SPI-Informed Drought Forecasts Integrating Advanced Signal Decomposition and Machine Learning Models

Abstract

Drought is an extremely terrifying environmental calamity, causing declining agricultural production, escalating food prices, water scarcity, soil erosion, increased wildfire risks, and changes in ecosystem. Drought data is noisy and poses challenges to accurate forecasts due to it being nonstationary and non-linear. This research aims to construct a contemporary and novel approach termed as TVFEMD-GPR, crossbreeding time varying filter-based empirical mode decomposition (TVFEMD) and gaussian process regression (GPR), to model multi-scaler standardized precipitation index (SPI) to forecast droughts. At first, the statistically significant lags at (t − 1) were computed via partial auto-correlation function (PACF). In the second step, the TVFEMD splits the (t − 1) lag into several factors named as intrinsic mode functions (IMFs) and residual components. The third step is the final step, where the GPR model took the IMFs and residual as input predictors to forecast one-month SPI (SPI1), three-months SPI (SPI3), six-months SPI (SPI6), and twelve-months SPI1 (SPI12) for Mackay and Springfield stations in Australia. To benchmark the new TVFEMD-GPR model, the long short-term memory (LSTM), boosted regression tree (BRT), and cascaded forward neural network (CFNN) were also developed to assess their accuracy in drought forecasting. Moreover, the TVFEMD was integrated to create TVFEMD-LSTM, TVFEMD-BRT, and TVFEMD-CFNN models to forecast multi-scaler SPI where the TVFEMD-GPR surpassed all comparable models in both stations. The outcomes proved that the TVFEMD-GPR outperformed comparable models by acquiring ENS = 0.5054, IA = 0.8082, U95% = 1.8943 (SPI1), ENS = 0.6564, IA = 0.8893, U95% = 1.5745(SPI3), ENS = 0.8237, IA = 0.9502, U95% = 1.1123 (SPI6), and ENS = 0.9285, IA = 0.9813, U95% = 0.7228 (SPI12) for Mackay Station. For Station 2 (Springfield), the TVFEMD-GPR obtained these metrics as ENS = 0.5192, IA = 0.8182, U95% = 1.9100 (SPI1), ENS = 0.6716, IA = 0.8953, U95% = 1.5163 (SPI3), ENS = 0.8289, IA = 0.9534, U95% = 1.1296 (SPI6), and ENS = 0.9311, IA = 0.9829, and U95% = 0.7695 (SPI12). The research exhibits the practicality of the TVFEMD-GPR model to anticipate drought events, minimize their impacts, and implement timely mitigation strategies. Moreover, the TVFEMD-GPR can assist in early warning systems, better water management, and reducing economic losses.

1. Introduction

Drought is a complex natural phenomenon that significantly impacts environmental factors such as food security, vegetation health, agricultural yield, and socioeconomic conditions in a given region. Severe droughts result in medium- to long-term declines in agricultural and food productivity [1]. Droughts disproportionately affect the agricultural sector, with over 83 percent of all damage and losses due to drought ascribed to agriculture. This contrasts with other disasters such as storms, earthquakes, tsunamis, and volcanic eruptions, which affect many sectors but do not largely damage agriculture to the same degree [2]. Consequently, agriculture-dependent nations such as Bangladesh, Nepal, Afghanistan, India, Pakistan, and Sri Lanka have repeatedly seen droughts over the past fifty years [3,4]. The incidence of severe drought events is anticipated to rise due to global warming [5]. In this setting, the timely prediction of drought conditions is essential for effective drought management.

Comprehending the susceptibility of these systems to climate variability has conventionally relied on the premise that historical hydrological extremes will recur, a premise that is now untenable due to human climate change [6]. The evaluation of hazards associated with anticipated prolonged hydrological anomalies, such as droughts, often relies on general circulation model (GCM) simulations of precipitation, temperature, and soil moisture. Nevertheless, climate model simulations of these variables have considerable biases over many time scales, resulting in differing capacities to accurately depict prolonged climate anomalies [7]. Droughts result from the interaction of various climatic factors, including prolonged low precipitation, elevated temperatures, strong winds, low humidity, and infrequent climate variations (e.g., El Niño-Southern Oscillation (ENSO), encompassing both warm (El Niño) and cold (La Niña) phases of ENSO) [8]. Consequently, it is essential to comprehend the interplay of GCM biases and uncertainties across each of these factors. Although GCMs effectively simulate large-scale atmospheric variables such as temperature and sea level pressure, their proficiency in depicting localized, sustained hydrological anomalies is constrained by their coarse resolution, incomplete model architectures, feedback mechanisms including albedo and land-atmosphere interactions, and the parameterization of clouds and convection in precipitation [9].

Drought exhibits diverse spatiotemporal attributes, such as severity, size, intensity, duration, and geographic extent [10]. Numerous drought indices have been proposed to quantify and assess the different facets of drought. Drought indicators are vital tools for the empirical evaluation and study of drought characteristics [11]. They are tasked with defining and analyzing the particulars of drought across various hydrological landscapes, offering a uniform metric to assess drought severity in different areas [12]. The analysis of drought consequences relies on individual and/or several hydrometeorological variables, including streamflow and precipitation [13]. Numerous traditional criteria were developed to mitigate the effects of drought. Nonetheless, no singular index exists that can adequately encapsulate and communicate the severity and intensity of such an incident [14]. Initially, negative precipitation anomalies served as an index to denote drought, although they failed to account for the impact of drought on hydrology and agriculture. In this context, various indices have been established, including the standardized precipitation index (SPI) [15], standardized precipitation evapotranspiration index (SPEI) [16], and Palmer drought severity index (PDSI) [17].

According to [18], the PDSI saw extensive use towards the end of the twentieth century. It considers things like soil moisture, streamflow, possible evapotranspiration, and precipitation that came before. Although PDSI has its uses, it is not very responsive to sudden changes in drought status because of an inherent time scale that makes it take longer to react to less severe drought circumstances. Contrarily, the SPI calls for fewer variables and has very straightforward definitions. To track multiple locations’ moisture conditions at once, the SPI makes use of multiscale characteristics. But it limits its applicability to the context of climate change because it ignores temperature and concentrates only on precipitation. A solution to this problem was the SPEI, which considers temperature and the disparity between evapotranspiration and precipitation to determine the severity of a drought and strikes a balance between the two [19].

The effectiveness of different drought indices has been the subject of various studies conducted in the last several decades. One study by [10] looked at how well various meteorological drought indices—such as the PDSI, modified PDSI, self-calibrating PDSI (scPDSI), surface wetness index, SPI, and SPEI—performed in different regions of China. The scPDSI was determined to be the best index for capturing the unique climate of China based on the results. But the scPDSI narrowed the value range a little bit more than the PDSI, thus wet/dry condition classification needs some tweaking. Departure from Normal, effective drought index, SPI, SPEI, and reconnaissance drought index (RDI) were all assessed in western India by [20]. Utilizing information obtained from four climate stations over a 25-year period, the indices were computed for several durations, including 1, 3, 6, 9, and 12 months. The results showed that in semi-arid climates, a SPEI lasting 9 months was the best indicator to use. Another work [21] conducted an evaluation of the SPI and SPEI in the Tigray Region of Northern Ethiopia. There was a reasonable amount of agreement between those indexes, according to the results. Traditional drought indices are widely used, although their accuracy is greatly affected by weather and the criteria used to calculate it. In [22], it was also noted that when applied to diverse geographies, these approaches can occasionally generate incorrect conclusions. Because of their low processing costs, ease of implementation, and minimum inputs, soft computing approaches have replaced their more traditional counterparts, which have their own set of limitations. Furthermore, drought can also be categorized via precipitation anomalies by utilizing SPI in terms of their timescales where the associated numbers represent months [23]. The SPI is a worldwide standard metric to examine drought events.

Drought has various damaging effects on socio-economic conditions of a society [24]. Countrysides are extremely vulnerable to drought due to their dependence on agriculture and water [25]. According to [26], droughts cause 83% of the agricultural losses globally by directly affecting crops and livestock production. Droughts are constantly impacting agricultural crops and livestock, decreasing water supply, increasing bushfires, and causing soil erosion and overall environmental degradation in Australia [27]. Globally, Australia places 5th in financial damage and 15th population-wise in countries poorly impacted by droughts [28]. Consequently, drought forecasting has received attention from policy makers to obtain greater awareness to improve future responses [29].

