{"id":"ee9c367f-fd8f-46f9-9c7f-7d481c9753f7","arxiv_id":"2502.09079","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Major cryptocurrency price series are indistinguishable from Brownian noise in complexity analysis, and no forecasting model beats a naive random-walk baseline.","lead":"This paper measures how random five major cryptocurrency prices are and finds they behave like Brownian noise, while simple forecasting models match or beat complex machine learning and deep learning models. The study suggests that univariate price forecasting of these cryptocurrencies has little room for improvement beyond naive baselines.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The analysis uses raw USD price levels, so the Brownian-noise complexity and naive-model forecasting results are generic properties of any I(1) series, not evidence of cryptocurrency-specific unpredictability; checking returns or log-prices would settle this.","rationale":"The reader's weakest assumption identifies exactly this issue: raw USD price levels are inappropriate for drawing conclusions about unpredictability, because any integrated series would produce the same Brownian-noise CH-plane position and 1/f^2 spectrum. My analysis confirms that this is the single most load-bearing concern. The complexity results (Section 4.1) are generic to I(1) processes, and the forecasting results (Section 4.2) are dominated by the trivial fact that a random walk's best predictor is its last value. Thus the paper's central claim is overstated unless the analysis is repeated on returns or log-prices. This does not change the reader's conditional verdict: the empirical work is valid as a demonstration of random-walk behavior in crypto prices, but the conclusion must be reframed and the additional analyses run. The DL MAPE values near 99% (e.g., VanillaRNN and LSTM for BTC and ETH in Table 3) are concerning, but they are not the primary load-bearing issue because the qualitative claim of naive superiority would likely survive even if those models were fixed. My concrete test directly addresses the core transformation issue and would settle whether the 'unpredictability' conclusion is specific to cryptocurrencies. I agree with the reader's assessment and therefore leave the verdict unchanged. No ad hominem is intended; the critique is on the argument's scope.","tokens_in":14500,"tokens_out":4431,"duration_ms":49575,"concrete_test":"Reproduce Sections 4.1 and 4.2 using daily log-returns r_t = log(P_t / P_{t-1}) instead of price levels, with the same time windows, d values, and model set, and add a constant-zero (or constant-mean) benchmark. Specifically: (1) Compute the CH-plane coordinates and PSD exponent for the five return series and for a simulated random-walk price series; if the returns plot near white noise (alpha ~ 0) and the simulated random-walk prices reproduce the paper's Brownian position, the complexity result is an artifact of integration. (2) Retrain all forecasting models on the last t_w returns and forecast the next f_h returns, comparing MAPE (or RMSE) against the constant-zero forecast.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that univariate cryptocurrency forecasting is 'essentially comparable to pure noise forecasting' rests on treating raw USD price levels as the univariate object for both the complexity analysis and the forecasting benchmark. For any non-stationary integrated process, such as a random walk or the cumulative sum of iid noise, the price level has a PSD ~1/f^2 and maps to the Brownian-noise region of the CH-plane (Figs. 3-4, Table 2), regardless of whether its increments are predictable. Thus the complexity results in Section 4.1 do not distinguish cryptocurrency prices from a generic random walk; they only confirm that prices are integrated. Likewise, the forecasting results in Tables 3-4 compare models on price levels, where the optimal short-horizon forecast for an I(1) process is the current price or a drift-adjusted version, so naive models should trivially dominate. This makes the conclusion a mathematical consequence of the price-level transformation rather than an empirical finding about cryptocurrency-specific unpredictability. The interpretation that shorter windows 'increasingly align with white noise' is also suspect, since finite-sample PSD estimates for an integrated process flatten as the window shrinks. To support the claim as stated, the analysis must be applied to returns or log-prices; if those are white-noise-like, the correct claim is 'returns are unpredictable,' not 'prices are pure noise.' This concern is load-bearing because both pillars of the paper -- complexity and forecasting -- are artifacts of the chosen object, so changing it collapses the Brownian resemblance to white noise and changes the meaning of the naive-model benchmark.