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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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filter.",[1793,1834,1836],{"id":1835},"_1-period-and-frequency-are-reciprocals","1. Period and frequency are reciprocals",[1798,1838,1839],{},"For a component",[1841,1842,1845],"span",{"className":1843},[1844],"katex-display",[1841,1846,1849,1914],{"className":1847},[1848],"katex",[1841,1850,1853],{"className":1851},[1852],"katex-mathml",[1854,1855,1858],"math",{"xmlns":1856,"display":1857},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[1859,1860,1861,1909],"semantics",{},[1862,1863,1864,1874,1878,1881,1884,1887,1891,1894,1896,1899,1902,1905],"mrow",{},[1865,1866,1867,1871],"msub",{},[1868,1869,1870],"mi",{},"x",[1868,1872,1873],{},"t",[1875,1876,1877],"mo",{},"=",[1868,1879,1880],{},"A",[1868,1882,1883],{},"cos",[1875,1885,1886],{},"⁡",[1875,1888,1890],{"stretchy":1889},"false","(",[1868,1892,1893],{},"ω",[1868,1895,1873],{},[1875,1897,1898],{},"+",[1868,1900,1901],{},"φ",[1875,1903,1904],{"stretchy":1889},")",[1875,1906,1908],{"separator":1907},"true",",",[1910,1911,1913],"annotation",{"encoding":1912},"application\u002Fx-tex","x_t=A\\cos(\\omega t+\\varphi),",[1841,1915,1918,1997,2037],{"className":1916,"ariaHidden":1907},[1917],"katex-html",[1841,1919,1922,1927,1985,1990,1994],{"className":1920},[1921],"base",[1841,1923],{"className":1924,"style":1926},[1925],"strut","height:0.5806em;vertical-align:-0.15em;",[1841,1928,1931,1935],{"className":1929},[1930],"mord",[1841,1932,1870],{"className":1933},[1930,1934],"mathnormal",[1841,1936,1939],{"className":1937},[1938],"msupsub",[1841,1940,1944,1976],{"className":1941},[1942,1943],"vlist-t","vlist-t2",[1841,1945,1948,1971],{"className":1946},[1947],"vlist-r",[1841,1949,1953],{"className":1950,"style":1952},[1951],"vlist","height:0.2806em;",[1841,1954,1956,1961],{"style":1955},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1841,1957],{"className":1958,"style":1960},[1959],"pstrut","height:2.7em;",[1841,1962,1968],{"className":1963},[1964,1965,1966,1967],"sizing","reset-size6","size3","mtight",[1841,1969,1873],{"className":1970},[1930,1934,1967],[1841,1972,1975],{"className":1973},[1974],"vlist-s","​",[1841,1977,1979],{"className":1978},[1947],[1841,1980,1983],{"className":1981,"style":1982},[1951],"height:0.15em;",[1841,1984],{},[1841,1986],{"className":1987,"style":1989},[1988],"mspace","margin-right:0.2778em;",[1841,1991,1877],{"className":1992},[1993],"mrel",[1841,1995],{"className":1996,"style":1989},[1988],[1841,1998,2000,2004,2007,2011,2015,2019,2023,2026,2030,2034],{"className":1999},[1921],[1841,2001],{"className":2002,"style":2003},[1925],"height:1em;vertical-align:-0.25em;",[1841,2005,1880],{"className":2006},[1930,1934],[1841,2008],{"className":2009,"style":2010},[1988],"margin-right:0.1667em;",[1841,2012,1883],{"className":2013},[2014],"mop",[1841,2016,1890],{"className":2017},[2018],"mopen",[1841,2020,1893],{"className":2021,"style":2022},[1930,1934],"margin-right:0.0359em;",[1841,2024,1873],{"className":2025},[1930,1934],[1841,2027],{"className":2028,"style":2029},[1988],"margin-right:0.2222em;",[1841,2031,1898],{"className":2032},[2033],"mbin",[1841,2035],{"className":2036,"style":2029},[1988],[1841,2038,2040,2043,2046,2050],{"className":2039},[1921],[1841,2041],{"className":2042,"style":2003},[1925],[1841,2044,1901],{"className":2045},[1930,1934],[1841,2047,1904],{"className":2048},[2049],"mclose",[1841,2051,1908],{"className":2052},[2053],"mpunct",[1798,2055,2056,2057,2087,2088,2172,2173,2273],{},"the angular frequency is ",[1841,2058,2060,2074],{"className":2059},[1848],[1841,2061,2063],{"className":2062},[1852],[1854,2064,2065],{"xmlns":1856},[1859,2066,2067,2071],{},[1862,2068,2069],{},[1868,2070,1893],{},[1910,2072,2073],{"encoding":1912},"\\omega",[1841,2075,2077],{"className":2076,"ariaHidden":1907},[1917],[1841,2078,2080,2084],{"className":2079},[1921],[1841,2081],{"className":2082,"style":2083},[1925],"height:0.4306em;",[1841,2085,1893],{"className":2086,"style":2022},[1930,1934]," radians per observation, frequency is ",[1841,2089,2091,2125],{"className":2090},[1848],[1841,2092,2094],{"className":2093},[1852],[1854,2095,2096],{"xmlns":1856},[1859,2097,2098,2122],{},[1862,2099,2100,2103,2105,2107,2111,2113,2117,2120],{},[1868,2101,2102],{},"f",[1875,2104,1877],{},[1868,2106,1893],{},[1868,2108,2110],{"mathvariant":2109},"normal","\u002F",[1875,2112,1890],{"stretchy":1889},[2114,2115,2116],"mn",{},"2",[1868,2118,2119],{},"π",[1875,2121,1904],{"stretchy":1889},[1910,2123,2124],{"encoding":1912},"f=\\omega\u002F(2\\pi)",[1841,2126,2128,2148],{"className":2127,"ariaHidden":1907},[1917],[1841,2129,2131,2135,2139,2142,2145],{"className":2130},[1921],[1841,2132],{"className":2133,"style":2134},[1925],"height:0.8889em;vertical-align:-0.1944em;",[1841,2136,2102],{"className":2137,"style":2138},[1930,1934],"margin-right:0.1076em;",[1841,2140],{"className":2141,"style":1989},[1988],[1841,2143,1877],{"className":2144},[1993],[1841,2146],{"className":2147,"style":1989},[1988],[1841,2149,2151,2154,2157,2160,2163,2166,2169],{"className":2150},[1921],[1841,2152],{"className":2153,"style":2003},[1925],[1841,2155,1893],{"className":2156,"style":2022},[1930,1934],[1841,2158,2110],{"className":2159},[1930],[1841,2161,1890],{"className":2162},[2018],[1841,2164,2116],{"className":2165},[1930],[1841,2167,2119],{"className":2168,"style":2022},[1930,1934],[1841,2170,1904],{"className":2171},[2049]," cycles