Providing a comprehensive understanding of missing data imputation processes in evapotranspiration-related research: a systematic literature review

Figures & data

Figure 1. Mechanisms and patterns of missing data. MAR = Missing at Random; MCAR = Missing Completely at Random; MNAR = Missing Not at Random.

Figure 2. The steps of the literature review conducted via Scopus.

Figure 3. Keyword networks generated by VOSviewer based on the Scopus database.

Table 1. Commonly applied missing data imputation methods.

Method	Basis	Rationale	Assumption	Complexity	HP	CC	Explanation
Deletion	N/A	N/A	N/A	Low	N/A	Low	It removes observations containing missing values from the dataset. Advantage: Easy to implement. Disadvantage: Information loss.
Simple imputation	Statistical	N/A	N/A	Low	N/A	Low	It replaces missing values with the estimations derived from the quantitative attribute of the available values. Advantage: Easy to implement. Disadvantage: Prone to change the original distribution of the variable.
Mean imputation	Statistical	N/A	N/A	Low	N/A	Low	It replaces the missing value with the arithmetic mean of the remaining dataset. Advantage: Easy to implement. Disadvantage: Potentially distorts imputations due to unequal data distribution and the influence of extreme values on the mean.
Regression imputation	Statistical	N/A	Normal distribution	Low	N/A	Low	It estimates the relationship between input and output variables, or between a data point and its corresponding variable. Advantage: Easy to implement. Disadvantage: Requires many assumptions due to its parametric nature.
Hot-deck imputation	Data based	N/A	N/A	Mid	N/A	High	It replaces missing data with values from complete data cases. Advantage: Being robust to misidentification and interaction effects as the imputations are generated using an implicit model. Disadvantage: Underestimates standard errors due to reduced variability that results from the selection of completed data from already observed data. In addition, when the dataset is not MCAR, data imputations can result in unpredictable bias.
Expectation–maximization	Statistical	N/A	Distribution	Mid	N/A	High	It is an iterative approach utilized to compute the maximum likelihood estimation of variables within statistical models. Advantage: Better than mean imputations as it preserves the relationship with other variables. Disadvantage: It does not provide standard error estimations.
Multiple imputation	Statistical	N/A	N/A	Mid	N/A	High	The regression assignment is re-operated more than once and each dataset is stored for later use, while the obtained errors are calculated. Then, the real dataset is attained by taking the average of the corresponding data. Advantage: It produces statistically valid estimates under the assumption of MAR and yields asymptotically consistent, efficient, and normally distributed estimates. Disadvantage: Due to its inherent variability, it often results in different estimates each time, leading to different outcomes among researchers working with the same dataset.
Linear interpolation	Statistical	Distance-based	Linearity	Mid	N/A	Low	It performs interpolation between a data point immediately preceding the gap and another data point shortly following the gap. Advantage: When properly utilized, it provides improved estimates for statistical inference. Disadvantage: It provides limited accuracy where continuous long missing data exists.
Kriging	Statistical	Distance-based	Stationarity and isotropy	Mid	N/A	Mid	It is a spatial estimation method that utilizes nearby data points to predict the values of unsampled points and uses semi-variogram structural features to make unbiased estimations of positional changes at unsampled locations. Advantage: Measuring the uncertainty of predictions and ensuring uncertainties to the user, enabling more reliable and informed decision making processes. Disadvantage: Explaining spatial relationships using a linear model, which may result in an inability to capture complex or non-linear relationships present in the dataset.
Inverse distance weighting	Statistical	Distance-based	N/A	Mid	N/A	Low	It is grounded on the estimation of the values of unsampled points by utilizing the values of sampled points. The estimated values are a function of both distance and the size of neighbouring points, and the importance and impact on the cell being estimated decrease as distance increases. Advantage: When used properly, it enhances the accuracy of statistical inference. Disadvantage: If the data has low density or is unevenly distributed, it can lead to inaccurate estimation of missing values in certain regions.
K-nearest neighbours	Machine learning	Distance-based	N/A	Low	Yes	Low	It works by identifying the closest k (referring to the number of neighbours) number of data points to the data point being evaluated, and assigning it to the same class or estimating its value based on the values of those k data points. Advantage: Easy to implement. Disadvantage: It does not work well with large datasets as calculating the distance between data instances is costly. Similarly, it is not compatible with high-dimensional data as the calculation process becomes more complex for each dimension. Additionally, the KNN algorithm is sensitive to noisy and missing data.
Decision trees	Machine learning	Tree-based	N/A	Mid	Yes	High	It uses a sequential approach to perform the classification or regression process, aiming to minimize the standard error to obtain an optimal decision tree. Advantage: Easy to understand, interpret, and visualize and does not require normalization or scaling of data. Disadvantage: It is prone to overfitting.
Artificial neural networks	Machine learning	Black box	N/A	High	Yes	High	Inspired by biological neural networks, ANNs are a set of mathematical operations aiming to solve inherent non-linear problems. Advantage: Non-linear and flexible, able to work with incomplete information. Disadvantage: High computational costs, requiring huge amounts of data, and limited explanations regarding its internal processes.
Adaptive neuro-fuzzy inference system	Machine learning	Black box	N/A	High	Yes	Very high	It defines the relationship between inputs and outputs in fuzzy space, while neural networks try to find the most suitable parameters of the model. Advantage: Less complex compared to other decision making systems. Disadvantage: Poses significant challenges in estimations having large number of inputs, including the curse of dimensionality, interpretability of rules, and parameter training.
Support vector machine	Machine learning	Distance-based and rule-based	N/A	High	Yes	High	It performs optimal classification with a single differential decision surface and is based on the principle of structural risk minimization. Advantage: It is more effective when there is a clear margin of separation between classes. Disadvantage: It has limited suitability for high-dimensional spaces and requires choosing an optimal kernel, which is a challenging task.

