## 4 Experiment
### Optimizing Network Parameters by the PSO  
In the pursuit of effectively updating the weight of the neural network, this study has employed the Adam optimizer for the optimization of model parameters, with a batch size established at 128. The ConvLSTM model's compatibility with our dataset was assured via the implementation of the particle swarm optimization (PSO) algorithm for ConvLSTM hyperparameter optimization.  
The PSO algorithm, by mimicking the communal behaviors exhibited by birds when locating food, leverages a combination of individual and collective experiences to progressively approach the target of interest. Continuous position updates, shaped by their personal optimum position in tandem with the overall flock's optimum position, consequently get folded into an optimal configuration [25]. PSO operates in iterations, allowing for swarm updates through alterations in each component's velocity and position in every cycle. These updates fundamentally depend on personal best values (pbest) and global best values (gbest). Therefore, the accurate tuning of the PSO parameters is of utmost importance.  
[PERSON] and [PERSON]'s research [22] posits that a decent solution success rate is achievable when a PSO algorithm deploys a particle range of 20 to 50. However, our experiments have illustrated that employing a particle size of 6 allows for quicker convergence of the PSO algorithm with fewer computational resource requirements. A comprehensive review of PSO-centric literature [26; 27; 28; 29] guided us in setting the particle size at 6 and iteration count at 10, with the upper and lower bounds of W being determined as 0.9 and 0.4, respectively.  
Furthermore, an examination of academic literature surrounding the application of the ConvLSTM network in prediction problems [30; 31], as well as relevant works, highlighted the pivotal role of the number of ConvLSTM layers and the size of the convolution kernel corroborated by the results of our experiments. Consequently, we selected the learning rate, size of the convolution kernel, the number of ConvLSTM layers, and the epoch as targets for optimization. A logical search range is essential to preventing issues such as excessive resource consumption associated with expansive search ranges during the search process. Following the analysis of related research, definitions for the search ranges were formulated as being [0.00001, 0.0005] for the learning rate, [1; 9] for convolution kernel size, and [2; 7] for layers of ConvLSTM. Besides, a few random epoch values were tested while keeping other parameters constant, revealing a suboptimal model performance when the epoch is less than 100 and an improved performance when it exceeds 300. However, to account for potential randomness, the epoch range was set between [1; 400].  
With the search range of the target optimization parameters, we will obtain the optimal parameters by the PSO. As delineated in Table 6, the optimal parameters for the ConvLSTM model, as yielded by application of the PSO, stand at a learning rate of 5.7399496072129 \(\times\) 10\({}^{5}\) an epoch count of 365, ConvLSTM layer featuring 6 neurons, and kernel-size of 6. Figure 6 shows the evolution of the loss function of the PSO Deep-ConvLSTM model during training and testing. Utilizing this optimal parameter configuration should potentialize the performance of the ConvLSTM model, particularly in the domain of arbitrage spread prediction.  
Figure 6: Evolution of loss function in PSO Deep-ConvLSTM model during training and testing.