Machine learning and AI-assisted trading have attracted growing interest for the past few years. Here, we use this approach to test the hypothesis that the inefficiency of the cryptocurrency market can be exploited to generate abnormal profits. We analyse daily data for 1,681 cryptocurrencies for the period between Nov. 2015 and Apr. 2018. We show that simple trading strategies assisted by state-of-the-art machine learning algorithms outperform standard benchmarks. Our results show that nontrivial, but ultimately simple, algorithmic mechanisms can help anticipate the short-term evolution of the cryptocurrency market.
过去几年，机器学习和人工智能辅助交易引起了越来越多的兴趣。在这里，我们使用这种方法来检验加密货币市场的低效率可以被利用来产生异常利润的假设。我们分析了 2015 年 11 月至 2018 年 4 月期间 1,681 种加密货币的每日数据。我们表明，在最先进的机器学习算法的辅助下，简单的交易策略优于标准基准。我们的结果表明，重要但最终简单的算法机制可以帮助预测加密货币市场的短期演变。

1. Introduction 一、简介

The popularity of cryptocurrencies has skyrocketed in 2017 due to several consecutive months of superexponential growth of their market capitalization [1], which peaked at more than $800 billions in Jan. 2018. Today, there are more than 1,500 actively traded cryptocurrencies. Between 2.9 and 5.8 millions of private as well as institutional investors are in the different transaction networks, according to a recent survey [2], and access to the market has become easier over time. Major cryptocurrencies can be bought using fiat currency in a number of online exchanges (e.g., Binance [3], Upbit [4], Kraken [5], etc.) and then be used in their turn to buy less popular cryptocurrencies. The volume of daily exchanges is currently superior to $15 billions. Since 2017, over 170 hedge funds specialised in cryptocurrencies have emerged and Bitcoin futures have been launched to address institutional demand for trading and hedging Bitcoin [6].
由于其市值连续几个月呈超指数增长，加密货币的受欢迎程度在 2017 年飙升[1]，并于 2018 年 1 月达到超过 8000 亿美元的峰值。如今，活跃交易的加密货币有超过 1,500 种。根据最近的一项调查 [2]，有 2.9 至 580 万私人和机构投资者处于不同的交易网络中，并且随着时间的推移，进入市场变得更加容易。主要的加密货币可以在许多在线交易所（例如 Binance [3]、Upbit [4]、Kraken [5] 等）中使用法定货币购买，然后依次用于购买不太受欢迎的加密货币。目前每日交易量超过 150 亿美元。自2017年以来，出现了超过170家专门从事加密货币的对冲基金，并推出了比特币期货，以满足机构对比特币交易和对冲的需求[6]。

The market is diverse and provides investors with many different products. Just to mention a few, Bitcoin was expressly designed as a medium of exchange [7, 8]; Dash offers improved services on top of Bitcoin’s feature set, including instantaneous and private transactions [9]; Ethereum is a public, blockchain-based distributed computing platform featuring smart contract (scripting) functionality, and Ether is a cryptocurrency whose blockchain is generated by the Ethereum platform [10]; Ripple is a real-time gross settlement system (RTGS), currency exchange, and remittance network Ripple [11], and IOTA is focused on providing secure communications and payments between agents on the Internet of Things [12].
市场是多元化的，为投资者提供了许多不同的产品。仅举几例，比特币被明确设计为一种交换媒介 [7, 8]；达世币在比特币功能集的基础上提供改进的服务，包括即时和私密交易[9]；以太坊是一个公共的、基于区块链的分布式计算平台，具有智能合约（脚本）功能，而以太币是一种加密货币，其区块链由以太坊平台生成[10]； Ripple是一个实时总结算系统（RTGS）、货币兑换和汇款网络Ripple[11]，而IOTA则专注于在物联网上的代理之间提供安全的通信和支付[12]。

The emergence of a self-organised market of virtual currencies and/or assets whose value is generated primarily by social consensus [13] has naturally attracted interest from the scientific community [8, 14–30]. Recent results have shown that the long-term properties of the cryptocurrency marked have remained stable between 2013 and 2017 and are compatible with a scenario in which investors simply sample the market and allocate their money according to the cryptocurrency’s market shares [1]. While this is true on average, various studies have focused on the analysis and forecasting of price fluctuations, using mostly traditional approaches for financial markets analysis and prediction [31–35].
价值主要由社会共识产生的虚拟货币和/或资产的自组织市场的出现[13]自然引起了科学界的兴趣[8,14-30]。最近的结果表明，所标记的加密货币的长期属性在 2013 年至 2017 年间保持稳定，并且与投资者简单地对市场进行抽样并根据加密货币的市场份额来分配资金的情况相兼容 [1]。虽然平均而言确实如此，但各种研究都集中在价格波动的分析和预测上，大多使用传统的金融市场分析和预测方法[31-35]。

The success of machine learning techniques for stock markets prediction [36–42] suggests that these methods could be effective also in predicting cryptocurrencies prices. However, the application of machine learning algorithms to the cryptocurrency market has been limited so far to the analysis of Bitcoin prices, using random forests [43], Bayesian neural network [44], long short-term memory neural network [45], and other algorithms [32, 46]. These studies were able to anticipate, to different degrees, the price fluctuations of Bitcoin, and revealed that best results were achieved by neural network based algorithms. Deep reinforcement learning was showed to beat the uniform buy and hold strategy [47] in predicting the prices of 12 cryptocurrencies over one-year period [48].
机器学习技术在股票市场预测方面的成功[36-42]表明这些方法在预测加密货币价格方面也可能有效。然而，到目前为止，机器学习算法在加密货币市场的应用仅限于使用随机森林[43]、贝叶斯神经网络[44]、长短期记忆神经网络[45]和比特币价格分析。其他算法 [32, 46]。这些研究能够在不同程度上预测比特币的价格波动，并表明基于神经网络的算法可以获得最佳结果。在预测一年期间 12 种加密货币的价格方面，深度强化学习被证明能够击败统一的买入并持有策略 [47][48]。

