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)
boostedstates an association- 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
plainstates a cause- 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
hedgedstates a cause- 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.
plainstates an association- 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
boostedstates a cause- 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
hedgedstates an association- 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)
hedgedstates an association- 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)
boostedstates an association- 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
plainstates a cause- 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
plainstates a cause- 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
plainstates a cause- 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 5,501–5,600 of 7,174
Predictability and Analysis of Weekly Temperature using Autoregressive Time Series Models
Kaur, S., & Rakshit, M. (2019)
Sudrimo, S. N., Sadik, K., & Sumertajaya, I. M. (2020) · doi:10.4108/eai.2-8-2019.2290519
Prediction of Biochemical Oxygen Demand Using Radial Basis Function Network
Noor, M. R. S., & Buliali, J. L. (2020) · Kinetik Game Technology Information System Computer Network Computing Electronics and Control · doi:10.22219/kinetik.v5i1.1006
Count data modelling application.
Ibeji, J. U. (2019) · ResearchSpace (University of KwaZulu-Natal)
Analysis of Cryptographic Communications Using Bit-Plane Measures and Fuzzy Computing
Arvind, & Ratan, R. (2020) · Advances in intelligent systems and computing · doi:10.1007/978-981-15-0751-9_90
Estimating Long Term Equity Implied Volatility
Crawford, D. A. (2019) · Open University of Cape Town (University of Cape Town)
Hanif, H., & Ahsan, A. (2019) · Mehran University Research Journal of Engineering and Technology · doi:10.22581/muet1982.1902.04
Movie Recommendations Based on a Recurrent Neural Network Model
Ng, Y. (2020) · CRC Press eBooks · doi:10.1201/9781003034971-4
Hällmark, K., & Baldesi, A. L. (2018) · KTH Publication Database DiVA (KTH Royal Institute of Technology)
New approach for local C-band weather radar precipitation calibration
Ávila-Carrasco, J. R., Herrera, G. S., Júnez-Ferreira, H. E., et al. (2020) · Atmósfera · doi:10.20937/atm.52763
Saleem, N., Mahadevan, R. R., Nandakumar, S. G., & Andy, R. B. (2020) · IEEE International Conference on Power Electronics, Smart Grid, and Renewable Energy (PESGRE) · doi:10.1109/pesgre45664.2020.9070644
Resistance Partitioning of Headwater Mountain Streams—A Case Study in Southern Ecuador
Cedillo, S., Timbe, L., Samaniego, E. P., et al. (2020) · World Environmental and Water Resources Congress 2020 · doi:10.1061/9780784482971.018
Suniada, K. I., Susilo, E., Siwi, W. E. R., et al. (2020) · International Journal of Remote Sensing and Earth Sciences (IJReSES) · doi:10.30536/j.ijreses.2019.v16.a3252
Model Performance Evaluation: Sales Prediction
Singh, A., & Srivastava, S. (2020) · Communications in computer and information science · doi:10.1007/978-981-15-5827-6_3
Adityo, S., Hendrato, H., Sucipto, S., et al. (2020) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1511/1/012074
Risk Prediction Assessment In Life Insurance Company Through Dimensionality
Dwivedi, R. M. S. K., Mishra, A., & Gupta, R. K. (2020) · International journal of scientific and technology research
Popović, B., Ruskovski, I., Milovanov, S., et al. (2019) · doi:10.7251/stp2014127p
Global Ocean Carbon Dioxide Flux Mapping Techniques: Evaluation, Development, and Discrepancies
Gloege, L. (2020) · Institutional Archive of Ifremer (French Research Institute for Exploitation of the Sea) · doi:10.7916/d8-j3p1-tf92
