Because there is a discrepancy between how individual investors and investment institutions choose Bitcoin and its new derivatives and Exchange-Traded Funds (ETFs), this paper used Bitcoin and ProShares Bitcoin Strategy ETF (BITO) data and a mixed variational mode decomposition and bidirectional gated cycle unit model to examine the interconnections between Bitcoin and its new derivative ETFs, from which actionable recommendations were developed. As well as conducting financial simulation trading using Bitcoin and BITO, the study expanded to examine other major ETFs. It was found that: (1) Bitcoin data could be employed to forecast and describe BITO; (2) under T+0 trading, Bitcoin was more volatile, profitable, and risky than BITO; and (3) under T+1 trading, Bitcoin was less volatile, profitable, and risky than BITO; however, the T+1 trading was found to have higher volatility, profits, and risk than T+0 trading. This study, therefore, builds a bridge from theory to practice for the prediction and description of new ETFs. Different from previous studies, this study explored the relationships between Bitcoin and BITO using Artificial Intelligence and quantitative financial simulations, which extends the practical and theoretical understanding of the Bitcoin market.