寫在前面

本博客為模式識別作業的記錄，實作批感知器演算法、Ho Kashyap演算法和MSE多類擴展方法，可參考教材 [ 1 ] \color{#0000FF}{[1]} [1]，所用資料如下如所示：

批感知器演算法

從 a = 0 \mathbf a=0 a=0開始迭代，分類 ω 1 \omega_1 ω1?和 ω 2 \omega_2 ω2?并計算最終的解向量，記錄下收斂的步數，

"""批感知器演算法"""

import matplotlib.pyplot as plt
import numpy as np
import math

data = np.loadtxt('data.txt')
trn_data1 = data[0:20, 0:2]   # 問題一（1）
trn_lable1 = data[0:20, 2]

# trn_data1 = data[10:30, 0:2]   # 問題一（1）
# trn_lable1 = data[10:30, 2]

plt.scatter(trn_data1[0:10, 0], trn_data1[0:10, 1], color='blue', marker='o', label=''r'$\omega_1$')
plt.scatter(trn_data1[10:20, 0], trn_data1[10:20, 1], color='red', marker='x', label=''r'$\omega_2$')
plt.xlabel(''r'$x_1$')
plt.ylabel(''r'$x_2$')
plt.legend(loc='upper left')
plt.title('Original Data')
plt.show()

X = np.hstack((np.ones((trn_data1.shape[0], 1)), trn_data1))

X1 = X[0:10, :]
X2 = -1 * X[10:20, :]
X = np.vstack((X1, X2))   # 規范化增廣樣本
w = np.zeros((3, 1), dtype='float32')


step = 0
for i in range(100):
    if i == 0:
        y_pred = np.where(np.dot(X, w) == 0)[0]  # 統計分類錯誤的點數
    else:
        y_pred = np.where(np.dot(X, w) < 0)[0]
    step += 1
    print('第%2d次更新，分類錯誤的點個數：%2d' % (step, len(y_pred)))
    if len(y_pred) > 0:
        w += (np.sum(X[y_pred, :], axis=0)).reshape((3, 1))
    else:
        break

print('解向量為：', w)
x1 = np.array([-5, 8])
x2 = -1 * (w[1]*x1 + w[0])/w[2]

plt.scatter(trn_data1[0:10, 0], trn_data1[0:10, 1], color='blue', marker='o', label=''r'$\omega_1$')
plt.scatter(trn_data1[10:20, 0], trn_data1[10:20, 1], color='red', marker='x', label=''r'$\omega_2$')
plt.plot([x1[0], x1[1]], [x2[0], x2[1]], 'black')
plt.xlabel(''r'$x_1$')
plt.ylabel(''r'$x_2$')
plt.legend(loc='upper left')
plt.title('classified Data')
plt.show()

演算法在第24次更新收斂，此時解向量 a = ( 34.0 , ? 30.4 , 34.1 ) \mathbf{a} = (34.0,?30.4, 34.1) a=(34.0,?30.4,34.1)

Ho Kashyap演算法

"""Ho Kashyap演算法"""

import matplotlib.pyplot as plt
import numpy as np
import math

data = np.loadtxt('data.txt')
trn_data1 = data[0:10, 0:2]   # 問題二(1)
trn_lable1 = data[0:10, 2]
temp = data[20:30, 0:2]
temp_lb = data[20:30, 2]

# trn_data1 = data[10:20, 0:2]   # 問題二(2)
# trn_lable1 = data[10:20, 2]
# temp = data[30:40, 0:2]
# temp_lb = data[30:40, 2]

trn_data = np.vstack((trn_data1, temp))
trn_lable = np.vstack((trn_lable1, temp_lb))

# print(trn_data.shape, '\n', trn_lable.shape)
# print(trn_data)

plt.scatter(trn_data[0:10, 0], trn_data[0:10, 1], color='blue', marker='o', label=''r'$\omega_2$')
plt.scatter(trn_data[10:20, 0], trn_data[10:20, 1], color='red', marker='x', label=''r'$\omega_4$')
plt.xlabel(''r'$x_1$')
plt.ylabel(''r'$x_2$')
plt.legend(loc='upper left')
plt.title('Original Data')
plt.show()

