The figure below shows two positive points (purple squares) and two negative poi
ID: 3884739 • Letter: T
Question
The figure below shows two positive points (purple squares) and two negative points (green circles): That is, the training data set constants of: (x_1, y_1) = ((5, 4), +1) (x_2, y_2) = ((8, 3), +1) (x_3, y_3) = ((7, 2), -1) (x_4, y_4) = ((3, 3)-1) This data set is separable. If we call the horizontal axis of the space u and the vertical axis v, then a form of the decision boundary is v = c^+au. Note that in this form, c is the intersection of the boundary with the vertical (v) axis, and a is the slope. In all cases, if (u, v) is one of the points x_1 or x_2, then the point will be above the boundary (that is, if x_1 = (u, v), then v greaterthanorequalto 2 c^+au, and similarly for x_2). Likewise, c_3 and x_4 must be below the boundary. Various values of a and c could be chosen, but there are significant constraints on what a and c can be. Deduce those constraints, and then find a possible value of a and c in the list below a) v=7 - 7u 8 b) v=8 - 3u 4 c) v=4 - u 9 d) v=7 - 2u 3Explanation / Answer
Consider (u,v) as x1(5,4).
For this v>=c+au
or 4>=c+5a
or c<=4-5a
Conside (u,v) as x2(8,3)
For this v>=c+au
or 3>=c+8a
or c<=3-8a
For (u,v) in x3(7,2), the point will be below the v axis
or v<=c+au
or 2<=c+7a
or c>=2-7a
For (u,v) in x4(3,3), the point will be below the v axis
or v<=c+au
or 3<=c+3a
or c>=3-3a
So the constarints are
c<=4-5a, c<=3-8a, c>=2-7a, c>=3-3a
Considering the given equations of the form v=c+au, lets check if the values of c, a are possible
a) c=7, a=-7/8
If this is a possible value, all constraints will be true
c<=4-5a, c<=3-8a, c>=2-7a, c>=3-3a
or 7<=4+35/8, 7<=3+7, 7>=2+49/8, 7>=3+21/8
The 3rd condition 7>=2+49/8 is false. Hence this point is not possible
b) c=8, a=-3/4
If this is a possible value, all constraints will be true
c<=4-5a, c<=3-8a, c>=2-7a, c>=3-3a
or 8<=4+15/4, 8<=3+6, 8>=2+21/4, 8>=3+9/4
1st condition is false. Hence it is not a possible value
c) c=4, a=-1/9
If this is a possible value, all constraints will be true
c<=4-5a, c<=3-8a, c>=2-7a, c>=3-3a
or 4<=4+5/9, 4<=3+8/9, 4>=2+7/9, 4>=3+3/9
2nd condition is false. Hence it is not a possible value
d) c=7, a=-2/3
If this is a possible value, all constraints will be true
c<=4-5a, c<=3-8a, c>=2-7a, c>=3-3a
or 7<=4+10/3, 7<=3+16/3, 7>=2+14/3, 7>=3+2
All condition are true. Hence (7,-2/3) is a possible value for c,a
So possible values of c and a is (7,-2/3)