{"format":"alysly-page/1","url":"https://alysly.com/learn/stationarity-and-memory/","json":"https://alysly.com/json/learn/stationarity-and-memory/index.json","lang":"en-US","title":"Fractional Differentiation: Stationarity Without Memory Loss","description":"Why returns throw away too much information and how fractional differentiation makes a price series stationary without erasing its memory.","published":"2026-08-02T15:54:51+02:00","modified":"2026-11-01T18:26:04+01:00","alternates":[{"lang":"de-DE","url":"https://alysly.com/de/lernen/stationaritaet-und-erinnerung/","json":"https://alysly.com/json/de/lernen/stationaritaet-und-erinnerung/index.json"}],"about":"Machine-readable version of the page at url. It contains the text, tables and key figures published there, but not the interactive charts and tools.","breadcrumb":[{"name":"Investing & Trading Basics","url":"https://alysly.com/learn/"},{"name":"Level 3","url":"https://alysly.com/learn/#level-3"},{"name":"Stationarity and Memory","url":"https://alysly.com/learn/stationarity-and-memory/"}],"content":[{"type":"paragraph","text":"Level 3 AI in Trading · Part 4/9"},{"type":"heading","level":1,"text":"Stationarity and Memory"},{"type":"paragraph","text":"By Alexander Bock, founder and data analyst","links":[{"text":"Alexander Bock","href":"https://alysly.com/about/"}]},{"type":"paragraph","text":"Statistical models assume that the properties of the data stay stable over time: the mean, the variance, the relationships. That property is called stationarity. Price series do not have it, because their mean drifts permanently."},{"type":"heading","level":2,"id":"the-usual-compromise-and-what-it-costs","text":"The Usual Compromise and What It Costs"},{"type":"paragraph","text":"The standard fix is to model returns instead of prices. That makes the series stationary, which is why almost all of financial econometrics works with returns."},{"type":"paragraph","text":"The cost is rarely stated. Taking differences erases nearly all of the memory in the series. A return series no longer knows whether the price sits near its two-year high or its two-year low. Yet that information – the price level compared with its own past – is the core of almost every charting feature."},{"type":"paragraph","text":"So you face a dilemma. The raw series has memory but no stationarity. The return series has stationarity but no memory."},{"type":"heading","level":2,"id":"the-middle-path","text":"The Middle Path"},{"type":"paragraph","text":"Differencing does not have to use whole numbers. Instead of “each value minus the previous one”, which is order 1, you can use a fractional order such as 0.4. Technically, the series is combined with a sequence of declining weights that does not set older values to zero but only reduces them."},{"type":"image","src":"https://alysly.com/img/ta/fractional-differentiation.fbd918d2d55ef0396d4e4b488e043e47da69cc3ecdeffa97375cb7ed58eb4d40.webp","alt":"Price series with a trend and the same series after fractional differentiation"},{"type":"paragraph","text":"(1) The price series carries memory — today depends on yesterday — but is not stationary: its mean drifts. Models need stationarity. The usual fix, taking returns, is an integer differentiation and erases almost all memory. (2) Fractional differentiation with an exponent between 0 and 1 removes only as much memory as stationarity requires — the rest survives as signal."},{"type":"paragraph","text":"The method is called fractional differentiation, and the procedure is straightforward:"},{"type":"list","ordered":true,"items":["Raise the order step by step, starting at zero.","After each step, run a stationarity test.","Take the smallest order that passes."]},{"type":"paragraph","text":"An order well below 1 is often enough. Many studies find values between 0.2 and 0.5. The remaining correlation with the original series is then still high: the series is stationary and still knows where it came from."