Root mean square error (RMSE) or mean absolute error (MAE)? – Arguments against avoiding RMSE in the literature
Open accessPublished version of a preprint
Abstract
Abstract. Both the root mean square error (RMSE) and the mean absolute error (MAE) are regularly employed in model evaluation studies. Willmott and Matsuura (2005) have suggested that the RMSE is not a good indicator of average model performance and might be a misleading indicator of average error, and thus the MAE would be a better metric for that purpose. While some concerns over using RMSE raised by Willmott and Matsuura (2005) and Willmott et al. (2009) are valid, the proposed avoidance of RMSE in favor of MAE is not the solution. Citing the aforementioned papers, many researchers chose MAE over RMSE to present their model evaluation statistics when presenting or adding the RMSE measures could be more beneficial. In this technical note, we demonstrate that the RMSE is not ambiguous in its meaning, contrary to what was claimed by Willmott et al. (2009). The RMSE is more appropriate to represent model performance than the MAE when the error distribution is expected to be Gaussian. In addition, we show that the RMSE satisfies the triangle inequality requirement for a distance metric, whereas Willmott et al. (2009) indicated that the sums-of-squares-based statistics do not satisfy this rule. In the end, we discussed some circumstances where using the RMSE will be more beneficial. However, we do not contend that the RMSE is superior over the MAE. Instead, a combination of metrics, including but certainly not limited to RMSEs and MAEs, are often required to assess model performance.
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Written from the paper’s full text.
TL;DR
This technical note argues against the recommendation by Willmott and Matsuura (2005) and Willmott et al. (2009) to abandon RMSE in favor of MAE, showing that RMSE is not ambiguous in meaning and is more appropriate when errors are expected to be Gaussian. The authors also demonstrate that RMSE satisfies the triangle inequality required of a distance metric, contrary to the earlier claims, while concluding that a combination of metrics is often needed.
Key findings
- RMSE is not ambiguous in its meaning, contrary to Willmott et al. (2009); it is more appropriate than MAE when the error distribution is expected to be Gaussian, whereas MAE suits uniformly distributed errors.
- Using randomly generated zero-mean, unit-variance Gaussian pseudo-errors, RMSEs converge to the true standard deviation (SE = 1) once n ≥ 100, with standard deviation within 5% of truth; MAEs converge to 0.8, approximating the expectation of √(2/π). Neither metric is robust at n = 4 or n = 10.
- RMSE satisfies the triangle inequality required for a distance metric: the example in Willmott et al. (2009) (d(a,c)=4, d(a,b)=2, d(b,c)=3) actually represents the MSE, not the RMSE, and the L2-norm proof in Eqs. (5)–(7) shows RMSE does satisfy it.
- RMSE has practical advantages: it avoids absolute values (useful for gradients/sensitivities and data assimilation cost functions) and is generally better at discriminating among model results by weighting unfavorable conditions more heavily.
Methods
- Technical note / commentary design: no new observational dataset; the argument is analytical, using norm theory (L1- and L2-norms linked to MAE and RMSE) and a proof of the triangle inequality for the L2-norm.
- A Monte Carlo-style demonstration: five sets of pseudo-errors of size n = 4, 10, 100, 1000, 10 000, and 1000 000 drawn from a zero-mean, unit-variance Gaussian distribution with different random seeds, comparing RMSEs and MAEs (Table 1).
- Reference to prior empirical example: Chai et al. (2009) reported RMSE-to-MAE ratios of 1.63 to 2.29 for model NO2 column predictions versus SCIAMACHY satellite observations.
Limitations
- Stated by the authors: the analysis assumes unbiased errors and does not consider uncertainties from observation errors or the method used to compare model and observations; all statistics are less useful with limited sample sizes (e.g., n = 4 or 10), where presenting the errors themselves may be more appropriate; for non-normal distributions, additional moments (mean, variance, skewness, flatness) are needed; outliers several orders larger than other samples may justifiably be removed when sample numbers are limited, and systematic errors may need removal before computing RMSEs.
- Evident from the text: the demonstration relies on synthetic Gaussian pseudo-errors rather than real model–observation comparisons, so the convergence results are conditional on the Gaussian assumption; the authors explicitly do not claim RMSE is superior to MAE, so the note offers guidance rather than a definitive metric selection rule.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
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Cites
Statistics
Claims the paper makes
4 of 8 cited sources checked · 4 supported · 0 flagged
- Premisecitation
Willmott and Matsuura (2005) demonstrated that while keeping the MAE constant at 2.0, the RMSE varies from 2.0 to 4.0.
“Using hypothetical sets of four errors, Willmott and Matsuura (2005) demonstrated that while keeping the MAE as a constant of 2.0, the RMSE varies from 2.0 to 4.0.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites Willmott and Matsuura (2005) source
- Findingreasoning
The RMSE satisfies the triangle inequality requirement for a distance metric.
“In addition, we show that the RMSE satisfies the triangle inequality requirement for a distance metric, whereas Willmott et al. (2009) indicated that the sums-ofsquares-based statistics do not satisfy this rule.”
- Also stated in: § 3 Triangle inequality of a metric · boosted
“This proves that RMSE satisfies the triangle inequality required for a distance function metric.”
