The DOI matches its Crossref record; 13 of the 13 references it cites were found in a registry or an index; 4 of the 8 sources its claims cite were checked for support.
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.
Overview
AI snapshot
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.
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Claims the paper makes
14 Own interpretation · 6 No basis given · 4 Supported · 3 Sources not found · 3 Own result · 1 No full text of the sources · 1 Not checked yet
Every source the paper cites for this claim was checked and supports it.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
What the sources say
- Supportedcites Willmott and Matsuura (2005) source
Table 1 identifies five hypothetical sets of four errors, and Section 3 says their RMSE increases as the error magnitudes become more variable. The paper gives the bounds as MAE ≤ RMSE ≤ √n × MAE; with four errors and an MAE of 2.0, those bounds are 2.0 and 4.0. Page numbers are unavailable in our copy. (LLM, conf=0.90)
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > Abstract
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 3 Triangle inequality of a metric · boosted
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 2 Interpretation of RMSE and MAE
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 2 Interpretation of RMSE and MAE
plainn = 10- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Rests on an assumption: Model errors are likely to have a normal distribution rather than a uniform distribution.
- Also stated in: § 1 Introduction · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Giving higher weighting to the unfavorable conditions, the RMSE usually is better at revealing model performance differences.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 4 Summary and discussion
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“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).”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 3 Triangle inequality of a metric
What the sources say
- Not matchedcites Horn and Johnson, 1990
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > Abstract
What the sources say
- Not matchedcites Willmott et al. (2009)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 4 Summary and discussion · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 4 Summary and discussion
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Similarly, one can readily show that, for several sets of errors with the same RMSE, the MAE would vary from set to set.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Probably not a substantial claim. 10
- It may state two separate claims as one. 2
- The quote only partly states this claim. 1
Every source the paper cites for this claim was checked and supports it.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
What the sources say
- Supportedcites Willmott and Matsuura (2005) source
The Concluding Remarks state the attributed point directly, including the qualification that the increase need not be monotonic. (LLM, conf=0.95)
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
What the sources say
- Supportedcites Willmott and Matsuura (2005) source
The Concluding Remarks state that RMSE depends on error-magnitude variability, MAE, and the square root of sample size. The claim says it varies with sample size n, which accurately conveys that dependence without asserting that the relationship is linear. (LLM, conf=0.93)
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“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).”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
What the sources say
- Supportedcites Willmott and Matsuura (2005) source
Section 3 describes an inconsistent relationship between MAE and RMSE, including cases where RMSE changes with error-magnitude variance. Section 4 explains that RMSE depends on MAE, the distribution of error magnitudes, and the number of errors. This supports the claim that the measures are not equivalent and that one cannot generally derive either value from the other alone. (LLM, conf=0.90)
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Table 1 shows that the MAEs converge to 0.8, an approximation to the expectation of 2 π .”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 2 Interpretation of RMSE and MAE
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“When both metrics are calculated, the RMSE is by definition never smaller than the MAE.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The MAE is suitable to describe uniformly distributed errors.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 2 Interpretation of RMSE and MAE
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 4 Summary and discussion
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
In the paper
“The ratio of RMSE to MAE ranged from 1.63 to 2.29 (see Table 1 of Chai et al., 2009).”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
What the sources say
- Not checked — no full text of the sourcecites Chai et al., 2009 source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 4 Summary and discussion
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence.
- Also stated in: § 4 Summary and discussion · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence.
- Also stated in: § 4 Summary and discussion · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial 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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 4 Summary and discussion
“| 4 | 0.92, 0.65, 1.48, 1.02, 0.79 0.70, 0.57, 1.33, 1.16, 0.76 | |”
§ Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature > 1 Introduction
“When both metrics are calculated, the RMSE is by definition never smaller than the MAE.”