Other attempts to use machine learning to predict the prices of cryptocurrencies other than Bitcoin come from nonacademic sources [49–54]. Most of these analyses focused on a limited number of currencies and did not provide benchmark comparisons for their results.
其他使用机器学习来预测比特币以外的加密货币价格的尝试来自非学术来源[49-54]。这些分析大多数集中于有限数量的货币，并且没有提供其结果的基准比较。

Here, we test the performance of three models in predicting daily cryptocurrency price for 1,681 currencies. Two of the models are based on gradient boosting decision trees [55] and one is based on long short-term memory (LSTM) recurrent neural networks [56]. In all cases, we build investment portfolios based on the predictions and we compare their performance in terms of return on investment. We find that all of the three models perform better than a baseline ‘simple moving average’ model [57–60] where a currency’s price is predicted as the average price across the preceding days and that the method based on long short-term memory recurrent neural networks systematically yields the best return on investment.
在这里，我们测试了三个模型在预测 1,681 种货币的每日加密货币价格方面的性能。其中两个模型基于梯度增强决策树 [55]，一个模型基于长短期记忆 (LSTM) 循环神经网络 [56]。在所有情况下，我们都会根据预测构建投资组合，并在投资回报方面比较它们的表现。我们发现，所有三个模型都比基线“简单移动平均线”模型[57-60]表现更好，其中货币价格被预测为前几天的平均价格，并且基于长期短期记忆循环的方法神经网络系统地产生最佳的投资回报。

The article is structured as follows: In Materials and Methods we describe the data (see Data Description and Preprocessing), the metrics characterizing cryptocurrencies that are used along the paper (see Metrics), the forecasting algorithms (see Forecasting Algorithms), and the evaluation metrics (see Evaluation). In Results, we present and compare the results obtained with the three forecasting algorithms and the baseline method. In Conclusion, we conclude and discuss results.
本文的结构如下：在材料和方法中，我们描述了数据（请参阅数据描述和预处理）、本文中使用的表征加密货币的指标（请参阅指标）、预测算法（请参阅预测算法）以及评估指标（参见评估）。在结果中，我们展示并比较了三种预测算法和基线方法获得的结果。在结论中，我们总结并讨论结果。

2. Materials and Methods
2。材料和方法

2.1. Data Description and Preprocessing
2.1.数据描述和预处理

Cryptocurrency data was extracted from the website Coin Market Cap [61], collecting daily data from 300 exchange markets platforms starting in the period between November 11, 2015, and April 24, 2018. The dataset contains the daily price in US dollars, the market capitalization, and the trading volume of 1,681 cryptocurrencies, where the market capitalization is the product between price and circulating supply, and the volume is the number of coins exchanged in a day. The daily price is computed as the volume weighted average of all prices reported at each market. Figure 1 shows the number of currencies with trading volume larger than V_min over time, for different values of V_min. In the following sections, we consider that only currencies with daily trading volume higher than 10⁵ USD (United States dollar) can be traded at any given day.
加密货币数据摘自 Coin Market Cap 网站[61]，收集了 2015 年 11 月 11 日至 2018 年 4 月 24 日期间从 300 个交易市场平台收集的每日数据。该数据集包含以美元计价的每日价格、市场市值，以及 1,681 种加密货币的交易量，其中市值是价格与流通供应量之间的乘积，交易量是一天内交换的硬币数量。每日价格计算为每个市场报告的所有价格的成交量加权平均值。图 1 显示了随着时间的推移，对于不同的 V _min 值，交易量大于 V _min 的货币数量。在以下部分中，我们认为只有每日交易量高于 10 ⁵ USD（美元）的货币才可以在任何一天进行交易。

Details are in the caption following the image — **Figure 1 图1**
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**Number of cryptocurrencies.** The cryptocurrencies with volume higher than V_min as a function of time, for different values of V_min. For visualization purposes, curves are averaged over a rolling window of 10 days.
加密货币的数量。对于不同的 V _min 值，作为时间函数的交易量高于 V _min 的加密货币。出于可视化目的，曲线是 10 天滚动窗口内的平均值。

The website lists cryptocurrencies traded on public exchange markets that have existed for more than 30 days and for which an API and a public URL showing the total mined supply are available. Information on the market capitalization of cryptocurrencies that are not traded in the 6 hours preceding the weekly release of data is not included on the website. Cryptocurrencies inactive for 7 days are not included in the list released. These measures imply that some cryptocurrencies can disappear from the list to reappear later on. In this case, we consider the price to be the same as before disappearing. However, this choice does not affect results since only in 28 cases the currency has volume higher than 10⁵ USD right before disappearing (note that there are 124,328 entries in the dataset with volume larger than 10⁵ USD).
该网站列出了在公共交易市场上交易的加密货币，这些货币已存在超过 30 天，并且可以使用 API 和公共 URL 显示总开采供应量。网站上不包含每周数据发布前 6 小时内未交易的加密货币的市值信息。 7天内不活跃的加密货币不包含在发布的列表中。这些措施意味着一些加密货币可能会从列表中消失，稍后又会重新出现。在这种情况下，我们认为价格与消失之前相同。但是，此选择不会影响结果，因为仅在 28 种情况下，货币在消失前的交易量高于 10 ⁵ 美元（请注意，数据集中有 124,328 个条目的交易量大于 10 ⁵

2.2. Metrics 2.2.指标

Cryptocurrencies are characterized over time by several metrics, namely,
随着时间的推移，加密货币有几个指标来表征，即：

