Numeric Prediction Task Performance Measure
An Numeric Prediction Task Performance Measure is an prediction error measure for a numeric prediction task (see: approximation task).
- AKA: Point Estimation Proximity Measure.
- Context:
- It can be an input to an Estimator Evaluation Task.
- It can be associated to an Approximation Error Derivative Function.
- It can range from being an Absolute Approximation Error to being a Relative Approximation Error.
- It can be a Forecasting Error Measure, for forecasting tasks.
- Example(s):
- a Scale-Dependent Approximation Performance Measure, such as:
- Mean Squared Error, MSE() = [math]\displaystyle{ \frac{1}{n}\sum_{i=1}^n(\hat{Y_i} - Y_i)^2. }[/math]
- Root Mean Square Error, RMSE() = [math]\displaystyle{ \sqrt{\operatorname{MSE}(\hat{\theta})} =\sqrt{\frac{\sum_{t=1}^n (y_t - \hat y_t)^2}{n}}. }[/math]
- Mean Absolute Error, MAE() = [math]\displaystyle{ \frac{1}{n}\sum_{i=1}^n \left| f_i-y_i\right| =\frac{1}{n}\sum_{i=1}^n \left| e_i \right|. }[/math]
- Median Absolute Error (MdAE)
- a Scale-Independent Approximation Performance Measure, such as:
- Mean Relative Error, MRE() = [math]\displaystyle{ \frac{1}{n}\sum_{t=1}^n \frac{f_t-a_t}{a_t} }[/math]
- Mean Percent Error, MPE() = [math]\displaystyle{ \frac{100%}{n}\sum_{t=1}^n \frac{f_t-a_t}{a_t} }[/math]
- Mean Absolute Percentage Error, MAPE() = [math]\displaystyle{ \frac{100%}{n}\sum_{t=1}^n \left| \frac{A_t-F_t}{A_t}\right| }[/math]
- Median Absolute Percentage Error (MdAPE).
- Root Mean Square Percentage Error (RMSPE).
- Root Median Square Percentage Error (RMdSPE).
- Symmetric Mean Absolute Percentage Error (sMAPE).
- Symmetric Median Absolute Percentage Error (sMdAPE).
- a Relative Errors-based Approximation Performance Measure, such as:
- a Relative Measure-based Approximation Performance Measure, such as:
- a Scaled Error-based Approximation Performance Measure, such as:
- …
- a Scale-Dependent Approximation Performance Measure, such as:
- Counter-Example(s):
- a Classification Error, such as F1 Error ..., and AUC Measure?
- a Ranking Error Measure.
- a Range Estimation Measure.
- See: Task Performance Measure, Numerical Analysis, Numerical Stability.
References
2016
- (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Approximation_error Retrieved:2016-4-11.
- 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 accurate reading of a piece of paper is 4.5 cm but since the ruler does not use decimals, you round it to 5 cm.) 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 error is propagated by the algorithm.
- The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because
2016
- (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Approximation_error#Formal_Definition Retrieved:2016-4-11.
- One commonly distinguishes between the relative error and the absolute error.
Given some value v and its approximation vapprox, the absolute error is : [math]\displaystyle{ \epsilon = |v-v_\text{approx}|\ , }[/math] where the vertical bars denote the absolute value.
If [math]\displaystyle{ v \ne 0, }[/math] the relative error is : [math]\displaystyle{ \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, }[/math] and the percent error is : [math]\displaystyle{ \delta = 100\%\times\eta = 100\%\times\frac{\epsilon}{|v|} = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. }[/math] In words, the absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.
- One commonly distinguishes between the relative error and the absolute error.
1998
- (Cowan, 1998) ⇒ Cowan. (1998). “Statistical Data Analysis." Oxford University Press, ISBN:0198501560
- QUOTE: An alternative (and often equivalent) method of reporting the statistical error of a measurement is with a confidence interval, which was first developed by Neyman (Ney37 ]. Suppose as above that one has [math]\displaystyle{ n }[/math] observations of a random variable [math]\displaystyle{ x }[/math] which can be used to evaluate_an estimator [math]\displaystyle{ \hat{\theta}(x_l,...,x_n) }[/math] for a parameter [math]\displaystyle{ \theta }[/math], and that the value obtained is <math>\hat{\theta}_{obs}<math>.