- Also stated in: § 4 Summary and discussion · boosted
“In addition, we demonstrate that the RMSE satisfies the triangle inequality required for a distance function metric.”
Quote in the paperSubstantial claim - Also stated in: § 3 Triangle inequality of a metric · boosted
- Findingown data
When the sample size reaches 100 or above, using calculated RMSEs one can re-construct the error distribution close to its truth with its standard deviation within 5% of its truth (SE = 1).
“When the sample size reaches 100 or above, using the calculated RMSEs one can re-construct the error distribution close to its “truth” or “exact solution”, with its standard deviation within 5 % to its truth (i.e., SE = 1).”
Quote in the paperSubstantial claim- The authors state this as an assumption: The errors are unbiased and follow a normal distribution when presenting the RMSE.
- The authors state this as an assumption: The uncertainties brought in by observation errors or the method used to compare model and observations are not considered.
- Findingown data
Neither the RMSEs nor the MAEs are robust when only 4 or 10 samples are used to calculate those values.
“For instance, Table 1 shows that neither the RMSEs nor the MAEs are robust when only 4 or 10 samples are used to calculate those values.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The errors are unbiased and follow a normal distribution when presenting the RMSE.
- The authors state this as an assumption: The uncertainties brought in by observation errors or the method used to compare model and observations are not considered.
- Interpretationreasoning
The RMSE is more appropriate to represent model performance than the MAE when the error distribution is expected to be Gaussian.
“The RMSE is more appropriate to represent model performance than the MAE when the error distribution is expected to be Gaussian.”
- Also stated in: § 1 Introduction · plain
“When the error distribution is expected to be Gaussian and there are enough samples, the RMSE has an advantage over the MAE to illustrate the error distribution.”
- Also stated in: § 4 Summary and discussion · plain
“We present that the RMSE is not ambiguous in its meaning, and it is more appropriate to use than the MAE when model errors follow a normal distribution.”
Quote in the paperSubstantial claim- Rests on an assumption: Model errors are likely to have a normal distribution rather than a uniform distribution.
- Also stated in: § 1 Introduction · plain
- Interpretationreasoning
Giving higher weighting to unfavorable conditions, the RMSE usually is better at revealing model performance differences.
“Giving higher weighting to the unfavorable conditions, the RMSE usually is better at revealing model performance differences.”
Quote in the paperSubstantial claim - Premisecitation
The L1-norm and L2-norm are closely related to the MAE and the RMSE, respectively.
“The L1-norm and L2-norm are closely related to the MAE and the RMSE, respectively, as shown in Eqs. ( 3) and ( 4): Horn and Johnson, 1990).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Horn and Johnson, 1990
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The RMSE is not ambiguous in its meaning, contrary to what was claimed by Willmott et al. (2009).
“In this technical note, we demonstrate that the RMSE is not ambiguous in its meaning, contrary to what was claimed by Willmott et al. (2009).”
- Also stated in: § 4 Summary and discussion · plain
“We present that the RMSE is not ambiguous in its meaning, and it is more appropriate to use than the MAE when model errors follow a normal distribution.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Willmott et al. (2009)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Also stated in: § 4 Summary and discussion · plain
- Interpretationreasoning
RMSEs avoid the use of absolute value, which is highly undesirable in many mathematical calculations.
“One distinct advantage of RMSEs over MAEs is that RM-SEs avoid the use of absolute value, which is highly undesirable in many mathematical calculations.”
Quote in the paperSubstantial claim - Premisenone given
For several sets of errors with the same RMSE, the MAE would vary from set to set.
“Similarly, one can readily show that, for several sets of errors with the same RMSE, the MAE would vary from set to set.”
Quote in the paperNot a substantial claim - Interpretationown data
All statistics are less useful when there are only a limited number of error samples.
“It should be noted that all statistics are less useful when there are only a limited number of error samples.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The example in Willmott et al. (2009) represents the mean square error (MSE), which cannot be used as a distance metric, rather than the RMSE.
“In fact, this example represents the mean square error (MSE), which cannot be used as a distance metric, rather than the RMSE.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Willmott et al. (2009)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The MAE might be affected by a large amount of average error values without adequately reflecting some large errors.
“In this regard, the MAE might be affected by a large amount of average error values without adequately reflecting some large errors.”
Quote in the paperSubstantial claim - Premisecitation
The RMSE tends to grow larger than the MAE with n because its lower limit is fixed at the MAE and its upper limit increases with n.
“The RMSE tends to grow larger than the MAE with n 1foot_1 since its lower limit is fixed at the MAE and its upper limit (n 1 2 • MAE) increases with n 1 2 .”
Quote in the paperSubstantial claim - Interpretationown data
When there are more samples, reconstructing the error distribution using RMSEs will be even more reliable.
“When there are more samples, reconstructing the error distribution using RMSEs will be even more reliable.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The errors are unbiased and follow a normal distribution when presenting the RMSE.
- The authors state this as an assumption: The uncertainties brought in by observation errors or the method used to compare model and observations are not considered.
- Interpretationreasoning
When evaluating different models using a single metric, differences in the error distributions become more important.
“When evaluating different models using a single metric, differences in the error distributions become more important.”