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies five daily USD cryptocurrency price series (LTC-USD, BNB-USD, BTC-USD, ETH-USD, XRP-USD) from 2020-07-03 to 2023-12-21. It characterizes their complexity using Bandt-Pompe permutation entropy, the Complexity-Entropy causality plane, the power spectral density, and permutation Jensen-Shannon distances to colored noises, and it benchmarks twelve forecasting models (statistical, machine learning, and deep learning) with a rolling-origin evaluation across three training-window lengths and forecast horizons of 1, 7, and 30 days. The paper concludes that cryptocurrency price levels resemble Brownian noise and that univariate forecasting of cryptocurrencies is 'essentially comparable to pure noise forecasting,' with simple Naive models consistently outperforming more complex ML and DL models.","tokens_in":14711,"tokens_out":6079,"duration_ms":59398,"significance":"The forecasting benchmark is a useful negative result: it confirms at daily frequency that simple random-walk-style baselines are hard to beat for these five assets, and it provides a cautionary counterweight to studies that report ML/DL gains without comparing to naive baselines. The rolling-origin evaluation and the inclusion of multiple horizons and time windows are strengths. However, the complexity analysis is applied to raw price levels, so the Brownian-noise resemblance is a generic property of integrated series rather than a cryptocurrency-specific finding; the forecasting comparison lacks formal significance tests; and the DL results contain MAPE values near 99% that look like implementation artifacts. The central claim is defensible only after reframing or after additional analysis on returns or log-prices.","major_comments":[{"comment":"The complexity analysis is performed on raw USD price levels, yet the conclusions are phrased as properties of cryptocurrency dynamics. For any non-stationary integrated series—for instance the cumulative sum of independent increments, regardless of whether those increments are predictable—the price-level PSD is approximately 1/f^2 and the CH-plane position falls near the Brownian noise line. The reported Brownian resemblance is therefore a mathematical consequence of the level transformation and does not distinguish cryptocurrencies from a generic random walk. This affects the paper's framing in the abstract and Section 5 ('pure noise'). The authors should either re-run the CH-plane/PJSD/PSD analysis on returns or log-prices, or explicitly reframe the result as 'price levels behave like an integrated process; returns are the relevant object for predictability.' Without this, the complexity pillar of the paper does not support cryptocurrency-specific unpredictability.","section":"Section 3.1 and 4.1 (Figs. 3-4, Table 2)"},{"comment":"The claim that sophisticated statistical models are 'not statistically distinguishable' from Naive models is not supported by evidence. Table 4 reports mean MAPE and standard deviation, and the text treats overlapping standard deviations as a statistical finding. Overlap of unconditional standard deviations is not a significance test, and the aggregation across assets and windows ignores the rolling-window dependence of the forecast errors. A paired comparison per asset and window (e.g., Diebold-Mariano tests on absolute or squared errors, or a Wilcoxon signed-rank test on MAPE differences) is needed. This matters because the paper's headline conclusion is that naive models are comparable or superior; without a formal test, small numerical differences such as NaiveSeasonal 1.636 vs AutoETS 1.634 at horizon 1 in Table 4 cannot be interpreted as evidence of equivalence.","section":"Section 4.2 (Tables 3 and 4)"},{"comment":"The deep-learning results contain MAPE values near 99% for BTC-USD and ETH-USD across all time windows and horizons (e.g., VanillaRNN 99.091 and LSTM 99.765 for BTC-USD in the 3-year window). Given price levels of tens of thousands of dollars, these values indicate that the models failed to produce usable forecasts rather than that the series is unpredictable. The paper states in Section 3.1 that 'no data preprocessing was applied' and provides no hyperparameters, architecture details, training epochs, or seeds for the DL models. Without this information, or without an evaluation on standardized returns or levels, the conclusion that 'DL models