H³O-LGBM: hybrid Harris hawk optimization based light gradient boosting machine model for real-time trading

Abstract

The gross domestic product (GDP) of a country is mainly dependent on its trade and external sector which improves the country's income. According to FY2021–2022, India's nominal GDP is estimated to be 3.12 trillion US dollars. Overall exports and imports have a year-over-year increase of 49.6% and 68% respectively. Machine learning techniques have the potential to improve India's Current gross value by up to 15% by the year 2035. The integration of data, Technology, and talent helps to create intelligent models that enhance artificial intelligence growth. This paper presents an optimized light gradient boosting machine (Light GBM) model using the hybrid Harris hawk optimization (H³O) algorithm for trade forecasting. The overfitting problem in the conventional Harris Hawk Optimization is overcome using the exclusive feature bundling (EFB) and the gradient-based one-side sampling (GOSS) methodologies. The H³O optimization algorithm offers fast convergence by optimizing different lightGBM parameters such as a number of training iterations, maximum depth, minimal data in the leaf, etc. To improve the performance, one step further, the residual errors of the optimized lightGBM model are corrected using the Markov Chain model. The main aim of the optimized lightGBM model is to extract the crucial input values of certain variables such as imports and exports of goods and services, service trade, and merchandise trade and predict the price movement decision. The proposed model identifies the interrelationship with the external market and future market growth along with analyzing the variation in market conditions. The prediction decision is mainly to hold, but, or sell the stocks. When evaluated using the precious metal price forecast and stock market datasets, the proposed methodology shows that the hybrid approach can enhance the prediction performance. The results show that the input parameters were efficient in predicting the economic growth regarding the Intermarket trading system (ITS) and services with higher accuracy.

Introduction

Trade is a fundamental economic term that involves products and services, buying and selling, with a buyer compensating a seller or the exchange of products or services between parties (Zreik 2020). Trade data consists of imports, trade balance, and exports. In general, international commerce refers to a country’s economic activity that is linked to another country’s economic relationship. A trade balance of the country is comprised of a succession of such actions. A country's trade volume reflects the aggregate influence of its macroeconomic policies. To assess the impact of international trade policy, statistics on the country's imports and exports, as well as the long-run equilibrium connection between these two variables, are required. Because changes in a country's export and import demand do not occur fast, there is always a time lag impact on trade volume and so trade data necessitate deep study (Sun et al. 2018). The emergence of mobile payment services in India to provide an instantaneous, 24/7 service and electronic fund transfer through networks using mobile phones was initiated in 2010 by the National corporation of india (NPCI) (Jakhiya et al. 2020). Due to the cooperative functioning of banks, mobile service providers, and online payment gateways, there is rapid progress in payment services since 2013 in India. In addition to this, it is expected that online transactions will reach over INR 1 trillion. After demonetization, India surpasses China in mobile wallet adoption with 55.4% according to survey respondents. Different mobile payment modes available in India are digital wallets, point of sale (POS), and unified payment interface (UPI); these payment systems are linked to credit card/debit cards for money transactions. Nowadays, digital wallets are becoming the leading factors used broadly for daily transactions at petrol stations, grocery shops, tea stalls, auto-rickshaws, and supermarkets. The functioning of the Indian payment industry is mostly restricted by cash-based transactions. Some of the benefits of using mobile-based payments are: saves time, reduces the risk of theft, avoids illegal cash float, and promotes loan repayment installment.

The majority of nations confront trade reversal restrictions. Furthermore, the latest epidemic has halted worldwide trade. However, the global pandemic cannot be seen as the particular reason for the international trade performance has deterioration, rather statistics show that most nations have experienced a decline in international commerce since the first half of the past decade. A thorough examination of global trade statistics reveals that trade as a share of global gross domestic product (GDP) falls dramatically from 2011 to 2016 and then begins to increase again. Throughout the decade, India has encountered a number of trade-regressive difficulties on the global stage (Das 2022). The model of trade and innovation is one example that predicts that positive stacks in exports help enterprises become more inventive and productive. The related rent from a company's innovation effort enhances the market size impact refers to a firm's market size. The GDP of a nation is a fundamental measure of the country's economy. GDP is the monetary value of all products and services formed and exported and imported over a certain time (Aslam and Awan 2018). This is calculated based on existing GDP. An enhancement in GDP indicates that a country's economy is growing, and this statistic is seen as a critical benchmark for the country's economy. The United States has the world's highest GDP, followed by Japan, the United Kingdom, India, China, Germany, and France. In most circumstances, calculating GDP is difficult. GDP is simply the revenue made and the expenditure spent by a nation, and imports and exports play a significant influence in determining a country's GDP as it relates to countries. The overall expense computed for every final product and service is called GDP which also incorporates additional details such as gross investments, government purchases, exports, imports, etc. (Belmechri, 2021).

Technologies play an important role in our daily lives. They have transformed nearly every area of people’s lives, including investing and financial trading. Investing and trading in the stock market has never been easier, and technological improvements have paved the way for Machine learning to be involved in investment operations and many trading. The objective of Machine learning, which replicates human intellect, is for robots to learn and solve problems in the same way that people do, but at a lower cost and in less time (Ellaji et al. 2021). Predicting international trade patterns is critical since these are fundamental factors of a country's economic development, growth, and macroeconomic stability (Saif et al. 2021). Exports also provide salary and employment in the domestic economy, therefore forecasting future export patterns or trends in international trade is a top concern for policymakers all over the world. Traditionally, trade policy research has concentrated on evaluating and projecting exports using a variety of classic statistical and econometric methodologies. In recent years, there has been widespread acceptance of machine learning and artificial intelligence algorithms that outperform classic econometric models in terms of forecasting and efficiency accuracy. Machine learning is a revolutionary and adaptable artificial intelligence method utilized for problem-solving and sophisticated statistical analysis. Machine learning is a massive data-driven technique for modeling unsupervised or supervised connections (Mirnaghi and Haghighat 2020).

More than ever before, the current generation utilized digital apps, which enhance the transformation and demand processes in practically every field and business. It has also had an impact on financial investment and trading (Talwar et al. 2021). Trading or investing in financial markets just takes a minute to invest, trade, and access utilizing a variety of digital platforms in a few steps. One reason for the ease of access and high demand is that there is a lot of emphasis on utilizing Machine learning to detect, execute, automate, and analyze successful transactions within a second of the split. The exciting trading path is similarly comparable, beginning with corporate finance courses and progressing to become fascinated with relevant materials and subjects before trading on a brokerage platform. Facing and understanding several problems and obstacles led to the completion of work connected to an important and appealing issue, trade and machine learning. The information gathered to get a better understanding of the subject is supplemented with more in-depth resources to get to the point and cross the gap between finance and information technology (Dagiene et al. 2022). The existing studies made use of the economic approaches to predict the financial market rate and these models have one major drawback: they model the complex and nonstationary nature of the stock market via linear and stationary assumptions (Sokolov-Mladenović et al. 2017; Li et al. 2021). These models mainly resulted in poor market prediction results. To minimize these complexities, in this paper we present an optimized light gradient boosting machine (Light GBM) architecture for financial market prediction. The major contributions of this paper are delineated as follows:

• This paper presents an H³O optimized lightGBM architecture for increasing the prediction performance of the stocks and exchange funds. The proposed model decomposes the actual financial time series data into different components to predict the buy, sell, and hold points for the stock exchange dataset.

