Social media signals have been successfully used to develop large-scale predictive and anticipatory analytics. For example, forecasting stock market prices and influenza outbreaks. Recently, social data has been explored to forecast price fluctuations of cryptocurrencies, which are a novel disruptive technology with significant political and economic implications. In this paper we leverage and contrast the predictive power of social signals, specifically user behavior and communication patterns, from multiple social platforms GitHub and Reddit to forecast prices for three cyptocurrencies with high developer and community interest – Bitcoin, Ethereum, and Monero.

We evaluate the performance of neural network models that rely on long short-term memory units (LSTMs) trained on historical price data and social data against price-only LSTMs and baseline autoregressive integrated moving average (ARIMA) models, commonly used to predict stock prices. Our results not only demonstrate that social signals reduce error when forecasting daily coin price, but also show that the language used in comments within the official communities on Reddit (r/Bitcoin, r/Ethereum, and r/Monero) are the best predictors overall. We observe that models are more accurate in forecasting price one day ahead for Bitcoin (4% root mean squared percent error) compared to Ethereum (7%) and Monero (8%).

§ INTRODUCTION

Cryptocurrencies, like Bitcoin, Ethereum, and Monero, are a new and disruptive technology that are often leveraged in highly volatile and fast-evolving environments.
As with stocks and other securities, cryptocurrencies are bought, held, and traded. Unlike traditional currencies and stocks, these digital currencies rely on decentralized systems and cryptographic technologies, blockchain ledgers, rather than a centralized institution, banks. In this new paradigm, money is moved more quickly, independently, and often anonymously or semi-anonymously. As a result, the wide adoption and historic volatility of cryptocurrencies have significant political and economic implications. Price speculation, where traders buy securities in the hopes that they will quickly rise in price, often occurs in these highly volatile markets. Speculative trading typically occurs with little (or no) regard to the asset's fundamental value, but rather in regard to patterns in the the asset's price movement, rumor, or other suppositious data.

Signals from social media have been extensively used to predict real world events such as election results [36, 29, 5, 9, 38, 12], movie sales [21, 2, 33, 1],
protests [17],
public health events [37, 7, 15, 25, 3], and  stock market activity [4, 6, 18, 22, 19, 20, 27, 23, 28, 42, 16, 41]. Ding  ding2014using introduced a deep neural network approach to predict the directionality of stock prices and the S&P 500 index using signals from related news events and Tetlock [34] highlighted the correlation between media pessimism and  market prices and volume. Bollen  bollen2011twitter predict relative differences in the daily Dow Jones industrial average using measures of collective mood states derived from Twitter activity and found the addition of some but not all possible states improved the predictive ability of their proposed models. Similar to Bollen , we analyze the benefit of including or excluding a range of social signals in our proposed models.

Like in stock markets and securities, cryptocurrency market activity has also been predicted using social signals. Previous work by Kim  kim2016predicting predicted the direction of price fluctuations for cryptocurrency coins utilizing social signals from online cryptocurrency forums. Other studies focus on predicting relative changes, the return, in coin prices [28, 39]. For example, Wang and Verne wang2017buzz proposed a model to predict the return, the change in price relative to the opening value. Phillips and Gorse phillips2017predicting predict the beginning and end of spikes in cryptocurrency prices, which they call  “price bubbles”, using hidden Markov models that were previously used for the detection of influenza outbreaks. Although these previous works were able to predict certain changes in a cryptocurrency's price, they were not able to predict the actual price of the asset. In the present work we evaluate the benefit of incorporating a variety of social signals into models in order to forecast the actual daily price high-values of three popular cryptocurrencies.

Coins plotted by developer and community interest. The present work will focus on three of the most popular: Bitcoin (BTC), Ethereum (ETH), and Monero (XMR)

Motivation: the alignment of price high (USD) and social interactions on Reddit (r/bitcoin) and GitHub (bitcoin/bitcoin).

With the increasing use and reliance on these digital currencies, price fluctuation forecasting is an interesting yet difficult challenge. We address this problem by leveraging the predictive power of social signals from GitHub and Reddit to forecast immediate and near future prices of three popular cryptocurrencies: Bitcoin, Monero and Ethereum. In summary, our main contributions are:

* we develop neural network models that incorporate both social and price signals to generate forecasts of coin prices,

* we present an in-depth analysis of model performance for coin price forecasts up to 3 days in advance for Bitcoin, Ethereum, and Monero when incorporating a variety of social signals and the relative improvement over models that rely solely on price history, and

* we report average performance of models for forecasts of coin price up to two weeks in advance.

§.§ Why Bitcoin, Ethereum, and Monero?

A preliminary analysis of potential cryptocurrencies identified Bitcoin, Ethereum, and Monero as the top three cryptocurrencies in terms of both developer interest on GitHub and community interest on social media platforms. To determine which coins to focus on in this study, we collected data from CoinGecko[<https://www.coingecko.com/en/coins/all>] for 1,742 cryptocurrencies and performed an initial analysis of coins in terms of the developer interest (a measure of activity in public repositories on GitHub and Bitbucket) and community interest (a measure of discussions and popularity on social media) features released by CoinGecko.
The key results of this analysis are illustrated in Figure <ref>, which shows that Bitcoin, Ethereum, and Monero have the highest degree of both developer and community interest.

§ SOCIAL AND FINANCIAL DATA COLLECTION

In this study, we focus on taking advantage of social data to forecast the price for three cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), and Monero (XMR). Bitcoin is the first decentralized cryptocurrency and holds the highest market capitalization (market cap)[<https://www.cryptocompare.com/>]. Ethereum holds the second highest market cap and relies on the same blockchain technology that underpins Bitcoin. Monero, while still popular, holds a much lower market cap than Bitcoin or Ethereum but focuses on privacy; transactions are private (with the origin, destination, and amounts obfuscated) and untraceable (transactions cannot be linked to a particular cyber- or real-world identity).

Are social signals relevant to cryptocurrency prices? To gain an initial understanding of this question, we illustrate the alignment of price, social interactions, and real-world events related to Bitcoin in Figure <ref>.

Historical price data (daily high, low, and price at market open and close) for each coin was collected from CryptoCompare<ref>. This resulted in a price history from:

* 2010/07/16 through 2018/05/21 for Bitcoin,

* 2015/08/06 through 2018/05/21 for Ethereum,

* 2015/01/28 through 2018/05/21 for Monero.

In addition to financial data, we collected publicly available data for two social platforms, GitHub and Reddit, from which we extracted social signals for each coin.

GitHub Dataset GitHub is a collaborative software social network primarily used to develop and share novel technologies and software. As of 2017, 67M repositories (repos) used by 24M users and 1.5M organizations were hosted on the site.[<https://octoverse.github.com/>] We collected interactions with the main repo for each coin (<bitcoin/bitcoin>, <ethereum/go-ethereum>, and <monero-project/monero>) from a public subset of the GitHub archive.[<https://www.githubarchive.org/>] These interactions can be divided into two main categories of event types: 1) indications of user interest or repo popularity such as watching or forking the repo and 2) direct contributions when a user reviews or directly contributes to code, reports issues, comments on issues or participates in code reviews.

Reddit Dataset Similarly, we collected all posts and comments submitted to the official subreddits[Reddit posts and comments are publicly available via an archive hosted at <https://files.pushshift.io/reddit/>.] for each of the cryptocurrencies of interest (<r/bitcoin, r/ethereum>, and <r/monero>) across the three years (2015 – 2017) for which we collected GitHub events data.
Reddit is a popular social news aggregator[The 18th most popular site globally and 5th most popular within the U.S. according to Alexa.com: <https://www.alexa.com/topsites>] that allows communities of users to share and discuss information, opinions, and entertainment media. Each of the subreddits has substantial traffic including 3.6K, 500, and 380 comments posted each day, on average, with community-sizes of 913K, 337K, and 137K subscribers (as of August 2018) for <r/bitcoin, r/ethereum>, and <r/monero>, respectively.

§ METHODOLOGY

In this section, we present the methodology used to evaluate the benefit of incorporating signals from social media into models that forecast the future price of a cryptocurrency. We trained and evaluated models that rely on 1) historical price alone, 2) historical price and each social signal, and 3) historical price and combinations of each of the social signals.

§.§ Forecasting Tasks

We define the forecasting tasks as predicting the daily price high of each cryptocurrency of interest $j$ days in advance, $Y_{t_i+j}$, focusing on the immediate and near future $Y_{t_i+1}$, $Y_{t_i+2}$, $Y_{t_i+3}$ (1, 2, or 3 days in the future, respectively) using predictive signals from $k$ days in the immediate past $X_{[t_{i-k},t_{i}]}$ and evaluate any benefits that arise from incorporating up to two weeks of signal history by varying $k$ from 1 to 14 days. We consider models that incorporate social signals and historical price values versus those that consider only historical pricing as predictive signals to identify the performance gains (if any exist) when social signals are included.
Models are trained and evaluated independently for each of the cryptocurrencies of interest.

§.§ Social Signals

Along with the daily price high, we use a variety of signals of popularity, activity, and language used in discussions across two popular social media platforms: GitHub and Reddit. Here, we describe the social signals we extracted from social media activity related to the three coins of interest.

For the GitHub platform, we consider two types of social signals:

* $GH_{Pop}$ – a vector representation of the daily totals for each popularity event: the Watch event, where users star a repo in order to receive notifications, and the Fork event, where users create a copy of the repo code. These event types provide a measure of how popular a given repo is among users who may or may not be direct contributors.

* $GH_{All}$ – a vector representation of the overall activity, daily counts for each popularity (Watch and Fork)
and direct contribution event types (CommitComment, Issue, IssueComment, PullRequest, PullRequestReviewComment, and Push).

For the Reddit platform, we consider four types of social signals:

* $R_{Vol}$ – the volume of comments posted each day, a signal for the size of discussion within the coin's official subreddit.

* $R_{Lang}$ – the language used in comments 	represented as 10k-dimensional vectors of word-level daily-normalized statistics that focus on the most frequent unigrams.

* $R_{Score}$ – a vector representation of the quartiles of Reddit scores (# upvotes - # downvotes) for comments posted each day, an indication of the range of popularity of comments for the given day.

* $R_{Sent}$ – quartiles for the subjectivity and polarity of comments each day  which provides a signal of the distribution of sentiment in discussions within the coin's official subreddit.