HP: hyperparameter; CC: computational cost.

Table 2. Details regarding the missing value imputation studies utilizing the Bowen-Ratio technique.

ID	Reference	Imputation methods	Method/data	Gap size	Performance metric
1.	Burakowski et al. (2018)	LI	Bowen Ratio/RSWin	<0.01%	N/A
2.	Sexstone et al. (2016)	LI	Bowen Ratio/Rn	N/A	N/A
3.	Balogun et al. (2009)	LI	Bowen Ratio/Rn	15%	N/A
4.	Guo et al. (2007)	PT, LR	Bowen Ratio/LE	1-7%	R^2

LI: Linear Interpolation; PT: Priestley-Taylor; LR: Linear Regression; RSWin: Incoming Shortwave Radiation; LE: Latent Heat Flux; R2: Coefficient of Determination.

Table 3. Details regarding the missing value imputation studies utilizing the eddy-covariance technique.

ID	Reference	Imputation methods	Method/data	Gap size	Performance metric
5.	Goodrich et al. (2021)	PLS, NN, KNN	Eddy covariance/N2O	15%	R^2
6.	Tang et al. (2019)	LR	Eddy covariance/H, LE	N/A	N/A
7.	Deb Burman et al. (2019)	REddyProc package	Eddy covariance/H, LE, Rn, RSWin, G, PPFD	N/A	N/A
8.	Abudu et al. (2010)	ANN	Eddy covariance/LE	5, 10, 20, 30, 40%	R^2, RMSE, MAE, MRE
9.	Chen et al. (2012)	MLR, KNN	Eddy covariance/LE	5, 10, 20, 40%	RMSE
10.	Hui et al. (2004)	MI	Eddy covariance/NE, LE, H	6–51%	R^2
11.	Alavi et al. (2006)	KF	Eddy covariance/LE	4–17%	R^2, RMSE, Rbias, R
12.	Kunwor et al. (2017)	Traditional, static window method, moving window method, parameter prediction method	Eddy covariance/NEE	35%	RMSE
13.	Schmidt et al. (2008)	NN	Eddy covariance/ CO2	13%	RMSE
14.	Mahabbati et al. (2021)	ANN, LR, RF, SVM, XGB, FFNN	Eddy covariance/NE, LE, H	1–32%	R^2, RMSE, MBE, VR, AI
15.	Li et al. (2007)	LI	Eddy covariance/H, LE	N/A	N/A
16.	Eamus et al. (2013)	Self-organizing linear output, ANN	Eddy covariance/E, C	N/A	RMSE
17.	Foltýnová et al. (2020)	REddyProc package	Eddy covariance/LE	1–100 half-hour	RMSE, MBE
18.	Boudhina et al. (2018)	REddyProc package, LR, MLR, evaporative fraction	Eddy covariance/LE	N/A	R^2, RMSE
19.	Zitouna-Chebbi et al. (2018)	REddyProc package	Eddy covariance/LE	10–70%	B, RB, RRMSE, R^2