Several models have been established to increase the forecasting abilities to forecast drought event accurately, including regression analysis, autoregressive integrated moving average (ARIMA), artificial neural networks (ANN), fuzzy logic (FL), and other hybrid models [30,31]. Machine learning (ML) models to forecast the future values have acquire substantial attention from the scientific community in forecasting droughts [32]. Wavelet transforms was integrated with ML to forecast short-term SPI in Ethiopia [33], while [34] led a study on droughts using deep learning to forecast SPI in a semi-arid climate region. The work in [35] forecasted meteorological droughts by designing ensemble ML models whereas [36,37] proposed a multilayer perceptron for SPI forecasting in the US and Iran, respectively. A novel implementation of pre-processing approaches and hybrid kernel-based models is for short- and long-term groundwater drought forecasting [38]. The literature shows that drought forecasting tendencies are based on climatic parameters using the regression and simple ML models. Moreover, the previous works have been conducted in larger areas and zones, but not for a small region and locality. The ML accuracy is substantially affected by failing to extract all the relevant characteristics of the data. A multiresolution analysis is effective in uncovering the embedded features by tackling this issue. Previous studies show that Fourier spectral analysis [39]; discrete wavelet transformation [40,41]; Empirical Mode Decomposition (EMD) [42]; Ensemble EMD [43]; complete ensemble EMD with adaptive noise [44]; and improved complete ensemble empirical mode decomposition with adaptive noise [45] are the commonly implemented techniques. However, these approaches experience the major issue of mode mixing and demands a sequential decomposition [46]. To tackle this trouble, the time varying filter-based empirical mode decomposition (TVFEMD) method is implemented, which offers better frequency separation and stability, especially under noisy or low-sampling rate.

The novelty of this work is based on the hybridization of the time varying filter-based empirical mode decomposition (TVFEMD) and gaussian process regression (GPR) to design a TVFEMD-GPR model to forecast multi-scaler SPIs (i.e., SPI1, SPI3, SPI6, and SPI12). The TVFEMD method efficiently improves the ability to decompose the SPI’s data into IMFs, stability under low sampling rates, and noise robustness. The TVFEMD method addresses the non-stationarity and noise in SPI time series, facilitating the separation of noise and short-term fluctuations from significant temporal structures. The GPR model offers various benefits in terms of flexibility to complex relationships and probabilistic forecasts with uncertainty quantification. The non-parametric nature of GPR is able to adjust to various data distributions without strict assumptions and can deal with sparse data. This work conducts a comparative analysis of various robust predictive models, including Cascaded Feedforward Neural Networks (CFNN), Long Short-Term Memory networks (LSTM), and Boosted regression tree (BRT), to appraise their usefulness to forecast droughts in the Mackay and Springfield stations from Australia.

The Mackay region’s high potential evaporation rate leads to soil moisture loss, which makes the drought-like conditions even worse. Climate change is causing alterations that go beyond natural rainfall patterns, which could make drought and water shortages happen more often and with more severity. On the other hand, Springfield has been devasted by droughts during 1997–2009 and 2017–2020, impacting farmers, livestock, and water resources. Both rural and urban areas have been significantly impacted by these droughts in terms of agriculture, water supply, and the economy. The following section presents the materials and methods, followed by model development (Section 3), application results and analysis (Section 4), further discussion (Section 5), and conclusion (Section 6).

2. Materials and Methods

2.1. Study Area and Data Description

Station 1: Mackay is situated on the coast of Queensland and is well-known for its diverse economy, including mining, agriculture, specifically sugarcane, and tourism. Mackay is experiencing two distinct seasons of tropical climate. The dry season is milder and less humid than the wet season, which brings high temperatures, humidity, and sporadic cyclones. The majority of Mackay’s 1585 mm of annual rainfall occurs between December and March. The weather is changing in Mackay as average temperatures in Queensland are already 1 °C higher than they were 100 years ago, and in the last few decades, there has been a clear warming trend. The area has both climate change and climate variability, which causes both natural changes and changes that go beyond what is normal for drought. The high potential evaporation rate makes the soil lose moisture, which makes the drought-like conditions even worse. Climate change is causing changes that go beyond natural rainfall patterns, which could make drought and water shortages happen more often and with more severity.

Station 2: Springfield is located in the City of Ipswich, Queensland, Australia. The climate of Springfield is defined by cool winters and warm-to-hot summers, with the winter and spring seasons seeing the most rainfall. The average summer high temperature is around 30 °C, usually hot and dry, while winters are cool, with highs in the north typically reaching 13 °C and in the high country, 4 °C. The winter and springtime see the most rainfall and there is a range of humidity levels, with summertime typically having higher humidity levels. The regions have suffered droughts, impacting farmers, livestock, and water resources. Both rural and urban areas have been significantly impacted by these droughts in terms of agriculture, water supply, and the economy.

The acquired datasets were obtained from the Scientific Information for Landowners (SILO), Queensland, Australia and were only available for the period of 1905 to 2024 for the selected locations in Queensland, Australia. Figure 1 shows the map of these two stations and

Table 1. Geographic and statistical description of the data.

Geographic Description	Springfield				Mackay
Longitude (°E)	152.9170				149.1868
Latitude (°S)	−27.6542				−21.1443
Elevation	69 m				11 m
Statistical Description	SPI1	SPI3	SPI6	SPI12	SPI1	SPI3	SPI6	SPI12
Minimum	−4.029	−3.300	−3.081	−3.236	−2.634	−2.765	−2.652	−2.322
Maximum	3.921	3.582	3.422	3.580	3.765	3.655	3.514	3.236
Mean	0.001	−0.001	−0.0005	−0.0001	−0.0009	−0.003	−0.002	−0.001
Std. Deviation	0.996	1.000	1.000	1.0003	0.988	0.999	0.999	1.0001
Skewness	0.044	0.119	0.114	0.051	0.320	0.302	0.324	0.352
Kurtosis	0.401	0.103	0.078	0.035	0.345	0.222	0.120	−0.198

describes the basic statistics of the data.

2.2. Time Varying Filter-Based Empirical Mode Decomposition (TVFEMD)

Enhancements to the empirical mode technique are implemented in the TVFEMD algorithm. This method outperforms existing techniques in terms of frequency division performance, stability at low sampling rate, and resistance to noise interference [47]. To get beyond the limitation of mode mixing with local narrowband signal output and minimize the occasional error of additional noise effect, TVFEMD uses adaptive filtering with shifting filtering techniques to replace the sifting process of traditional EMD. Once again, it is shown that short signals or signals with low sample rates are unsuitable for the classic EMD [48]. Screening for EMD, adjusting the local cut-off frequency, and screening for time-varying filtering are the three main components of TVFEMD [49]. The main steps of the TVFEMD are as follows:

• For each signal $i=1,2,\dots ,$ evaluate the maximum possible location and assign it the label $u_i$.
• Locate all occurrences of intermittency that meet the following requirements:

$${\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\phi }_{bis}^{\prime }\left(u_i:u_{i+1}\right)\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left({\phi }_{bis}^{\prime }\left(u_i:u_{i+1}\right)\right)}{\mathrm{m}\mathrm{i}\mathrm{n}\left({\phi }_{bis}^{\prime }\left(u_i:u_{i+1}\right)\right)}}>\rho$$

(1)

${\phi }_{bis}^{\prime }$ stands for the bisecting frequency, and the exact method for calculating it may be found in [47]; $\rho$ is the specified threshold for the rate of frequency shift between two consecutive maxima. Because of this requirement, we can rewrite $u_i$ as $e_j$ for all $j=1,2,\dots ,$ and treat its location as an interlude.