Quote in the paperSubstantial claim - Interpretationreasoning
Penalizing large errors through defined least-square terms proves to be very effective in improving model performance.
“In such applications, penalizing large errors through the defined least-square terms proves to be very effective in improving model performance.”
Quote in the paperNot a substantial claim - Interpretationnone given
The existence of outliers and their probability of occurrence is well described by the normal distribution underlying the use of the RMSE.
“In fact, the existence of outliers and their probability of occurrence is well described by the normal distribution underlying the use of the RMSE.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The errors are unbiased and follow a normal distribution when presenting the RMSE.
- Interpretationnone given
It might be difficult to calculate the gradient or sensitivity of the MAEs with respect to certain model parameters.
“For instance, it might be difficult to calculate the gradient or sensitivity of the MAEs with respect to certain model parameters.”
Quote in the paperSubstantial claim - Interpretationreasoning
Under the circumstances of calculating model error sensitivities or data assimilation applications, MAEs are definitely not preferred over RMSEs.
“Under the circumstances of calculating model error sensitivities or data assimilation applications, MAEs are definitely not preferred over RMSEs.”
Quote in the paperSubstantial claim
- Classified as probably not a substantial claim. 10
- May combine two claims. 2
- The quote only partly states this claim. 1
Classified as probably not a substantial claim.
PremisecitationThe RMSE tends to become increasingly larger than the MAE as the distribution of error magnitudes becomes more variable.
“They summarized that the RMSE tends to become increasingly larger than the MAE (but not necessarily in a monotonic fashion) as the distribution of error magnitudes becomes more variable.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites Willmott and Matsuura (2005) source
Classified as probably not a substantial claim. May combine two claims.
PremisecitationThe RMSE varies with the variability of the error magnitudes, the total-error or average-error magnitude (MAE), and the sample size n.
“They concluded that the RMSE varies with the variability of the the error magnitudes and the total-error or averageerror magnitude (MAE), and the sample size n.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites Willmott and Matsuura (2005) source
May combine two claims.
PremisecitationWillmott and Matsuura (2005) proved that the RMSE is not equivalent to the MAE, and one cannot easily derive the MAE value from the RMSE (and vice versa).
“Willmott and Matsuura (2005) have simply proved that the RMSE is not equivalent to the MAE, and one cannot easily derive the MAE value from the RMSE (and vice versa).”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites Willmott and Matsuura (2005) source
The quote only partly states this claim.
Findingown dataThe MAEs converge to 0.8, an approximation to the expectation of sqrt(2/pi).
“Table 1 shows that the MAEs converge to 0.8, an approximation to the expectation of 2 π .”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The errors are unbiased and follow a normal distribution when presenting the RMSE.
- The authors state this as an assumption: The uncertainties brought in by observation errors or the method used to compare model and observations are not considered.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenWhen both metrics are calculated, the RMSE is by definition never smaller than the MAE.
“When both metrics are calculated, the RMSE is by definition never smaller than the MAE.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenThe MAE is suitable to describe uniformly distributed errors.
“The MAE is suitable to describe uniformly distributed errors.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).
InterpretationreasoningFor other kinds of distributions, more statistical moments of model errors, such as mean, variance, skewness, and flatness, are needed to provide a complete picture of the model error variation.
“For other kinds of distributions, more statistical moments of model errors, such as mean, variance, skewness, and flatness, are needed to provide a complete picture of the model error variation.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe ratio of RMSE to MAE ranged from 1.63 to 2.29 in Chai et al. (2009) model NO2 column predictions compared to SCIAMACHY satellite observations.
“The ratio of RMSE to MAE ranged from 1.63 to 2.29 (see Table 1 of Chai et al., 2009).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites Chai et al., 2009 source
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone givenIn many model sensitivity studies that use only RMSE, a detailed interpretation is not critical because variations of the same model will have similar error distributions.
“In many of the model sensitivity studies that use only RMSE, a detailed interpretation is not critical because variations of the same model will have similar error distributions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable).
PremisereasoningAny single metric provides only one projection of the model errors and emphasizes a certain aspect of the error characteristics.
“As every statistical measure condenses a large number of data into a single value, it only provides one projection of the model errors emphasizing a certain aspect of the error characteristics of the model performance.”
- Also stated in: § 4 Summary and discussion · plain
“As stated earlier, any single metric provides only one projection of the model errors and, therefore, only emphasizes a certain aspect of the error characteristics.”
Quote in the paperNot a substantial claim- Also stated in: § 4 Summary and discussion · plain
Probably not a substantial claim: may not pass S2 (checkable).
InterpretationreasoningA combination of metrics, including but not limited to RMSEs and MAEs, are often required to assess model performance.
“A combination of metrics, including but certainly not limited to RMSEs and MAEs, are often required to assess model performance.”
- Also stated in: § 4 Summary and discussion · plain
“A combination of metrics, including but certainly not limited to RMSEs and MAEs, are often required to assess model performance.”
Quote in the paperNot a substantial claim- Also stated in: § 4 Summary and discussion · plain
Classified as probably not a substantial claim.
Premisenone givenThe RMSE penalizes variance as it gives errors with larger absolute values more weight than errors with smaller absolute values.
