Autoencoder-based composite drought indices

Depending on the type, drought events are described using different indices, such as meteorological, agricultural, and hydrological. The use of different indices often causes confusion for making water-related management decisions. One simple summarized index which can describe the different aspects of drought is desired. Several methods have therefore been proposed, especially with the linear combination method which does not adequately describe drought characteristics. Meanwhile, autoencoders, nonlinear transformation in dimensional reduction, have been applied in the deep learning literature. The objective of this study, therefore, was to derive autoencoder-based composite drought indices (ACDIs). First, a basic autoencoder was directly applied as ACDI, illustrating a negative relation with the observed drought indices which was further multiplied by a negative. Also, the hyperbolic tangent function was adopted instead of the sigmoid transfer function due to its higher sensitivity to drought conditions. For better expression of drought indices, positive and unity constraints were applied for weights, denoted as ACDI-C. Further simplification was made as sACDI by excluding the decoding module since it was not necessary. All applied weights of different sites over a country can be unified into one weight, and the same weights were made for all the sites, called as sACDI1. In the context of model evaluation, a comprehensive analysis was undertaken employing metrics as root mean squared error (RMSE), mean absolute error (MAE), and correlation coefficients. The collective findings underscore the superior performance of both the sACDI and sACDI1 models over their counterparts. Notably, these simplified models manifestly diminished RMSE and MAE values, indicating their enhanced predictive capabilities. Of particular note, sACDI1 exhibited a discernibly lower MAE in comparison to alternative models. Further alarm performance metrics was conducted including the false alarm ratio, probability of detection, and accuracy (ACC). The investigations revealed superiority of the simplified models in terms of alarm ACC, especially in the case of SRSI(A). The developed ACDI can comprehensively summarize multiple drought aspects and provide summarized information about drought conditions.


Introduction
Drought is caused by low amount of precipitation occurring over an extended period of time.Its frequency is being intensified by climate change.To mitigate mitigation measures, drought monitoring, assessment, analysis, and forecasting are done using different indices (Niemeyer 2008).Selection of an appropriate drought index is therefore essential, which depends upon the classification of drought indices.Drought is generally classified into three fundamental categories: meteorological, hydrological, and agricultural (Wilhite and Glantz 1985, Mishra and Singh 2010, Eslamian et al 2017, Balti et al 2020).Meteorological drought is a consequence of precipitation deficit (Mishra andSingh 2011, Wei et al 2021), and to monitor and assess meteorological drought, several drought indices have been developed, such as standardized precipitation index (SPI), standardized precipitation evapotranspiration index, Palmer drought severity index, and Z-index (Mishra and Singh 2011).
Hydrological drought is related with reduced streamflow, and low reservoir storage, and its indices include standardized reservoir supply index (SRSI), Palmer hydrological drought index, streamflow drought index, and surface water supply index (SWSI).Agricultural drought occurs due to decreased soil moisture (Mishra and Singh 2010) and representative indices are soil moisture index and standardized soil moisture index (SSI) (Yang et al 2023).
Drought conditions are generally represented using an index with a single type of drought (Balint et al 2013, Rajsekhar et al 2015, Chen et al 2020) and many drought studies using a single index have been carried out.For example, for monitoring drought Hayes et al (1999) concluded that SPI was a valuable tool because of its flexibility to observe different time scales.Hayes et al (2000) further concluded that SPI was able to identify the intensity and duration of drought and illustrate the near real-time monitoring.While assessing drought in Romania, Ionita et al (2016) found SPI having the capability to represent the anomalies of precipitation.
While a single type of drought is mostly assessed, a region may be affected by more than one type of drought (Liu et al 2019).For this reason, indices compositing different types of droughts have been proposed, such as aggregate drought index (ADI), combined drought index (CDI), and multivariate drought index.Keyantash and Dracup (2004) developed ADI aggregating three types of droughts, including meteorological, hydrological, and agricultural.They adopted six variables of precipitation, evapotranspiration, streamflow, reservoir storage, soil moisture content, and snow water content, employing principle component analysis (PCA) to determine the importance of each variable.
Another composite index, named CDI, was developed by Balint et al (2013) who calculated three indices, including precipitation drought index (PDI), vegetation drought index, and temperature drought index using rainfall, temperature, and NDVI data, respectively.Weights were assigned subjectively as 50% for PDI and 25% for each of the other two.Correlating SPI and CDI with crop yield, Al-Bakri et al (2019) showed that CDI had a higher correlation with crop yield than had SPI, indicating that the combined index was more reasonable than a single index.
Further development of a composite drought index was conducted by Rajsekhar et al (2015) as MDI integrating precipitation, runoff, evapotranspiration, and soil moisture data.They indicated that PCA has the limitations due to the assumption of linearity in data reduction, and the supposition that most information is included in the direction where the maximum data variance exists (Hao andSingh 2015, Waseem et al 2015).For these reasons, the kernel entropy component analysis was applied to enhance the performance of PCA and they concluded that the method overcoming the limits, reasonably reflects the multivariate drought conditions.Liu et al (2019) and Bageshree et al (2022) conducted a drought index composition study using PCA and copulas.Both studies compared the index developed with each PCA and copulas, and the results showed that the composite index using copulas better caught drought events.Regarding the limitations of PCA, the current study intended to propose a novel artificial neural network (ANN)-based method for drought index composition.
Autoencoders, as a type of ANNs, have been applied to reduce the dimensionality with encoding and decoding functions by learning an efficient representation (Kramer 1991, 1992, Goodfellow et al 2016, Lee et al 2021).Autoencoders can be comparable to traditional dimensionality reduction technique such as PCA because it can apply nonlinear transformation to project data in a lower dimensional space.The capability to capture nonlinear relationships within data sets distinguishes autoencoders from linear PCA, enabling a deeper comprehension of patterns and variable interactions.Unlike PCA, which primarily relies on the variance of its principal component (PC1), autoencoders, being a type of ANNs, possess the inherent capacity to autonomously learn features.This suggests that autoencoders are adept at extracting more informative and representative features for the development of composite drought indices (CDI).Since linear PCA has been mostly applied for compositing multiple drought indices and autoencoder can overcome the limits of PCA, the objective of the current study is to develop an autoencoder-based composite technique for combining drought indices so that multiple drought conditions can be appropriately summarized into one specific index.
As regions are affected by more than one type of drought, indices of three drought types (i.e.meteorological, hydrological, and agricultural) were composited in the current study.SPI was selected for meteorological drought and SRSI was adopted for hydrological and agricultural droughts.Using autoencoder, the weight for each index was assessed and the composite drought index was further developed as autoencoder-based CDI (ACDI).

