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📜  2位二进制输入与非逻辑门感知器算法的实现

📅  最后修改于: 2022-05-13 01:54:50.132000             🧑  作者: Mango

2位二进制输入与非逻辑门感知器算法的实现


在机器学习领域,感知器是一种用于二元分类器的监督学习算法。感知器模型实现以下函数:

    \[ \begin{array}{c} \hat{y}=\Theta\left(w_{1} x_{1}+w_{2} x_{2}+\ldots+w_{n} x_{n}+b\right) \\ =\Theta(\mathbf{w} \cdot \mathbf{x}+b) \\ \text { where } \Theta(v)=\left\{\begin{array}{cc} 1 & \text { if } v \geqslant 0 \\ 0 & \text { otherwise } \end{array}\right. \end{array} \]

对于权重向量的特定选择$\boldsymbol{w}$和偏置参数$\boldsymbol{b}$ ,模型预测输出$\boldsymbol{\hat{y}}$对于相应的输入向量$\boldsymbol{x}$ .

2位二进制变量的NAND逻辑函数真值表,即输入向量$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$和相应的输出$\boldsymbol{y}$

$\boldsymbol{x_{1}}$$\boldsymbol{x_{2}}$$\boldsymbol{y}$
001
011
101
110

我们可以观察到, $NAND(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}) = NOT(AND(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}))$
现在对于相应的权重向量$\boldsymbol{w} : (\boldsymbol{w_{1}}, \boldsymbol{w_{2}})$输入向量的$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$到 AND 节点,相关的感知函数可以定义为:

    \[$\boldsymbol{\hat{y}\prime} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{AND}\right)$ \]

稍后,AND节点的输出$\boldsymbol{\hat{y}\prime}$是具有权重的 NOT 节点的输入$\boldsymbol{w_{NOT}}$ .然后对应的输出$\boldsymbol{\hat{y}}$是 NAND 逻辑函数的最终输出,相关的感知器函数可以定义为:

    \[$\boldsymbol{\hat{y}} = \Theta\left(w_{NOT}  \boldsymbol{\hat{y}\prime}+b_{NOT}\right)$\]


对于实现,考虑的权重参数是$\boldsymbol{w_{1}} = 1, \boldsymbol{w_{2}} = 1, \boldsymbol{w_{NOT}} = -1$和偏置参数是$\boldsymbol{b_{AND}} = -1.5, \boldsymbol{b_{NOT}} = 0.5$ .

Python实现:

# importing Python library
import numpy as np
  
# define Unit Step Function
def unitStep(v):
    if v >= 0:
        return 1
    else:
        return 0
  
# design Perceptron Model
def perceptronModel(x, w, b):
    v = np.dot(w, x) + b
    y = unitStep(v)
    return y
  
# NOT Logic Function
# wNOT = -1, bNOT = 0.5
def NOT_logicFunction(x):
    wNOT = -1
    bNOT = 0.5
    return perceptronModel(x, wNOT, bNOT)
  
# AND Logic Function
# w1 = 1, w2 = 1, bAND = -1.5
def AND_logicFunction(x):
    w = np.array([1, 1])
    bAND = -1.5
    return perceptronModel(x, w, bAND)
  
# NAND Logic Function
# with AND and NOT  
# function calls in sequence
def NAND_logicFunction(x):
    output_AND = AND_logicFunction(x)
    output_NOT = NOT_logicFunction(output_AND)
    return output_NOT
  
# testing the Perceptron Model
test1 = np.array([0, 1])
test2 = np.array([1, 1])
test3 = np.array([0, 0])
test4 = np.array([1, 0])
  
print("NAND({}, {}) = {}".format(0, 1, NAND_logicFunction(test1)))
print("NAND({}, {}) = {}".format(1, 1, NAND_logicFunction(test2)))
print("NAND({}, {}) = {}".format(0, 0, NAND_logicFunction(test3)))
print("NAND({}, {}) = {}".format(1, 0, NAND_logicFunction(test4)))
输出:
NAND(0, 1) = 1
NAND(1, 1) = 0
NAND(0, 0) = 1
NAND(1, 0) = 1

这里,模型预测输出( $\boldsymbol{\hat{y}}$ ) 每个测试输入都与 NAND 逻辑门常规输出 ( $\boldsymbol{y}$ ) 根据 2 位二进制输入的真值表。
因此,验证了 NAND 逻辑门的感知器算法是正确实现的。