本文将以周志华《机器学习》中的习题8.3的要求和数据,用python完成一个基于决策树的AdaBoost。
从网上下载或自己编程实现AdaBoost,以不剪枝决策树为基学习器,在西瓜数据集3.0α上训练一个AdaBoost集成,并与图8.4进行比较。
西瓜数据集3.0α如下(第一列表示密度,第二列表示含糖量,第三列为数据的标签,表示是否是好瓜):
0.697,0.460,是
0.774,0.376,是
0.634,0.264,是
0.608,0.318,是
0.556,0.215,是
0.403,0.237,是
0.481,0.149,是
0.437,0.211,是
0.666,0.091,否
0.243,0.267,否
0.245,0.057,否
0.343,0.099,否
0.639,0.161,否
0.657,0.198,否
0.360,0.370,否
0.593,0.042,否
0.719,0.103,否
关于adaboost的理论可以参考这篇博客:Adaboost 算法实例解析
这里直接进行实现。
首先,根据题目要求,要实现一个基于不剪枝决策树的Adaboost学习器,学习器可以对西瓜数据进行二分类。
所以,第一步实现决策树,这里按照信息熵来寻找最优的划分属性。另外,我们还需要注意,因为集成学习过程中有权重的概念,所以实现的决策树也要将权重参与计算。
首先是获取数据部分,从txt读入数据,具体见代码:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import math
import copy
def get_data():
x = []
y = []
with open("./xigua.txt", 'r') as f:
for line in f.readlines():
words = line.split(',')
instance = []
for word in words[:2]:
instance.append(float(word))
x.append(instance)
if '是' in words[2]:
y.append(1)
elif '否' in words[2]:
y.append(0)
return x, y
然后,是基于信息熵的决策树中比较重要的部分,求信息熵。信息熵代表数据集的数据纯度,信息熵越高表示数据的类别越杂,反之越纯。
求信息熵的公式:
(pk表示第k类数据的占比,Y等于类别总数)
def ent(D):
"""
求信息熵。
Parameters:
D : D是一个元组,格式为(w, x, y)。表示数据集,w为权重,x为数据,y为标签。
Returns:
一个实数,表示信息熵。
"""
w, x, y = D
h = 1e-17
y = np.array(y)
p0 = np.sum(w * (y == 0)) / np.sum(w)
p1 = np.sum(w * (y == 1)) / np.sum(w)
return -p0*np.log2(p0 + h) - p1*np.log2(p1 + h)
接下来是信息增益,简单来说,就是通过一次划分信息熵减少了多少。
信息增益公式如下:
(a表示划分属性,Dv表示将D划分后的第v个数据集,V等于划分后的数据集个数)
另外,我这里的代码没有严格跟公式一样,有个不同的地方:
公式是传入划分属性,然后在函数体内进行划分。这里直接传入的划分好的数据,划分工作交给外部了,反正效果是一样的。
def gain(D, DS):
"""
求信息增益。
Parameters:
D : D是一个元组,格式为(w, x, y)。表示划分前的数据集,w为权重,x为数据,y为标签。
DS :DS是一个列表,元素为元组,每个元组格式为(w, x, y)。表示划分后的数据集组成的列表。
Returns:
一个实数gain,表示信息增益。
"""
dw, dx, dy = D
original_size = np.sum(dw)
ent_d = ent(D)
gain = ent_d
for di in DS:
w, x, y = di
gain -= (np.sum(w) / original_size) * ent(di)
return gain
接下来就是按照信息增益来找最佳划分属性(best_divider)了,我们需要注意这里的西瓜属性都是离散型数据,所以我们要按照离散型属性的套路来实现。
即对于每个属性,先将训练集按照该属性排序,然后取两两间的中间值看作待选择的divider(比如在本题中就是,对于密度,就将17个西瓜按照密度排序,然后取出16个中间值,同样对于含糖量,将17个西瓜按照含糖量排序,取出16个间隔值,最后能得到32个间隔值,这32个间隔值就是32个divider(divider结构形如 (“密度”, 0.542)))。
然后在得到所有可能的划分依据divider后,按照信息增益来选择增益最大的divider。
def find_best_divider(D, A):
"""
按照信息增益最大的准则寻找最佳划分。
Parameters:
D : 同上。
Returns:
(划分属性,划分属性值)
"""
w, x, y = D
best_divider = None
gains = []
dividers = []
for attr in A:
sorted_D = sort_D_by_attr(D, attr)
sorted_w, sorted_x, sorted_y = sorted_D
for i in range(len(sorted_x)-1):
mid_val = (sorted_x[i][attr] + sorted_x[i+1][attr]) / 2.0
dividers.append((attr, mid_val))
for divider in dividers:
DS = divide(D, divider)
gains.append(gain(D, DS))
max_gain = 0
for i in range(len(gains)):
if gains[i] > max_gain:
max_gain = gains[i]
best_divider = dividers[i]
return best_divider
