Information Entropy-Based Hybrid Models Improve the Accuracy of Reference Evapotranspiration Forecast

. Accurate forecasting of reference crop evapotranspiration (ET 0 ) is vital for sustainable water resource management. In this study, four popularly used single models were selected to forecast ET 0 values, including support vector regression, Bayesian linear regression, ridge regression, and lasso regression models, respectively. Tey all had advantages of low requirement of data input and good capability of data ftting. However, forecast errors inevitably existed in those forecasting models due to data noise or overftting. In order to improve the forecast accuracy of models, hybrid models were proposed to integrate the advantages of the single models. Before the construction of hybrid models, each single model’s weight was determined based on two weight determination methods, namely, the variance reciprocal and information entropy weighting methods. To validate the accuracy of the proposed hybrid models, 1–30d forecast data from January 2 to February 1, 2022, were used as a test set in Xinxiang, North China Plain. Te results confrmed the feasibility of the information entropy-based hybrid model. In detail, the information entropy model generated the mean absolute percentage errors of 11.9% or a decrease by 48.9% compared to the single and variance reciprocal hybrid models. Moreover, the model generated a correlation coefcient of 0.90 for 1–30d ET 0 forecasting or an increase by 13.6% compared to other models. Te standard deviation and the root mean square error of the information entropy model were 1.65 mm · d − 1 and 0.61mm · d − 1 or had a decrease by 16.4% and 23.7%. Te maximum precision and the F 1 score were 0.9618 and 0.9742 for the information entropy model. It was concluded that the information entropy-based hybrid model had the best midterm (1–30 d) ET 0 forecasting performance in the North China Plain.


Introduction
With the fast growth of world population, people's requirements for both food and water resources are dramatically increasing [1].To cope with the problems, intensive and water-saving agriculture has been rapidly developing to meet the demand on the planet [2].It has been well-known that water resources used for agricultural sector have occupied 70% of the groundwater withdrawn in China [3,4].Furthermore, abiotic drought stress happens more often than before in the context of global warming, resulting in yield stagnation or failure in drought-stressed areas [5].Timely and precision irrigation is one of the most efective approaches to meet the dual goal of high yields and water-saving.With the intensifcation of global water shortage, it is crucial to develop a high-efcient water-saving irrigation technique [6].Te forecast of reference evapotranspiration (ET 0 ) is the basis for developing this technique [7], as crop water requirement can be estimated using ET 0 and crop coefcients.Te improvement in ET 0 forecast accuracy will greatly improve the accuracy of irrigation forecasting.
Due to stochastic changes in weather systems, accurate ET 0 forecast still remains a challenge [8].To improve ET 0 forecast accuracy, diferent types of forecasting models have been developed, including physical models, statistical models, and combined hybrid models [9].Physical models achieve ET 0 forecast based on future meteorological data via simulating the relationships among the atmosphere, land surface, and waters [10].However, the accuracy of numerical weather prediction (NWP) in forecasting long-term meteorological parameters limits the accuracy of other models based on weather forecasts.Statistical models mainly include linear regression models, time-series models, and machine learning models [11].Due to low requirement of data input and good capability of data ftting, those models have been widely adopted to ET 0 forecast [12].With a limited amount of meteorological factors, linear regression models such as Bayesian linear regression and ridge regression have shown advantages in ET 0 forecast in China [13], Mediterranean zones [14], and US High Plains [15].Besides, several neural network models were introduced to forecast ET 0 , including BP neural networks and support vector machine models [16].In Turkey, monthly mean ET 0 was estimated using adaptive network-based fuzzy inference system (ANFIS) and artifcial neural network (ANN) models [17].It was found that both the ANFIS and ANN methods were superior to Hargreaves and Ritchie methods in estimation of ET 0 .Regarding the complexity of ET 0 forecast, the applicability of most statistical models was limited, so more novel models have been attempted in recent years [18,19].To well simulate the dynamics of ET 0 trends, researchers combined the physical and statistical models [20].Tese hybrid models were adopted to predict nonstationary data series [21].In Peninsular Malaysia, a mixed multifractal forecasting model was adopted to forecast ET 0 trends by combining the light gradient boosting machine, decision forest regression, and artifcial neural network models [22].A number of studies also indicated that the performance of hybrid forecasting models outperformed that of single models, and the forecast accuracy was greatly improved by hybrid models [23][24][25].For example, in Atakum, Turkey, a hybrid model was constructed for ET 0 forecast based on the autoregressive integrated moving average model and generalized regression neural networks, and the hybrid model efectively improved ET 0 forecast accuracy [26].In Brazil, a hybrid model was established for ET 0 forecast based on support vector machine and artifcial neural network models, and the results showed that the hybrid model had the highest ET 0 forecast efciency and accuracy [27].Although time-series models have also been applied to ET 0 forecast, those models cannot refect the internal correlation among factors, compared to hybrid models [28].Because time-series models usually did not consider external factors, it would induce forecast errors when encountering signifcant external changes [29].
Till now, how to determine each single model's weight for a hybrid model is still a challenging task [30].Research on weight assignment based on diferent weight decomposition methods is little conducted in ET 0 forecasting [31].In this study, two hybrid ET 0 forecasting models were proposed based on variance reciprocal and information entropy algorithms.We hypothesized that the combined hybrid models were able to achieve more accurate ET 0 forecast values than single forecasting models.Te purposes of this study were as follows: (I) to select the optimal weight determining method for the construction of hybrid ET 0 forecasting models, (II) to identify the optimal hybrid ET 0 model by comparing the accuracy of diferent single and hybrid models, and (III) to explain the reason why the proposed hybrid model has advantages over other models.

