Precious metals such as gold, silver, palladium, and platinum have been highly esteemed for their scarcity, industrial uses, and historical importance as a means of preserving wealth. Gaining a comprehensive understanding of the factors that impact the values of these metals is of utmost importance for investors, policymakers, and academics. The emergence of machine learning techniques provides novel and potent tools for analyzing and forecasting the behavior of precious metal markets. Hence, the significance of this research is in utilizing these sophisticated methods to get a more profound understanding of the price fluctuations of valuable metals, ultimately assisting in improved decision-making and strategy development for different participants in the market.

The structure of the study is as follows:

Introduction: This section elucidates the importance of valuable metals, namely gold, silver, palladium, and platinum. The text explores the historical significance of these assets, their practical uses in industry, and their value as investments. The introduction emphasizes the significance of employing machine learning methods to analyze and forecast the pricing of these metals.

Literature Review: This section examines prior research, conventional analytical techniques, and the recent incorporation of machine learning in financial analysis. The study conducts a meticulous assessment of previous research and identifies areas that need further investigation, which this study intends to fill.

Gaussian Mixture Model (GMM):

This section explains the Gaussian Mixture Model (GMM) and its use in the analysis of financial data. The text discusses the mathematical definition of GMM, the utilization of the Expectation-Maximization (EM) algorithm for parameter estimation, and the benefits of employing GMM to represent intricate price distributions of precious metals.

Results and discussion:

This section focuses on the application of the Gaussian Mixture Model (GMM) and other advanced machine learning techniques to analyze the price data of gold, silver, palladium, and platinum from the years 2017 to 2023. The method involves several steps, including data preprocessing, model training and validation, and the assessment of model performance using metrics such as BIC, AIC, ICL, log-likelihood, NEC, and entropy. The discoveries and patterns identified in the price dynamics of the selected precious metals are thoroughly addressed, emphasizing crucial results.

Conclusion:

The concluding part provides a concise overview of the study’s discoveries and their significance for investors, legislators, and scholars. The text explores the advantages of employing machine learning techniques to analyze the precious metal markets and proposes prospective avenues for future research. The conclusion also evaluates the study’s impact on the current pool of knowledge and its practical implications in making financial decisions and developing strategies.

The study by Çelik and Başarır (2017) examines the use of ANN for predicting the prices of precious metals and shows encouraging outcomes. The research utilized RapidMiner software to analyze the daily prices of gold, silver, palladium, and platinum, together with 15 macroeconomic factors, spanning from January 2010 to December 2015. The performance of the model was assessed using metrics like root mean squared error, absolute error, relative error, Spearman’s Rho, and Kendall’s Tau. The results suggest that ANN can accurately and effectively forecast the prices of gold and palladium. However, the forecasts for silver and platinum were less accurate. The study highlights the potential of ANN as a prediction tool for forecasting precious metal prices. It suggests that future research could enhance accuracy by including additional data and making comparisons with other machine learning algorithms and time series analysis approaches.

The objective of the study by Varshini et al. (2024) is to forecast the future prices of mineral commodities, including gold, silver, copper, platinum, palladium, and aluminum, by employing a blend of machine learning and deep learning models. The covered models include stacked LSTM, bidirectional LSTM, SVR with EGB, and gated recurrent units. The evaluation of model performance relies on three metrics: RMSE, MAE, and MAPE. The selection of mineral, duration of the sample period, and choice of input data significantly impact the precision of forecasts.

The study by Zhou and Xu (2023) introduces an innovative three-stage hybrid learning method to precisely predict the values of silver, palladium, and platinum. The hybrid model surpasses other models, emphasizing the significance of integrating diverse components to improve prediction accuracy and generalization power. These findings provide useful information for investing in and producing precious metals.

The study by Ulsami and Junejo (2017) provides evidence that machine learning is successful in forecasting gold prices, revealing that the stock valuation of a renowned company has a more significant influence on gold rates than the condition of the US economy. The study conducted by Sarangi et al. (2021) assesses the effectiveness of a hybrid model driven by machine learning in predicting gold prices. This model utilizes an ANN-PSO model, which integrates Particle Swarm Optimisation. The study applied the ANN-PSO model to monthly gold prices in India from January 2012 to June 2021. The findings demonstrate that this model can effectively forecast future gold prices.

The study by Oztoprak and Orman (2021) specifically examines the application of Explainable Artificial Intelligence (XAI) in forecasting the values of precious metals. It offers a thorough survey of existing research in this field. The study by Çelik and Başarır (2017) uses an ANN model created with Rapid Miner to evaluate the capacity to forecast the prices of gold, silver, palladium, and platinum. The model uses daily data from Jan. 2010 to Dec. 2015. The results suggest precise and efficient forecasts for gold and palladium, whereas they are less accurate for silver and platinum. This model is designed to forecast the pricing of materials utilized by jewelry manufacturers.

The study by Phitthayanon and Rungreunganun (2019) presents two separate deep-learning models for predicting the prices of gold, silver, and diamonds. It highlights that the proposed approach can be simply implemented in small jewelry enterprises in Thailand, as pricing data is readily available. The model attains a level of accuracy that is comparable to what is found in existing literature, while using less data. However, there is a notable difference in accuracy between the models for precious metals and diamond prices. This difference is related to the greater availability of data for precious metals.