Before evaluating forecasting models,  we explore the relationships between coin-related social signals and their respective price high time-series. To do so, we first examine the correlation of social signals ($x$) and coin-price ($y$) over a sample of $N$ days for each of the three coins of interest using Pearson R correlation to examine the linear relationships between social signals and coin price. Pearson $R$ correlation ranges from -1 (perfectly inversely correlated) to 1 (perfectly correlated), where a score of 0 indicates linearly independent variables, no correlation. Next, we consider non-linear correlations between signals ($x$) and price ($y$) using distance correlation [30, 32, 31]:

dCorr(x,y) = {
dCov(x,y)/√(dVar(x)dVar(y))  ,     dVar(x)dVar(y) > 0

0,                 dVar(x)dVar(y) = 0

where distance covariance ($dCov$) and distance variance ($dVar$) are defined as:

dCov(x,y) = √(∑_k,l=1^n A_kl B_kl/n^2),
dVar(x) = dCov(x,x)

and    A_kl = a_kl - 1/n∑_l=1^n a_kl - 1/n∑_k=1^n a_kl + 1/n^2∑_k,l=1^na_kl.

$B_{kl}$ is defined similarly with $b_{kl}$ in place of $a_{kl}$ where $a_{kl}$ and $b_{kl}$ are Euclidean distance matrices of $x$ and $y$, respectively, defined as

a_kl = |x_k-x_l|,     b_kl = |y_k-y_l|.

Distance correlation varies from 0 to 1, where a distance correlation of 0 indicates independence of variables.
Finally, we also examine the variation of each signal ($x$) by the interquartile range (IQR) and standard deviation $\sigma$
for each of the signals of interest, identifying those that remain consistent over the time period of interest (and thus are potentially less informative).

§.§ Forecasting Models

Neural network models with long short-term memory (LSTM) layers have previously been used to effectively forecast influenza dynamics from a combination of clinical and social media signals, outperforming models that did not incorporate the social signals [37]. LSTMs are a type of recurrent neural networks with built in memory cells that store information and can exploit long range context [11]. These networks are surrounded by gating units that allow or prohibit the reset, read, and writing of such information. Inspired by the influenza dynamic models, we propose a neural network model, shown in Figure <ref>, that also utilizes LSTM layers.

The proposed neural network architecture consists of a 400-dimensional LSTM layer followed by an 800-dimensional LSTM then a dense layer with a single unit. We do not employ an activation layer or activation function in the final dense output layer because we have defined each forecasting task as a regression task to predict numerical high values so an activation transformation is not needed. The optimizer used by the model is the ADAM optimizer (a parameter specific adaptive learning rate method)[14] and we optimized performance by minimizing the mean squared error of a 20% subset of the training data which is used as a validation set.
To avoid overfitting, we employed early stopping callbacks with a maximum limit of 20 epochs. As a result, although models could have trained for up to 20 epochs, our models typically needed, on average, between five and six epochs only. The described architecture of the proposed model enables a relatively short training time for an expedient and scalable framework.

Signals used as input, e.g., social signals and historical price data, are fed to the network as a single concatenated vector. Before concatenation, each signal value is normalized to range between 0 and 1 using min-max normalization for the given feature across the entire dataset.
Target price values ($T$) and signal input vectors ($I$) are defined as:

T(Y,j) = {Yj, Y_j+1, ...Y_|Y|}

I(X,k) = { <X_i-k,...,X_i-1,X_i> for     i ∈[k,|X|]  }

where $X$ is a 2D array of the min-max normalized signals of interest, $Y$ is the historical price high time series, $j$ is the size of the forecasting window (how many days in advance to predict the price), and $k$ is the number of days of signal history used.

[scale=0.45, transform shape]

[->,ultra thick] (0.5,-0.6) – (0.5,0);
[above] at (0.5,0) Price Forecast;

(-0.2,-0.8) rectangle (1.2,-1.7);

[right] at (3,-1.4) Dense Layer (1);

(-1.25,-2.4) edge (0.1,-1.7) ;
(2.25,-2.4) edge (0.9,-1.7) ;

(1.75,-2.4) edge (0.3,-1.7);
(-0.75,-2.4) edge (0.7,-1.7);

(-1.75,-2.4) rectangle (-0.75,-3.2);
(1.75,-2.4) rectangle (2.75,-3.2);
[right] at (3,-2.9) LSTM Layer (800);

[->,thick] (-0.4,-2.8) – (0,-2.8);
[loosely dotted, ultra thick] (0.1,-2.8) edge (0.9,-2.8);
[->,thick] (0.9,-2.8) – (1.4,-2.8);

(-1.75,-3.8) rectangle (-0.75,-4.6);
(1.75,-3.8) rectangle (2.75,-4.6);
[right] at (3,-4.3) LSTM Layer (400);

(-1.25,-3.8) edge (-1.25,-3.2) ;
(2.25,-3.8) edge (2.25,-3.2) ;

(-1,-3.2) edge (2,-3.8);
(-1,-3.8) edge (2,-3.2);

[->,thick] (-0.4,-4.2) – (0,-4.2);
[loosely dotted, ultra thick] (0.1,-4.2) edge (0.9,-4.2);
[->,thick] (0.9,-4.2) – (1.4,-4.2);

[->,thick] (0.5,-5) – (0.5,-4.5);
[below] at (3.75,-5) Input;

(-1,-5) rectangle (-0.5,-5.5);
(-0.5,-5) rectangle (0,-5.5);
(0,-5) rectangle (0.5,-5.5);
(0.5,-5) rectangle (1,-5.5);
(1,-5) rectangle (1.5,-5.5);
(1.5,-5) rectangle (2,-5.5);

Neural network model architecture.

Parameter Tuning To identify our final model configuration, we performed a series of parameter tuning. We varied several of the model parameters: batch sizes of 16, 32, and 64; learning rates of 0.1, 0.01, 0.001, 0.0001, and 0.00001; and combinations of LSTM layers ranging from 10 to 400-dimensional layers followed by 20 to 800-dimensional layers. We found best performance when using a batch size of 16, learning rate of 0.001, and 400 and 800 units for the first and second LSTM layers, respectively.

§.§.§ Baseline

The autoregressive integrated moving average (ARIMA) model is a commonly used architecture when forecasting the price or return of stocks [24, 40]. Essentially ARIMA models treat the future value of a variable (price)
as a linear combination of past values and errors.  We train a variety of ARIMA models and identify the top performing ARIMA model for each of our forecasting windows of interest
to use as baselines. To identify these baseline models, we fix the size of the moving average window to 0, the difference order to 1 (to make the time series stationary), and perform autocorrelation analysis
to identify a range of appropriate lag parameters. We found that models with a lag order of 0 achieved the best performance overall across coins.
$R$ results for these baseline models were at least 0.95 ($p<0.001$) for Bitcoin, at least 0.92 ($p <0.001$) for Ethereum, and at least 0.91 ($p<0.001$) for Monero.

§.§ Evaluation

To ensure performance comparison across identical train and test dates for all combinations of training and forecasting window sizes, the train and test periods were identified using the largest possible window sizes. We first restricted the dataset to the period for which we had data for all signals across all coins. Then, input vectors and target prices were formatted for the largest training (14) and forecasting (3) window sizes and split into 80% for train and 20% for test. This resulted in a training period from 2015/11/11 through 2017/04/18 (525 days) and a test period from 2017/05/04 through 2017/08/31 (120 days) used for all model configurations.

Formatting Input and Target Data
[1]
Price and Social Signal Input Vectors X, Price Time Series Y
Model Input
$\textit{stringlen} \gets \text{length of }\textit{string}$
$i \gets \textit{patlen}$
$i > \textit{stringlen}$ false
$j \gets \textit{patlen}$
$\textit{string}(i) = \textit{path}(j)$
$j \gets j-1$.
$i \gets i-1$.
goto loop.
$i \gets i+\max(\textit{delta}_1(\textit{string}(i)),\textit{delta}_2(j))$.
goto top.

Model performance is evaluated using the following error measurements: root mean squared error (RMSE), mean squared percentage error (MSPE), mean absolute percentage error (MAPE), max absolute percentage error (MaxAPE), and root mean squared percentage error (RMSPE). For a set of $N$ predictions ($\hat{y}$) and true price values ($y$), MAPE, also known as mean absolute percentage deviation, is defined as:

MAPE(ŷ,y) =1/N∑_i=0^N|ŷ_̂î - y_i|/y_i

maximum absolute percentage error is defined as:

MaxAPE(ŷ,y) =max(  |ŷ_̂î - y_i|/y_i )

and mean squared percentage error is defined as:

MSPE(ŷ,y) =  1/N∑_i=0^N(ŷ_̂î - y_i)/y_i^2

Prices vary widely between coins and, for some, within coins over time. Therefore, we primarily report the performance metrics that consider percentage errors as they allow a relative comparison across the three coins of interest.

§ SOCIAL SIGNAL ANALYSIS

In this section we explore the relationships between daily price and signals of popularity and direct contributions to GitHub repositories and volume, sentiment, and popularity of cryptocurrency-related discussions on Reddit for our three coins of interest.
To identify if an informative relationship exists, we examine person and distance correlation of social signals with coin price and the variance of all features to identify which signals to include in the battery of model configurations we consider in our ablation experiments.

In Table <ref>, we see that price is correlated ($p<0.001$) with daily volume for each of the GitHub event types, although to varying degrees within and across coins. We also see that both Fork and Watch events are highly correlated for Bitcoin while Watch events are highly correlated and Fork events are moderately correlated for the other two coins.
For the Reddit platform, we see that daily comment volume and comment scores are both correlated with daily price highs across all three coins of interest. While there is no significant linear correlation between subjectivity and polarity of the daily batch of comments and price highs for Bitcoin, these features have varying levels of correlation across the remaining two coins of interest. As we saw with GitHub features, there is some variation across coins of interest in the relationships between the various social signals and price timeseries.

Pearson $R$ correlation and distance correlation ($DC$) of daily price and social signals.
Pearson results are significant ($p<0.001$) unless indicated
with a dash `—'  ($p\ge 0.05$).