per observation, and period is ",[1841,2174,2176,2210],{"className":2175},[1848],[1841,2177,2179],{"className":2178},[1852],[1854,2180,2181],{"xmlns":1856},[1859,2182,2183,2207],{},[1862,2184,2185,2188,2190,2193,2195,2197,2199,2201,2203,2205],{},[1868,2186,2187],{},"P",[1875,2189,1877],{},[2114,2191,2192],{},"1",[1868,2194,2110],{"mathvariant":2109},[1868,2196,2102],{},[1875,2198,1877],{},[2114,2200,2116],{},[1868,2202,2119],{},[1868,2204,2110],{"mathvariant":2109},[1868,2206,1893],{},[1910,2208,2209],{"encoding":1912},"P=1\u002Ff=2\\pi\u002F\\omega",[1841,2211,2213,2233,2255],{"className":2212,"ariaHidden":1907},[1917],[1841,2214,2216,2220,2224,2227,2230],{"className":2215},[1921],[1841,2217],{"className":2218,"style":2219},[1925],"height:0.6833em;",[1841,2221,2187],{"className":2222,"style":2223},[1930,1934],"margin-right:0.1389em;",[1841,2225],{"className":2226,"style":1989},[1988],[1841,2228,1877],{"className":2229},[1993],[1841,2231],{"className":2232,"style":1989},[1988],[1841,2234,2236,2239,2243,2246,2249,2252],{"className":2235},[1921],[1841,2237],{"className":2238,"style":2003},[1925],[1841,2240,2242],{"className":2241},[1930],"1\u002F",[1841,2244,2102],{"className":2245,"style":2138},[1930,1934],[1841,2247],{"className":2248,"style":1989},[1988],[1841,2250,1877],{"className":2251},[1993],[1841,2253],{"className":2254,"style":1989},[1988],[1841,2256,2258,2261,2264,2267,2270],{"className":2257},[1921],[1841,2259],{"className":2260,"style":2003},[1925],[1841,2262,2116],{"className":2263},[1930],[1841,2265,2119],{"className":2266,"style":2022},[1930,1934],[1841,2268,2110],{"className":2269},[1930],[1841,2271,1893],{"className":2272,"style":2022},[1930,1934]," observations.",[2275,2276,2277,2353],"table",{},[2278,2279,2280],"thead",{},[2281,2282,2283,2287,2319,2350],"tr",{},[2284,2285,2286],"th",{},"Pattern",[2284,2288,2290,2291],{"align":2289},"right","Frequency ",[1841,2292,2294,2307],{"className":2293},[1848],[1841,2295,2297],{"className":2296},[1852],[1854,2298,2299],{"xmlns":1856},[1859,2300,2301,2305],{},[1862,2302,2303],{},[1868,2304,2102],{},[1910,2306,2102],{"encoding":1912},[1841,2308,2310],{"className":2309,"ariaHidden":1907},[1917],[1841,2311,2313,2316],{"className":2312},[1921],[1841,2314],{"className":2315,"style":2134},[1925],[1841,2317,2102],{"className":2318,"style":2138},[1930,1934],[2284,2320,2321,2322],{"align":2289},"Angular frequency ",[1841,2323,2325,2338],{"className":2324},[1848],[1841,2326,2328],{"className":2327},[1852],[1854,2329,2330],{"xmlns":1856},[1859,2331,2332,2336],{},[1862,2333,2334],{},[1868,2335,1893],{},[1910,2337,2073],{"encoding":1912},[1841,2339,2341],{"className":2340,"ariaHidden":1907},[1917],[1841,2342,2344,2347],{"className":2343},[1921],[1841,2345],{"className":2346,"style":2083},[1925],[1841,2348,1893],{"className":2349,"style":2022},[1930,1934],[2284,2351,2352],{"align":2289},"Period",[2354,2355,2356,2441,2529],"tbody",{},[2281,2357,2358,2362,2398,2438],{},[2359,2360,2361],"td",{},"annual cycle in monthly data",[2359,2363,2364],{"align":2289},[1841,2365,2367,2386],{"className":2366},[1848],[1841,2368,2370],{"className":2369},[1852],[1854,2371,2372],{"xmlns":1856},[1859,2373,2374,2383],{},[1862,2375,2376,2378,2380],{},[2114,2377,2192],{},[1868,2379,2110],{"mathvariant":2109},[2114,2381,2382],{},"12",[1910,2384,2385],{"encoding":1912},"1\u002F12",[1841,2387,2389],{"className":2388,"ariaHidden":1907},[1917],[1841,2390,2392,2395],{"className":2391},[1921],[1841,2393],{"className":2394,"style":2003},[1925],[1841,2396,2385],{"className":2397},[1930],[2359,2399,2400],{"align":2289},[1841,2401,2403,2422],{"className":2402},[1848],[1841,2404,2406],{"className":2405},[1852],[1854,2407,2408],{"xmlns":1856},[1859,2409,2410,2419],{},[1862,2411,2412,2414,2416],{},[1868,2413,2119],{},[1868,2415,2110],{"mathvariant":2109},[2114,2417,2418],{},"6",[1910,2420,2421],{"encoding":1912},"\\pi\u002F6",[1841,2423,2425],{"className":2424,"ariaHidden":1907},[1917],[1841,2426,2428,2431,2434],{"className":2427},[1921],[1841,2429],{"className":2430,"style":2003},[1925],[1841,2432,2119],{"className":2433,"style":2022},[1930,1934],[1841,2435,2437],{"className":2436},[1930],"\u002F6",[2359,2439,2440],{"align":2289},"12 months",[2281,2442,2443,2446,2482,2526],{},[2359,2444,2445],{},"weekly cycle in daily data",[2359,2447,2448],{"align":2289},[1841,2449,2451,2470],{"className":2450},[1848],[1841,2452,2454],{"className":2453},[1852],[1854,2455,2456],{"xmlns":1856},[1859,2457,2458,2467],{},[1862,2459,2460,2462,2464],{},[2114,2461,2192],{},[1868,2463,2110],{"mathvariant":2109},[2114,2465,2466],{},"7",[1910,2468,2469],{"encoding":1912},"1\u002F7",[1841,2471,2473],{"className":2472,"ariaHidden":1907},[1917],[1841,2474,2476,2479],{"className":2475},[1921],[1841,2477],{"className":2478,"style":2003},[1925],[1841,2480,2469],{"className":2481},[1930],[2359,2483,2484],{"align":2289},[1841,2485,2487,2507],{"className":2486},[1848],[1841,2488,2490],{"className":2489},[1852],[1854,2491,2492],{"xmlns":1856},[1859,2493,2494,2504],{},[1862,2495,2496,2498,2500,2502],{},[2114,2497,2116],{},[1868,2499,2119],{},[1868,2501,2110],{"mathvariant":2109},[2114,2503,2466],{},[1910,2505,2506],{"encoding":1912},"2\\pi\u002F7",[1841,2508,2510],{"className":2509,"ariaHidden":1907},[1917],[1841,2511,2513,2516,2519,2522],{"className":2512},[1921],[1841,2514],{"className":2515,"style":2003},[1925],[1841,2517,2116],{"className":2518},[1930],[1841,2520,2119],{"className":2521,"style":2022},[1930,1934],[1841,2523,2525],{"className":2524},[1930],"\u002F7",[2359,2527,2528],{"align":2289},"7 