Pratomo, D. G., & Aziz, M. R. F. (2020) · Geoid · doi:10.12962/j24423998.v15i2.7129
Cho, Y., & Yoo, K.-E. (2020) · Journal of the Korean Society for Aviation and Aeronautics · doi:10.12985/ksaa.2020.28.2.053
Time Series Modelling of Oil Price Fluctuations: Applications to Libya and Nigeria
Awidan, R. H. M. (2019) · Sheffield Hallam University · doi:10.7190/shu-thesis-00292
REAL-TIME PREDICTION OF SHIMS DIMENSIONS IN POWER TRANSFER UNITS USING MACHINE LEARNING
Jansson, D., & Blomstrand, R. (2019) · DiVA (Mälardalen University College)
Adaptive Reinforcement Learning Model for Simulation of Urban Mobility during CrisesPreprint
Fan, C. H., Jiang, X., & Mostafavi, A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.01359
Urban Traffic Flow Forecast Based on FastGCRNNPreprint
Zhang, Y., Lu, M., & Li, H. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.08087
Handling uncertainty in eHealth sensors using fuzzy system modeling
Gupta, A., & Mukherjee, N. (2020) · Health and Technology · doi:10.1007/s12553-020-00465-y
Rashid, N. N. M., Mansor, M. A., Kasihmuddin, M. S. M., et al. (2020) · AIP conference proceedings · doi:10.1063/5.0018147
Estimating socioeconomic indicators using online data
Lochanachit, S. (2026) · Warwick Research Archive Portal (University of Warwick) · doi:10.82444/warw.33808891
López-Restrepo, S., Botero, A. Y., Pinel, N., Montoya, O. L. Q., Segers, A. J., & Heemink, A. W. (2020) · doi:10.31223/x56882
Weather Variability Control in Three Colombian Airports
Vargas-Daza, K., Misat-Gómez, G., & Rodado, D. N. (2020) · Smart innovation, systems and technologies · doi:10.1007/978-981-33-4256-9_37
Simulation of Nd:YAG laser welding of thin SS sheets by using MATLAB program
Mahmood, E. J. (2020) · AIP conference proceedings · doi:10.1063/5.0028120
Distribution-based Regression Models for Semi-Bounded Data Analysis
Koochemeshkian, P., Zamzami, N., & Bouguila, N. (2020) · Conference proceedings/Conference proceedings - IEEE International Conference on Systems, Man, and Cybernetics · doi:10.1109/smc42975.2020.9282858
Monatisa, L. (2020) · Unisa Institutional Repository (University of South Africa)
Analyzing the Impact of COVID 19 on Global Trends and Predicting Future Cases
Han, G., Han, J., Han, S., & Jeong, H. (2021) · American Journal of Theoretical and Applied Statistics · doi:10.11648/j.ajtas.20211001.11
Assessment of Online Weather Platform Data for PV Output Power Forecasting
Zulkifly, Z., Baharin, K. A., & Gan, C. K. (2020) · IEEE International Conference on Power and Energy · doi:10.1109/pecon48942.2020.9314381
Suresh, I. (2016) · doi:10.5194/bg-2016-195-ac2
Ha, S., Liu, D., & Mu, L. (2021) · Research Square · doi:10.21203/rs.3.rs-227553/v1
Time-series sales forecasting for an Enterprise Resource Planning system
Malila, T. (2019) · Theseus (Ammattikorkeakoulujen)
COVID-19 India Outbreak Trend Analysis and Predictions using Multi Time Series Models
Khurana, J., & Ashri, A. (2021) · International Journal of Computer Applications · doi:10.5120/ijca2021921119
Forecasting Electricity Consumption in a Moroccan Educational InstitutionPreprint
Daki, H., Hannani, A. E., & Ouahmane, H. (2021) · Research Square · doi:10.21203/rs.3.rs-248534/v1
Cedillo, S., Timbe, L., Samaniego, E. P., et al. (2021) · La Granja · doi:10.17163/lgr.n33.2021.08
A dynamical system model for predicting gene expression from the epigenomePreprint
Brunner, J. D., Kim, J., Downing, T. L., Mjolsness, E., & Kober, K. M. (2020) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.08.03.234740
Medeiros, T. G. G., de Andrade, C. F., & Lins, D. R. (2023) · Anais Congresso Brasileiro de Energia Solar · doi:10.59627/cbens.2020.946
MPC based Caster Wheel Aware Motion Planning for Differential Drive Robots
Aguirregomezcorta, J. A. (2020)
Agent-Based Modeling of Phytoplankton in Experimental Water Column Reactors.
