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一、集成学习综述
-
- 集成方法或元算法是对其他算法进行组合的一种方式，下面的博客中主要关注的是AdaBoost元算法。将不同的分类器组合起来，而这种组合结果被称为集成方法/元算法。使用集成算法时会有很多的形式，如：
- 不同算法的集成
- 同一种算法在不同设置下的集成
- 数据集不同部分分配给不同分类器之后的集成
-
- AdaBoost算法优缺点
- 优点：泛化错误率低，易编码，可以应用在大部分分类器上，无参数调整
- 缺点：对离群点敏感
- 适用数据类型：数值型和标称型数据
二、基于数据集多重采样的分类器
-
- bagging方法(bootstrap aggregating)
- 主要思想：
- (1). 从原始数据集中抽取新的训练集。每次从原始数据集中使用有放回采样数据的方法,抽取n个样本(在原始数据集中，有些样本可能被重复采样，而有些样本可能一次都未被采样到)。共进行k次抽取，得到k个新的数据集(k个新训练集之间是相互独立的)，新的数据集的大小和原始数据集的大小相等。
- (2). 每次使用一个新的训练集得到一个模型，k个新的训练集总共可以得到k个新的模型
- (3). 对分类问题：将(2)中得到的k个模型采用投票方式得到分类结果；对回归问题：计算(2)中模型的均值作为最后的结果(所有模型的重要性相同！！！)
-
- boosting方法
- 不论是在bagging还是boosting当中，所使用的多个分类器的类型都是一样的。但是，在boosting中，不同的分类器通过串行训练来获得的，每个新分类器都根据已训练出的分类器的性能来进行训练。boosting是通过关注被已有分类器错分的那些数据来获得新的分类器，，boosting方法有多个版本，下面介绍的是最流行的一个版本AdaBoosting算法。
- 主要思想：
- (1). 对每一次的训练数据样本赋予一个权重，并且每一次样本的权重分布依赖上一次的分类结果。
- (2). 基分类器之间采用序列的线性加权方式来组合。
三、bagging方法与boosting方法对比
-
- 样本选择上：
- bagging方法：新的训练集是在原始训练集中采用有放回的方式采样样本的，从原始训练集中选取的每个新的训练集之间是相互独立的。
- boosting方法：每一次的训练集不变，只是训练集中的每个样本在分类器中的权重发生变化，而权重是根据上一次的分类结果进行调整的。
-
- 样本权重上：
- bagging方法：使用均匀选取样本，每个样本的权重相同。
- boosting方法：根据错误率不断调整样本的权重，错误率越大，权重越大。
-
- 预测函数：
- bagging方法：所有预测函数的权重相等
- boosting方法：每个弱分类器都有相应的权重，对分类误差小的分类器会有更大的权重
-
- 并行计算：
- bagging方法：各个预测函数可以并行计算
- boosting方法：各个预测函数只能顺序生成，因为后一个模型需要前一个模型的输出结果
四、集成学习的常见应用
-
- 常见算法
- Bagging + 决策树 = 随机森林
- AdaBoost + 决策树 = 提升树
- Gradient Boosting + 决策树 = GBDT
-
- 基于错误率提升分类器的性能(AdaBoost算法原理介绍)
- 2.1 AdaBoost算法介绍
集成学习算法思想:使用弱分类器和多个样本来构建一个强分类器。AdaBoost是adaptive boosting的缩写，主要运行过程是：首先，对训练数据集中的每个样本进行训练，并赋予每个样本一个权重，这些权重构成一个向量D。一开始，这些样本的初始权重都是相同的！然后，在训练数据上训练出一个弱分类器并计算弱分类器的错误率。接着，在相同的训练数据上再次训练弱分类器。在分类器的第二次训练过程中，将会重新调整每个样本的权重！其中对第一次中分对的样本降低其权重，在第一次中分错的样本提高其权重。为了从所有弱分类器中得到最终的分类结果，AdaBoost还会对每个弱分类器都分配一个权重值alpha,这些alpha值是基于每个弱分类器的错误率进行计算出来的。 -
2.2 错误率ε的定义如下：
错误率计算公式.png -
2.3 alpha的计算公式
alpha的计算公式.png -
2.4 AdaBoost算法流程如下
AdaBoost算法流程.png - 2.5 对上图的解释如下：
- 首先，对训练数据集中的每个样本进行初始化权重，此时每个样本的权重是相同的，这些权重构成了权重向量D；然后，经过第一个弱分类器后，训练集中每个样本的权重发生变化，根据第一个弱分类器的分类结果计算其错误率ε;接着，计算出alpha的值；计算出aplha值之后，可以对权重向量D进行更新，使得对第一个分类器分类结果中分类错误的样本，提高其权重。对分类正确的样本，降低其权重。权重向量D的更新方法如下：