days",[2281,2530,2531,2534,2569,2600],{},[2359,2532,2533],{},"alternating observations",[2359,2535,2536],{"align":2289},[1841,2537,2539,2557],{"className":2538},[1848],[1841,2540,2542],{"className":2541},[1852],[1854,2543,2544],{"xmlns":1856},[1859,2545,2546,2554],{},[1862,2547,2548,2550,2552],{},[2114,2549,2192],{},[1868,2551,2110],{"mathvariant":2109},[2114,2553,2116],{},[1910,2555,2556],{"encoding":1912},"1\u002F2",[1841,2558,2560],{"className":2559,"ariaHidden":1907},[1917],[1841,2561,2563,2566],{"className":2562},[1921],[1841,2564],{"className":2565,"style":2003},[1925],[1841,2567,2556],{"className":2568},[1930],[2359,2570,2571],{"align":2289},[1841,2572,2574,2588],{"className":2573},[1848],[1841,2575,2577],{"className":2576},[1852],[1854,2578,2579],{"xmlns":1856},[1859,2580,2581,2585],{},[1862,2582,2583],{},[1868,2584,2119],{},[1910,2586,2587],{"encoding":1912},"\\pi",[1841,2589,2591],{"className":2590,"ariaHidden":1907},[1917],[1841,2592,2594,2597],{"className":2593},[1921],[1841,2595],{"className":2596,"style":2083},[1925],[1841,2598,2119],{"className":2599,"style":2022},[1930,1934],[2359,2601,2602],{"align":2289},"2 observations",[1798,2604,2605],{},"Always state the sampling interval. “Frequency 0.1” means ten observations per cycle, not automatically ten days or years.",[1793,2607,2609],{"id":2608},"_2-spectrum-and-autocovariance-contain-the-same-second-order-information","2. Spectrum and autocovariance contain the same second-order information",[1798,2611,2612],{},"For a weakly stationary process with suitable regularity,",[1841,2614,2616],{"className":2615},[1844],[1841,2617,2619,2701],{"className":2618},[1848],[1841,2620,2622],{"className":2621},[1852],[1854,2623,2624],{"xmlns":1856,"display":1857},[1859,2625,2626,2698],{},[1862,2627,2628,2630,2632,2634,2636,2638,2649,2670,2673,2675,2677,2679,2696],{},[1868,2629,2102],{},[1875,2631,1890],{"stretchy":1889},[1868,2633,1893],{},[1875,2635,1904],{"stretchy":1889},[1875,2637,1877],{},[2639,2640,2641,2643],"mfrac",{},[2114,2642,2192],{},[1862,2644,2645,2647],{},[2114,2646,2116],{},[1868,2648,2119],{},[2650,2651,2652,2655,2668],"munderover",{},[1875,2653,2654],{},"∑",[1862,2656,2657,2660,2662,2665],{},[1868,2658,2659],{},"h",[1875,2661,1877],{},[1875,2663,2664],{},"−",[1868,2666,2667],{"mathvariant":2109},"∞",[1868,2669,2667],{"mathvariant":2109},[1868,2671,2672],{},"γ",[1875,2674,1890],{"stretchy":1889},[1868,2676,2659],{},[1875,2678,1904],{"stretchy":1889},[2680,2681,2682,2685],"msup",{},[1868,2683,2684],{},"e",[1862,2686,2687,2689,2692,2694],{},[1875,2688,2664],{},[1868,2690,2691],{},"i",[1868,2693,1893],{},[1868,2695,2659],{},[1875,2697,1908],{"separator":1907},[1910,2699,2700],{"encoding":1912},"f(\\omega)=\\frac{1}{2\\pi}\\sum_{h=-\\infty}^{\\infty}\n\\gamma(h)e^{-i\\omega h},",[1841,2702,2704,2731],{"className":2703,"ariaHidden":1907},[1917],[1841,2705,2707,2710,2713,2716,2719,2722,2725,2728],{"className":2706},[1921],[1841,2708],{"className":2709,"style":2003},[1925],[1841,2711,2102],{"className":2712,"style":2138},[1930,1934],[1841,2714,1890],{"className":2715},[2018],[1841,2717,1893],{"className":2718,"style":2022},[1930,1934],[1841,2720,1904],{"className":2721},[2049],[1841,2723],{"className":2724,"style":1989},[1988],[1841,2726,1877],{"className":2727},[1993],[1841,2729],{"className":2730,"style":1989},[1988],[1841,2732,2734,2738,2812,2815,2894,2897,2901,2904,2907,2910,2951],{"className":2733},[1921],[1841,2735],{"className":2736,"style":2737},[1925],"height:3.0118em;vertical-align:-1.3604em;",[1841,2739,2741,2745,2809],{"className":2740},[1930],[1841,2742],{"className":2743},[2018,2744],"nulldelimiter",[1841,2746,2748],{"className":2747},[2639],[1841,2749,2751,2800],{"className":2750},[1942,1943],[1841,2752,2754,2797],{"className":2753},[1947],[1841,2755,2758,2774,2785],{"className":2756,"style":2757},[1951],"height:1.3214em;",[1841,2759,2761,2765],{"style":2760},"top:-2.314em;",[1841,2762],{"className":2763,"style":2764},[1959],"height:3em;",[1841,2766,2768,2771],{"className":2767},[1930],[1841,2769,2116],{"className":2770},[1930],[1841,2772,2119],{"className":2773,"style":2022},[1930,1934],[1841,2775,2777,2780],{"style":2776},"top:-3.23em;",[1841,2778],{"className":2779,"style":2764},[1959],[1841,2781],{"className":2782,"style":2784},[2783],"frac-line","border-bottom-width:0.04em;",[1841,2786,2788,2791],{"style":2787},"top:-3.677em;",[1841,2789],{"className":2790,"style":2764},[1959],[1841,2792,2794],{"className":2793},[1930],[1841,2795,2192],{"className":2796},[1930],[1841,2798,1975],{"className":2799},[1974],[1841,2801,2803],{"className":2802},[1947],[1841,2804,2807],{"className":2805,"style":2806},[1951],"height:0.686em;",[1841,2808],{},[1841,2810],{"className":2811},[2049,2744],[1841,2813],{"className":2814,"style":2010},[1988],[1841,2816,2819],{"className":2817},[2014,2818],"op-limits",[1841,2820,2822,2885],{"className":2821},[1942,1943],[1841,2823,2825,2882],{"className":2824},[1947],[1841,2826,2829,2854,2867],{"className":2827,"style":2828},[1951],"height:1.6514em;",[1841,2830,2832,2836],{"style":2831},"top:-1.8479em;margin-left:0em;",[1841,2833],{"className":2834,"style":2835},[1959],"height:3.05em;",[1841,2837,2839],{"className":2838},[1964,1965,1966,1967],[1841,2840,2842,2845,2848,2851],{"className":2841},[1930,1967],[1841,2843,2659],{"className":2844},[1930,1934,1967],[1841,2846,1877],{"className":2847},[1993,1967],[1841,2849,2664],{"className":2850},[1930,1967],[1841,2852,2667],{"className":2853},[1930,1967],[1841,2855,2857,2860],{"style":2856},"top:-3.05em;",[1841,2858],{"className":2859,"style":2835},[1959],[1841,2861,2862],{},[1841,2863,2654],{"className":2864},[2014,2865,2866],"op-symbol","large-op",[1841,2868,2870,2873],{"style":2869},"top:-4.3em;margin-left:0em;",[1841,2871],{"className":2872,"style":2835},[1959],[1841,2874,2876],{"className":2875},[1964,1965,1966,1967],[1841,2877,2879],{"className":2878},[1930,1967],[1841,2880,2667],{"className":2881},[1930,1967],[1841,2883,1975],{"className":2884},[1974],[1841,2886,2888],{"className":2887},[1947],[1841,2889,2892],{"className":2890,"style":2891},[1951],"height:1.3604em;",[1841,2893],{},[1841,2895],{"className":2896,"style":2010},[1988],[1841,2898,2672],{"className":2899,"style":2900},[1930,1934],"margin-right:0.0556em;",[1841,2902,1890],{"className":2903},[2018],[1841,2905,2659],{"className":2906},[1930,1934],[1841,2908,1904],{"className":2909},[2049],[1841,2911,2913,2916],{"className":2912},[1930],[1841,2914,2684],{"className":2915},[1930,1934],[1841,2917,2919],{"className":2918},[1938],[1841,2920,2922],{"className":2921},[1942],[1841,2923,2925],{"className":2924},[1947],[1841,2926,2929],{"className":2927,"style":2928},[1951],"height:0.8991em;",[1841,2930,2932,2935],{"style":2931},"top:-3.113em;margin-right:0.05em;",[1841,2933],{"className":2934,"style":1960},[1959],[1841,2936,2938],{"className":2937},[1964,1965,1966,1967],[1841,2939,2941,2944,2948],{"className":2940},[1930,1967],[1841,2942,2664],{"className":2943},[1930,1967],[1841,2945,2947],{"className":2946,"style":2022},[1930,1934,1967],"iω",[1841,2949,2659],{"className":2950},[1930,1934,1967],[1841,2952,1908],{"className":2953},[2053],[1798,2955,2956],{},"and",[1841,2958,2960],{"className":2959},[1844],[1841,2961,2963,3031],{"className":2962},[1848],[1841,2964,2966],{"className":2965},[1852],[1854,2967,2968],{"xmlns":1856,"display":1857},[1859,2969,2970,3028],{},[1862,2971,2972,2974,2976,2978,2980,2982,2996,3008,3010,3012,3014,3016,3020,3023,3025],{},[1868,2973,2672],{},[1875,2975,1890],{"stretchy":1889},[1868,2977,2659],{},[1875,2979,1904],{"stretchy":1889},[1875,2981,1877],{},[2983,2984,2985,2988,2994],"msubsup",{},[1875,2986,2987],{},"∫",[1862,2989,2990,2992],{},[1875,2991,2664],{},[1868,2993,2119],{},[1868,2995,2119],{},[2680,2997,2998,3000],{},[1868,2999,2684],{},[1862,3001,3002,3004,3006],{},[1868,3003,2691],{},[1868,3005,1893],{},[1868,3007,2659],{},[1868,3009,2102],{},[1875,3011,1890],{"stretchy":1889},[1868,3013,1893],{},[1875,3015,1904],{"stretchy":1889},[3017,3018,3019],"mtext",{}," ",[1868,3021,3022],{},"d",[1868,3024,1893],{},[1868,3026,3027],{"mathvariant":2109},".",[1910,3029,3030],{"encoding":1912},"\\gamma(h)=\\int_{-\\pi}^{\\pi}e^{i\\omega h}f(\\omega)\\,d\\omega.",[1841,3032,3034,3061],{"className":3033,"ariaHidden":1907},[1917],[1841,3035,3037,3040,3043,3046,3049,3052,3055,3058],{"className":3036},[1921],[1841,3038],{"className":3039,"style":2003},[1925],[1841,3041,2672],{"className":3042,"style":2900},[1930,1934],[1841,3044,1890],{"className":3045},[2018],[1841,3047,2659],{"className":3048},[1930,1934],[1841,3050,1904],{"className":3051},[2049],[1841,3053],{"className":3054,"style":1989},[1988],[1841,3056,1877],{"className":3057},[1993],[1841,3059],{"className":3060,"style":1989},[1988],[1841,3062,3064,3068,3133,3136,3171,3174,3177,3180,3183,3186,3189,3192],{"className":3063},[1921],[1841,3065],{"className":3066,"style":3067},[1925],"height:2.3846em;vertical-align:-0.9703em;",[1841,3069,3071,3075],{"className":3070},[2014],[1841,3072,2987],{"className":3073,"style":3074},[2014,2865,2866],"margin-right:0.4445em;position:relative;top:-0.0011em;",[1841,3076,3078],{"className":3077},[1938],[1841,3079,3081,3124],{"className":3080},[1942,1943],[1841,3082,3084,3121],{"className":3083},[1947],[1841,3085,3088,3106],{"className":3086,"style":3087},[1951],"height:1.4143em;",[1841,3089,3091,3094],{"style":3090},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[1841,3092],{"className":3093,"style":1960},[1959],[1841,3095,3097],{"className":3096},[1964,1965,1966,1967],[1841,3098,3100,3103],{"className":3099},[1930,1967],[1841,3101,2664],{"className":3102},[1930,1967],[1841,3104,2119],{"className":3105,"style":2022},[1930,1934,1967],[1841,3107,3109,3112],{"style":3108},"top:-3.8129em;margin-right:0.05em;",[1841,3110],{"className":3111,"style":1960},[1959],[1841,3113,3115],{"className":3114},[1964,1965,1966,1967],[1841,3116,3118],{"className":3117},[1930,1967],[1841,3119,2119],{"className":3120,"style":2022},[1930,1934,1967],[1841,3122,1975],{"className":3123},[1974],[1841,3125,3127],{"className":3126},[1947],[1841,3128,3131],{"className":3129,"style":3130},[1951],"height:0.9703em;",[1841,3132],{},[1841,3134],{"className":3135,"style":2010},[1988],[1841,3137,3139,3142],{"className":3138},[1930],[1841,3140,2684],{"className":3141},[1930,1934],[1841,3143,3145],{"className":3144},[1938],[1841,3146,3148],