“While the MAE gives the same weight to all errors, the RMSE penalizes variance as it gives errors with larger absolute values more weight than errors with smaller absolute values.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
The errors are unbiased and follow a normal distribution when presenting the RMSE.
“The underlying assumption when presenting the RMSE is that the errors are unbiased and follow a normal distribution.”
“We also assume the error sample set is unbiased.”
The paper relies on this without support.
Model errors are likely to have a normal distribution rather than a uniform distribution.
“Because model errors are likely to have a normal distribution rather than a uniform distribution, the RMSE is a better metric to present than the MAE for such a type of data.”
The authors state this as an assumption.
The uncertainties brought in by observation errors or the method used to compare model and observations are not considered.
“The uncertainties brought in by observation errors or the method used to compare model and observations are not considered here.”
The paper relies on this without support.
There are often hundreds of observations available to calculate model statistics.
“Fortunately, there are often hundreds of observations available to calculate model statistics, unlike the examples with n = 4 (Willmott and Matsuura, 2005) and n = 10 ( Willmott et al., 2009).”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Table reports an MAE larger than its RMSE
MinorFor the fourth set of four errors, the table lists RMSE = 1.02 and MAE = 1.16. This contradicts the paper's statement that RMSE is never smaller than MAE when both are calculated for the same errors.
“| 4 | 0.92, 0.65, 1.48, 1.02, 0.79 0.70, 0.57, 1.33, 1.16, 0.76 | |”
“When both metrics are calculated, the RMSE is by definition never smaller than the MAE.”
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An Overview of Voice Conversion and Its Challenges: From Statistical Modeling to Deep Learning
Şişman, B., Yamagishi, J., King, S., & Li, H. (2020) · IEEE/ACM Transactions on Audio Speech and Language Processing · doi:10.1109/taslp.2020.3038524 · Cited by 374 · see citing works
“The common objective evaluation metrics include Mel-cepstral distortion (MCD) [246] for spectrum, and PCC [247] and RMSE [248]–[250] for prosody.”
BackgroundInfluentialBui, D. T., Pradhan, B., Nampak, H., Bui, Q., Tran, Q.-A., & Nguyen, Q.-P. (2016) · Journal of Hydrology · doi:10.1016/j.jhydrol.2016.06.027 · Cited by 356 · see citing works
Tran, V. Q., Dang, V. Q., & Ho, L. S. (2022) · Construction and Building Materials · doi:10.1016/j.conbuildmat.2022.126578 · Cited by 351 · see citing works
Artificial Intelligence Approaches for Prediction of Compressive Strength of Geopolymer Concrete
Dao, D. V., Ly, H., Trinh, S. H., et al. (2019) · Materials · doi:10.3390/ma12060983 · Cited by 344 · see citing works
“These measures can be calculated by the following equations [55]:”
BackgroundMethodInfluentialNiu, M., Kebreab, E., Hristov, A. N., Oh, J., Arndt, C., Bannink, A., Bayat, A. R., Brito, A. F., Boland, T. M., Casper, D. P., et al. (2018) · Global Change Biology · doi:10.1111/gcb.14094 · Cited by 343 · see citing works
“The MAE was calculated as shown in Equation (vi) to quantify the prediction error as suggested by Chai and Draxler (2014): In both cases, smaller RMSPE or MAE implies better model performance.”
Building Auto-Encoder Intrusion Detection System based on random forest feature selection
Li, X., Chen, W., Zhang, Q., & Wu, L. (2020) · Computers & Security · doi:10.1016/j.cose.2020.101851 · Cited by 340 · see citing works
On Mean Absolute Error for Deep Neural Network Based Vector-to-Vector Regression
Qi, J., Du, J., Siniscalchi, S. M., et al. (2020) · IEEE Signal Processing Letters · doi:10.1109/lsp.2020.3016837 · Cited by 331 · see citing works
“An exception occurs when the expected error satisfies Gaussian-distributed and enough training samples are available [3].”
BackgroundForecasting air quality time series using deep learning
Freeman, B. S., Taylor, G. W., Gharabaghi, B., & Thé, J. V. G. (2018) · Journal of the Air & Waste Management Association · doi:10.1080/10962247.2018.1459956 · Cited by 321 · see citing works
“MAE and RMSE are widely used measures of continuous variables, with RMSE criticized for overbiasing toward large errors (Chai and Draxler 2014; Willmott and Matsuura 2005).”
MethodAL‐Musaylh, M. S., Deo, R. C., Adamowski, J., & Li, Y. (2017) · Advanced Engineering Informatics · doi:10.1016/j.aei.2017.11.002 · Cited by 302 · see citing works
Zhang, Z., Ye, L., Qin, H., Liu, Y., Wang, C., Yu, X. J., Yin, X., & Li, J. (2019) · Applied Energy · doi:10.1016/j.apenergy.2019.04.047 · Cited by 299 · see citing works
Khalil, M., McGough, A. S., Pourmirza, Z., Pazhoohesh, M., & Walker, S. L. (2022) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2022.105287 · Cited by 307 · see citing works
Comparative Analysis of Machine Learning Techniques for Predicting Air Quality in Smart Cities
Ameer, S., Shah, M. A., Khan, A., et al. (2019) · IEEE Access · doi:10.1109/access.2019.2925082 · Cited by 293 · see citing works
Vafaei, S., Soosani, J., Adeli, K., Fadaei, H., Naghavi, H., Pham, T. D., & Bui, D. T. (2018) · Remote Sensing · doi:10.3390/rs10020172 · Cited by 280 · see citing works
“However, the equation is significantly influenced by large values and outliers [84].”
