## 使用機器學習做類神經網路分類
#### 此次的目的是:
嘗試生成一張各顏色在特定範圍內均勻分布的點陣圖，透過機器學習來分類出特定標籤的點之間的分界。

需要的函式庫

import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense
from sklearn.model_selection import train_test_split

此次選定點陣分布的大概形狀為一同心圓，為了能夠讓每個點在圓中能夠均勻分布，需先定義

def generate_circle_points(r_min, r_max, num_points): # 使用均勻的角度和均勻的半徑生成坐標
    angles = np.random.uniform(0, 2 * np.pi, num_points)
    radii = np.sqrt(np.random.uniform(r_min**2, r_max**2, num_points)) 
    x = radii * np.cos(angles)
    y = radii * np.sin(angles)
    return x, y

接著繪製出散點圖

# 生成數據點
R0 = 0
R1 = 2
R2 = 1
R3 = 3
x_o, y_o = generate_circle_points(R0, R1, N)  #在半徑0 to 2 之間生成N個點，N=200
x_x, y_x = generate_circle_points(R2, R3, N)  #在半徑1 to 3 之間生成N個點，N=200

# 繪製散點圖
plt.figure(figsize=(8, 8))
plt.scatter(x_o, y_o, c='blue', marker='o', label='Positive O') #令生成在半徑 0 to 2的點藍色，標籤為"positive o" 
plt.scatter(x_x, y_x, c='orange', marker='o', label='Negative O')#令生成在半徑 1 to 3的點黃色，標籤為"negative o"

# 整合並標記數據點
x:將{[x_o,y_o以列合併後],[x_x,y_x以列合併後]}用vstack併起來
y:[200個1,200個-1]
X = np.vstack((np.column_stack((x_o, y_o)), np.column_stack((x_x, y_x))))
y = np.array([1] * N + [-1] * N)  # 1代表o，-1代表x


plt.xlabel('X')
plt.ylabel('Y')
plt.title('Scatter Plot with Different Classes')
plt.legend()
plt.grid(True)
plt.axis('equal')
plt.show()

產生的結果為:

define一個使用模型分別對測試集、訓練集的ｘ,ｙ進行預測的function

def plot_predictions(X_train, y_train, X_test, y_test, model):
    # 使用模型對訓練集進行預測
    y_pred_train_continuous = model.predict(X_train)
    y_pred_train = np.where(y_pred_train_continuous.flatten() >= 0, 1, -1)

    # 使用模型對測試集進行預測
    y_pred_test_continuous = model.predict(X_test)
    y_pred_test = np.where(y_pred_test_continuous.flatten() >= 0, 1, -1)
    plt.figure(figsize=(8, 8))

    # 訓練集 - 預測正確
    #positive預測正確:產生藍色'o',label="Train Correct"
    #negative預測正確:產生橘色'o',label="Train Correct"
    correct_train = (y_train == y_pred_train)
    oc = (y_train == 1) & correct_train
    xc = (y_train == -1) & correct_train
    plt.scatter(X_train[oc, 0], X_train[oc, 1], c='blue', marker='o', label='Train Correct (Positive)')
    plt.scatter(X_train[xc, 0], X_train[xc, 1], c='orange', marker='o', label='Train Correct (Negative)')

    # 訓練集 - 預測錯誤
    #positive預測錯誤:產生藍色'x',label="Train InCorrect"
    #negative預測錯誤:產生橘色'x',label="Train InCorrect"
    incorrect_train = (y_train != y_pred_train)
    oi = (y_train == 1) & incorrect_train
    xi = (y_train == -1) & incorrect_train
    plt.scatter(X_train[oi, 0], X_train[oi, 1], c='blue', marker='x', label='Train Incorrect (Positive)')
    plt.scatter(X_train[xi, 0], X_train[xi, 1], c='orange', marker='x', label='Train Incorrect (Negative)')

    # 測試集 - 預測正確
    #positive預測正確:產生青色'o',label="Test Correct"
    #negative預測正確:產生粉色'o',label="Test Correct"
    correct_test = (y_test == y_pred_test)
    oc = (y_test == 1) & correct_test
    xc = (y_test == -1) & correct_test
    plt.scatter(X_test[oc, 0], X_test[oc, 1], c='cyan', marker='o', label='Test Correct (Positive)')
    plt.scatter(X_test[xc, 0], X_test[xc, 1], c='pink', marker='o', label='Test Correct (Negative)')

    # 測試集 - 預測錯誤
    #positive預測錯誤:產生青色'x',label="Test InCorrect"
    #negative預測錯誤:產生粉色'x',label="Test InCorrect"
    incorrect_test = (y_test != y_pred_test)
    oi = (y_test == 1) & incorrect_test
    xi = (y_test == -1) & incorrect_test
    plt.scatter(X_test[oi, 0], X_test[oi, 1], c='cyan', marker='x', label='Test Incorrect (Positive)')
    plt.scatter(X_test[xi, 0], X_test[xi, 1], c='pink', marker='x', label='Test Incorrect (Negative)')

    plt.xlabel('X Coordinate')
    plt.ylabel('Y Coordinate')
    plt.title('Model Predictions')
    plt.legend()
    plt.grid(True)
    plt.show()    #匯出加上預測結果的分布圖