{"className":3147},[1942],[1841,3149,3151],{"className":3150},[1947],[1841,3152,3154],{"className":3153,"style":2928},[1951],[1841,3155,3156,3159],{"style":2931},[1841,3157],{"className":3158,"style":1960},[1959],[1841,3160,3162],{"className":3161},[1964,1965,1966,1967],[1841,3163,3165,3168],{"className":3164},[1930,1967],[1841,3166,2947],{"className":3167,"style":2022},[1930,1934,1967],[1841,3169,2659],{"className":3170},[1930,1934,1967],[1841,3172,2102],{"className":3173,"style":2138},[1930,1934],[1841,3175,1890],{"className":3176},[2018],[1841,3178,1893],{"className":3179,"style":2022},[1930,1934],[1841,3181,1904],{"className":3182},[2049],[1841,3184],{"className":3185,"style":2010},[1988],[1841,3187,3022],{"className":3188},[1930,1934],[1841,3190,1893],{"className":3191,"style":2022},[1930,1934],[1841,3193,3027],{"className":3194},[1930],[1798,3196,3197,3198,3227],{},"The time domain asks how observations ",[1841,3199,3201,3214],{"className":3200},[1848],[1841,3202,3204],{"className":3203},[1852],[1854,3205,3206],{"xmlns":1856},[1859,3207,3208,3212],{},[1862,3209,3210],{},[1868,3211,2659],{},[1910,3213,2659],{"encoding":1912},[1841,3215,3217],{"className":3216,"ariaHidden":1907},[1917],[1841,3218,3220,3224],{"className":3219},[1921],[1841,3221],{"className":3222,"style":3223},[1925],"height:0.6944em;",[1841,3225,2659],{"className":3226},[1930,1934]," lags apart co-move. The frequency domain asks how variance is distributed across slow and fast oscillations. The total area is variance:",[1841,3229,3231],{"className":3230},[1844],[1841,3232,3234,3285],{"className":3233},[1848],[1841,3235,3237],{"className":3236},[1852],[1854,3238,3239],{"xmlns":1856,"display":1857},[1859,3240,3241,3282],{},[1862,3242,3243,3255,3257,3259,3261,3263,3265,3267,3269,3271,3273,3275,3278,3280],{},[2983,3244,3245,3247,3253],{},[1875,3246,2987],{},[1862,3248,3249,3251],{},[1875,3250,2664],{},[1868,3252,2119],{},[1868,3254,2119],{},[1868,3256,2102],{},[1875,3258,1890],{"stretchy":1889},[1868,3260,1893],{},[1875,3262,1904],{"stretchy":1889},[3017,3264,3019],{},[1868,3266,3022],{},[1868,3268,1893],{},[1875,3270,1877],{},[1868,3272,2672],{},[1875,3274,1890],{"stretchy":1889},[2114,3276,3277],{},"0",[1875,3279,1904],{"stretchy":1889},[1868,3281,3027],{"mathvariant":2109},[1910,3283,3284],{"encoding":1912},"\\int_{-\\pi}^{\\pi}f(\\omega)\\,d\\omega=\\gamma(0).",[1841,3286,3288,3387],{"className":3287,"ariaHidden":1907},[1917],[1841,3289,3291,3294,3354,3357,3360,3363,3366,3369,3372,3375,3378,3381,3384],{"className":3290},[1921],[1841,3292],{"className":3293,"style":3067},[1925],[1841,3295,3297,3300],{"className":3296},[2014],[1841,3298,2987],{"className":3299,"style":3074},[2014,2865,2866],[1841,3301,3303],{"className":3302},[1938],[1841,3304,3306,3346],{"className":3305},[1942,1943],[1841,3307,3309,3343],{"className":3308},[1947],[1841,3310,3312,3329],{"className":3311,"style":3087},[1951],[1841,3313,3314,3317],{"style":3090},[1841,3315],{"className":3316,"style":1960},[1959],[1841,3318,3320],{"className":3319},[1964,1965,1966,1967],[1841,3321,3323,3326],{"className":3322},[1930,1967],[1841,3324,2664],{"className":3325},[1930,1967],[1841,3327,2119],{"className":3328,"style":2022},[1930,1934,1967],[1841,3330,3331,3334],{"style":3108},[1841,3332],{"className":3333,"style":1960},[1959],[1841,3335,3337],{"className":3336},[1964,1965,1966,1967],[1841,3338,3340],{"className":3339},[1930,1967],[1841,3341,2119],{"className":3342,"style":2022},[1930,1934,1967],[1841,3344,1975],{"className":3345},[1974],[1841,3347,3349],{"className":3348},[1947],[1841,3350,3352],{"className":3351,"style":3130},[1951],[1841,3353],{},[1841,3355],{"className":3356,"style":2010},[1988],[1841,3358,2102],{"className":3359,"style":2138},[1930,1934],[1841,3361,1890],{"className":3362},[2018],[1841,3364,1893],{"className":3365,"style":2022},[1930,1934],[1841,3367,1904],{"className":3368},[2049],[1841,3370],{"className":3371,"style":2010},[1988],[1841,3373,3022],{"className":3374},[1930,1934],[1841,3376,1893],{"className":3377,"style":2022},[1930,1934],[1841,3379],{"className":3380,"style":1989},[1988],[1841,3382,1877],{"className":3383},[1993],[1841,3385],{"className":3386,"style":1989},[1988],[1841,3388,3390,3393,3396,3399,3402,3405],{"className":3389},[1921],[1841,3391],{"className":3392,"style":2003},[1925],[1841,3394,2672],{"className":3395,"style":2900},[1930,1934],[1841,3397,1890],{"className":3398},[2018],[1841,3400,3277],{"className":3401},[1930],[1841,3403,1904],{"className":3404},[2049],[1841,3406,3027],{"className":3407},[1930],[3409,3410,3412],"h3",{"id":3411},"finite-fourier-matrix","Finite Fourier matrix",[1798,3414,3415,3416,3645],{},"For ",[1841,3417,3419,3471],{"className":3418},[1848],[1841,3420,3422],{"className":3421},[1852],[1854,3423,3424],{"xmlns":1856},[1859,3425,3426,3468],{},[1862,3427,3428,3431,3433,3435,3441,3443,3446,3448,3461],{},[1868,3429,1870],{"mathvariant":3430},"bold",[1875,3432,1877],{},[1875,3434,1890],{"stretchy":1889},[1865,3436,3437,3439],{},[1868,3438,1870],{},[2114,3440,3277],{},[1875,3442,1908],{"separator":1907},[1875,3444,3445],{},"…",[1875,3447,1908],{"separator":1907},[1865,3449,3450,3452],{},[1868,3451,1870],{},[1862,3453,3454,3457,3459],{},[1868,3455,3456],{},"n",[1875,3458,2664],{},[2114,3460,2192],{},[2680,3462,3463,3465],{},[1875,3464,1904],{"stretchy":1889},[1868,3466,3467],{"mathvariant":2109},"⊤",[1910,3469,3470],{"encoding":1912},"\\mathbf