Data Establishment.
Experimental data were collected from Xinxiang Meteorological Station, North China Plain (35 °08′ N, 113 °45′ E, a.s.l.73 m).Tis paper selected the dataset from January 1, 2020, to December 31, 2022, including maximum air temperature (T max ), minimum air temperature (T min ), mean air temperature (T mean ), and relative humidity (RH).Te four parameters have shown signifcant correlations with ET 0 variations in the temperate monsoon climate of China [16].Tis study extracted the features of these data on the same historical days in each year.

Feature Extracting.
To extract daily features of meteorological data, we supposed that there were H-related meteorological factors on each single day.Based on the assumption, daily eigenvectors on days i and j were expressed as (u i1 , u i2 , . . ., u iH ) T and (u j1 , u j2 , . . ., u jH ) T .Feature similarity on days i and j was defned as follows: where O ij represents daily feature similarity, H is the number of meteorological factors, u iH and u jH are eigenvectors on days i and j, and h represents the number of current meteorological factors.

Data Preprocessing.
Due to diferent dimensions of data features, data normalization was needed in data preprocessing, which was a step down-scaling raw data to desired scope for further processes.In this study, the minmax normalization was adopted to normalize the target parameters.Te expression was as follows: where x ′ is the normalized dimensionless data, x is the original data, x min is the minimum value in the original data, and x max is the maximum value in the original data.

Data Training and
Test.Tis study divided the dataset into the training set and the test set at an 8 : 2 ratio.To obtain as much efective information as possible from the 2020-2022 learning data, a cross-validation method was used to segment the dataset, and a 5-fold cross validation was chosen to obtain the best estimate.

Selection of Single Models
2.5.1.Support Vector Regression (SVR).When a support vector regression model (SVR) was used for forecast analysis, its core was to establish an optimal classifcation surface 2 Advances in Meteorology using an insensitive loss function [32].In this way, the mean square error of all training sets from this optimal classifcation surface can be minimized.
where X is the observed data, y is the corresponding target value, N is the number of data samples, f (X) is the Bayesian linear regression model, w is the weight coefcient, and ε is the residual.In the model, the weight coefcients (w) are independent of the observed data (X), and ε values are independent and identically distributed.Bayesian linear regression assumes that the residual follows a normal distribution.

Ridge Regression.
Ridge regression is an improved least squares estimation method used for the analysis of collinear data.In ridge regression, regression coefcient values are introduced to reduce the efect of the covariance of independent variables [34].Te regression is more suitable to ft poor-conditioned data than the least squares method [35].It is more suitable to solve the problem of collinearity of independent variable data and the lack of explanatory parameters in multiple linear regression [36].[36].In the regression, if the interpreted variable Y i is set to be independent with given observed values, Y i will be considered independent with respect to standardized observed values (x ij ).Lasso regression was expressed as follows:

Lasso Regression. Lasso regression focuses on the multiple regression and performs feature selection by restricting absolute values for target models. It has a strong ability to attenuate the regression coefcient vector via selecting useful data features and obtaining reliable variable selection function
where x ij is standardized observed values and t is the harmonic parameter (t ≥ 0).When t gradually decreases, regression coefcients will also decrease and gradually tend to zero.When t approximates zero, it will be eliminated at the time i and j.
2.6.Construction of Hybrid Models.To construct hybrid forecasting models, reasonable weights should be assigned to each single model.Te following steps describe the weight determination process for hybrid models.

