Retinal Prosthesis Dissertation - Student Home Pages

x2
Linear separation
1.00
0.90
0.80
0.70
0.60
0.50
0.40
x2
Class A
Class B
Linear (x2)
0.30
0.20
0.10
0.00
-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00
x1
Figure 13 A hyperplane separating two classes/clusters of data.
Based on the equation for a straight line i.e. y = mx +c we can say that the equation
for a line separating the data clusters of class A and class B is x 2 = m x 1 +c. This can
be re-arranged to give x 2 – mx 1 – c = 0 and we can find the value for c by noting
where the line passes through the x 2 axis (the y of the fundamental straight line
equation), i.e. at x 2 = 0.235. Therefore: when x 1 = 0, x 2 = 0.235. Substituting this into
x 2 = m x 1 +c, this gives 2.35 – m*0 – c = 0 and thus c = 2.35.
Likewise m can be found by observing where the line crosses the x 1 axis, at x = -0.4,
so when x 2 = 0, then x 1 will equal -0.4. Substituting gets 0 – m (-0.4) – 2.35 = 0, thus
0 – m (-0.4) = 2.35 which is 0.4m = 2.35, therefore m = 2.35/0.4 = 5.875.
The equation for the straight line (hyperplane) of figure 10 is:
x 2 – 5.875x 1 – 2.35 = 0. (5)
In terms of neural networks this hyperplane is fixed by the weights and biases of the
neuron. The bias defines the position of the plane in terms of its perpendicular
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