Study area
Along with the global increase of drought intensity, South Korea is also experiencing water deficiency especially in recent years (Kyoung et al 2011, Kim et al 2015).The drought occurrence and intensity over South Korea is expected to increase significantly (Nam et al 2015, Kwak et al 2016).Therefore, an appropriate drought index development can play a critical role for drought monitoring for this region.Also, while the country experiences multiple types of droughts concurrently and a composite approach to drought warnings becomes imperative, alerts in South Korea are issued based on three distinct types of droughts, each tailored to specific usage purposes: meteorological, agricultural, and residential and industrial.With the increase of drought intensity and the circumstance of issuing separate alerts for different drought types, South Korea was selected as the study area for the composite methodology application.
The models for combining multiple drought indices were applied in South Korea.The country is located between 34 • -37 • latitude and 124 This region experienced the longest drought (lasting as long as 227 d) in the observed period.Imminent governmental actions had been taken, such as limiting water supplies to domestic industries.

Data
Three indices representing agricultural, hydrological and meteorological droughts for 167 governmental sectors are shown in figure 1 as well as their time series in figure 2. To represent the meteorological drought over South Korea, SPI was estimated with the observed precipitation for all 167 stations with the period from 1975 to 2023.The locations of the weather stations available from Korea Meteorological Association (http://sts.kma.go.kr/) are shown in figure 1(c) with blue dots, and Thiessen polygons (Diskin 1970, Song andPark 2021) were drawn to assign precipitation values of the employed stations to governmental sectors.
For agricultural and hydrological droughts, SRSI (Hasegawa et al 2016, Schilstra et al 2024) available from National Drought Information Portal (www.drought.go.kr) was employed.Like SPI, monthly SRSI values were estimated with the standardization of the reservoir monthly volume into a normal variate by weighting the total reservoir volume in each sector.For the construction of agricultural index, SRSI(A) values were driven by the reservoirs for agricultural purposes.Among 167 governmental sectors, 131 sectors of SRSI(A) were available, while 36 sectors were not applicable.
For hydrological drought, the SRSI(H) values were estimated with different hydrological sources

Study workflow
The study workflow, as depicted in figure 4, unfolds as follows: (1) Derivation of three drought indices-SPI, SRSI(A), and SRSI(H)-corresponding to meteorological, agricultural, and hydrological droughts, respectively; (2) Development of autoencoder-based composite drought index (ACDI) alongside its constrained variant (ACDI-C) and simplified version (sACDI); (3) Evaluation of ACDIs and traditional PCA model performance using metrics such as root mean squared error (RMSE), mean absolute error (MAE), and correlation, as well as assessment of alarm performance through false alarm ratio (FAR), probability of detection (POD), and accuracy (ACC); and finally, (4) Synthesizing the findings to draw conclusive insights.