struggle' is based on potentially broken training runs. The authors should report the exact Darts configuration for each DL model and either preprocess inputs (e.g., scaling) or explain why the reported MAPE values are plausible.","section":"Section 3.3 and Table 3"},{"comment":"The statement that the PSD 'consistently follows the power law decay exponent of the Brownian motion' is based on a visual comparison with a 1/f^2 reference line. No power-law exponent is estimated from the data, no confidence intervals are given, and the PSD for shorter time windows is not shown. A quantitative fit of the log-log slope (e.g., via linear regression on the log PSD or a periodogram-based estimator) with uncertainty bounds is needed to support the claim that the exponent is indistinguishable from 2 rather than, say, 1.5 or 2.5. This is load-bearing for the complexity conclusion.","section":"Section 4.1 (Fig. 4)"}],"minor_comments":[{"comment":"The caption reads 'Permutation Jehnsenn-Shannon distances'; this should be 'Permutation Jensen-Shannon distances'.","section":"Table 2 caption"},{"comment":"The three subfigures (a), (b), and (c) appear to display the same three-panel image; please replace them with the actual CH-plane plots for each of the three time windows.","section":"Figure 3"},{"comment":"The description of C_JS[P] as 'multiplying this divergence by the entropy of P' omits the required normalization; the standard formula is C_JS[P] = Q_J[P, P_e] * H[P] / H_max, where Q_J includes a normalization constant.","section":"Section 3.2"},{"comment":"The phrase 'no data preprocessing was applied' is ambiguous: clarify whether it refers only to cleaning/missing values, and specify whether input features were standardized for the ML and DL models.","section":"Section 3.1"},{"comment":"The term 'pure noise forecasting' is never defined; specify the noise forecast benchmark used (e.g., a random walk without drift, or a bootstrap of returns) so the comparison is explicit.","section":"Section 5"},{"comment":"Reference [23] is cited twice with different descriptions in the same paragraph; please review the citation to avoid duplication and clarify which findings come from which study.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a conference-style empirical study with a modest but useful negative result. The main issues are fixable: re-run or reframe the complexity analysis on returns/log-prices, add formal significance tests for the forecast comparisons, and document the DL experimental setup. The duplicate Figure 3 panels and several typos also suggest the arXiv version needs editorial care before it can be considered for journal publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a careful, internally consistent benchmark of univariate crypto forecasting, and the headline—Naive models beat ML/DL on MAPE across five coins, three windows, and multiple horizons—holds up for the data as analyzed. But the companion claim, that crypto prices 'resemble Brownian noise' and are 'pure noise,' is largely an artifact of analyzing raw price levels rather than returns. Any integrated series lands near the Brownian-noise locus in the CH-plane and has a 1/f^2 PSD, and the optimal forecast for a random walk is the last price. So the complexity result is not crypto-specific evidence, and the forecasting result is mostly a restatement of the random walk property.\n\nWhat the paper does well: it is a broad benchmark implemented in Darts, with rolling retraining, multiple horizons, and explicit comparison to naive baselines. Tables 3 and 4 give a useful snapshot of how far ML/DL methods fall behind simple drift and seasonal models. The authors also correctly note that many prior ML papers skipped naive comparisons, which is a real gap that this study helps fill.\n\nThe soft spots are mostly about framing and inference. There are no formal significance tests; 'not statistically distinguishable' is inferred from overlapping standard deviations, which is not a valid test. The DL results for BTC and ETH are pathological (MAPE around 99%), which suggests scaling or implementation problems that are not discussed. No code or data is provided, so the numbers cannot be reproduced. The PJSD distances in Table 2 are all fairly close, and the XRP exception (pink noise) is noted but not explored. Most importantly, the conclusion that 'forecasting future prices based solely on historical data may be unfeasible' is too broad, since the analysis never looks at returns or log prices.