BackgroundNguyen, K. T., Nguyen, Q. D., Lê, A. T., et al. (2020) · Construction and Building Materials · doi:10.1016/j.conbuildmat.2020.118581 · Cited by 272 · see citing works
Meta-learning on Heterogeneous Information Networks for Cold-start Recommendation
Lu, Y., Fang, Y., & Shi, C. (2020) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/3394486.3403207 · Cited by 279 · see citing works
“performance [2], it is intuitive that the larger the loss value in a semantic space, the smaller the corresponding weight should be.”
BackgroundNam, K., Hwangbo, S., & Yoo, C. (2020) · Renewable and Sustainable Energy Reviews · doi:10.1016/j.rser.2020.109725 · Cited by 271 · see citing works
Novel approach for streamflow forecasting using a hybrid ANFIS-FFA model
Yaseen, Z. M., Ebtehaj, I., Bonakdari, H., Deo, R. C., Danandeh Mehr, A., Mohtar, W. H. M. W., Diop, L., El-shafie, A., & Singh, V. P. (2017) · Journal of Hydrology · doi:10.1016/j.jhydrol.2017.09.007 · Cited by 267 · see citing works
Review of medical image quality assessment
Chow, L. S., & Paramesran, R. (2016) · Biomedical Signal Processing and Control · doi:10.1016/j.bspc.2016.02.006 · Cited by 266 · see citing works
Low-Cost Outdoor Air Quality Monitoring and Sensor Calibration
Concas, F., Mineraud, J., Lagerspetz, E., Varjonen, S., Liu, X., Puolamäki, K., Nurmi, P., & Tarkoma, S. (2021) · ACM Transactions on Sensor Networks · doi:10.1145/3446005 · Cited by 275 · see citing works
“ge distance of points from the regression model. RMSE and MSE weigh errors proportionally to their magnitude, whereas MAE weighs all errors equally. This makes RMSE and MSE more sensitive to outliers [125], suggesting that MAE is a better measure for measuring the average performance, whereas (R)MSE is useful for meas…”
BackgroundGramatica, P. (2020) · International Journal of Quantitative Structure-Property Relationships · doi:10.4018/ijqspr.20200701.oa1 · Cited by 260 · see citing works
“The different quality of these two parameters is commented in the literature (Chai & Draxler, 2014; Roy et al., 2016).”
BackgroundMohammadi, B., & Mehdizadeh, S. (2020) · Agricultural Water Management · doi:10.1016/j.agwat.2020.106145 · Cited by 255 · see citing works
Bhagat, S. K., Tung, T. M., & Yaseen, Z. M. (2019) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2019.119473 · Cited by 252 · see citing works
Deep learning as a tool for ecology and evolution
Borowiec, M. L., Dikow, R. B., Frandsen, P. B., McKeeken, A. J., Valentini, G., & White, A. E. (2022) · Methods in Ecology and Evolution · doi:10.1111/2041-210x.13901 · Cited by 253 · see citing works
Hannan, M. A., Lipu, M. S. H., Hussain, A., Ker, P. J., Mahlia, T. M. I., Mansor, M., Ayob, A., Saad, M. H. M., & Dong, Z. Y. (2020) · Scientific Reports · doi:10.1038/s41598-020-61464-7 · Cited by 244 · see citing works
“In this study, RMSE was chosen as the objective function because of the large number of sample variables and randomness behavior of SOC errors 27 .”
MethodVerification of deterministic solar forecasts
Yang, D., Alessandrini, S., Antonanzas, J., Antoñanzas-Torres, F., Bădescu, V., Beyer, H. G., Blaga, R., Boland, J. W., Bright, J. M., Coimbra, C. F. M., et al. (2020) · Solar Energy · doi:10.1016/j.solener.2020.04.019 · Cited by 244 · see citing works
“Furthermore, since there are countless publications that discuss and conclude that one metric is better than the other, it is not difficult to cite those articles that support any choice the author wishes to make (Chai and Draxler, 2014).”
BackgroundWang, X., Tolksdorf, V., Otto, M., et al. (2020) · International Journal of Climatology · doi:10.1002/joc.6686 · Cited by 238 · see citing works
“Since RMSE gives more weight to errors with larger absolute values and is more sensitive to outliers, it provides information about the variability of the error distribution (Chai and Draxler, 2014; Willmott and Matsuura, 2005).”
BackgroundDynamic Neural Network Modelling of Soil Moisture Content for Predictive Irrigation Scheduling
Adeyemi, O., Grove, I. G., Peets, S., et al. (2018) · Sensors · doi:10.3390/s18103408 · Cited by 236 · see citing works
“The RMSE and MAE quantify the prediction errors in the same units of the variables.”