• In order to address the complex optimization of the stock market future price forecasting and trading, the Harris Hawk Optimization is integrated with the Elite fractional derivative mutation strategy to form the Hybrid HHO optimization algorithm also known as the H³O algorithm and its main role is to improve the exploitation capability of the lightGBM architecture.

• The H³O optimization algorithm is mainly used to optimize the crucial parameters such as maximum tree depth, minimum data in leaf, number of training iterations, and number of leaves to minimize the overfitting capability of the Light GBM architecture.

• The Markov chain model is used in this study to implement vectorized backtesting trading strategy and to identify the stock's future price log returns.

• The findings obtained suggest that the proposed model is effective in providing improved trading performances when compared to the existing methodologies. The effectiveness of the proposed model is validated using two datasets, namely the precious metal price forecast and the stock market.

The rest of this paper is arranged accordingly. Section 2 presents the existing literary works and Sect. 3 describes the overall working of the proposed framework. Section 4 displays the experimental results in detail and Sect. 5 concludes the paper.

Related works

HAGEMANN et al. (2019) illustrated a hybrid artificial intelligent system for designing the highly-automated production systems based on real industrial scenarios. The production system design was improved for the automotive industry during the process complexity and product variations. The missing data connection problems were overcome by using a hybrid approach. The limitation was that the analysis of dataset issues was performed at various levels of data quality. Eberhard et al. (2019) proposed to predict the trading interaction with the online marketplace via three diverse networks (a) trading (b) online (c)location-based networks. In online-based networks, the user data was acquired through specialized bots. The trading network purchases were extracted and the information was crawled into the online networks. For predicting the trading interactions, 57 homophilic and topological features were utilized and generated in the various constellations. The results showed that the accuracy based on supervised learning was 92.5%.

Zhang et al. (2022) discussed the online trading issues based on available products. Here, numerous time series for search issues during the initial period were studied. The product employed in the study was homogeneous and the price rate was investigated for every period to decide the total number of selling products that were available during trading. The significant intention of this paper was to enhance the total revenue. In addition to this, an online technique was presented to evaluate the competitive ratio. Finally, the experimentation was carried out to determine the lower bound as well as the competitive ratio. Tenyakov et al. (2017) presented the Kalman and hidden Markov model (HMM) filtering method to implement the pairs trading in real-time. The Kalman and the HMM design methods were blended together for providing a powerful approach to the actualization of pairs-trading. Then, the new hybrid design was outperformed in every individual filtration technique. But, this paper failed to explain the production of filtered metrics.

Fister et al. (2021) explained the two robust long short-term memory (TRLSTM) techniques for stocks trading. In this paper, two LSTM neural networks were used for trading stocks and the trading performances were compared. The tests were conducted to prove that the LSTM was a robust tool and very reliable. The experimental results showed that both LSTM networks were outperformed the other state-of-art methods. Meanwhile, this technique failed to implement in the real-time trading system. Taghian et al. (2022) formulated Deep reinforcement learning (DRL) techniques to learn the trading asset-specific rules. Different feature phases were extracted in the DRL design. The design performances based on the various input representations were evaluated and also the asset situations were analyzed. The results revealed that the scheme outperformed the other designs. However, reducing the portfolios was hard for human experts.

Shen et al. (2021) elaborated on the long short–term memory(LSTM) recurrent neural networks (RNN) based on effective multinational trade forecasting. The LSTM-based multivariate approaches were employed for providing efficient trade forecasting and extracting the temporal changes with trade data. In this paper, this approach acquired perfect performance of forecasting in the foreign trade information. Meanwhile, this technique does not describe the various economic perspectives. Samuel et al. (2022) proposed an energy trading system based on blockchain and artificial intelligence (AI) for homes. To create the blocks and select the miners proof-of-computational closeness (PoCC) consenses are developed. In the distributed trading environment, existing energy pricing policy issues are solved by the developed analytical energy pricing policy. During energy trading, privacy is maintained by dynamic multi-pseudonym. The cloud servers’ computation offloading is performed by an improved sparse neural network (ISNN). In INSS, the error convergence rate is accelerated by the Jaya optimization algorithm which minimizes the parameters used for training.

Trading system design

The overview of the proposed design is presented in Fig. 1. The H³O optimization technique achieves rapid convergence by optimizing several lightGBM parameters such as the number of training iterations, maximum depth, minimal data in the leaf, etc. The main aim of this work is to improve the prediction accuracy by using the Light GBM and Harris Hawk optimization algorithm for stock market prediction is proposed. The proposed method is completely data-driven and needs fewer assumptions. Based on the input parameters, the proposed model generates three outputs namely buy, hold, or sell. The two main steps in the proposed model are training and prediction. Initially, the dataset is partitioned into two, where 80% is used for training whereas the remaining 20% is used for testing. To examine the prediction model’s practicability, with the use of prediction results the trading strategies are generated. Then the comparison of trading results is made with other strategies. For the prediction model, the inputs such as stock features price information, various external market information, detected structural breakpoints, and technical indicators are used. Then the financial performance and the prediction accuracy are evaluated. According to the daily closing price and asset’s log return, three labels such as “Buy”, “Sell”, and “Hold” are used for labeling every data point. The generated one-day ahead prediction is utilized for the formulation of trading strategies. This is formulated based on the data labeling. To increase performance even further, the residual errors of the optimized lightGBM model are rectified using the Markov Chain model. Then the comparison is made with formulated strategy and other several common trading strategies to examine the financial performance. The data points in the test set are labeled with the use of prediction output then the trading strategies are generated. With the generated strategies the trades are started and the performance is compared with various conventional strategies.

Fig. 1.