2cBitcoin       2cEthereum       2cMonero

Social Signal    $R$     $DC$       $R$     $DC$       $R$     $DC$

Watch   teal!870.87      teal!86 	0.86

teal!68 0.68      teal!73 	0.73

teal!72 0.72      teal!68 	0.68

Fork   teal!750.75      teal!72 	0.72

teal!40 0.40      teal!38 	0.38

teal!41 0.41      teal!48 	0.48

Issues   teal!5 0.05      teal!9 	0.09

teal!22 0.22      teal!5 	0.05

teal!00 0.00      teal!36 	0.36

IssueComment   teal!13 0.13      teal!14 	0.14

teal!360.36      teal!29 	0.29

teal!25 0.25     teal!54 0.54

Push   teal!06 0.06      teal!9 0.09

teal!060.06      teal!17 	0.17

teal!07 0.07      teal!	11	 	0.11

CommitComment   teal!06 0.06      teal!	9	 	0.09

teal!04 0.04      teal!	12	 	0.12

teal!08 0.08      teal!	8	 	0.08

PullRequest (PR)   teal!180.18      teal!	20	 	0.20

teal!14 0.14       teal!	22	 	0.22

teal!15 0.15      teal!	21	 	0.21

PRReviewComment   teal!39 0.39    teal!	37	 	0.37

teal!29 0.29     teal!	36	 	0.36

teal!22 0.22     teal!	44	 	0.44

Comment Volume      teal!58   0.58 	     teal!	62	 	0.62

teal!48  0.48       teal!	51	 	0.51

teal!67 0.67       teal!	78	 	0.78

Subjectivity         — 	    teal!	0	 	0.00

—				    teal!	5	 	0.05

teal!16 0.16       teal!	25	 	0.25

Polarity        — 		    teal!	2	 	0.02

teal!13 0.13       teal!	15	 	0.15

teal!31  0.31       teal!	43	 	0.43

Score      teal!34 0.34     teal!	37	 	0.37

teal!47  0.47      teal!	59	 	0.59

teal!69 0.69     teal!	79	 	0.79

Standard deviations ($\sigma$) and Inter-Quartile Range (IQR) of daily price and social signals.

2cBitcoin			   	2cEthereum			   	2cMonero

Signal	   	$\sigma$	    $IQR$	   	$\sigma$	   $IQR$	   	$\sigma$	   $IQR$

Price High	   	868.00	   	626.85	   	99.44	   	34.67	   	19.71	   	11.96

Watch	   	10.34	   	7.00	   	2.09	   	2.00	   	1.60	   	1.00

Fork	   	4.55	   	4.00	   	0.66	   	0.00	   	0.94	   	1.00

Issues	   	3.34	   	3.00	   	27.42	   	1.00	   	2.47	   	2.00

IssueComment	   	24.98	   	34.00	   	7.93	   	7.00	   	9.51	   	10.00

Push	   	3.57	   	5.00	   	1.06	   	0.00	   	3.80	   	1.00

CommitComment	   	0.90	   	0.00	   	0.18	   	0.00	   	0.26	   	0.00

PullRequest (PR)	   	5.92	   	8.00	   	1.14	   	0.00	   	5.15	   	4.00

PRReviewComment	   	13.25	   	15.00	   	2.57	   	0.00	   	5.50	   	1.00

Comment Volume	   	1990.43	   	1981.00	   	695.86	   	369.00	   	376.34	   	466.00

Score	   	242.97	   	129.00	   	80.00	   	47.00	   	18.34	   	19.00

Subjectivity	   	0.00	   	0.00	   	0.00	   	0.00	   	0.13	   	0.00

Polarity	   	0.01	   	0.00	   	0.06	   	0.00	   	0.23	   	0.25

When we consider how the signals vary within themselves in Table <ref>, we find simaliar patterns across and wihtin coins. The most highly correlated social signals have larger, relative to other signals, variation as summarized by the standard deviations and inter-quartile ranges of the signal vectors. We find that the subjectivity and polarity signals from Reddit comments linked to Bitcoin and Ethereum that held no significant correlation with price also show little to no variation. However, we see they vary slightly to moderately for the third coin, Monero. As each of our social signals indicate a relationship with price for at least one of our coins of interest, we include all GitHub and Reddit social signals as well as combinations of signals from both the GitHub and Reddit platforms in our ablation experiments and highlight the results in the following section.

§ FORECASTING RESULTS

In this section, we describe the performance of models that rely on historical price alone, historical price and each signal from GitHub or Reddit, and historical price and combinations of each GitHub and Reddit signal. In particular, we highlight models which incorporated social signals that achieved high performance relative to the baseline ARIMA models and LSTMs that relied on historical price alone.

First, we explored the benefit of increased signal history with a comparison of model performance using signals from one to fourteen days in the past in Figure <ref>. We plot MSPE as a function of training window size, the number of days of signal history to rely on, for neural network (LSTM) models that rely on each of the combinations of predictive signals when forecasting price one to two days in advance. Interestingly, we see that models with the smallest window size (1) achieved the best performance.  Therefore, we use a signal history window size of 1 day for subsequent model evaluation.

Next we focused on identifying model configurations that achieve the best performance to evaluate the benefit of incorporating social signals from a variety of platforms and the benefit of a variety of combinations of social signals. To do so, we perform an ablation study where we compare models that incorporate each combination of price and social signals with models (LSTM and ARIMA baselines) that relied solely on historical prices. To identify the best, overall model, we then averaged percentage errors across the three coins and ranked social signal-infused models with the baseline ARIMA and neural network models that did not incorporate social signals.

We present a summary of the averaged error for the top performing models in Table <ref>, ordered by the mean of mean RMSPE over the three forecasting tasks. The top performing model is the proposed LSTM that relies on price history and $R_{Lang}$, the representation of the language used in comments within the official subreddit. Here, we see that LSTM models that incorporate social signals outperform the price only baselines when averaged across the three coins and forecasting windows. If we only consider the immediate forecasting window of one day in advance, the proposed LSTM model that relies solely on price history outperforms the others.

We then consider performance for each coin individually in Table <ref>. Here we see that, in most cases, neural network models that incorporated social signals slightly outperformed both ARIMA baselines and neural network models that relied solely on price history, but these results are not statistically significant. However, we see that our proposed LSTM neural network architecture, and especially neural network models that incorporate social signals, minimize the maximum absolute percentage error (MaxAPE). That is, in worst-case prediction performance, we see a much lower error rate for our proposed models that rely on social signals alongside price history. We see the best performance is achieved when models forecast the price one day in the future (FW = 1 Day); unsurprisingly, it is easiest to predict the next day's price using signals from the day before.

If we expand the range of forecasting windows beyond the near and immediate future, we see that, again, the LSTM model that incorporates the $R_{Lang}$ social signal alongside historical price has the best performance across the three coins. Table <ref> presents the RMSPE of top performing models when forecasting price up to two weeks in advance.  Figure <ref> illustrates how this top performing model, performs similarly to price-only LSTM and ARIMA models, on average, outperforms the price-only models in respect to the worst-case prediction errors (MaxAPE).

MSPE results for neural network models for each coin plotted by varying training window sizes for the first two forecasting windows, when predicting one or two days in advance. Jitter has been added to x-axis for enhanced readability.

RMSPE averaged over the three coins of interest (Bitcoin, Ethereum, and Monero) when predicting price up to 3 days in advance for ARIMA baseline and top performing neural network (LSTM) models. Lowest values highlighted in bold.

4cForecasting Window (days)

Model   Signal      1      2      3      mean

LSTM                $\$+R_{Lang}$
6.70         9.88          12.06      9.55

LSTM       $\$+GH_{Pop}+R_{Lang}$
6.64                9.99               12.40           9.68

LSTM                 $\$+R_{Vol}$
6.78                9.98               12.48           9.75

LSTM                         $\$$
6.60               10.46               12.32           9.79

ARIMA                         $\$$
7.30               10.56               13.10          10.32

MAPE, RMSPE, and MaxAPE results for baseline ARIMA, neural network models that rely solely on historical price, and the top performing social signal enhanced neural network models, as identified during ablation experiments.
Results for neural network models (LSTM) reported used the top performing training window size of 1 day. Lowest error rates are highlighted in bold.

3c|Bitcoin      3c|Ethereum

FI     Model      Signals       1cMAPE     1cRMSPE      1c|MaxAPE      1cMAPE     1cRMSPE      1c|MaxAPE       1cMAPE     1cRMSPE      1cMaxAPE

50em90$ in 1 day
ARIMA     $\$$      3.24 $\pm$ 0.65
4.83 $\pm$ 3.35
5.13 $\pm$ 1.18
8.29 $\pm$ 6.50
5.84 $\pm$ 1.19
8.78 $\pm$ 6.26

LSTM     $\$$      3.11 $\pm$ 0.55
4.35 $\pm$ 2.69
4.71 $\pm$ 0.96
7.08 $\pm$ 4.48
5.71 $\pm$ 1.12
8.39 $\pm$ 5.96

LSTM     $\$+R_{Lang}$      3.26 $\pm$ 0.55
4.45 $\pm$ 2.69
4.76 $\pm$ 0.97
7.16 $\pm$ 4.54
5.94 $\pm$ 1.11
8.49 $\pm$ 5.85

LSTM     $\$+GH_{Pop}+R_{Lang}$      3.19 $\pm$ 0.56
4.44 $\pm$ 2.68
4.73 $\pm$ 0.97
7.10 $\pm$ 4.50
5.71 $\pm$ 1.12
8.38 $\pm$ 5.97

LSTM     $\$+R_{Vol}$      3.35 $\pm$ 0.55
4.52 $\pm$ 2.71
5.06 $\pm$ 0.99
7.44 $\pm$ 4.68
5.69 $\pm$ 1.12
8.38 $\pm$ 5.99

50em90$ in 2 days   ARIMA     $\$$      5.35 $\pm$ 0.87
7.18 $\pm$ 4.13
8.44 $\pm$ 1.56
12.05 $\pm$ 7.83
8.74 $\pm$ 1.61
12.44 $\pm$ 7.56

LSTM     $\$$      5.14 $\pm$ 0.76
6.61 $\pm$ 3.68
9.81 $\pm$ 1.49
12.78 $\pm$ 6.84
8.44 $\pm$ 1.55
11.99 $\pm$ 7.40

LSTM     $\$+R_{Lang}$      5.38 $\pm$ 0.79
6.90 $\pm$ 3.82
7.67 $\pm$ 1.34
10.61 $\pm$ 6.07
8.90 $\pm$ 1.50
12.12 $\pm$ 7.26

LSTM     $\$+GH_{Pop}+R_{Lang}$      5.40 $\pm$ 0.78
6.89 $\pm$ 3.78
7.87 $\pm$ 1.43
11.11 $\pm$ 6.33
8.32 $\pm$ 1.56
11.95 $\pm$ 7.48

LSTM     $\$+R_{Vol}$      5.11 $\pm$ 0.76
6.59 $\pm$ 3.67
8.11 $\pm$ 1.44
11.34 $\pm$ 6.38
8.44 $\pm$ 1.56
12.02 $\pm$ 7.48