MethodInfluentialA Hybrid LSTM Neural Network for Energy Consumption Forecasting of Individual Households
Yan, K., Li, W., Ji, Z., Qi, M., & Du, Y. (2019) · IEEE Access · doi:10.1109/access.2019.2949065 · Cited by 225 · see citing works
Kouadio, L. A., Deo, R. C., Byrareddy, V. M., Adamowski, J., Mushtaq, S., & Nguyen, V. P. (2018) · Computers and Electronics in Agriculture · doi:10.1016/j.compag.2018.10.014 · Cited by 232 · see citing works
Méndez-Morales, M., Maathuis, B. H. P., Hein‐Griggs, D., & Alvarado-Gamboa, L.-F. (2020) · Water · doi:10.3390/w12020482 · Cited by 223 · see citing works
Autopilot control unmanned aerial vehicle system for sewage defect detection using deep learning
Pandey, B. K., Pandey, D., & Sahani, K. (2024) · Engineering Reports · doi:10.1002/eng2.12852 · Cited by 220 · see citing works
Temporal Convolutional Networks with RNN approach for chaotic time series prediction
Düdükçü, H. V., Taşkıran, M., Taşkıran, Z. G. Ç., & Yıldırım, T. (2022) · Applied Soft Computing · doi:10.1016/j.asoc.2022.109945 · Cited by 221 · see citing works
Arora, A., Arabameri, A., Pandey, M. K., Siddiqui, M. A., Shukla, U. K., Bui, D. T., Mishra, V. N., & Bhardwaj, A. (2020) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2020.141565 · Cited by 220 · see citing works
Ning, Y. D., Kazemi, H., & Tahmasebi, P. (2022) · Computers & Geosciences · doi:10.1016/j.cageo.2022.105126 · Cited by 218 · see citing works
Boukerche, A., Tao, Y., & Sun, P. (2020) · Computer Networks · doi:10.1016/j.comnet.2020.107484 · Cited by 215 · see citing works
Lee, S. M., & Lee, D.-H. (2018) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph15071322 · Cited by 208 · see citing works
“…the accuracy of prediction between the models was examined It is determined by the difference between the predicted and actual values measured by the model and is a good indicator of the average error [17,37,38]. by comparing the RMSE values commonly used as a measure of model performance [36].”
MethodPrasad, R., Deo, R. C., Li, Y., & Maraseni, T. (2018) · Geoderma · doi:10.1016/j.geoderma.2018.05.035 · Cited by 208 · see citing works
Sentiment analysis methods, applications, and challenges: A systematic literature review
Mao, Y., Liu, Q., & Zhang, Y. (2024) · Journal of King Saud University - Computer and Information Sciences · doi:10.1016/j.jksuci.2024.102048 · Cited by 286 · see citing works
“Such as Mean Square Error(Verma et al., 2017), Ranking loss (Moghaddam and Ester, 2010), Receiver Operator Characteristic (ROC) (Krupinski, 2017), Area Under the Curve (AUC) (Bowers and Zhou, 2019), Root Mean Square Error (RMSE) (Chai and Draxler, 2014a) and mean absolute error (MAE) (Chai and Draxler, 2014b; Willmott…”
Nguyen, H., & Bui, X.-N. (2018) · Natural Resources Research · doi:10.1007/s11053-018-9424-1 · Cited by 206 · see citing works
DeRousseau, M. A., Laftchiev, E., Kasprzyk, J. R., Rajagopalan, B., & Srubar, W. V. (2019) · Construction and Building Materials · doi:10.1016/j.conbuildmat.2019.08.042 · Cited by 213 · see citing works
Zhang, J., Li, D., & Wang, Y. (2020) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2020.120665 · Cited by 204 · see citing works
A Deep Learning Method for Short-Term Residential Load Forecasting in Smart Grid
Ye, H., Zhou, Y., Li, Q., Xu, W., & Zheng, X. (2020) · IEEE Access · doi:10.1109/access.2020.2981817 · Cited by 203 · see citing works
Bartlett, J. D., O’Connor, F. K., Pitchford, N. W., Torres-Ronda, L., & Robertson, S. J. (2016) · International Journal of Sports Physiology and Performance · doi:10.1123/ijspp.2015-0791 · Cited by 201 · see citing works
PV power forecast using a nonparametric PV model
Almeida, M. P., Perpiñán, O., & Narvarte, L. (2015) · Solar Energy · doi:10.1016/j.solener.2015.03.006 · Cited by 208 · see citing works
“The Mean Absolute Error, MAE, (Equation 10), is less influenced by outliers, although the relative influence depends on the number of samples and the error distribution (Chai and Draxler, 2014).”
BackgroundIndividual mangrove tree measurement using UAV-based LiDAR data: Possibilities and challenges
Yin, D., & Wang, L. (2019) · Remote Sensing of Environment · doi:10.1016/j.rse.2018.12.034 · Cited by 197 · see citing works
Generative Semi-supervised Learning for Multivariate Time Series Imputation
Miao, X., Wu, Y., Wang, J., Gao, Y., Mao, X., & Yin, J. (2021) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v35i10.17086 · Cited by 210 · see citing works
“In the experiments, we utilize root mean square error (RMSE) (Jeffery, Garofalakis, and Franklin 2006) and mean absolute error (MAE) (Chai and Draxler 2014) to measure the imputation performance.”