Outline of the proposed methodology

The purpose of this paper is to implement the Light BGM with optimization. The real-time trading is satisfied using the H³O optimized lightGBM architecture because it requires minimal assumptions and is completely data-driven. The existing model primarily models stock trades in the portfolio based on the portfolio's overall behavior (Taghian et al. 2022). The proposed model is based on the following assumptions: The way each stock in the portfolio is traded is determined by its performance. The common strategy followed by existing models for stock prediction in the portfolio is as follows. The objective of a common strategy is to capture the stocks’ general or average behavior held in the portfolio. The complexity is reduced by this method, which means, for a wide range of stocks single-model is dedicated hardware. The disadvantage of this single model is, that for a complete update the sequence of underlying financial data should be needed, the contradictory behavior of two or more stocks may lead to canceling each other.

Formulation of the H³O optimization algorithm

The H³O optimization algorithm is formulated by integrating the Elite fractional derivative mutation (EFDM) strategy to improve the exploitation capability of the Harris Hawk optimization (HHO) algorithm.

Harris hawks optimization-based elite fractional derivative mutation algorithm

Harris Hawk optimization (HHO) is a population-based gradient-free optimization algorithm, simulated by prey searching capability, surprise attack, and diverse chasing styles of hawk (Guo et al. 2022). The HHO algorithm solves multi-objective optimization problems, local optima issues, and convergence problems. However, to further enhance the effectiveness and performance of an algorithm, the elite fractional mutation strategy is introduced with HHO thus enhancing exploitation capability. The adoption of this strategy in HHO is named Harris hawk's optimization-based elite fractional derivative mutation (HHO-EFDM) algorithm. Like other optimization algorithms, the HHO algorithm initializes the population members randomly as $X^{(0)}=\left(P_1^{(0)},P_2^{(0)},P_3^{(0)},...,P_n^{(0)}\right)$ here $n$ depicts the total hawk population; $y^{\mathit{th}}$ population individual is given by $P_y^{(0)}=\left\{p_{y1}^{(0)},p_{y2}^{(0)},...,p_{\mathit{yh}}^{(0)}\right\}$, where $h$ represent decision variable dimension. The two significant steps in HHO are exploitation and exploration which are numerically described as follows,

Phase I: Exploration

In HHO, the hawk population is considered as a candidate solution in which they search the prey randomly on the search space. The $y^{\mathit{th}}$ individual in the population is updated as,

$$P_y^{(x+1)}=\left\{\begin{array}{c}{P}_{\Re }^{(x)}-R_1\left|P_{\Re }^{(x)}-2R_2{P}_y^{(x)}\right|,d_1\ge 0.5\\ P_Q^{(x)}-P_s^x-R_3\left(LB+R_4\left(UB-LB\right),d_1<0.5\right)\\ \end{array}\right)$$ 1

The term $P_y^{(x+1)}$ indicates hawk’s position vector for next iteration, $P_{\Re }^{(x)}$ depicts randomly chosen hawk, $P_y^{(x)}$ represents position vector of hawk in current iteration, $P_Q^{(x)}$ signifies position of prey and $P_s^x$ implies mean position of $x^{\mathit{th}}$ generation hawk. Also,$R_1$, $R_2$, $R_3$ and $R_4$ signifies random numbers; $\mathit{UB}$ and $\mathit{LB}$ denotes upper bounds and lower bounds of the decision variable.

Phase II: Exploitation

In this phase, the hawk approaches the prey by surprise and pounces, while the prey tries to escape from a such dangerous situation. But, the prey will lose its energy when trying to escape from the sight of the hawk. This escaping behavior of prey is modeled in two ways as soft besiege $(\epsilon \ge 0.5)$ and hard besiege $(\epsilon <0.5)$ with progressive rapid dives. This process is updated based on the below expression as follows,

$$P_y^{(x+1)}=\left\{\begin{array}{c}{P}_Q^{(x)}-P_y^{(x)}-\epsilon \left|S_j·P_Q^{(x)}-P_y^{(x)}\right|,\left|\epsilon \right|\ge 0.5,d_2\ge 0.\\ P_Q^{(x)}-\epsilon \left|S_j·P_Q^{(x)}-P_y^{(x)}\right|,\left|\epsilon \right|<0.5,d_2\ge 0.5\\ \end{array}\right)$$ 2

$$P_y^{(x+1)}=\left\{\begin{array}{c}Mf(M)<f(P_y^{(x)})\\ Nf(N)<f(P_y^{(x)})\\ \end{array}\right),d_2<0.5$$ 3

In the above equation,$\epsilon =2{\epsilon }_0\left(1-x/S\right)$, the uniformly distributed random number is represented as ${\epsilon }_0$ in which they lie within the interval [− 1,1], the total and current iteration is depicted as $x$ and $S$ respectively, prey’s jump strength is denoted as $S_j=2\left(1-R_5\right)$ and the random numbers $R_5$ and $d_2$ lies between [0,1].

Elite fractional derivative mutation strategy

The optimal solution highly relies on the exploitation capability of the swarm intelligence algorithm greatly. To efficiently enhance the exploitation capability of the algorithm, the features of fractional order derivatives such as memory, storage and heritability are utilized and thereby preventing premature convergence problems. Initially, the n population individuals are organized based on the fitness value as excellent to poor; then initial $E^{\mathit{th}}$ individuals are chosen as elites set $P_{\mathit{ELITE}}^{(x)}=\left\{P_{(1)}^{(x)},P_{(2)}^{(x)},...,P_{(E)}^{(x)}\right\}$, here $E=\left[\frac{n}{2}-\frac{n-2}{2}·\frac{x}{n}\right]$ in which they lower from $\frac{n}{2}$ to 1and $\left[·\right]$ represent integer function. By the execution of fractional derivative mutation for $E$ elites, the exploitation capability of the HHO algorithm will increase. In order to build appropriate elite mutation, 1D $\alpha$ order GL fractional derivative for the function $g\left(y\right)$ is described as,

$$G_{\vartheta }^{\alpha }g\left(y\right)=e^{-q\vartheta \alpha }\underset{\left|k\right|\to 0}{lim}\frac{{\sum }_{t=0}^{+\infty }{(-1)}^t\left(\begin{array}{c}c\\ t\\ \end{array}\right)g\left(y-tk\right)}{{\left|k\right|}^{\alpha }}$$ 4

$$\left(\begin{array}{c}c\\ t\\ \end{array}\right)=\frac{\alpha \left(\alpha -1\right)...\left(\alpha -t+1\right)}{t!}$$ 5

The term $\vartheta \in (-\pi ,\pi ]$ and $\left(\begin{array}{c}c\\ t\\ \end{array}\right)$ depicts a binomial coefficient. When $\vartheta =\pi$ it is subject to backward GL derivative and when $\vartheta =0$ it is subject to forwarding GL derivative. The truncation of order $\delta$ with respect to both left and right derivatives with the limit $k$ as $k_1$ and $k_2$ is obtained as,

$$L{G}_g^{\alpha }g\left(y\right)=\frac{1}{k_1^{\alpha }}\sum _{t=0}^{\delta }{(-1)}^t\left(\begin{array}{c}c\\ t\\ \end{array}\right)g\left(y-t{k}_1\right)$$ 6

$$R{G}_g^{\alpha }g\left(y\right)=\frac{1}{k_2^{\alpha }}\sum _{t=0}^{\delta }{(-1)}^t\left(\begin{array}{c}c\\ t\\ \end{array}\right)g\left(y-t{k}_2\right)$$ 7

