 LFD Book Forum Problem 3.3 Typo
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#1
 erezarnon Junior Member Join Date: Oct 2012 Posts: 3 Problem 3.3 Typo

In part e, I believe it should ask us to repeat parts b and c
#2 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595 Re: Problem 3.3 Typo

Yes, thank you.

Quote:
 Originally Posted by erezarnon In part e, I believe it should ask us to repeat parts b and c
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#3
 rpistu Member Join Date: Oct 2012 Posts: 10 Re: Problem 3.3 Typo

For part e, when plotting the data and the final hypothesis learnt from a 3rd order polynomial feature transform, the shape of the nonlinear hypothesis is almost like a line. Is that true? Actually, I didn't find an effective way to plot the nonlinear hypothesis.
#4 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595 Re: Problem 3.3 Typo

A simple way to plot a nonlinear classification hypothesis, which is actually the way used for all the figures in the book, is the brute force approach. Construct a grid of points, for example:

x1={0,0.01,0.02,...,1}
x2={0,0.01,0.02,...,1}

So every pair (x1,x2) from each set is a point in 2 dimensions. Now, for every such pair, evaluate the hypothesis h(x1,x2) and plot a red point if h=-1 and a blue point if h=+1. Note h can be any hypothesis, 3rd order polynomial, etc.

It is slow but it works.

Quote:
 Originally Posted by rpistu For part e, when plotting the data and the final hypothesis learnt from a 3rd order polynomial feature transform, the shape of the nonlinear hypothesis is almost like a line. Is that true? Actually, I didn't find an effective way to plot the nonlinear hypothesis.
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#5
 rpistu Member Join Date: Oct 2012 Posts: 10 Re: Problem 3.3 Typo

Thanks. It seems like what you said is to plot the data that classified based on the hypothesis. However, the problem requires us to plot the original data and the final hypothesis. For linear classification, the hypothesis is a line, so it's easy to plot. But for nonliear classification, for example the 3rd order polynomial feature transform here, I think it really hard to find the corresponding x2 if given x1 for h(x1, x2): x->ᶲ3. I cannot imagine by using the brute force approach.

Quote:
 Originally Posted by magdon A simple way to plot a nonlinear classification hypothesis, which is actually the way used for all the figures in the book, is the brute force approach. Construct a grid of points, for example: x1={0,0.01,0.02,...,1} x2={0,0.01,0.02,...,1} So every pair (x1,x2) from each set is a point in 2 dimensions. Now, for every such pair, evaluate the hypothesis h(x1,x2) and plot a red point if h=-1 and a blue point if h=+1. Note h can be any hypothesis, 3rd order polynomial, etc. It is slow but it works.
#6 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595 Re: Problem 3.3 Typo

The hypothesis is not a line. The hypothesis is a region on the 2-dimensional space that is +1 and the complement of that region is -1. The boundary that separates the +1 from the -1 region is a line for linear classification. For nonlinear classification the boundary is something else. In either case you can identify the boundary by identifying the regions which are +1 and the regions which are -1.

For nonlinear boundary, there may be more than one x2 for a given x1 or even no x2 for a given x1. This is why the brute force approach is the simplest for identifying the boundary.

Quote:
 Originally Posted by rpistu Thanks. It seems like what you said is to plot the data that classified based on the hypothesis. However, the problem requires us to plot the original data and the final hypothesis. For linear classification, the hypothesis is a line, so it's easy to plot. But for nonliear classification, for example the 3rd order polynomial feature transform here, I think it really hard to find the corresponding x2 if given x1 for h(x1, x2): x->ᶲ3. I cannot imagine by using the brute force approach.
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Have faith in probability
#7
 rpistu Member Join Date: Oct 2012 Posts: 10 Re: Problem 3.3 Typo

I used the brute force method as you said, however, it still seems like the two regions are separated by a line (maybe a little curve). Also, I get the same plot by using the Matlab function fsolve. It's a little strange.

Quote:
 Originally Posted by magdon The hypothesis is not a line. The hypothesis is a region on the 2-dimensional space that is +1 and the complement of that region is -1. The boundary that separates the +1 from the -1 region is a line for linear classification. For nonlinear classification the boundary is something else. In either case you can identify the boundary by identifying the regions which are +1 and the regions which are -1. For nonlinear boundary, there may be more than one x2 for a given x1 or even no x2 for a given x1. This is why the brute force approach is the simplest for identifying the boundary.
#8
 BojanVujatovic Member Join Date: Jan 2013 Posts: 13 Re: Problem 3.3 Typo

It is also possible to draw these hypothesis (if one is using Matlab as a programming language) using symbolic variables and ezplot:
Code:
```syms x y;
ezplot('curve for plotting as a string');```
And also similar options exist if working with other programming languages.

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