futher intro perceptron

Feed forward and learning algorithms as further intro to single-layer perceptron.

Info:

• Further intro to perceptron: Datasets generation, classification, andlearning
  • url https://osf.io/9vxfz
  • version 20241119_v7
• Outline
  • Intro 3
  • Linear separable dataset generation 6
  • Feed forward in single-layer perceptron 23
  • Binary classification ability 29
  • Decision boundary line 47
  • Learning process 60
  • Error estimation 76
  • Additional topic as intermezzo 91
  • Closing 98

Sketch:

$$\tag{1} f(x) = \left\{ \begin{array}{lr} 1, & x \ge 0, \newline 0, & x < 0. \end{array} \right.$$

$$\tag{2} ax + by + c = 0.$$

$$\tag{3} z_1 = w_{11} x_1 + w_{12} x_2 + b_1$$

$$\tag{4} y_1 = f_{\rm bs}(z_1)$$

$$\tag{5} y_1 = f\left( \left[ \begin{array}{ccc} w_{11} & w_{12} & b_1 \end{array} \right] \left[ \begin{array}{c} x_1 \newline x_2 \newline 1 \end{array} \right] \right)$$

$$\tag{6} x^{n+1}_i = y^{n}_i$$

$$\tag{7} x_i^{n+1} = f^n\left( \left[ \begin{array}{ccc} w_{ij}^n & w_{ij}^n & b_i^n \end{array} \right] \left[ \begin{array}{c} x_j^n \newline x_j^n \newline 1 \end{array} \right] \right)$$

flowchart RL
subgraph n[" "]
  S1
  A1
end
I1 --"w11"--> S1
I2 --"w12"--> S1
I0 --"b1"--> S1
S1 --> A1 --> O1
O1((y1))
I1((x1))
I2((x2))
I0((1))
A1((fbs))
S1(["Σ wx+b"])

$$\tag{8} w_{11} x_1 + w_{12} x_2 + b = 0.$$

$$\tag{9} x_2 = - \left(\frac{w_{11}}{w_{12}}\right) x_1 - \left(\frac{b}{w_{12}}\right).$$

$$\tag{10} {\rm SSE} = \sum_{i = 1}^n (y_i - \hat{y}_i)^2.$$

$$\tag{11} {\rm MCE} = \frac{1}{n} \sum_{i = 1}^n \delta(y_i,\hat{y}_i).$$

$$\tag{12} \delta(a, b) = \left\{ \begin{array}{cc} 1 & a = b, \newline 0 & a \ne b. \end{array} \right.$$

flowchart RL
I1 --"w11"--> H1
I2 --"w12"--> H1
I1 --"w21"--> H2
I2 --"w22"--> H2
H1 --"u11"--> O1
H2 --"u12"--> O1
I1((x1))
I2((x2))
H1(["y1|tanh"])
H2(["y2|tanh"])
O1(["z1|bstep"])