求信息增益前,需要根据每个divider得到划分结果,划分函数如下:
def divide(D, divider):
"""
按照divider内容对数据集D进行划分。
Parameters:
D : 同上。
divider :(划分属性,划分值)按照划分属性大于和小于划分值情况将D分别划分为D0和D1,并包含进DS
Returns:
划分完的数据集列表,包含D0和D1
"""
w, x, y = D
DS = []
divider_attr, divider_attr_val = divider
w0, x0, y0 = [],[],[]
w1, x1, y1 = [],[],[]
for i in range(len(x)):
if x[i][divider_attr] < divider_attr_val:
w0.append(w[i])
x0.append(x[i])
y0.append(y[i])
else:
w1.append(w[i])
x1.append(x[i])
y1.append(y[i])
D0 = (w0, x0, y0)
D1 = (w1, x1, y1)
DS = [D0, D1]
return DS
按照属性进行排序:
def sort_D_by_attr(D, attr):
sorted_D = copy.deepcopy(D)
w, x, y = sorted_D
size = len(x)
for i in range(size-1):
for j in range(size-i-1):
if x[j][attr] > x[j+1][attr]:
t = x[j]
x[j] = x[j+1]
x[j+1] = t
t = w[j]
w[j] = w[j+1]
w[j+1] = t
t = y[j]
y[j] = y[j+1]
y[j+1] = t
return sorted_D
然后,是决策树的节点类:
class Node:
def __init__(self):
self.attr = None
self.attr_val = None
self.less_than_val_child = None
self.bigger_than_val_child = None
self.result_category = None
def is_leaf(self):
if self.attr is None:
return True
else:
return False
def __str__(self):
return "划分属性:" + str(self.attr) + ' ' + "划分值:" + str(self.attr_val) + ', ' + "结果:" + str(self.result_category)
def show(self):
if self is None:
return
print(self)
if self.less_than_val_child is not None:
self.less_than_val_child.show()
if self.bigger_than_val_child is not None:
self.bigger_than_val_child.show()
然后,就可以实现决策树的训练过程了,整个过程是一个递归过程,递归返回的情况有三种,一种是当前数据集类别相同,则不用划分。第二种是当前数据集的所有属性都相同,没法划分,第三种是当前的数据集是空的,那么不能划分。
def tree_generate(D, A):
'''
Parameters:
D为数据集,A为属性集
'''
w, x, y = D
node = Node()
if is_same_y(D):
node.result_category = y[0]
return node
if A is None or len(A) == 0 or is_same_val_in_A(D, A):
most_possible_y = get_most_possible_y(D)
node.result_category = most_possible_y
return node
attr, attr_val = find_best_divider(D, A)
divider = (attr, attr_val)
DS = divide(D, divider)
node.attr = attr
node.attr_val = attr_val
new_A = A
# new_A.remove(attr)
if DS[0] is None:
node0 = Node()
node0.result_category = get_most_possible_y(D)
node.less_than_val_child = node0
else:
node.less_than_val_child = tree_generate(DS[0], new_A)
if DS[1] is None:
node1 = Node()
node1.result_category = get_most_possible_y(D)
node.bigger_than_val_child = node1
else:
node.bigger_than_val_child = tree_generate(DS[1], new_A)
return node
其他函数:
def get_most_possible_y(D):
w, x, y = D
y = np.array(y)
cnt_0 = np.sum(w * (y == 0))
cnt_1 = np.sum(w * (y == 1))
if cnt_0 > cnt_1:
return 0
else:
return 1
def is_same_val_in_A(D, A):
w, x, y = D
for a in A:
first_val = x[0][a]
for xi in x:
if xi[a] != first_val:
return False
return True
def is_same_y(D):
w, x, y = D
for yi in y:
if yi != y[0]:
return False
return True
决策树训练函数:
def train_1(D):