3

On the rising or falling edge of ${\phi }_{bis}^{\prime }\left(t\right)$, there are two possible places for each instance of $e_j$; if ${\phi }_{bis}^{\prime }\left(u_{i+1}\right)>{\phi }_{bis}^{\prime }\left(u_i\right)$, then ${\phi }_{bis}^{\prime }\left(u_i\right)$ can be considered a floor. An alternative is to observe ${\phi }_{bis}^{\prime }\left(u_i\right)$ at its falling edge if ${\phi }_{bis}^{\prime }\left(u_{i+1}\right)<{\phi }_{bis}^{\prime }\left(u_i\right)$.

4

To make local adjustments to the cut-off frequency, interpolate between the peaks. The local narrow-band signal is then obtained by passing the input signal through time-varying filters.

5

When applying a filter to the signal $x\left(t\right)$, the B-spline approximation is used, which uses the extreme timing of $h\left(t\right)$:

$$h(t)=\mathrm{c}\mathrm{o}\mathrm{s}\left[\int {\phi }_{bis}^{\prime }(t)dt\right]$$

(2)

$Q\left(t\right)$ stands for the suitable approximation result. An important property of time-varying filters is their roll-off, which is defined by the order of a B-spline approximation and so affects its performance.

6

The incoming signal’s compliance with the stop criterion $\tau \left(t\right)$ can be determined by calculating it:

$$\tau (t)={\displaystyle \frac{B_{\mathrm{Loughlin}}(t)}{{\phi }_{\mathrm{avg}}(t)}}\le \mathsf{\varsigma }$$

(3)

In this context, ${\phi }_{\mathrm{avg}}(t)$ denotes the weighted average of the immediate frequencies for each component, while $B_{\mathrm{Loughlin}}(t)$ refers to the Loughlin immediate bandwidth. The symbol $\mathsf{\varsigma }$ indicates the bandwidth threshold, which is used to decide if the incoming signal requires filtering. If $\tau \left(t\right)$ satisfies this condition, the signal x(t) can be regarded as IMF; if not, then $x\left(t\right)$ is adjusted to $x\left(t\right)=x\left(t\right)-Q\left(t\right)$, and steps 1 through 6 are reiterated. By following the above steps, the original signal $x\left(t\right)$ can be broken down into several distinct IMFs.

$$x(t)={\sum }_{v=1}^K\hspace{0.17em}{c}_v(t)$$

(4)

The residual item is denoted as $c_K\left(t\right)$, the total number of decomposition levels is $K$, and the IMF at level $v$ is $c_v\left(t\right)$, where $i=1,2,\dots ,K-1$.

2.3. Gaussian Process Regression (GPR)

According to Williams and Rasmussen [50], GPR is a nonparametric non-linear regression method which has a solid theoretical foundation. It does a good job even when dealing with tiny datasets that have non-linear regression concerns. In terms of practical and theoretical achievements, GPR has been a strong competitor in the field of supervised learning for the last several decades. Direct definition of prior probability distributions over latent functions is achieved by means of a GPR. Both the mean and the covariance functions provide a complete description of GPR [51]. A GP blended with Gaussian likelihood generates the posterior GP over the new model output. $D\left(X,Y\right)$ represents the number of observations given the training data.

$$X={\left[x_1^T,x_2^T,\dots ,x_n^T\right]}^T$$

(5)

$$Y={\left[y_1,y_2,\dots ,y_n\right]}^T$$

(6)

This is how the observation model is put together:

$$y=f\left(x\right)+{\epsilon }_n,{\epsilon }_n~N\left(0,{\sigma }_n^2\right)$$

(7)

In this case, ${\sigma }_n^2$ stands for the noise variance and $f\left(x\right)$ for the latent variable. $\mathrm{f}[\mathrm{f}=f(X\left)\right]$ can have its distribution specified using:

$$\mathrm{f}=N\left(0,C\left(X,X\right)\right)$$

(8)

Let $f_{\ast }$ denote the anticipated value of an invisible point $x_{\ast }$. The GP prior specifies that the joint distribution of $\mathrm{f}$ and $f_{\ast }$ has a Gaussian representation expressed as

$$p\left(\mathrm{f},f_{\mathrm{*}}\right)=N\left(\left[\begin{array}{l}0\\ 0\end{array}\right],\left[\begin{array}{ll}C& C^{\ast }\\ C_{\ast }^T& \tilde{C}\end{array}\right]\right)$$

(9)

where

$$C_{\ast }=C\left(x_{\ast },X\right)$$

(10)

$$\tilde{C}=C\left(x_{\ast },x_{\ast }\right)$$

(11)

With the use of the training set, Bayesian inference can be expressed as the posterior distribution of the objective prediction $f_{\mathrm{*}}$ as

$$p\left(f_{\mathrm{*}}\mid \mathrm{Y}\right)=N\left({\mu }_{f_{\mathrm{*}}},{\sigma }_{f_{\mathrm{*}}}^2\right)$$

(12)

Whereas the data for the mean and standard deviation are given by

$${\mu }_{f_{\ast }}=C_{\ast }{K}^{-1}Y$$

(13)

$${\sigma }_{f_{\ast }}^2=\tilde{C}-C_{\ast }^T{K}^{-1}{C}_{\ast }$$

(14)

The equation $\mathrm{K}=\mathrm{C}+{\sigma }_n^2{\mathrm{I}}_n$ states that ${\mathrm{I}}_n$ is the identity matrix of size $n\times n$.

For the GPR to be effective in its modeling, the covariance function is crucial. For the purpose of defining the GPR, the widely used squared exponential (SE) function was selected for its outstanding adaptability and clear interpretation:

$$\mathrm{C}\left(\mathbf{x},{\mathbf{x}}^{\prime }\right)={\eta }^2\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{1}{2}}{\sum }_{k=1}^d\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{x_k-x_k^{\prime }}{l_k}}\right)}^2\right]$$

(15)

The signal variance is denoted by ${\eta }^2$, while the length scale is represented by $l$. To determine the hyperparameters $\mathsf{\Theta }=\{\eta ,l\}$, the likelihood function is maximized as:

$$\mathcal{L}(\mathsf{\Theta })={\displaystyle \frac{1}{2}}{\mathrm{Y}}^{\mathrm{T}}{\mathrm{K}}^{-1}\mathrm{Y}+{\displaystyle \frac{1}{2}}\mathrm{l}\mathrm{o}\mathrm{g}|\mathrm{K}|+{\displaystyle \frac{n}{2}}\mathrm{l}\mathrm{o}\mathrm{g}(2\pi )$$

(16)

2.4. Long Short-Term Memory (LSTM)

A variant of the RNN, the Long Short-Term Memory (LSTM) network, was developed by [52]. A dedicated storage unit and a technique for controlling the network’s data stream are features of the LSTM [53]. According to [54], LSTM arrives at the best solution by optimizing the error function with gate cells and enabling inter-neuron communication. With its long-term memory and ability to learn from past data, the LSTM model is perfect for capturing non-linear trends in time series. Consequently, many time series challenges have been effectively resolved using the LSTM model. The three gates that make up the conventional construction of an LSTM are forget, input, and output [54]. The vanishing gradient problem with RNN is fixed by the LSTM, which learns from data held over a lengthy period. Removing unnecessary data from the cell state is the responsibility of the first layer of the memory gate, which is defined as follows:

$$f_t=\sigma \left(w_f\times X_t+Y_f\times h_{t-1}+p_f\right)$$

(17)

In this context, $f_t$ represents the forgetting threshold at time t, $w_f$ and $Y_f$ denote the weights, $\sigma$ denotes the sigmoid activation function, $h_{t-1}$ denotes the output value at time t, $X_t$, the input value, and $p_f$, the bias term, are all defined. The second input gate takes the data from the present set of inputs and decides what should be stored in the cell state [55]. Together, the choice $i_t$ that changed the tanh layer and the value to generate a new state value $C_t$ make up this. The expression for it is

$$i_t=\sigma \left(w_i\times X_i+Y_i\times h_{t-1}+p_i\right)$$

(18)

$$C_t=\sigma \left(w_c\times X_c+Y_c\times h_{t-1}+p_c\right)$$

(19)

There are bias terms $p_i$ and $p_c$, weights $w_i,w_c,Y_i$ and $Y$, where $i_t$ is the input threshold at time $t$. For each given time $t$, we can change the cell’s state using the following expression:

$$C_t=f_t\times C_t+i_t\times p_o$$

(20)

The output data for the current time step constitute the third layer, which is represented as

$$O_t=\sigma \left(w_0\times X_c+Y_o\times h_{t-1}+p_o\right)$$

(21)

When $O_t$ denotes the output threshold at time t, the cell’s output value is described as

$$h_t=O_t\times \mathit{tanh}\left(C_t\right)$$

(22)

Here, $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ represents the activation function, while $h_t$ indicates the output value of the cell at time $t$. The data undergoes processing through all three gates, resulting in the output of significant information while invalid information is discarded.

