where is the indicator function that equals 1 when the condition is true and 0 otherwise. DA ranges from 0 to 1, with a higher value indicating better directional prediction.
Prior to implementing the full MLP, we assessed predictive performance with simpler models. Specifically, we applied XGBoost and logistic regression to price prediction, but both models performed poorly. The detailed results for these models are presented in Table 2. As the table shows, incorporating multiple indicators into these simpler models is insufficient for accurately predicting price fluctuations, indicating the necessity of employing more complex deep-learning approaches.
In this study, 70% of the data was split to train the model and remained 30% of the data was for prediction. The model performance of the prediction is shown in Table 3. Figure 2 shows the loss trained by using data from Cryptopunks into the model. At about the 5000th epoch, the loss stopped decreasing and an early stopping cut-off was applied. Figure 3 depicts the loss resulting from feeding BAYC data into the model. The loss ceased decreasing around the 3400th epoch, triggering an early stopping cut-off.
As shown in Table 3, Cryptopunks analysis results show that the model explains and predicts price volatility relatively well. The R Score is 0.7473, which can explain approximately 74.73% of the variability in the data. The Index of Agreement is high at 0.9339, indicating that the predictions and actual values match well, and Theil’s U Index is low at 0.1983, meaning that the error in the predicted values is small. Directional Accuracy is 0.8786, indicating that the direction of price rise and fall can be predicted with approximately 87.86% accuracy.
The results of the BAYC analysis show that it has lower predictive performance than Cryptopunks. The R Score is 0.2539, and it suggests that either BAYC’s price volatility is more erratic, or the model does not capture that volatility as well as Cryptopunks’ price prediction. The Index of Agreement is 0.8149, indicating that the predicted and actual values are relatively consistent, but the Theil’s U Index is high at 0.6670, indicating a large error in the predicted value. In addition, the Directional Accuracy is 0.8380.
In the analysis results of Cryptopunks and BAYC, the main reasons for the difference in model performance can be seen as differences in the features of the data and the suitability of the model. In the case of Cryptopunks, the model was able to explain and predict price volatility relatively well, while in the case of BAYC, performance was likely relatively low due to data irregularity or model limitations. As shown earlier in Fig. 1, the price fluctuations of BAYC were greater than those of Cryptopunks, and the trading volume and the number of social media posts related to BAYC were significantly higher, leading to data imbalance and irregularity. Because this study focused on predicting the average price, there was a substantial deviation from the actual prices, which emerged as a limitation of the model.
In this study, deep learning was used to predict the price of PFP NFTs and the results were analyzed using eXplainable artificial intelligence (XAI) methods to interpret the model and how each feature affected the price. XAI is useful for understanding the process that the model went through to get the result and how each feature contributed to the model. Therefore, this study utilizes shapley additive explanation (SHAP) to interpret the results by measuring the SHAP value of each feature’s contribution to the prediction model.
Figure 4 displays the most significant features for predicting Cryptopunks, listed in descending order of importance. Figure 5 shows a visualization of the impact of each feature on the model’s output, with each feature positively and negatively labeled. In Fig. 5, higher values (red) tend to increase the predicted price, while lower values (blue) show a negative impact.
Analyzing the features that contributed to Cryptopunks’ price prediction, in order of importance, the 10-day moving average of Cryptopunks’ price was the most influential, and as the 10-day moving average increased, the price of Cryptopunks increased. The “Discussion_Count” attribute shows that as the amount of discussions per day in Cryptopunks’ online community increases, the price is negatively affected, but the interaction term with “SMA_5” has a positive effect on the price. The interaction term between “Discussion_Count” and “US6MT” means that the interaction term between the amount of discussion in the online community and the 6-month US Treasury yield had a positive impact on the price. “US6MT” denotes the U.S. 6-Month Treasury yield, with MT standing for “Month Treasury yield”. The standalone effect of “Discussion_Count” is negative, because excessive discussion can provoke FUD (fear, uncertainty, and doubt) and depress prices. By contrast, when “Discussion_Count” interacts with “SMA_5”, which signals an upward market trend, or with “US6MT”, which reflects an accommodative short-term interest-rate environment, high discussion volume acts as a cue for trend confirmation or heightened risk appetite, thereby exerting a positive impact on price.
On the other hand, the features that influenced BAYC’s price were completely different from Cryptopunks. Figure 6 shows the top features that contributed to the model predicting the price of BAYC. In particular, the price of BAYC was strongly influenced by the interaction term between Bitcoin’s price and US long-term bond yields.