-
2.5.1 如果某个样本被第一个弱分类器分类正确，那么该样本的权重更新公式是：
分类正确情况.png -
2.5.2 如果某个样本被第一个弱分类器分类错误，那么该样本的权重更新公式是：
分类错误情况.png
-
- 在计算出D后，AdaBoost又开始进行下一轮的迭代，AdaBoost算法会不断的重复训练和调整权重，直到训练错误率为0或弱分类器的数目达到用户的指定值为止。
- 首先，对训练数据集中的每个样本进行初始化权重，此时每个样本的权重是相同的，这些权重构成了权重向量D；然后，经过第一个弱分类器后，训练集中每个样本的权重发生变化，根据第一个弱分类器的分类结果计算其错误率ε;接着，计算出alpha的值；计算出aplha值之后，可以对权重向量D进行更新，使得对第一个分类器分类结果中分类错误的样本，提高其权重。对分类正确的样本，降低其权重。权重向量D的更新方法如下：
-
-
AdaBoost算法实战(基于单层决策树构建弱分类器)
数据集可视化.png
-
- 3.1从上图可以看出，试着从某个坐标轴上选择一个值（即选择一条与坐标轴平行的直线）来将所有的蓝色圆点和橘色圆点分开，这显然是不可能的。这就是单层决策树难以处理的一个著名问题。通过使用多颗单层决策树，我们可以构建出一个能够对该数据集完全正确分类的分类器。
#################数据集的可视化#####################
def loadSimData():
"""
创建单层决策树的数据集
"""
dataMat = np.matrix([[1., 2.1],
[1.5, 1.6],
[1.3, 1.],
[1., 1.],
[2., 1.]])
classLabels = [1.0, 1.0, -1.0, -1.0, 1.0]
return dataMat, classLabels
def showDataSet(dataMat, labelMat):
"""
数据可视化
"""
data_plus = [] # 正样本
data_minus = [] # 负样本
for i in range(len(dataMat)):
if labelMat[i] > 0:
data_plus.append(dataMat[i])
else:
data_minus.append(dataMat[i])
data_plus_np = np.array(data_plus)
data_minus_np = np.array(data_minus) # 转化成numpy中的数据类型
plt.scatter(np.transpose(data_plus_np)[0], np.transpose(data_plus_np)[1])
plt.scatter(np.transpose(data_minus_np)[0], np.transpose(data_minus_np)[1])
plt.title("Dataset Visualize")
plt.xlabel("x1")
plt.ylabel("x2")
plt.show()
if __name__ == '__main__':
data_Arr, classLabels = loadSimData()
showDataSet(data_Arr, classLabels)
数据集可视化2.png
- 3.2蓝色横线上的是一个类别，蓝色横线下边是一个类别。显然，此时有一个蓝点分类错误，计算此时的分类误差错误率为1/5 = 0.2。这个横线与坐标轴的y轴的交点，就是我们设置的阈值，通过不断改变阈值的大小，找到使单层决策树的分类误差最小的阈值。同理，竖线也是如此，找到最佳分类的阈值，就找到了最佳单层决策树。
#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Date : 2019-05-12 21:31:41
# @Author : cdl ([email protected])
# @Link : https://github.com/cdlwhm1217096231/python3_spider
# @Version : $Id$
import numpy as np
import matplotlib.pyplot as plt
# 数据集可视化
def loadSimpleData():
dataMat = np.matrix([[1., 2.1],
[1.5, 1.6],
[1.3, 1.],
[1., 1.],
[2., 1.]])