With an increasing number of iterations, step size of mutation $(z_{(p,q)}^{(x)})$ will decrease and the algorithm attains an optimal solution. According to this, the mutation step sizes $k_1$ and $k_2$ in terms of $p^{\mathit{th}}$ individual and $q^{\mathit{th}}$ dimension are defined as,

$$k_1=\frac{U{B}_{(q)}^{(x)}-z_{(p,q)}^{(x)}}{\delta +1}·\left(1-\frac{x}{S}\right)$$ 8

$$k_2=\frac{z_{(p,q)}^{(x)}-L{B}_{(q)}^{(x)}}{\delta +1}·\left(1-\frac{x}{S}\right)$$ 9

Moreover, a greedy selection strategy, an algorithmic paradigm is employed to better conserve the optimal individuals from the actual elite individuals. The annualized return is set as the fitness of the H³O optimization algorithm and it is computed as shown below.

$$X=\left({\left(\frac{C_{\mathit{total}}}{C_{\mathit{initial}}}\right)}^{\frac{1}{N}}-1\right)\times 100$$ 10

where X represents the annual return, $C_{\mathit{total}}$ is the total capital, $C_{\mathit{initial}}$ is the initial capital, and N is the total number of years.

H³O algorithm

Figure 2 illustrates the combined flow diagram of Light GBM and Harris hawk optimization-based elite fractional derivative mutation (HHO-EFDM) framework. Initially, the precious metal price forecast and stock market datasets are trained using the light GBM model. In this, the increased number of data dimensions reduces the accuracy of prediction results so that it infuses GOSS and EFB concepts. Here, the optimal nodes are split via variance gain using GOSS and the training process is accelerated using EFB. The light GBM model enhances computational effectiveness and minimizes loss function without influencing prediction accuracy. However, to further increase the system performance and to neglect the losses in effective means, the output of the light GBM model is introduced into the HHO-EFDM algorithm. In this, the exploration capability of HHO is further improved by adopting an elite fractional derivative mutation strategy. The HHO-EFDM algorithm searches for better solutions iteratively until it reaches an optimal solution. Thus, the HHO-EFDM algorithm ensures an optimal solution by performing repetitive iterations.

Fig. 2.

Combined flow diagram of Light GBM and HHO-EFDM

LightGBM for trade prediction

Each stock adopts a proposed strategy to the stock's unique characteristics. For each stock company, only one model is initialized, trained, and then simulated. So, various financial instruments are allowed by this proposed model like cryptocurrencies, commodities, and bonds. The advantage of this proposed model is that it is less time-consuming when compared to the existing models and provides various financial instruments which are distributed over many computers. To yield better performance and to eliminate different dimensions’ effects, the normalization is done from the range 0 to 1, which is shown below,

$$\tilde{y}=\frac{y-miny}{maxy-miny}$$ 11

The data’s normalized value is denoted as $\tilde{y}$, the data’s true value is denoted as y, and the minimum and maximum data values are denoted as min y and max y. One of the most popular classifier algorithms is the Gradient boosting decision tree (GBDT) (Chen et al. 2019). Let us consider the training set $\left\{\left(y_1,z_1\right),\left(y_2,z_2\right),M,\left(y_o,z_o\right)\right\}$, the data samples are denoted by $y$, the class labels are represented by $z$ and the estimated function is represented by $G\left(y\right)$. The main objective of the GBDT is to reduce the loss of functions $M\left(z,G\left(y\right)\right)$:

$$\widehat{G}=AR\underset{G}{\mathit{MIN}}{F}_{y,z}\left[M\left(z,G\left(Y\right)\right)\right]$$ 12

Then, the GBDT iterative criterion (ke et al. 2017) is acquired by employing line search for reducing the loss of functions

$$G_n\left(y\right)=G_{n-1}\left(y\right)+{\lambda }_n{i}_n\left(y\right)$$ 13

From the above equation, ${\lambda }_n=AR\underset{\lambda }{\mathit{MIN}}{\sum }_{j=1}^{o}M\left(z_j,G_{n-1}\left(y_j\right)+\lambda i_n\left(y_j\right)\right)$, the iteration number is indicated by $n$, the base decision trees are represented by $i_n\left(y\right)$. The GBDT's accuracy and efficiency cannot produce satisfactory results if either the feature dimension or the total number of samples is increased.

The ensemble algorithm is the GBDT then the base classifiers are considered as the decision tree, then the split point’s identification is the main cost in decision tree learning. The highly effective GBDT employing the exclusive feature bundling (EFB) and the gradient-based one-side sampling (GOSS) is known as LightGBM. In GBDT, for splitting every node commonly information gain is utilized. LightGBM utilizes GOSS for determining the split points through computing the variance gain (Sun et al. 2020). Then the mathematical expressions for splitting the points are expressed as;

$$W_k\left(e\right)=\frac{1}{o}\left(\frac{{\left({\sum }_{y_j\in B_m}{h}_j+\frac{1-b}{c}{\sum }_{y_j\in B_m}{h}_j\right)}^2}{o_m^k\left(e\right)}+\frac{{\left({\sum }_{y_j\in B_s}{h}_j\_+\frac{1-b}{c}{\sum }_{y_j\in C_s}{h}_j\right)}^2}{o_s^k\left(e\right)}\right)$$ 14

From the above equation, $B_m=\left\{y_j\in B:y_{\mathit{jk}}\le e\right\}$, $B_s=\left\{y_j\in B:y_{\mathit{jk}}>e\right\}$, $C_j=\left\{y_j\in C:y_{\mathit{jk}}\le e\right\}$,$C_s=\left\{y_j\in C:y_{\mathit{jk}}>e\right\}$, the loss function negative gradients are represented by $h_j$, for normalizing the gradients $\frac{1-b}{c}$ is utilized.

