Perceptron

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graph LR
%% Inputs
X1([x₁])
X2([x₂])
B([1])

%% Neuron and output
SUM([∑])
F([f])
Y([yᵢ])

%% Labeled edges only
X1 -- w₁ --> SUM
X2 -- w₂ --> SUM
B  -- b  --> SUM

SUM --> F --> Y

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Parts

• This is the summation \(z = w_1x_1 + w_2x_2 + b\)
• In Activation Function We can use any kind of functions like Step Function, Sigmoid Function

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Formula

Info

The summation for a simple model can be expressed as: \(z = w_1x_1 + w_2x_2 + b\)

Activation Function : In Activation Function We can use any kind of functions like Step Function, Sigmoid Function

Summation Examples

iq	cgpa	Placed
78	55	1
45	34	0

Let's assume the following weights and bias:

• \(w_1 = 1\)
• \(w_2 = 2\)

• \(b = 6\)

  • If the value is \(\ge\) then placed
  • If the value is \(<\) then not placed

Row 1Row 2

(78 * 1) + (55 * 2) + 6

Output

78 + 110 + 6 = 194 

The value 194 is \(\ge\) the placement condition, so the predicted outcome is Placed (1).

(45 * 1) + (34 * 2) + 6

Output

45 + 68 + 6 = 119

The value 119 is \(\ge\) the placement condition, so the predicted outcome is Placed (1).

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Neuron Vs Perceptron

Interpretation

Geometric Intuition

Explanation

1. Linear Combination / Pre-activation (\(Z\)):

   \[Z = w_1 x_1 + w_2 x_2 + b\]

2. Step Activation Function (\(Y\)):

   \[Y = f(Z) = \begin{cases} 1 & Z \geq 0 \\ 0 & Z < 0 \end{cases}\]

3. General Form of a Line (Substitution):

   Let \(w_1 = A, w_2 = B, b = C\) and \(x_1 = x, x_2 = Y\),

   then:

   \[\text{General form of line: } Ax + By + C = 0\]

4. Region Classification based on the Line:

   \[\begin{cases} \geq 0 & \text{+ve Region} \\ < 0 & \text{-ve Region} \end{cases}\]

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Limitation

• It works only Linear and sort of linear
• Not work on non-linear line

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