PLS: Partial Least Square; NN: Neural Network; KNN: K-Nearest Neighbors; H: Sensible Heat Flux; G: Soil Heat Flux; PPFD: Photosynthetic Photon Flux Densities; ANN: Artificial Neural Networks; RMSE: Root Mean Square Error; MAE: Mean Absolute Error; MRE: Mean Relative Error; MLR: Multiple Linear Regression; MI: Mean Imputation; KF: Kalman Filter; Rbias: Relative Bias; RF: Random Forest; SVM: Support Vector Machine; XGB: Extra Gradient Boosting; FFNN: Feed Forward Neural Network; MBE: Mean Bias Error; VR: Ratio of Variance; AI: Index of Agreement; E: Expectation Stage; Rrmse: Relative Root Mean Square Error; NEE: Net Ecosystem Exchange.

Table 4. Details regarding the studies devoted to imputing missing humidity data.

ID	Reference	Imputation methods	Method/data	Gap size	Performance metric
20.	Costa et al. (2021)	MICE	Humidity	17–54%	RMSE
21.	Hasanpour Kashani and Dinpashoh (2012)	AA, ID, NR, single best estimator, MLR, UK traditional method, CSM, ANN, ANFIS, GP,	Humidity	N/A	MAE, NSE, SS
22.	Afrifa-Yamoah et al. (2020)	Kalman smoothing, ARIMA, MLR	Humidity	10%	MAE, RMSE, SMAPE
23.	Körner et al. (2018)	GTB	Humidity	50%	RMSE, MAE, R^2
24.	Anjomshoaa and Salmanzadeh (2019)	LI, CI	Humidity	1–374 h	VR, MAE, RMSE
25.	Lara-Estrada et al. (2018)	Bayesian network	Humidity	20%	RMSE, Bias
26.	Lee et al. (2010)	LI	Humidity	0.1%	N/A

MICE: Multiple Imputations by Chained Equations; NSE: Nash–Sutcliffe Model Efficiency Coefficient; SS: Skill Score; AA: Simple Arithmetic Averaging; ID: Inverse Distance Interpolation; NR: Normal Ratio; CSM: Closest Station Method; ANFIS: Adaptive Neuro-Fuzzy Inference System; GP: Genetic Programming; ARIMA: Autoregressive Integrated Moving Average; SMAPE: Symmetric Mean Absolute Percentage Error; GTB: Gradient Tree Boosting; CI = Cubic Splines Interpolation.

Table 5. Details regarding the studies devoted to imputing missing solar radiation data.

ID	Reference	Imputation methods	Method/data	Gap size	Performance metric
27.	Zainudin et al. (2015)	LI, SI, CI, KNN	Solar radiation	10, 20, 30, 40, 50%	RMSE, R^2
28.	Ogunsola and Song (2014)	SSA, SASR, TBA	Solar radiation	5, 10, 15, 20, 25 days	ME, MAE, RMSE
29.	Žliobaitė et al. (2014)	LR	Solar radiation	1–25%	MSE
30.	Kim et al. (2019)	LI, MI, MICE, KNN	Solar radiation	10, 15, 20%	RMSE, MRE, RMSD, MRD
31.	Turrado et al. (2014)	MICE, IDW, MLR	Solar radiation	0.002–1%	RMSE, MAE
32.	Demirhan and Renwick (2018)	36 imputation methods	Solar radiation	5, 10, 25, 50%	rMAE, rRMSE, MASE
33.	Schwandt et al. (2014)	Basic gap filling, satellite-based gap filling	Solar radiation	0.001–1.6%	NA
34.	Ernst and Gooday (2019)	DNI and DHI	Solar radiation	17%	NA

SI: Simple Imputation; SSA: Singular Spectrum Analysis; SASR: Statistically Adjusted Solar Radiation; TBA: Temperature-Based Approach; MSE: Mean Square Error; IDW: Inverse Distance Weighting; DNI: Direct Normal Irradiance; DHI: Horizontal Irradiance; MASE: Mean Absolute Standard Error.