The higher-dimensional features are sparse and the several sparse features are exclusive. The LightGBM has bundled the exclusive feature into a single feature. The feature scanning technique is modeled for constructing identical feature histograms with feature bundles. The lightGBM computational complexity is minimized $P\left(D{}_a\ast F_e\right)$. The LightGBM is the efficient GBDT implementation through EFB and GOSS for enhancing the computation efficiency. The GOSS is utilized for splitting the optimum node by computing the variance gain. The EFB is used for accelerating the training mechanism of GBDT by bundling several exclusive features into lower dense features. The LightGBM design $G_N\left(y\right)$ is acquired by the weighted combinations method.

$$G_v\left(y\right)=\sum _{n=1}^v{\lambda }_n{i}_n\left(y\right)$$ 15

The maximum number of the iteration is denoted by $N$, base decision trees are denoted by $i_n\left(y\right)$ To evaluate the performance of finance and accuracy prediction of the proposed approach with the use of assets’ daily closing price labeling is done to all data points like “Hold”, “Buy”, or “Sell”. Here, “Buy” and “Sell” are labeled in pair-wise form by strategy marks. If the time p is labeled as “Buy” then the price of daily closing time p + 1 is higher when compared to time p but “Sell” is used to label p + 1 time. Otherwise, the term “Hold” is used to label the time p, then to reach p + 1 the algorithm proceeds. Based on the data labeling protocol, the trading strategies are created from the prediction results. Based on generated strategies, the trades are started after the automatic labeling. According to the predicted labels, stock features are bought, sold, or hold. The stock features are bought at a timestamp when the predicted label is “buy” with all available capital. The stock features are sold at timestamp when the predicted label is “Sell”, and no transactions are done when the predicted label is “Hold”. The initial capital of this simulation is set to $10,00,000.

Stock future price log returns using the Markov chain

Open price represents the price at which the security initially trades (i e. morning) at the start of the business day. In a similar way, close price depicts the price paid at last (i e. evening) during usual business hours. Moreover, the closing price adjusted for right offerings, dividends, and stock splits is referred to as the adjustable close price. These adjustments in closing price make things easier to estimate the stock performance. Analogously, the high price and low price defines the highest and lowest stock prices traded during business hours. The number of transactions on a particular day is found by adding the total transactions during business hours. The adoption of these six fundamental financial data aspects develops some technical indicators thereby expanding the dataset and adding some more data content to the dataset. The obtained prices exhibit a high level of multi-collinearity. This will severely affect the considerations of traditional regression analysis and thereby raising the question of whether the traditional strategy is feasible or not. Subsequent to the completion of the online training process, the mechanical trading system (MTS) simulation is carried out with its generated array. At last, the solutions for the concluded out-of-sample instances are stated and visualized to the end-users.

A mechanical trading system (MTS) operates as a backtesting strategy based on the Intermarket trading system (ITS) and the stock portfolio to buy, sell or hold. Depending on the financial information and real market, the stocks are downloaded from the web gateway. Generally, the MTS is utilized in two different ways: (i) daily basis (sequential) and (ii) one-time deal (only once at the time when trading is completed). Because of certain discomforts, this paper utilized the Markov chain to implement vectorized backtesting, a backtest trading strategy. The assumptions made for MTS are described as follows: (i) MTS avoids dealing with stock market short selling, (ii) acquiring close price just a moment prior to closing market and this facilitates to carry out the orders, (iii) the requested market orders are executed by MTS when the market is liquid, (iv) the market is fairly large to execute the orders so that it does not influence the market prices and predicted market behaviors, (v) large stocks are mostly preferred for purchasing when the buy signal is triggered and the whole stocks are sold when they sell trading signal is triggered; thus, after purchasing, several free cash assets persist in the stock portfolio and (vi) no transaction is permitted after the apparatus is purchased or sold in N-days time frame. The Markov chain mainly decomposes the actual stock price into different frequencies to reflect the price variations in the stock market. The Markov chain model can effectively obtain the different factors to predict the stock futures to yield meaningful insights at different time intervals. The Markov chain is a completely random process with no after-effects that are used to solve prediction issues with volatility (Gao et al. 2021). With conditions known at a given moment, the Markov chain solves the probability distribution at the following moment. The situation in the next moment is related only to the present moment and has nothing to do with other moments. The data obtained from each stock market is classified into three groups namely to form a 3 × 3 matrix with decrease, increase, and stable values. The Markov chain's residual correction stages are as follows.

Step 1: Solve the error matrix $\overline{B}\left(p\right)$ using the optimal neural network prediction matrix $\stackrel{⌢}{B}\left(p\right)$ and the real value matrix $B\left(p\right)$.The error matrix is defined as follows:

$$\overline{B}(p)=\left[\begin{array}{cc}-m^{(1)}(2)& 1\\ \begin{array}{c}-m^{(1)}(2)\\ \begin{array}{c}\cdots \\ -m^{(1)}(p)\end{array}\end{array}& \begin{array}{c}1\\ \begin{array}{c}\cdots \\ 1\end{array}\end{array}\end{array}\right]$$ 16

The time series input is considered as the state s which arrived at time p. The state s residual interval median value is taken as $\left[r_{s-},r_{s+}\right]$ with an optimal predictive value ${\widehat{a}}^{(0)}(p)$.${\widehat{a}}^{(0)}$ is the actual sequence modeled for the Markov chain model. Finally, the optimal predictive value of the Markov chain is formulated as $\widehat{B}(p+1)$ as shown in the below equation.

$$\widehat{B}={\widehat{a}}^{(0)}(p)\left[1+\frac{r_{s-},r_{s+}}{2}\right]$$ 17

Equations (17) and (18) are used to calculate the error matrix's mean value $\overline{Y}$ and the standard deviation $R$. Using the mean–variance approach, calculate the error state interval.

$$\overline{Y}=\frac{1}{m}\sum _{p=1}^m\overline{B}\left(p\right)$$ 18

$$R=\sqrt{\frac{1}{m-1}}\sum _{p=1}^m\left(\overline{B}\left(p\right)-\overline{Y}\right)$$ 19

Step 2: Using the obtained error state interval in Step 1 and Eq. (20), compute the matrix of transition probabilities.

$$t_{\mathit{ji}}^{\left(p\right)}=\frac{m_{\mathit{ji}}^p}{M_j}$$ 20

where $t_{\mathit{ji}}^{\left(p\right)}$ denotes the transition probability matrix and $M_j$ denotes the total number of occurrences of error in state $j$ and $m_{\mathit{ji}}^p$ is the amount of error in the state $j$,$p$ step is used to move to the state $i$.

