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.
Integrity signals
Registry
Cites
Statistics
Verification
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).”
Method
Cited by
Sorted by citations
Items 6,101–6,200 of 7,174
Gaussian Process Regression (GPR) Forecasting Electricity Consumption for Mascara City
Gherib, I. D. E., Aichouba, M., Alik, B., et al. (2024) · doi:10.1109/afros62115.2024.11036942
Yield Curve Modeling: Applicability of the Traditional Factor Models for Sri Lanka Government Bonds
Dayarathne, K. P. N. S., & Thayasivam, U. (2025) · SN Computer Science · doi:10.1007/s42979-025-04075-1
UdN: A Bio-Inspired Data Network for Significant Pattern Extraction in Cognitive Internet of Things
Jha, V., & Tripathi, P. (2025) · Operations Research Forum · doi:10.1007/s43069-025-00492-3
Tumuluru, J. S., Gottula, J., Hidalgo, M. A., et al. (2025) · The journal of cotton science/Journal of cotton science · doi:10.56454/qohs1717
Kayode, Y. O., Ikuemonisan, F. E., Garba, L. G., et al. (2025) · Kinematika i Fizika Nebesnykh Tel · doi:10.15407/kfnt2025.02.047
Bali, A., Alzbergaitė, G., Vargas, E. K. M., et al. (2025) · Journal of Food Process Engineering · doi:10.1111/jfpe.70166
Fan, Y., Ma, W., Liu, Z., Liang, Y., Shi, Z., & Liu, H. (2025) · Asia Conference on Power and Electrical Engineering (ACPEE) · doi:10.1109/acpee64358.2025.11041223
Cardiac Measurement Calculation on Point-of-Care Ultrasonography with Artificial IntelligencePreprint
Mercaldo, S., Bizzo, B. C., Sadore, T., Halle, M. A., MacDonald, A. L., Newbury‐Chaet, I., L’Italien, E., Schultz, A. S., Tam, V., Hegde, S. M., et al. (2025) · medRxiv · doi:10.1101/2025.06.27.25330438
Hydrogeologic GIS Modeling of East Contra Costa Groundwater Subbasin
Krinks, P. N. (2025) · doi:10.46569/8k71ns744
Subseasonal to Seasonal (S2S) Prediction Algorithms Using Hybrid Machine Learning Techniques
Kim, H., Zhou, S., Bienkowski, A., & Pattipati, K. R. (2025) · Artificial Intelligence for the Earth Systems · doi:10.1175/aies-d-23-0108.1 · Cited by 1
Moerkens, R., Chan, K. H., Bellinkx, S., et al. (2025) · Journal of Applied Entomology · doi:10.1111/jen.13464
Adebayo, S. A., & Sathasivam, S. (2025) · Contemporary Mathematics · doi:10.37256/cm.6420256263
A Novel Dynamic Model Selection Method for VSIMM Filter
Fu, R., Tian, Y., Liu, Y., et al. (2025) · doi:10.1109/icaisisas64483.2025.11051660
Carnier, R., Luiz, A. J. B., Ayer, J. E. B., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5339302
Adjusted CBA-Wheat model for predicting aboveground biomass in winter wheat from hyperspectral data
Chen, J., Yu, J., Yang, M., et al. (2025) · Field Crops Research · doi:10.1016/j.fcr.2025.110060
Liu, S., Zhang, L., Tan, T., et al. (2025) · Transactions in GIS · doi:10.1111/tgis.70091
Radu, A. I., Tolea, B., Beles, H., et al. (2025) · Sensors · doi:10.3390/s25144291
Geng, H., Wang, W., Liu, J., et al. (2025) · Research Square · doi:10.21203/rs.3.rs-6969350/v1
Liquid State Machine - Based Temporal Prediction Model for ETH Price Dynamics
Patel, M. K., Makhijani, P., & Darji, M. (2025) · International Conference on Inventive Computing and Informatics (ICICI) · doi:10.1109/icici65870.2025.11069665