Kangalli Uyar et al.’s (2024) study outlines a two-step approach for forecasting price bubbles in precious metals. The technique commences by conducting a unit root test on the right side in order to identify the presence of bubbles. Subsequently, a variety of machine learning techniques, such as logistic regression, SVM, CART, random forests, EGB, and neural networks, are employed to detect and assess the factors that contribute to the formation of bubbles. The study found that macroeconomic factors influenced the monthly price swings of gold, silver, palladium, and platinum from 1990 to 2022, resulting in the emergence of price bubbles. The study affirms that the US consumer confidence index exerts a favorable influence on determining the speculative expansion in the prices of gold, platinum, and silver. The paper by Ling and Zhu (2017) utilized GMM to determine the probability of precipitation occurring on a specific day and time based on historical meteorological data.

The objective of the study by Varshini et al. (2024) is to forecast future metal prices in commodity markets by employing machine learning and deep learning models such as Stacked LSTM, Convolutional LSTM, Bidirectional LSTM, SVR EGB, and Gated Recurrent Unit. Performance is assessed using three indicators: RMSE, MAE, and MAPE. The accuracy of predictions can vary significantly based on the particular mineral, the time period of sampling, and the unique input data. Finance, healthcare, education, transportation, and others have used AI to innovate and advance ( Liu & Zheng, 2020 ). Forecasting in financial markets is difficult because of its non-linear, dynamic nature and the presence of several uncertainties ( Vaidya, 2020 ). Technical analysis indicators are effective instruments for quickly earning gains, especially when dealing with intricate data that has non-linear relationships, all without the need for specialized financial knowledge ( Chandar, 2022 ), the study by Díaz et al. (2022) investigated the influence of the COVID-19 pandemic on the volatility of the global stock market, revealing a notable and adverse impact. An in-depth investigation of classified stock markets helps clarify the correlation between worldwide markets ( Li et al., 2022 ). Machine learning, a subset of artificial intelligence, is exceptionally efficient at tackling a wide range of difficulties in different domains ( Chan et al., 2022 ). Researchers have made predictions about the potential failure of certain firms in the wake of the economic slowdown caused by the coronavirus pandemic ( Elhoseny et al., 2022 ).

The Gaussian Mixture Model (GMM) is a statistical model that describes the likelihood distribution of a dataset by combining various Gaussian (normal) distributions ( Bishop, 2006 ). It is commonly employed for clustering and estimating density in the fields of machine learning and statistics ( Lee & McLachlan, 2013 ).

The probability density function of a Gaussian Mixture Model (GMM) is expressed as a summation of distinct Gaussian distributions, each weighted by a specific factor ( Du Roy de Chaumaray & Marbac, 2024 ):

$p\left(x\right)={\displaystyle {\sum }_{k=1}^k{\text{π}}_k\cdot N\left(x|{\mu }_k,{\sum }_k\right)}$ (1)

where ${\text{π}}_k$ denotes the weight of the k-th Gaussian component, ${\mu }_k$ represents the mean (centers of Gaussian components).

$0\le {\mu }_k\le 1$ (2)

${\displaystyle {\sum }_{k=1}^k{\text{π}}_k}=1$ (3)

where ${\sum }_k$ represents the covariance.

The mixing coefficients are denoted by ${\mu }_k$ relative magnitudes of each Gaussian component in the mixture.

$N\left(x|{\mu }_k,{\sum }_k\right)$, the Gaussian distribution, with mean and covariance:

$0\le {\mu }_k\le 1$ (4)

Expectation-Maximization (EM) algorithm

The Expectation-Maximization (EM) algorithm is a commonly employed technique for determining the maximum probability estimates of parameters in statistical models that involve hidden variables ( Dempster et al., 1977 ). The EM technique is highly efficient when used to the Gaussian Mixture Model (GMM) ( McLachlan & Krishnan, 2007 ).

Maximum A Posteriori (MAP)

MAP classification is a statistical method employed in pattern recognition and machine learning to make choices using observable data ( Murphy, 2012 ). The approach is based on Bayesian probability theory and its objective is to maximize the posterior probability of a class, given the available data ( Müller et al., 2015 ).

The study, which utilized machine learning models, analyzed the price trends of gold, silver, palladium, and platinum from 2017 to 2023, yielding various insights and findings. The model of Palladium showed the most favorable fit among the precious metals, as shown by the lowest BIC, AIC, and ICL values. In contrast, the model based on platinum had the highest levels of entropy and NEC values, suggesting greater levels of uncertainty and complexity. The Cumulative Distribution Function (CDF) plots for each metal yielded valuable insights into the probability distribution and price volatility within the selected time frame. The strong agreement between the empirical and estimated Cumulative Distribution Functions (CDFs) confirmed the assumptions and dependability of the selected models. During the COVID-19 epidemic, the values of precious metals, namely gold and silver, experienced substantial rises. This demonstrates their classification as safe-haven investments during times of economic turmoil. Palladium and platinum prices also exhibited volatility; however, the patterns were more diverse and less prominent in comparison to gold and silver. The results emphasize the need to employ sophisticated machine-learning methodologies for financial modeling and forecasting. Investors and policymakers can utilize these observations to make well-informed judgments regarding investments in precious metals and measures for managing risks.