Fitness Function Selection in PSO-VMD Decomposition: Implications for LSTM-Based Oil Price Forecasting

Predicting the price of crude oil is one of the most difficult tasks in modern economics. The market does not move in straight lines or predictable patterns; instead, it behaves like a chaotic storm, driven by a mix of economic data, political events, and sudden shifts in supply and demand. Because the price fluctuates so wildly and unpredictably, traditional mathematical tools often fail to capture its true nature. To make sense of this chaos, researchers have turned to a strategy called decomposition. Imagine trying to understand a complex song by listening to the entire orchestra at once; it is nearly impossible to hear the melody. However, if you could separate the instruments so you could listen to the strings, then the brass, and then the percussion individually, the structure of the music would become clear. In the world of data, this means breaking a single, messy stream of numbers into smaller, simpler pieces that are easier to study. Once these pieces are understood, they can be reassembled to create a much clearer picture of what might happen next.

A recent study by researcher Abdelkader Aouar tackles a specific problem within this decomposition strategy. The research focuses on a method called Variational Mode Decomposition, or VMD, which is a sophisticated way of splitting time series data into those simpler pieces. While VMD is powerful, it requires two critical settings to be adjusted correctly before it can work: the number of pieces the data should be cut into, and a factor that controls how strictly the pieces are separated. If these settings are wrong, the data might be cut into too few pieces, leaving important details hidden, or too many pieces, creating unnecessary noise. In the past, scientists have used computer algorithms to automatically find the best settings, but they have relied on a specific rule to judge what "best" means. This rule, known as envelope entropy, was originally designed to find faults in machine parts, like a broken bearing in an engine. It works by looking for sharp, sudden spikes in a signal. Because this rule worked well for machines, many researchers assumed it would also work well for oil prices. Aouar's study challenges this assumption by asking a simple question: does a rule designed for broken gears actually help predict the price of oil?

To answer this, the researcher built a hybrid forecasting system that combines three distinct tools. First, the system uses an optimization algorithm called Particle Swarm Optimization, which mimics how a flock of birds searches for food, to automatically test thousands of different settings for the VMD tool. Second, it uses the VMD method to break the daily oil prices into their component parts. Third, it feeds these parts into a type of artificial intelligence called a Long Short-Term Memory network, or LSTM, which is excellent at remembering patterns over time. The crucial part of the experiment was not the tools themselves, but the rule used to tell the computer which settings were the best. The study tested four different rules: three that measure the complexity or irregularity of the data (called sample entropy, permutation entropy, and envelope entropy) and one that simply measures how much information is lost when the data is broken apart and put back together (called reconstruction error).

The results of the experiment were striking and overturned the prevailing habit of using the machine-fault rule. When the researchers used the envelope entropy rule, which had been the standard choice in many previous studies, the model performed poorly, yielding a coefficient of determination (R²) of -0.3519. This negative value indicates that the model's predictions were less accurate than simply using the average price as a baseline, a significant underperformance compared to the other methods. The other two complexity-based rules performed better but still missed the mark. The clear winner was the reconstruction error rule. This rule, which focuses on keeping the original information intact rather than trying to make the signal look "clean" or "sparse," allowed the model to achieve a level of accuracy that was far superior to the others. The final prediction model using this rule was able to forecast oil prices with an error rate of just 1.37 percent, a figure that is remarkably close to the actual values. In contrast, the models using the other rules had error rates that were several times higher.

The study also revealed why the winning rule worked so well. When the computer was told to minimize the loss of information, it chose to break the oil price data into nine distinct pieces. This allowed the system to capture long-term trends, medium-term cycles, and short-term spikes all at once. In contrast, the rules based on complexity told the computer to break the data into only two pieces, which was too simple to capture the full story of the market. Furthermore, the winning rule selected a setting that allowed the data pieces to retain more of their original shape, whereas the other rules forced the data into a smoother, more rigid form that stripped away valuable details. The findings suggest that for financial markets, which are defined by their volatility and sudden jumps, the goal should not be to find the simplest possible signal, but to preserve as much of the original, messy reality as possible.

This research does more than just improve a single prediction model; it highlights a critical flaw in how many scientific studies are conducted. For years, researchers have applied rules from one field, such as mechanical engineering, to another, like finance, without testing if those rules actually fit. This study proves that what works for a vibrating machine part does not necessarily work for the price of a barrel of oil. By systematically testing different rules under the exact same conditions, the researcher showed that the choice of the evaluation rule is just as important as the computer algorithm itself. The study concludes that for forecasting complex financial data, the most effective strategy is to prioritize keeping the original information intact rather than trying to force the data into a pre-conceived notion of what a "clean" signal should look like. This insight offers a new path forward for anyone trying to understand the unpredictable rhythms of global markets.