w, x, y = D
n = len(x[0])
A = list(range(n))
node = tree_generate(D, A)
# node = tree_stool_generate(D, A)
return node
决策树预测函数(后面AdaBoost实现要用到):
def predict(node, x):
while(True):
if node.is_leaf():
return node.result_category
attr = node.attr
attr_val = node.attr_val
if x[attr] > attr_val:
node = node.bigger_than_val_child
else:
node = node.less_than_val_child
至此,决策树的代码就写完了,可以通过以下代码简单测试一下:
x, y = get_data()
w = [1/17 for _ in range(17)]
A = list(range(2))
D = (w, x, y)
node = tree_generate(D, A)
node.show()
就是对训练数据进行一次划分(权重均分),可以看到,生成的决策树和《机器学习》P90 图4.3 中的结果一致:
(这里没有对树进行画图展示,直接按照先序遍历循序打印了节点)
参数T表示集成学习包含几个基学习器,也就是几个决策树。首先获取数据,然后进入for循环,循环T次,每一次将训练数据,训练标签,权重包含进D中,然后进行训练,得到学习器ht,然后计算一下ht的错误率et。这里主要要避免et=0的情况,防止一会除0。如果et>0.5则停止训练。然后将学习器ht存入学习器列表h_list,然后计算该学习器的权重alpha[t],表示该学习器最终的"话语权"。
然后就是adaboost的重点,更新权重,adaboost通过 “降低预测正确样本的权重,提高预测错误样本的权重” 来保证这些基学习器的多样性,更新权重的公式如下:
Dt(x)表示数据x在第t轮训练的权重(对应代码中的变量w,w[i]表示第i条数据的权重,注意没有下标t,因为每次训练直接覆盖上次的权重了)。
Zt就是一个规范化因子,用于保证权重加和等于1(对应代码中的变量Z)。
def adaboost_1_train(T):
alpha = []
h_list = []
x, y = get_data()
N = len(y)
w = [1/N for _ in range(N)]
for t in range(T):
print('w[', t, '] : \n', w, '\n')
D = (w, x, y)
ht = train_1(D)
et = loss(ht, D)
if et == 0:
et = 0.00001
if et > 0.5:
break
h_list.append(ht)
alpha.append(0.5 * math.log((1-et)/et))
Z = 0
for i in range(N):
predict_y = predict(ht, x[i])
if predict_y == y[i]:
Z += w[i] * math.exp(-alpha[t])
else:
Z += w[i] * math.exp(alpha[t])
for i in range(N):
w[i] = w[i] / Z
predict_y = predict(ht, x[i])
if predict_y == y[i]:
w[i] *= math.exp(-alpha[t])
else:
w[i] *= math.exp(alpha[t])
return h_list, alpha
计算错误率:
def loss(h, D):
w, x, y = D
loss = 0
for i in range(len(x)):
predict_y = predict(h, x[i])
if predict_y != y[i]:
loss += w[i]
return loss
集成学习器预测:
def adaboost_predict(model, x):
h_list, alpha = model
p_pos = 0
p_neg = 0
for i in range(len(h_list)):
predict_y = predict(h_list[i], x)
p_pos += predict_y * alpha[i]
p_neg += (1-predict_y) * alpha[i]
return p_pos > p_neg
决策树的分类边界的可视化:
def model_plot(h):
x, y = get_data()
train_label = y
train_value = x
x1 = [mapi[0] for mapi in train_value]
x2 = [mapi[1] for mapi in train_value]
x = np.c_[x1,x2]
np_x = np.asarray(x)
np_y = np.asarray(train_label)
N, M = 100, 100
x1_min, x2_min = np_x.min(axis=0)
x1_max, x2_max = np_x.max(axis=0)
x1_min -= 0.1
x2_min -= 0.1
x1_max += 0.1
x2_max += 0.1
t1 = np.linspace(x1_min, x1_max, N)
t2 = np.linspace(x2_min, x2_max, M)
grid_x, grid_y = np.meshgrid(t1,t2)
grid = np.stack([grid_x.flat, grid_y.flat], axis=1)
y_fake = np.zeros((N*M,))
y_predict = []
for xi in grid:
y_predict.append(predict(h, xi))
cm_light = mpl.colors.ListedColormap(['#A0FFA0', '#FFA0A0'])