Autoencoder
An autoencoder can be defined as a type of ANN, which efficiently represents input variables by encoding them into latent variables as dimensional reduction.The autoencoder is related to PCA in that the latent variables of the autoencoder correspond to the principal components (PCs).A simple autoencoder with one hidden layer and one neuron in the middle with three input variables can be depicted in figure 3.With three input variables x = [x 1 , x 2 , x 3 ], the encoding model can be defined as where ] and b I are the weight and bias parameters.σ is an element-wise activation function for which a sigmoid or a rectified linear unit (ReLU) has been commonly applied.Decoding can also be described as: where It is worthwhile to mention that autoencoders are related to PCA in that the hidden variable z (equation ( 1)) directly connects to the PCs by taking the activation function (σ) as a linear function.In a linear combination of PCs, the first component (PC 1 ) can be estimated by maximizing the variance as: Note that the hidden variable with one hidden node (z) can be used as a drought index.Since the hidden variable as a drought index is in the range of [0 and 1] from the sigmoid transfer function, it is more convenient to apply the hidden variable (z) and transform it into a standard normal variate (z N ) so that its scale is relevant with three drought indices as (5) One of the main reasons for this transformation is to scale the drought index as SPI and SRSI.Unlike other indices, the process of fitting into a probabilistic distribution is unnecessary, since the z variable in the sigmoid transfer function has already been worked as the cumulative distribution function (CDF).

Developing autoencoder-based composite drought index (ACDI)
In the current study, three drought indices for the input variables in an autoencoder [x 1 , x 2 , x 3 ] were applied as SRSI(A), SRSI(H), and SPI.The hidden variable z was used as an autoencoder-based composite drought index (ACDI).The sigmoid activation function was applied in (equation ( 1)) as where, y = W T I x + b I .In this case, the input variable ranged as [0,1] and preliminary results indicated that the activation function was more sensitive to high values.The objective of the current study was to develop a composite drought index for drought status that had extreme negative values.Therefore, the hyperbolic tangent function (tanh) can be more attractive as where the range of tanh is [−1, 1].

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In order to provide the same range as other variables, the hidden variable z with the tanh activation function should be transformed into a standard normal variate as where Φ −1 is the inverse of the CDF of the standard normal variate.Note that (z + 1)/2 was applied to make the tanh function into a probability distribution function with the range [0,1] and nonnegative.Note that the tanh function monotonically increases and by changing (z + 1)/2, it is nonnegative with ranges as [0,1].Instead of the gamma CDF transformation, the modified tanh function as (z + 1)/2 can be applied.

ACDI with constraints (ACDI-C)
Though an autoencoder is a type of ANN and its model structure is not related with physical characteristics of input variables, it is profitable to develop the ACDI with a physical relation.Therefore, we further developed ACDI by adding two constraints as: and w Ii > 0 for i = 1, 2, and 3.
The unity constraint in equation ( 9) was included, since three drought indices should be weighted proportionally and the sum of weights must be unity.Furthermore, negative weight in equation ( 10) should be avoided, because drought indices cannot be negatively related with each other.

Simplified version of ACDI (sACDI)
For training the autoencoders, both input and output weight parameters (W I and W O ) were estimated.However, it is not necessary to estimate the output weight parameters which were not employed in the current study.This superfluous estimation might reduce the model performance especially with limited observations.The estimation of the unnecessary parameter set of the output weights can be excluded by a simple modification as follows.
(2) Estimate the input weight parameters (W I ) with the distance measurement with the hidden variable z rather than the output x as: Note that the same constraints as in equations ( 9) and ( 10) were applied for the current model.
The final sACDI model can be fully described as where, x A , x H , and x M represent the agricultural, hydrological, and meteorological drought indices, respectively.

