在时间序列预测中，评价指标的多样性为模型性能的评估带来多角度的参考意义。该篇推文列举了当前已知的27种确定性预测评估指标及其Python的实现，其中Python的评估指标函数实现基于numpy库（调用方法：import numpy as np）。

01 误差

def _error(actual: np.ndarray, predicted: np.ndarray):""" Simple error """return actual - predicted

02 误差百分比

def _percentage_error(actual: np.ndarray, predicted: np.ndarray):""" Percentage error """return (actual - predicted) / actual

03 均方误差 MSE

def mse(actual: np.ndarray, predicted: np.ndarray):""" Mean Squared Error """return np.mean(np.square(_error(actual, predicted)))

04 均方根误差 RMSE

def rmse(actual: np.ndarray, predicted: np.ndarray):""" Root Mean Squared Error """return np.sqrt(mse(actual, predicted))

05 标准化均方根误差 NRMSE

def nrmse(actual: np.ndarray, predicted: np.ndarray):""" Normalized Root Mean Squared Error """return rmse(actual, predicted) / (actual.max() - actual.min())

06 平均误差 ME

def me(actual: np.ndarray, predicted: np.ndarray):""" Mean Error """return np.mean(_error(actual, predicted))

07 平均绝对误差 MAE

def mae(actual: np.ndarray, predicted: np.ndarray):""" Mean Absolute Error """return np.mean(np.abs(_error(actual, predicted)))

08 中位数绝对误差 MedAE

def mdae(actual: np.ndarray, predicted: np.ndarray):""" Median Absolute Error """return np.median(np.abs(_error(actual, predicted)))

09 平均百分比误差 MPE

def mpe(actual: np.ndarray, predicted: np.ndarray):""" Mean Percentage Error """return np.mean(_percentage_error(actual, predicted))

10 平均绝对百分比误差 MAPE

def mape(actual: np.ndarray, predicted: np.ndarray):"""Mean Absolute Percentage ErrorProperties:+ Easy to interpret+ Scale independent- Biased, not symmetric- Undefined when actual[t] == 0Note: result is NOT multiplied by 100"""return np.mean(np.abs(_percentage_error(actual, predicted)))

11 中位数绝对误差百分比 MedAPE

def mdape(actual: np.ndarray, predicted: np.ndarray):"""Median Absolute Percentage ErrorNote: result is NOT multiplied by 100"""return np.median(np.abs(_percentage_error(actual, predicted)))

12 对称平均绝对误差百分比 SMAPE

def smape(actual: np.ndarray, predicted: np.ndarray):"""Symmetric Mean Absolute Percentage ErrorNote: result is NOT multiplied by 100"""return np.mean(2.0 * np.abs(actual - predicted) / ((np.abs(actual) + np.abs(predicted)) + EPSILON))

13 对称中位数绝对误差百分比 SMDAPE

def smdape(actual: np.ndarray, predicted: np.ndarray):"""Symmetric Median Absolute Percentage ErrorNote: result is NOT multiplied by 100"""return np.median(2.0 * np.abs(actual - predicted) / (np.abs(actual) + np.abs(predicted)))

14 平均反正切绝对百分比误差 MAAPE

def maape(actual: np.ndarray, predicted: np.ndarray):"""Mean Arctangent Absolute Percentage ErrorNote: result is NOT multiplied by 100"""return np.mean(np.arctan(np.abs((actual - predicted) / actual))

15 平均绝对比例误差 MASE

def mase(actual: np.ndarray, predicted: np.ndarray, seasonality: int = 1):"""Mean Absolute Scaled ErrorBaseline (benchmark) is computed with naive forecasting (shifted by @seasonality)"""return mae(actual, predicted) / mae(actual[seasonality:], _naive_forecasting(actual, seasonality))

16 标准化绝对误差 NAE

def std_ae(actual: np.ndarray, predicted: np.ndarray):""" Normalized Absolute Error """__mae = mae(actual, predicted)return np.sqrt(np.sum(np.square(_error(actual, predicted) - __mae))/(len(actual) - 1))

17 标准化绝对百分比误差 NAPE

def std_ape(actual: np.ndarray, predicted: np.ndarray):""" Normalized Absolute Percentage Error """__mape = mape(actual, predicted)return np.sqrt(np.sum(np.square(_percentage_error(actual, predicted) - __mape))/(len(actual) - 1))

18 均方根误差百分比 RMSPE

def rmspe(actual: np.ndarray, predicted: np.ndarray):"""Root Mean Squared Percentage ErrorNote: result is NOT multiplied by 100"""return np.sqrt(np.mean(np.square(_percentage_error(actual, predicted))))

19 中位数方根误差百分比 RMedSPE

def rmdspe(actual: np.ndarray, predicted: np.ndarray):"""Root Median Squared Percentage ErrorNote: result is NOT multiplied by 100"""return np.sqrt(np.median(np.square(_percentage_error(actual, predicted))))

20 均方根比例误差 RMSSE

def rmsse(actual: np.ndarray, predicted: np.ndarray, seasonality: int = 1):""" Root Mean Squared Scaled Error """q = np.abs(_error(actual, predicted)) / mae(actual[seasonality:], _naive_forecasting(actual, seasonality))return np.sqrt(np.mean(np.square(q)))

21 积分标准方根误差 INRSE

def inrse(actual: np.ndarray, predicted: np.ndarray):""" Integral Normalized Root Squared Error """return np.sqrt(np.sum(np.square(_error(actual, predicted))) / np.sum(np.square(actual - np.mean(actual))))

22 相对方根误差 RRSE

def rrse(actual: np.ndarray, predicted: np.ndarray):""" Root Relative Squared Error """return np.sqrt(np.sum(np.square(actual - predicted)) / np.sum(np.square(actual - np.mean(actual))))

23 平均相对误差 MRE

def mre(actual: np.ndarray, predicted: np.ndarray, benchmark: np.ndarray = None):""" Mean Relative Error """return np.mean(_relative_error(actual, predicted, benchmark))

24 相对绝对值误差 RAE

def rae(actual: np.ndarray, predicted: np.ndarray):""" Relative Absolute Error (aka Approximation Error) """return np.sum(np.abs(actual - predicted)) / (np.sum(np.abs(actual - np.mean(actual))) + EPSILON)

25 平均相对绝对值误差 MRAE

def mrae(actual: np.ndarray, predicted: np.ndarray, benchmark: np.ndarray = None):""" Mean Relative Absolute Error """return np.mean(np.abs(_relative_error(actual, predicted, benchmark)))

26 中位数相对绝对值误差 MedAE

def mdrae(actual: np.ndarray, predicted: np.ndarray, benchmark: np.ndarray = None):""" Median Relative Absolute Error """return np.median(np.abs(_relative_error(actual, predicted, benchmark)))

27 平均方向精度 MDA

def mda(actual: np.ndarray, predicted: np.ndarray):""" Mean Directional Accuracy """return np.mean((np.sign(actual[1:] - actual[:-1]) == np.sign(predicted[1:] - predicted[:-1])).astype(int))

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