{"paper_id":"062929da-a31a-4e0c-928d-1f1dd2b09f5f","body_text":"Improved K–R Reservoir Architecture for Long-Memory Time Series Prediction: Multi-Timescale Dynamics, Adaptive Delay Embedding, and a Principled Operating Regime | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Improved K–R Reservoir Architecture for Long-Memory Time Series Prediction: Multi-Timescale Dynamics, Adaptive Delay Embedding, and a Principled Operating Regime RamaKrishna Pasupuleti This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9448229/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Echo State Networks (ESNs) are powerful for temporal sequence modelling, yet performance degrades substantially on tasks requiring memory beyond 30 time steps. This paper presents the Improved K–R Reservoir Architecture combining: (1) a multi-timescale dual reservoir with fast (α = 1.0) and slow (α = 0.02–0.1) dynamics; (2) adaptive input delay embedding matched to task memory order M; and (3) state normalisation before ridge regression. Evaluated against a properly tuned ESN baseline across eight benchmarks with 10 seeds each, K–R yields statistically significant NRMSE improvements of 24–85% on six long-memory aperiodic tasks (p ≤ 0.005). Ablation confirms delay embedding as the dominant contributor (+ 29%); scaling experiments show the largest gains at small N (+ 36% at N = 50). Three characterisation experiments establish a quantitative operating regime: a hybrid signal crossover shows K–R improvement grows continuously with aperiodic fraction γ (+ 11% at γ = 0.15, + 37% at γ = 1.0); a real-world decomposition confirms that electricity demand residuals retain daily periodicity, so ESN dominates; and a noise robustness test reveals K–R degrades approximately 3.6× faster than ESN below SNR = 10 dB, explained analytically by delay-feature noise amplification. K–R remains competitive on synthetic periodic signals, but real-world periodic datasets favour ESN due to noise and structural correlations in residuals. K–R is most effective when the signal is genuinely aperiodic and input SNR ≥ 15 dB; a well-tuned standard ESN is preferred otherwise. This framework transforms reservoir method selection from heuristic to principled. Computational Neuroscience reservoir computing echo state network long-memory time series multi-timescale reservoir adaptive delay embedding noise robustness operating regime NARMA Mackey-Glass sunspot prediction household power consumption Full Text Additional Declarations The authors declare no competing interests. Supplementary Files KRSupplementaryMaterial1.docx Supplementary Material benchmarks.py Code 1 esnbaseline.py Code2 generateallfigures.py Code3 krreservoir.py code4 runadvancedexperiments.py code5 runelectricity.py code6 runmainexperiments.py code7 Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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interests.\",\"formattedTitle\":\"\\u003cp\\u003e\\u003cstrong\\u003eImproved K–R Reservoir Architecture for Long-Memory Time Series Prediction: Multi-Timescale Dynamics, Adaptive Delay Embedding, and a Principled Operating 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Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true},\"keywords\":\"reservoir computing, echo state network, long-memory time series, multi-timescale reservoir, adaptive delay embedding, noise robustness, operating regime, NARMA, Mackey-Glass, sunspot prediction, household power consumption\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-9448229/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-9448229/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eEcho State Networks (ESNs) are powerful for temporal sequence modelling, yet performance degrades substantially on tasks requiring memory beyond 30 time steps. This paper presents the Improved K\\u0026ndash;R Reservoir Architecture combining: (1) a multi-timescale dual reservoir with fast (α\\u0026thinsp;=\\u0026thinsp;1.0) and slow (α\\u0026thinsp;=\\u0026thinsp;0.02\\u0026ndash;0.1) dynamics; (2) adaptive input delay embedding matched to task memory order M; and (3) state normalisation before ridge regression. Evaluated against a properly tuned ESN baseline across eight benchmarks with 10 seeds each, K\\u0026ndash;R yields statistically significant NRMSE improvements of 24\\u0026ndash;85% on six long-memory aperiodic tasks (p\\u0026thinsp;\\u0026le;\\u0026thinsp;0.005). Ablation confirms delay embedding as the dominant contributor (+\\u0026thinsp;29%); scaling experiments show the largest gains at small N (+\\u0026thinsp;36% at N\\u0026thinsp;=\\u0026thinsp;50). Three characterisation experiments establish a quantitative operating regime: a hybrid signal crossover shows K\\u0026ndash;R improvement grows continuously with aperiodic fraction γ (+\\u0026thinsp;11% at γ\\u0026thinsp;=\\u0026thinsp;0.15, +\\u0026thinsp;37% at γ\\u0026thinsp;=\\u0026thinsp;1.0); a real-world decomposition confirms that electricity demand residuals retain daily periodicity, so ESN dominates; and a noise robustness test reveals K\\u0026ndash;R degrades approximately 3.6\\u0026times; faster than ESN below SNR\\u0026thinsp;=\\u0026thinsp;10 dB, explained analytically by delay-feature noise amplification. K\\u0026ndash;R remains competitive on synthetic periodic signals, but real-world periodic datasets favour ESN due to noise and structural correlations in residuals. K\\u0026ndash;R is most effective when the signal is genuinely aperiodic and input SNR\\u0026thinsp;\\u0026ge;\\u0026thinsp;15 dB; a well-tuned standard ESN is preferred otherwise. This framework transforms reservoir method selection from heuristic to principled.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Improved K–R Reservoir Architecture for Long-Memory Time Series Prediction: Multi-Timescale Dynamics, Adaptive Delay Embedding, and a Principled Operating Regime\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2026-04-20 08:51:31\",\"doi\":\"10.21203/rs.3.rs-9448229/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true}}],\"origin\":\"\",\"ownerIdentity\":\"dafb1023-9678-48dc-92fe-bcfd0983ee2d\",\"owner\":[],\"postedDate\":\"April 20th, 2026\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[{\"id\":66517204,\"name\":\"Computational Neuroscience\"}],\"tags\":[],\"updatedAt\":\"2026-04-20T08:51:33+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2026-04-20 08:51:31\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-9448229\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-9448229\",\"identity\":\"rs-9448229\",\"version\":[\"v1\"]},\"buildId\":\"XKTyCvWXoU3ODBz1xrDgd\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}