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Day3 : Perceptron
Study Language

πŸ“Œ Perceptronλž€?


νΌμ…‰νŠΈλ‘ (Perceptron)은 생물학적 λ‰΄λŸ°μ„ μˆ˜ν•™μ μœΌλ‘œ λͺ¨λΈλ§ν•œ 인곡 λ‰΄λŸ° λͺ¨λΈλ‘œ, μ—¬λŸ¬ μž…λ ₯ μ‹ ν˜Έλ₯Ό λ°›μ•„ 각 μž…λ ₯에 λŒ€ν•œ κ°€μ€‘μΉ˜(Weight)λ₯Ό κ³±ν•œ ν›„, μ΄λ“€μ˜ 가쀑합(Weighted Sum)을 κ³„μ‚°ν•˜κ³ , ν™œμ„±ν™” ν•¨μˆ˜(Activation Function)λ₯Ό 톡해 μ΅œμ’… 좜λ ₯을 κ²°μ •ν•˜λŠ” ꡬ쑰이닀.


πŸ”§ ꡬ쑰 (Perceptron Structure)

μž…λ ₯(x) β†’ κ°€μ€‘μΉ˜(w) β†’ 가쀑합(βˆ‘) β†’ ν™œμ„±ν™” ν•¨μˆ˜(f) β†’ 좜λ ₯(y)
  • μž…λ ₯ (Input): AND, OR λ“± 논리 연산을 μœ„ν•œ μž…λ ₯ μ‹ ν˜Έ.
  • κ°€μ€‘μΉ˜ (Weight): μž…λ ₯ μ‹ ν˜Έμ˜ μ€‘μš”λ„λ₯Ό κ²°μ •ν•˜λ©°, ν•™μŠ΅μ„ 톡해 쑰정됨.
  • 가쀑합 (Weighted Sum): 각 μž…λ ₯κ³Ό 그에 λŒ€μ‘ν•˜λŠ” κ°€μ€‘μΉ˜μ˜ 곱을 λͺ¨λ‘ λ”ν•œ κ°’.
  • ν™œμ„±ν™” ν•¨μˆ˜ (Activation Function): 가쀑합이 μž„κ³„κ°’μ„ λ„˜μœΌλ©΄ 1, λ„˜μ§€ λͺ»ν•˜λ©΄ 0을 좜λ ₯ν•˜λŠ” ν•¨μˆ˜. λŒ€ν‘œμ μœΌλ‘œ λ‹¨μœ„ 계단 ν•¨μˆ˜ μ‚¬μš©.
  • 좜λ ₯ (Output): μ΅œμ’… κ²°κ³Όκ°’ (보톡 0 λ˜λŠ” 1의 이진 좜λ ₯).

🎯 μš”μ•½

  • νΌμ…‰νŠΈλ‘ μ€ 이진 λΆ„λ₯˜ 문제λ₯Ό ν•΄κ²°ν•  수 μžˆλŠ” κ°€μž₯ 기본적인 신경망 ꡬ쑰이닀.
  • ν•™μŠ΅μ„ 톡해 μž…λ ₯ μ‹ ν˜Έμ˜ μ€‘μš”λ„λ₯Ό λ‚˜νƒ€λ‚΄λŠ” κ°€μ€‘μΉ˜κ°€ μ‘°μ •λœλ‹€.
  • 단측 νΌμ…‰νŠΈλ‘ μ€ μ„ ν˜• 뢄리 κ°€λŠ₯ν•œ 문제만 ν•΄κ²°ν•  수 μžˆλ‹€.

πŸ‘¨β€πŸ’» μ‹€μŠ΅


πŸ’‘ Code : AND & OR & NAND & XOR Gate Perceptron

# AND & OR & NAND & XOR Gate Perceptron
import numpy as np
import matplotlib.pyplot as plt

class Perceptron:
    def __init__(self, input_size, lr=0.1, epochs=10):
        self.weights = np.zeros(input_size)
        self.bias = 0
        self.lr = lr
        self.epochs = epochs
        self.errors = []

    def activation(self, x):
        return np.where(x > 0, 1, 0)

    def predict(self, x):
        linear_output = np.dot(x, self.weights) + self.bias
        return self.activation(linear_output)

    def train(self, X, y):
        for epoch in range(self.epochs):
            total_error = 0
            for xi, target in zip(X, y):
                prediction = self.predict(xi)
                update = self.lr * (target - prediction)
                self.weights += update * xi
                self.bias += update
                total_error += int(update != 0.0)
            self.errors.append(total_error)
            print(f"Epoch {epoch+1}/{self.epochs}, Errors: {total_error}")