},{"type":"heading","level":2,"id":"why-this-matters-for-this-course","text":"Why This Matters for This Course"},{"type":"paragraph","text":"It gives a formal version of what Levels 1 and 2 claim informally. If the price level compared with its own past meant nothing, then support, resistance and trend would all be superstition. If raw prices could be used directly, no statistics would be needed."},{"type":"paragraph","text":"Fractional differentiation makes the question measurable: how much memory do you have to give up in order to model at all? The answer depends on the data – and as a rule it is considerably less than the automatic switch to returns assumes."},{"type":"heading","level":2,"id":"practical-notes","text":"Practical Notes"},{"type":"list","ordered":false,"items":["The weight sequence has to be cut off somewhere. Common practice is to stop once the weights fall below a small threshold. The first observations are then dropped.","The order is determined on the training period, not on all of the data. Otherwise it becomes the next hidden look into the future.","Every series needs its own order. A value that fits an equity index rarely fits an exchange rate."]},{"type":"heading","level":2,"id":"next","text":"Next"},{"type":"paragraph","text":"The inputs are usable now. The harder question follows: what exactly should the model predict?"},{"type":"list","ordered":false,"items":[{"text":"Previous Data and Bars: The Sample Decides","href":"https://alysly.com/learn/data-and-bars/"},{"text":"Next Labels: Triple Barrier and Meta-Labeling","href":"https://alysly.com/learn/labeling/"}]},{"type":"heading","level":2,"text":"Indices"},{"type":"table","caption":"Indices, performance over the selected period","columns":["Index","1M"],"rows":[[{"text":"Dow Jones","href":"https://alysly.com/markets/indices/dow-jones/"},{"text":"+1.0%","data":{"m1":"+1.0%","m3":"-4.3%","m6":"+4.9%","ytd":"+7.6%","max":"+30.0%"}}],[{"text":"S&P 500","href":"https://alysly.com/markets/indices/sp500/"},{"text":"-2.6%","data":{"m1":"-2.6%","m3":"-2.2%","m6":"+4.2%","ytd":"+10.6%","max":"+35.6%"}}],[{"text":"MSCI World","href":"https://alysly.com/markets/indices/msci-world/"},{"text":"-2.0%","data":{"m1":"-2.0%","m3":"-2.4%","m6":"+3.7%","ytd":"+10.0%","max":"+35.0%"}}],[{"text":"MSCI ACWI","href":"https://alysly.com/markets/indices/msci-acwi/"},{"text":"-3.3%","data":{"m1":"-3.3%","m3":"-3.0%","m6":"+2.3%","ytd":"+10.2%","max":"+41.5%"}}]]},{"type":"heading","level":2,"text":"ETFs"},{"type":"table","caption":"ETFs, performance over the selected period","columns":["Fund","1M"],"rows":[[{"text":"SPDR S&P 500 ETF Trust SPY","links":[{"text":"SPDR S&P 500 ETF Trust","href":"https://alysly.com/markets/etfs/us/spy/"}]},{"text":"-2.6%","data":{"m1":"-2.6%","m3":"-2.2%","m6":"+4.2%","ytd":"+10.6%","max":"+35.6%"}}],[{"text":"Vanguard S&P 500 ETF VOO","links":[{"text":"Vanguard S&P 500 ETF","href":"https://alysly.com/markets/etfs/us/voo/"}]},{"text":"-2.6%","data":{"m1":"-2.6%","m3":"-2.2%","m6":"+4.2%","ytd":"+10.6%","max":"+34.1%"}}],[{"text":"iShares Core MSCI World UCITS ETF USD (Acc) EUNL","links":[{"text":"iShares Core MSCI World UCITS ETF USD (Acc)","href":"https://alysly.com/markets/etfs/ishares-core-msci-world-ucits-etf-usd-acc/"}]},{"text":"-2.0%","data":{"m1":"-2.0%","m3":"-2.3%","m6":"+4.3%","ytd":"+10.6%","max":"+487.5%"}}],[{"text":"Xtrackers MSCI Emerging Markets UCITS ETF 1c XMME","links":[{"text":"Xtrackers MSCI Emerging Markets UCITS ETF 1c","href":"https://alysly.com/markets/etfs/xtrackers-msci-emerging-markets-ucits-etf-1c/"}]},{"text":"-9.4%","data":{"m1":"-9.4%","m3":"-4.1%","m6":"-3.3%","ytd":"+14.0%","max":"+93.0%"}}]]},{"type":"heading","level":2,"text":"Stocks"},{"type":"table","caption":"Stocks, performance over the selected period","columns":["Stock","1M"],"rows":[[{"text":"NVIDIA Corporation NVDA","links":[{"text":"NVIDIA Corporation","href":"https://alysly.com/markets/stocks/nvidia-corporation/"},{"text":"United