50em90$ in 3 days   ARIMA     $\$$      6.80 $\pm$ 1.03
8.85 $\pm$ 4.72
10.85 $\pm$ 1.83
14.81 $\pm$ 8.74
11.14 $\pm$ 1.99
15.64 $\pm$ 9.17

LSTM     $\$$      6.33 $\pm$ 0.92
8.08 $\pm$ 4.29
10.36 $\pm$ 1.66
13.79 $\pm$ 7.35
10.77 $\pm$ 1.92
15.07 $\pm$ 8.90

LSTM     $\$+R_{Lang}$      6.37 $\pm$ 0.92
8.12 $\pm$ 4.31
9.86 $\pm$ 1.53
12.96 $\pm$ 6.90
10.96 $\pm$ 1.90
15.10 $\pm$ 8.85

LSTM     $\$+GH_{Pop}+R_{Lang}$      6.36 $\pm$ 0.92
8.13 $\pm$ 4.30
9.84 $\pm$ 1.53
12.92 $\pm$ 6.87
11.77 $\pm$ 2.02
16.16 $\pm$ 9.28

LSTM     $\$+R_{Vol}$      6.36 $\pm$ 0.92
8.13 $\pm$ 4.28
10.38 $\pm$ 1.67
13.84 $\pm$ 7.37
10.97 $\pm$ 1.99
15.46 $\pm$ 9.07

50em901 day
ARIMA                         $\$$
3.24 $\pm$ 0.65       0.48 $\pm$ 0.34               23.19
5.13 $\pm$ 1.18       0.83 $\pm$ 0.65                48.19
5.84 $\pm$ 1.19       0.88 $\pm$ 0.63              42.71

LSTM                         $\$$
3.11 $\pm$ 0.55      0.43 $\pm$ 0.27               17.16
4.71 $\pm$ 0.96      0.71 $\pm$ 0.45         26.18
5.71 $\pm$ 1.12      0.84 $\pm$ 0.60              42.53

LSTM                $\$+R_{Lang}$
3.26 $\pm$ 0.55       0.44 $\pm$ 0.27               16.66
4.76 $\pm$ 0.97       0.72 $\pm$ 0.45                26.53
5.94 $\pm$ 1.11       0.85 $\pm$ 0.59         41.83

LSTM       $\$+GH_{Pop}+R_{Lang}$
3.19 $\pm$ 0.56       0.44 $\pm$ 0.27               16.77
4.73 $\pm$ 0.97      0.71 $\pm$ 0.45                26.35
5.71 $\pm$ 1.12      0.84 $\pm$ 0.60              42.70

LSTM                 $\$+R_{Vol}$
3.35 $\pm$ 0.55       0.45 $\pm$ 0.27         16.49
5.06 $\pm$ 0.99       0.74 $\pm$ 0.47                27.36
5.69 $\pm$ 1.12      0.84 $\pm$ 0.60              42.72

50em902 days          ARIMA                         $\$$
5.35 $\pm$ 0.87       0.72 $\pm$ 0.41               23.35
8.44 $\pm$ 1.56       1.21 $\pm$ 0.78                48.35
8.74 $\pm$ 1.61       1.24 $\pm$ 0.76              41.13

LSTM                         $\$$
5.14 $\pm$ 0.76      0.66 $\pm$ 0.37               22.03
9.81 $\pm$ 1.49       1.28 $\pm$ 0.68                37.30
8.44 $\pm$ 1.55       1.20 $\pm$ 0.74              41.35

LSTM                $\$+R_{Lang}$
5.38 $\pm$ 0.79       0.69 $\pm$ 0.38               23.79
7.67 $\pm$ 1.34      1.06 $\pm$ 0.61        33.64
8.90 $\pm$ 1.50       1.21 $\pm$ 0.73          39.80

LSTM       $\$+GH_{Pop}+R_{Lang}$
5.40 $\pm$ 0.78       0.69 $\pm$ 0.38               23.23
7.87 $\pm$ 1.43       1.11 $\pm$ 0.63                35.01
8.32 $\pm$ 1.56        1.20 $\pm$ 0.75              41.73

LSTM                 $\$+R_{Vol}$
5.11 $\pm$ 0.76      0.66 $\pm$ 0.37         21.78
8.11 $\pm$ 1.44       1.13 $\pm$ 0.64                35.31
8.44 $\pm$ 1.56        1.20 $\pm$ 0.75              41.80

50em903 days          ARIMA                         $\$$
6.80 $\pm$ 1.03       0.89 $\pm$ 0.47               23.94
10.85 $\pm$ 1.83       1.48 $\pm$ 0.87                50.03
11.14 $\pm$ 1.99       1.56 $\pm$ 0.92              49.89

LSTM                         $\$$
6.33 $\pm$ 0.92      0.81 $\pm$ 0.43        23.01
10.36 $\pm$ 1.66       1.38 $\pm$ 0.73                37.17
10.77 $\pm$ 1.92      1.51 $\pm$ 0.89              50.23

LSTM                $\$+R_{Lang}$
6.37 $\pm$ 0.92     0.81 $\pm$ 0.43               23.92
9.86 $\pm$ 1.53        1.30 $\pm$ 0.69                35.41
10.96 $\pm$ 1.90      1.51 $\pm$ 0.89        49.36

LSTM       $\$+GH_{Pop}+R_{Lang}$
6.36 $\pm$ 0.92       0.81 $\pm$ 0.43               23.13
9.84 $\pm$ 1.53      1.29 $\pm$ 0.69         35.31
11.77 $\pm$ 2.02       1.62 $\pm$ 0.93              52.58

LSTM                 $\$+R_{Vol}$
6.36 $\pm$ 0.92      0.81 $\pm$ 0.43               23.08
10.38 $\pm$ 1.67       1.38 $\pm$ 0.74                37.31
10.97 $\pm$ 1.99       1.55 $\pm$ 0.91              51.53

4.0         ARIMA                         $\$$        8.1 $\pm$ 1.18       1.04 $\pm$ 0.54               29.90      12.66 $\pm$ 2.03       1.69 $\pm$ 0.96                49.21      13.08 $\pm$ 2.22       1.79 $\pm$ 0.99              49.36

4.0        LSTM                         $\$$       7.43 $\pm$ 1.06       0.94 $\pm$ 0.49               29.79      12.84 $\pm$ 1.99       1.69 $\pm$ 0.89                50.29      12.87 $\pm$ 2.18       1.76 $\pm$ 0.98              50.85

4.0        LSTM                $\$+R_{Lang}$       7.36 $\pm$ 1.06       0.94 $\pm$ 0.49               28.89       11.42 $\pm$ 1.8       1.51 $\pm$ 0.83                47.75      12.79 $\pm$ 2.12       1.73 $\pm$ 0.96              49.14

4.0        LSTM       $\$+GH_{Pop}+R_{Lang}$       7.48 $\pm$ 1.05       0.94 $\pm$ 0.49               30.10      11.84 $\pm$ 1.88       1.57 $\pm$ 0.85                49.05      12.71 $\pm$ 2.17       1.74 $\pm$ 0.97              50.58

4.0        LSTM                 $\$+R_{Vol}$       7.36 $\pm$ 1.06       0.94 $\pm$ 0.49               28.42       12.88 $\pm$ 2.0       1.69 $\pm$ 0.89                50.43       13.11 $\pm$ 2.2       1.78 $\pm$ 0.99              51.87

5.0         ARIMA                         $\$$       9.29 $\pm$ 1.29       1.17 $\pm$ 0.58               30.21      14.45 $\pm$ 2.13       1.86 $\pm$ 0.99                50.28      14.93 $\pm$ 2.29       1.95 $\pm$ 1.03              50.65

5.0        LSTM                         $\$$        8.5 $\pm$ 1.14       1.05 $\pm$ 0.51               27.48      14.59 $\pm$ 2.18       1.89 $\pm$ 0.98                53.68      14.54 $\pm$ 2.41       1.96 $\pm$ 1.06              53.10

5.0        LSTM                $\$+R_{Lang}$        8.6 $\pm$ 1.14       1.06 $\pm$ 0.52               28.37      13.75 $\pm$ 2.05       1.78 $\pm$ 0.94                52.55      14.37 $\pm$ 2.29       1.91 $\pm$ 1.03              51.49

5.0        LSTM       $\$+GH_{Pop}+R_{Lang}$       8.74 $\pm$ 1.15       1.08 $\pm$ 0.52               29.01      14.33 $\pm$ 2.15       1.86 $\pm$ 0.96                53.23      14.32 $\pm$ 2.32       1.92 $\pm$ 1.04              51.91

5.0        LSTM                 $\$+R_{Vol}$       8.51 $\pm$ 1.13       1.05 $\pm$ 0.51               27.35      14.14 $\pm$ 2.12       1.83 $\pm$ 0.96                53.25      15.87 $\pm$ 2.48        2.09 $\pm$ 1.1              54.35

6.0         ARIMA                         $\$$      10.15 $\pm$ 1.39       1.27 $\pm$ 0.62               30.46      16.43 $\pm$ 2.26       2.06 $\pm$ 1.05                51.70      16.23 $\pm$ 2.45       2.11 $\pm$ 1.11              61.72

6.0        LSTM                         $\$$       9.36 $\pm$ 1.16       1.13 $\pm$ 0.53               27.61      16.01 $\pm$ 2.27       2.03 $\pm$ 1.02                54.90       15.78 $\pm$ 2.4        2.06 $\pm$ 1.1              61.92

6.0        LSTM                $\$+R_{Lang}$       9.33 $\pm$ 1.21       1.15 $\pm$ 0.54               28.31      16.37 $\pm$ 2.31       2.07 $\pm$ 1.04                55.45       15.78 $\pm$ 2.4        2.05 $\pm$ 1.1              61.88

6.0        LSTM       $\$+GH_{Pop}+R_{Lang}$       9.26 $\pm$ 1.19       1.13 $\pm$ 0.53               27.92      16.12 $\pm$ 2.29       2.04 $\pm$ 1.03                55.21      16.01 $\pm$ 2.54       2.12 $\pm$ 1.15              63.53

6.0        LSTM                 $\$+R_{Vol}$       9.44 $\pm$ 1.21       1.15 $\pm$ 0.54               28.88      16.48 $\pm$ 2.32       2.08 $\pm$ 1.04                55.71      16.89 $\pm$ 2.63       2.22 $\pm$ 1.18              64.88

7.0         ARIMA                         $\$$      10.87 $\pm$ 1.47       1.36 $\pm$ 0.65               34.05      18.47 $\pm$ 2.37        2.26 $\pm$ 1.1                54.98      17.56 $\pm$ 2.61       2.27 $\pm$ 1.18              60.22