# AND 게이트 데이터 및 ν•™μŠ΅
X_and = np.array([[0,0],[0,1],[1,0],[1,1]])
y_and = np.array([0,0,0,1])

print(" AND Gate Training")
ppn_and = Perceptron(input_size=2)
ppn_and.train(X_and, y_and)

print("\n AND Gate Test:")
for x in X_and:
    print(f"Input: {x}, Predicted Output: {ppn_and.predict(x)}")

# OR 게이트 데이터 및 ν•™μŠ΅
X_or = np.array([[0,0],[0,1],[1,0],[1,1]])
y_or = np.array([0,1,1,1])

print("\n OR Gate Training")
ppn_or = Perceptron(input_size=2)
ppn_or.train(X_or, y_or)

print("\n OR Gate Test:")
for x in X_or:
    print(f"Input: {x}, Predicted Output: {ppn_or.predict(x)}")

# NAND 게이트 데이터 및 ν•™μŠ΅
X_nand = np.array([[0,0],[0,1],[1,0],[1,1]])
y_nand = np.array([1,1,1,0])  # AND와 λ°˜λŒ€

print("\n NAND Gate Training")
ppn_nand = Perceptron(input_size=2)
ppn_nand.train(X_nand, y_nand)

print("\n NAND Gate Test:")
for x in X_nand:
    print(f"Input: {x}, Predicted Output: {ppn_nand.predict(x)}")

# XOR 게이트 데이터 및 ν•™μŠ΅
X_xor = np.array([[0,0],[0,1],[1,0],[1,1]])
y_xor = np.array([0,1,1,0])  # μ„ ν˜• 뢄리 λΆˆκ°€λŠ₯

print("\n XOR Gate Training")
ppn_xor = Perceptron(input_size=2)
ppn_xor.train(X_xor, y_xor)

print("\n XOR Gate Test:")
for x in X_xor:
    print(f"Input: {x}, Predicted Output: {ppn_xor.predict(x)}")

βœ… Result : AND & OR & NAND & XOR Gate Perceptron

 AND Gate Training
Epoch 1/10, Errors: 1
Epoch 2/10, Errors: 3
Epoch 3/10, Errors: 3
Epoch 4/10, Errors: 2
Epoch 5/10, Errors: 1
Epoch 6/10, Errors: 0
Epoch 7/10, Errors: 0
Epoch 8/10, Errors: 0
Epoch 9/10, Errors: 0
Epoch 10/10, Errors: 0

 AND Gate Test:
Input: [0 0], Predicted Output: 0
Input: [0 1], Predicted Output: 0
Input: [1 0], Predicted Output: 0
Input: [1 1], Predicted Output: 1

 OR Gate Training
Epoch 1/10, Errors: 1
Epoch 2/10, Errors: 2
Epoch 3/10, Errors: 1
Epoch 4/10, Errors: 0
Epoch 5/10, Errors: 0
Epoch 6/10, Errors: 0
Epoch 7/10, Errors: 0
Epoch 8/10, Errors: 0
Epoch 9/10, Errors: 0
Epoch 10/10, Errors: 0

 OR Gate Test:
Input: [0 0], Predicted Output: 0
Input: [0 1], Predicted Output: 1
Input: [1 0], Predicted Output: 1
Input: [1 1], Predicted Output: 1

 NAND Gate Training
Epoch 1/10, Errors: 2
Epoch 2/10, Errors: 3
Epoch 3/10, Errors: 3
Epoch 4/10, Errors: 0
Epoch 5/10, Errors: 0
Epoch 6/10, Errors: 0
Epoch 7/10, Errors: 0
Epoch 8/10, Errors: 0
Epoch 9/10, Errors: 0
Epoch 10/10, Errors: 0

 NAND Gate Test:
Input: [0 0], Predicted Output: 1
Input: [0 1], Predicted Output: 1
Input: [1 0], Predicted Output: 1
Input: [1 1], Predicted Output: 0

 XOR Gate Training
Epoch 1/10, Errors: 2
Epoch 2/10, Errors: 3
Epoch 3/10, Errors: 4
Epoch 4/10, Errors: 4
Epoch 5/10, Errors: 4
Epoch 6/10, Errors: 4
Epoch 7/10, Errors: 4
Epoch 8/10, Errors: 4
Epoch 9/10, Errors: 4
Epoch 10/10, Errors: 4

 XOR Gate Test:
Input: [0 0], Predicted Output: 1
Input: [0 1], Predicted Output: 1
Input: [1 0], Predicted Output: 0
Input: [1 1], Predicted Output: 0

πŸ’‘ Code : 경계 κ²°μ • μ‹œκ°ν™” ν•¨μˆ˜ (AND, OR, NAND, XOR)