Camacho, M. C. (2020) · NCSU Libraries Repository (North Carolina State University Libraries)
Calsi, B. X., Conde, L. A., Angulo, J. R., et al. (2021) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1841/1/012001
Iftikhar, A., Musa, S., Alam, M. M., et al. (2021) · Journal of Computer Science · doi:10.3844/jcssp.2021.67.89
Le, T. (2021) · Contributions to finance and accounting · doi:10.1007/978-3-030-71242-6_3
Neill, A. J., Birkel, C., Maneta, M., et al. (2021) · doi:10.5194/hess-2021-158-rc1
Regional modeling of surface and sub-surface dynamics in the Bay of Bengal using Modular Ocean ModelPreprint
Tirodkar, S., Sarkar, M., Chauhan, R., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2105.02430
Ebissa, T. N. (2021) · Journal of Water Resources and Ocean Science · doi:10.11648/j.wros.20211002.12
Potential Threats of Social Engineering Practices to Social Work
Tuna, G., & EFEOĞLU, E. (2021) · Advances in public policy and administration (APPA) book series · doi:10.4018/978-1-7998-7772-1.ch016
Flash Cache Hybrid Storage System For Video-On-Demand Server
Al-Wesabi, O. A. M. (2018)
Karaoke Key Recommendation Via Personalized Competence-Based Rating Prediction
Wang, Y., Tanaka, S., Yokoyama, K., Wu, H.-T., & Fang, Y. (2021) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp39728.2021.9414524
Porównanie wybranych metod pomiaru prędkości liniowej taśmy
Bortnowski, P., Gładysiewicz, L., Król, R., et al. (2020) · Transport Przemysłowy i Maszyny Robocze
The ANN Design Based On PMU Readings For Fault Location Detection
AzriyenniAzhariZakri, RoisKhumaini, M., Syaibi, H., et al. (2021) · International journal of scientific and technology research
Potential Use of Artificial Neural Networks in Prediction of Dengue Abundance in Agra District
Sant, T. G., & Satsangi, N. (2021) · IOP Conference Series Materials Science and Engineering · doi:10.1088/1757-899x/1116/1/012201
Neill, A. J. (2021) · doi:10.5194/hess-2021-158-ac2
Hair Identification Extension for the Application of Automatic Face Detection Systems
Agarwal, S., Misra, S., Srivastava, V., et al. (2021) · Advances in intelligent systems and computing · doi:10.1007/978-981-33-4367-2_75
Neill, A. J. (2021) · doi:10.5194/hess-2021-158-ac1
Efficient Data-specific Model Search for Collaborative FilteringPreprint
Gao, C., Yao, Q., Jin, D., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2106.07453
Improving Learning Performance in Neural Networks
Al-akashi, F. H. A. (2021) · International Journal of Hybrid Information Technology · doi:10.21742/ijhit.2021.14.1.02
Zaidi, S. M. (2019) · UMP Institutional Repository (Universiti Malaysia Pahang)
Navarro, M. A., Oliva, D. A., Zaldívar, D., et al. (2021) · Studies in computational intelligence · doi:10.1007/978-3-030-70542-8_22
EVT in Risk Estimation on Commodities Exchanges
Ganczarek-Gamrot, A., Krężołek, D., & Trzpiot, G. (2021) · Scientific Publications / University of Economics in Katowice
Zuzani, P. N., Ngongondo, C., Mwale, F., et al. (2021) · Research Square · doi:10.21203/rs.3.rs-418688/v1
Sánchez, D. E. (2021) · Revista de Estudios Andaluces · doi:10.12795/rea.2021.i42.01
The perfect forecast for cycling
Beekman, Y. A. (2021)
Transformed WLS-Based Data Reconciliation for a Large-Scale Process Network
