【模式识别】判别函数和感知机算法

给定两个类别的一共8个样本，通过感知机算法求解判别函数

def bi_perception():
    X1 = np.array([
        [0, 0, 0],
        [1, 0, 0],
        [1, 0, 1],
        [1, 1, 0]
    ])
    X2 = np.array([
        [0, 0, 1],
        [0, 1, 1],
        [0, 1, 0],
        [1, 1, 1]
    ])
    # 增广
    X1 = np.concatenate((X1, np.ones((len(X1), 1))), axis=1)
    X2 = np.concatenate((X2, np.ones((len(X2), 1))), axis=1)
    X1 = np.concatenate((X1, -X2), axis=0)
    C = 1
    w = np.zeros(X1.shape[1])
    # w = np.array([-1., -1., 1., 1.])
    cnt = 0
    ind = 0
    np.random.shuffle(X1)
    while cnt < len(X1):
        cnt = 0
        for dat in X1:
            print(dat, '·', w, '=', np.matmul(dat, w))
            if np.matmul(dat, w) <= 0:
                w += C*dat
            else:
                cnt += 1
        ind += 1
    print("迭代次数：", ind)
    print("return w=", w)

算得：

迭代次数： 3
return w= [ 2. -2. -2.  1.]

即$d(x)=2x_1-2x_2-2x_3+1$

给定三个类别的各1个样本，通过感知机算法求解判别函数

def tri_perception():
    X = np.array([
        [-1, -1],
        [0, 0],
        [1, 1]
    ])
    X = np.concatenate((X, np.ones((3, 1))), axis=1)
    w = np.zeros((3, X.shape[1]))
    c = 1
    ind = 0
    cnt = 0
    while cnt < len(X)-1:
        print(w)
        tmp = np.zeros(len(X))
        lind = ind % 3
        for i in range(len(tmp)):
            tmp[i] = c*np.matmul(w[i], X[lind])
        for i in range(len(tmp)):
            tmp[i] = tmp[lind]-tmp[i]
            if tmp[i] > 0 and i != lind:
                cnt += 1
        if cnt == len(X)-1:
            continue
        for i in range(len(X)):
            if i == lind:
                w[i] += X[lind]
            else:
                w[i] -= X[lind]
        cnt = 0
        ind += 1
    print(w)

算得：

W=[[-2. -2. -1.]
 [ 0.  0. -1.]
 [ 2.  2. -1.]]

即

$d_1(x)=-2x_1-2x_2-1$

$d_2(x)=-1$

$d_3(x)=2x_1+2x_2-1$