2.5. Boosted Regression Tree (BRT)

The BRT functions as a non-parametric model and does not assume prior relationships between input and objective variables [56]. Instead, it combines boosting techniques with regression trees [56]. The BRT serves as a model that enhances performance accuracy through the integration of multiple individual models. The BRT approach primarily relies on $\left(\mathrm{a}\right)$ CART regression trees and $\left(\mathrm{b}\right)$ the development and integration of multiple models through a boosting procedure, resulting in a more accurate and resilient model. Mathematically, BRT is based on sequentially adding simple regression trees:

$$f\left(x\right)=f_o\left(x\right)+\sum \left(\mu \times f_m\left(x\right)\right)$$

(23)

Here $f_o\left(x\right)$ denotes the initial prediction, m defines the no. of iterations, $f_m\left(x\right)$ is the prediction of the m-th tree, and the learning rate is $\mu$.

The BRT method addresses the limitations of a single decision tree, which constructs only the initial tree from the training data, while subsequent data is employed to develop the following trees [57]. Boosting techniques are utilized to improve the predictive performance of the regression tree. The process is similar to model averaging, where the results from multiple models are combined. However, it employs a boosting operation that incrementally adjusts the models to align with a subset of the training set [58]. The effectiveness of the BRT is significantly dependent on two regularization parameters: (i) the count of additive terms or trees ($n_t$) and (ii) the learning rate (LR). The learning rate parameter is utilized to reduce the influence of each individual tree within the model, with a range of 0.1 to 0.0001. A smaller learning rate results in a reduced loss function; however, this necessitates the inclusion of additional trees ($n_t$) in the model [59]. This methodology offers multiple benefits, such as the ability to quickly evaluate a large dataset that is less prone to overfitting.

2.6. Cascaded Forward Neural Network (CFNN)

The CFNN model represents a variant of artificial neural network (ANN) models [60]. The system utilizes a parallel information processing architecture that comprises three distinct layers of neurons: input, hidden, and output. CFNN exhibits an architecture akin to that of FFNN; however, the input signal is interconnected with each subsequent concealed layer via a weight matrix. The differentiation is found within the neurons of the concealed layer. A hidden neuron is incorporated into these networks at each subsequent stage. Each new neuron receives information from the input neurons as well as all previously activated hidden neurons before transmitting to the input of each output neuron. The input and output neurons are interconnected, in addition to the interactions occurring between the hidden neurons. All hidden layers in a CFNN, with the exception of the first hidden layer, consist of a minimum of two weight matrices. These matrices serve to control the output signal of the top layer and the input signal of the network, respectively. This topology offers increased flexibility in the training process, thereby improving the network’s ability to perform non-linear mappings. The BP learning algorithm optimizes the weight matrix and bias matrix of a CFNN throughout the training process. The objective is to align the actual output of the network closely with the predicted output, as quantified by the mean square error. Prior to modeling, it is essential to define the network topology, which encompasses the number of hidden layers and the number of neurons within each layer, for traditional neural networks like MLP. Consequently, the reliable identification of optimal design is often challenging and generally necessitates a trial-and-error approach [61]. The mathematical expression of the MLP architecture can be formulated as written as:

$$y=g^o\left({\sum }_{j=1}^k{w}_j^o{g}_j^h\left({\sum }_{i=1}^n{w}_{ji}^h{x}_i\right)\right)$$

(24)

where $g^o$ defines the activation function on the output layer and $g_j^h$ is the activation function on the hidden layer. Equation (1) reduces to the following form by adding a bias:

$$y=g^o\left(w^b+{\sum }_{j=1}^k{w}_j^o{g}_j^h\left(w_j^b+{\sum }_{i=1}^n{w}_{ji}^h{x}_i\right)\right)$$

(25)

Here b shows the weight from bias to output. If the connection made on the perceptron and multilayer network is joined, then the network with direct connection between the input layer and the output layer is formed. This formation refers to the CFNN. Mathematically then, CFNN can be expressed as:

$$y={\sum }_{i=1}^n{g}^i{w}_i^i{x}_i+g^o\left({\sum }_{j=1}^k{w}_j^o{g}_j^h\left({\sum }_{i=1}^n{w}_{ji}^h{x}_i\right)\right)$$

(26)

Here $g^i$ denotes the activation function and $w_i^i$ is weight from the input layer to the output layer. By adding bias to the input layer and the activation function of each neuron in the hidden layer, then

$$y={\sum }_{i=1}^n{g}^i{w}_i^i{x}_i+g^o\left(w^b+{\sum }_{j=1}^k{w}_j^o{g}^h\left(w_j^b+{\sum }_{i=1}^n{w}_{ji}^h{x}_i\right)\right)$$

(27)

The first step involves training cascade networks with input and output neurons, analogous to traditional networks. Training will conclude if the error is deemed acceptable following a specified number of repetitions. If not, the model will undergo re-execution at each stage by incorporating a new neuron and systematically training the network to minimize residual error [60]. The training process will persist until the error rate decreases to a level below the specified target threshold or until the rate of change diminishes.

2.7. Model Performance Evaluation

A vital step in building a model is evaluating its performance, which means comparing the models’ predictions with their actual values using statistical metrics to see how accurately the model matches the output. The following metrics are used in this research to evaluate the accuracy of the models and to compare them: R (Correlation Coefficient), RMSE (Root Mean Square Error), MAE (Mean Absolute Error), IA (Willmott’s Index of agreement) [62], ENS (Nash-Sutcliffe efficiency) [63], KGE (Kling-Gupta efficiency) [64], and the uncertainty coefficient with a 95% confidence level. These metrics can be described mathematically using the following equations:

$$R={\displaystyle \frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}\left({ObservedSPI}_{o,i}-\overline{{ObservedSPI}_o}\right)\left({ForecastedSPI}_{o,i}-\overline{{ForecastedSPI}_o}\right)}{\sqrt{\sum _{\mathrm{i}=1}^{\mathrm{N}}({{ObservedSPI}_{o,i}-\overline{{ObservedSPI}_o})}^2\sum _{\mathrm{i}=1}^{\mathrm{N}}({{ForecastedSPI}_{o,i}-\overline{{ForecastedSPI}_o})}^2}}}$$

(28)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N({ObservedSPI}_{o,i}-{ForecastedSPI}_{o,i}{)}^2}$$

(29)

$$IA=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left({ObservedSPI}_{o,i}-{ForecastedSPI}_{o,i}\right)}^2}{{\sum }_{i=1}^N{\left(\left|{ObservedSPI}_{o,i}-{\overline{ObservedSPI}}_o\right|+\left|{ObservedSPI}_{o,i}-{\overline{ObservedSPI}}_o\right|\right)}^2}}$$

(30)

$$MAE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|{ObservedSPI}_{o,i}-{ForecastedSPI}_{o,i}\right|$$

(31)

$$ENS=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left({ObservedSPI}_{o,i}-{ForecastedSPI}_{o,i}\right)}^2}{{\sum }_{i=1}^N{\left({ObservedSPI}_{o,i}-{\overline{ObservedSPI}}_o\right)}^2}}$$

(32)

$$KGE=1-\sqrt{{\left(R-1\right)}^2+{\left(\alpha -1\right)}^2+{\left(\beta -1\right)}^2}$$

(33)

$$U_{95\%}=1.96\sqrt{{Standarddeviation}^2-{RMSE}^2}$$

(34)

The predicted value of the flood index is denoted as ${ForecastedSPI}_{o,i}$ while the actual value is represented by ${ObservedSPI}_{o,i.}\overline{ObservedSPI}$ represents the mean of real values, whereas $\overline{{ForecastedSPI}_o}$ represents the mean of predicted outcomes. A total of $N$ data points have been gathered, $\alpha$ shows how different the actual and forecasted values are in terms of variability, and $\beta$ is the ratio of the two sets of mean values. The IA can take on values between 0 and 1. The best value for the ENS, which can vary from (range from $-\infty$ to +1), is 1, and it is used to compare the performance of models. This statistic ranks the model’s performance as follows: excellent (ENS > 0.75), (0.65 < ENS < 0.75), satisfactory (0.50 < ENS < 0.65), acceptable (0.40 < ENS < 0.50), and inadequate (ENS < 0.4). KGE ranges from $-\infty$ to 1, with values close to 1 indicating accurate predictions from the model.

