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 Error
Properties:
+ Easy to interpret
+ Scale independent
- Biased, not symmetric
- Undefined when actual[t] == 0
Note: 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 Error
Note: 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 Error
Note: 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 Error
Note: 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 Error
Note: 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 Error
Baseline (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 Error
Note: 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 Error
Note: 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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