7.0        LSTM                         $\$$       9.98 $\pm$ 1.19       1.19 $\pm$ 0.54               30.32      17.88 $\pm$ 2.37       2.21 $\pm$ 1.09                58.03      16.88 $\pm$ 2.63       2.22 $\pm$ 1.19              61.53

7.0        LSTM                $\$+R_{Lang}$       10.0 $\pm$ 1.26       1.21 $\pm$ 0.56               27.86      17.68 $\pm$ 2.34       2.18 $\pm$ 1.08                58.08      16.87 $\pm$ 2.62       2.22 $\pm$ 1.19              61.48

7.0        LSTM       $\$+GH_{Pop}+R_{Lang}$       9.99 $\pm$ 1.23       1.21 $\pm$ 0.55               28.02       17.6 $\pm$ 2.35       2.18 $\pm$ 1.08                58.23      18.01 $\pm$ 2.77       2.36 $\pm$ 1.25              64.15

7.0        LSTM                 $\$+R_{Vol}$      10.22 $\pm$ 1.27       1.24 $\pm$ 0.57               29.21      18.37 $\pm$ 2.42       2.27 $\pm$ 1.11                58.86      16.91 $\pm$ 2.63        2.22 $\pm$ 1.2              62.10

8.0         ARIMA                         $\$$      11.59 $\pm$ 1.54       1.44 $\pm$ 0.69               34.19      20.16 $\pm$ 2.51       2.44 $\pm$ 1.14                58.02      18.71 $\pm$ 2.74        2.4 $\pm$ 1.24              66.52

8.0        LSTM                         $\$$      10.64 $\pm$ 1.22       1.26 $\pm$ 0.57               31.60      19.57 $\pm$ 2.51       2.39 $\pm$ 1.14                60.98       18.2 $\pm$ 2.78       2.37 $\pm$ 1.27              68.07

8.0        LSTM                $\$+R_{Lang}$      10.58 $\pm$ 1.19       1.24 $\pm$ 0.57               31.99      20.67 $\pm$ 2.63       2.52 $\pm$ 1.18                62.30      18.21 $\pm$ 2.77       2.37 $\pm$ 1.27              67.81

8.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      10.52 $\pm$ 1.23       1.25 $\pm$ 0.57               31.85      21.09 $\pm$ 2.65       2.56 $\pm$ 1.19                63.08       18.0 $\pm$ 2.77       2.36 $\pm$ 1.27              67.93

8.0        LSTM                 $\$+R_{Vol}$      10.82 $\pm$ 1.29       1.29 $\pm$ 0.59               33.02       22.02 $\pm$ 2.7       2.65 $\pm$ 1.21                63.34      18.14 $\pm$ 2.79       2.37 $\pm$ 1.27              67.72

9.0         ARIMA                         $\$$      12.39 $\pm$ 1.61       1.52 $\pm$ 0.72               36.15      21.72 $\pm$ 2.68        2.63 $\pm$ 1.2                57.24      19.56 $\pm$ 2.83        2.5 $\pm$ 1.29              66.16

9.0        LSTM                         $\$$      11.43 $\pm$ 1.29       1.34 $\pm$ 0.62               35.21      22.26 $\pm$ 2.76       2.69 $\pm$ 1.24                61.52      19.47 $\pm$ 2.97       2.54 $\pm$ 1.35              68.73

9.0        LSTM                $\$+R_{Lang}$       11.41 $\pm$ 1.3       1.34 $\pm$ 0.62               35.22       21.1 $\pm$ 2.62       2.55 $\pm$ 1.19                59.92      18.81 $\pm$ 2.91       2.47 $\pm$ 1.32              67.73

9.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      11.55 $\pm$ 1.37       1.38 $\pm$ 0.64               36.09      21.33 $\pm$ 2.66        2.58 $\pm$ 1.2                60.20      19.36 $\pm$ 2.96       2.53 $\pm$ 1.33              68.07

9.0        LSTM                 $\$+R_{Vol}$      11.91 $\pm$ 1.47       1.44 $\pm$ 0.66               37.46       21.4 $\pm$ 2.69        2.6 $\pm$ 1.21                60.43      19.44 $\pm$ 2.95       2.53 $\pm$ 1.34              68.64

10.0         ARIMA                         $\$$      13.07 $\pm$ 1.66       1.59 $\pm$ 0.74               35.43      23.33 $\pm$ 2.78       2.79 $\pm$ 1.23                62.42      20.17 $\pm$ 2.88       2.57 $\pm$ 1.32              67.86

10.0        LSTM                         $\$$      12.03 $\pm$ 1.43       1.44 $\pm$ 0.66               34.89       23.2 $\pm$ 2.82       2.79 $\pm$ 1.26                59.74      19.18 $\pm$ 2.95       2.51 $\pm$ 1.34              68.58

10.0        LSTM                $\$+R_{Lang}$      12.08 $\pm$ 1.44       1.44 $\pm$ 0.66               35.04      22.77 $\pm$ 2.76       2.73 $\pm$ 1.24                59.29      19.16 $\pm$ 2.93        2.5 $\pm$ 1.34              68.33

10.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      12.24 $\pm$ 1.34       1.43 $\pm$ 0.65               34.80      24.05 $\pm$ 2.89       2.88 $\pm$ 1.29                61.03       20.3 $\pm$ 3.07       2.64 $\pm$ 1.39              70.17

10.0        LSTM                 $\$+R_{Vol}$      12.63 $\pm$ 1.55       1.52 $\pm$ 0.69               36.52      22.88 $\pm$ 2.79       2.75 $\pm$ 1.25                59.47       21.7 $\pm$ 3.11       2.76 $\pm$ 1.42              71.09

11.0         ARIMA                         $\$$      13.72 $\pm$ 1.67       1.65 $\pm$ 0.76               34.96      24.89 $\pm$ 2.84       2.94 $\pm$ 1.27                62.38       20.55 $\pm$ 2.9        2.6 $\pm$ 1.34              67.53

11.0        LSTM                         $\$$      12.55 $\pm$ 1.45       1.49 $\pm$ 0.68               33.69      24.52 $\pm$ 2.91       2.93 $\pm$ 1.29                61.67      20.07 $\pm$ 3.07       2.62 $\pm$ 1.39              69.17

11.0        LSTM                $\$+R_{Lang}$      12.65 $\pm$ 1.43       1.49 $\pm$ 0.68               33.74       24.47 $\pm$ 2.9       2.92 $\pm$ 1.29                61.61      19.99 $\pm$ 3.05       2.61 $\pm$ 1.39              69.02

11.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      12.46 $\pm$ 1.28       1.43 $\pm$ 0.65               35.92      24.53 $\pm$ 2.94        2.93 $\pm$ 1.3                61.51       22.2 $\pm$ 3.23       2.84 $\pm$ 1.45              71.30

11.0        LSTM                 $\$+R_{Vol}$      12.99 $\pm$ 1.52        1.54 $\pm$ 0.7               34.83       24.29 $\pm$ 2.9        2.9 $\pm$ 1.29                61.38       21.09 $\pm$ 3.2       2.74 $\pm$ 1.43              70.87

12.0         ARIMA                         $\$$       14.45 $\pm$ 1.7       1.72 $\pm$ 0.77               35.28      26.41 $\pm$ 2.94        3.1 $\pm$ 1.31                67.58      20.94 $\pm$ 2.93       2.64 $\pm$ 1.36              67.73

12.0        LSTM                         $\$$      13.43 $\pm$ 1.53       1.58 $\pm$ 0.71               35.38      25.52 $\pm$ 2.97       3.03 $\pm$ 1.32                61.54       20.0 $\pm$ 2.92       2.56 $\pm$ 1.35              67.97

12.0        LSTM                $\$+R_{Lang}$       13.4 $\pm$ 1.52       1.58 $\pm$ 0.71               35.25      25.48 $\pm$ 2.97       3.02 $\pm$ 1.32                61.50      19.97 $\pm$ 2.92       2.56 $\pm$ 1.35              68.02

12.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      14.17 $\pm$ 1.75       1.71 $\pm$ 0.75               37.56      25.36 $\pm$ 2.97       3.02 $\pm$ 1.32                61.15       21.53 $\pm$ 3.1       2.74 $\pm$ 1.43              70.65

12.0        LSTM                 $\$+R_{Vol}$      13.74 $\pm$ 1.53       1.61 $\pm$ 0.72               36.56      26.05 $\pm$ 3.03       3.09 $\pm$ 1.34                62.24       23.85 $\pm$ 3.2       2.96 $\pm$ 1.47              72.51

13.0         ARIMA                         $\$$      15.25 $\pm$ 1.72        1.8 $\pm$ 0.78               36.32       27.9 $\pm$ 3.01       3.25 $\pm$ 1.35                69.05       20.95 $\pm$ 2.9       2.63 $\pm$ 1.36              66.88

13.0        LSTM                         $\$$      14.29 $\pm$ 1.58       1.67 $\pm$ 0.73               36.69      27.66 $\pm$ 3.09       3.24 $\pm$ 1.38                64.69      20.01 $\pm$ 2.96       2.58 $\pm$ 1.38              68.08

13.0        LSTM                $\$+R_{Lang}$      14.26 $\pm$ 1.56       1.66 $\pm$ 0.72               36.58      27.52 $\pm$ 3.08       3.23 $\pm$ 1.38                64.55      20.96 $\pm$ 3.03       2.68 $\pm$ 1.41              69.22

13.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      14.39 $\pm$ 1.58       1.68 $\pm$ 0.72               36.14      28.15 $\pm$ 3.11        3.29 $\pm$ 1.4                65.40      22.71 $\pm$ 3.17       2.86 $\pm$ 1.46              70.86

13.0        LSTM                 $\$+R_{Vol}$      15.39 $\pm$ 1.81       1.83 $\pm$ 0.78               38.98      29.41 $\pm$ 3.23       3.43 $\pm$ 1.44                66.49      20.88 $\pm$ 3.04       2.67 $\pm$ 1.41              68.60

14.0         ARIMA                         $\$$      15.85 $\pm$ 1.78        1.86 $\pm$ 0.8               37.31      29.34 $\pm$ 3.02       3.37 $\pm$ 1.39                74.70      21.36 $\pm$ 2.89       2.66 $\pm$ 1.37              67.05

14.0        LSTM                         $\$$       15.35 $\pm$ 1.7       1.79 $\pm$ 0.76               37.15      30.39 $\pm$ 3.29       3.54 $\pm$ 1.47                66.65      21.39 $\pm$ 2.95        2.68 $\pm$ 1.4              69.26

14.0        LSTM                $\$+R_{Lang}$      15.29 $\pm$ 1.69       1.79 $\pm$ 0.76               37.05      30.31 $\pm$ 3.29       3.53 $\pm$ 1.47                66.59      21.44 $\pm$ 2.95        2.69 $\pm$ 1.4              69.19

14.0        LSTM       $\$+GH_{Pop}+R_{Lang}$      14.45 $\pm$ 1.48       1.66 $\pm$ 0.69               33.68      30.91 $\pm$ 3.33       3.59 $\pm$ 1.48                66.83       22.12 $\pm$ 3.0       2.76 $\pm$ 1.42              69.75

14.0        LSTM                 $\$+R_{Vol}$      15.81 $\pm$ 1.78       1.86 $\pm$ 0.78               37.99      30.84 $\pm$ 3.34       3.59 $\pm$ 1.48                67.13      21.17 $\pm$ 2.97        2.67 $\pm$ 1.4              69.42

RMSPE averaged over the three coins of interest (Bitcoin, Ethereum, and Monero) when predicting price up to 14 days in advance for ARIMA baseline and top performing neural network (LSTM) models.