classLabels = [1.0, 1.0, -1.0, -1.0, 1.0]
return dataMat, classLabels
def showDataSet(dataMat, labelMat):
data_plus = []
data_minus = []
for i in range(len(dataMat)):
if labelMat[i] > 0:
data_plus.append(dataMat[i])
else:
data_minus.append(dataMat[i])
data_plus_np = np.array(data_plus)
data_minus_np = np.array(data_minus)
plt.scatter(np.transpose(data_plus_np)[0], np.transpose(data_plus)[1])
plt.scatter(np.transpose(data_minus_np)[0], np.transpose(data_minus)[1])
plt.title("dataset visualize")
plt.xlabel("x")
plt.ylabel("y")
plt.show()
# 构建单层决策树分类函数
def stumpClassify(dataMat, dimen, threshval, threshIneq):
"""
dataMat:数据矩阵
dimen:第dimen列，即第几个特征
threshval:阈值
threshIneq:标志
返回值retArray：分类结果
"""
retArray = np.ones((np.shape(dataMat)[0], 1)) # 初始化retArray为1
if threshIneq == "lt":
retArray[dataMat[:, dimen] <= threshval] = -1.0 # 如果小于阈值,则赋值为-1
else:
retArray[dataMat[:, dimen] > threshval] = 1.0 # 如果大于阈值,则赋值为-1
return retArray
# 找到数据集上最佳的单层决策树，单层决策树是指只考虑其中的一个特征，在该特征的基础上进行分类，寻找分类错误率最低的阈值即可。例如本文中的例子是，如果以第一列特征为基础，阈值选择1.3，并且设置>1.3的为-1,<1.3的为+1,这样就构造出了一个二分类器
def buildStump(dataMat, classLabels, D):
"""
dataMat:数据矩阵
classLabels:数据标签
D：样本权重
返回值是:bestStump:最佳单层决策树信息;minError:最小误差;bestClasEst：最佳的分类结果
"""
dataMat = np.matrix(dataMat)
labelMat = np.matrix(classLabels).T
m, n = np.shape(dataMat)
numSteps = 10.0
bestStump = {} # 存储最佳单层决策树信息的字典
bestClasEst = np.mat(np.zeros((m, 1))) # 最佳分类结果
minError = float("inf")
for i in range(n): # 遍历所有特征
rangeMin = dataMat[:, i].min()
rangeMax = dataMat[:, i].max()
stepSize = (rangeMax - rangeMin) / numSteps # 计算步长
for j in range(-1, int(numSteps) + 1):
for inequal in ["lt", "gt"]:
threshval = (rangeMin + float(j) * stepSize) # 计算阈值
predictVals = stumpClassify(
dataMat, i, threshval, inequal) # 计算分类结果
errArr = np.mat(np.ones((m, 1))) # 初始化误差矩阵
errArr[predictVals == labelMat] = 0 # 分类完全正确，赋值为0
# 基于权重向量D而不是其他错误计算指标来评价分类器的，不同的分类器计算方法不一样
weightedError = D.T * errArr # 计算弱分类器的分类错误率---这里没有采用常规方法来评价这个分类器的分类准确率，而是乘上了样本权重D
print("split: dim %d, thresh %.2f, thresh ineqal: %s, the weighted error is %.3f" % (
i, threshval, inequal, weightedError))
if weightedError < minError:
minError = weightedError
bestClasEst = predictVals.copy()
bestStump["dim"] = i
bestStump["thresh"] = threshval
bestStump["ineq"] = inequal
return bestStump, minError, bestClasEst