Step 3: Using Eq. (21), compute the state vector $T\left(p\right)$ at $p$ steps. The state vector is in the range of 0–1. Determine the state interval at $p$ step.

$$T\left(p\right)=T^{\left(0\right)}\times T^{\left(p\right)}=T^{\left(0\right)}\times {\left(T^{\left(1\right)}\right)}^p$$ 21

$T^{\left(0\right)}$ represents the starting state vector matrix, $T^{\left(1\right)}$ represents the one-step transition probability matrix, and for computation $T^{\left(1\right)}$ is presented in Eq. (22).

$$t^{\left(1\right)}=\left[\begin{array}{c}{t}_{11}{t}_{12}\dots t_{1i}\\ t_{21}{t}_{22}\dots t_{2i}\\ ⋮⋮⋮⋮\\ t_{j1}{t}_{j2}\dots t_{\mathit{ji}}\\ \end{array}\right]$$ 22

Step 4: Using the transition probability matrix and state vector acquired in Steps 2 and 4, calculate Eqs. (23) and (24) for the residual correction value at the projected time.

$$N\left(p+1\right)=n_1{u}_1+n_2{u}_2+....+n_m{u}_m$$ 23

$$n_j=\frac{x_j}{x_1+x_2+\dots +x_m}$$ 24

where $n_j$ represents the state weight, $u_1$ represents the transition interval's midpoint, and $x_j$ represents the state probability matrix's $j$ ^th probability value.

Step 5: Using Eq. (25), calculate the residual correction value after the Markov chain correction in Step 4.

$${\widehat{y}}_2^{\left(0\right)}\left(p+1\right)={\widehat{y}}_1^{\left(0\right)}\left(p+1\right)+N\left(p+1\right)$$ 25

${\widehat{y}}_2^{\left(0\right)}\left(p+1\right)$ represents the updated value at the time $p+1$, whereas ${\widehat{y}}_1^{\left(0\right)}\left(p+1\right)$ represents the unrevised value at the time $p+1$.

The error reference matrix is used to model the stock future price log returns and the error points mainly to indicate the change in the stock market return volatility. The stock market state is represented here using the dummy variable created. For each instance, the error points are continuously monitored and the value of the binary variable remains unaltered. If an error occurs, the binary variable is changed which represents the stock market shift. For every sample in the dataset, this step is repeated. The M error points represent the M + 1 stock market condition values.

Result and discussion

The stock market futures are predicted using the proposed methodology and the efficiency is tested by comparing it with different state-of-art techniques. The evaluations are obtained for both prediction and financial trading. The LightGBM algorithm converges fast since it uses a leaf-wise strategy but this model is prone to overfitting. To overcome this problem, an H³O algorithm is used to adjust the hyperparameters of the architecture to improve the training performance by identifying the optimal parameter configuration. The hyperparameters are initially mapped to the cost metric which is also the loss function. The main hyperparameters that when optimized can reduce the complexity of the model are maximum depth (MD), number of leaves (NL), Number of training iterations (NTI), and minimum data in the leaf (MDL). If the value of ML increases it results in overfitting and too many NTI will also cause the same problem. If the NL values are minimized, the complexity will be automatically reduced. Initially, the values of MD, NL, NTI, and MDL is set as (1,2,..,12), (20–256), (10,20,…450), and (20,40,..,250). The optimized parameters are presented in Table 1.

Table 1.

Optimized parameters of LightGBM

Parameters	Optimal Value
MD	10
NL	230
MDL	40
NTI	200

Dataset description

The datasets (Roy 2017) used in this study are presented as follows:

Dataset 1: Precious metal price forecast

This dataset https://www.statista.com/statistics/254547/precious-metal-price-forecast/ mainly comprises the statistical details associated with the precious metal prices from 2022 to 2025 computed worldwide. The platinum price is estimated to rise up to 975 US dollars per troy ounce in the year 2025. The precious metals are the gold, silver, and platinum group metals. The silver price falls in the range of 23.5 US dollars in the year 2023 and the gold price is to drop near 1663 US dollars per ounce by the year 2023.

Dataset 2: Stock market dataset

This dataset (Roy 2017) mainly comprises the list of Indian stock information provided by the National Stock Exchange (NSE). The information regarding the well-performing stocks along with the buy, hold, or sell options are also provided in the dataset. The category column in the dataset provides the information regarding every stock that the classifier wants to predict as buy, hold, or sell.

Performance evaluation metrics

The mean square error (MSE) is mainly used to evaluate the performance of the proposed model in terms of loss. Since MSE is the loss function, a minimal value represents increased robustness. The MSE is computed as shown below:

$$MSE=\frac{1}{M}\sum _{i=1}^X{\left(\widehat{A}(i)-A(i)\right)}^2$$ 26

The mean average percentage error (MAPE) is computed as shown in equation () and it is mainly used to access the performance accuracy of the proposed model.

$$MAPE=\frac{1}{M}\sum _{i=1}^X\left|\frac{{\widehat{A}}_i-A_i}{A_i}\right|$$ 27

The root mean square error (RMSE) is computed using the below equation

$$RMSE=\sqrt{\frac{1}{M}\sum _{i=1}^X{\left({\widehat{A}}_i-A_i\right)}^2}$$ 28

The mean absolute error (MAE) is computed using the below equation and it mainly identifies the deviation of the actual value from the predicted value.

$$MAE=\sqrt{\frac{1}{M}\sum _{i=1}^X\left|{\widehat{A}}_i-A_i\right|}$$ 29

Relative absolute error (RAE) is the fraction of the relative error divided by the actual value and the difference magnitude of the actual and approximate value is expressed as the absolute error and is computed as follows:

$$RAE=\left|\frac{{\widehat{A}}_i-A_i}{A_i}\right|$$ 30

The total number of instances present in the dataset is represented as M, ${\widehat{A}}_i$ is the predicted value, and $A_i$ is the real value at the time i.

Performance analysis

The two datasets described above are used as input datasets. The tenfold cross-validation methods are employed for picking testing and training datasets. Table 2 represents an analysis of correlation coefficients for four types of datasets with different methods such as Kalman and hidden Markov model (HMM) filtering method (Tenyakov and Mamon 2017), two robust long short-term memory (TRLSTM) (Fister et al. 2021), long short–term memory(LSTM) recurrent neural networks (RNN) (Shen et al. 2021), variational mode decomposition (VMD)-iterated cumulative sums of squares (ICSS)-bidirectional gated recurrent unit (BiGRU) (VMD-ICSS-BiGRU) (Li et al. 2021), and Deep reinforcement learning (DRL) technique (Taghian et al. 2022). From this table, the proposed method has high correlation coefficients compared with other methods.

Table 2.

Comparative analysis of correlation coefficient

Dataset	Proposed	RNN-LSTM (Shen et al. 2021)	DRL (Taghian et al. 2022)	TRLSTM (Fister et al. 2021)	Kalman-HMM (Tenyakov and Mamon 2017)	VMD-ICSS-BiGRU (Li et al. 2021)
Dataset 1	0.99	0.92	0.94	0.94	0.87	0.94
Dataset 2	0.99	0.91	0.93	0.92	0.85	0.92

The precious metal price forecast and stock market datasets are considered input datasets. Table 3 denotes the analysis of relative absolute error for four datasets with various methods like RNN-LSTM (Shen et al. 2021), DRL (Taghian et al. 2022), TRLSTM (Fister et al. 2021), Kalman-HMM (Tenyakov and Mamon 2017), VMD-ICSS-BiGRU (Li et al. 2021), and proposed. Among all those methods, the proposed method has a very low relative absolute error value and which shows the good performance of the proposed method.