3. Model Development

During the model development phase, several alternative models were proposed, including TVFEMD-GPR, TVFEMD-LSTM, TVFEMD-BRT, and TVFEMD-CFNN, aimed at forecasting multi-scaler SPI for the Springfield and Mackay stations. Additionally, standalone models such as GPR, LSTM, BRT, and CFNN were utilized to evaluate their efficiency in comparison. The models were executed in the MATLAB R2023a environment on a system equipped with an Intel Core i5-8400 CPU, operating at 2.80 GHz, and 8 GB of RAM. The following outlines the steps involved in the model development process:

• The TVFEMD is sensitive to noisy and non-stationary signals, particularly in the case of SPI’s. Moreover, the TVFEMD is also sensitive to the choice of hyperparameters such as the filter strength, the no. of IMFs to extract, and the regularization parameter. By acquiring the IMFs and residuals, the TVFEMD approach starts to demarcate the SPI1, SPI3, SPI6, and SPI12 data all at once. Through the use of trial and error, the optimal number of IMFs for the Springfield station was determined to be [SPI1 = 21, SPI3 = 20, SPI6 = 23, SPI12 = 20], while for Mackay it was [SPI1 = 25, SPI3 = 19, SPI6 = 25, SPI12 = 20]. Several numbers of IMFs were acquired and then selected the best numbers for which the models generated the best performance. The design parameters of the TVFEMD method during the decomposition of SPI indices are B-spline order, End flag parameter, Bandwidth threshold criteria, and no. of IMFs. When breaking down the data into individual station IMFs and residuals, the TVFEMD method’s design parameters are listed in

  Table 2. Design parameters of TVFEMD method during the decomposition of the data into IMFs and residuals for each station.

  Springfield					Mackay
  	B-Spline Order	End Flag Parameter	Bandwidth Threshold Criteria	No. of IMFs	B-Spline Order	End Flag Parameter	Bandwidth Threshold Criteria	No. of IMFs
  SPI1	26	0	0.1	21	26	0	0.1	25
  SPI3	26	0	0.1	20	26	0	0.1	19
  SPI6	26	0	0.1	23	26	0	0.1	25
  SPI12	26	0	0.1	20	26	0	0.1	20

  .
• As seen in Figure 2, the statistically significant lags of each IMF at one month, three months, six months, and twelve months ahead (i.e., t − 1) SPI for the Springfield and Mackay stations were determined using the partial autocorrelation function (PACF). Strong correlations between the IMFs at lags (t − 1) were observed.
• We subsequently fed the statistically significant deconstructed IMFs straight into the model at this point, and we used the GPR model to build the hybrid TVFEMD-GPR method, which uses the large PACF delays at (t − 1) of SPI1 to predict the SPI one month from now. To predict drought indices for the stations in Springfield and Mackay, the procedure was performed for SPI3, SPI6, and SPI12 using the TVFEMD-GPR model. In order to build models, the data was split into two sets: training and testing. Data used to train the GPR model makes up 70% of the training set, whereas data used to validate the model makes up 30% of the testing set. In addition, the models were normalized and denormalized inside the [0, 1] interval to speed up their convergence. This study developed several benchmarking models to evaluate the TVFEMD-GPR model. The hybrid TVFEMD-LSTM, TVFEMD-BRT, and TVFEMD-CFNN models were created by fusing the standalone models LSTM, BRT, and CFNN with the TVFEMD. These models were used to forecast multi-scaler SPI drought indices. The suggested modeling framework is schematically shown in Figure 3.
• Improving the model’s accuracy during development is mostly dependent on fine-tuning and adjusting the hyperparameters. Finding the best hyperparameters can be performed in a number of ways; in this case, the trial-and-error method was adopted. Several sets of combinations of these hyperparameters were used and then we selected the optimum set for which the models generated highest precision to forecast SPI indices. To find the best hyperparameters in MATLAB, the RMSE served as the convergence criterion. The hyperparameters, which include the log likelihood, basis function, kernel function, beta, iteration, and more, are given in

  Table 3. Parameter setting of the models forecasting flood index.

  Stations	Models	Tuning Parameters
  Springfield Station	GPR	Hybrid and Standalone Structure Log Likelihood = [654.5; 365.4]-SPI1: [818.6; 600.3]-SPI3: [1036.6; 768.6]-SPI6: [1370.9; 1078.8]-SPI12 Basis Function = Linear Kernel Function = Squared Exponential Sigma = [0.087; 0.154]-SPI1: [0.001; 0.116]-SPI3: [0.001; 0.094]-SPI6: [0.0018; 0.064]-SPI12 Active Set Size = 820; Verbose = 0 Optimizer = Quasi newton
  	LSTM	Hidden units = [50-Hybrid model; 10-Standalone model]-SPIs Optimizer = Adam, Verbose = 0 Gradient Threshold = 1, Initial Learn Rate = 0.005 Learn Rate Drop period = 200; Batch Size = 32 Learn Rate Drop Factor = 0.1; Epochs = 250
  	BRT	Learn rate = 0.194, Method = LSBoost N Learn = 100, Learners Weight = 0 Learner name = Tree
  	CFNN	Hybrid Structure: (20-29-1)-SPI1, (19-21-1)-SPI3, (25-21-1)-SPI6, (21-21-1)-SPI12; Standalone Structure: (1-21-1)-SPI1; (1-21-1)-SPI3; (1-21-1)-SPI6; (1-21-1)-SPI12 Validation checks = 6, Training = Levenberg-Marquadt Mu = 0.001
  Mackay Station	GPR	Hybrid and Standalone Structure Log Likelihood = [654.5; 365.4]-SPI1: [818.6; 600.3]-SPI3: [1036.6; 768.6]-SPI6: [1370.9; 1078.8]-SPI12 Basis Function = Linear Kernel Function = Squared Exponential Sigma = [0.087; 0.154]-SPI1: [0.001; 0.116]-SPI3: [0.001; 0.094]-SPI6: [0.0018; 0.064]-SPI12 Active Set Size = 820; Verbose = 0 Optimizer = Quasi newton
  	LSTM	Hidden units = [50-Hybrid model; 10-Standalone model]-SPIs Optimizer = Adam, Verbose = 0 Gradient Threshold = 1, Initial Learn Rate = 0.005 Learn Rate Drop period = 200; Batch Size = 32 Learn Rate Drop Factor = 0.1; Epochs = 250
  	BRT	Learn rate = 0.194, Method = LSBoost N Learn = 100, Learners Weight = 0 Learner name = Tree
  	CFNN	Hybrid Structure: (25-31-1)-SPI1, (19-30-1)-SPI3, (25-32-1)-SPI6, (20-31-1)-SPI12; Standalone Structure: (1-20-1)-SPI1; (1-20-1)-SPI3; (1-20-1)-SPI6; (1-21-1)-SPI12 Validation checks = 6, Training = Levenberg-Marquadt Mu = 0.001

  . The GPR model relied on these parameters. Key hyperparameters for the LSTM model include hidden units, optimizer, verboseness, batch size, gradient threshold, and epochs. The learn rate value and ensemble method (i.e., LSBoost) were the most crucial factors for BRT, whereas the number of neurons in the hidden layer and training procedure were the most critical for CFNN.