- 3.3 通过遍历，改变不同的阈值，计算最终的分类误差，找到分类误差最小的分类方式，即为我们要找的最佳单层决策树。这里lt表示less than，表示分类方式，对于小于阈值的样本点赋值为-1，gt表示greater than，也是表示分类方式，对于大于阈值的样本点赋值为-1。经过遍历，我们找到，训练好的最佳单层决策树的最小分类误差为0，就是对于该数据集，无论用什么样的单层决策树，分类误差最小就是0。这就是我们训练好的弱分类器。接下来，使用AdaBoost算法提升分类器性能，将分类误差缩短到0，看下AdaBoost算法是如何实现的。
# 使用Adaboost算法提升弱分类器性能
def adbBoostTrainDS(dataMat, classLabels, numIt=40):
"""
dataMat:数据矩阵
classLabels:标签矩阵
numIt:最大迭代次数
返回值：weakClassArr 训练好的分类器 aggClassEst:类别估计累计值
"""
weakClassArr = []
m = np.shape(dataMat)[0]
D = np.mat(np.ones((m, 1)) / m) # 初始化样本权重D
aggClassEst = np.mat(np.zeros((m, 1)))
for i in range(numIt):
bestStump, error, classEst = buildStump(
dataMat, classLabels, D) # 构建单个单层决策树
# 计算弱分类器权重alpha,使error不等于0,因为分母不能为0
alpha = float(0.5 * np.log((1.0 - error) / max(error, 1e-16)))
bestStump["alpha"] = alpha # 存储每个弱分类器的权重alpha
weakClassArr.append(bestStump) # 存储单个单层决策树
print("classEst: ", classEst.T)
expon = np.multiply(-1 * alpha *
np.mat(classLabels).T, classEst) # 计算e的指数项
D = np.multiply(D, np.exp(expon))
D = D / D.sum()
# 计算AdaBoost误差,当误差为0时，退出循环
aggClassEst += alpha * classEst # 计算类别估计累计值--注意这里包括了目前已经训练好的每一个弱分类器
print("aggClassEst: ", aggClassEst.T)
aggErrors = np.multiply(np.sign(aggClassEst) != np.mat(
classLabels).T, np.ones((m, 1))) # 目前集成分类器的分类误差
errorRate = aggErrors.sum() / m # 集成分类器分类错误率，如果错误率为0，则整个集成算法停止，训练完成
print("total error: ", errorRate)
if errorRate == 0.0:
break
return weakClassArr, aggClassEst
- 3.4 使用AdaBoost提升分类器性能
# Adaboost分类函数
def adaClassify(dataToClass, classifier):
"""
dataToClass:待分类样本
classifier:训练好的强分类器
"""
dataMat = np.mat(dataToClass)
m = np.shape(dataMat)[0]
aggClassEst = np.mat(np.zeros((m, 1)))
for i in range(len(classifier)): # 遍历所有分类器，进行分类
classEst = stumpClassify(
dataMat, classifier[i]["dim"], classifier[i]["thresh"], classifier[i]["ineq"])
aggClassEst += classifier[i]["alpha"] * classEst
print(aggClassEst)
return np.sign(aggClassEst)
- 3.5 整个Adaboost提升算法的代码如下：
#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Date : 2019-05-12 21:31:41
# @Author : cdl ([email protected])
# @Link : https://github.com/cdlwhm1217096231/python3_spider
# @Version : $Id$
import numpy as np
import matplotlib.pyplot as plt
# 数据集可视化
def loadSimpleData():
dataMat = np.matrix([[1., 2.1],
[1.5, 1.6],
[1.3, 1.],
[1., 1.],
[2., 1.]])