Table 3.

Comparative analysis of relative absolute error

Dataset	Proposed hybrid HHO-EFDM optimized LightGBM model	RNN-LSTM (Shen et al. 2021)	DRL (Taghian et al. 2022)	TRLSTM (Fister et al. 2021)	Kalman-HMM (Tenyakov & Mamon 2017)	VMD-ICSS-BiGRU (Li et al. 2021)
Precious metal price forecast	9.61	20.46	30.12	29.67	16.07	11.23
Stock market	13.63	23.68	35.67	19.02	28.37	14.65

The analysis of root means square error (RMSE) for different methods such as RNN-LSTM (Shen et al. 2021), DRL (Taghian et al. 2022), TRLSTM (Fister et al. 2021), Kalman-HMM (Tenyakov and Mamon 2017), VMD-ICSS-BiGRU [12], and proposed Hybrid HHO-EFDM optimized LightGBM model are established in Table 4. The proposed method has a very low RMSE value related to other methods. The next higher performance is achieved by the VMD-ICSS-BiGRU (Li et al. 2021) model.

Table 4.

Comparative analysis of RMSE in precious metal price forecast dataset

Categories	Proposed	RNN-LSTM (Shen et al. 2021)	DRL (Taghian et al. 2022)	TRLSTM (Fister et al. 2021)	Kalman-HMM (Tenyakov and Mamon 2017)	VMD-ICSS-BiGRU (Li et al. 2021)
Gold	78.36	112.14	100.76	149.35	125.84	85.67
Platinum	106.47	187.49	262.94	164.07	224.18	123.90
Silver	86.60	100.49	187.8	160.82	99.38	99.07

At first, different metrics were used for comparing various existing methods and the proposed method. The data from eight stock markets are taken as input data for comparison and the output of different sharemarkets and metal price predictions are explained in Tables 5 and 6. From these tables, the correlation coefficients are set as close to 1 which shows a strong positive linear relationship. In a few cases, the RMSE values are slightly high and the proposed model gives low RMSE and best correlation values. The various existing and proposed methods are implemented for the analysis of precious metal price forecasts and stock market datasets.

Table 5.

Comparative analysis of trade decisions using different share markets

Share market name	RMSE	MAE	Correlation coefficient	RAE(%)
Aarti drugs	120.49	15.376	0.9764	54.749
ADFFOODS	46.73	149.75	0.9124	48.175
DHANBANK	10.674	9.754	0.8531	41.029
Axis bank	213.957	185.76	0.9134	32.837
Bajaj auto	90.106	59.278	0.9847	25.413
Bajaj corp	197.16	10.918	0.9673	27.581
Bajaj finsv	30.47	18.690	0.8349	30.145
DABUR	80.38	162.471	0.9237	25.204

Table 6.

Performance evaluation using the precious metal price forecast dataset

Country	RMSE	MAE	Correlation coefficient	RAE(%)
Gold	130.3386	100.5828	0.9261	60.9887
Platinum	49.8584	45.4828	0.9674	25.2276
Silver	12.1234	10.7013	0.8778	50.8481

Evaluation tests

The financial beta ${\beta }_F$ is denoted as a standardized financial indicator that is defined as the ratio of covariance of asset and index prices to the covariance of index prices. The beta is expressed as below,

$${\beta }_F=\frac{\text{cov}\left(s_j,s_n\right)}{\text{cov}\left(s_n\right)}$$ 24

where $s_j$, $s_n$ represents the daily returns of market and investments respectively. The Sharpe ratio is also called a return-to-risk financial indicator. The Sharpe ratio is measured as the proportion of expected return per unit of risk is calculated as,

$$T_b=\sqrt{M}·\frac{\mathrm{E}\left(s_j,s_n\right)}{\sqrt{\text{var}\left(s_j,s_n\right)}}$$ 25

where $T_b$, $M$ are the annualized asset of Sharpe ratio and the number of days respectively. The Sharpe ratio is calculated from the daily data and the positive values are indicates, that the expected profit is higher than the market return and the negative values represent, that the expected profit is lower than the market return. Jensen’s alpha is an indicator that measures the final values rather than the daily data value and also it described as a measurement of excess return than the expected return. The Jensen's alpha is given below,

$$\alpha =P_j-\left(P_g+{\beta }_{FjQ}·\left(P_Q-P_g\right)\right)$$ 26

where $P_j$, $P_g$, $P_Q$ are realized return, risk-free rate, and overall market return respectively. the positive Jensen’s alpha value shows the best performance in investments rather than expected and the negative Jensen’s alpha represents the worst performance in investment. The Treynor ratio is measured as an excess amount of reward in investments compared with the risk-free rate per given unit of volatility. This is calculated from the difference between the risk-free rate and realized return and it is formulated as,

$$S=\frac{P_j-P_g}{{\beta }_{FjQ}}$$ 27

The greater difference between the risk-free rate and realized return gives the high Treynor ratio. The positive value shows the investments are more suitable than risk-free investments and the negative value shows less suitableness in investments. The existing and proposed methods have a low-risk value compared with the benchmark and the best investment is selected for each LSTM-ITS. The standard deviation, mean value, maximum, and minimum values are calculated from the investments are shown in Table 7. The proposed model has maximally performed 34 trades and minimally performed 0 trades in the investments. In the proposed model, the average investment has 7.15 trades and the standard deviation has 5.69 trades. The more frequent trading was incorporated with high transaction costs, more trades, and trading costs.

Table 7.