Research on Cluttered Scene Grasping Based on Rapid Point Cloud Registration Algorithm
Zeng, H., Yu, C., Shen, Z., et al. (2025) · International Conference on Control, Automation and Robotics (ICCAR) · doi:10.1109/iccar64901.2025.11073011
LSTMR: Spatio-Temporal 3D Multiscale ResNet Model for Cellular Network Traffic Prediction
Chen, G., Wang, W., & Zeng, Q. (2025) · Communications in computer and information science · doi:10.1007/978-981-96-7191-5_9
Cong, R., Matsumoto, T., & FUJIYAMA, A. (2025) · Waste · doi:10.3390/waste3030023
Software Architecture Influence on Software Effort Estimation
Hai, V. V., Tin, N. T., & Prokopová, Z. (2025) · Lecture notes in networks and systems · doi:10.1007/978-3-031-94770-4_25
Kayacik, O., Doğan, A. H., & Erdoğan, B. (2025) · Indian Journal of Physics · doi:10.1007/s12648-025-03724-8
Climate change projected to exacerbate the economic costs of biological invasionsPreprint
Gu, S., Chen, S., Tu, W., Han, L., Zhang, Q., Hong, Y., Lin, Z., Du, Y., Liao, W. B., & Liu, X. (2025) · Research Square · doi:10.21203/rs.3.rs-7029011/v1
Temporal resolution enhancement of NDVI time series using machine learning approaches
Park, S., Kim, C., & Han, H. (2025) · Modeling Earth Systems and Environment · doi:10.1007/s40808-025-02551-4
Han, J., Zeng, X., & Xiang, Y. (2026) · International Journal of Machine Learning and Cybernetics · doi:10.1007/s13042-026-03217-y
Forecasting road traffic accidents in India: a SARIMAX approach incorporating COVID-19 effects
Singh, S. K., & Singh, V. L. (2025) · International Journal of Injury Control and Safety Promotion · doi:10.1080/17457300.2025.2541662
Sobhani, A., Laimighofer, J., Ortner, R., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5376267
Ayalew, G. G., & Ayalew, G. M. (2025) · International Journal of Management and Fuzzy Systems · doi:10.11648/j.ijmfs.20251102.12
Nicolae, R., Niewenhuis, D., Talluri, S., et al. (2026) · ACM International Conference on Computing Frontiers (CF) · doi:10.1145/3801487.3806065
Zheng, C., Liu, Y., Zhu, J., Chaffart, D., & Yuan, Y. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5379005
Salazar, P. G., González, P. S., McIntyre, N., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5379058
Wie, Y. M., Lee, K. G., Lee, K. H., & Lee, K. H. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5380129
Viscoelastic relaxation and afterslip following the 2008 M 7.9 Wenchuan earthquake
Gong, W., & Sun, Y. (2025) · Journal of Geodynamics · doi:10.1016/j.jog.2025.102111
SpaNN: Spatial Transcriptomic Data Enhancement Using Deep Neural Network
Chen, S., Wang, W., Zheng, R., & Li, M. (2025) · IEEE Transactions on Computational Biology and Bioinformatics · doi:10.1109/tcbbio.2025.3594349
Sya'bana, M. R., Sisinggih, D., & SAJALI, M. A. (2025) · Jurnal Teknologi dan Rekayasa Sumber Daya Air · doi:10.21776/ub.jtresda.2025.005.02.088
İnce, M., & Özkavak, H. V. (2025) · Applied Sciences · doi:10.3390/app15147997
Predicting Surface Temperature at Indian Ports Using Machine Learning Models
Sarkar, P., & Kumar, P. (2025) · Global Conference in Emerging Technology (GINOTECH) · doi:10.1109/ginotech63460.2025.11077004