Determination of Target Attributes.
To determine each model's target attribute, a decision matrix was established.
Te matrix was expressed as follows: where c ij is predicted values of the ith model on the jth similar day, m is the number of single forecasting models, and s is the total number of the similar days.

Construction of Eigenvalue Matrix.
Eigenvalue is the transformation of a linear transformation represented by a matrix into a numerical transformation.Te feature vector corresponding to the feature value is the key.Te properties of a complex matrix can be transformed into the feature of eigenvectors.In this way, the complex data can be simplifed to be analyzed.Te eigenvalue matrix was expressed as follows: where λ is the coefcient matrix of the forecasting models, m is the number of single forecasting models, and s is the total number of the similar days.

Normalization of Eigenvalue Matrix.
To make different meteorological parameters comparable and easy to be adopted in the calculation of weights, eigenvalues were normalized using equation ( 2).

Construction of Matrix R.
Te normalized r was used to obtain the matrix R. Te calculation formula was expressed as follows: where s is the total number of the similar days and r ij is the normalized eigenvalue of the ith model on the jth similar day.
Information entropy was adopted to measure how cluttered the system data were.Te information entropy method was usually used to evaluate the amount of information carried by the dataset through characterizing the complexity and quantifying the amount of uncertainty in a system.It is a metric that describes the degree of chaos in a system to determine the diversity of data.In this study, information entropy was expressed as follows: where E i is the information entropy of the matrix R, s is the total number of the similar days, and r ij is the normalized value of the eigenvalue of the ith model on the jth similar day.
Considering the properties of the logarithmic functions in equation ( 8), we defned that, when r ij was equal to zero, r ij ln r ij also became zero.Te function assumes that the weight of a model may approximate to zero when E i is extremely small.
Te magnitude of the weight vector (ω it ) represents the importance of the corresponding model m in a hybrid model.Te larger ω i is, the more important a single model is in the hybrid model, and vice versa.In this study, the weight vector was calculated based on the values of E i as follows: where ω it is the weight vector on days t, m is the number of single forecasting models, and E i is the information entropy of the matrix R.

Variance Reciprocal-Based Weight Determination.
Te variance reciprocal method refers to determining the weight using the proportion of the reciprocal of the sum of error squares of a single model to that of the total sum of error squares.Tis method avoids the appearance of negative weight values and distributes greater weights to more accurate forecasting models.Te model was expressed as follows: where e i is the square of the forecast error of the ith single model, y t is the total sum of error squares, and y i is the sum of error squares of the ith single model.
Based on the values of e i for each single forecasting model, the weight (w it ) of the ith single model in a hybrid model was expressed as follows: where m is the number of single forecasting models and e i is the squares of the forecast error of the ith single forecasting model.We assumed that there were m diferent single forecasting models to be integrated in a hybrid model.According to each model's weight and predicted values, the hybrid forecasting model was expressed as follows: where y t is the predicted value from a hybrid model at time t, w it is the weight of the ith single model at time t, and y it is the predicted value from the ith single model at time t.In the model, the sum of all the w it is 1.00.
To obtain the predicted ET 0 values on days t, predicted results of ET 0 from each single model should be multiplied by ω it .Terefore, the fnal results of ET 0 were a product of each allocated weight and the single predicted value of ET 0 .

