For a 3-class problem with three discriminant functions $g_{12}, g_{13},$ and $g_{32}$, we can determine the class of a feature vector by evaluating these functions. Here's a step-by-step solution:

Given Discriminant Functions:

1. $g_{12} = x_1 = 0$

   • This function tells us the boundary between Class 1 and Class 2 lies along the line $x_1 = 0$.

2. $g_{13} = x_2 = 0$

   • This function tells us the boundary between Class 1 and Class 3 lies along the line $x_2 = 0$.

3. $g_{32} = x_1 + x_2 = 0$

   • This function tells us the boundary between Class 3 and Class 2 lies along the line $x_1 + x_2 = 0$.

To classify a feature vector, we check the position of the vector relative to these boundaries.

Part (a): Classifying the Feature Vector $(6, 4)$

The feature vector $(6, 4)$ has:

• $x_1 = 6$ and $x_2 = 4$.

1. Check with $g_{12} = x_1 = 0$:

   • Since $x_1 = 6 > 0$, the vector lies in the region for Class 1 or Class 3.

2. Check with $g_{13} = x_2 = 0$:

   • Since $x_2 = 4 > 0$, the vector lies in the region for Class 1 or Class 2.

3. Check with $g_{32} = x_1 + x_2 = 0$:

   • $x_1 + x_2 = 6 + 4 = 10 > 0$, so the vector lies in the region for Class 1.

Since the vector satisfies all conditions for Class 1, the classification for the feature vector $(6, 4)$ is:

$\text{Class 1}$

Part (b): Classifying for Class 2

To determine where Class 2 would lie, we need:

1. $x_1 \leq 0$,
2. $x_1 + x_2 \leq 0$, and
3. $x_2 > 0$.

Part (c): Classifying for Class 3

To determine where Class 3 would lie, we need:

1. $x_2 \leq 0$,
2. $x_1 + x_2 \geq 0$, and
3. $x_1 > 0$.

Part (d): Ambiguous Point

An ambiguous point would satisfy all boundaries, meaning it lies precisely on the boundaries between classes:

• For example, a point like $(0, 0)$ satisfies $g_{12} = 0$, $g_{13} = 0$, and $g_{32} = 0$, thus it lies on the intersection of all three boundaries and is ambiguous.

Summary:

• (6, 4) belongs to Class 1.
• Class 2 points satisfy $x_1 \leq 0$, $x_2 > 0$, $x_1 + x_2 \leq 0$.
• Class 3 points satisfy $x_2 \leq 0$, $x_1 > 0$, $x_1 + x_2 \geq 0$.
• Ambiguous point example: $(0, 0)$.

Would you like further details or explanations?

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1. How do discriminant functions determine class boundaries?
2. Can a feature vector lie on a boundary and still belong to a class?
3. What types of discriminant functions are used for multi-class problems?
4. How does ambiguity arise in classification boundaries?
5. What methods can be used to resolve ambiguity in classification?

Tip: When working with multiple discriminant functions, sketching the boundaries helps visualize class regions and intersections.