Optimization of Indonesia's Trade Predictions Through Penalized Regression Models
Wijaya, P. A., Margaretha, H., Cahyadi, L., et al. (2025) · International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON) · doi:10.1109/ecti-con64996.2025.11100850
Doğan, H., Temel, F. A., Yolcu, Ö. C., et al. (2025) · Biomass and Bioenergy · doi:10.1016/j.biombioe.2025.108265
Çelik, İ., Kahraman, Ö. F., Kılıç, M., et al. (2025) · Soil and Tillage Research · doi:10.1016/j.still.2025.106797
Fine-Grained Visual Aspects in Genre Prediction
Mondal, P., Kushum, Kumari, T., Saha, S., Singh, J. P., Biswas, P., Singh, B., & Pedanekar, N. (2025) · IEEE Transactions on Computational Social Systems · doi:10.1109/tcss.2025.3591607
Yang, X., & Yu, L. (2025) · Symmetry · doi:10.3390/sym17081298
APPLICATION OF HEURISTIC OPTIMIZATION APPROACHES TO EVALUATE COMPACTION PARAMETERS
Rahman, M. M. (2025) · Civil Engineering Beyond Limits · doi:10.36937/cebel.2025.1995
Moncada, A. F., & Arif, M. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5384168
omar, Z., & Liyanage, L. H. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5382326
Tanasa, I., Cazacu, M. M., Botan, D., et al. (2025) · Environments · doi:10.3390/environments12080285
Human-Robot Collaboration: An ANFIS-Based Model for Ergonomic Risk and Stress Evaluation
Bozkus, E., Hubbard, E., & Guo, W. (2025) · Lecture notes in networks and systems · doi:10.1007/978-3-032-00839-8_16
Neural Network Approaches for Enhanced Nitrate Management in Recirculating Aquaculture SystemsPreprint
Fan, X., Li, J., Cui, Z., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5393589
Aitouhanni, I., & Berqia, A. A. (2024) · IEEE International Conference on Engineering Technologies and Applied Sciences (ICETAS) · doi:10.1109/icetas62372.2024.11120239
Short-Term EV Load Forecasting Using Kolmogorov Arnold Networks
Fahim, B. M., Akbar, M. K., & Amayri, M. (2025) · IEEE International Symposium on Industrial Electronics (ISIE) · doi:10.1109/isie62713.2025.11124625
A fog-assisted group-based truth discovery framework over mobile crowdsensing data streams
Alamri, B. H. S., Monowar, M. M., Alshehri, S., & Zafar, M. H. (2025) · PLoS ONE · doi:10.1371/journal.pone.0330656 · Cited by 1
Yan, P., Zheng, L., Li, W., et al. (2025) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2025.114421
CRISPR-FMC: a dual-branch hybrid network for predicting CRISPR-Cas9 on-target activity
Li, C. T., Li, J., Zou, Q., & Feng, H. (2025) · Frontiers in Genome Editing · doi:10.3389/fgeed.2025.1643888
Brown, A., & Kissinger, A. (2025) · SPE Offshore Europe Conference (Offshore Europe) · doi:10.2118/226785-ms · Cited by 1
Deep Learning-Driven Automated Data Generation for Enhanced Anomaly Detection in Cybersecurity
Gough, M., Dewan, M. A. A., & Wang, H. (2025) · Intelligent Cybersecurity Conference (ICSC) · doi:10.1109/icsc65596.2025.11140556
Crop Yield Estimation in Indiana Using a Semi-Physical Model
Sen, K., Gautam, V., & Pathak, S. (2025) · Advances in geographic information science · doi:10.1007/978-3-031-94078-1_9
Bagheri, S., Karımzadeh, S., Feizizadeh, B., & Samadianfard, S. (2025) · Advances in Space Research · doi:10.1016/j.asr.2025.09.018