MethodGervin, K., Salas, L. A., Bakulski, K. M., van Zelm, M. C., Koestler, D. C., Wiencke, J. K., Duijts, L., Moll, H. A., Kelsey, K. T., Kobor, M. S., et al. (2019) · Clinical Epigenetics · doi:10.1186/s13148-019-0717-y · Cited by 195 · see citing works
“For those investigators interested in predicting absolute cell 388 proportions from DNA methylation data, as RMSE measures how accurately the model predicts the 389 response, for this specific scenario with few outliers and a sample size >100, RMSE is the most 390 important criterion for fit[30] and IDOL is the prefer…”
MethodReview of Data Fusion Methods for Real-Time and Multi-Sensor Traffic Flow Analysis
Kashinath, S. A., Mostafa, S. A., Mustapha, A., Mahdin, H., Lim, D., Mahmoud, M. A., Mohammed, M. A., Al‐rimy, B. A. S., Fudzee, M. F. M., & Yang, T. J. (2021) · IEEE Access · doi:10.1109/access.2021.3069770 · Cited by 203 · see citing works
“This evaluation technique is scale-independent [140] and extremely useful for model evaluations and gives the same weight to the error [136].”
BackgroundMethodInfluentialGhorbani, M. A., Deo, R. C., Yaseen, Z. M., et al. (2017) · Theoretical and Applied Climatology · doi:10.1007/s00704-017-2244-0 · Cited by 192 · see citing works
“The results are evaluated using five performance criteria metrics: root mean square error (RMSE), mean absolute error (MAE), Nash-Sutcliffe efficiency (NS), and the Willmott’s Index (WI).”
InfluentialAssessing the accuracy of predictive models for numerical data: Not r nor r2, why not? Then what?
Li, J. (2017) · PLoS ONE · doi:10.1371/journal.pone.0183250 · Cited by 192 · see citing works
A study on water quality prediction by a hybrid CNN-LSTM model with attention mechanism
Yang, Y., Xiong, Q., Wu, C., Zou, Q., Yu, Y., Yi, H., & Gao, M. (2021) · Environmental Science and Pollution Research · doi:10.1007/s11356-021-14687-8 · Cited by 192 · see citing works
Goodness-of-fit criteria for hydrological models: Model calibration and performance assessment
Althoff, D., & Rodrigues, L. N. (2021) · Journal of Hydrology · doi:10.1016/j.jhydrol.2021.126674 · Cited by 193 · see citing works
Ghimire, S., Deo, R. C., Downs, N. J., et al. (2019) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2019.01.158 · Cited by 189 · see citing works
A New Criterion for Model Selection
Pham, H. (2019) · Mathematics · doi:10.3390/math7121215 · Cited by 190 · see citing works
H2O2 treatment enhanced the heavy metals removal by manure biochar in aqueous solutions
Wang, Y., & Liu, R. (2018) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2018.02.137 · Cited by 184 · see citing works
Aghighi, H., Azadbakht, M., Ashourloo, D., Shahrabi, H. S., & Radiom, S. (2018) · IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing · doi:10.1109/jstars.2018.2823361 · Cited by 187 · see citing works
A systematic review of personal thermal comfort models
Martins, L. A., Soebarto, V. I., & Williamson, T. (2021) · Building and Environment · doi:10.1016/j.buildenv.2021.108502 · Cited by 188 · see citing works
Content-Noise Complementary Learning for Medical Image Denoising
Geng, M., Meng, X., Yu, J., Zhu, L., Jin, L., Jiang, Z., Qiu, B., Li, H., Kong, H., Yuan, J., et al. (2021) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2021.3113365 · Cited by 184 · see citing works
An overview of performance evaluation metrics for short-term statistical wind power forecasting
Sopeña, J. M. G., Pakrashi, V., & Ghosh, B. (2020) · Renewable and Sustainable Energy Reviews · doi:10.1016/j.rser.2020.110515 · Cited by 182 · see citing works
Hossain Lipu, M. S., Hannan, M. A., Hussain, A., Saad, M. H., Ayob, A., & Uddin, M. N. (2019) · IEEE Transactions on Industry Applications · doi:10.1109/tia.2019.2902532 · Cited by 188 · see citing works
“paper, the root mean square error (RMSE) is chosen as a fitness function due to high sample dataset and random distribution of error estimation [24].”
Hai, T., Diop, L., Bodian, A., et al. (2018) · Agricultural Water Management · doi:10.1016/j.agwat.2018.06.018 · Cited by 178 · see citing works
Alslaity, A., & Orji, R. (2022) · Behaviour and Information Technology · doi:10.1080/0144929x.2022.2156387 · Cited by 177 · see citing works
Prasad, R., Deo, R. C., Li, Y., & Maraseni, T. (2017) · Atmospheric Research · doi:10.1016/j.atmosres.2017.06.014 · Cited by 173 · see citing works
Lin, H., Long, E., Ding, X., et al. (2018) · PLoS Medicine · doi:10.1371/journal.pmed.1002674 · Cited by 173 · see citing works
“Evaluation metrics Three evaluation metrics—the coefficient of determination (R(2)), the root mean square error (RMSE), and the mean absolute error (MAE)—were used to assess the performance of the regression algorithm in predicting a targeted SE [12].”