classLabels = [1.0, 1.0, -1.0, -1.0, 1.0]
return dataMat, classLabels
def showDataSet(dataMat, labelMat):
data_plus = []
data_minus = []
for i in range(len(dataMat)):
if labelMat[i] > 0:
data_plus.append(dataMat[i])
else:
data_minus.append(dataMat[i])
data_plus_np = np.array(data_plus)
data_minus_np = np.array(data_minus)
plt.scatter(np.transpose(data_plus_np)[0], np.transpose(data_plus)[1])
plt.scatter(np.transpose(data_minus_np)[0], np.transpose(data_minus)[1])
plt.title("dataset visualize")
plt.xlabel("x")
plt.ylabel("y")
plt.show()
# 构建单层决策树分类函数
def stumpClassify(dataMat, dimen, threshval, threshIneq):
"""
dataMat:数据矩阵
dimen:第dimen列，即第几个特征
threshval:阈值
threshIneq:标志
返回值retArray：分类结果
"""
retArray = np.ones((np.shape(dataMat)[0], 1)) # 初始化retArray为1
if threshIneq == "lt":
retArray[dataMat[:, dimen] <= threshval] = -1.0 # 如果小于阈值,则赋值为-1
else:
retArray[dataMat[:, dimen] > threshval] = 1.0 # 如果大于阈值,则赋值为-1
return retArray
# 找到数据集上最佳的单层决策树，单层决策树是指只考虑其中的一个特征，在该特征的基础上进行分类，寻找分类错误率最低的阈值即可。例如本文中的例子是，如果以第一列特征为基础，阈值选择1.3，并且设置>1.3的为-1,<1.3的为+1,这样就构造出了一个二分类器
def buildStump(dataMat, classLabels, D):
"""
dataMat:数据矩阵
classLabels:数据标签
D：样本权重
返回值是:bestStump:最佳单层决策树信息;minError:最小误差;bestClasEst：最佳的分类结果
"""
dataMat = np.matrix(dataMat)
labelMat = np.matrix(classLabels).T
m, n = np.shape(dataMat)
numSteps = 10.0
bestStump = {} # 存储最佳单层决策树信息的字典
bestClasEst = np.mat(np.zeros((m, 1))) # 最佳分类结果
minError = float("inf")
for i in range(n): # 遍历所有特征
rangeMin = dataMat[:, i].min()
rangeMax = dataMat[:, i].max()
stepSize = (rangeMax - rangeMin) / numSteps # 计算步长
for j in range(-1, int(numSteps) + 1):
for inequal in ["lt", "gt"]:
threshval = (rangeMin + float(j) * stepSize) # 计算阈值
predictVals = stumpClassify(
dataMat, i, threshval, inequal) # 计算分类结果
errArr = np.mat(np.ones((m, 1))) # 初始化误差矩阵
errArr[predictVals == labelMat] = 0 # 分类完全正确，赋值为0
# 基于权重向量D而不是其他错误计算指标来评价分类器的，不同的分类器计算方法不一样
weightedError = D.T * errArr # 计算弱分类器的分类错误率---这里没有采用常规方法来评价这个分类器的分类准确率，而是乘上了样本权重D
print("split: dim %d, thresh %.2f, thresh ineqal: %s, the weighted error is %.3f" % (
i, threshval, inequal, weightedError))
if weightedError < minError:
minError = weightedError
bestClasEst = predictVals.copy()
bestStump["dim"] = i
bestStump["thresh"] = threshval
bestStump["ineq"] = inequal
return bestStump, minError, bestClasEst
# 使用Adaboost算法提升弱分类器性能
def adbBoostTrainDS(dataMat, classLabels, numIt=40):
"""
dataMat:数据矩阵
classLabels:标签矩阵
numIt:最大迭代次数
返回值：weakClassArr 训练好的分类器 aggClassEst:类别估计累计值
"""
weakClassArr = []
m = np.shape(dataMat)[0]
D = np.mat(np.ones((m, 1)) / m) # 初始化样本权重D
aggClassEst = np.mat(np.zeros((m, 1)))
for i in range(numIt):
bestStump, error, classEst = buildStump(
dataMat, classLabels, D) # 构建单个单层决策树
# 计算弱分类器权重alpha,使error不等于0,因为分母不能为0
alpha = float(0.5 * np.log((1.0 - error) / max(error, 1e-16)))
bestStump["alpha"] = alpha # 存储每个弱分类器的权重alpha
weakClassArr.append(bestStump) # 存储单个单层决策树
print("classEst: ", classEst.T)
expon = np.multiply(-1 * alpha *
np.mat(classLabels).T, classEst) # 计算e的指数项
D = np.multiply(D, np.exp(expon))
D = D / D.sum()
# 计算AdaBoost误差,当误差为0时，退出循环
aggClassEst += alpha * classEst # 计算类别估计累计值--注意这里包括了目前已经训练好的每一个弱分类器
print("aggClassEst: ", aggClassEst.T)
aggErrors = np.multiply(np.sign(aggClassEst) != np.mat(
classLabels).T, np.ones((m, 1))) # 目前集成分类器的分类误差
errorRate = aggErrors.sum() / m # 集成分类器分类错误率，如果错误率为0，则整个集成算法停止，训练完成
print("total error: ", errorRate)
if errorRate == 0.0:
break
return weakClassArr, aggClassEst
# Adaboost分类函数
def adaClassify(dataToClass, classifier):
"""
dataToClass:待分类样本
classifier:训练好的强分类器
"""
dataMat = np.mat(dataToClass)
m = np.shape(dataMat)[0]
aggClassEst = np.mat(np.zeros((m, 1)))
for i in range(len(classifier)): # 遍历所有分类器，进行分类
classEst = stumpClassify(
dataMat, classifier[i]["dim"], classifier[i]["thresh"], classifier[i]["ineq"])
aggClassEst += classifier[i]["alpha"] * classEst
print(aggClassEst)
return np.sign(aggClassEst)
if __name__ == "__main__":
dataMat, classLabels = loadSimpleData()
showDataSet(dataMat, classLabels)
weakClassArr, aggClassEst = adbBoostTrainDS(dataMat, classLabels)
print(adaClassify([[0, 0], [5, 5]], weakClassArr))
- 结果如下：
