Decision Boundary Line (Linear/Non-Linear)

In summary, by extending the feature space with quadratic terms, the non-linear decision boundary line can be represented as a linear equation in terms of X1, X1^2, X2, and X2^2. This is because the equation (1 + X1)^2 + (2 − X2)^2 = 4 can be simplified to X1^2 - 2X1 + X2^2 - 4X2 + 1 = 0, and by letting Y1 = X1^2 and Y2 = X2^2, the equation becomes linear in terms of X1, X2, Y1, and Y2.
  • #1
brojesus111
39
0

Homework Statement



Given a non-linear decision boundary line: (1 + X1)^2 + (2 − X2)^2 = 4

Argue that while the decision boundary is not linear in terms of X1 and X2, it is linear in terms of X1,X1^2 , X2, and X2^2 .

The Attempt at a Solution



I'm honestly not sure. I realize the curve is a circle, but I don't understand how it could be turned linear by having it terms of X1,X1^2 , X2, and X2^2
 
Physics news on Phys.org
  • #2
Is it because we are extending the feature space by including quadratic terms that can address this non-linearity?
 
  • #3
It's pretty basic algebra that (1+ X1)^2+ (2- X2)^2= X1^2- 2X1+ 1+ X2^2- 4X2+ 4= 4
so X1^2- 2X1+ X2^2- 4X2+ 1= 0.

If you let Y1= X1^2 and Y2= X2^2, then you have Y1- 2X1+ Y2- 4Y1+ 1= 0 which is 'linear in X1, X2, Y1, and Y2".
 

1. What is a decision boundary line?

A decision boundary line is a line that separates different classes or categories in a dataset. It is used in classification algorithms to determine which class a data point belongs to based on its features.

2. How is a decision boundary line determined?

A decision boundary line is determined by the algorithm used for classification. In linear classification, the boundary line is a straight line that maximally separates the classes. In non-linear classification, more complex boundary lines can be used, such as curves or higher order polynomials.

3. What is the role of a decision boundary line in machine learning?

The decision boundary line is a crucial component in machine learning as it helps to classify data points and make predictions. It is used to determine the class of new data points based on their features and the learned relationship between the features and classes.

4. Can a decision boundary line change?

Yes, a decision boundary line can change depending on the type of data and the algorithm used for classification. For example, in non-linear classification, the boundary line can change based on the complexity of the model or the number of features used.

5. How does the choice of algorithm affect the decision boundary line?

The choice of algorithm can greatly affect the decision boundary line. Different algorithms have different ways of determining the boundary line, and some may be better suited for certain types of data or classes. It is important to choose the right algorithm to ensure an accurate and efficient decision boundary line for classification tasks.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
4K
  • Calculus and Beyond Homework Help
Replies
14
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
Back
Top