(i)
Price, the exchange rate, determined by supply and demand dynamics.
价格，即汇率，由供需动态决定。
(ii) (二)
Market capitalization, the product of the circulating supply and the price.
市值，流通供应量与价格的乘积。
(iii) (三)
Market share, the market capitalization of a currency normalized by the total market capitalization.
市场份额，按总市值标准化的货币市值。
(iv) （四）
Rank, the rank of currency based on its market capitalization.
排名，根据其市值对货币进行排名。
(v)
Volume, coins traded in the last 24 hours.
过去 24 小时内的交易量、代币交易量。
(vi) (六)
Age, lifetime of the currency in days.
年龄，货币的生命周期（以天为单位）。

The profitability of a currency c over time can be quantified through the return on investment (ROI), measuring the return of an investment made at day t_i relative to the cost [62]. The index i rolls across days and it is included between 0 and 895, with t₀ = November 11, 2015, and t₈₉₅ = April 24, 2018. Since we are interested in the short-term performance, we consider the return on investment after 1 day defined as
货币 c 随着时间的推移的盈利能力可以通过投资回报率 (ROI) 来量化，衡量 t 日相对于成本的投资回报率 [62]。索引 i 跨天滚动，包含在 0 到 895 之间，其中 t ₀ = 2015 年 11 月 11 日，t ₈₉₅ = 2018 年 4 月 24 日。短期表现，我们考虑1天后的投资回报率，定义为

$mathematical equation$ (1)

In Figure 2, we show the evolution of the ROI over time for Bitcoin (orange line) and on average for currencies whose volume is larger than V_min = 10⁵ USD at t_i − 1 (blue line). In both cases, the average return on investment over the period considered is larger than 0, reflecting the overall growth of the market.
在图 2 中，我们显示了比特币（橙色线）以及在 t 时交易量大于 V _min = 10 ⁵ 美元的货币的平均投资回报率随时间的演变 - 1（蓝线）。在这两种情况下，所考虑期间的平均投资回报率都大于 0，反映了市场的整体增长。

2.3. Forecasting Algorithms
2.3.预测算法

We test and compare three supervised methods for short-term price forecasting. The first two methods rely on XGBoost [63], an open-source scalable machine learning system for tree boosting used in a number of winning Kaggle solutions (17/29 in 2015) [64]. The third method is based on the long short-term memory (LSTM) algorithm for recurrent neural networks [56] that have demonstrated to achieve state-of-the-art results in time-series forecasting [65].
我们测试并比较了三种短期价格预测的监督方法。前两种方法依赖于 XGBoost [63]，这是一种用于树提升的开源可扩展机器学习系统，用于许多获胜的 Kaggle 解决方案（2015 年 17/29）[64]。第三种方法基于循环神经网络的长短期记忆（LSTM）算法[56]，该算法已被证明可以在时间序列预测中实现最先进的结果[65]。

Method 1. The first method considers one single regression model to describe the change in price of all currencies (see Figure 3). The model is an ensemble of regression trees built by the XGBoost algorithm. The features of the model are characteristics of a currency between time t_j − w and t_j − 1 and the target is the ROI of the currency at time t_j, where w is a parameter to be determined. The characteristics considered for each currency are price, market capitalization, market share, rank, volume, and ROI (see (1)). The features for the regression are built across the window between t_j − w and t_j − 1 included (see Figure 3). Specifically, we consider the average, the standard deviation, the median, the last value, and the trend (e.g., the difference between last and first value) of the properties listed above. In the training phase, we include all currencies with volume larger than 10⁵ USD and t_j between t_i − W_training and t_i. In general, larger training windows do not necessarily lead to better results (see results section), because the market evolves across time. In the prediction phase, we test on the set of existing currencies at day t_i. This procedure is repeated for values of t_i included between January 1, 2016, and April 24, 2018.
方法1. 第一种方法考虑一个单一回归模型来描述所有货币的价格变化（见图3）。该模型是由 XGBoost 算法构建的回归树的集合。模型的特征是货币在时间 t − w 和 t − 1 之间的特征，目标是货币在时间 t 的 ROI，其中 w 是待确定的参数。每种货币考虑的特征是价格、市值、市场份额、排名、交易量和投资回报率（参见（1））。回归特征是在 t − w 和 t − 1 之间的窗口内构建的（见图 3）。具体来说，我们考虑上面列出的属性的平均值、标准差、中位数、最后一个值和趋势（例如，最后一个值和第一个值之间的差异）。在训练阶段，我们包括交易量大于 10 ⁵ 美元且 t 在 t − W _training 和 t 之间的所有货币。一般来说，较大的训练窗口不一定会带来更好的结果（参见结果部分），因为市场会随着时间的推移而发展。在预测阶段，我们在 t 天测试现有货币集。对于 2016 年 1 月 1 日至 2018 年 4 月 24 日期间包含的 t 值，重复此过程。

Method 2. Also the second method relies on XGBoost, but now the algorithm is used to build a different regression model for each currency c_i (see Figure 4). The features of the model for currency c_i are the characteristics of all the currencies in the dataset between t_j − w and t_j − 1 included and the target is the ROI of c_i at day t_j (i.e., now the algorithm learns to predict the price of the currency i based on the features of all the currencies in the system between t_j − w and t_j − 1). The features of the model are the same used in Method 1 (e.g., the average, standard, deviation, median, last value, and difference between last and first value of the following quantities: price, market capitalization, market share, rank, volume, and ROI) across a window of length w. The model for currency c_i is trained with pairs features target between times t_i − W_training and t_i − 1. The prediction set includes only one pair: the features (computed between t_i − w and t_i − 1) and the target (computed at t_i) of currency c_i.
方法2。第二种方法也依赖于XGBoost，但现在该算法用于为每种货币c构建不同的回归模型（见图4）。货币 c 的模型的特征是包含在 t − w 和 t − 1 之间的数据集中所有货币的特征，目标是 c 在第 t 天的 ROI（即，现在算法学习预测货币 i 基于 t − w 和 t − 1 之间系统中所有货币的特征。该模型的特征与方法 1 中使用的相同（例如，平均值、标准、偏差、中值、最后一个值以及以下数量的最后一个值与第一个值之间的差值：价格、市值、市场份额、排名、交易量和 ROI）跨越长度为 w 的窗口。货币 c 的模型使用时间 t − W _training 和 t − 1 之间的对特征目标进行训练。预测集仅包含一对：特征（在 t − w 和 t − 1 之间计算）和货币的目标（在 t 计算） c．