split: dim 0, thresh 0.90, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 0.90, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.00, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.00, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.10, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.10, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.20, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.20, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.30, thresh ineqal: lt, the weighted error is 0.200 split: dim 0, thresh 1.30, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.40, thresh ineqal: lt, the weighted error is 0.200 split: dim 0, thresh 1.40, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.50, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.50, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.60, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.60, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.70, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.70, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.80, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.80, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 1.90, thresh ineqal: lt, the weighted error is 0.400 split: dim 0, thresh 1.90, thresh ineqal: gt, the weighted error is 0.400 split: dim 0, thresh 2.00, thresh ineqal: lt, the weighted error is 0.600 split: dim 0, thresh 2.00, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 0.89, thresh ineqal: lt, the weighted error is 0.400 split: dim 1, thresh 0.89, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.00, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.00, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.11, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.11, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.22, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.22, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.33, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.33, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.44, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.44, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.55, thresh ineqal: lt, the weighted error is 0.200 split: dim 1, thresh 1.55, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.66, thresh ineqal: lt, the weighted error is 0.400 split: dim 1, thresh 1.66, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.77, thresh ineqal: lt, the weighted error is 0.400 split: dim 1, thresh 1.77, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.88, thresh ineqal: lt, the weighted error is 0.400 split: dim 1, thresh 1.88, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 1.99, thresh ineqal: lt, the weighted error is 0.400 split: dim 1, thresh 1.99, thresh ineqal: gt, the weighted error is 0.400 split: dim 1, thresh 2.10, thresh ineqal: lt, the weighted error is 0.600 split: dim 1, thresh 2.10, thresh ineqal: gt, the weighted error is 0.400 classEst: [[-1. 1. -1. -1. 1.]] aggClassEst: [[-0.69314718 0.69314718 -0.69314718 -0.69314718 0.69314718]] total error: 0.2 split: dim 0, thresh 0.90, thresh ineqal: lt, the weighted error is 0.250 split: dim 0, thresh 0.90, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.00, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.00, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.10, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.10, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.20, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.20, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.30, thresh ineqal: lt, the weighted error is 0.500 split: dim 0, thresh 1.30, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.40, thresh ineqal: lt, the weighted error is 0.500 split: dim 0, thresh 1.40, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.50, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.50, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.60, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.60, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.70, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.70, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.80, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.80, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 1.90, thresh ineqal: lt, the weighted error is 0.625 split: dim 0, thresh 1.90, thresh ineqal: gt, the weighted error is 0.250 split: dim 0, thresh 2.00, thresh ineqal: lt, the weighted error is 0.750 split: dim 0, thresh 2.00, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 0.89, thresh ineqal: lt, the weighted error is 0.250 split: dim 1, thresh 