Add LnQ Performance measure

[antoinecarme]

According to :

https://en.wikipedia.org/wiki/Symmetric_mean_absolute_percentage_error

A limitation to SMAPE is that if the actual value or forecast value is 0, the value of error will boom up to the upper-limit of error. (200% for the first formula and 100% for the second formula).

Provided the data are strictly positive, a better measure of relative accuracy can be obtained based on the log of the accuracy ratio: log(Ft / At) This measure is easier to analyse statistically, and has valuable symmetry and unbiasedness properties. When used in constructing forecasting models the resulting prediction corresponds to the geometric mean (Tofallis, 2015).

[antoinecarme]

Reference article :

Tofallis, C (2015) "A Better Measure of Relative Prediction Accuracy for Model Selection and Model Estimation", Journal of the Operational Research Society, 66(8),1352-1362.

http://uhra.herts.ac.uk/bitstream/handle/2299/16654/A_Better_Measure_of_Relative_prediction_accuracy_for_model_selection_and_fitting_Preprint_JORS_.pdf?sequence=3

[antoinecarme]

Can be used through the option :

self.mModelSelection_Criterion = "LnQ";

[antoinecarme]

from

http://uhra.herts.ac.uk/bitstream/handle/2299/16654/A_Better_Measure_of_Relative_prediction_accuracy_for_model_selection_and_fitting_Preprint_JORS_.pdf?sequence=3

[antoinecarme]

Also :

http://uhra.herts.ac.uk/bitstream/handle/2299/16654/A_Better_Measure_of_Relative_prediction_accuracy_for_model_selection_and_fitting_Preprint_JORS_.pdf?sequence=3

[antoinecarme]

L. Tornqvist, P. Vartia and Y. O. Vartia (1985), How Should Relative Changes Be Measured? The American Statistician, 39, 43-46 http://www.jstor.org/stable/2683905

[antoinecarme]

https://stats.stackexchange.com/questions/154918/calculating-percentage-difference-between-maximum-and-current-value

[antoinecarme]

Performance Metrics (Error Measures) in Machine Learning Regression, Forecasting and Prognostics: Properties and Typology Alexei Botchkarev

https://arxiv.org/abs/1809.03006

Botchkarev, A. (2019) A new typology design of performance metrics to measure errors in machine learning regression algorithms. Interdisciplinary Journal of Information, Knowledge, and Management, 14, 45-79.

https://doi.org/10.28945/4184 https://www.gsrc.ca/metrics_typology2019.pdf

[antoinecarme]

Hyndman, R. J., & Koehler, A. B. (2006). Another look at measures of forecast accuracy. International journal of forecasting, 22(4), 679-688. https://doi.org/10.1016/j.ijforecast.2006.03.001

[antoinecarme]

Provided that predicted and actual values are strictly positive :

LnQ_i = ln(predicted_i / actual_i) = ln(predicted_i) – ln(actual_i)

LnQ = sum_i {LnQ_i ^ 2}

[antoinecarme]

Ozone model (model selection based on LnQ) :

[antoinecarme]

Air Passengers model (model selection based on LnQ):

[antoinecarme]

Closing