\n\nWho this is for: practitioners and researchers doing crypto forecasting benchmarks, as a caution that complex models require naive baselines. It deserves a serious referee because the empirical effort is real, but the paper needs major revision: either move the analysis to returns, or frame the conclusions as 'daily price levels behave like a random walk.' I would engage with it constructively, but I would not take the Brownian-noise and pure-noise claims as new results about crypto specifically.","headline":"A solid univariate forecasting benchmark whose Brownian-noise conclusion is mostly an artifact of using price levels rather than returns; worth refereeing, but needs reframing.","tokens_in":15328,"tokens_out":2787,"would_cite":true,"duration_ms":26478,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Five major cryptocurrencies show Brownian-noise complexity, and naive models beat machine-learning and deep-learning forecasters on daily USD prices.","keywords":["cryptocurrency forecasting","univariate time series","permutation entropy","complexity-entropy causality plane","Brownian noise","naive forecasting models","machine learning forecasting","deep learning forecasting"],"falsifier":"Compute the same CH-plane, PSD, and model benchmark on daily returns or log-returns instead of raw prices; if the series move to the white-noise region and the spectral slope flattens to $\\alpha\\approx 0$, while naive models still win, the Brownian-noise claim is an artifact of cumulative price levels rather than a discovery about crypto dynamics.","tokens_in":14250,"feed_emoji":"📉","tokens_out":7795,"duration_ms":70239,"temperature":0.7,"pith_summary":"Using daily USD prices for Litecoin, Binance Coin, Bitcoin, Ethereum, and XRP between 2020 and 2023, this paper argues that univariate forecasting of cryptocurrencies is essentially comparable to pure noise forecasting. In the Complexity–Entropy plane all five coins sit on the colored-noise line, their power spectra follow roughly $1/f^2$ (Brownian) decay, and permutation Jensen–Shannon distances are smallest to Brownian noise for most coins. Across three training windows and 1-, 7-, and 30-day horizons, naive models match or beat sophisticated statistical, machine-learning, and deep-learning models on MAPE; on shorter windows the series move toward white noise. The conclusion a sympathetic reader takes away is that simple benchmarks are the right yardstick and that price history alone does not support complex forecasting.","feed_headline":"Crypto prices look like pure noise to forecasters","feed_subtitle":"Five major coins match Brownian-noise complexity, and naive models beat ML and deep learning.","key_machinery":"The central measuring device is the Complexity–Entropy causality plane (CH-plane), which plots permutation entropy (the Shannon entropy of ordinal patterns of length $d$) against the intensive Jensen–Shannon statistical complexity. A time series that sits on the 'noises line' of this plane is structurally similar to white, pink, Brownian, or $1/f^{5/2}$ noise; the paper adds the permutation Jensen–Shannon distance to quantify which noise is nearest, and the power spectral density to estimate the exponent $\\alpha$ in $S(f)\\propto 1/f^{\\alpha}$. On the forecasting side, the load-bearing machinery is a rolling-window backtest that retrains each model as the training set grows, and the MAPE metric evaluated only at the forecast horizon, with naive models serving as the baseline that complex models must beat.","core_discovery":"The paper's central claim is that the five cryptocurrencies' univariate dynamics closely resemble noise processes when measured with permutation entropy, statistical complexity, and power spectral density. All series lie on the noises line of the CH-plane, with a spectral exponent close to $\\alpha=2$, the Brownian-motion value; XRP is closest to pink ($1/f$) noise. Consistently, no statistical, machine-learning, or deep-learning model reliably outperforms NaiveDrift or NaiveSeasonal across all settings, and the deep recurrent models often fail badly, with MAPEs near 100 percent on Bitcoin and Ethereum. The authors interpret this as evidence that apparent complexity in cryptocurrency prices is mostly noise, and that forecasting future prices from historical prices alone is largely unfeasible.","pith_inferences":["The Brownian-noise result is likely a generic property of cumulative price levels: any integrated random walk would occupy the same CH-plane region, so a natural extension is to rerun the analysis on returns, where the expected signature would shift to white noise.","A testable extension is to check whether volatility clustering or jump dynamics in returns are forecastable even though