--- title: Forking-Sequences — Part I: Statistically and... description: Forking-sequences is a training and inference paradigm for multi-horizon time series forecasting that encodes the entire series across all forecast creation... canonical: https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Forking-Sequences — Part I: Statistically and Computationally Efficient Multi-Horizon Forecasting | daily.dev og:description: Forking-sequences is a training and inference paradigm for multi-horizon time series forecasting that encodes the entire series across all forecast creation... og:url: https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh og:image: https://api.daily.dev/og/posts/0SJ00yDhh.png og:image:alt: Forking-Sequences — Part I: Statistically and Computationally Efficient Multi-Horizon Forecasting og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Forking-Sequences — Part I: Statistically and Computationally Efficient Multi-Horizon Forecasting **[ML CMU](https://daily.dev/sources/mlcmu)** · 11 min read · 3 upvotes · 0 comments ## Summary Forking-sequences is a training and inference paradigm for multi-horizon time series forecasting that encodes the entire series across all forecast creation dates (FCDs) in a single forward pass, rather than encoding one window at a time as in standard window-sampling. The approach is encoder-agnostic, validated across MLP, RNN, LSTM, CNN, Transformer, and State Space models. Theoretically, it reduces gradient variance at O(1/T) and improves SNR linearly with T, directly mitigating vanishing gradients in recurrent architectures. Empirically, LSTM encoders saw up to 49.3% sCRPS improvement, RNNs 46.2%, and CNNs 28.6% over window-sampling on M1/M3/M4/Tourism benchmarks. Computationally, reusing encoder outputs across FCDs drops cross-validation complexity from O(T²) to O(T) for LSTMs, RNNs, and CNNs, and from O(T³) to O(T²) for Transformers. Part II will cover forecast volatility reduction through ensembling. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### What is the difference between window-sampling and forking-sequences in multi-horizon time series forecasting? Window-sampling encodes one fixed-length history window at a time and decodes a single forecast per window, treating forecast creation dates independently. Forking-sequences instead jointly encodes and decodes the entire series across all forecast creation dates in one forward pass, producing a full grid of forecasts, which pools gradient information and reuses encoder computation across dates. _See how architectural training choices affect forecasting accuracy through curated machine learning research on daily.dev._ ### How much does forking-sequences reduce inference computational complexity compared to window-sampling for LSTM and CNN encoders? Forking-sequences drops cross-validation inference complexity from O(T squared) to O(T) for LSTM, RNN, and CNN encoders by encoding the full series once and propagating hidden states forward instead of recomputing the encoder from scratch at every forecast creation date. For Transformers, it reduces unrestricted self-attention cost from O(T cubed) to O(T squared), the same T-fold gain. _Track efficiency gains like this across model architectures by following ML research on daily.dev._ ### Why does forking-sequences improve accuracy more for LSTM and RNN encoders than for Transformer encoders? Forking-sequences pools gradient information across all forecast creation dates, decreasing gradient variance linearly and mitigating vanishing gradients in recurrent architectures like LSTMs and RNNs that suffer from long backpropagation-through-time chains. Transformers use self-attention letting every timestep attend directly to others, so they largely avoid vanishing gradients already, leaving less room for improvement: LSTM sCRPS improved 49.3% versus only 24.7% for Transformers and 6.4% for State Space encoders. _Compare architecture-specific tradeoffs like these when choosing a forecasting encoder on daily.dev._ ## Similar posts on daily.dev - [Forking-Sequences — Part II: Multi-Horizon Forecast Ensembling with Reduced Volatility](https://daily.dev/posts/forking-sequences-part-ii-multi-horizon-forecast-ensembling-with-reduced-volatility-lp5meah1m) · ML CMU · 1 upvotes · 0 comments - [Five Questions About Chronos-2, the Time Series Foundation Model](https://daily.dev/posts/five-questions-about-chronos-2-the-time-series-foundation-model-owv5sq0op) · Towards Data Science · 0 upvotes · 0 comments - [Google solved an Old RNN Problem](https://daily.dev/posts/google-solved-an-old-rnn-problem-xsfjvxaga) · Daily Dose of Data Science \| Avi Chawla \| Substack · 44 upvotes · 3 comments - [The Unreasonable Difficulty of Time Series Forecasting](https://daily.dev/posts/the-unreasonable-difficulty-of-time-series-forecasting-yozpddbsw) · Hacker News · 0 upvotes · 0 comments - [Understanding Seq2Seq Models: From RNNs to LSTMs, GRUs, and Beyond](https://daily.dev/posts/understanding-seq2seq-models-from-rnns-to-lstms-grus-and-beyond-omsfw8ieq) · Medium · 0 upvotes · 0 comments --- Tags: [#deep-learning](https://daily.dev/tags/deep-learning), [#neural-networks](https://daily.dev/tags/neural-networks), [#time-series-forecasting](https://daily.dev/tags/time-series-forecasting) [View this post on daily.dev](https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Forking-Sequences — Part I: Statistically and Computationally Efficient Multi-Horizon Forecasting","url":"https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh"},"datePublished":"2026-08-10T21:51:29.304Z","dateModified":"2026-09-14T08:32:30.115Z","description":"Forking-sequences is a training and inference paradigm for multi-horizon time series forecasting that encodes the entire series across all forecast creation...","image":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/101d9823bf171d51265a1ce47b2a9651?_a=AQAEuop","thumbnailUrl":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/101d9823bf171d51265a1ce47b2a9651?_a=AQAEuop","isAccessibleForFree":true,"articleSection":"ML CMU","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"ML CMU","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/f6a0448c0a90422d9a19c706a926c0ad","url":"https://daily.dev/sources/mlcmu"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":3},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"deep-learning,neural-networks,time-series-forecasting","timeRequired":"PT11M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"ML CMU","item":"https://daily.dev/sources/mlcmu"},{"@type":"ListItem","position":3,"name":"Forking-Sequences — Part I: Statistically and Computationally Efficient Multi-Horizon Forecasting"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/forking-sequences-part-i-statistically-and-computationally-efficient-multi-horizon-forecasting-0sj00ydhh#faq","mainEntity":[{"@type":"Question","name":"What is the difference between window-sampling and forking-sequences in multi-horizon time series forecasting?","acceptedAnswer":{"@type":"Answer","text":"Window-sampling encodes one fixed-length history window at a time and decodes a single forecast per window, treating forecast creation dates independently. 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