14c|Forecasting Window (days)

Model      Signals      1      2      3      4      5      6      7      8      9      10      11      12      13      14      mean

$\$+R_{Lang}$                6.70       9.88         12.06         13.92          15.83               17.57             18.72               20.45              21.22          22.25            23.38              23.87                25.23                26.67          18.41

$\$$         6.60               10.46               12.32               14.63               16.35               17.38               18.74               20.08               21.92                22.44                23.43                23.92         24.96                26.71          18.57

$\$+GH_{Pop}$                6.80               10.27               12.76               14.50               16.19               17.36               18.75         19.97               21.40                23.13                23.76                24.26                25.66                27.32          18.72

$\$+GH_{Pop}+R_{Lang}$                6.64                9.99               12.40               14.19               16.17               17.66               19.14               20.56               21.62                23.16                24.01                24.90                26.11                26.67          18.80

$\$$                7.30               10.56               13.10               15.07               16.62               18.15               19.62               20.95               22.17                23.17                23.99                24.88                25.58            26.32          19.11

MaxAPE (above) and MAPE with 95% confidence intervals (below) for each coin plotted by varying forecasting window sizes (1 to 14 days in advance). Jitter has been added to x-axis of MAPE plots for enhanced readability.

Predictions from the overall top performing neural network model (incorporating price and $R_{Lang}$, language used in comments posted to the official subreddit communities). Solid gray lines represent the true price high values for each day across the test period (2017/05/04-2017/08/31) and colored lines with markers (representing individual predictions) plot predictions.

We illustrate the actual predictions of the top performing social signal enhanced neural network model against the true price values over the test period in Fig. <ref>. This allows us to visualize not only the performance of each model relative to the true price by day but also relative to the trends in true price values across coins. As we saw in Table <ref>, the range of true price values differs greatly by coin. While Bitcoin ranges from $1000 to $5000, Ethereum ranges from $100 - $400 and Monero ranges between $30 and $50 for the majority of the test period with a spike to between $100 and $150 in the final 10 days of the test period. The differences in the variance of true price parallel the differences in model performance. These results indicate that variations in price or the range of price values may heavily affect predictability of coin price; coins with lower price values or variance are more difficult to predict.

§ DISCUSSION

In each of our analyses it is apparent that models perform best when forecasting the price for Bitcoin then Ethereum and worst when forecasting for Monero. In Fig. <ref>, we saw evidence that this may be attributable to variations in true price but we hypothesize it may also be tied to the size of activity on social media or market cap – Monero has both the smallest social activity from which we drew social signals (GitHub event and comment counts noted above when we described data collection) and the lowest market cap (as is shown in Fig. <ref>). As we saw in Fig. <ref>, Monero also had the smallest range of price values. This suggests that activity volumes may have a direct effect on cryptocurrency prices, or at the least, performance of models that forecast coin price.

As we saw above, the best model performance is achieved for forecast windows of 1 day in advance, an intuitive result as this is an easier task than longer windows of 2 and 3 days. However, we also find better performance across all forecasting windows for Bitcoin. Bitcoin is both the oldest and most established coin with the largest market cap. Within our three coins of interest, Ethereum has the next highest market cap but Monero has the next longest lifetime. As a result, we plot MAPE as a function of lifetime and as a function of market cap[Market cap for each coin collected from CryptoCompare.com on September 4th 2018.] in Fig. <ref> to explore whether one of these characteristics may have an effect on price values or model performance. We calculate lifetime as the number of days from when the genesis block was mined (the beginning of the “ledger” of transactions) to the end of the testing period (2017/08/31).

When we examine Fig. <ref>,  we find that MaxAPE tends to decrease as market cap increases among top performing models but we do not find the same pattern when we plotted MAPE by lifetime. It is important to note that the sample size of coins is small (N=3) and these patterns may not hold among a larger sample of coins or among a set of coins with more variation in lifetimes or market cap. However, these results combined with the increased performance of models that rely on indications of popularity (event counts, comment volume, etc.) suggest that the popularity of a coin affects the performance of predictive models that forecast coin prices. Intuitively, popularity should affect price – a coin that no one knows about or that is less popular probably has a lower price.

hide axis,
legend style=draw=none,legend cell align=left,legend columns = -1, column sep = 3mm
arima, only marks, mark size=3,mark=+,thick;
price, only marks,mark size=3,mark=x,thick;
LSTM (Price);
github, only marks, mark=o,thick;
LSTM (Price+GitHub);

hide axis,
legend style=draw=none,legend cell align=left,legend columns = -1, column sep = 3mm
reddit, only marks, mark=o,thick;
LSTM (Price+Reddit);
githubAndReddit, only marks, mark=o,thick;
LSTM (Price+GitHub+Reddit);

MaxAPE as a function of lifetime (# days between mining of genesis block and end of test period), as a function of market cap (in billion USD), and standard deviations of coin price high (USD) for ARIMA and LSTM price only  and top performing price and social signal LSTM models.

There are several potential impacts of identifying models that reliably forecast the actual prices of cryptocurrencies, beyond the obvious use of identifying which coin to purchase and when. One such use would be as indications of irregular and potentially fraudulent or deceptive activity. For example, a “pump and dump” where a group artificially inflates the price of a coin with excessive transactions and then sells to unsuspecting speculators outside the group at the peak. Although illegal in stocks and securities, this behavior is not yet regulated in cryptocurrencies and is openly advertised [10, 35]. Similarly, fabricated or misleading news stories and social media postings have been used to artifically inflate or decrease coin prices.  Say there is a model that reliably predicts the price of a cryptocurrency and it begins to fail at an unexpected magnitude, could this be used as an indication of “pump and dump” activity or other manipulation?

§ CONCLUSIONS AND FUTURE WORK

In the current work, we have presented models that incorporate social signals for improved forecasts of cryptocurrency price values. Rather than predicting the returns (relative price difference) or the direction of fluctuations (increase versus decrease in price) – both are much easier tasks compared to our task formulation, our models predict the daily price high in USD.  We focused on social signals from the language, volume, and sentiment of discussions on Reddit and indications of popularity and direct contribution activity on GitHub.  Our analysis of model performance and comparisons to baselines that rely solely on price history have identified the benefit of social signals.
Although performance improvements, on average, are not statistically significant, in a task such as price value prediction in a market as volatile as cryptocurrencies, even a modest performance improvement can have a notable impact. More so, the minimization of worst-case error (MaxAPE) when using our proposed social signal-infused model is of significant benefit.

With the speed and volatility of cryptocurrency markets, it may be more valuable to model price in granularities of hours or minutes rather than days. However, a limitation of this approach is the accessibility of such fine-grained pricing data.  As the price data available was provided in daily increments, we presented the results of models for predictions of more coarse-grained daily price high-values. The proposed neural network models and framework of social signal extraction could be easily adapted to a more fine-grained approach.

Due to data collection (API) constraints, we performed a similar analysis of models using Twitter social signals for a subset of the time period of interest. Results were inconclusive but future work will also consider such social signals from Twitter and other related platforms alongside GitHub and Reddit.
We are also in the process of collecting data to cover not only the time period considered in the current work but also a second dataset that considers the periods of historic spikes and dips in Bitcoin pricing, the period covering the significant spike in the price of Bitcoin at the end of 2017 and steep drops that occurred over the first few months of 2018.

Our analysis and discussion have also identified several other avenues of future work. One such avenue is the potential to generalize the top performing models for the most popular coin or coins, Bitcoin, to forecast price values for other coins, such as new or less popular coins. Can we combine the social signals for Bitcoin or cryptocurrencies in general with a coin's historical price features to predict coins that do not have a sufficient volume of social activity? Another direction for future work is the adaption of these models for use in anomaly detection — can models be adapted to identify coins that may be compromised (through targeted deception in news stories and social media discussions or fraudulent behavior like “pump and dump” schemes)? Finally, a third avenue is an expanded exploration into the trends identified as potential drivers of price predictability: social activity volumes, market share, coin lifetimes, variance, etc. Our analysis has identified several trends and potential explanatory variables however we focused on the three coins with the highest community and developer interest. An expanded analysis that focuses on a larger variety of coins along both axes would provide a more rigorous evaluation.

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# Dynamically induced magnetism in KTaO3

R. Matthias Geilhufe1 Vladimir Juričić1,2 Stefano Bonetti3,4 Jian-Xin Zhu5
Alexander V. Balatsky1,6 1Nordita, KTH Royal Institute of Technology and
Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
2Departamento de Física, Universidad Técnica Federico Santa María, Casilla
110, Valparaíso, Chile
3Department of Physics, Stockholm University, 10691 Stockholm, Sweden
4 Department of Molecular Sciences and Nanosystems, Ca’ Foscari University of
Venice, 30172 Venice, Italy
5 Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA 6Department of Physics,
University of Connecticut, Storrs, CT 06269, USA

###### Abstract

Dynamical multiferroicity features entangled dynamic orders: fluctuating
electric dipoles induce magnetization. Hence, the material with paraelectric
fluctuations can develop magnetic signatures if dynamically driven. We
identify the paraelectric KTaO3 (KTO) as a prime candidate for the observation
of the dynamical multiferroicity. We show that when a KTO sample is exposed to
a circularly polarized laser pulse, the dynamically induced ionic magnetic
moments are of the order of 5% of the nuclear magneton per unit cell. We
determine the phonon spectrum using ab initio methods and identify T1u as
relevant soft phonon modes that couple to the external field and induce
magnetic polarization. We also predict a corresponding electron effect for the
dynamically induced magnetic moment which is enhanced by several orders of
magnitude due to the significant mass difference between electron and ionic
nucleus.