For the Mackay and Springfield stations, spanning four SPI time zones (SPI1, SPI3, SPI6, and SPI12), Figure 4 displays the training accuracy of eight models: GPR, CFNN, LSTM, BRT, and their variants TVFEMD-GPR, TVFEMD-CFNN, TVFEMD-LSTM, and TVFEMD-BRT. The models are evaluated using six performance metrics: R, RMSE, MAE, IA, and U95%. When the values of R, ENS, and IA are high, it means that there is a good relationship between the variables, and when the values of RMSE, MAE, and U95% are low, it means that there is less prediction error and uncertainty. A summary of the results showing which models performed the best at each SPI in each training period and station is presented here.

For SPI1, the best method is TVFEMD-BRT, for both the MacKay and Springfield stations, while for SPI3 and others, the best method is TVFEMD-GPR, which generated the best assessment metrics in the training period in Figure 4. Similarly, the TVFEMD-GPR acquired the best performance for the MacKay and Springfield stations to predict SPI6 as compared to other models. Again, for SPI12 forecasting, the best method appeared to be TVFEMD-GPR for the MacKay and Springfield stations in Figure 4 by acquiring R, RMSE, MAE, ENS, IA, and U95%. Thus, Mackay and Springfield stations benefit significantly from the use of TVFEMD-enhanced models, especially TVFEMD-GPR for mid- to long-term forecasts, while TVFEMD-BRT is best for short-term predictions.

4. Application Results and Analysis

Table 4. Testing performance of the TVFEMD-GPR vs. TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN models using R, RMSE, and MAE.

Station 1: Mackay					Station 2: Springfield
	R	RMSE	MAE	RAE	R	RMSE	MAE	RAE
SPI1
GPR	0.1033	0.9668	0.7528	0.9940	0.0540	0.9937	0.7708	0.9967
TVFEMD-GPR	0.7113	0.6831	0.5335	0.7044	0.7206	0.6887	0.5432	0.7024
CFNN	0.0519	0.9876	0.7690	1.0154	0.0314	1.0294	0.7984	1.0324
TVFEMD-CFNN	0.5285	0.9321	0.7297	0.9635	0.5235	0.9568	0.7453	0.9637
LSTM	0.0895	1.4475	1.2002	1.5847	0.0027	1.9384	1.7033	2.2026
TVFEMD-LSTM	0.9519	1.1795	1.0851	1.4327	0.9461	1.7038	1.6326	2.1112
BRT	−0.0167	1.0714	0.8502	1.1226	−0.0227	1.0695	0.8300	1.0734
TVFEMD-BRT	0.5936	0.7846	0.6301	0.8320	0.6206	0.7837	0.6068	0.7847
SPI3
GPR	0.6814	0.7090	0.5357	0.7250	0.6774	0.7022	0.5497	0.7260
TVFEMD-GPR	0.8105	0.5679	0.4336	0.5868	0.8197	0.5468	0.4310	0.5693
CFNN	0.6684	0.7225	0.5454	0.7382	0.6734	0.7508	0.5967	0.7881
TVFEMD-CFNN	0.7517	0.6948	0.5356	0.7249	0.5645	1.1190	0.8565	1.1312
LSTM	0.7001	1.3373	1.1497	1.5559	0.7057	1.4861	1.3127	1.7337
TVFEMD-LSTM	0.9644	1.1628	1.0748	1.4546	0.9726	1.3909	1.3189	1.7419
BRT	0.6493	0.7456	0.5755	0.7789	0.6273	0.7553	0.5830	0.7699
TVFEMD-BRT	0.6602	0.7298	0.5668	0.7671	0.6339	0.7423	0.6063	0.8008
SPI6
GPR	0.8157	0.5528	0.4054	0.5408	0.8369	0.5393	0.3981	0.5043
TVFEMD-GPR	0.9076	0.4011	0.3009	0.4014	0.9114	0.4074	0.3154	0.3996
CFNN	0.8143	0.5555	0.4135	0.5515	0.8312	0.5478	0.4071	0.5157
TVFEMD-CFNN	0.5853	0.9625	0.7781	1.0380	0.4706	1.6590	1.3601	1.7229
LSTM	0.8134	1.2459	1.0922	1.4570	0.8364	1.3798	1.2406	1.5715
TVFEMD-LSTM	0.9395	1.1841	1.0823	1.4437	0.9758	1.3148	1.2374	1.5675
BRT	0.7889	0.5899	0.4331	0.5777	0.8048	0.5916	0.4313	0.5463
TVFEMD-BRT	0.7204	0.6650	0.5240	0.6991	0.7107	0.6970	0.5580	0.7068
SPI12
GPR	0.9439	0.3223	0.2039	0.2602	0.9438	0.3497	0.2455	0.2801
TVFEMD-GPR	0.9637	0.2608	0.1768	0.2256	0.9666	0.2776	0.2019	0.2304
CFNN	0.9416	0.3294	0.2108	0.2690	0.9377	0.3696	0.2600	0.2967
TVFEMD-CFNN	0.8471	0.5493	0.4007	0.5113	0.8375	0.7352	0.5906	0.6738
LSTM	0.9460	1.0344	0.9172	1.1704	0.9372	1.4143	1.3032	1.4869
TVFEMD-LSTM	0.9801	1.0085	0.9050	1.1548	0.9655	1.4034	1.3045	1.4884
BRT	0.9247	0.3715	0.2324	0.2966	0.9350	0.3756	0.2684	0.3063
TVFEMD-BRT	0.8078	0.5944	0.4783	0.6104	0.7550	0.7056	0.5585	0.6372

evaluates the testing accuracy of eight models—TVFEMD-GPR, TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN—based on four performance metrics (R, RMSE, MAE, and RAE) for two stations (Mackay (Station 1) and Springfield (Station 2)) across four time zones (SPI1, SPI3, SPI6, SPI12). The best performing models for station 1 (Mackay) is TVFEMD-LSTM under different time zone SPI1 (R = 0.9519, RMSE = 1.1795, MAE = 1.0851, RAE = 1.4327), SPI3 (R = 0.9644, RMSE = 1.1628, MAE = 1.0748, RAE = 1.4546), and SPI12 (R = 0.9801, RMSE = 1.0085, MAE = 0.9050, RAE = 1.1548), while the TVFEMD-GPR model is best for SPI6 (R = 0.9076, RMSE = 0.4011, MAE = 0.3009, RAE = 0.4014). Correspondingly for Springfield (Station 2), the best model is TVFEMD-LSTM under SPI1 (R = 0.9461, RMSE = 1.7038, MAE = 1.6326, RAE = 2.1112), SPI3 (R = 0.9726, RMSE = 1.3909, MAE = 1.3189, RAE = 1.7419), and SPI6 (R = 0.9758, RMSE = 1.3148, MAE = 1.2374, RAE = 1.5675), while for SPI12 the best model is TVFEMD-GPR (R = 0.9666, RMSE = 0.2776, MAE = 0.2019, RAE = 0.2304). Mackay (Station 1) performed best primarily under the TVFEMD-LSTM method for SPI1, SPI3, and SPI12, and TVF-EMD-GPR for SPI6. Springfield (Station 2) showed best performances predominantly under the TVFEMD-LSTM method for SPI1, SPI3, and SPI6, with TVF-EMD-GPR performing best for SPI12. Thus, TVFEMD-LSTM generally showed superior performance across most SPI zones and both stations, except for specific cases (SPI6 at Mackay and SPI12 at Springfield), where TVFEMD-GPR excelled.

Table 5. The performance of TVFEMD-GPR vs. TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN models based on assessment metrics ENS, IA, and U95%.