x=(x_0,\\ldots,x_{n-1})^\\top",[1841,3472,3474,3494],{"className":3473,"ariaHidden":1907},[1917],[1841,3475,3477,3481,3485,3488,3491],{"className":3476},[1921],[1841,3478],{"className":3479,"style":3480},[1925],"height:0.4444em;",[1841,3482,1870],{"className":3483},[1930,3484],"mathbf",[1841,3486],{"className":3487,"style":1989},[1988],[1841,3489,1877],{"className":3490},[1993],[1841,3492],{"className":3493,"style":1989},[1988],[1841,3495,3497,3501,3504,3545,3548,3551,3555,3558,3561,3564,3614],{"className":3496},[1921],[1841,3498],{"className":3499,"style":3500},[1925],"height:1.0991em;vertical-align:-0.25em;",[1841,3502,1890],{"className":3503},[2018],[1841,3505,3507,3510],{"className":3506},[1930],[1841,3508,1870],{"className":3509},[1930,1934],[1841,3511,3513],{"className":3512},[1938],[1841,3514,3516,3537],{"className":3515},[1942,1943],[1841,3517,3519,3534],{"className":3518},[1947],[1841,3520,3523],{"className":3521,"style":3522},[1951],"height:0.3011em;",[1841,3524,3525,3528],{"style":1955},[1841,3526],{"className":3527,"style":1960},[1959],[1841,3529,3531],{"className":3530},[1964,1965,1966,1967],[1841,3532,3277],{"className":3533},[1930,1967],[1841,3535,1975],{"className":3536},[1974],[1841,3538,3540],{"className":3539},[1947],[1841,3541,3543],{"className":3542,"style":1982},[1951],[1841,3544],{},[1841,3546,1908],{"className":3547},[2053],[1841,3549],{"className":3550,"style":2010},[1988],[1841,3552,3445],{"className":3553},[3554],"minner",[1841,3556],{"className":3557,"style":2010},[1988],[1841,3559,1908],{"className":3560},[2053],[1841,3562],{"className":3563,"style":2010},[1988],[1841,3565,3567,3570],{"className":3566},[1930],[1841,3568,1870],{"className":3569},[1930,1934],[1841,3571,3573],{"className":3572},[1938],[1841,3574,3576,3605],{"className":3575},[1942,1943],[1841,3577,3579,3602],{"className":3578},[1947],[1841,3580,3582],{"className":3581,"style":3522},[1951],[1841,3583,3584,3587],{"style":1955},[1841,3585],{"className":3586,"style":1960},[1959],[1841,3588,3590],{"className":3589},[1964,1965,1966,1967],[1841,3591,3593,3596,3599],{"className":3592},[1930,1967],[1841,3594,3456],{"className":3595},[1930,1934,1967],[1841,3597,2664],{"className":3598},[2033,1967],[1841,3600,2192],{"className":3601},[1930,1967],[1841,3603,1975],{"className":3604},[1974],[1841,3606,3608],{"className":3607},[1947],[1841,3609,3612],{"className":3610,"style":3611},[1951],"height:0.2083em;",[1841,3613],{},[1841,3615,3617,3620],{"className":3616},[2049],[1841,3618,1904],{"className":3619},[2049],[1841,3621,3623],{"className":3622},[1938],[1841,3624,3626],{"className":3625},[1942],[1841,3627,3629],{"className":3628},[1947],[1841,3630,3633],{"className":3631,"style":3632},[1951],"height:0.8491em;",[1841,3634,3636,3639],{"style":3635},"top:-3.063em;margin-right:0.05em;",[1841,3637],{"className":3638,"style":1960},[1959],[1841,3640,3642],{"className":3641},[1964,1965,1966,1967],[1841,3643,3467],{"className":3644},[1930,1967],", define",[1841,3647,3649],{"className":3648},[1844],[1841,3650,3652,3755],{"className":3651},[1848],[1841,3653,3655],{"className":3654},[1852],[1854,3656,3657],{"xmlns":1856,"display":1857},[1859,3658,3659,3752],{},[1862,3660,3661,3663,3670,3681,3683,3697,3700,3702,3724,3726,3729,3731,3733,3735,3737,3739,3741,3743,3745,3747,3749],{},[1875,3662,1890],{"stretchy":1889},[1865,3664,3665,3668],{},[1868,3666,3667],{},"F",[1868,3669,3456],{},[1865,3671,3672,3674],{},[1875,3673,1904],{"stretchy":1889},[1862,3675,3676,3679],{},[1868,3677,3678],{},"j",[1868,3680,1873],{},[1875,3682,1877],{},[2680,3684,3685,3687],{},[1868,3686,3456],{},[1862,3688,3689,3691,3693,3695],{},[1875,3690,2664],{},[2114,3692,2192],{},[1868,3694,2110],{"mathvariant":2109},[2114,3696,2116],{},[1868,3698,3699],{},"exp",[1875,3701,1886],{},[1862,3703,3704,3706,3708,3710,3712,3714,3716,3718,3720,3722],{},[1875,3705,1890],{"fence":1907},[1875,3707,2664],{},[2114,3709,2116],{},[1868,3711,2119],{},[1868,3713,2691],{},[1868,3715,3678],{},[1868,3717,1873],{},[1868,3719,2110],{"mathvariant":2109},[1868,3721,3456],{},[1875,3723,1904],{"fence":1907},[1875,3725,1908],{"separator":1907},[1988,3727],{"width":3728},"2em",[1868,3730,3678],{},[1875,3732,1908],{"separator":1907},[1868,3734,1873],{},[1875,3736,1877],{},[2114,3738,3277],{},[1875,3740,1908],{"separator":1907},[1875,3742,3445],{},[1875,3744,1908],{"separator":1907},[1868,3746,3456],{},[1875,3748,2664],{},[2114,3750,3751],{},"1.",[1910,3753,3754],{"encoding":1912},"(F_n)_{jt}\n=n^{-1\u002F2}\\exp\\left(-2\\pi ijt\u002Fn\\right),\n\\qquad