levels are not; the current study's MAPE on levels would not detect such structure.","The naive-model dominance implies that practitioners should evaluate trading strategies net of transaction costs; a random-walk benchmark may already be the best achievable baseline.","Adding exogenous covariates such as sentiment or volume is the paper's own suggested escape route, but the evidence here suggests the gains would come from those covariates, not from more elaborate univariate models."],"forward_implications":["If the claim holds, reported gains from univariate crypto price forecasting should be treated skeptically unless they beat a naive baseline on out-of-sample data.","Model selection in this domain should default to simple baselines; complex deep architectures add cost and variance without accuracy gains.","Shorter training windows push the series closer to white noise, so short-horizon forecasts are not more reliable than long-horizon ones for these assets.","The poor performance of Prophet in this setting shows that models assuming seasonality or holiday structure are mismatched to cryptocurrency dynamics."],"supporting_citations":[{"why":"Defines permutation entropy, the randomness measure used for all series.","marker":"[3]"},{"why":"Introduces the complexity-entropy causality plane and its noise-line benchmark.","marker":"[21]"},{"why":"Provides the permutation Jensen-Shannon distance used to compare each cryptocurrency to colored noises.","marker":"[27]"},{"why":"Supplies the naive, drift, ARIMA, and exponential-smoothing baselines that the complex models must beat.","marker":"[14]"},{"why":"Provides the software implementation of the models and the rolling-window backtesting procedure.","marker":"[12]"},{"why":"Earlier complexity-entropy analysis of cryptocurrencies that this study extends toward forecasting.","marker":"[23]"},{"why":"Defines the XGBoost model included as a machine-learning representative.","marker":"[8]"},{"why":"Defines the N-BEATS architecture used as a deep-learning representative.","marker":"[20]"}],"fun_headline_variants":["Crypto prices look like pure noise to forecasters","Five coins mimic Brownian noise, defying forecasters","Naive models beat ML for crypto forecasting","Crypto unpredictability: complexity is just noise","Forecasting crypto? It's like predicting Brownian motion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on treating raw USD price levels as the univariate series; any non-stationary random walk would show the same Brownian-like complexity profile, so the noise conclusion may be a property of integration rather than of cryptocurrencies specifically.","fun_headline_variants_meta":{"raw":{"variants":["Crypto prices look like pure noise to forecasters","Five coins mimic Brownian noise, defying forecasters","Naive models beat ML for crypto forecasting","Crypto unpredictability: complexity is just noise","Forecasting crypto? It's like predicting Brownian motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000156,"raw_usage":{"total_tokens":1197,"prompt_tokens":906,"completion_tokens":291,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":216}},"tokens_in":522,"tokens_out":291,"duration_ms":3216,"temperature":1.0,"reasoning_tokens":216,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:41:58.875295+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same CH-plane, PSD, and model benchmark on daily returns or log-returns instead of raw prices; if the series move to the white-noise region and the spectral slope flattens to $\\alpha\\approx 0$, while naive models still win, the Brownian-noise claim is an artifact of cumulative price levels rather than a discovery about crypto dynamics.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the complexity-entropy causality plane and its noise-line benchmark."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines permutation entropy, the randomness measure used for all series."},{"cited_title":"Ribeiro, and Osvaldo A","cited_arxiv_id":null,"evidence_quote":"Provides the permutation Jensen-Shannon distance used to compare each cryptocurrency to colored noises."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the naive, drift, ARIMA, and exponential-smoothing baselines that the complex models must beat."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the software implementation of the models and the rolling-window backtesting procedure."},{"cited_title":"Ludermir, and Tatijana Stosic","cited_arxiv_id":null,"evidence_quote":"Earlier complexity-entropy analysis of cryptocurrencies that this study extends toward forecasting."}],"review_version":1}