Statistical methods for model performance
Model performance was further conducted using RMSE, MAE, and correlation between the observed data of three indices and the composite drought index.RMSE and MAE are two widely used standard metrics for model evaluation (Hodson 2022).The two metrics have been adopted in various fields including meteorology and climate research studies (Chai and Draxler 2014), and can be calculated as: where n is the number of observations, x i is the observed value of three indices, and xi is the composite drought index value.Correlation coefficient which is another famous method for comparing the original data with the predicted was also tested with the formula as: Correlation = Cov(x,y) σxσy .From the formula, Cov (x, y) is the variance of x and y.Also, σ x and σ y are the standard deviations of each x and y.
Alarm performance was also tested for model comparison.First, four alarm conditions were set as true positive, true negative (TN), false positive (FP), and false negative (FN).For example, TP in the current study is the case that the estimated ACDI was lower than a preset alarm level while one of three observed indices (i.e.SASI(A), SRSI(H), and SPI6) was also lower than the level to issue a warning.The alarm performances were further defined as FAR = FP TP+FP , POD = TP TP+FP , and ACC = TP+TN TP+FP+TN+FN .Through the results, the models can be compared regarding that smaller FAR indicates a better model, and larger POD and ACC illustrate a better performance.

Results of the basic ACDI
The ACDI was developed with a typical autoencoder as in equations ( 1) and ( 5) with the sigmoid transfer function and no constraints for the weight parameters.One example of the result is presented in the T Lee et al A simple remedy for this is to take negative values as The negative ACDI result with equation ( 13) is presented in the right panels of figure 4. The represented series were more positively related than the original one.Though the negative series seemed better related with the original one, this estimated ACDI z N tended to be higher than the individual drought indices.

Results of the modified ACDI with constraints (ACDI-C)
The original ACDI model performance presented two substantial problems for using it as a composite drought index.First, the hidden variable was inversely related with three input drought indices.The negative modification in equation ( 13), however, is not reliable since this negative solution cannot be applied when only part of three weights w I is negative and the others are not.Second, the further negative series shown in the right panels of figure 4 still presented the tendency of overestimation.Since the objective of the current study was to develop a composite method to represent drought conditions, its overestimated index often downsized the significance of current drought conditions.This overestimation might be originated from the drawback of the sigmoid transfer function, that is nonzero-centered with the range of [0,1] shifting its mean value (Chen et al 2024).
Therefore, the original ACDI was modified by adding two constraints in equations ( 9) and ( 10) and adopting the tanh function instead of sigmoid for the encoding module.This modified ACDI version with constraints was denoted as ACDI-C.The estimated ACDI and ACDI-C are compared with scatterplot in figure 5 for randomly selected four sites.The range of the original ACDI (about −1.5 and 1.5) was shorter than the one of ACDI-C (about −2.5 and 2).Especially, the extremes were relatively suppressed in a shorter range for the original ACDI.Note that the T Lee et al  13)) shown in the right panels.Note that the observed data is presented with a black dash-dotted line while the ACDI result if presented with a red solid line.Also, the simulated output of the autoencoder in equation ( 2) is shown with blue dotted line.
values in about ±[1.0 and 1.5] were densely plotted in the ACDI, while the estimated index for the ACDI-C model was much widely spread.As mentioned in table 1, the drought index with less than −2.0 was classified as severe drought.From this implication, the ACDI series did not produce any severe drought in its estimation, while the ACDI-C series generated a portion of severe droughts.Also, the time series shown in figure 6 showed that the combined index of the ACDI-C had smaller values than the one of the ACDI and closely related with the original drought indices.Therefore, the result implied that the adaptation of the tanh function and the constraints was appropriate for producing a composite drought index.

Results of the simplified version of ACDI (sACDI)
The model was developed to minimize the error between the observed three drought indices and the  output of the autoencoder x in equation ( 2).However, the limitation of this autoencoder-based model is to optimize the output values that are not employed in application.The final ACDI is abstracted from the hidden variable, not the output.Therefore, we further developed a simplified version of ACDI as sACDI by eliminating the decoding module and directly connecting the minimization of the loss function with the observed data and the hidden variable z which is worked as the ACDI (see equation ( 11)).In this setting, the input weights with two constraints in equations ( 9) and (10) were estimated by producing the hidden variable z with the standard normal transformation in equation ( 12) as close as possible to three drought indices.
The time series of the estimated sACDIs for two sites are presented in figure 7 and the estimates reasonably followed the patterns of all three drought indices.Exceptionally low values in one variable (black dash dotted line) were, however, not captured by this one summarized drought index (red line).For example, the observed SRSI(A) values much lower T Lee et al