Method 3. The third method is based on long short-term memory networks, a special kind of recurrent neural networks, capable of learning long-term dependencies. As for Method 2, we build a different model for each currency. Each model predicts the ROI of a given currency at day t_i based on the values of the ROI of the same currency between days t_i − w and t_i − 1 included.
方法3。第三种方法基于长短期记忆网络，这是一种特殊的循环神经网络，能够学习长期依赖性。对于方法2，我们为每种货币建立了不同的模型。每个模型根据 t − w 天和 t − 1 天之间相同货币的 ROI 值来预测 t 天给定货币的 ROI。

Baseline Method. As baseline method, we adopt the simple moving average strategy (SMA) widely tested and used as a null model in stock market prediction [57–60]. It estimates the price of a currency at day t_i as the average price of the same currency between t_i − w and t_i − 1 included.
基线方法。作为基线方法，我们采用简单移动平均策略（SMA），该策略已被广泛测试并用作股市预测中的零模型[57-60]。它将 t 天的货币价格估计为 t − w 和 t − 1 之间相同货币的平均价格。

2.4. Evaluation 2.4.评估

We compare the performance of various investment portfolios built based on the algorithms predictions. The investment portfolio is built at time t_i − 1 by equally splitting an initial capital among the top n currencies predicted with positive return. Hence, the total return at time t_i is
我们比较基于算法预测构建的各种投资组合的表现。投资组合是在时间 t − 1 时通过将初始资本平均分配给预测为正回报的前 n 个货币来构建的。因此，时间 t 的总回报为

$mathematical equation$ (2)

The portfolios performance is evaluated by computing the Sharpe ratio and the geometric mean return. The Sharpe ratio is defined as
通过计算夏普比率和几何平均回报来评估投资组合的表现。夏普比率定义为

$mathematical equation$ (3)

where $mathematical equation$ is the average return on investment obtained between times 0 and t_i and s_R is the corresponding standard deviation.
其中 $mathematical equation$ 是时间 0 到 t 之间获得的平均投资回报率，s _R 是相应的标准差。

The geometric mean return is defined as
几何平均回报定义为

$mathematical equation$ (4)

where t_i corresponds to the total number of days considered. The cumulative return obtained at t_i after investing and selling on the following day for the whole period is defined as G(t_i) ².
其中 t 对应于所考虑的总天数。整个期间在次日投资并卖出后在 t 时刻获得的累计收益定义为 G(t) ² 。

The number of currencies n to include in a portfolio is chosen at t_i by optimising either the geometric mean G(t_i − 1) (geometric mean optimisation) or the Sharpe ratio S(t_i − 1) (Sharpe ratio optimisation) over the possible choices of n. The same approach is used to choose the parameters of Method 1 (w and W_training), Method 2 (w and W_training), and the baseline method (w).
通过在可能的选择上优化几何平均值 G(t − 1)（几何平均值优化）或夏普比率 S(t − 1)（夏普比率优化），在 t 处选择投资组合中包含的货币数量 n的 n.使用相同的方法选择方法 1（w 和 W _training ）、方法 2（w 和 W _training ）和基线方法 (w) 的参数。

3. Results 3. 结果

We predict the price of the currencies at day t_i, for all t_i included between Jan 1, 2016, and Apr 24, 2018. The analysis considers all currencies whose age is larger than 50 days since their first appearance and whose volume is larger than $100000. To discount for the effect of the overall market movement (i.e., market growth, for most of the considered period), we consider cryptocurrencies prices expressed in BTC (Bitcoin). This implies that Bitcoin is excluded from our analysis.
我们预测 2016 年 1 月 1 日至 2018 年 4 月 24 日期间所有 t 日货币的价格。该分析考虑了自首次出现以来超过 50 天且交易量大于 100000 美元的所有货币。为了抵消整体市场变动的影响（即在所考虑的大部分时间段内的市场增长），我们考虑以 BTC（比特币）表示的加密货币价格。这意味着比特币被排除在我们的分析之外。

3.1. Parameter Setting 3.1.参数设置

First, we choose the parameters for each method. Parameters include the number of currencies n to include the portfolio as well as the parameters specific to each method. In most cases, at each day t_i we choose the parameters that maximise either the geometric mean G(t_i − 1) (geometric mean optimisation) or the Sharpe ratio S(t_i − 1) (Sharpe ratio optimisation) computed between times 0 and t_i.
首先，我们为每个方法选择参数。参数包括包含投资组合的货币数量 n 以及特定于每种方法的参数。在大多数情况下，在每一天 t，我们选择最大化在时间 0 和 t 之间计算的几何平均值 G(t − 1)（几何平均优化）或夏普比率 S(t − 1)（夏普比率优化）的参数。

Baseline Strategy. We test the performance of the baseline strategy for choices of window w ≥ 2 (the minimal requirement for the ROI to be different from 0) and w < 30. We find that the value of w mazimising the geometric mean return (see Appendix Section A) and the Sharpe ratio (see Appendix Section A) fluctuates especially before November 2016 and has median value 4 in both cases. The number of currencies included in the portfolio oscillates between 1 and 11 with median at 3, both for the Sharpe ratio (see Appendix Section A) and the geometric mean return (see Appendix Section A) optimisation.
基线策略。我们测试了窗口 w ≥ 2（ROI 不同于 0 的最低要求）和 w < 30 的基线策略的性能。我们发现 w 的值使几何平均回报最大化（参见附录 A 部分））和夏普比率（参见附录 A 部分）尤其在 2016 年 11 月之前波动，并且在两种情况下中值为 4。投资组合中包含的货币数量在 1 到 11 种之间波动，中位数为 3，这都是为了夏普比率（参见附录 A 部分）和几何平均回报（参见附录 A 部分）优化。