X = np.hstack((np.ones((trn_data.shape[0], 1)), trn_data))
X1 = X[0:10, :]
X2 = -1 * X[10:20, :]
X = np.vstack((X1, X2))   # 規范化增廣樣本
w = np.random.randn(3, 1)
b = np.random.rand(X.shape[0], 1)
eta = 0.1
MAX_iteration = 1000

step = 0
acc = 0
for i in range(MAX_iteration):
    e = np.dot(X, w) - b
    e_p = 0.5 * (e + abs(e))
    eta_k = eta/(i+1)
    # if (e == 0).all():
    if max(abs(e)) <= min(b):
        print(e)
        break
    else:
        b += 2 * eta_k * e_p
        Y_p = np.linalg.pinv(X)
        w = np.dot(Y_p, b)
    step += 1
    acc = (np.sum(np.dot(X, w) > 0)) / X.shape[0]
    print('第%2d次更新, 分類準確率%f' % (step, acc))
    # print(e, '\n')
    if step == MAX_iteration and acc != 1:
        print("未找到解，線性不可分！")


x1 = np.array([-8, 8])
x2 = -1 * (w[1]*x1 + w[0])/w[2]
plt.scatter(trn_data[0:10, 0], trn_data[0:10, 1], color='blue', marker='o', label=''r'$\omega_2$')
plt.scatter(trn_data[10:20, 0], trn_data[10:20, 1], color='red', marker='x', label=''r'$\omega_4$')
plt.plot([x1[0], x1[1]], [x2[0], x2[1]], 'black')
plt.xlabel(''r'$x_1$')
plt.ylabel(''r'$x_2$')
plt.text(3, -2, 'Accuracy: %.3f' % acc)
plt.legend(loc='upper left')
plt.title('classified Data')
plt.show()

MSE多類擴展方法

首先根據訓練樣本計算出權值W，然后根據此權值和測驗樣本計算出預測值，并對該值取argmax 得到最終的輸出標簽，從而得到分類正確率為100%，

"""MSE多類擴展方法"""

import numpy as np

data = np.loadtxt('data.txt')
train_data = []
train_label = []
test_data = []
test_label = []

# 資料預處理
for i in [0, 10, 20, 30]:
    temp_data = data[i:i+8, 0:2]
    temp_label = data[i:i+8, 2]
    train_data.append(temp_data)
    train_label.append(temp_label)
    test_data.append(data[i+8:i+10, 0:2])
    test_label.append(data[i+8:i+10, 2])

train_data = np.array(train_data).reshape((32, 2))
train_label = np.array(train_label).reshape(-1).astype(int) - 1
test_data = np.array(test_data).reshape((8, 2))
test_label = np.array(test_label).reshape(-1).astype(int) - 1

train_label = np.eye(4)[train_label]      # 32x4大小，one-hot向量矩陣
test_label = np.eye(4)[test_label]        # 8x4大小


train_data = np.hstack((train_data, np.ones((train_data.shape[0], 1))))
test_data = np.hstack((test_data, np.ones((test_data.shape[0], 1))))

print(train_data.shape, train_label.shape)
print(test_data.shape, test_label.shape)

train_data = train_data.T     # 按照課件的公式調整資料維度
train_label = train_label.T
test_data = test_data.T
test_label = test_label.T

print(train_data.shape, train_label.shape)
print(test_data.shape, test_label.shape)

lam = 1e-9
temp = np.linalg.inv(np.dot(train_data, train_data.T) + lam)
W = np.dot(np.dot(temp, train_data), train_label.T)   # 3x4維

y_pred = np.dot(W.T, test_data)
index = np.argmax(y_pred, axis=0)   # 得到每個樣本對應類別的值最大的索引
y_pred = np.eye(y_pred.shape[0])[index].T   # 轉換為one-hot向量

wrong_num = sum(sum(y_pred - test_label))
acc = (1 - wrong_num / len(test_label)) * 100
print('分類正確率為：%.2f%% ' % acc)

Ref.

[1]Richard O. Duda. 模式分類[M]. 李宏東等譯. 機械工業出版社 中信出版社, 2003.