Model comparison
Model performance was further evaluated using RMSE, MAE, and correlation between three observed drought indices and the composite drought index, here denoted as z N .In figure 8, RMSE, MAE, and correlation are illustrated with boxplots for all 119 sites.RMSE and MAE indicated that sACDI better performed than ACDI and ACDI-C.In correlation, they were similar to each other except that ACDI and ACDI-C had substantially low values and negative values, while sACDI did not have very low correlation.This indicated that the sACDI estimate were more reliable than the others, which is critical for drought monitoring and alarming systems.
In addition, the distribution of weight parameters was investigated, since a highly biased weight into only one variable should not be avoided and composite drought conditions have to be extracted harmoniously from all three drought indices.12) for two sites as site1 and site100.
The ACDI-C constraining the weights to be positive and their sum to be unity as in equations ( 9) and ( 10) are presented in the (a-2), (b-2), and (c-2) panels.Results showed that w 1 was mostly distributed in lower than 0.4 range, while w 2 and w 3 were mostly in the range of 0.3 and 0.6.This indicated that the lower weighting is placed on SRSI(A) than the other two indices.In a field application, agricultural drought is one of the major factors handled by water managers.Therefore, a smaller weight to SRSI(A) is often not desirable and large variability of weights might hinder the application of the estimate as a drought index in the field.
By contrast, results of sACDI showed that the weights were mostly distributed over the range of 0.25 and 0.4 for all three weights.Slightly higher weights were observed for SRSI(A) than for SRSI(H) and SPI6.This short range and relatively high peak indicated that sACDI was reliable for application.Furthermore, it might be wise to suggest one set of input weights for all sites in South Korea.The mean value of weights for all 119 sites was [0.3488, 0.3287, 0.3183] and was adjusted to be unity as [0.3503, 0.3301, 0.3197].This final set of input weight (W I ) was tested by applying to all sites, called as sACDI1.Its performance is indicated in figure 8 and presents that RMSE was similarly low to the one of sACDI and had the lowest MAE.The correlation of sACDI was also very close to sACDI1.The proposed model was further compared with PCA.Results in figure 8 show that the performance was better than ACDI, but worse than ACDI-C and sACDI.The weights of PCA were also checked and found to contain negative weights which is not desirable for application.
Figure 10 and the original ACDI model excluded show a result which is not comparable with a weird behavior.Note that smaller FAR and larger POD and ACC indicate better performance.In FAR shown in the top panels of figure 11, sACDI and sACDI-1 outperformed ACDI-C and PCA, especially in SRSI(A) (see the panel (a-1)).POD and ACC presented the same outperformance of sACDI and sACDI1 with higher values.ACCs of SRSI(H) and SPI6 did not have much difference between the estimated models.Overall, the alarm performance results indicated that sACDI and sACDI1 outperformed the other models of PCA, especially for SRSI(A).

Limitations and discussions
A comprehensive drought index, aiming encapsulation of various drought characteristics, should adequately represent all pertinent aspects and exhibit sensitivity to extreme drought scenarios, further enabling water managers to draw conclusive assessments.Thus far, a prevalent approach involves the linear combination method of PCA to amalgamate distinct drought indices (Keyantash and Dracup 2004, Liu et al 2019), along with its modified versions.However, a few critical considerations must be addressed: (1) The output variable of the first principal component (PC1) exhibits diverse variances, necessitating normalization and exclusively accommodating linear correlations; (2) Negative weights are often computed, particularly when applied drought indices lack linear correlation; and (3) The weights are not collectively summed to one, requiring further investigation into the estimated weights.
These limitations have deterred water managers from directly utilizing estimated CDIs based on PCA.
T Lee et al In contrast, ACDIs, rooted in recent ANN methodologies, offer a straightforward approach, accommodating nonlinearity.By employing critical constraints such as non-negativity and unity of summed weights during parameter estimation, the resulting index emerges as reliable and stable, devoid of negative values-a crucial consideration given as the unintended implications of negative weights.For instance, a negative weight assigned to an agricultural drought index with an extremely low value diminishes the drought's significance in the composite index, thereby underscoring the significance of constraints applied in this study (see equation ( 10)).
Additionally, the unity condition (equation ( 9)), coupled with the non-negativity constraint, restricts estimated weights between 0 and 1, ensuring realistic weight distributions.The autoencoder network's transfer function maps the composite variable into a probability domain, obviating the need for further normalization.Stable weight estimation is evident from the consistent distribution of estimated weights across different stations in South Korea, facilitating the adoption of a representative weight set for composite drought index estimation across various stations.This stability fosters acceptance among water managers and local authorities, aiding in future drought management planning.
Moreover, the proposed ACDI methodology can be extended to integrate other hydro-environmental variables, consolidating multiple variables into a concise representation, such as flood indices and water-quality indices.However, it is important to note that the simplified version of the sACDI model accommodates only a single variable and cannot abstract multiple latent variables.Furthermore, the proposed model does not account for temporal effects of drought characteristics but amalgamates contemporaneously estimated values of