Method 1. We explore values of the window w in {3,5, 7,10} days and the training period W_training in {5,10,20} days (see Appendix Section A). We find that the median value of the selected window w across time is 7 for both the Sharpe ratio and the geometric mean optimisation. The median value of W_training is 5 under geometric mean optimisation and 10 under Sharpe ratio optimisation. The number of currencies included in the portfolio oscillates between 1 and 43 with median at 15 for the Sharpe ratio (see Appendix Section A) and 9 for the geometric mean return (see Appendix Section A) optimisation.
方法 1. 我们探索 {3,5,7,10} 天的窗口 w 值和 {5,10,20} 天的训练周期 W _training （参见附录 A 部分）。我们发现，对于夏普比率和几何平均优化，所选窗口 w 随时间变化的中值为 7。 W _training 的中值在几何平均优化下为 5，在夏普比率优化下为 10。投资组合中包含的货币数量在 1 到 43 种之间波动，夏普比率（参见附录 A 部分）的中位数为 15，几何平均回报（参见附录 A 部分）优化的中位数为 9。

Method 2. We explore values of the window w in {3,5, 7,10} days and the training period W_training in {5,10,20} days (see Appendix, Figure 10). The median value of the selected window w across time is 3 for both the Sharpe ratio and the geometric mean optimisation. The median value of W_training is 10 under geometric mean and Sharpe ratio optimisation. The number of currencies included has median at 17 for the Sharpe ratio and 7 for the geometric mean optimisation (see Appendix Section A).
方法2.我们探索{3,5,7,10}天内的窗口w值和{5,10,20}天内的训练周期W _training （参见附录，图10）。对于夏普比率和几何平均优化，所选窗口 w 在时间上的中值为 3。在几何平均和夏普比率优化下，W _training 的中值为10。所包含的货币数量的夏普比率中位数为 17，几何平均优化中位数为 7（参见附录 A 部分）。

Method 3. The LSTM has three parameters: The number of epochs, or complete passes through the dataset during the training phase; the number of neurons in the neural network, and the length of the window w. These parameters are chosen by optimising the price prediction of three currencies (Bitcoin, Ripple, and Ethereum) that have on average the largest market share across time (excluding Bitcoin Cash that is a fork of Bitcoin). Results (see Appendix Section A) reveal that, in the range of parameters explored, the best results are achieved for w = 50. Results are not particularly affected by the choice of the number of neurones nor the number of epochs. We choose 1 neuron and 1000 epochs since the larger these two parameters, the larger the computational time. The number of currencies to include in the portfolio is optimised over time by mazimising the geometric mean return (see Appendix Section A) and the Sharpe ratio (see Appendix Section A). In both cases the median number of currencies included is 1.
方法 3. LSTM 具有三个参数： epoch 的数量，或者说在训练阶段对数据集的完整遍历；神经网络中神经元的数量以及窗口 w 的长度。这些参数是通过优化三种货币（比特币、瑞波币和以太坊）的价格预测来选择的，这三种货币在一段时间内平均拥有最大的市场份额（不包括作为比特币分叉的比特币现金）。结果（参见附录 A 部分）表明，在探索的参数范围内，w = 50 时可获得最佳结果。结果并没有特别受到神经元数量和 epoch 数量选择的影响。我们选择 1 个神经元和 1000 个 epoch，因为这两个参数越大，计算时间就越长。随着时间的推移，通过最大化几何平均回报（参见附录 A 部分）和夏普比率（参见附录 A 部分），投资组合中包含的货币数量得到优化。在这两种情况下，包含的货币数量中位数均为 1。

3.2. Cumulative Return 3.2.累计回报

In Figure 5, we show the cumulative return obtained using the 4 methods. The cumulative returns achieved on April 24 under the Sharpe ratio optimisation are ~65 BTC (Baseline), ~1.1 · 10³ BTC (Method 1), ~95 BTC (Method 2), ~1.2 · 10⁹ BTC (Method 3). Under geometric mean optimisation we obtain ~25 BTC (Baseline), ~19 · 10³ BTC (Method 1), ~1.25 BTC (Method 2), ~3.6 · 10⁸ BTC (Method 3). The cumulative returns obtained in USD are higher (see Appendix Section D). This is expected, since the Bitcoin price has increased during the period considered. While some of these figures appear exaggerated, it is worth noticing that (i) we run a theoretical exercise assuming that the availability of Bitcoin is not limited and (ii) under this assumption the upper bound to our strategy, corresponding to investing every day in the most performing currency results in a total cumulative return of 6 · 10¹²³ BTC (see Appendix Section B). We consider also the more realistic scenario of investors paying a transaction fee when selling and buying currencies (see Appendix Section C). In most exchange markets, the fee is typically included between 0.1% and 0.5% of the traded amount [66]. For fees up to 0.2%, all the investment methods presented above lead, on average, to positive returns over the entire period (see Appendix Section C). The best performing method, Method 3, achieves positive gains also when fees up to 1% are considered (see Appendix Section C).
在图 5 中，我们显示了使用 4 种方法获得的累积回报。夏普比率优化下 4 月 24 日实现的累积回报为 ~65 BTC（基线）、~1.1 · 10 ³ BTC（方法 1）、~95 BTC（方法 2）、~1.2 · 10 < b1> BTC（方法 3）。在几何平均优化下，我们获得 ~25 BTC（基线）、~19 · 10 ³ BTC（方法 1）、~1.25 BTC（方法 2）、~3.6 · 10 ⁸ BTC （方法3）。以美元计算获得的累积回报更高（参见附录 D 部分）。这是预料之中的，因为比特币价格在所考虑的时期内有所上涨。虽然其中一些数字似乎有些夸大，但值得注意的是（i）我们进行了一项理论练习，假设比特币的可用性不受限制，以及（ii）在此假设下，我们策略的上限，对应于每天投资于表现最好的货币的总累积回报为 6 · 10 ¹²³ BTC（请参阅附录 B 部分）。我们还考虑了投资者在买卖货币时支付交易费的更现实的情况（参见附录 C 部分）。在大多数交易市场中，费用通常包含在交易金额的 0.1% 到 0.5% 之间 [66]。对于高达 0.2% 的费用，上述所有投资方法平均而言会在整个期间带来正回报（参见附录 C 部分）。表现最佳的方法（方法 3）在考虑高达 1% 的费用时也能实现正收益（参见附录 C 部分）。