Statistical Evaluation Metrics.
In order to evaluate the forecasting performance of proposed models, this paper used the root mean square error (RMSE), mean absolute percentage error (MAPE), and coefcient of determination (R 2 ) to analyze the accuracy of ET 0 forecasting models from the perspective of the error ratio and goodness of ft.Te RMSE and MAPE can be used to represent the average error of the predicted result with respect to the ground-truth result.Lower RMSE and MAPE indicate smaller errors between the predicted and observed values.R 2 is used to

Inputs Outputs
Input layer Hidden layer Output layer Figure 1: Single-layer neural network structure of a support vector regression model. 4 Advances in Meteorology quantify the correlation between the forecasts and observations.Higher R 2 indicates better forecast performance of a forecasting model.Te mathematical equations of the statistical indices were described as follows: where 2.9.Statistical Analysis.In this study, support vector regression, Bayesian linear regression, ridge regression, and lasso regression were conducted using the Python 3.7 programming language.By taking 80% of the historical data as the training set, the data from the rest of the months were used for testing.Data were subjected to analysis of variance (ANOVA) using SPSS 20.0 (SPSS Inc., Chicago, IL, USA).
Signifcance was declared at the probability level of 0.05, unless otherwise stated.Graphs were plotted using Sigmaplot 12.0 (Systat Software, San Jose, CA, USA).

Normalization and Correlation Analysis.
Most machine learning algorithms required the variables to satisfy a normal distribution.Tis paper performed a normalization test for the interpreted variables through plotting the data probability distribution.Te probability plot indicated the degree to which the actual distribution of the variables was in line with the theoretical normal distribution.Te test was used to examine whether the data were in agreement with a normal distribution pattern.If the data followed a normal distribution, the data were regarded coinciding with the theoretical straight line (Figure 2(a)).Based on the distribution of ET 0 values, our results showed that after processing of the raw data, the processed ET 0 data fell into the −3 to 3 quantiles, suggesting the data conformed to a normal distribution, and were able to be applied to the machine learning algorithms.Tis result was in agreement with the previous studies conducted in China [38], India [39], Turkey [40], and North America [41].Before using a model to predict target values, it was usually necessary to perform correlation analysis to remove unrelated variables.Tis method was used to reduce computational complexity and improve the interpretability of the model [16].In this study, the Pearson coefcient method was adopted to perform correlation analysis, which was popularly adopted by previous studies [42,43].Tis method mainly measured the linear correlation between variables, with the correlation coefcients from −1 to 1.In the present study, the correlation coefcient between RH and ET 0 was less than 0.25, implying a very weak correlation between the two variables (Figure 2(b)).Te coefcients among T max , T min , T mean , and ET 0 were greater than 0.70, among which T max had the highest correlation coefcient of 0.84.In a humid subtropical climate of China, it was also observed that T max was the most correlated parameters to ET 0 , followed by T min and T mean [20].In Quebec, Canada, a noticeable exponential relationship between air temperature and ET 0 was observed in a humid continental climate [44].In northeast China, T max was considered the greatest contributor to ET 0 fuctuations related to low radiation conditions [45].According to correlation analysis, RH was excluded as an input factor for data feature extraction.

Forecast Performance of Single Models.
In this study, four single models were selected according to the recommendation from previous literature [16,20], including support vector regression (SVR), Bayesian linear regression, ridge regression, and lasso regression (Figure 3).Te four single models showed good capacity to ft the linear relationships between observed and predicted ET 0 values.Tey produced similar ET 0 trends to the observed ET 0 changes in 2022.Te observed ET 0 values were from 0.26 to 7.32 mm•d −1 from the P-M model, whereas the predicted ET 0 ranges were from 0.24 to 7.48 mm•d −1 , 0.45 to 7.54 mm•d −1 , 0.09 to 7.12 mm•d −1 , and 0.02 to 6.98 mm•d −1 for SVR, Bayesian, lasso, and ridge regression models, respectively.Te highest values of ET 0 appeared during 160-180 Julian days (corresponding to mid-June), while the lowest values were observed in 1-10 Julian days (corresponding to early January).On average, mean ET 0 predicted by SVR was 3.04 mm•d −1 , or a decrease by 6.2% compared to the real observations.Similarly, average values of Bayesian linear regression, ridge regression, and lasso regression were 3.01, 2.77, and 2.97 mm•d −1 , respectively, or had a decrease by 6.8-17.2%.Te annual ET 0 values of the four single models were 1002.9-1110.3mm•yr −1 or had a decrease by 8.1-18.4% compared to the real accumulated ET 0 value from the P-M model.It can be concluded that all the single models generated lower averaged and accumulated ET 0 values than did the P-M model.However, both the averaged and accumulated ET 0 predicted by SVR models were much closer to the Advances in Meteorology real observations than did the linear regression models.In previous studies, Piotrowski et al. found that the SVR model had higher prediction accuracy than linear regression models, such as the ridge regression model [46].Moreover, among those linear regression models, the Bayesian regression model had higher accuracy than the other two proposed linear models.Te reason may lie in that, through establishment of a payof function, the Bayesian model is able to generate an optimal iteration algorithm to obtain desired predicted values [47].