# 分割數據集, 把 N 中的 20% 做為測試
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# 創建神經網路模型
N1 = 3
model = Sequential()
model.add(Dense(N1, input_dim=2, activation='tanh'))  # 第一層有N1個神經元，使用tanh激活函數
model.add(Dense(2, input_dim=2, activation='tanh'))  # 第二層有2個神經元使用tanh
model.add(Dense(1, activation='tanh'))  # 第三層有1個神經元使用tanh

# 編譯模型
model.compile(loss='mean_squared_error', optimizer='adam', metrics=['accuracy'])

接下來就開始訓練

# 訓練模型並記錄訓練過程
#次數為100次，每批量5個
#verbose設置為0表示不顯示訓練過程的詳細信息
#validation_data指定了驗證數據的特徵和標籤。
history = model.fit(X_train, y_train, epochs=100, batch_size=5, verbose=0, validation_data=(X_test, y_test))

# 獲取損失函數值
loss_values = history.history['loss']

<font color="#f00">**損失函數（Loss Function）是在機器學習中用來評估模型預測與實際目標之間差異的一個指標。損失函數值是這個指標的具體數值，它表示模型在一次訓練迭代中對訓練數據的預測效果有多好或多差。**

**在深度學習中，損失函數通常是一個根據模型預測輸出和實際標籤計算的數值。目標是使這個損失函數的值越小越好，因為它表示模型的預測越接近實際標籤。一般而言，損失函數越小，模型的性能越好。**</font>
**model.fit函數會根據訓練數據進行多次訓練迭代，每次迭代都會計算一次損失函數值。這些損失函數值被存儲在history.history['loss']中，並在後續的程式碼中用來繪製損失函數值的變化趨勢。**

# 繪製損失函數圖
plt.figure(figsize=(8, 4))
plt.plot(loss_values, label='Training Loss')
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.title('Loss Function During Training')
plt.legend()
plt.grid(True)
plt.show()

# 使用 plot_predictions 繪製訓練集和測試集中預測正確和預測錯誤的點
plot_predictions(X_train, y_train, X_test, y_test, model)

# 獲取第一層的權重和偏置
weights, biases = model.layers[0].get_weights()


# 生成更多隨機點進行預測
x_a, y_a = generate_circle_points(R0, R3, Np)
additional_X = np.column_stack((x_a, y_a))
y_pred_additional = model.predict(additional_X)
y_pred_additional = np.where(y_pred_additional.flatten() >= 0, 1, -1)

# 繪製測試數據和預測結果
plt.figure(figsize=(8, 8))
plt.scatter(x_o, y_o, c='blue', marker='o', label='Class Positive')
plt.scatter(x_x, y_x, c='orange', marker='o', label='Class Negative')
plt.scatter(additional_X[y_pred_additional == 1, 0], additional_X[y_pred_additional == 1, 1], c='blue', marker='o', label='Predicted Positive')
plt.scatter(additional_X[y_pred_additional == -1, 0], additional_X[y_pred_additional == -1, 1], c='orange', marker='o', label='Predicted Negative')

# 繪製第一層的N1條直線
x_values = np.linspace(-2, 2, 100)
y_values = np.linspace(-2, 2, 100)
for i in range(N1):
a, b = weights[:, i]
c = biases[i]
if abs(a) > abs(b):
x_line = (-b * y_values - c) / a
plt.plot(x_line, y_values, linestyle='--', label=f'Neuron {i+1} Boundary')
else:
y_line = (-a * x_values - c) / b
plt.plot(x_values, y_line, linestyle='--', label=f'Neuron {i+1} Boundary')

plt.xlabel('X')
plt.ylabel('Y')
plt.title('Additional Points Classification with Neuron Boundaries')
plt.legend()
plt.grid(True)
plt.axis('equal')
plt.show()
結果:

算出所有預測錯誤與預測正確的結果，並計算accuracy

y_pred_train_continuous = model.predict(X_train)
y_pred_train = np.where(y_pred_train_continuous.flatten() >= 0, 1, -1)

TP = sum((y_pred_train==1) & (y_train == 1))#true positive 原本是1，預測出來是1
TN = sum((y_pred_train==-1) & (y_train == -1))#true negative 原本是-1，預測出來是-1
FP = sum((y_pred_train==1) & (y_train == -1))#false positive 原本是-1，預測出來是1
FN = sum((y_pred_train==-1) & (y_train == 1))
print("In training set:")
TPR = TP/(TP+FN)
TNR = TN/(TN+FP)
FPR = 1-TNR
FNR = 1-TPR
PPV = TP/(TP+FP)
NPV = TN/(TN+FN)
F1 = 2*PPV*TPR/(PPV+TPR)
ACC = (TP+TN)/(TP+TN+FP+FN)
print("TP=",TP,"TN=",TN,"FP=",FP,"FN=",FN)
print("Accuracy=",ACC)