Comparative analsis of Sharpe ratio, Treynor ratio, Jensen’s alpha, and Beta

		Proposed hybrid HHO-EFDM optimized LightGBM model	RNN-LSTM (Shen et al. 2021)	DRL (Taghian et al. 2022)	TRLSTM (Fister et al. 2021)	Kalman-HMM (Tenyakov and Mamon 2017)
Value	Euros	311,123.08	264,693.44	116,634.19	242,469.22	258,631.60
Beta	${\beta }_F$	− 0.0095	− 0.0300	0.4953	0.1612	0.7564
	Max	34	28	85	9	0
Profit	$P_j$(%)	14.40	− 3.83	− 58.21	− 11.72	− 5.92
Sharpe ratio	$T_b$	0.2148	0.0035	− 3.5792	− 2.3316	0.0755
Treynor ratio	$S$	− 18.77	0.61	− 3.31	− 2.06	− 2.01
Jensen’s alpha	$\alpha$(%)	12.23	− 5.36	55.70	− 11.03	0.11
No. of transactions	Min	2	1	32	1	0
	Mean	7.15	10.58	66.05	5.73	0
	Stdev	5.69	4.75	7.99	0.10	0.001

Figure 3 depicts the analysis of the mean square error (MSE) of the proposed method. For a different number of samples, the MSE rate of the proposed method is graphically represented. As the number of samples increases, the mean square error rate decreases. Therefore, the proposed method has better efficiency than other methods. Figure 4 represents the Root mean square error rate analysis of the proposed method with respect to different trade parameters. The graph was plotted for training and checking data. The analysis shows that the optimized parameter has better accuracy compared to unoptimized parameters i.e.the optimized lightGBM model parameter has achieved a lower RMSE value.

Fig. 3.

MSE analysis

Fig. 4.

RMSE analysis

Figure 5 shows the recognition rate analysis of the proposed method. The graph is plotted for different numbers of iterations and recognition rates. As the number of iterations increases the recognition rate of the proposed method also increases.

Fig. 5.

Analysis of recognition state

The training and validation loss analysis of a various number of sample sequences is represented by Fig. 6 which has a high correlation rate with original data but it has a great impact on the performance of the final model if whether the high correlation is used before training. The training and validation sets are having severe fluctuations so the performance of the unoptimized lightGBM model is unstable. This is mainly due to the overfitting.

Fig. 6.

Training and validation loss analysis

Figure 7 shows the gross domestic product (GDP) growth rate of two selected input values. The predicted values and the actual values are denoted as a selected input value. From this figure, the optimized light-GBM model is used for selected input parameter variations. A very high non-linearity can be obtained in the GDP growth rate based on selected input parameters.

Fig. 7.

GDP growth rate of two selected inputs

Discussion

The efficiency of the proposed model is computed by running a total of 10 trials for each stock separately. In this way, the final stock value was derived. We have also computed the statistical analysis via the mean, minimum, and maximum values obtained. This step was mainly done to ensure that the proposed model is robust to the existing techniques. A larger min–max value shows a poor prediction performance for the investment trading strategy. Table 7 shows the comparative analysis in terms of Sharpe ratio, Treynor ratio, Jensen’s alpha, and Beta. Based on the results shown we can observe that the state-of-art models such as showed higher disparities for several stocks.

For a total of four stocks, the proposed model achieved a higher performance of 1,999,821 euros. The risk associated with the different datasets increases the difference between the proposed and existing model. The generalization ability of the proposed model can be noticed via the variance and mean values achieved. To help the investors select an appropriate trade strategy we have also incorporated the strong sell and strong buy policy. In this way, the proposed model is capable of identifying the current trends and offering optimal decisions. The proposed model also offered the highest sharp ratio when compared to the RNN-LSTM (Shen et al. 2021), DRL (Taghian et al. 2022), TRLSTM (Fister et al. 2021), and Kalman-HMM (Tenyakov and Mamon 2017) models scored negative sharp values which indicates that the results were not meaningful.

The performance of the proposed Hybrid HHO-model is compared with different architectures such as LightGBM-based Particle Swarm Optimization (PSO) (Tang et al. 2021), Improved Harris hawk optimization (IHHO-LightGBM) (Tang et al. 2021), and Bayesian optimization algorithm (BOA) based LightGBM model (Huang et al. 2022). The comparison is mainly held for a single-step horizontal prediction in terms of RMSE, MAPE, and MAE. As per the results shown in the Table 8, we can conclude that the proposed model offers improved performance when compared to the state-of-art predictive models.

Table 8.

Horizontal prediction performance evaluation

Models	RMSE	MAPE	MAE
LightGBM-PSO (Tang et al. 2021)	0.0104	0.0125	0.105
IHHO-LightGBM (Tang et al. 2021)	0.0063	0.0079	0.0075
BOA-based LGBM model (Huang et al. 2022)	0.0059	0.0069	0.0058
Proposed Hybrid HHO-EFDM optimized LightGBM model	0.0047	0.0059	0.0021

For a specific period (2005–2014), the performance of the stock market as well. The low-interest rate of the stock market attracted more consumers and served as a substitute for bank deposits and bonds. The stock market faced a time of intense difficulty when COVID-19 struck. Even though it didn’t affect the financial institutions' status, it caused potential losses and delays in real-time. The stock markets offered an immediate solution via the monetary policy drivers to handle the plunge in the stock market and solve the critical financial issues. The empirical investigation demonstrates that the proposed model offers superior prediction performance when compared to the conventional stock trade prediction models. The additional experiments conducted which are presented in Figs. 5, 6, 7 show the proposed model's efficiency in controlling parameters and time constraints to offer improved performance. This is made possible due to the incorporation of the H3O algorithm and Markov model along with the LightGBM architecture. In the future, the proposed model can be extended by incorporating the details of different stock markets present in the USA, Canada, China, etc. along with different trading instruments such as blockchain and cryptocurrencies. We also plan to design real-time trading applications that can be deployed in lightweight mobile devices to offer stock predictions and different stock recommendations.

Conclusion

This paper presents an optimized Light GBM model for trade prediction using the H³O algorithm. The EFB and GOSS approaches are used to alleviate the overfitting problem in conventional Harris hawk optimization. The H³O optimization technique achieves rapid convergence by optimizing several lightGBM parameters such as the number of training iterations, maximum depth, and minimal data in the leaf. The Markov chain model can effectively obtain the many components to anticipate stock futures and give significant insights at various time intervals. The error reference matrix is used to simulate stock future price log returns, and the error points represent changes in stock market return volatility. The precious metal price forecast and stock market datasets are the two world trade organization dataset used in this work. Different measures such as RMSE, MAE, Correlation coefficient, and RAE are used to assess performance and the comparison techniques employed include RNN-LSTM, DRL, TRLSTM, Kalman-HMM, AND VMD-ICSS-BiGRU. The optimized light-GBM model is utilized for selected input parameter variations based on the gross domestic product (GDP) growth rate of two specified input values. Based on selected input parameters, a very high non-linearity in the GDP growth rate can be generated. The optimized lightGBM model parameter has a lower RMSE value than the unoptimized lightGBM model parameter, indicating that it is more accurate than the unoptimized lightGBM model. The mean square error rate reduces as the number of samples increases. The proposed method's recognition rate increases as the number of iterations grow. As a result, the proposed strategy is more efficient than others.

Funding

Not applicable.

Data availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Code availability

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Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Research involving in human and animal participants

This article does not contain any studies with human or animal subjects performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Consent for publication

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