The cumulative return in Figure 5 is obtained by investing between January 1st, 2016 and April 24th, 2018. We investigate the overall performance of the various methods by looking at the geometric mean return obtained in different periods (see Figure 6). Results presented in Figure 6 are obtained under Sharpe ratio optimisation for the baseline (Figure 6(a)), Method 1 (Figure 6(b)), Method 2 (Figure 6(c)), and Method 3 (Figure 6(d)). Note that, while in this case the investment can start after January 1, 2016, we optimised the parameters by using data from that date on in all cases. Results are considerably better than those achieved using geometric mean return optimisation (see Appendix Section E). Finally, we observe that better performance is achieved when the algorithms consider prices in Bitcoin rather than USD (see Appendix Section D).
图5中的累积回报是通过2016年1月1日至2018年4月24日之间的投资获得的。我们通过观察不同时期获得的几何平均回报来调查各种方法的整体表现（见图6）。图 6 中显示的结果是在基线夏普比率优化下获得的（图 6(a)）、方法 1（图 6(b)）、方法 2（图 6(c)）和方法 3（图 6(d) ））。请注意，虽然在这种情况下投资可以在 2016 年 1 月 1 日之后开始，但我们在所有情况下都使用该日期以来的数据来优化参数。结果比使用几何平均回报优化所获得的结果要好得多（参见附录 E 部分）。最后，我们观察到，当算法考虑比特币价格而不是美元价格时，可以获得更好的性能（参见附录 D 部分）。

3.3. Feature Importance 3.3.特征重要性

In Figure 7, we illustrate the relative importance of the various features in Method 1 and Method 2. For Method 1, we show the average feature importance. For Method 2, we show the average feature importance for two sample currencies: Ethereum and Ripple.
在图 7 中，我们说明了方法 1 和方法 2 中各个特征的相对重要性。对于方法 1，我们显示了平均特征重要性。对于方法 2，我们显示了两种示例货币的平均特征重要性：以太坊和瑞波币。

3.4. Portfolio Composition
3.4.投资组合构成

The 10 most selected currencies under Sharpe ratio optimisation are the following:
夏普比率优化中最常选择的 10 种货币如下：

Baseline. Factom (91 days), E-Dinar Coin (89 days), Ripple (76 days), Ethereum (71 days), Steem (70 days), Lisk (70 days), MaidSafeCoin (69 days), Monero (58 days), BitShares (55 days), EDRCoin (52 days).
基线。 Factom（91 天）、E-Dinar Coin（89 天）、Ripple（76 天）、Ethereum（71 天）、Steem（70 天）、Lisk（70 天）、MaidSafeCoin（69 天）、Monero（58 天）、比特股（55 天）、EDRCoin（52 天）。

Method 1. Ethereum (154 days), Dash (128 days), Monero (111 days), Factom (104 days), Ripple (94 days), Litecoin (93 days), Dogecoin (92 days), Maid Safe Coin (86 days), BitShares (73 days), Tether (59 days)
方法1.以太坊（154天）、达世币（128天）、门罗币（111天）、Factom（104天）、瑞波币（94天）、莱特币（93天）、狗狗币（92天）、Maid Safe Coin（86天）天）、比特股（73 天）、Tether（59 天）

Method 2. Ethereum (63 days), Monero (61 days), Factom (51 days), Ripple (42 days), Dash (40 days), Maid Safe Coin (40 days), Siacoin (30 days), NEM (26 days), NXT (26 days), Steem (23 days).
方法 2. 以太坊（63 天）、门罗币（61 天）、Factom（51 天）、Ripple（42 天）、Dash（40 天）、Maid Safe Coin（40 天）、Siacoin（30 天）、NEM（26 天）天）、NXT（26 天）、Steem（23 天）。

Method 3. Factom (48 days), Monero (46 days), Ethereum (39 days), Lisk (36 days), Maid Safe Coin (32 days), E-Dinar Coin (32 days), BitShares (26 days), B3 Coin (26 days), Dash (25 days), Cryptonite (22 days).
方法 3. Factom（48 天）、Monero（46 天）、Ethereum（39 天）、Lisk（36 天）、Maid Safe Coin（32 天）、E-Dinar Coin（32 天）、BitShares（26 天）、 B3 Coin（26 天）、Dash（25 天）、Cryptonite（22 天）。

4. Conclusion 4。结论

We tested the performance of three forecasting models on daily cryptocurrency prices for 1,681 currencies. Two of them (Method 1 and Method 2) were based on gradient boosting decision trees and one is based on long short-term memory recurrent neural networks (Method 3). In Method 1, the same model was used to predict the return on investment of all currencies; in Method 2, we built a different model for each currency that uses information on the behaviour of the whole market to make a prediction on that single currency; in Method 3, we used a different model for each currency, where the prediction is based on previous prices of the currency.
我们测试了三种预测模型对 1,681 种货币的每日加密货币价格的性能。其中两种（方法 1 和方法 2）基于梯度提升决策树，一种基于长短期记忆循环神经网络（方法 3）。方法一中，使用相同的模型来预测所有币种的投资回报率；在方法2中，我们为每种货币建立了不同的模型，该模型使用整个市场行为的信息来对该单一货币进行预测；在方法 3 中，我们对每种货币使用不同的模型，其中预测基于货币的先前价格。