Chakradhar, B., Shanmugam, G., Datta, K. L. N. S., et al. (2021) · Advances in intelligent systems and computing · doi:10.1007/978-981-16-0730-1_8
Shapley values and machine learning to characterize metamaterials for seismic applicationsPreprint
Oniz, D., Mo, Y. L., & Nakshatrala, K. B. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2108.00493
Predicting diarrhoea outbreak with climate change
Abdullahi, T. (2021) · Open University of Cape Town (University of Cape Town)
Retweet Prediction Based on User Behavior
Firdaus, S. N. (2021) · doi:10.32920/ryerson.14657001.v1
Evaluation of Long-Term Degree-Days Estimated with Several Methods for Corn in Nebraska, USAPreprint
Dai, S., Shulski, M. D., Yang, H. S., et al. (2021) · Research Square · doi:10.21203/rs.3.rs-284619/v1
A Two-Parameter Model for Water-Lubricated Pipeline Transportation of Unconventional Crudes
Rushd, S., Ahmed, E., Mahmud, S., & Hossain, S. S. (2021) · Energies · doi:10.3390/en14185665
Woldemariam, Y. D. (2021) · SN Computer Science · doi:10.1007/s42979-021-00832-0
Development of Detection and Prediction System for Induction Motor Faults Using Linear Regression
Sani, N. N., & Noh, F. H. B. M. (2021) · Progress in Engineering Application and Technology
Epidemiological Investigation Model of Novel Coronavirus Pneumonia Based on Data Mining Technology
Chen, K., Yang, B. X., Hao, M., et al. (2021) · E3S Web of Conferences · doi:10.1051/e3sconf/202129203026
alfahady, N. H., Abbo, K. K., & Mahmood, E. M. N. (2021) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1999/1/012089
Sharma, P., & Qanungo, K. (2021) · Asian Journal of Chemistry · doi:10.14233/ajchem.2021.23322
Penilaian Harga Wajar Saham Pada Emiten Yang Terdaftar Di Indeks Lq45 Periode 2017 – 2019
Toman, G. (2021) · Jurnal Indonesia Sosial Teknologi · doi:10.36418/jist.v2i9.232
High-definition processing of remote sensing images based on CUT-CycleGAN
Deng, L., Zhang, Y., & Wang, X. (2021) · Chinese Control Conference (CCC) · doi:10.23919/ccc52363.2021.9549656
QoS Evaluation for Wireless Networks Using K-Nearest Neighbor (k-NN) Algorithm
Goutam, S., & Unnikrishnan, S. (2021) · Lecture notes in networks and systems · doi:10.1007/978-981-16-3807-7_18
Dissolved organic carbon dynamics in a salt marsh creek
Codden, C. J. (2020) · doi:10.17760/d20382054
Pose-aware monocular localization of occluded pedestrians in 3D scene space
Rahimi, M. M., Khoshelham, K., Stevenson, M., & Winter, S. (2021) · ISPRS Open Journal of Photogrammetry and Remote Sensing · doi:10.1016/j.ophoto.2021.100006
Proposal of Bot Detection Method by Multi-Source Access Log Analysis
Tanaka, T., & Tsuda, K. (2021) · IEEJ Transactions on Electronics Information and Systems · doi:10.1541/ieejeiss.141.1205
Modelos de Previsão das Taxas de Juros dos Títulos IPCA do Tesouro Direto
Oliveira, S. L. R., Gomes, R. M., Santos, B. A., & Silva, F. R. (2021) · Congresso Brasileiro de Inteligência Computacional (CBIC) · doi:10.21528/cbic2021-11
HIZLI, H., & KILIÇALP, N. (2021) · Osmaniye Korkut Ata Üniversitesi Fen Bilimleri Enstitüsü Dergisi · doi:10.47495/okufbed.979558
Urban City: Crime Attractors and Offender Mobility
Nguyen, V. T. (2021) · SIAM Undergraduate Research Online · doi:10.1137/21s1420551
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