plt.pcolormesh(grid_x, grid_y, np.array(y_predict).reshape(grid_x.shape), cmap=cm_light)
plt.scatter(x[:,0], x[:,1], s=30, c=train_label, marker='o')
plt.show()
至此,已经完成了adaboost的全部实现,可以开始测试。
令基学习器的数量为11,进行训练,然后分别打印权重的变化过程,并绘制11个决策树的分类边界。
# 主程序
# (不剪枝决策树, 集成训练, 分开展示)
model = adaboost_1_train(11)
h_list, alpha = model
for h in h_list:
model_plot(h)
观察结果:
w[ 0 ] :
[0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705]
w[ 1 ] :
[0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684]
w[ 2 ] :
[0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647]
w[ 3 ] :
[0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684]
w[ 4 ] :
[0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647]
w[ 5 ] :
[0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684]w[ 6 ] : [0.0588235294117647, 0.0588235294117647,
0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647]w[ 7 ] : [0.058823529411764684, 0.058823529411764684,
0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684]w[ 8 ] : [0.0588235294117647, 0.0588235294117647,
0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647]w[ 9 ] : [0.058823529411764684, 0.058823529411764684,
0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684, 0.058823529411764684]w[ 10 ] : [0.0588235294117647, 0.0588235294117647,
0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647, 0.0588235294117647]
可以看到,每轮训练的权重没有任何变化,并且训练出的决策树都是相同的。这种现象发生的原因就是,第一个基学习器训练后,其准确率已经达到100%,也就是说没有预测错误的样本,那么权重也就不会发生改变,进而后面10个基学习器的训练结果就和第一个一样了。这也验证了集成学习的特点,集成学习可以将多个弱学习器集成变成一个强学习器,但是如果基学习器就是强学习器(或者说在当前数据集上表现足够好),那么集成学习的效果就不显著了。
为了展示出adaboost效果,这里将基学习器改成弱学习器,把不剪枝决策树改为决策树桩(只进行一次划分的决策树),重新实验。
决策树桩的实现:
def tree_stool_generate(D, A):
node = Node()
attr, attr_val = find_best_divider(D, A)
node.attr = attr
node.attr_val = attr_val
DS = divide(D, (attr, attr_val))
child_l = Node()
child_l.result_category = get_most_possible_y(DS[0])
child_b = Node()
child_b.result_category = get_most_possible_y(DS[1])
node.less_than_val_child = child_l
node.bigger_than_val_child = child_b
return node
决策树桩训练:
def train_2(D):
w, x, y = D
n = len(x[0])
A = list(range(n))
# node = tree_generate(D, A)
node = tree_stool_generate(D, A)
return node
决策树桩版的adaboost:
def adaboost_2_train(T):
alpha = []
h_list = []
x, y = get_data()
N = len(y)
w = [1/N for _ in range(N)]
for t in range(T):
print('w[', t, '] : \n', w, '\n')
D = (w, x, y)
ht = train_2(D)
et = loss(ht, D)
if et == 0:
et = 0.00001
if et > 0.5:
break
h_list.append(ht)
alpha.append(0.5 * math.log((1-et)/et))
Z = 0
for i in range(N):
predict_y = predict(ht, x[i])
if predict_y == y[i]:
Z += w[i] * math.exp(-alpha[t])
else:
Z += w[i] * math.exp(alpha[t])
for i in range(N):
w[i] = w[i] / Z
predict_y = predict(ht, x[i])
if predict_y == y[i]:
w[i] *= math.exp(-alpha[t])