We built investment portfolios based on the predictions of the different method and compared their performance with that of a baseline represented by the well-known simple moving average strategy. The parameters of each model were optimised for all but Method 3 on a daily basis, based on the outcome of each parameters choice in previous times. We used two evaluation metrics used for parameter optimisation: The geometric mean return and the Sharpe ratio. To discount the effect of the overall market growth, cryptocurrencies prices were expressed in Bitcoin. All strategies produced profit (expressed in Bitcoin) over the entire considered period and for a large set of shorter trading periods (different combinations of start and end dates for the trading activity), also when transaction fees up to 0.2% are considered.
我们根据不同方法的预测构建投资组合，并将其表现与众所周知的简单移动平均策略代表的基线进行比较。根据之前每个参数选择的结果，每天对除方法 3 之外的所有模型的参数进行优化。我们使用了两个用于参数优化的评估指标：几何平均回报率和夏普比率。为了抵消整体市场增长的影响，加密货币价格以比特币表示。所有策略在整个考虑期间和大量较短交易期间（交易活动开始和结束日期的不同组合）都产生了利润（以比特币表示），并且考虑了高达 0.2% 的交易费用。

The three methods performed better than the baseline strategy when the investment strategy was ran over the whole period considered. The optimisation of parameters based on the Sharpe ratio achieved larger returns. Methods based on gradient boosting decision trees (Methods 1 and 2) worked best when predictions were based on short-term windows of 5/10 days, suggesting they exploit well mostly short-term dependencies. Instead, LSTM recurrent neural networks worked best when predictions were based on ~50 days of data, since they are able to capture also long-term dependencies and are very stable against price volatility. They allowed making profit also if transaction fees up to 1% are considered. Methods based on gradient boosting decision trees allow better interpreting results. We found that the prices and the returns of a currency in the last few days preceding the prediction were leading factors to anticipate its behaviour. Among the two methods based on random forests, the one considering a different model for each currency performed best (Method 2). Finally, it is worth noting that the three methods proposed perform better when predictions are based on prices in Bitcoin rather than prices in USD. This suggests that forecasting simultaneously the overall cryptocurrency market trend and the developments of individual currencies is more challenging than forecasting the latter alone.
当投资策略在整个考虑期间运行时，这三种方法的表现优于基准策略。基于夏普比率的参数优化取得了较大的回报。当预测基于 5/10 天的短期窗口时，基于梯度提升决策树的方法（方法 1 和 2）效果最佳，这表明它们充分利用了大部分短期依赖性。相反，当基于约 50 天的数据进行预测时，LSTM 循环神经网络效果最佳，因为它们还能够捕获长期依赖性，并且对于价格波动非常稳定。如果考虑高达 1% 的交易费用，他们也可以盈利。基于梯度提升决策树的方法可以更好地解释结果。我们发现预测前最后几天货币的价格和回报是预测其行为的主要因素。在基于随机森林的两种方法中，考虑每种货币不同模型的方法表现最佳（方法 2）。最后，值得注意的是，当预测基于比特币价格而不是美元价格时，所提出的三种方法表现更好。这表明，同时预测整体加密货币市场趋势和个别货币的发展比单独预测后者更具挑战性。

It is important to stress that our study has limitations. First, we did not attempt to exploit the existence of different prices on different exchanges, the consideration of which could open the way to significantly higher returns on investment. Second, we ignored intraday price fluctuations and considered an average daily price. Finally, and crucially, we run a theoretical test in which the available supply of Bitcoin is unlimited and none of our trades influence the market. Notwithstanding these simplifying assumptions, the methods we presented were systematically and consistently able to identify outperforming currencies. Extending the current analysis by considering these and other elements of the market is a direction for future work.
需要强调的是，我们的研究有其局限性。首先，我们并没有试图利用不同交易所存在的不同价格，考虑到这一点可能会为显着提高投资回报开辟道路。其次，我们忽略了日内价格波动并考虑了日均价格。最后，也是至关重要的，我们进行了一项理论测试，其中比特币的可用供应是无限的，并且我们的交易不会影响市场。尽管有这些简化的假设，我们提出的方法仍然能够系统地、一致地识别表现优异的货币。通过考虑市场的这些因素和其他因素来扩展当前的分析是未来工作的方向。

A different yet promising approach to the study cryptocurrencies consists in quantifying the impact of public opinion, as measured through social media traces, on the market behaviour, in the same spirit in which this was done for the stock market [67]. While it was shown that social media traces can be also effective predictors of Bitcoin [68–74] and other currencies [75] price fluctuations, our knowledge of their effects on the whole cryptocurrency market remain limited and is an interesting direction for future work.
研究加密货币的一种不同但有前途的方法是量化公众舆论对市场行为的影响（通过社交媒体痕迹衡量），这与股票市场的做法相同[67]。虽然研究表明社交媒体痕迹也可以有效预测比特币[68-74]和其他货币[75]价格波动，但我们对其对整个加密货币市场影响的了解仍然有限，这是未来工作的一个有趣方向。