0.89, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.00, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.00, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.11, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.11, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.22, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.22, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.33, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.33, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.44, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.44, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.55, thresh ineqal: lt, the weighted error is 0.125 split: dim 1, thresh 1.55, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.66, thresh ineqal: lt, the weighted error is 0.250 split: dim 1, thresh 1.66, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.77, thresh ineqal: lt, the weighted error is 0.250 split: dim 1, thresh 1.77, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.88, thresh ineqal: lt, the weighted error is 0.250 split: dim 1, thresh 1.88, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 1.99, thresh ineqal: lt, the weighted error is 0.250 split: dim 1, thresh 1.99, thresh ineqal: gt, the weighted error is 0.250 split: dim 1, thresh 2.10, thresh ineqal: lt, the weighted error is 0.750 split: dim 1, thresh 2.10, thresh ineqal: gt, the weighted error is 0.250 classEst: [[ 1. 1. -1. -1. -1.]] aggClassEst: [[ 0.27980789 1.66610226 -1.66610226 -1.66610226 -0.27980789]] total error: 0.2 split: dim 0, thresh 0.90, thresh ineqal: lt, the weighted error is 0.143 split: dim 0, thresh 0.90, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.00, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.00, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.10, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.10, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.20, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.20, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.30, thresh ineqal: lt, the weighted error is 0.286 split: dim 0, thresh 1.30, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.40, thresh ineqal: lt, the weighted error is 0.286 split: dim 0, thresh 1.40, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.50, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.50, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.60, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.60, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.70, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.70, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.80, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.80, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 1.90, thresh ineqal: lt, the weighted error is 0.357 split: dim 0, thresh 1.90, thresh ineqal: gt, the weighted error is 0.143 split: dim 0, thresh 2.00, thresh ineqal: lt, the weighted error is 0.857 split: dim 0, thresh 2.00, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 0.89, thresh ineqal: lt, the weighted error is 0.143 split: dim 1, thresh 0.89, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.00, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.00, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.11, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.11, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.22, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.22, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.33, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.33, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.44, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.44, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.55, thresh ineqal: lt, the weighted error is 0.500 split: dim 1, thresh 1.55, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.66, thresh ineqal: lt, the weighted error is 0.571 split: dim 1, thresh 1.66, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.77, thresh ineqal: lt, the weighted error is 0.571 split: dim 1, thresh 1.77, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.88, thresh ineqal: lt, the weighted error is 0.571 split: dim 1, thresh 1.88, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 1.99, thresh ineqal: lt, the weighted error is 0.571 split: dim 1, thresh 1.99, thresh ineqal: gt, the weighted error is 0.143 split: dim 1, thresh 2.10, thresh ineqal: lt, the weighted error is 0.857 split: dim 1, thresh 2.10, thresh ineqal: gt, the weighted error is 0.143 classEst: [[1. 1. 1. 1. 1.]] aggClassEst: [[ 1.17568763 2.56198199 -0.77022252 -0.77022252 0.61607184]] total error: 0.0 [[-0.69314718] [ 0.69314718]] [[-1.66610226] [ 1.66610226]] [[-2.56198199] [ 2.56198199]] [[-1.] [ 1.]] [Finished in 2.5s]