Miguel González

Statistical errors

Mean Absolute Error (MAE)

MAE=1n∑i=1n∣yi−y^i∣\text{MAE}=\frac{1}{n}\sum_{i=1}^n|y_i-\hat{y}_i|

Mean Squared Error (MSE)

MSE=1n∑i=1n(yi−y^i)2\text{MSE}=\frac{1}{n}\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2

Residual Sum of Squares (RSS)

RSS=∑i=1n(yi−y^i)2\text{RSS}=\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2

Root Mean Square [Error/Deviation] (RMSE, RMS)

RMS=1n∑i=1nxi2\text{RMS}=\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2} RMSD=MSE=1n∑i=1n(yi−y^i)2\text{RMSD}=\sqrt{\text{MSE}}=\sqrt{\frac{1}{n}\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2}

Mean Absolute Percentage Error (MAPE)

MAPE=100%n∑i=1n∣yi−y^iyi∣\text{MAPE}=\frac{100\%}{n}\sum_{i=1}^n\left|\frac{y_i-\hat{y}_i}{y_i}\right|
MAPEInterpretation
<10%Highly accurate forecasting
10-20%Good forecasting
20-50%Reasonable forecasting
>50%Inaccurate forecasting

(Lewis, 1982, p.40)

Mean Square Percentage Error (MSPE)

MSPE=1n∑i=1n(yi^−yiyi)2\text{MSPE}=\frac{1}{n}\sum_{i=1}^n\left(\frac{\hat{y_i}-y_i}{y_i}\right)^2

Root Mean Square Percentage Error (RMSPE)

Swanson et al., Fomby, Shcherbakov et al.

RMSPE=MSPE=1n∑i=1n(yi^−yiyi)2\text{RMSPE}=\sqrt{\text{MSPE}}=\sqrt{\frac{1}{n}\sum_{i=1}^n\left(\frac{\hat{y_i}-y_i}{y_i}\right)^2}

Mean Absolute Scaled Error (MASE)

MASE=MAEQ\text{MASE}=\frac{\text{MAE}}{Q}

with QQ as scaling constant.

Median Absolute Deviation (MAD)

MAD=median(∣yi−median(y)∣)\text{MAD}=\text{median}\left(|y_i-\text{median}(y)|\right)

Symmetric Mean Absolute Percentage Error (sMAPE)

sMAPE=100n∑i=1n∣y^i−yi∣(∣yi∣+∣y^i∣)/2\text{sMAPE}=\frac{100}{n}\sum_{i=1}^{n}\frac{|\hat{y}_i-y_i|}{(|y_i|+|\hat{y}_i|)/2}

Resources