Normalized Absolute Error Wiki
Contents |
may be challenged and removed. (December 2009) (Learn how and when to remove this template message) The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of prediction accuracy of a forecasting method in statistics, for example in trend estimation. relative absolute error It usually expresses accuracy as a percentage, and is defined by the formula: M =
Mean Absolute Error Formula
100 n ∑ t = 1 n | A t − F t A t | , {\displaystyle {\mbox{M}}={\frac {100}{n}}\sum _{t=1}^{n}\left|{\frac {A_{t}-F_{t}}{A_{t}}}\right|,} where At
Mean Absolute Percentage Error
is the actual value and Ft is the forecast value. The difference between At and Ft is divided by the Actual value At again. The absolute value in this calculation is summed for every forecasted point in time and
Mean Absolute Error Excel
divided by the number of fitted pointsn. Multiplying by 100 makes it a percentage error. Although the concept of MAPE sounds very simple and convincing, it has major drawbacks in practical application [1] It cannot be used if there are zero values (which sometimes happens for example in demand data) because there would be a division by zero. For forecasts which are too low the percentage error cannot exceed 100%, but for forecasts which are too high there is mean absolute error example no upper limit to the percentage error. When MAPE is used to compare the accuracy of prediction methods it is biased in that it will systematically select a method whose forecasts are too low. This little-known but serious issue can be overcome by using an accuracy measure based on the ratio of the predicted to actual value (called the Accuracy Ratio), this approach leads to superior statistical properties and leads to predictions which can be interpreted in terms of the geometric mean.[1] Contents 1 Alternative MAPE definitions 2 Issues 3 See also 4 External links 5 References Alternative MAPE definitions[edit] Problems can occur when calculating the MAPE value with a series of small denominators. A singularity problem of the form 'one divided by zero' and/or the creation of very large changes in the Absolute Percentage Error, caused by a small deviation in error, can occur. As an alternative, each actual value (At) of the series in the original formula can be replaced by the average of all actual values (Āt) of that series. This alternative is still being used for measuring the performance of models that forecast spot electricity prices.[2] Note that this is the same as dividing the sum of absolute differences by the sum of actual values, and is sometimes referred to as WAPE. Issues[edit] While MAPE is one of the most popular measures for forecasting error, there are many studies
neutral. Please help improve it by replacing them with more appropriate citations to reliable, independent, third-party sources. (April 2011) (Learn how and when to remove this template message) In statistics, the mean absolute mean relative error scaled error (MASE) is a measure of the accuracy of forecasts . It was mean absolute scaled error proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as mean percentage error a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements."[1] The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such https://en.wikipedia.org/wiki/Mean_absolute_percentage_error as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.[2] Contents 1 Rationale 2 Non seasonal time series 3 Seasonal time series 4 See also 5 References Rationale[edit] The mean absolute scaled error has the following desirable properties:[3] Scale invariance: The mean absolute scaled error is independent of the scale of the data, so can be used to compare forecasts across data sets with https://en.wikipedia.org/wiki/Mean_absolute_scaled_error different scales. Predictable behavior as y t → 0 {\displaystyle y_{t}\rightarrow 0} : Percentage forecast accuracy measures such as the Mean absolute percentage error (MAPE) rely on division of y t {\displaystyle y_{t}} , skewing the distribution of the MAPE for values of y t {\displaystyle y_{t}} near or equal to 0. This is especially problematic for data sets whose scales do not have a meaningful 0, such as temperature in Celsius or Fahrenheit, and for intermittent demand data sets, where y t = 0 {\displaystyle y_{t}=0} occurs frequently. Symmetry: The mean absolute scaled error penalizes positive and negative forecast errors equally, and penalizes errors in large forecasts and small forecasts equally. In contrast, the MAPE and median absolute percentage error (MdAPE) fail both of these criteria, while the "symmetric" sMAPE and sMdAPE[4] fail the second criterion. Interpretability: The mean absolute scaled error can be easily interpreted, as values greater than one indicate that in-sample one-step forecasts from the naïve method perform better than the forecast values under consideration. Asymptotic normality of the MASE: The Diebold-Mariano test for one-step forecasts is used to test the statistical significance of the difference between two sets of forecasts. To perform hypothesis
1 ( x ) = 1 + x {\displaystyle P_{1}(x)=1+x} (red) at a = 0. The approximation https://en.wikipedia.org/wiki/Approximation_error error is the gap between the curves, and it increases for x values further from 0. The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because the measurement of the data is not precise due to the instruments. (e.g., the absolute error accurate reading of a piece of paper is 4.5cm but since the ruler does not use decimals, you round it to 5cm.) or approximations are used instead of the real data (e.g., 3.14 instead of π). In the mathematical field of numerical analysis, the numerical stability of an algorithm in numerical analysis indicates how the mean absolute error error is propagated by the algorithm. Contents 1 Formal Definition 1.1 Generalizations 2 Examples 3 Uses of relative error 4 Instruments 5 See also 6 References 7 External links Formal Definition[edit] One commonly distinguishes between the relative error and the absolute error. Given some value v and its approximation vapprox, the absolute error is ϵ = | v − v approx | , {\displaystyle \epsilon =|v-v_{\text{approx}}|\ ,} where the vertical bars denote the absolute value. If v ≠ 0 , {\displaystyle v\neq 0,} the relative error is η = ϵ | v | = | v − v approx v | = | 1 − v approx v | , {\displaystyle \eta ={\frac {\epsilon }{|v|}}=\left|{\frac {v-v_{\text{approx}}}{v}}\right|=\left|1-{\frac {v_{\text{approx}}}{v}}\right|,} and the percent error is δ = 100 % × η = 100 % × ϵ | v | = 100 % × | v − v approx v | . {\displaystyle \delta =100\%\times \eta =100\%\times {\frac {\epsilon }{|v|}}=100\%\times \left|{\frac {v-v_{\text{approx}}}{v}}\right|.} In