Citations
Cites
Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203
David, F. N., & Tukey, J. W. (1977) · Biometrics · doi:10.2307/2529486 · Cited by 12,707
Willmott, C., & Matsuura, K. (2005) · Climate Research · doi:10.3354/cr030079 · Cited by 6,512 · cited 4×
“Further, Willmott et al. (2009) concluded that the sums-of-squaresbased error statistics such as the RMSE and the standard error have inherent ambiguities and recommended the use of alternates such as the MAE.”
BackgroundMethodInfluentialHuber, P. J., & Ronchetti, E. M. (2009) · Wiley series in probability and statistics · doi:10.1002/9780470434697 · Cited by 1,823
Ambiguities inherent in sums-of-squares-based error statistics
Willmott, C. J., Matsuura, K., & Robeson, S. M. (2008) · Atmospheric Environment · doi:10.1016/j.atmosenv.2008.10.005 · Cited by 212 · cited 1×
“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).”
BackgroundMcKeen, S. A., Wilczak, J. M., Grell, G., et al. (2005) · Journal of Geophysical Research Atmospheres · doi:10.1029/2005jd005858 · Cited by 208
Savage, N., Agnew, P., Davis, L. S., Ordóñez, C., Thorpe, R., Johnson, C. E., O’Connor, F. M., & Dalvi, M. (2013) · Geoscientific model development · doi:10.5194/gmd-6-353-2013 · Cited by 128
Taylor, M. H., Lösch, M., Wenzel, M., et al. (2013) · Journal of Climate · doi:10.1175/jcli-d-13-00089.1 · Cited by 93
Chai, T., Kim, H. C., Lee, P., et al. (2013) · Geoscientific model development · doi:10.5194/gmd-6-1831-2013 · Cited by 87
Self-citationA multi-physics ensemble of present-day climate regional simulations over the Iberian Peninsula
Jerez, S., Montávez, J. P., Jiménez‐Guerrero, P., et al. (2012) · Climate Dynamics · doi:10.1007/s00382-012-1539-1 · Cited by 80 · cited 2×
“(2009) (e.g., Taylor et al., 2013; Chatterjee et al., 2013; Jerez et al., 2013).”
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Su, H., Ma, S., & Xu, X. (2024) · PLoS ONE · doi:10.1371/journal.pone.0300142
Chirwa, A. C., Ngongondo, C., & Vunain, E. (2024) · Research Square · doi:10.21203/rs.3.rs-4252531/v1
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Agostino, Í. R. S., Berger, S. L. T., & Frazzon, E. M. (2021) · International Conference on Industrial Engineering and Operations Management (IEOM) · doi:10.46254/eu04.20210098
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Namara, A. F., Nurjaya, I. W., & Hartanto, M. T. (2024) · BIO Web of Conferences · doi:10.1051/bioconf/202410603004
Wang, L., Wen, W., & Qin, J. (2023) · International Conference on Internet of Things, Automation and Artificial Intelligence (IoTAAI) · doi:10.1145/3653081.3653116
Josué, D. F., Virginie, T. A., Ntamack, G. E., & Kenmeugne, B. (2024) · Geomechanics and Geoengineering · doi:10.1080/17486025.2024.2347283 · Cited by 1
Tiozon, R. N., Buenafe, R. J. Q., Herath, A., Sreenivasulu, N., & Butardo, V. M. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4820100
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Bai, B. M., Baditha, S. N., Patta, M., et al. (2024) · Lecture notes in networks and systems · doi:10.1007/978-981-97-2079-8_22
Khoshkehbar, M. G., Idris, N., & Taklimi, Z. M. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4782069 · Cited by 1
Ekanayake, A., & Chandrasekara, N. V. (2024) · i-manager s Journal on Artificial Intelligence & Machine Learning · doi:10.26634/jaim.2.1.20181
Bonita, J. D., Gomez, N. C. F., Licnachan, L. O., Ignacio, P. S., Mayes, A. G., & Onda, D. F. L. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4830438