The Extended Goodwin Model and Wage–Labor Paradoxes Metric in South Africa
Chikore, T., Makobe, M. T., & Nyabadza, F. (2025) · Mathematical and Computational Applications · doi:10.3390/mca30050098
Shariatifar, M., Pazanialenjareghi, F., & Lin, H. (2025) · Advanced Membranes · doi:10.1016/j.advmem.2025.100171
Microplastics can significantly affect dry soil aggregate stability and wind erosionPreprint
Luo, X., He, J., zhao, X., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5472515
NGUYEN, S. T., Do-Trung, H., & Nguyen-Hoang, N. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5472213
Comparing LSTM, GRU, and CNN for forecasting the density of mobile network users and flows
Bastos, Â., Silva, B., Arruda, D., et al. (2025) · AIP conference proceedings · doi:10.1063/5.0286102
Bihon, N. H., Yadav, B. K., & Abhishek, A. (2025) · AQUA - Water Infrastructure Ecosystems and Society · doi:10.2166/aqua.2025.047
Çadırcı, S., & Tekdemir, İ. G. (2025) · Journal of Innovative Science and Engineering (JISE) · doi:10.38088/jise.1635104
A new intelligent model selection approach
Bal, Ç., & Aladağ, Ç. H. (2025) · Soft Computing · doi:10.1007/s00500-025-10910-7
Time Series Modeling for Smart Energy Consumption in Industry 4.0
Ferreira, L., Lopes, J., Duarte, J., et al. (2025) · Lecture notes in computer science · doi:10.1007/978-3-032-05176-9_18
Yao, W., Fen, X., Xiao, Y., Li, Z., Chen, X., & Wen, W. (2025) · Lecture notes in computer science · doi:10.1007/978-3-032-04549-2_15
YILDIZ, E. A., & Kabasakal, K. A. (2025) · Gazi Üniversitesi Gazi Eğitim Fakültesi Dergisi · doi:10.17152/gefad.1700937
Özsoy, K., & Aksoy, B. (2025) · Fırat Üniversitesi Mühendislik Bilimleri Dergisi · doi:10.35234/fumbd.1739022
Enhancing Path Loss Prediction Accuracy in Wireless Communication Using Deep Neural Networks
Tuama, M. H., Mashloosh, W. M., Albehadili, H. M., et al. (2025) · Lecture notes in networks and systems · doi:10.1007/978-981-96-7134-2_32
Evaluation of Different Statistical Models in Simulation of Rainfall for the West Bengal, India
Raha, S., Mondal, M., Gayen, S. K., et al. (2025) · Springer eBooks · doi:10.1007/978-3-031-93335-6_5
Baudraz, M. E. A., Childs, D. Z., Kelly, R., et al. (2025) · Journal of Ecology · doi:10.1111/1365-2745.70148
Hossain, M. D., Morshed, N., & Siam, S. I. (2025) · doi:10.1109/qpain66474.2025.11171883
Hasan, T., Khatun, M. M., Hosen, M. F., & Morshed, M. (2025) · IEEE International Conference on Quantum Photonics, Artificial Intelligence & Networking (QPAIN) · doi:10.1109/qpain66474.2025.11172123
Thas, O., & Jaspers, S. (2025) · Journal of Statistical Computation and Simulation · doi:10.1080/00949655.2025.2558863
Tatsumi, S., Ando, K., & Asai, T. (2025) · Nonlinear Theory and Its Applications IEICE · doi:10.1587/nolta.16.848
Zhao, C., Yang, H., Zhang, Y., Yan, W. C., Li, Q., Chen, A., & Liu, X. (2025) · BioEnergy Research · doi:10.1007/s12155-025-10895-z
Reconstruction of a Digital Model of Objects Base on Aerial Laser Scanning Data
Gabov, V. S., Zinkevich, A. V., & Barhatov, K. A. (2025) · International Russian Automation Conference (RusAutoCon) · doi:10.1109/rusautocon65989.2025.11177344
Hybrid Approaches and Frameworks in Clustering