Conflicts of Interest 利益冲突

The authors declare that they have no conflicts of interest.
作者声明他们没有利益冲突。

Appendix 附录

A. Parameter Optimisation
A. 参数优化

In Figure 8, we show the optimisation of the parameters w (a, c) and n (b, d) for the baseline strategy. In Figure 9, we show the optimisation of the parameters w (a, d), W_training (b, e), and n (c, f) for Method 1. In Figure 10, we show the optimisation of the parameters w (a, d), W_training (b, e), and n (c, f) for Method 2. In Figure 11, we show the median squared error obtained under different training window choices (a), number of epochs (b) and number of neurons (c), for Ethereum, Bitcoin and Ripple. In Figure 12, we show the optimisation of the parameter n (c, f) for Method 3.
在图 8 中，我们显示了基线策略的参数 w (a, c) 和 n (b, d) 的优化。在图 9 中，我们显示了方法 1 的参数 w (a, d)、W _training (b, e) 和 n (c, f) 的优化。在图 10 中，我们显示了对方法 2 的参数 w (a, d)、W _training (b, e) 和 n (c, f) 进行优化。在图 11 中，我们显示了在不同训练下获得的中值平方误差以太坊、比特币和瑞波币的窗口选择 (a)、纪元数 (b) 和神经元数 (c)。在图 12 中，我们展示了方法 3 的参数 n (c, f) 的优化。

B. Return under Full Knowledge of the Market Evolution
B. 充分了解市场演变情况下的回报

In Figure 13, we show the cumulative return obtained by investing every day in the top currency, supposing one knows the prices of currencies on the following day.
在图 13 中，我们显示了每天投资顶级货币所获得的累积回报，假设人们知道第二天的货币价格。

C. Return Obtained Paying Transaction Fees
C. 支付交易费用获得的回报

In this section, we present the results obtained including transaction fees between 0.1% and 1% [66]. In general, one can not trade a given currency with any given other. Hence, we consider that each day we trade twice: We sell altcoins to buy Bitcoin, and we buy new altcoins using Bitcoin. The mean return obtained between Jan. 2016 and Apr. 2018 is larger than 1 for all methods, for fees up to 0.2% (see Table 1). In this period, Method 3 achieves positive returns for fees up to 1%. The returns obtained with a 0.1% (see Figure 14) and 0.2% (see Figure 15) fee during arbitrary periods confirm that, in general, one obtains positive gains with our methods if fees are small enough.
在本节中，我们介绍获得的结果，包括 0.1% 到 1% 之间的交易费用 [66]。一般来说，一个人不能将一种给定的货币与任何给定的另一种货币进行交易。因此，我们认为每天我们交易两次：我们出售山寨币来购买比特币，然后我们使用比特币购买新的山寨币。 2016 年 1 月至 2018 年 4 月期间，所有方法获得的平均回报均大于 1，费用高达 0.2%（见表 1）。在此期间，方法3实现了高达1%的费用正回报。在任意时期内以 0.1%（见图 14）和 0.2%（见图 15）费用获得的回报证实，一般来说，如果费用足够小，则使用我们的方法可以获得正收益。

Table 1. Daily geometric mean return for different transaction fees. Results are obtained considering the period between Jan. 2016 and Apr. 2018.
表1.不同交易费用的每日几何平均收益。结果是根据 2016 年 1 月至 2018 年 4 月期间得出的。

	no fee 不收费	0.1%	0.2%	0.3%	0.5%	1%
Baseline 基线	1.005	1.003	1.001	0.999	0.995	0.985
Method 1 方法一	1.008	1.006	1.004	1.002	0.998	0.988
Method 2 方法2	1.005	1.003	1.001	0.999	0.995	0.985
Method 3 方法三	1.025	1.023	1.021	1.019	1.015	1.005

D. Results in USD
D. 结果以美元计算

In this section, we show results obtained considering prices in USD. The price of Bitcoin in USD has considerably increased in the period considered. Hence, gains in USD (Figure 16) are higher than those in Bitcoin (Figure 5). Note that, in Figure 16, we have made predictions and computed portfolios considering prices in Bitcoin. Then, gains have been converted to USD (without transaction fees). In Table 2, we show instead the gains obtained running predictions considering directly all prices in USD. We find that, in most cases, better results are obtained from prices in BTC.
在本节中，我们显示考虑美元价格而获得的结果。在此期间，以美元计价的比特币价格大幅上涨。因此，美元收益（图 16）高于比特币收益（图 5）。请注意，在图 16 中，我们考虑了比特币的价格进行了预测并计算了投资组合。然后，收益已转换为美元（不含交易费用）。在表 2 中，我们显示了直接考虑所有美元价格的运行预测所获得的收益。我们发现，在大多数情况下，从 BTC 价格中可以获得更好的结果。

Table 2. Geometric mean returns in USD. Results are obtained for the various methods by running the algorithms considering prices in BTC (left column) and USD (right column).
表 2. 以美元计算的几何平均回报。通过运行考虑 BTC（左列）和 USD（右列）价格的算法，获得各种方法的结果。

	Geometric mean in USD (from BTC prices) 美元几何平均值（来自 BTC 价格）	Geometric mean in USD (from USD prices) 美元几何平均值（来自美元价格）
Baseline 基线	1.0086	1.0141
Method1 方法1	1.0121	1.0085
Method2 方法2	1.0091	1.0086
Method3 方法3	1.0289	1.0134

E. Geometric Mean Optimisation
E. 几何平均优化

In Figure 17, we show the geometric mean return obtained by between two arbitrary points in time under geometric mean return optimisation for the baseline (Figure 17(a)), Method 1 (Figure 17(b)), Method 2 (Figure 17(c)), and Method 3 (Figure 17(d)).
在图17中，我们展示了在基线的几何平均收益优化下，两个任意时间点之间获得的几何平均收益（图17（a）），方法1（图17（b）），方法2（图17（ c)) 和方法 3（图 17(d)）。

Open Research 开放研究

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Issue Online: 04 November 2018
Version of Record online: 04 November 2018
Manuscript accepted: 17 October 2018
Manuscript revised: 28 September 2018
Manuscript received: 29 May 2018

Anticipating Cryptocurrency Prices Using Machine Learning
使用机器学习预测加密货币价格

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Abstract 抽象的

1. Introduction 一、简介

2. Materials and Methods
2。材料和方法