Station 1: Mackay				Station 2: Springfield
	ENS	IA	U95%	ENS	IA	U95%
SPI1
GPR	0.0091	0.1707	2.6806	−0.0007	0.1261	2.7546
TVFEMD-GPR	0.5054	0.8082	1.8943	0.5192	0.8182	1.9100
CFNN	−0.0338	0.2501	2.7370	−0.0739	0.2274	2.8545
TVFEMD-CFNN	0.0790	0.7258	2.5849	0.0722	0.7165	2.6312
LSTM	−1.2208	0.4384	3.4135	−2.8075	0.4069	4.2730
TVFEMD-LSTM	−0.4744	0.6653	2.4887	−1.9417	0.5532	3.4748
BRT	−0.2167	0.3014	2.9712	−0.1592	0.2797	2.9631
TVFEMD-BRT	0.3475	0.7285	2.1753	0.3776	0.7593	2.1733
SPI3
GPR	0.4644	0.7898	1.9663	0.4585	0.7931	1.9472
TVFEMD-GPR	0.6564	0.8893	1.5745	0.6716	0.8953	1.5163
CFNN	0.4439	0.7898	2.0032	0.3808	0.7997	2.0460
TVFEMD-CFNN	0.4857	0.8625	1.9241	−0.3751	0.7018	2.9534
LSTM	−0.9053	0.5439	2.9995	−1.4254	0.5298	3.2385
TVFEMD-LSTM	−0.4404	0.6671	2.4409	−1.1245	0.6058	2.8607
BRT	0.4077	0.7856	2.0677	0.3735	0.7737	2.0943
TVFEMD-BRT	0.4326	0.7691	2.0210	0.3947	0.7599	2.0544
SPI6
GPR	0.6652	0.8926	1.5331	0.7001	0.9079	1.4957
TVFEMD-GPR	0.8237	0.9502	1.1123	0.8289	0.9534	1.1296
CFNN	0.6619	0.8937	1.5403	0.6906	0.9040	1.5192
TVFEMD-CFNN	−0.0148	0.7445	2.6080	−1.8366	0.6119	4.5025
LSTM	−0.7003	0.5950	2.7406	−0.9623	0.5867	2.9725
TVFEMD-LSTM	−0.5359	0.6485	2.5058	−0.7817	0.6354	2.7207
BRT	0.6187	0.8796	1.6360	0.6392	0.8920	1.6393
TVFEMD-BRT	0.5155	0.8071	1.8442	0.4992	0.8235	1.9306
SPI12
GPR	0.8908	0.9701	0.8938	0.8907	0.9702	0.9697
TVFEMD-GPR	0.9285	0.9813	0.7228	0.9311	0.9829	0.7695
CFNN	0.8859	0.9687	0.9123	0.8779	0.9678	1.0248
TVFEMD-CFNN	0.6829	0.9134	1.4972	0.5171	0.8994	2.0150
LSTM	−0.1241	0.6967	2.2457	−0.7866	0.6341	2.9745
TVFEMD-LSTM	−0.0684	0.7142	2.1609	−0.7591	0.6386	2.9321
BRT	0.8549	0.9600	1.0300	0.8739	0.9661	1.0418
TVFEMD-BRT	0.6287	0.8571	1.6456	0.5552	0.8295	1.9450

evaluates the testing accuracy of eight models—TVFEMD-GPR, TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN—based on three performance metrics: ENS, IA, and U95% for two stations, Mackay (Station 1) and Springfield (Station 2) across four SPI time zones (SPI1, SPI3, SPI6, and SPI12). Here we give a clear and concise explanation highlighting the best-performing models for each station across these SPI zones. For Mackay Station 1, the best method is TVFEMD-GPR for SPI1 to acquire highest ENS = 0.5054, highest IA = 0.8082, and lowest uncertainty U95% = 1.8943. Likewise, for SPI3, TVFEMD-GPR has the best ENS = 0.6564, IA = 0.8893, and lowest U95% = 1.5745. For SPI6, clearly TVFEMD-GPR is superior across all three metrics: ENS = 0.8237, IA = 0.9502, and U95% = 1.1123. Similarly, for SPI12, highest performance in terms of ENS = 0.9285, IA = 0.9813, and low uncertainty U95% = 0.7228 was achieved by TVFEMD-GPR.

For Station 2 (Springfield), the best method for all time zones is TVFEMD-GPR. For instance, the TVFEMD-GPR has the best efficiency ENS = 0.5192, IA = 0.8182, and U95% = 1.9100 (SPI1), and ENS = 0.6716, IA = 0.8953, and U95% = 1.5163 (SPI3). For SPI6, the highest values of ENS = 0.8289, IA = 0.9534, and U95% = 1.1296 were generated by TVFEMD-GPR. Similarly, for SPI12, TVFEMD-GPR is best with ENS = 0.9311, IA = 0.9829, and U95% = 0.7695. TVFEMD-GPR consistently delivered the highest efficiency (ENS), strongest agreement (IA), and lowest uncertainty (U95%) across all SPI time zones and both stations. The model demonstrates robustness and reliability for drought prediction across varying temporal scales, clearly surpassing other evaluated methods (TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN). Overall, the results from Table 5 confirm that TVF-EMD-GPR is the best-performing predictive model, highly suitable for precise drought forecasting at both the Mackay and Springfield stations.

The swarm plots in Figure 5a,b illustrate the range of actual and forecasted SPI (Standardized Precipitation Index) across various forecasting models and methods, including TVFEMD-GPR, TVFEMD-CFNN, TVFEMD-LSTM, TVFEMD-BRT, GPR, CFNN, LSTM, and BRT, at forecast lead times of 1, 3, 6, and 12 months. At both stations, TVFEMD-GPR demonstrated superior performance, exhibiting the lowest Forecast Error (FE) compared to all other models. The TVFEMD-GPR model demonstrated superior accuracy in predicting the SPI across all lead times compared to the other models. At both stations, TVFEMD-GPR demonstrated superior performance relative to all other models, including individual models (GPR, CFNN, LSTM, BRT) and hybrid models such as TVFEMD-CFNN, TVFEMD-LSTM, and TVFEMD-BRT. The predictions of TVFEMD-GPR for the SPI exhibit significant variability, demonstrating distinct trends during specific timeframes while becoming increasingly complex in others.

Figure 5 demonstrates that TVFEMD-GPR consistently yields SPI values that closely correspond with the observed data, particularly at the 1-month, 3-month, and 6-month lead times. The TVFEMD-CFNN and TVFEMD-BRT models exhibit deviations from the observed data; however, they generally uphold a satisfactory level of accuracy across all timeframes. The TVFEMD-LSTM model demonstrates a wider range in its predictions, especially at extended lead times (6 months and 12 months), suggesting increased uncertainty in its forecasting capability for longer periods. Traditional models, including GPR, CFNN, LSTM, and BRT, exhibit greater variability in their forecasts. Certain models, such as GPR and LSTM, demonstrate superior performance at shorter lead times but encounter difficulties in accurately representing the true distribution at extended lead times. The results indicate that the TVFEMD-GPR model surpasses others in aligning with observed data, while hybrid models like TVFEMD-CFNN and TVFEMD-BRT demonstrate competitive performance, especially for short-term forecasts.

The TVFEMD-GPR model exhibits enhanced efficacy in forecasting the SPI over multiple time horizons, establishing it as a dependable method for predicting precipitation anomalies at both stations. Nonetheless, the performance of alternative models, including TVFEMD-CFNN and TVFEMD-BRT, warrants consideration, particularly in contexts where computational efficiency and model diversity are critical for forecasting tasks.

An exhaustive assessment of the model’s efficacy is provided by Figure 6, which displays the empirical cumulative distribution function (ECDF) for the predicted and observed daily flood index. As shown in Figure 6, the model’s performance can be thoroughly evaluated by looking at the empirical cumulative distribution function (ECDF) for both the observed and anticipated daily flood index. The performance of the TVFEMD-GPR model closely aligns with the observed data across multiple forecast horizons (SPI1, SPI3, SPI6, and SPI12), as indicated by the ECDF curves for the Mackay station and Springfield station (Figure 6).

Across the various SPI timeframes, TVFEMD-GPR demonstrates a clear advantage by generating an ECDF curve that aligns most closely with the observed data in comparison to other models. This includes hybrid models such as TVFEMD-LSTM, TVFEMD-BRT, and TVFEMD-CFNN, as well as standalone models like GPR, LSTM, BRT, and CFNN. This indicates that TVFEMD-GPR demonstrates superior performance compared to its counterparts in effectively capturing the features of the recorded flood indices. The TVFEMD-LSTM model exhibits significant deviations from the observed empirical cumulative distribution function (ECDF) during certain forecast periods. Generally, the hybrid TVFEMD-based models show a tendency of converging to the observed distribution. The results indicate that, in the context of drought index forecasting, the TVFEMD-GPR model consistently provides more accurate and reliable forecasts compared to alternative models, thereby demonstrating its superior predictive capability.