Summary and conclusions
The objective of the current study was to derive a novel approach compositing agricultural, hydrological, and meteorological drought indices.We tested autoencoder-based models in the current study, since autoencoders are successfully applied in dimensionality reduction.The direct application of autoencoder with the sigmoid function often presents a negative relation with three observed drought indices.A simple remedy by multiplying negative as in equation ( 13) was used, and the sigmoid transfer function was changed into tanh to yield a better representation, especially extreme events, called ACDI.Since three input weights may be desirable with the condition of positive and unity, these constraints were applied, denoted as ACDI-C.Further simplification was made by excluding the decoding module from ACDI, since its module is not necessary in the current application, and optimization is directly made with the hidden variable as in equation ( 11), denoted as sACDI.
Additionally, the alarm performance measurements of FAR, POD, and ACC with different levels for warning were made.The sACDI and sACDI1 were superior to the other models in alarm performance with less false alarm and higher accuracy.Overall, results indicated that the simplified version (sACDI) outperformed the other versions (i.e.ACDI and ACDI-C) as well as the common linear method of dimensionality reduction (PCA).Also, sACDI was stable throughout the applied sites, indicating that one overall mean value of the weights (sACDI1) can be similarly utilized.It is desirable for water managers and governors to identify drought conditions with one simple weight parameter set.The developed autoencoder-based composite drought index models can be useful to combine other drought indices as well.Moreover, any number of multiple variables can be adopted for ACDI models to summarize multiple characteristics into one variable, not only limited to drought indices.However, care should be taken not to release critical characteristics of each variable, since it is inevitable to lose certain information by squeezing multiple variables into one variable.

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Figure 1.Map of the 167 governmental sectors with the information of data existence for (a) agricultural, (b) hydrological, (c) meteorological droughts.Note that (a) the agricultural data are in 131 sectors based on the reservoir rate and 36 sectors with no data; (b) the hydrological data are in 136 sectors from different hydrological sources and 21 sectors with no data; and (c) the meteorological data were made with 60 weather stations marked with blue dots for all 167 sectors whose precipitation values are assigned according to the Theissen boundaries.

Figure 4 .
Figure 4. Overall study workflow of the current study.

Figure 5 .
Figure 5.Time series of three drought indices as (a) SRSI(A), (b) SRSI(H), and (c) SPI6 for observations and the ACDI with (1) the original model shown in the left panels and (2) the negative model (see equation (13)) shown in the right panels.Note that the observed data is presented with a black dash-dotted line while the ACDI result if presented with a red solid line.Also, the simulated output of the autoencoder in equation (2) is shown with blue dotted line.

Figure 6 .
Figure 6.Scatterplot of the estimated drought index of ACDI and its constraints version as ACDI-C for randomly selected four sites as [1, 10, 50 100].

Figure 7 .
Figure 7. Time series of three drought indices and the combined drought index with ACDI and ACDI-C for site 1.
Figure 9 presents histograms and kernel density estimates (Lall et al 1993, Salas and Lee 2010) of probability density function (f ) for three weights of SRSI(A), SRSI(H), SPI6.As expected, the weights of ACDI model were rather double-normally distributed throughout [−1 ∼ 1] with two peaks as shown in the (a-1), (b-1), and (c-1) panels of figure 9. Positive and negative parts were almost evenly distributed.

Figure 8 .
Figure 8.Time series of three drought indices as (a) SRSI(A), (b) SRSI(H), and (c) SPI6 of the observed data (dark dash-dotted line) and the sACDI (red solid line) in equation (12) for two sites as site1 and site100.

TFigure 9 .
Figure 9. Performance measurement of RMSE (a), MAE (b), and Correlation between the observed three drought indices and the autoencoder-based models as ACDI, ACDI-C, sACDI, sACDI1 and PCA.Note that the correlation measurement is presented at each observed drought index(the panels of (c)-(e)).

Table 1 .
Status and alert according to the magnitude of the drought indices (SPI, SRSI).
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