2 位二进制输入与非门感知器算法的实现

原文:https://www . geeksforgeeks . org/2 位二进制输入与非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进制分类器的监督学习算法。感知器模型实现以下功能:

[ \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$\boldsymbol{w}$和偏差参数$\boldsymbol{b}$的特定选择，模型预测相应输入向量$\boldsymbol{x}$的输出$\boldsymbol{\hat{y}}$。

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

$\boldsymbol{x_{1}}$ $\boldsymbol{x_{2}}$ $\boldsymbol{y}$
Zero Zero one
Zero one one
one Zero one
one one Zero

我们可以观察到，$NAND(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}) = NOT(AND(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}))$ 现在对于输入向量$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$到 AND 节点的对应权重向量$\boldsymbol{w} : (\boldsymbol{w_{1}}, \boldsymbol{w_{2}})$，关联的感知器函数可以定义为:

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

稍后，“与”节点$\boldsymbol{\hat{y}\prime}$的输出是权重为$\boldsymbol{w_{NOT}}$的“非”节点的输入。那么相应的输出$\boldsymbol{\hat{y}}$就是与非门逻辑函数的最终输出，相关的感知器函数可以定义为:

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

实施时，考虑的权重参数为$\boldsymbol{w_{1}} = 1, \boldsymbol{w_{2}} = 1, \boldsymbol{w_{NOT}} = -1$，偏差参数为$\boldsymbol{b_{AND}} = -1.5, \boldsymbol{b_{NOT}} = 0.5$。

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)))

Output:

NAND(0, 1) = 1
NAND(1, 1) = 0
NAND(0, 0) = 1
NAND(1, 0) = 1

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

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