As shown in Fig. 7, the interaction term between US long-term bond yields and the price of Bitcoin had a negative impact on the price of BAYC. In addition, the interaction term between US short-term Treasury yields and the 10-day lower band also had a negative impact on the price of BAYC. On the other hand, the Nasdaq Composite Index had a positive impact on the price of BAYC. However, the interaction term between the subjective sentiment score and the 10-year US Treasury yield is negatively related to the price of BAYC, suggesting that the sentiment of social media users’ comments about the PFP NFT in the online community and the yield of long-term Treasuries collectively influence the price. The variable name for the subjective sentiment score is “Subjectivity_Avg,” and the 10-year US Treasury yield is labeled as “US10YT”.
In this study, PFP NFT prices were predicted using various indicators, but this study used sentiment analysis of online communities, which was not included in previous studies to predict the price of PFP NFTs. In addition, sentiment was not just divided into positive and negative, but subjective sentiment was analyzed to evaluate the impact of subjective or objective remarks on the price. Also, by predicting the price not only by sentiment alone, but by predicting the sentiment score and the interaction term with other features, the price can be predicted more accurately through complex and diverse relationships with other features.
Figure 8 shows the contribution of polarity to Cryptopunks’ price prediction in terms of SHAP value. From this figure, it can be seen that polarity alone has a negative impact on price, but the interaction term with other attributes may have the opposite effect.
This study used the same methodology to predict the price of these PFP NFTs, but found that the results for BAYC were different from those of Cryptopunks. As shown in Fig. 9, the polarity of BAYC was the opposite of that of Cryptopunks, which means that polarity had a positive effect on the price of BAYC, and most of the interaction terms with other features had a positive effect on the price.
This study utilized the degree of subjective attitude and polarity of sentiment to predict the price of PFP NFTs. Figure 10 shows the relationship between the price of Cryptopunks and sentiment scores, and Fig. 11 shows the relationship between the price of BAYC and sentiment scores. Figure 10 is a three-dimensional scatterplot of the scores that influenced the price increase and decrease of Cryptopunks, and Fig. 11 is a three-dimensional scatterplot of the scores that influenced the price increase and decrease of BAYC. By contrasting the two figures, it is able to see that positive emotional and objective attitudes influenced the price increase, while relatively subjective attitudes influenced the price decrease.
Overall, the sentiment scores exerted a comparatively smaller influence on price than the technical and interest-rate indicators, yet their impact was not negligible. In particular, Figs. 5 and 7 show that the interaction effects between sentiment scores and technical indicators made a substantial contribution to price prediction. The table containing the specific SHAP values for BAYC is presented as Table 4, while the detailed SHAP values for Cryptopunks are summarized in Table 5.
The cross-correlation analysis revealed that both the BAYC and Cryptopunks collections exhibit numerous statistically significant relationships between sentiment indicators and NFT prices. For BAYC, Polarity_Avg showed significant correlations at both negative and positive lags, with the strongest effect observed at lag -8 (Corr: -0.193, p < 0.001), suggesting that changes in price tend to precede shifts in sentiment polarity by about eight days. Other significant values for Polarity_Avg and Subjectivity_Avg were observed at lags -24 and -23, although the correlation coefficients were relatively low. Subjectivity_Avg demonstrated significant positive and negative associations at various lags, most notably a positive correlation at lag -8 (Corr: 0.163, p < 0.001) and a negative correlation at lag 25 (Corr: -0.149, p < 0.001). These results indicate that price and sentiment interact in a complex manner across different time horizons. Notably, for BAYC, the results for Polarity_Avg and Subjectivity_Avg correspond to Fig. 12(a) and Fig. 12(b), respectively.
In the case of Cryptopunks, there were significant positive correlations between Polarity_Avg and price across a variety of lag intervals, with the most pronounced association at lag -22 (Corr: 0.149, p < 0.001), implying that a rise in price may forecast an increase in sentiment polarity after a considerable period. Additional significant correlations were observed for Polarity_Avg at both negative and positive lags, underscoring the highly dynamic nature of sentiment-price interactions in the NFT market. For Subjectivity_Avg, significant negative correlations were found at lags 13 and 26, though the magnitude of these associations was modest. Overall, these findings demonstrate that sentiment indicators and NFT prices are interrelated in both the short and long term, and that the direction and strength of these associations can vary substantially depending on the lag considered. For Cryptopunks, the results for Polarity_Avg and Subjectivity_Avg correspond to Fig. 12(c) and Fig. 12(d), respectively.
Note: *** Significant at the 1% level, ** Significant at the 5% level, * Significant at the 10% level.
