用最小二乘法OLS做回归，并解读结果

使用OLS做回归

#使用OLS做多元线性回归拟合
from sklearn import linear_model,cross_validation, feature_selection,preprocessing
import statsmodels.formula.api as sm
from statsmodels.tools.eval_measures import mse 
from statsmodels.tools.tools import add_constant 
from sklearn.metrics import mean_squared_error
X = b_data.values.copy()
X_train, X_valid, y_train, y_valid =
cross_validation.train_test_split( X[:, :-1], X[:, -1],train_size=0.80)
result = sm.OLS( y_train, add_constant(X_train) ).fit()
result.summary()

https://www.datarobot.com/blog/ordinary-least-squares-in-python/

解读summary结果

summary结果中提供了很多关于拟合的信息，下面是这些描述信息的含义：

第一个表左边部分是关于拟合的基本信息:

Element	Description	中文描述（个人翻译，如有错误请留言指出）
Dep. Variable	Which variable is the response in the model	响应变量/独立变量 y
Model	What model you are using in the fit	用来做拟合的模型
Method	How the parameters of the model were calculated	模型参数的计算方法
No. Observations	The number of observations (examples)	观察样本个数
DF Residuals	Degrees of freedom of the residuals. Number of observations – number of parameters	残差*度 = 样本数-模型参数个数
DF Model	Number of parameters in the model (not including the constant term if present)	模型*度 = 模型参数个数(不包括常数项)

第一个表的右边部分显示的是拟合的好坏情况：

Element	Description	中文描述（个人翻译，如有错误请留言指出）
R-squared	The coefficient of determination. A statistical measure of how well the regression line approximates the real data points	R决定系数，描述回归曲线对真实数据点拟合程度的统计量，取值[0，1]之间，越接近于1 越好
Adj. R-squared	The above value adjusted based on the number of observations and the degrees-of-freedom of the residuals	调整R决定系数，是R决定系数加上了残差*度后的重新计算的结果，通过比R小。 计算公式：
F-statistic	A measure how significant the fit is. The mean squared error of the model divided by the mean squared error of the residuals	F统计值是对总体线性的显著性假设检验值，衡量拟合的重要性。 计算公式：
Prob (F-statistic)	The probability that you would get the above statistic, given the null hypothesis that they are unrelated	概率F值
Log-likelihood	The log of the likelihood function.	似然函数的对数
AIC	The Akaike Information Criterion. Adjusts the log-likelihood based on the number of observations and the complexity of the model.	AIC(赤池信息准则)，是统计模型选择中用于评判模型优劣的一个非常广泛应用的信息量准则。 AIC值越小，模型越好。
BIC	The Bayesian Information Criterion. Similar to the AIC, but has a higher penalty for models with more parameters.	BIC准则，对模型参数考虑的更多，是对AIC模型的弥补。它弥补了AIC准则不能给出模型阶数的相容估计问题。

第二个表显示的是拟合系数信息：

Element	Description	中文描述（个人翻译，如有错误请留言指出）
coef	The estimated value of the coefficient	模型系数估计值
std err	The basic standard error of the estimate of the coefficient. More sophisticated errors are also available.	系数的标准差
t	The t-statistic value. This is a measure of how statistically significant the coefficient is.	t统计量，衡量系数的统计显著性
P > |t|	P-value that the null-hypothesis that the coefficient = 0 is true. If it is less than the confidence level, often 0.05, it indicates that there is a statistically significant relationship between the term and the response.	P值，显示统计量和响应变量间是否具有显著线性相关性。
[95.0% Conf. Interval]	The lower and upper values of the 95% confidence interval	95%置信区间的上下边界值

最后一个表显示的是对残差分布的统计检验评估：

Element	Description	中文描述（个人翻译，如有错误请留言指出）
Skewness	A measure of the symmetry of the data about the mean. Normally-distributed errors should be symmetrically distributed about the mean (equal amounts above and below the line).	偏度系数，描述数据关于均值的对称情况。
Kurtosis	A measure of the shape of the distribution. Compares the amount of data close to the mean with those far away from the mean (in the tails).	峰度系数，通过比较接*均值的数据量 和远离平均值的数据量来 描述数据分布形状的度量值。
Omnibus	D’Angostino’s test. It provides a combined statistical test for the presence of skewness and kurtosis.	对偏度和峰度进行组合统计检验的一个概括值
Prob(Omnibus)	The above statistic turned into a probability	把上面的统计值变成概率值
Jarque-Bera	A different test of the skewness and kurtosis	另一个检验偏度和峰度的统计测试
Prob (JB)	The above statistic turned into a probability	把上面的统计值变成概率值
Durbin-Watson	A test for the presence of autocorrelation (that the errors are not independent.) Often important in time-series analysis	对自相关性的假设检验。通常在时间序列分析中很重要。
Cond. No	A test for multicollinearity (if in a fit with multiple parameters, the parameters are related with each other).	多重共线性的假设检验