先谈一下个人对多元逐步回归的理解：多元逐步回归的最本质的核心是最小二乘原理，本方法中调用smf方法。
# encoding: utf-8
"""
功能：多元逐步回归
描述：基于python实现多元逐步回归的功能
作者：CHEN_C_W （草木陈）
时间：2019年4月12日(星期五) 凌晨
地点：杭州
参考：
"""
import numpy as np
import pandas as pd
from sklearn.preprocessing import StandardScaler
from statsmodels.formula import api as smf
class FeatureSelection():
def stepwise(self, df, response, intercept=True, normalize=False, criterion='bic',
f_pvalue_enter=.05, p_value_enter=.05, direction='backward', show_step=True,
criterion_enter=None, criterion_remove=None, max_iter=200, **kw):
"""
逐步回归。
参数
----
df : dataframe
分析用数据框，response为第一列。
response : str
回归分析相应变量。
intercept : bool, 默认是True
模型是否有截距项。
criterion : str, 默认是'bic'
逐步回归优化规则。
f_pvalue_enter : float, 默认是.05
当选择criterion=’ssr‘时，模型加入或移除变量的f_pvalue阈值。
p_value_enter : float, 默认是.05
当选择derection=’both‘时，移除变量的pvalue阈值。
direction : str, 默认是'backward'
逐步回归方向。
show_step : bool, 默认是True
是否显示逐步回归过程。
criterion_enter : float, 默认是None
当选择derection=’both‘或'forward'时，模型加入变量的相应的criterion阈值。
criterion_remove : float, 默认是None
当选择derection='backward'时，模型移除变量的相应的criterion阈值。
max_iter : int, 默认是200
模型最大迭代次数。
"""
criterion_list = ['bic', 'aic', 'ssr', 'rsquared', 'rsquared_adj']
if criterion not in criterion_list:
raise IOError('请输入正确的criterion, 必须是以下内容之一：', '\n', criterion_list)
direction_list = ['backward', 'forward', 'both']
if direction not in direction_list:
raise IOError('请输入正确的direction, 必须是以下内容之一：', '\n', direction_list)
# 默认p_enter参数
p_enter = {'bic': 0.0, 'aic': 0.0, 'ssr': 0.05, 'rsquared': 0.05, 'rsquared_adj': -0.05}
if criterion_enter: # 如果函数中对p_remove相应key传参，则变更该参数
p_enter[criterion] = criterion_enter
# 默认p_remove参数
p_remove = {'bic': 0.01, 'aic': 0.01, 'ssr': 0.1, 'rsquared': 0.05, 'rsquared_adj': -0.05}
if criterion_remove: # 如果函数中对p_remove相应key传参，则变更该参数
p_remove[criterion] = criterion_remove
if normalize: # 如果需要标准化数据
intercept = False # 截距强制设置为0
df_std = StandardScaler().fit_transform(df)
df = pd.DataFrame(df_std, columns=df.columns, index=df.index)
""" forward """
if direction == 'forward':
remaining = list(df.columns) # 自变量集合
remaining.remove(response)
selected = [] # 初始化选入模型的变量列表
# 初始化当前评分,最优新评分
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, remaining[0])
else:
formula = "{} ~ {} - 1".format(response, remaining[0])
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
current_score = eval（'result.' + criterion)
best_new_score = eval（'result.' + criterion)
if show_step:
print('\nstepwise starting:\n')
iter_times = 0
# 当变量未剔除完，并且当前评分更新时进行循环
while remaining and (current_score == best_new_score) and (iter_times < max_iter):
scores_with_candidates = [] # 初始化变量以及其评分列表
for candidate in remaining: # 在未剔除的变量中每次选择一个变量进入模型，如此循环
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected + [candidate]))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected + [candidate]))
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
fvalue = result.fvalue
f_pvalue = result.f_pvalue
score = eval（'result.' + criterion)
scores_with_candidates.append((score, candidate, fvalue, f_pvalue)) # 记录此次循环的变量、评分列表
if criterion == 'ssr': # 这几个指标取最小值进行优化
scores_with_candidates.sort(reverse=True) # 对评分列表进行降序排序
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop() # 提取最小分数及其对应变量
if ((current_score - best_new_score) > p_enter[criterion]) and (
best_new_f_pvalue < f_pvalue_enter): # 如果当前评分大于最新评分
remaining.remove(best_candidate) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.append(best_candidate) # 将最新最优分对应的变量放入已选变量列表
current_score = best_new_score # 更新当前评分
if show_step: # 是否显示逐步回归过程
print('Adding %s, SSR = %.3f, Fstat = %.3f, FpValue = %.3e' %
(best_candidate, best_new_score, best_new_fvalue, best_new_f_pvalue))
elif (current_score - best_new_score) >= 0 and (
best_new_f_pvalue < f_pvalue_enter) and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (best_new_f_pvalue < f_pvalue_enter) and iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
elif criterion in ['bic', 'aic']: # 这几个指标取最小值进行优化
scores_with_candidates.sort(reverse=True) # 对评分列表进行降序排序
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop() # 提取最小分数及其对应变量
if (current_score - best_new_score) > p_enter[criterion]: # 如果当前评分大于最新评分
remaining.remove(best_candidate) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.append(best_candidate) # 将最新最优分对应的变量放入已选变量列表
current_score = best_new_score # 更新当前评分
