  In this paper, we present a measure of time irreversibility using trend
pattern statistics. We define the irreversibility index as the Kullback-Leibler
divergence between the distribution of uptrends subsequences (increasing
trends) and the corresponding downtrends subsequences distribution (decreasing
trends) in a time series. We use this index to analyze the degree of
irreversibility in log return series over time, specifically focusing on five
cryptocurrencies: Bitcoin, Ethereum, Ripple, Litecoin, and Bitcoin Cash. Our
analysis reveals a strong indication of irreversibility in all these
cryptocurrencies and the characteristic evolves over time. We additionally
evaluate the market efficiency for these cryptocurrencies based on a recently
proposed information-theoretic measure. By comparing inefficiency and
irreversibility, we explore the relationship between these statistical
features. This comparison provides insight into the non-trivial relationship
between inefficiency and irreversibility.