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Yang, T., Wu, C., Chen, J., et al. (2024) · IOP Conference Series Earth and Environmental Science · doi:10.1088/1755-1315/1337/1/012030
Islam, M. A., Ali, M. M., Benitez, I. B., et al. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4837474
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Yao, Z., Li, Z., Wu, X., et al. (2023) · International Conference on Engineering Technology and Technopreneurship (ICE2T) · doi:10.1109/ice2t58637.2023.10540554
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Idi, D., Jibasen, D., Okolo, A., et al. (2024) · International Journal of Development Mathematics (IJDM) · doi:10.62054/ijdm/0102.10
Setiawan, A., Saputra, A. H., Kristianto, A., et al. (2024) · Springer proceedings in physics · doi:10.1007/978-981-97-0740-9_18
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Du, C., & Xiong, S. (2024) · IEEE Transactions on Consumer Electronics · doi:10.1109/tce.2024.3410989
Committee-Based Threshold Classification Algorithm with Reinforcement Active Learning
Zhao, D., & Shen, H. (2024) · IEEE International Conference on Control, Electronics and Computer Technology (ICCECT) · doi:10.1109/iccect60629.2024.10546099
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Andre, A., & Handhayani, T. (2024) · Jurnal Esensi Infokom Jurnal Esensi Sistem Informasi dan Sistem Komputer · doi:10.55886/infokom.v8i1.836 · Cited by 2
Trustworthy AI: Deciding What to Decide
Wu, C., Li, Y.-F., Li, J., Xu, J., & Bouvry, P. (2024) · Lecture notes in networks and systems · doi:10.1007/978-3-031-62281-6_8 · Cited by 2
CRSP: Emulating Human Cooperative Reasoning for Intelligible Story Point Estimation
Han, R., Han, W., Han, Z., Tian, Y., Chen, L., & Han, R. (2024) · IEEE International Conference on Program Comprehension (ICPC) · doi:10.1145/3643916.3644417
Haoming, C., Hao, G., Zhu, H., et al. (2024) · Journal of Microwave Power and Electromagnetic Energy · doi:10.1080/08327823.2024.2341342
Giri, M. R., & Tamrakar, G. B. S. (2024) · International Journal on Engineering Technology · doi:10.3126/injet.v1i2.66697
Investigation on 3D-Lens Correction for Multispectral Camera Systems with Actuated Image Sensor
Rosenberger, M., Eilhauer, M. A., & Nopper, C.-B. (2024) · IEEE International Instrumentation and Measurement Technology Conference · doi:10.1109/i2mtc60896.2024.10560735
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Medhi, S., & Gogoi, M. (2024) · Journal of Applied and Natural Science · doi:10.31018/jans.v16i2.5510
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Bureš, L., Roub, R., Poláková, L., et al. (2024) · Vodohospodářské technicko-ekonomické informace · doi:10.46555/vtei.2024.01.001
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Harlow, D. G. (2024) · Materials · doi:10.3390/ma17143383
Introduction of Enso Index in Iod Forecast Based on an Improved Res-Lstm ModelPreprint
Yu, N., Hu, C., Wang, J., Rao, W., Wan, Z., & Li, J. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4890446
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Chirwa, A. C., Ngongondo, C., & Vunain, E. (2024) · Discover Atmosphere · doi:10.1007/s44292-024-00007-5
Petrone, T. T. (2024) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/d.3.2024.tde-12072024-095714
Talent Prediction Based on Switching Strategy Algorithm of Residual Grey Model
Pan, Y., Fei, X., Shan, B., et al. (2024) · Chinese Control and Decision Conference (CCDC) · doi:10.1109/ccdc62350.2024.10587918
Ameen, M. (2024) · Journal Port Science Research · doi:10.36371/port.2024.3.2
Ghosh, I., Singh, K., Jayaprakash, V., & Jayapalan, S. (2024) · Journal of Research in Pharmacy · doi:10.29228/jrp.795