# print(iter_times)
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (current_score - best_new_score) >= 0 and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
else:
scores_with_candidates.sort()
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop()
if (best_new_score - current_score) > p_enter[criterion]:
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
print(iter_times, flush=True)
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (best_new_score - current_score) >= 0 and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
iter_times += 1
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected))
self.stepwise_model = smf.ols(formula, df).fit() # 最优模型拟合
if show_step: # 是否显示逐步回归过程
print('\nLinear regression model:', '\n ', self.stepwise_model.model.formula)
print('\n', self.stepwise_model.summary())
""" backward """
if direction == 'backward':
remaining, selected = set(df.columns), set(df.columns) # 自变量集合
remaining.remove(response)
selected.remove(response) # 初始化选入模型的变量列表
# 初始化当前评分,最优新评分
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected))
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
current_score = eval（'result.' + criterion)
worst_new_score = eval（'result.' + criterion)
if show_step:
print('\nstepwise starting:\n')
iter_times = 0
# 当变量未剔除完，并且当前评分更新时进行循环
while remaining and (current_score == worst_new_score) and (iter_times < max_iter):
scores_with_eliminations = [] # 初始化变量以及其评分列表
for elimination in remaining: # 在未剔除的变量中每次选择一个变量进入模型，如此循环
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected - set(elimination)))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected - set(elimination)))
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
fvalue = result.fvalue
f_pvalue = result.f_pvalue
score = eval（'result.' + criterion)
scores_with_eliminations.append((score, elimination, fvalue, f_pvalue)) # 记录此次循环的变量、评分列表
if criterion == 'ssr': # 这几个指标取最小值进行优化
scores_with_eliminations.sort(reverse=False) # 对评分列表进行降序排序
worst_new_score, worst_elimination, worst_new_fvalue, worst_new_f_pvalue = scores_with_eliminations.pop() # 提取最小分数及其对应变量
if ((worst_new_score - current_score) < p_remove[criterion]) and (
worst_new_f_pvalue < f_pvalue_enter): # 如果当前评分大于最新评分
remaining.remove(worst_elimination) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.remove(worst_elimination) # 从已选变量列表中剔除最新最优分对应的变量
current_score = worst_new_score # 更新当前评分
if show_step: # 是否显示逐步回归过程
print('Removing %s, SSR = %.3f, Fstat = %.3f, FpValue = %.3e' %
(worst_elimination, worst_new_score, worst_new_fvalue, worst_new_f_pvalue))
elif criterion in ['bic', 'aic']: # 这几个指标取最小值进行优化
scores_with_eliminations.sort(reverse=False) # 对评分列表进行降序排序
worst_new_score, worst_elimination, worst_new_fvalue, worst_new_f_pvalue = scores_with_eliminations.pop() # 提取最小分数及其对应变量
if (worst_new_score - current_score) < p_remove[criterion]: # 如果评分变动不显著
remaining.remove(worst_elimination) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.remove(worst_elimination) # 从已选变量列表中剔除最新最优分对应的变量
current_score = worst_new_score # 更新当前评分
if show_step: # 是否显示逐步回归过程
print('Removing %s, %s = %.3f' % (worst_elimination, criterion, worst_new_score))
else:
scores_with_eliminations.sort(reverse=True)
worst_new_score, worst_elimination, worst_new_fvalue, worst_new_f_pvalue = scores_with_eliminations.pop()
if (current_score - worst_new_score) < p_remove[criterion]:
remaining.remove(worst_elimination)
selected.remove(worst_elimination)
current_score = worst_new_score
if show_step: # 是否显示逐步回归过程
print('Removing %s, %s = %.3f' % (worst_elimination, criterion, worst_new_score))
iter_times += 1
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected))
self.stepwise_model = smf.ols(formula, df).fit() # 最优模型拟合
if show_step: # 是否显示逐步回归过程
print('\nLinear regression model:', '\n ', self.stepwise_model.model.formula)
print('\n', self.stepwise_model.summary())
""" both """
if direction == 'both':
remaining = list(df.columns) # 自变量集合
remaining.remove(response)
selected = [] # 初始化选入模型的变量列表
# 初始化当前评分,最优新评分
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, remaining[0])
else:
formula = "{} ~ {} - 1".format(response, remaining[0])
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
current_score = eval（'result.' + criterion)
best_new_score = eval（'result.' + criterion)
if show_step:
print('\nstepwise starting:\n')
# 当变量未剔除完，并且当前评分更新时进行循环
iter_times = 0