Introduction. Dynamical multiferroicity Juraschek  _et al._ (2017a), the
phenomenon where the fluctuating electrical dipoles induce magnetization,
represents the dynamical counterpart of the Dzyaloshinskii-Moriya mechanism
Katsura  _et al._ (2005). The origin of this effect lies in the duality
between the electric and magnetic properties Jackson (1999). Quite generally,
the effect features entangled quantum orders. Most notably, displacive
paraelectrics (PE) exhibiting a ferroelectric (FE) phase transition
Khmel’nitskii and Shneerson (1973); Rowley  _et al._ (2014); Chandra  _et al._
(2017); Roussev and Millis (2003); Edge  _et al._ (2015); Rischau  _et al._
(2017); Narayan  _et al._ (2018); Arce-Gamboa and Guzman-Verri (2018) can
display an elevated magnetic response induced by either quantum Dunnett  _et
al._ (2019) or thermal fluctuations Khaetskii  _et al._ (2021) close to the
critical point. On the other hand, the dynamical magnetization can be induced
by externally driving the material, e.g. by applying the light or a lattice
strain Juraschek  _et al._ (2017a). Dynamic multiferroicity is an example of
the nonlinear phononics phenomenology Juraschek  _et al._ (2017b), where a two
phonon process induces magnetization. From the perspective of the materials
where dynamical multiferrroicity can be realized, the prime candidate to
search for the effect is SrTiO3 (STO), the paradigmatic quantum critical
paraelectric where ferroelectricity is induced by displacive fluctuations. It
has been recently predicted that the dynamically induced magnetization both by
external means and intrinsically, close to the FE QCP in this material, may be
in a measurable range Dunnett  _et al._ (2019); Khaetskii  _et al._ (2021).

In contrast to STO, KTaO3 (KTO) is a quantum disordered paraelectric at low
temperatures with a significantly gapped transverse optical mode Rowley  _et
al._ (2014). At zero stress, KTO retains its cubic structure down to helium
temperatures Lines and Glass (2001). The transition into a ferroelectric phase
in KTO can be induced as well, e.g., by impurities Golovina  _et al._ (2012);
Prater  _et al._ (1981) or strain Tyunina  _et al._ (2010). It is assumed that
the ground state of KTO is a quantum disordered phase and significantly away
from quantum critical fluctuations. Since KTO behaves as a regular quantum
paraelectric quantum critical modes are gapped. Furthermore, on the
paraelectric side of the quantum critical point, the fluctuations of the
polarization are expected to be stronger and might give rise to a more
dominant signal of a dynamically induced magnetic moment. So far, no
prediction regarding the effect of a dynamically induced magnetization has
been made for KTO, and this is precisely the aim of the current paper.

Following the formalism of dynamical multiferroicity Juraschek  _et al._
(2017a); Dunnett  _et al._ (2019); Khaetskii  _et al._ (2021), we investigate
the induction of magnetic moments by applying circularly polarized terahertz
radiation resonant with the phonon frequency that yield fluctuating local
electric dipoles, according to

$\mathbf{M}=\alpha\mathbf{P}\times\frac{\partial}{\partial
t}\mathbf{P}=\gamma\mathbf{u}\times m\frac{\partial}{\partial t}\mathbf{u}.$
(1)

Here, $\mathbf{M}$ denotes the local magnetic moment, $\mathbf{P}$ the
electric polarization, $\mathbf{u}$ the atomic displacement (associated with
the relevant phonon mode in our analysis), $m$ the particle mass, while
$\alpha$ and the gyromagnetic ratio $\gamma$ are the respective coupling
constants. By performing an ab initio analysis of the phonon spectrum (see
Fig. 1), we single out ${\rm T}_{1u}$ soft phonon modes as relevant for the
dynamical multiferroicity. As we show, using both single mode approximation
and the full dynamical matrix approach, when the system is subjected to a
resonant circularly polarized laser pulse (Fig. 2), one obtains a measurable
magnetic signal. Taking a realistic value of the damping for the mode, we find
that the induced magnetic moment per unit cell can reach the values of $\sim
0.05~{}\mu_{N}$, where $\mu_{N}$ is the nuclear magneton. We also predict an
enhancement of the effect due to the coupling of the ion dynamics with the
electronic one, which should be detectable experimentally.

Phonon spectrum: First-principles calculation. KTO crystallizes in a cubic
lattice with space group Pm$\overline{3}$m (Fig 1(a)). We chose the
experimental lattice constants as determined by Zhurova et al. Zhurova  _et
al._ (2000), with a unit cell volume of 63.44 Å3. The phonon spectrum was
calculated using Phonopy Togo and Tanaka (2015). The related force matrix was
obtained from a $2\times 2\times 2$ supercell with automatically generated
displacements, where forces were calculated using the Vienna ab initio
simulation package VASP Kresse and Furthmüller (1996). The exchange
correlation functional was approximated by the PBE functional Perdew  _et al._
(1996). We chose $8\times 8\times 8$ points for the Brillouin zone integration
which corresponds to a $\mathbf{k}$-mesh density of $\approx
1050\,\mathbf{k}\text{-points}/\AA^{-3}$. We used a cut-off energy of $700$
eV. Additionally, we calculated the Hessian matrix for the energy landscape
using density functional perturbation theory. This approach also provides a
force matrix and phonon frequencies at the $\Gamma$ point, which we used to
estimate the dynamically induced magnetization, as explained below.

Figure 1: (a) Unit cell of KTaO3. (b) Calculated phonon spectrum and phonon
density of states.

The KTO unit cell contains 5 inequivalent sites, resulting in 15 phonon modes.
We studied the symmetry of the phonon modes using GTPack Geilhufe and Hergert
(2018); Hergert and Geilhufe (2018). Constructing a five-dimensional
permutation representation $\Gamma_{p}$ for the point group Oh and the 5 unit
cell sites and computing the direct product with the vector representation
$\Gamma_{v}=T_{1u}$ we obtain $\Gamma_{p}\otimes\Gamma_{v}\simeq 4T_{1u}\oplus
T_{2u}$ corresponding to the expected modes at the $\Gamma$-point in the
Brillouin zone Hergert and Geilhufe (2018). Using Phonopy we verify 4 T1u
modes at frequencies 0.0 THz, 3.02 THz, 6.16 THz, and 16.38 THz, as well as
one T2u mode at 7.94 THz. The former modes being soft but finite-frequency
modes are instrumental for the dynamical multiferroicity, as shown below. The
full phonon spectrum showing 3 acoustical and 12 optical modes is plotted in
Fig. 1(b). The values are in good agreement with previous experiments on KTO
Farhi  _et al._ (2000). These frequencies slightly change when calculated by
using the density functional perturbation theory, giving 0.0 THz, 3.17 THz,
6.19 THz, 8.05 THz, and 16.53 THz. We notice that, in contrast to STO, KTO
does not give rise to negative energy modes in the phonon spectrum for the
cubic phase, indicating the absence of a structural phase transition at low
temperatures. After identifying the T1u soft phonon modes, we analyze the
magnetic signal resulting from the exposure of the KTO system to an externally
applied circularly polarized laser pulse.

| K | Ta | O
---|---|---|---
charge [e $\approx 1,602\times 10^{-19}$ C] | 0.867 | 4.954 | -1.940
mass [u $\approx 1,66\times 10^{-27}$ kg] | 39.1 | 180,95 | 16.0
Table 1: Site parameters. Charge values according to DFT calculations
performed in this study.

Dynamical Multiferroicity. The polarization contains an ionic and an
electronic contribution and can be written as

$P_{i\alpha}=Z_{i\alpha\beta}^{*}u_{i\beta}+\epsilon_{0}\left(\epsilon_{\alpha\beta}-\delta_{\alpha\beta}\right)E_{\beta}.$
(2)

Here $u_{i\alpha}$ denotes a displacement of atom $i$ along the Cartesian
coordinate $\alpha$. The Born effective charge $Z_{i\alpha\beta}^{*}$
describes the response of the macroscopic polarization per unit cell to the
displacement of atom $i$, $Z^{*}_{i\alpha\beta}=\left.\Omega\frac{\partial
P_{\beta}}{\partial u_{i\alpha}}\right|_{\mathbf{E}=0}$, with $\Omega$ the
unit cell volume Ghosez  _et al._ (1998). The calculated Born effective
charges for KTO are given in Tab. 2. The electronic response of the
polarization to the electric field is approximated in terms of the static
dielectric tensor $\epsilon_{ij}$. Due to the cubic symmetry of the unit cell
the dielectric tensor is diagonal and we obtain

$\epsilon_{xx}=\epsilon_{yy}=\epsilon_{zz}=5.4.$ (3)

This value is sensitive to the chosen computational parameters, but consistent
with other references Persson (2014). $\epsilon_{0}\approx 5.52~{}$e2 keV-1
Å-1 is the vacuum permittivity.

| $Z^{*}_{xx}$ | $Z^{*}_{yy}$ | $Z^{*}_{zz}$
---|---|---|---
K | 1.13 | 1.13 | 1.13
O | -6.58 | -1.69 | -1.69
O | -1.69 | -6.58 | -1.69
O | -1.69 | -1.69 | -6.58
Ta | 8.83 | 8.83 | 8.83
Table 2: Calculated Born effective charges in units of the elementary charge
$e$.

We calculate atomic displacements $\mathbf{u}_{i}$ at the site $i$ using
classical equations of motion,

$\ddot{u}_{i\alpha}(t)+\eta\,\dot{u}_{i\alpha}(t)+\sum_{j\beta}K_{i\alpha\,j\beta}u_{j\beta}(t)=\frac{Z_{i\alpha}}{m_{i}}E^{*}_{\alpha}(t).$
(4)

Here, $Z_{i\alpha}=Z_{i}^{0}+\sum_{\beta}Z^{*}_{i\alpha\beta}u_{i\beta}$, with
$Z_{i}^{0}$ being the bare charge of the ion (see Tab. 1). $m_{i}$ the mass of
atom $i$, $\eta$ is a damping factor, and $\mathrm{K}$ is the dynamical
matrix. The electric field within the medium $\mathbf{E}^{*}$ is related to
the vacuum electric field $\mathbf{E}$ by

$\mathbf{E}^{*}=\epsilon^{-1}\mathbf{E}.$ (5)

In experiments an additional loss in the field strength has to be taken into
account due the polarization process. In our approach, the electric field
induces a collective displacement of the ionic positions by coupling to the
charge. Note that we do not include higher order corrections to the dielectric
screening Cartella  _et al._ (2018).