j,t=0,\\ldots,n-1.",[1841,3756,3758,3868,3987,4027],{"className":3757,"ariaHidden":1907},[1917],[1841,3759,3761,3765,3768,3810,3859,3862,3865],{"className":3760},[1921],[1841,3762],{"className":3763,"style":3764},[1925],"height:1.0361em;vertical-align:-0.2861em;",[1841,3766,1890],{"className":3767},[2018],[1841,3769,3771,3774],{"className":3770},[1930],[1841,3772,3667],{"className":3773,"style":2223},[1930,1934],[1841,3775,3777],{"className":3776},[1938],[1841,3778,3780,3802],{"className":3779},[1942,1943],[1841,3781,3783,3799],{"className":3782},[1947],[1841,3784,3787],{"className":3785,"style":3786},[1951],"height:0.1514em;",[1841,3788,3790,3793],{"style":3789},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1841,3791],{"className":3792,"style":1960},[1959],[1841,3794,3796],{"className":3795},[1964,1965,1966,1967],[1841,3797,3456],{"className":3798},[1930,1934,1967],[1841,3800,1975],{"className":3801},[1974],[1841,3803,3805],{"className":3804},[1947],[1841,3806,3808],{"className":3807,"style":1982},[1951],[1841,3809],{},[1841,3811,3813,3816],{"className":3812},[2049],[1841,3814,1904],{"className":3815},[2049],[1841,3817,3819],{"className":3818},[1938],[1841,3820,3822,3850],{"className":3821},[1942,1943],[1841,3823,3825,3847],{"className":3824},[1947],[1841,3826,3829],{"className":3827,"style":3828},[1951],"height:0.3117em;",[1841,3830,3831,3834],{"style":1955},[1841,3832],{"className":3833,"style":1960},[1959],[1841,3835,3837],{"className":3836},[1964,1965,1966,1967],[1841,3838,3840,3844],{"className":3839},[1930,1967],[1841,3841,3678],{"className":3842,"style":3843},[1930,1934,1967],"margin-right:0.0572em;",[1841,3845,1873],{"className":3846},[1930,1934,1967],[1841,3848,1975],{"className":3849},[1974],[1841,3851,3853],{"className":3852},[1947],[1841,3854,3857],{"className":3855,"style":3856},[1951],"height:0.2861em;",[1841,3858],{},[1841,3860],{"className":3861,"style":1989},[1988],[1841,3863,1877],{"className":3864},[1993],[1841,3866],{"className":3867,"style":1989},[1988],[1841,3869,3871,3875,3911,3914,3917,3920,3953,3956,3959,3963,3966,3969,3972,3975,3978,3981,3984],{"className":3870},[1921],[1841,3872],{"className":3873,"style":3874},[1925],"height:1.188em;vertical-align:-0.25em;",[1841,3876,3878,3881],{"className":3877},[1930],[1841,3879,3456],{"className":3880},[1930,1934],[1841,3882,3884],{"className":3883},[1938],[1841,3885,3887],{"className":3886},[1942],[1841,3888,3890],{"className":3889},[1947],[1841,3891,3894],{"className":3892,"style":3893},[1951],"height:0.938em;",[1841,3895,3896,3899],{"style":2931},[1841,3897],{"className":3898,"style":1960},[1959],[1841,3900,3902],{"className":3901},[1964,1965,1966,1967],[1841,3903,3905,3908],{"className":3904},[1930,1967],[1841,3906,2664],{"className":3907},[1930,1967],[1841,3909,2556],{"className":3910},[1930,1967],[1841,3912],{"className":3913,"style":2010},[1988],[1841,3915,3699],{"className":3916},[2014],[1841,3918],{"className":3919,"style":2010},[1988],[1841,3921,3923,3928,3931,3934,3937,3941,3944,3947,3950],{"className":3922},[3554],[1841,3924,1890],{"className":3925,"style":3927},[2018,3926],"delimcenter","top:0em;",[1841,3929,2664],{"className":3930},[1930],[1841,3932,2116],{"className":3933},[1930],[1841,3935,2119],{"className":3936,"style":2022},[1930,1934],[1841,3938,3940],{"className":3939,"style":3843},[1930,1934],"ij",[1841,3942,1873],{"className":3943},[1930,1934],[1841,3945,2110],{"className":3946},[1930],[1841,3948,3456],{"className":3949},[1930,1934],[1841,3951,1904],{"className":3952,"style":3927},[2049,3926],[1841,3954],{"className":3955,"style":2010},[1988],[1841,3957,1908],{"className":3958},[2053],[1841,3960],{"className":3961,"style":3962},[1988],"margin-right:2em;",[1841,3964],{"className":3965,"style":2010},[1988],[1841,3967,3678],{"className":3968,"style":3843},[1930,1934],[1841,3970,1908],{"className":3971},[2053],[1841,3973],{"className":3974,"style":2010},[1988],[1841,3976,1873],{"className":3977},[1930,1934],[1841,3979],{"className":3980,"style":1989},[1988],[1841,3982,1877],{"className":3983},[1993],[1841,3985],{"className":3986,"style":1989},[1988],[1841,3988,3990,3994,3997,4000,4003,4006,4009,4012,4015,4018,4021,4024],{"className":3989},[1921],[1841,3991],{"className":3992,"style":3993},[1925],"height:0.8389em;vertical-align:-0.1944em;",[1841,3995,3277],{"className":3996},[1930],[1841,3998,1908],{"className":3999},[2053],[1841,4001],{"className":4002,"style":2010},[1988],[1841,4004,3445],{"className":4005},[3554],[1841,4007],{"className":4008,"style":2010},[1988],[1841,4010,1908],{"className":4011},[2053],[1841,4013],{"className":4014,"style":2010},[1988],[1841,4016,3456],{"className":4017},[1930,1934],[1841,4019],{"className":4020,"style":2029},[1988],[1841,4022,2664],{"className":4023},[2033],[1841,4025],{"className":4026,"style":2029},[1988],[1841,4028,4030,4034],{"className":4029},[1921],[1841,4031],{"className":4032,"style":4033},[1925],"height:0.6444em;",[1841,4035,3751],{"className":4036},[1930],[1798,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d=F_n(\\mathbf x-\\bar x\\mathbf1)",[1841,4087,4089,4107,4168],{"className":4088,"ariaHidden":1907},[1917],[1841,4090,4092,4095,4098,4101,4104],{"className":4091},[1921],[1841,4093],{"className":4094,"style":3223},[1925],[1841,4096,3022],{"className":4097},[1930,3484],[1841,4099],{"className":4100,"style":1989},[1988],[1841,4102,1877],{"className":4103},[1993],[1841,4105],{"className":4106,"style":1989},[1988],[1841,4108,4110,4113,4153,4156,4159,4162,4165],{"className":4109},[1921],[1841,4111],{"className":4112,"style":2003},[1925],[1841,4114,4116,4119],{"className":4115},[1930],[1841,4117,3667],{"className":4118,"style":2223},[1930,1934],[1841,4120,4122],{"className":4121},[1938],[1841,4123,4125,4145],{"className":4124},[1942,1943],[1841,4126,4128,4142],{"className":4127},[1947],[1841,4129,4131],{"className":4130,"style":3786},[1951],[1841,4132,4133,4136],{"style":3789},[1841,4134],{"className":4135,"style":1960},[1959],[1841,4137,4139],{"className":4138},[1964,1965,1966,1967],[1841,4140,3456],{"className":4141},[1930,1934,1967],[1841,4143,1975],{"className":4144},[1974],[1841,4146,4148],{"className