Marzi, F., d’Aloisio, G., Marco, A. D., et al. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-66326-0_11
Choi, H., Kim, Y., Kang, H., Seo, H., Kim, M., Han, J., Kee, G., Park, S., Ko, S., Jung, H., et al. (2024) · Scientific Reports · doi:10.1038/s41598-024-68663-6
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Huang, T., Zhai, C., Zhang, Y., et al. (2024) · International Conference on Electronic Materials and Information Engineering (EMIE) · doi:10.1109/emie61984.2024.10616938
Ahern, Z. (2024) · Queensland University of Technology · doi:10.5204/thesis.eprints.251345
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Ismael, Y. M., & Awchi, T. A. (2024) · IOP Conference Series Earth and Environmental Science · doi:10.1088/1755-1315/1374/1/012063
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Li, F., Xu, Y., Lemus, O. D., et al. (2024) · Physica Medica · doi:10.1016/j.ejmp.2024.104504
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Nakamoto, R., Flanagan, B., Dai, Y., et al. (2024) · IEEE International Conference on Advanced Learning Technologies (ICALT) · doi:10.1109/icalt61570.2024.00028
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Nakamoto, R., Flanagan, B., Dai, Y., et al. (2024) · IEEE International Conference on Advanced Learning Technologies (ICALT) · doi:10.1109/icalt61570.2024.00054
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Çığşar, B., Ünal, D., Boulou, A.-H. B., et al. (2024) · Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi · doi:10.53433/yyufbed.1432156
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Abdullah, A. Z., & Setiawan, E. B. (2024) · INTENSIF Jurnal Ilmiah Penelitian dan Penerapan Teknologi Sistem Informasi · doi:10.29407/intensif.v9i1.22999
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Braimah, J. O., Makoni, T., & Correa, F. M. (2024) · F1000Research · doi:10.12688/f1000research.153978.1 · Cited by 1
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Zekri, D., Snoun, A., Delot, T., & Thilliez, M. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-70819-0_8
Nguyen, Q.-D., Nguyen, T. A. H., Pham, N. T. D., et al. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-70816-9_24
Mahdi, H. A. (2024) · Energy Reports · doi:10.1016/j.egyr.2024.09.012
Nguyen, M. H., Vinh, V. D., Nguyen, D. T., et al. (2024) · Regional Studies in Marine Science · doi:10.1016/j.rsma.2024.103828
Udomsup, R., Nuchkum, S., Srisertpol, J., Donjaroennon, N., & Leeton, U. (2024) · Machines · doi:10.3390/machines12090661
Maqbool, S., Khan, M. I., Raza, A., et al. (2024) · Research Square · doi:10.21203/rs.3.rs-4951035/v1
Nguyen, B., Baryskievic, A., Baryskievic, I. A., et al. (2024) · Research Square · doi:10.21203/rs.3.rs-4974214/v1
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Mirshafie, A., bazgir, S., & Ghasem, A. (2024) · Journal of Spatial Analysis Environmental Hazards · doi:10.61186/jsaeh.11.2.117
Choi, I., Kim, J. H., Lee, S. J., et al. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4974506
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Yang, N., Zhou, M., Luan, X., Qi, L., Tang, Y., & Han, Z. J. (2024) · IEEE Transactions on Computational Social Systems · doi:10.1109/tcss.2024.3447692
Yimer, A. K., Hatiye, S. D., & Haile, A. T. (2024) · Earth Sciences Research Journal · doi:10.15446/esrj.v28n2.111726
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Nguyen, X. T., Nguyen, D.-Q., & Nguyen, T. D. (2024) · AIP conference proceedings · doi:10.1063/5.0227913
Han, Y. (2024) · Applied and Computational Engineering · doi:10.54254/2755-2721/92/20241728