while remaining and (current_score == best_new_score) and (iter_times < max_iter):
scores_with_candidates = [] # 初始化变量以及其评分列表
for candidate in remaining: # 在未剔除的变量中每次选择一个变量进入模型，如此循环
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected + [candidate]))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected + [candidate]))
result = smf.ols(formula, df).fit() # 最小二乘法回归模型拟合
fvalue = result.fvalue
f_pvalue = result.f_pvalue
score = eval（'result.' + criterion)
scores_with_candidates.append((score, candidate, fvalue, f_pvalue)) # 记录此次循环的变量、评分列表
if criterion == 'ssr': # 这几个指标取最小值进行优化
scores_with_candidates.sort(reverse=True) # 对评分列表进行降序排序
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop() # 提取最小分数及其对应变量
if ((current_score - best_new_score) > p_enter[criterion]) and (
best_new_f_pvalue < f_pvalue_enter): # 如果当前评分大于最新评分
remaining.remove(best_candidate) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.append(best_candidate) # 将最新最优分对应的变量放入已选变量列表
current_score = best_new_score # 更新当前评分
if show_step: # 是否显示逐步回归过程
print('Adding %s, SSR = %.3f, Fstat = %.3f, FpValue = %.3e' %
(best_candidate, best_new_score, best_new_fvalue, best_new_f_pvalue))
elif (current_score - best_new_score) >= 0 and (
best_new_f_pvalue < f_pvalue_enter) and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (best_new_f_pvalue < f_pvalue_enter) and iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
elif criterion in ['bic', 'aic']: # 这几个指标取最小值进行优化
scores_with_candidates.sort(reverse=True) # 对评分列表进行降序排序
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop() # 提取最小分数及其对应变量
if (current_score - best_new_score) > p_enter[criterion]: # 如果当前评分大于最新评分
remaining.remove(best_candidate) # 从剩余未评分变量中剔除最新最优分对应的变量
selected.append(best_candidate) # 将最新最优分对应的变量放入已选变量列表
current_score = best_new_score # 更新当前评分
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (current_score - best_new_score) >= 0 and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
else:
scores_with_candidates.sort()
best_new_score, best_candidate, best_new_fvalue, best_new_f_pvalue = scores_with_candidates.pop()
if (best_new_score - current_score) > p_enter[criterion]: # 当评分差大于p_enter
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif (best_new_score - current_score) >= 0 and iter_times == 0: # 当评分差大于等于0，且为第一次迭代
remaining.remove(best_candidate)
selected.append(best_candidate)
current_score = best_new_score
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (best_candidate, criterion, best_new_score))
elif iter_times == 0: # 当评分差小于p_enter，且为第一次迭代
selected.append(remaining[0])
remaining.remove(remaining[0])
if show_step: # 是否显示逐步回归过程
print('Adding %s, %s = %.3f' % (remaining[0], criterion, best_new_score))
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected))
result = smf.ols(formula, df).fit() # 最优模型拟合
if iter_times >= 1: # 当第二次循环时判断变量的pvalue是否达标
if result.pvalues.max() > p_value_enter:
var_removed = result.pvalues[result.pvalues == result.pvalues.max()].index[0]
p_value_removed = result.pvalues[result.pvalues == result.pvalues.max()].values[0]
selected.remove(result.pvalues[result.pvalues == result.pvalues.max()].index[0])
if show_step: # 是否显示逐步回归过程
print('Removing %s, Pvalue = %.3f' % (var_removed, p_value_removed))
iter_times += 1
if intercept: # 是否有截距
formula = "{} ~ {} + 1".format(response, ' + '.join(selected))
else:
formula = "{} ~ {} - 1".format(response, ' + '.join(selected))
self.stepwise_model = smf.ols(formula, df).fit() # 最优模型拟合
if show_step: # 是否显示逐步回归过程
print('\nLinear regression model:', '\n ', self.stepwise_model.model.formula)
print('\n', self.stepwise_model.summary())
# 最终模型选择的变量
if intercept:
self.stepwise_feat_selected_ = list(self.stepwise_model.params.index[1:])
else:
self.stepwise_feat_selected_ = list(self.stepwise_model.params.index)
return self
def main():
print('main方法进入了......')
# url = "http://data.princeton.edu/wws509/datasets/salary.dat"
# data = pd.read_csv(url, sep='\\s+')
data = np.random.rand(200, 4)
df = pd.DataFrame({'A': data[:, 0], 'B': data[:, 1], 'C': data[:, 2], 'D': data[:, 3]})
print('测试data大小', df.shape)
print(df.head())
FeatureSelection().stepwise(df=df, response='A', max_iter=200, criterion='ssr') # criterion='ssr'是为了移除不合适特征
# 注：输入数据的结果如下图所示
if __name__ == "__main__":
main()
注：本文所用的测试是用随机数所求，所以逐步回归的结果大部分并没有通过检验
方法是基于python3.6版本开发，里面的py包不同版本可能存在差异，使用时需要谨慎。
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