else:
w[i] *= math.exp(alpha[t])
return h_list, alpha
然后,编写测试程序,进行测试和展示(同样是11个基学习器):
# 主程序
# (决策树桩, 集成训练, 分开展示)
model = adaboost_2_train(11)
h_list, alpha = model
for h in h_list:
model_plot(h)
x, y = get_data()
x = np.array(x)
y = np.array(y)
for i in range(len(y)):
plt.scatter(x[:, 0], x[:, 1], c=y, marker='o')
for h in h_list:
if h.attr == 0:
p_a = [h.attr_val, 0.01]
p_b = [h.attr_val, 0.6]
else:
p_a = [0.1, h.attr_val]
p_b = [0.8, h.attr_val]
plt.plot([p_a[0], p_b[0]], [p_a[1], p_b[1]], 'g-')
plt.show()
运行结果:
权重变化过程:
w[ 0 ] :
[0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705, 0.058823529411764705]w[ 1 ] : [0.03846153846153847, 0.03846153846153847,
0.03846153846153847, 0.03846153846153847, 0.03846153846153847, 0.03846153846153847, 0.03846153846153847, 0.03846153846153847, 0.03846153846153847, 0.12500000000000003, 0.03846153846153847, 0.03846153846153847, 0.12500000000000003, 0.12500000000000003, 0.12500000000000003, 0.03846153846153847, 0.03846153846153847]w[ 2 ] : [0.062499999999999986, 0.062499999999999986,
0.062499999999999986, 0.062499999999999986, 0.062499999999999986, 0.062499999999999986, 0.062499999999999986, 0.062499999999999986, 0.02777777777777778, 0.09027777777777778, 0.02777777777777778, 0.02777777777777778, 0.09027777777777778, 0.09027777777777778, 0.09027777777777778, 0.02777777777777778, 0.02777777777777778]w[ 3 ] : [0.042452830188679236, 0.042452830188679236,
0.042452830188679236, 0.042452830188679236, 0.042452830188679236, 0.042452830188679236, 0.042452830188679236, 0.042452830188679236, 0.05263157894736844, 0.06132075471698114, 0.01886792452830189, 0.01886792452830189, 0.1710526315789474, 0.1710526315789474, 0.06132075471698114, 0.05263157894736844, 0.05263157894736844]w[ 4 ] : [0.025423728813559313, 0.025423728813559313,
0.025423728813559313, 0.025423728813559313, 0.025423728813559313, 0.025423728813559313, 0.1285714285714285, 0.025423728813559313, 0.031519476657746064, 0.1857142857142857, 0.011299435028248588, 0.011299435028248588, 0.1024382991376747, 0.1024382991376747, 0.1857142857142857, 0.031519476657746064, 0.031519476657746064]w[ 5 ] : [0.018145161290322575, 0.018145161290322575,
0.018145161290322575, 0.018145161290322575, 0.018145161290322575, 0.018145161290322575, 0.09176267281105986, 0.018145161290322575, 0.052631578947368425, 0.13254608294930875, 0.008064516129032258, 0.008064516129032258, 0.17105263157894737, 0.17105263157894737, 0.13254608294930875, 0.052631578947368425, 0.052631578947368425]w[ 6 ] : [0.04146919431279622, 0.04146919431279622,