Weights Assigned to Hybrid Models.
Te determination of goodness of ft for single models helped calculate each single model's weight assigned to hybrid models [48].In this study, RMSE values of SVR and Bayesian linear regression models were within 0.152-0.168mm•d −1 for both training and test datasets, while R values of the two models were greater than 0.78, showing better forecast performance (Table 1).Te SVR and Bayesian models generated the mean absolute percentage errors (MAPEs) of 23.4% for training and test sets or had a decrease by 29.3% compared to the lasso and ridge models.Terefore, the SVR and Bayesian models were given higher weights (26.3-29.9%)than hybrid models (Figure 4).On average, the weights assigned to lasso and ridge models were 20.3% lower than those of SVR and Bayesian models.Tis fnding was similar to the results observed by Liu et al. [49].Based on the algorithms of information entropy, the SVR model had the highest weights of 0.299, followed by 0.274 for Bayesian linear regression, 0.203 for lasso regression, and 0.224 for ridge regression, respectively.Te information entropy method assigned more weights to SVR and Bayesian models than did the variance reciprocal method.

Forecast Performance of Hybrid Models.
In this study, hybrid forecasting models were established based on the SVR model, Bayesian linear regression, ridge regression, and Lasso regression models (Figure 5).Previous results indicated that hybrid models made good use of information from single models, which efectively increased their forecast accuracy [50].In this study, variance reciprocal and information entropy methods were adopted to construct hybrid models.Te four single forecasting models were incorporated into the hybrid forecasting models according to their assigned weights.Te predicted ET 0 ranges were from 0.38 to 7.12 mm•d −1 and 0.67 to 7.53 mm•d −1 for information entropy and variance reciprocal models.On average, mean ET 0 predicted by the information entropy model was 3.19 mm•d −1 or had a decrease by 1.8% compared to the real observations.Similarly, the average value of the variance reciprocal model was 3.04 mm•d −1 or had a decrease by 6.5%.Te annual ET 0 values of the two hybrid models were 1157.4-1196.3mm•yr −1 or had a decrease by 3.1-7.4% compared to the real accumulated ET 0 value from the P-M model.Our results indicated that the hybrid forecasting models signifcantly improved the forecasting accuracy when the advantages of single models were comprehensively incorporated.Te hybrid model was more accurate for predicting both daily ET 0 dynamics and annual accumulated ET 0 values in the North China Plain.Our fnding was in agreement with the previous results conducted in the Mediterranean climate of Iran [51].

Correlation Analysis of Forecasting Models.
Compared with the variance reciprocal hybrid model, correlation coefcients (R) were signifcantly increased, while RMSE values were appreciably decreased by the information entropy hybrid model (Table 1).Te information entropy hybrid model generated the mean absolute percentage errors (MAPEs) of 11.9% for training and test sets or a decrease by 39.7%-58.1% compared to the single models and the variance reciprocal model.In this study, R, RMSE, and MAPE values of the reciprocal hybrid model were not signifcantly diferent from those of SVR and Bayesian models.
Correlation analysis showed that the information entropy hybrid-based model had the highest coefcient of determination (R 2 ) of 0.922 in 2022, followed by the SVR and Bayesian regression models (Figure 6).Te ridge      Advances in Meteorology variance reciprocal-based hybrid model.Te reason why the variance reciprocal-based model had lower forecasting accuracy might be that the model did not guarantee the errors of the hybrid models were small enough at each time node [52].Te excessive errors at single abnormal moments might result in the failure of the entire model [53].In this study, we recommended the information entropy-based hybrid model.Te advantage of the information entropy weight method was that it determined weights based on the data itself, which had strong objectivity and reduced the infuence of subjectivity on forecasted results [54].Te information entropybased method considered multiple indicators simultaneously, and it was not limited by the evaluation of a single indicator, which was why the information entropy-based hybrid model was preferable to the variance reciprocal model [55].

