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 --"w<sub>11</sub>"--> S1
I2 --"w<sub>12</sub>"--> S1
I0 --"b<sub>1</sub>"--> S1
S1 --> A1 --> O1
O1((y<sub>1</sub>))
I1((x<sub>1</sub>))
I2((x<sub>2</sub>))
I0((1))
A1((f<sub>bs</sub>))
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 --"w<sub>11</sub>"--> H1
I2 --"w<sub>12</sub>"--> H1
I1 --"w<sub>21</sub>"--> H2
I2 --"w<sub>22</sub>"--> H2
H1 --"u<sub>11</sub>"--> O1
H2 --"u<sub>12</sub>"--> O1
I1((x<sub>1</sub>))
I2((x<sub>2</sub>))
H1(["y<sub>1</sub>|tanh"])
H2(["y<sub>2</sub>|tanh"])
O1(["z<sub>1</sub>|bstep"])