States","href":"https://alysly.com/markets/stocks/browse/?country=United%20States"}]},{"text":"-13.5%","data":{"m1":"-13.5%","m3":"-2.5%","m6":"+5.2%","ytd":"+10.8%","max":"+62.2%"}}],[{"text":"Apple Inc. AAPL","links":[{"text":"Apple Inc.","href":"https://alysly.com/markets/stocks/apple-inc/"},{"text":"United States","href":"https://alysly.com/markets/stocks/browse/?country=United%20States"}]},{"text":"-0.3%","data":{"m1":"-0.3%","m3":"+7.3%","m6":"+16.8%","ytd":"+22.1%","max":"+45.8%"}}],[{"text":"Tesla Inc. TSLA","links":[{"text":"Tesla Inc.","href":"https://alysly.com/markets/stocks/tesla-inc/"},{"text":"United States","href":"https://alysly.com/markets/stocks/browse/?country=United%20States"}]},{"text":"+5.2%","data":{"m1":"+5.2%","m3":"+21.7%","m6":"+2.3%","ytd":"-11.4%","max":"+65.3%"}}]]},{"type":"paragraph","text":"Data note 1","href":"#dn-1"},{"type":"heading","level":2,"text":"More Studies"},{"type":"list","ordered":false,"items":[{"text":"Jul 21, 2026 Collectibles ×5.4 for the same cards, matched-model index, Mar 2022 – Aug 2026 Is the Pokémon Card Market in a Bubble? 4,159 Sales Analyzed Prices for PSA-graded cards multiplied. Measured on the same cards they rose 438%, while the raw median shows 786% because pricier cards came up for sale later.","href":"https://alysly.com/analysis/pokemon-card-market-bubble/"},{"text":"May 19, 2026 Crypto 45.6% Bitcoin’s annualized volatility, Jul 2024 – Sep 2026 Bitcoin in Investment Portfolios: The Real Numbers Bitcoin is calmer than it was and still about three times as volatile as the stock market. It moves with stocks, not with gold.","href":"https://alysly.com/analysis/bitcoin-investment-portfolios/"},{"text":"Mar 17, 2026 Wealth €1.2m needed for €3,000 a month indefinitely at a 3% real return Living Off Your Returns: What It Really Takes The capital you need is arithmetic. The return you assume is a guess, and it can triple the answer.","href":"https://alysly.com/analysis/living-off-returns/"}]},{"type":"paragraph","text":"View all studies","href":"https://alysly.com/analysis/#studies"},{"type":"heading","level":2,"text":"Learn"},{"type":"list","ordered":false,"items":[{"text":"Labels: Triple Barrier and Meta-Labeling Fixed horizons produce unusable labels – the triple barrier fixes that.","href":"https://alysly.com/learn/labeling/"},{"text":"Overlapping Examples and Sample Weights Overlapping labels fake independence – and how to correct for it.","href":"https://alysly.com/learn/sample-weights/"}]},{"type":"paragraph","text":"Course overview","href":"https://alysly.com/learn/"},{"type":"paragraph","text":"Updated Nov 1, 2026 · Content and figures checked Oct 19, 2026"},{"type":"heading","level":2,"id":"note-legend-h","text":"Data Notes"},{"type":"paragraph","text":"All calculations run automatically and are not investment advice."},{"type":"list","ordered":true,"items":[{"id":"dn-1","text":"Returns computed automatically from the price series, excluding dividends, taxes and costs."}]},{"type":"paragraph","text":"Data and methodology: Alysly · Responsible for content: Alexander Bock","links":[{"text":"Alysly","href":"https://alysly.com/methodology/"},{"text":"Alexander Bock","href":"https://alysly.com/about/"}]},{"type":"paragraph","text":"View all data notes","href":"https://alysly.com/notes/"}],"structured_data":[{"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://alysly.com/#org","name":"Alysly","url":"https://alysly.com/","logo":{"@type":"ImageObject","url":"https://alysly.com/img/icon-512.png","width":512,"height":512},"description":"Data-driven insights, transparent analysis and powerful tools for long-term investors.","email":"mail@alysly.com","contactPoint":{"@type":"ContactPoint","contactType":"customer 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Not investment advice. All data without guarantee. Financial information is analyzed and presented partly with AI assistance."}