# 경계 κ²°μ • μ‹œκ°ν™” ν•¨μˆ˜ (AND, OR, NAND, XOR)
from matplotlib.colors import ListedColormap
import matplotlib.pyplot as plt
import numpy as np

def plot_decision_boundary(X, y, model, title='Perceptron Decision Boundary'):
    cmap_light = ListedColormap(['#FFDDDD', '#DDDDFF'])  # λ°°κ²½ 색상
    cmap_bold = ListedColormap(['#FF0000', '#0000FF'])   # 점 색상

    h = .02  # mesh grid 간격

    x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
    y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
                         np.arange(y_min, y_max, h))

    Z = model.predict(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)

    plt.figure(figsize=(6, 5))
    plt.contourf(xx, yy, Z, cmap=cmap_light)
    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=cmap_bold,
                edgecolor='k', s=100, marker='o')
    plt.xlabel('Input 1')
    plt.ylabel('Input 2')
    plt.title(title)
    plt.grid(True)
    plt.show()

# AND 게이트 κ²°μ • 경계 μ‹œκ°ν™”
plot_decision_boundary(X_and, y_and, ppn_and, title='AND Gate Decision Boundary')

# OR 게이트 κ²°μ • 경계 μ‹œκ°ν™”
plot_decision_boundary(X_or, y_or, ppn_or, title='OR Gate Decision Boundary')

# NAND 게이트 κ²°μ • 경계 μ‹œκ°ν™”
plot_decision_boundary(X_nand, y_nand, ppn_nand, title='NAND Gate Decision Boundary')

# XOR 게이트 κ²°μ • 경계 μ‹œκ°ν™”
plot_decision_boundary(X_xor, y_xor, ppn_xor, title='XOR Gate Decision Boundary')

βœ… Result : 경계 κ²°μ • μ‹œκ°ν™” ν•¨μˆ˜ (AND, OR, NAND, XOR)

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πŸ’‘ Code : # 였λ₯˜ μ‹œκ°ν™” (AND, OR, NAND, XOR)

# 였λ₯˜ μ‹œκ°ν™” (AND, OR, NAND, XOR)
plt.figure(figsize=(8, 5))
plt.plot(range(1, len(ppn_and.errors) + 1), ppn_and.errors, marker='o', label='AND Gate')
plt.plot(range(1, len(ppn_or.errors) + 1), ppn_or.errors, marker='s', label='OR Gate')
plt.plot(range(1, len(ppn_nand.errors) + 1), ppn_nand.errors, marker='^', label='NAND Gate')
plt.plot(range(1, len(ppn_xor.errors) + 1), ppn_xor.errors, marker='x', label='XOR Gate')
plt.xlabel('Epochs')
plt.ylabel('Number of Errors')
plt.title('Perceptron Learning Error Over Epochs')
plt.legend()
plt.grid(True)
plt.show()

βœ… Result : 였λ₯˜ μ‹œκ°ν™” (AND, OR, NAND, XOR)

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πŸ’¬ Comment

  • νΌμ…‰νŠΈλ‘ : μž…λ ₯ 벑터에 κ°€μ€‘μΉ˜λ₯Ό κ³±ν•œ 합이 κΈ°μ€€(0)을 λ„˜λŠ”μ§€ νŒλ‹¨ν•˜κ³ , ν•™μŠ΅ κ³Όμ •μ—μ„œλŠ” ν‹€λ¦° 만큼만 μ‘°μ •ν•˜λ©° μ„ ν˜• 뢄리λ₯Ό λ°°μš°λŠ” ꡬ쑰
  • XOR은 μ„ ν˜• 뢄리 λΆˆκ°€λŠ₯ν•œ 문제이기 λ•Œλ¬Έμ—
    단측 νΌμ…‰νŠΈλ‘ μœΌλ‘œλŠ” ν•΄κ²°ν•  수 μ—†λ‹€.
  • 이λ₯Ό ν•΄κ²°ν•˜λ €λ©΄ λ‹€μΈ΅ νΌμ…‰νŠΈλ‘ (MLP)μ΄λ‚˜ λΉ„μ„ ν˜• λ³€ν™˜μ΄ ν•„μš”ν•˜λ‹€.