The boxplots presented in Figure 7a,b compare the observed and forecasted Standardized Precipitation Index (SPI) values at two stations, Mackay and Springfield. Each plot contrasts the performance of different forecasting models, including TVFEMD-GPR, TVFEMD-CFNN, TVFEMD-LSTM, TVFEMD-BRT, GPR, CFNN, LSTM, and BRT.

In Figure 7a (Mackay Station), the TVFEMD-GPR model regularly yields projected values that closely align with the observed SPI, exhibiting reasonably narrow interquartile ranges and little outliers. The TVFEMD-CFNN model demonstrates robust performance, but with some variation in SPI1 and SPI12 values. Conversely, models like GPR and CFNN have greater variability in their predictions, resulting in wider boxplots and an increased number of outliers, especially for SPI values with extended timeframes (e.g., SPI12).

Figure 7b (Springfield station) demonstrates analogous trends, with the TVFEMD-GPR model surpassing the other models in forecast accuracy. The TVFEMD-LSTM and TVFEMD-BRT models closely trail, exhibiting marginally higher discrepancies in their predictions for SPI1 and SPI3. The LSTM and BRT models exhibit greater variability in their predictions, whereas the TVFEMD-CFNN model maintains acceptable accuracy with comparatively narrow boxplots.

The findings indicate that TVFEMD-GPR offers the most dependable and precise SPI forecasts for both stations, with minimum fluctuation relative to alternative models. Alternative models, like TVFEMD-CFNN, LSTM, and BRT, provide valuable forecasts but demonstrate differing degrees of forecast uncertainty, particularly concerning extended SPI values. The findings underscore the efficacy of the TVFEMD-GPR model for both Mackay and Springfield stations, while indicating potential enhancements for the other models.

Figure 8a,b display scatter plots that juxtapose observed and predicted SPI values for the Mackay and Springfield stations, respectively, utilizing multiple models: TVFEMD-GPR, TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN. In the Mackay station (Figure 8a), the TVFEMD-GPR model consistently surpasses others across all SPI time scales, with RAEs varying from 0.704 (SPI1) to 0.225 (SPI12), demonstrating remarkable accuracy for both short- and long-term predictions. In contrast, the TVFEMD-CFNN model exhibits superior RAEs, ranging from 0.832 to 0.669, whilst the TVFEMD-BRT model demonstrates inferior performance with RAEs between 1.584 and 1.457. Among conventional models, GPR and CFNN outperform LSTM and BRT; however, they remain inferior to the TVFEMD-GPR model. In the Springfield station (Figure 8b), the TVFEMD-GPR model exhibits enhanced performance, with RAEs of 0.702 for SPI1 and 0.230 for SPI12. The TVFEMD-CFNN and TVFEMD-BRT models have elevated RAEs, spanning from 0.784 to 1.732, signifying diminished prediction accuracy. Among conventional models, GPR yields more precise forecasts than LSTM and CFNN; nonetheless, it remains inferior to the TVFEMD-GPR model. The TVFEMD-GPR model consistently delivers the most precise SPI forecasts at both stations over multiple time horizons.

5. Further Discussion

The TVFEMD-GPR model was developed and evaluated in this work to forecast multistep-ahead droughts for Mackay and Springfield stations in Australia. In comparison to the TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN models, the TVFEMD-GPR model is assessed and benchmarked. Its supremacy for drought forecasting is recorded, demonstrating that the TVFEMD-GPR model has superior forecasting capacity based on several evaluation criteria.

The results of this study indicate that the TVFEMD-GPR model effectively decomposes the inputs through the TVFEMD technique, thereby enhancing the accuracy of the GPR model for drought forecasting at 1, 3, 6, and 12 months ahead. The TVFEMD-GPR model outperforms the comparable methods to forecast droughts by efficiently capturing non-linear, complex relationships and spatio-temporal dependencies in the multiscale SPI data. Moreover, the TVFEMD-GPR model excels in handling large, high-dimensional, and multi-modal SPI datasets, leading to more accurate and robust drought forecasts.

The novel TVFEMD-GPR model demonstrated effectiveness in multi-scaler SPI forecasts for Australian regions, specifically Mackay and Springfield, compared to other models. However, additional recommendations and avenues for future research require exploration. This study utilized only the significant lags of the multi-scaler SPI derived in the TVFEMD-GPR model for forecasting. Nevertheless, accuracy could be enhanced by incorporating additional climatic, meteorological, and hydrological data as input predictors. Additionally, satellite-derived data can serve as an alternative that substantially improves the forecasting capabilities of the TVFEMD-GPR model; thus, integrating more physical data components into the multi-scaler SPI forecasts may represent a viable strategy.

Another weakness of the current study is that the proposed TVFEMD-GPR model was tested using data from only two stations in Australia. Although these stations provide diverse conditions, the geographical range rests locally constrained. The limited geographical exposure could impact the generalizability of the proposed TVFEMD-GPR model; therefore, datasets from other stations are important to completely evaluate the robustness. Although the TVFEMD-GPR model is efficient in SPI forecasting, the black-box nature restricted the ability to understand the relationship and associations of the inputs during the learning process. Therefore, the hybridization of the explainable AI models (i.e., Local Interpretable Model-Agnostic Explanations (LIME)) [65] and Shapley Additive explanations (SHAP) [66] can be beneficial to provide the model’s prediction explainability and interpretability.

Even though there are advanced ML models that are frequently utilized for forecasting, their opaque nature hinders their capabilities and makes it challenging to comprehend and assess the complex interrelationships among the inputs while they learn. As a result, studying how numerical weather prediction models include machine learning can be an intriguing endeavor. Bootstrapping methods [67] and Bayesian Model Averaging [68] can also be used to optimize the TVFEMD-GPR model, which takes into account the inherent uncertainty in the model.

The TVFEMD enhances the accuracy of the GPR model by concurrently capturing non-stationary and non-linear characteristics within the drought data, while also resolving mode mixing challenges. It has been demonstrated that TVFEMD-GPR is a viable, data-driven model for hydrological and climatological sciences, capable of offering valuable insights for water resource management. This can assist Australia in developing more effective and proactive preventative measures.

6. Conclusions

A complementary data decomposition-based framework was designed to forecast multistep-ahead droughts in the Mackay and Springfield stations, Australia. The framework innovatively combines Gaussian process regression (GPR) with time-varying filter-based empirical mode decomposition (TVFEMD) to create the TVFEMD-GPR model to forecast SPI1, SPI3, SPI6, and SPI12. The findings indicate that the TVFEMD-GPR model markedly improves the predictions of multi-scaler SPI for the Mackay and Springfield stations in Australia. Furthermore, at both stations, the TVFEMD-GPR model demonstrates superior predictive accuracy compared to the TVFEMD-LSTM, TVFEMD-BRT, TVFEMD-CFNN, GPR, LSTM, BRT, and CFNN models. Valuation scores such as R, RMSE, MAE, ENS, KGE, IA, and U95% were employed to predict droughts on monthly, quarterly, semi-annual, and annual intervals. The TVFEMD segmented the input into IMFs and then fed them into the GPR model during the development phase of the TVFEMD-GPR model to finally forecast multiscale droughts.

The TVFEMD enhances forecasting accuracy by effectively addressing the non-stationarity and non-linearity resulting from the intricate and complex nature of drought conditions. The TVFEMD-GPR model developed in this study represents a cutting-edge integration of the TVFEMD and GPR models. The results indicate that the TVFEMD-GPR model demonstrates superior performance compared to the comparison models in multi-scale drought forecasting for both stations. The proposed TVFEMD-GPR model can be applied in environmental, hydrology, climate change, renewable energy, and agriculture sectors in the future. Its implementation aims to enhance decision-making processes and broaden its potential and scope. As the future direction, the authors intend to further incorporate climate projections, along with other hybrid methods.