Roobaert, L., & Claeys, D. (2024) · SHS Web of Conferences · doi:10.1051/shsconf/202419802003
Taha, A. H., Ajel, A. R., Kadhim, M. A., Abed, J. K., & Ali, Z. B. (2024) · AIP conference proceedings · doi:10.1063/5.0236258
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Conejo, A. N. (2024) · Springer eBooks · doi:10.1007/978-981-97-4053-6_3
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Hrimech, A., & Mahmud, J. S. (2024) · Lecture notes in electrical engineering · doi:10.1007/978-981-97-3442-9_5
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Gemechu, T. M., Chen, B., Zhang, H., et al. (2024) · Remote Sensing · doi:10.3390/rs16213924
Carvalho, C. P. D., Costa, J. L. S., Fonte-Boa, T., Novo, T. A., de Oliveira, M. J. C., & Costa, F. M. (2024) · Brazilian Journal of Geophysics · doi:10.22564/brjg.v42i2.2315
Devi, U., & Selvakumari, N. A. S. (2024) · ShodhKosh Journal of Visual and Performing Arts · doi:10.29121/shodhkosh.v5.i6.2024.2371
Zhang, A., Lv, J., Geng, Y., et al. (2024) · Sensors · doi:10.3390/s24216980
Dong, C., Huang, G., Cheng, G., et al. (2024) · Ocean & Coastal Management · doi:10.1016/j.ocecoaman.2024.107382
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Chang, J., Yue, L., & Liu, Q. (2024) · Lecture notes in computer science · doi:10.1007/978-981-97-8505-6_12
Mocellin, A. P. G., Paiva, M. D. S., Koch, A. H. D. S., et al. (2024) · Ciência e Natura · doi:10.5902/2179460x87168
Paiva, M. D. S., Mocellin, A. P. G., Koch, A. H. D. S., et al. (2024) · Ciência e Natura · doi:10.5902/2179460x87167
Martínez, I., Astudillo, C. A., & Núñez, D. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-76604-6_17
Yu, C., Tong, H., Huang, D., Lu, J., Huang, J., Zhou, D., & Zheng, J. (2024) · Agriculture · doi:10.3390/agriculture14112076 · Cited by 1
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Neilson, D., Widagda, I. K. G. P., Agatsya, G. N. R., Qomariyah, N. N., & Bahana, R. (2024) · International Conference on ICT for Smart Society (ICISS) · doi:10.1109/iciss62896.2024.10751645
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Kai-di, S., De-zhi, D., Guo-qiang, Y., et al. (2024) · IEEE International Symposium on Product Compliance Engineering - Asia (ISPCE-ASIA) · doi:10.1109/ispce-asia64773.2024.10756223
Improving prediction accuracy in agricultural markets through the CIMA-AttGRU model
Jiang, Y., Liu, J., & Li, X. (2024) · PLoS ONE · doi:10.1371/journal.pone.0313066
Zaman, T., Gutmann, E. D., Wang, G., et al. (2024) · Journal of Hydrometeorology · doi:10.1175/jhm-d-24-0075.1
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Neyestani, A., Toma, D. M., Abarghouee, A. F., Daponte, P., Fernández, J. D. R., & Vito, L. D. (2024) · IEEE International Workshop on Metrology for the Sea; Learning to Measure Sea Health Parameters (MetroSea) · doi:10.1109/metrosea62823.2024.10765714
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Kaur, J., Swathi, P., & Manjunath, S. (2024) · Contributions to management science · doi:10.1007/978-3-031-67531-7_52
Janabi, M. A. M. A. (2024) · Springer eBooks · doi:10.1007/978-3-031-71503-7_8
Tran, D. T., Luong, N. D., & Nguyen, D. H. (2024) · Geodesy and Cartography · doi:10.3846/gac.2024.20237
Metadata
- 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
About these checks
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.
A language model drew the claims from the paper on 2026-10-17, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-17 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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