πŸ’‘ Code : MLP둜 XOR 문제 ν•΄κ²°

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap

class MultiLayerPerceptron:
    def __init__(self, input_size=2, hidden_size=4, output_size=1, lr=0.5, epochs=1000):
        self.W1 = np.random.uniform(-1, 1, (input_size, hidden_size))
        self.b1 = np.zeros((1, hidden_size))
        self.W2 = np.random.uniform(-1, 1, (hidden_size, output_size))
        self.b2 = np.zeros((1, output_size))
        self.lr = lr
        self.epochs = epochs
        self.losses = []

    def sigmoid(self, x):
        return 1 / (1 + np.exp(-np.clip(x, -250, 250)))

    def sigmoid_derivative(self, x):
        return x * (1 - x)

    def forward(self, X):
        self.z1 = np.dot(X, self.W1) + self.b1
        self.a1 = self.sigmoid(self.z1)
        self.z2 = np.dot(self.a1, self.W2) + self.b2
        self.a2 = self.sigmoid(self.z2)
        return self.a2

    def backward(self, X, y, output):
        m = X.shape[0]
        dZ2 = output - y
        dW2 = (1 / m) * np.dot(self.a1.T, dZ2)
        db2 = (1 / m) * np.sum(dZ2, axis=0, keepdims=True)
        dZ1 = np.dot(dZ2, self.W2.T) * self.sigmoid_derivative(self.a1)
        dW1 = (1 / m) * np.dot(X.T, dZ1)
        db1 = (1 / m) * np.sum(dZ1, axis=0, keepdims=True)

        self.W2 -= self.lr * dW2
        self.b2 -= self.lr * db2
        self.W1 -= self.lr * dW1
        self.b1 -= self.lr * db1

    def train(self, X, y):
        for epoch in range(self.epochs):
            output = self.forward(X)
            loss = np.mean((output - y) ** 2)
            self.losses.append(loss)
            self.backward(X, y, output)
            #if epoch % 200 == 0:
            #    print(f"Epoch {epoch}/{self.epochs}, Loss: {loss:.6f}")

    def predict(self, X):
        output = self.forward(X)
        return (output > 0.5).astype(int)

    def predict_prob(self, X):
        return self.forward(X).ravel()  # κ²°μ • κ²½κ³„μš©


# === XOR 데이터 ===
X_xor = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y_xor = np.array([[0], [1], [1], [0]])

# === ν•™μŠ΅ ===
print("\n=== XOR Gate Multi-Layer Perceptron Training ===")
mlp = MultiLayerPerceptron(input_size=2, hidden_size=2, lr=0.5, epochs=10000)
mlp.train(X_xor, y_xor)

# === 예츑 κ²°κ³Ό 좜λ ₯ ===
print("\nXOR GATE Test (Multi-Layer Perceptron):")
xor_predictions = mlp.predict(X_xor)
for i, x in enumerate(X_xor):
    predicted = xor_predictions[i][0]
    actual = y_xor[i][0]
    result = "βœ“" if predicted == actual else "βœ—"
    print(f"Input: {x}, Predicted: {predicted}, Actual: {actual}, {result}")


# === κ²°μ • 경계 μ‹œκ°ν™” ν•¨μˆ˜ ===
def plot_decision_boundary(X, y, model, title="Decision Boundary"):
    cmap_light = ListedColormap(['#FFDDDD', '#DDDDFF'])
    cmap_bold = ListedColormap(['#FF0000', '#0000FF'])

    h = .01
    x_min, x_max = X[:, 0].min() - 0.5, X[:, 0].max() + 0.5
    y_min, y_max = X[:, 1].min() - 0.5, X[:, 1].max() + 0.5
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
                         np.arange(y_min, y_max, h))

    grid = np.c_[xx.ravel(), yy.ravel()]
    Z = model.predict_prob(grid)
    Z = Z.reshape(xx.shape)

    plt.figure(figsize=(6, 5))
    plt.contourf(xx, yy, Z > 0.5, cmap=cmap_light)
    plt.scatter(X[:, 0], X[:, 1], c=y.ravel(), cmap=cmap_bold,
                edgecolor='k', s=120)
    plt.title(title)
    plt.xlabel("Input 1")
    plt.ylabel("Input 2")
    plt.grid(True)
    plt.show()


# === κ²°μ • 경계 μ‹œκ°ν™” ===
plot_decision_boundary(X_xor, y_xor, mlp, title="XOR MLP Decision Boundary")

# === 손싀 곑선 μ‹œκ°ν™” ===
plt.figure(figsize=(8, 5))
plt.plot(range(mlp.epochs), mlp.losses, color='purple')
plt.title("MLP Training Loss on XOR Problem")
plt.xlabel("Epochs")
plt.ylabel("Mean Squared Error")
plt.grid(True)
plt.show()

βœ… Result : MLP둜 XOR 문제 ν•΄κ²°

=== XOR Gate Multi-Layer Perceptron Training ===

XOR GATE Test (Multi-Layer Perceptron):
Input: [0 0], Predicted: 0, Actual: 0, βœ“
Input: [0 1], Predicted: 1, Actual: 1, βœ“
Input: [1 0], Predicted: 1, Actual: 1, βœ“
Input: [1 1], Predicted: 0, Actual: 0, βœ“

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