier that outputs the probability of a given category rather\n",
|
||||
"than a single value. For example, given $x_i$, the classifier\n",
|
||||
"outputs the probability of being in a category $k$. Logistic regression\n",
|
||||
@@ -2030,32 +2040,23 @@
|
||||
"\n",
|
||||
"Till now we have mainly focused on two classes, the so-called binary\n",
|
||||
"system. Suppose we wish to extend to $K$ classes. Let us for the sake\n",
|
||||
"of simplicity assume we have only two predictors. We have then\n",
|
||||
"following model"
|
||||
"of simplicity assume we have only two predictors. We have then following model"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"4\n",
|
||||
"0\n",
|
||||
" \n",
|
||||
"<\n",
|
||||
"<\n",
|
||||
"<\n",
|
||||
"!\n",
|
||||
"!\n",
|
||||
"M\n",
|
||||
"A\n",
|
||||
"T\n",
|
||||
"H\n",
|
||||
"_\n",
|
||||
"B\n",
|
||||
"L\n",
|
||||
"O\n",
|
||||
"C\n",
|
||||
"K"
|
||||
"$$\n",
|
||||
"\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"and"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -2145,13 +2146,13 @@
|
||||
"discussed in the material on [optimization\n",
|
||||
"methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n",
|
||||
"\n",
|
||||
"This will be discussed next week.\n",
|
||||
"This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that **Scikit-learn** offers. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Cancer Data again now with Decision Trees and other Methods"
|
||||
"## Wisconsin Cancer Data"
|
||||
]
|
||||
},
|
||||
{
|
||||
|
||||
Reference in New Issue
Block a user