Table 6. Details regarding the studies devoted to imputing missing wind speed data.

ID	Reference	Imputation methods	Method/data	Gap size	Performance metric
35.	Yang et al. (2011)	ANFIS	Wind speed	N/A	RMSE, R^2, MAE
36.	Liu et al. (2018)	GPR, MI	Wind speed	5, 10, 15, 20, 25, 30%	RMSE, MAE
37.	Mao and Jian (2016)	NN	Wind power	N/A	N/A
38.	Du et al. (2019)	NN	Wind speed	N/A	MSE, MAPE, MAE
39.	Li et al. (2019)	RNN	Wind speed	N/A	N/A
40.	He et al. (2013)	Mahalanobis distance	Wind speed	N/A	N/A
41.	Hocaoglu et al. (2009)	ANFIS	Wind speed	N/A	RMSE, R^2
42.	Xie and Sun (2018)	Low-rank matrix approximation, KNN, MPI, MI	Wind speed	N/A	RMSE
43.	Chen et al. (2016)	LSSVM	Wind speed	10%	MAPE, RMSE,
44.	Wesonga (2015)	MICE	Wind speed	28%	N/A
45.	Lin and Wang (2014)	CEM	Wind speed	1–12 h	MBE, MAPE, RMSE, r
46.	de Oliveira et al. (2021)	MissForest	Wind speed	1.69–43.75%	Mean_RMSE

GPR: Gaussian Process Regression; RNN: Recurrent Neural Network; MPI: Multiple Imputation; LSSVM: Least-Squares Support Vector Machine; CEM: Correlated Echelon Model; MAPE: Mean Absolute Percentage Error; r: Correlation Coefficient

Table 7. Details regarding the studies devoted to imputing missing temperature data.

#	Reference	Imputation methods	Method/data	Gap size	Performance metric
47.	Xu et al. (2013)	P-BSHADE, IDW, SRT	Temperature	N/A	RMSE, MAE
48.	Coulibaly and Evora (2007)	MLP, RBF, RNN, TLFN, TDRNN, CFNN	Temperature	2.1–65.2%	MAE, r
49.	Park et al. (2019)	LSTM	Temperature	6, 12, 24 h	RMSE, MBE
50.	López et al. (2021)	NN, KNN	Temperature	N/A	R^2
51.	Thevakaran and Sonnadara (2018)	Between-station technique	Temperature	0.55–1.22%	N/A
52.	Altan and Ustundag (2012)	Wavelet	Temperature	50%	MSE, MAE
53.	Katipoğlu (2022)	SVM, ANFIS, DT	Temperature	20%	RMSE, MAE, R^2
54.	Shtiliyanova et al. (2017)	Kriging	Temperature	16, 17.69%	NSE, MAPE, MAE, BIAS
55.	Cheval et al. (2020)	DINEOF	Land surface temperature	1.1–58.4%	R^2
56.	Kadow et al. (2020)	DL	Temperature	12.5%	P, RMSE
57.	Dumitrescu et al. (2020)	MLR, GAM	Land surface temperature	N/A	MAE, RMSE, r
58.	Zhou et al. (2020)	MLR	GIS/land surface temperature	N/A	RMSE
59.	Williams et al. (2018)	EMD, Bayesian compressive sensing	GIS/temperature	1, 5, 10, 20%	RMSE
60.	Lompar et al. (2019)	Debiasing procedure	GIS/temperature	5 days	RMSE
61.	Koksal et al. (2017)	ANN	Air temperature, relative humidity	30%	R^2, RMSE, MAE, MAPE

P-BSHADE: Biased Sentinel Hospitals-Based Area Disease Estimation; SRT: Spatial Regression Testing; MLP: Multi-Layer Perceptron; RBF: Radial Basis Function; TLFN: Time-Lagged Feedforward Network; TDRNN: Time Delay Recurrent Neural Network; CFNN: Counter Propagation Fuzzy-Neural Network; LSTM: Long Short-Term Memory; DT: Decision Trees; DINEOF: Data Interpolating Empirical Orthogonal Functions; DL: Deep Learning; GAM: Generalized Additive Model; EMD: Empirical Mode Decomposition; P: Precipitation.

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