0.04146919431279622, 0.04146919431279622, 0.04146919431279622, 0.04146919431279622, 0.20971563981042654, 0.04146919431279622, 0.0336854523863116, 0.0848326205574399, 0.005161480607580002, 0.005161480607580002, 0.1094777202555127, 0.1094777202555127, 0.0848326205574399, 0.0336854523863116, 0.0336854523863116]w[ 7 ] : [0.030492583673288998, 0.030492583673288998,
0.030492583673288998, 0.030492583673288998, 0.030492583673288998, 0.030492583673288998, 0.15420535171920433, 0.030492583673288998, 0.05263157894736842, 0.06237800910865321, 0.003795272175233137, 0.003795272175233137, 0.17105263157894735, 0.17105263157894735, 0.06237800910865321, 0.05263157894736842, 0.05263157894736842]w[ 8 ] : [0.0619860413620309, 0.0619860413620309,
0.02021956227836733, 0.02021956227836733, 0.02021956227836733, 0.02021956227836733, 0.10225321495060045, 0.02021956227836733, 0.03489987925383093, 0.12680348422154747, 0.007715118553244956, 0.007715118553244956, 0.11342460757495053, 0.11342460757495053, 0.12680348422154747, 0.10699071172635338, 0.03489987925383093]w[ 9 ] : [0.03890423725611639, 0.03890423725611639,
0.0497159090909091, 0.0497159090909091, 0.0497159090909091, 0.0497159090909091, 0.25142045454545453, 0.0497159090909091, 0.021904176373691363, 0.07958554420736225, 0.0048422321551632825, 0.0048422321551632825, 0.07118857321449694, 0.07118857321449694, 0.07958554420736225, 0.06715047358633955, 0.021904176373691363]w[ 10 ] : [0.026052036683798175, 0.026052036683798175,
0.033292021094723095, 0.033292021094723095, 0.033292021094723095, 0.033292021094723095, 0.16836250667902816, 0.033292021094723095, 0.043231476633221015, 0.053294079602197446, 0.0032425776376825907, 0.0032425776376825907, 0.1405022990579683, 0.1405022990579683, 0.053294079602197446, 0.1325324486176212, 0.043231476633221015]
11个决策树桩的分类边界(最后一个图是11个边界放在一张图中):
可以看到,使用决策树桩后,基学习器的训练结果发生了变化,同时权重也发生了变化。
(其中出现了全是绿色的结果,这种结果表明决策树按照某个值划分数据集,然后形成左右节点后,认为左右节点都是正例。这种情况就像是,密度>0.5的是好瓜,密度<0.5的也是好瓜。其原因就是,我在判断叶子节点的类别结果使用的算法是,看哪个类的带权占比大,而如果正例的权重都比较大,而反例都比较小的时候,就会发生这个现象)
然后,对集成学习器进行集成测试:
# 主程序
# (决策树桩, 集成展示)
x, y = get_data()
train_label = y
train_value = x
x1 = [mapi[0] for mapi in train_value]
x2 = [mapi[1] for mapi in train_value]
x = np.c_[x1,x2]
np_x = np.asarray(x)
np_y = np.asarray(train_label)
N, M = 100, 100
x1_min, x2_min = np_x.min(axis=0)
x1_max, x2_max = np_x.max(axis=0)
x1_min -= 0.1
x2_min -= 0.1
x1_max += 0.1
x2_max += 0.1
t1 = np.linspace(x1_min, x1_max, N)
t2 = np.linspace(x2_min, x2_max, M)
grid_x, grid_y = np.meshgrid(t1,t2)
grid = np.stack([grid_x.flat, grid_y.flat], axis=1)
y_fake = np.zeros((N*M,))
y_predict = []
for xi in grid:
y_predict.append(adaboost_predict(model, xi))
cm_light = mpl.colors.ListedColormap(['#A0FFA0', '#FFA0A0'])
plt.pcolormesh(grid_x, grid_y, np.array(y_predict).reshape(grid_x.shape), cmap=cm_light)
plt.scatter(x[:,0], x[:,1], s=30, c=train_label, marker='o')
plt.show()