Siddiqui, S., Chaudhuri, N. B., Akbar, S. M. T., et al. (2025) · Artificial intelligence-enhanced software and systems engineering · doi:10.1007/978-3-031-99882-9_21
Machine Learning Model Uncertainty in GNSS Positioning
Dobre, P., Nie, S., & Yang, H. (2025) · Proceedings of the Satellite Division's International Technical Meeting (Online)/Proceedings of the Satellite Division's International Technical Meeting (CD-ROM) · doi:10.33012/2025.20252
Ocampo-Marulanda, C., Lozano-Franco, L. K., Rodríguez, B. J. G., et al. (2025) · Open Agriculture · doi:10.1515/opag-2025-0465
Lateef, M. B. (2025) · Research Square · doi:10.21203/rs.3.rs-7199339/v1
Taşkın Verilerinin Modellenmesinde MEP Yönteminin Kullanılabilirliğinin Araştırılması
Kizilkaya, A., Bağatur, T., & Fidan, H. (2025) · Dicle Üniversitesi Fen Bilimleri Enstitüsü Dergisi · doi:10.55007/dufed.1680507
Qi, Y., Zhang, C., Yu, H., et al. (2025) · Environmental Modelling & Software · doi:10.1016/j.envsoft.2025.106712
Chang, L., El‐Meligy, M., & Alnowibet, K. A. (2025) · Measurement · doi:10.1016/j.measurement.2025.119154
Country-Wise Demand Forecasting in Indian Aviation: A Deep Learning Approach
Mangipudi, M. N., & Sabarmathi, G. (2025) · Algorithms for intelligent systems · doi:10.1007/978-981-96-5265-5_41
Yahaya, B. S., Mati, S., Kalmaz, D. B., & Wada, I. (2025) · Computational Economics · doi:10.1007/s10614-025-11104-x
Hustim, M., Alimuddin, H., Syafruddin, N. A., et al. (2025) · Engineering Technology & Applied Science Research · doi:10.48084/etasr.11378
Prediction of 3D Airspace Occupancy Using Machine Learning
Tafur, C. L., Orduy, J. E., Daza, P. F. M., et al. (2025) · Forecasting · doi:10.3390/forecast7040056
Akıner, M. E. (2025) · Pure and Applied Geophysics · doi:10.1007/s00024-025-03840-w · Cited by 1
Arregocés, H. A., Estrada, E., & Moscote, C. D. (2025) · Earth · doi:10.3390/earth6040122
PRPOS: A Periodicity-Aware Resource Prediction Framework for Online Services
Liang, Y., Zhou, H., Li, T., & Shen, H. (2025) · Applied Sciences · doi:10.3390/app152010967
Babbar, R., Misra, A., Srivastava, M., et al. (2025) · The International Journal of Advanced Manufacturing Technology · doi:10.1007/s00170-025-16761-8
Assessment of Future Grassland Fire Susceptibility Changes in Qinghai Province Based on CMIP6
Zhang, Z., Liu, F., Zhou, Q., et al. (2025) · Earth and Space Science · doi:10.1029/2025ea004400
Estimating large birds’ radar cross-section for aeroecology studies using T-matrix modelling
Reznikov, K., Melnikov, V. M., & Sapir, N. (2025) · Journal of The Royal Society Interface · doi:10.1098/rsif.2025.0510
Altarabsheh, A. (2025) · Engineering Construction & Architectural Management · doi:10.1108/ecam-04-2025-0710
Comparative Study on the Different Downscaling Methods for GPM Products in Complex Terrain Areas
Liu, J., Shi, X., Fang, Y., et al. (2025) · Earth · doi:10.3390/earth6040129
HIPACO: An RSSI Indoor Positioning Algorithm Based on Improved Ant Colony Optimization Algorithm
Jin, B., Zhao, Y., Tu, Q., et al. (2025) · Algorithms · doi:10.3390/a18100654
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 ›
Are you an author, or do you see a problem with this page? Request a review