Evaluation of Model Forecast Performance.
To validate the accuracy of forecasting models, both single and hybrid models were applied to forecast 1-30 lead day ET 0 trends using independent datasets from January 2 to February 1, 2022 (Table 2).Moreover, the Taylor diagram was plotted using observed and forecasted data in 2022 for a visual comparison test among diferent models (Figure 7).[56].Tey found that the hybrid model resulted in a reduction in MAE and RMSE.Te reason why hybrid models had the ability to lower prediction errors lied in that the innovative weight assignment method reduced the possibility of models outperformance and overftting by optimizing the weight assignments to models [57].Tis increased the generalizability of hybrid models in diferent climatic zones.Te study confrmed that the Kruskal-Wallis method was obvious to do better for accuracy evaluation of models when   10 Advances in Meteorology data pooled did not follow a normalized distribution [37].

Advances in Meteorology
Tis study provides insights on the optimal algorithms of weight determination for the construction of hybrid ET 0 forecasting models.

Conclusions
To achieve precise ET 0 forecast, this study proposed two hybrid models based on variance reciprocal and information entropy algorithms.Te two algorithms were used to assign weight of each single model to hybrid models.As a result, hybrid models signifcantly improved the forecast accuracy compared to the single models.To further investigate the general ability of the hybrid models, forecasted weather data were used to forecast ET 0 in 1-30 d lead days in 2022.It was observed that the information entropy-based hybrid model outperformed other forecasting models in improving ET 0 forecast performance.Tis study confrmed that the information entropy-based hybrid model was the one of the most efective hybrid models in midterm (1-30 d) ET 0 forecasting in the North China Plain.In future works, more attention should be paid on how to extend the generalizability of hybrid models to other climatic types and to improve the accuracy of long-term (>30 d) ET 0 forecasting through integrating the advantages of diferent regression and machine leaning models.

Figure 2 :
Figure 2: (a) Probability plot of reference evapotranspiration after data processing and (b) correlation coefcients between meteorological variables.RH: relative humidity; T max : maximum air temperature; T mean : mean air temperature; T min : minimum air temperature.

Figure 3 :
Figure 3: Comparison between the observed reference evapotranspiration (ET 0 ) values and the forecasted ET 0 values from (a) support vector regression, (b) Bayesian linear regression, (c) lasso regression, and (d) ridge models in 2022.

Figure 4 :
Figure 4: Weights assigned to each hybrid model from each single model based on the algorithms of information entropy (blue lines) and variance reciprocal (red lines) methods.

Figure 5 :
Figure 5: Comparison between the observed reference evapotranspiration (ET 0 ) values and the forecasted ET 0 values of hybrid models based on (a) information entropy and (b) variance reciprocal weighting methods in 2022.

Figure 6 :
Figure 6: Correlation analysis between observed and forecasted reference evapotranspiration (ET 0 ) values from the (a) support vector regression model, (b) Bayesian linear regression model, (c) ridge regression model, (d) lasso regression model, (e) information entropybased hybrid model, and (f ) variance reciprocal-based hybrid model in 2022.R 2 is the coefcient of determination.

Figure 7 :
Figure 7: Taylor diagram for the visual comparison among single and hybrid models.
[37] the number of observations, P i and O i are the predicted and observed values of the ith day, respectively, and P and O are the average values of P i and O i for the observation periods, respectively.2.8.Kruskal-Wallis Test.Te Kruskal-Wallis test was used to evaluate the accuracy of forecasted results from single and hybrid models.Diferent from parametric tests, the Kruskal-Wallis test was a nonparametric test without the data requirement of assumptions of normality and homogeneity of variance.With more than two data groups, it examined the medians of the data groups to determine if the predictions were from distinct populations with the same distribution.It used data ranks to calculate the accuracy instead of using numerical values.More detailed description about the Kruskal-Wallis test can be found in Clark et al.[37].

Table 1 :
Evaluation of forecasting accuracy of single and hybrid forecasting models.Note.R, correlation coefcient; RMSE, root mean square error; MAPE, mean absolute percentage error.Diferent letters in each column stand for signifcant diferences at p < 0.05.

Table 2 :
Performance metrics of single and hybrid models from January to February 1, 2022, using the Kruskal-Wallis test.Bold values indicate the best performance among SVR, Bayesian, ridge, lasso, information entropy, and variance reciprocal models.MAD is the median absolute deviation around the median. Note.