We continue by discussing the size of the dynamically induced magnetic moment
using a simplified analytical model. The full set of coupled differential
equations is solved numerically afterwards. We start by solving Eq. (4) within
a single-mode approximation, by considering one relevant mode $\omega_{i}=2\pi
f_{i}$, corresponding to one relevant site,

$\ddot{u}_{\alpha}(t)+\eta\,\dot{u}_{\alpha}(t)+\omega^{2}_{i}u_{\alpha}(t)=\frac{q}{m}\,E^{*}_{\alpha}(t).$
(6)

We choose circularly polarized light, i.e.,
$\mathbf{E}^{*}(t)=E^{*}_{0}\left(\sin(\omega t),\cos(\omega t),0\right)$. In
a coarse approximation, from (6), we notice that the displacement scales
linearly with the applied field,
$\mathbf{u}\approx\frac{q\mathbf{E}^{*}}{m\omega^{2}}$. For a harmonic
displacement, we can estimate the corresponding time derivative as
$\dot{\mathbf{u}}\approx\omega\mathbf{u}$. Using equation (1) and replacing
the gyromagnetic ratio by $\gamma=\frac{q}{2m}$, we can estimate the
asymptotic behavior for the dynamically induced magnetic moment by

$M_{z}\sim\frac{q^{3}{\mathbf{E}^{*}}^{2}}{m^{2}\omega^{3}}.$ (7)

Hence, the effect increases quadratically in the field strength, but decreases
with $\omega^{-3}$ in the driving frequency. The corresponding values for the
charge $q$ and the mass $m$ for KTO are given in Tab. 1. The charges
calculated using DFT are close to the chemistry picture of an ionic crystal,
with integer oxydation states O-2, K+1, and Ta+5.

Equation (6) can be solved exactly. As we are solely interested in the
contribution to the atomic displacement emerging due to exposure to an
external laser field, we only keep the inhomogeneous part of the solution of
Eq. (4) that can be written as

$\mathbf{u}(t)=\frac{1}{\Delta_{\omega}^{4}+4\eta^{2}\omega^{2}}\left(\begin{array}[]{cc}\Delta_{\omega}^{2}&-2\eta\omega\\\
2\eta\omega&\Delta_{\omega}^{2}\end{array}\right)\frac{q}{m}\mathbf{E}^{*},$
(8)

with $\Delta_{\omega}^{2}=\omega^{2}_{i}-\omega^{2}$. Evaluating the
polarization as $\mathbf{P}=\frac{q}{V}\mathbf{u}$, the $\omega$-dependent
part of Eq. (8) can be interpreted as the susceptibility $\chi$, by
transforming it into the well-known expression
$\mathbf{P}=\chi\epsilon_{0}\mathbf{E}$. Hence, we obtain for the
magnetization

$M_{z}=\frac{q^{3}\omega{\mathbf{E}^{*}}^{2}}{2m^{2}\left(\eta^{2}\omega^{2}+\Delta_{\omega}^{4}\right)}.$
(9)

In the limit $\omega\gg\omega_{i}$, we obtain
$\Delta_{\omega}^{4}\approx\omega^{4}$. Neglecting the damping term
$\eta^{2}\omega^{2}\ll\omega^{4}$ gives a similar expression to Eq. (7).

System driven with a terahertz pulse. Next, we consider a more realistic
terahertz pulse and solve Eq. (4) numerically. Such terahertz pulses are
nowadays available Salén  _et al._ (2019) and allow for large peak electric
field to drive phonons, but with an average deposited energy which is not
enough to melt the sample. We set
$\mathbf{u}_{\alpha}(0)=\dot{\mathbf{u}}_{\alpha}(0)=0$. The pulse is modeled
by a Gaussian embedding as follows,

$\mathbf{E}(t)=E_{0}\,e^{-\frac{(t-t_{0})^{2}}{2\sigma}}\left(\begin{array}[]{c}\sin(\omega
t)\\\ \cos(\omega t)\\\ 0\end{array}\right).$ (10)

The considered driving frequencies are 3.17 THz and 6.19 THz, being resonant
with the phonon modes. We choose a total width of $2$ ps with a peak at $2$ ps
and obtain the solution for a window up to 16 ps. The dynamically induced
magnetic moments are shown in Fig. 2 for various values of the damping
parameter ($\eta$ = 0.05 THz, 0.10 THz, 0.15 THz). Depending on the damping
factor we observe a slow decay of the dynamically induced magnetic moment. The
maximal total dynamically induced magnetic moment is $\approx 0.7\mu_{N}$ for
small damping of $\eta<0.1$ THz. The dynamically induced magnetic moment
decreases by about one order of magnitude for a driving frequency in resonance
with the $T_{1u}$ mode at 6.19 THz.

Figure 2: Dynamically induced total moment per unit cell for a laser pulse
with driving frequencies 3.17 THz (left panel) and 6.19 THz (right panel).

Due to the opposite local charges of the ions, the induced moments have
opposite strength for O, compared to Ta and K. The site resolved dynamically
induced moments due to local displacements are shown in Fig. 3. We observe
that the main contributions to the total induced magnetization per unit cell
come from Ta and O, being of the order of $0.2\mu_{N}$ and $-0.1\mu_{N}$ for a
small value of the damping parameter, $\eta=0.05$ THz.

Figure 3: Site resolved dynamically induced moments within the unit cell. We
used the same pulse as in Fig. 2, a driving frequency of 3.17 THz and a
damping of $\eta=0.05$ THz.

Conclusion and Outlook. We showed that KTO is a prominent candidate for the
observation of the dynamical multiferroicity. By performing an ab initio
analysis, we first find that the T1u soft phonon modes may be relevant for the
observation of the effect. We suggest an experimental setup where the KTO
sample is exposed to a circularly polarized laser field in the terahertz range
to excite phonons resonantly. The dynamically induced magnetization due to
locally oscillating dipoles could be measured by the time-resolved Faraday
effect using a femtosecond laser pulse in the visible range. The estimated
scale of the effect for an experimentally feasible setup is in the order of
$10^{-2}~{}\mu_{N}$ per unit cell, with $\mu_{N}$ being the nuclear magneton.
In Eq. (7) we show that in an asymptotic limit, the induced moment scales
quadratically with the electric field strength and to the the third power in
the charge. It also scales inversely with the third power in driving frequency
and the mass squared. In particular the latter feature could be of interest.

Here we discussed the ionic movement as a driver for the induced magnetism. We
now point out an interesting possibility of induced electron motion that also
would produce the magnetic moment. We expect the angular momentum transfer
from the moving ions to the electronic charge cloud in the solid. While the
exact microscopic details need to be worked out the qualitative argument goes
as follows. To estimate the gyromagnetic ratio for the coupling we follow
Refs. Rebane (1983); Khaetskii  _et al._ (2021) in a modified form. The
position of a charged ion is denoted by $\mathbf{u}_{+}$, the average
displacement of the electron cloud is $\mathbf{u}_{-}$. The respective masses
are $m_{+}$ and $m_{-}$. We introduce average and relative coordinates
$\mathbf{U}=(m_{+}\mathbf{u}_{-}+m_{-}\mathbf{u}_{+})/(m_{+}+m_{-})$ and
$\mathbf{u}=\mathbf{u}_{+}-\mathbf{u}_{-}$. We focus on the relative
coordinate, having the momentum $p=\mu\dot{\mathbf{u}}$ with
$\mu=m_{+}m_{-}/(m_{+}+m_{-})$. It follows for the angular momentum of the
relative coordinate

$\mathbf{L}=\mathbf{u}\times\mathbf{p}=\frac{m_{+}m_{-}}{m_{+}+m_{-}}\mathbf{u}\times\dot{\mathbf{u}}.$
(11)

Setting $m_{-}\mathbf{u}_{+}=m_{+}\mathbf{u}_{-}$ we obtain for the
dynamically induced moment according to Eq. (1)

$\mathbf{M}=\mathbf{m}_{+}+\mathbf{m}_{-}=\frac{q}{2}\frac{m_{+}-m_{-}}{m_{+}+m_{-}}\mathbf{u}\times\dot{\mathbf{u}}.$
(12)

Taking $\mathbf{M}=\gamma\mathbf{L}$, we obtain for the gyromagnetic ratio

$\gamma=\frac{q}{2}\left(\frac{1}{m_{-}}-\frac{1}{m_{+}}\right).$ (13)

For nonequal charges, this equation generalizes to

$\gamma=\frac{m_{+}}{m_{-}}\frac{q_{+}}{m_{+}+m_{-}}-\frac{m_{-}}{m_{+}}\frac{q_{-}}{m_{+}+m_{-}}.$
(14)

Hence, from Eqs. (13) and (14) it becomes apparent that the total gyromagnetic
ratio of ion and electron is dominated by the electron mass ($m_{i}/m_{e}\sim
10^{3}\dots 10^{5}$). Here we need to distinguish between a direct coupling of
the electron to the external field
$\sim\epsilon_{0}\left(\epsilon_{\alpha\beta}-\delta_{\alpha\beta}\right)E_{\beta}$
as well as an induced motion of the electrons due to the ionic movement. While
the former contribution to the total magnetization should vanish with
vanishing electric field, the latter should be present as long as the ionic
movement persists. More precise analysis will be a topic of a separate
publication.

We propose KTO as a prominent candidate for the observation of the dynamical
multiferroicity. Our findings open up a route for the experimental detection
of the entangled dynamical orders. They should also motivate further studies
of the candidate materials for the realization of the effect.

Acknowledgment. We are grateful to G. Aeppli, U. Aschauer, M. Basini, M.
Pancaldi, O. Tjernberg, I. Sochnikov, N. Spaldin and J. Weissenrieder for
useful discussions. We acknowledge support from VILLUM FONDEN via the Centre
of Excellence for Dirac Materials (Grant No. 11744), the European Research
Council under the European Union Seventh Framework ERS-2018-SYG 810451 HERO,
the Knut and Alice Wallenberg Foundation KAW 2018.0104. V.J. acknowledges the
support of the Swedish Research Council (VR 2019-04735) and J.-X.Z. was
supported by the Los Alamos National Laboratory LDRD Program. SB acknowledges
support from the Swedish Research Council (VR 2018-04611). The computational
resources were provided by the Swedish National Infrastructure for Computing
(SNIC) via the High Performance Computing Centre North (HPC2N) and the Uppsala
Multidisciplinary Centre for Advanced Computational Science (UPPMAX).

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