MethodPurity homophily in social networks.
Dehghani, M., JOHNSON, K., Hoover, J., Sagi, E., Garten, J., Parmar, N., Vaisey, S., Iliev, R. I., & Graham, J. (2016) · Journal of Experimental Psychology General · doi:10.1037/xge0000139 · Cited by 163 · see citing works
“To test the hypothesis that purity concerns are the best predictor of social distance, compared to other moral concerns, we compared the root-mean-squared errors (RMSE), a common metric of model fit (Chai & Draxler, 2014; Moriasi et al., 2007), generated from random permutation tests.”
Generating 3D city models without elevation data
Biljecki, F., Ledoux, H., & Stoter, J. (2017) · Computers, Environment and Urban Systems · doi:10.1016/j.compenvurbsys.2017.01.001 · Cited by 165 · see citing works
“The latter metric is sensitive to outliers, but we include it as well following the reasoning of Chai and Draxler (2014).”
BackgroundKim, J. R., Shim, W. H., Yoon, H. M., Hong, S. H., Lee, J. S., Cho, Y. A., & Kim, S. (2017) · American Journal of Roentgenology · doi:10.2214/ajr.17.18224 · Cited by 168 · see citing works
Himeur, Y., Alsalemi, A., Al‐Kababji, A., Bensaali, F., Amira, A., Sardianos, C., Dimitrakopoulos, G. J., & Varlamis, I. (2021) · Information Fusion · doi:10.1016/j.inffus.2021.02.002 · Cited by 171 · see citing works
“The simplest and most popular error are Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) [105], which are defined as follows: where r ui is the actual rating of user u for item i and ˆ r ui is the predicted rating.”
BackgroundInfluentialComparison of unstructured kinetic bacterial growth models.
Muloiwa, M., Nyende‐Byakika, S., & Dinka, M. O. (2020) · South African Journal of Chemical Engineering · doi:10.1016/j.sajce.2020.07.006 · Cited by 166 · see citing works
Best Practices for Elevation-Based Assessments of Sea-Level Rise and Coastal Flooding Exposure
Gesch, D. B. (2018) · Frontiers in Earth Science · doi:10.3389/feart.2018.00230 · Cited by 164 · see citing works
Liu, Y., Jing, W., Wang, Q., & Xia, X. (2020) · Advances in Water Resources · doi:10.1016/j.advwatres.2020.103601 · Cited by 163 · see citing works
Near real-time wind speed forecast model with bidirectional LSTM networks
Joseph, L. P., Deo, R. C., Prasad, R., Salcedo‐Sanz, S., Raj, N., & Soar, J. (2023) · Renewable Energy · doi:10.1016/j.renene.2022.12.123 · Cited by 163 · see citing works
“The absolute computation in 𝑀𝐴𝐸 reduces the biases, making it reliable over 𝑟 and 𝑅𝑀𝑆𝐸 [87].”
Hosseini, M., McNairn, H. E., Merzouki, A., et al. (2015) · Remote Sensing of Environment · doi:10.1016/j.rse.2015.09.002 · Cited by 162 · see citing works
Liu, C., Sun, B. N., Zhang, C., et al. (2020) · Applied Energy · doi:10.1016/j.apenergy.2020.115383 · Cited by 162 · see citing works
AL‐Musaylh, M. S., Deo, R. C., Li, Y., et al. (2018) · Applied Energy · doi:10.1016/j.apenergy.2018.02.140 · Cited by 159 · see citing works
Abd-Alhamid, F., Kent, M. G., Bennett, C., Calautit, J. K., & Wu, Y. (2019) · Building and Environment · doi:10.1016/j.buildenv.2019.106278 · Cited by 158 · see citing works
- Authors
- Chai, T., & Draxler, R. R.
- Affiliations
- National Oceanic and Atmospheric Administration (US)
- Venue
- Geoscientific Model Development
- Publisher
- Copernicus GmbH
- Published
- 2014-06-30
- Volume
- 7
- Issue
- 3
- Pages
- 1247-1250
- ISSN
- 1991-959X (ISSN-L)
- 1991-9603
- OpenAlex
- W2102148524
- Semantic Scholar
- 8c42db1839098b46f18265907464c16e49bb7a1c
- Cited by
- 7,178
- References
- 12
- Citations by year
- 2026: 1015 · 2025: 1120 · 2024: 980 · 2023: 873 · 2022: 751 · 2021: 683 · 2020: 538 · 2019: 411 · 2018: 324 · 2017: 235 · 2016: 175 · 2015: 59 · 2014: 13
- FWCI
- 106.31
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Geoscientific model development (publishedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Funders
- National Oceanic and Atmospheric Administration (NA09NES4400006)
- Topics
- Physical Sciences › Earth and Planetary Sciences › Atmospheric Science › Meteorological Phenomena and Simulations
- Physical Sciences › Environmental Science › Water Science and Technology › Hydrology and Watershed Management Studies
- Social Sciences › Decision Sciences › Management Science and Operations Research › Forecasting Techniques and Applications
- Keywords
- root mean square error, mean absolute error, model evaluation metrics, error distribution, Gaussian errors, triangle inequality, distance measures, model evaluation, hydrological model evaluation
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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