":4147},[1947],[1841,4149,4151],{"className":4150,"style":1982},[1951],[1841,4152],{},[1841,4154,1890],{"className":4155},[2018],[1841,4157,1870],{"className":4158},[1930,3484],[1841,4160],{"className":4161,"style":2029},[1988],[1841,4163,2664],{"className":4164},[2033],[1841,4166],{"className":4167,"style":2029},[1988],[1841,4169,4171,4174,4210,4213],{"className":4170},[1921],[1841,4172],{"className":4173,"style":2003},[1925],[1841,4175,4178],{"className":4176},[1930,4177],"accent",[1841,4179,4181],{"className":4180},[1942],[1841,4182,4184],{"className":4183},[1947],[1841,4185,4188,4197],{"className":4186,"style":4187},[1951],"height:0.5678em;",[1841,4189,4191,4194],{"style":4190},"top:-3em;",[1841,4192],{"className":4193,"style":2764},[1959],[1841,4195,1870],{"className":4196},[1930,1934],[1841,4198,4199,4202],{"style":4190},[1841,4200],{"className":4201,"style":2764},[1959],[1841,4203,4207],{"className":4204,"style":4206},[4205],"accent-body","left:-0.2222em;",[1841,4208,4078],{"className":4209},[1930],[1841,4211,2192],{"className":4212},[1930,3484],[1841,4214,1904],{"className":4215},[2049],[1798,4217,4218],{},"contains discrete Fourier coefficients. The matrix is unitary,",[1841,4220,4222],{"className":4221},[1844],[1841,4223,4225,4263],{"className":4224},[1848],[1841,4226,4228],{"className":4227},[1852],[1854,4229,4230],{"xmlns":1856,"display":1857},[1859,4231,4232,4260],{},[1862,4233,4234,4240,4249,4251,4258],{},[1865,4235,4236,4238],{},[1868,4237,3667],{},[1868,4239,3456],{},[2983,4241,4242,4244,4246],{},[1868,4243,3667],{},[1868,4245,3456],{},[1875,4247,4248],{},"∗",[1875,4250,1877],{},[1865,4252,4253,4256],{},[1868,4254,4255],{},"I",[1868,4257,3456],{},[1875,4259,1908],{"separator":1907},[1910,4261,4262],{"encoding":1912},"F_nF_n^*=I_n,",[1841,4264,4266,4376],{"className":4265,"ariaHidden":1907},[1917],[1841,4267,4269,4273,4313,4367,4370,4373],{"className":4268},[1921],[1841,4270],{"className":4271,"style":4272},[1925],"height:0.9857em;vertical-align:-0.247em;",[1841,4274,4276,4279],{"className":4275},[1930],[1841,4277,3667],{"className":4278,"style":2223},[1930,1934],[1841,4280,4282],{"className":4281},[1938],[1841,4283,4285,4305],{"className":4284},[1942,1943],[1841,4286,4288,4302],{"className":4287},[1947],[1841,4289,4291],{"className":4290,"style":3786},[1951],[1841,4292,4293,4296],{"style":3789},[1841,4294],{"className":4295,"style":1960},[1959],[1841,4297,4299],{"className":4298},[1964,1965,1966,1967],[1841,4300,3456],{"className":4301},[1930,1934,1967],[1841,4303,1975],{"className":4304},[1974],[1841,4306,4308],{"className":4307},[1947],[1841,4309,4311],{"className":4310,"style":1982},[1951],[1841,4312],{},[1841,4314,4316,4319],{"className":4315},[1930],[1841,4317,3667],{"className":4318,"style":2223},[1930,1934],[1841,4320,4322],{"className":4321},[1938],[1841,4323,4325,4358],{"className":4324},[1942,1943],[1841,4326,4328,4355],{"className":4327},[1947],[1841,4329,4332,4344],{"className":4330,"style":4331},[1951],"height:0.7387em;",[1841,4333,4335,4338],{"style":4334},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[1841,4336],{"className":4337,"style":1960},[1959],[1841,4339,4341],{"className":4340},[1964,1965,1966,1967],[1841,4342,3456],{"className":4343},[1930,1934,1967],[1841,4345,4346,4349],{"style":2931},[1841,4347],{"className":4348,"style":1960},[1959],[1841,4350,4352],{"className":4351},[1964,1965,1966,1967],[1841,4353,4248],{"className":4354},[2033,1967],[1841,4356,1975],{"className":4357},[1974],[1841,4359,4361],{"className":4360},[1947],[1841,4362,4365],{"className":4363,"style":4364},[1951],"height:0.247em;",[1841,4366],{},[1841,4368],{"className":4369,"style":1989},[1988],[1841,4371,1877],{"className":4372},[1993],[1841,4374],{"className":4375,"style":1989},[1988],[1841,4377,4379,4383,4425],{"className":4378},[1921],[1841,4380],{"className":4381,"style":4382},[1925],"height:0.8778em;vertical-align:-0.1944em;",[1841,4384,4386,4390],{"className":4385},[1930],[1841,4387,4255],{"className":4388,"style":4389},[1930,1934],"margin-right:0.0785em;",[1841,4391,4393],{"className":4392},[1938],[1841,4394,4396,4417],{"className":4395},[1942,1943],[1841,4397,4399,4414],{"className":4398},[1947],[1841,4400,4402],{"className":4401,"style":3786},[1951],[1841,4403,4405,4408],{"style":4404},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1841,4406],{"className":4407,"style":1960},[1959],[1841,4409,4411],{"className":4410},[1964,1965,1966,1967],[1841,4412,3456],{"className":4413},[1930,1934,1967],[1841,4415,1975],{"className":4416},[1974],[1841,4418,4420],{"className":4419},[1947],[1841,4421,4423],{"className":4422,"style":1982},[1951],[1841,4424],{},[1841,4426,1908],{"className":4427},[2053],[1798,4429,4430,4431,1908],{},"so it rotates the centred data into orthogonal frequency coordinates without changing total squared length. At Fourier frequency ",[1841,4432,4434,4464],{"className":4433},[1848],[1841,4435,4437],{"className":4436},[1852],[1854,4438,4439],{"xmlns":1856},[1859,4440,4441,4461],{},[1862,4442,4443,4449,4451,4453,4455,4457,4459],{},[1865,4444,4445,4447],{},[1868,4446,1893],{},[1868,4448,3678],{},[1875,4450,1877],{},[2114,4452,2116],{},[1868,4454,2119],{},[1868,4456,3678],{},[1868,4458,2110],{"mathvariant":2109},[1868,4460,3456],{},[1910,4462,4463],{"encoding":1912},"\\omega_j=2\\pi j\u002Fn",[1841,4465,4467,4524],{"className":4466,"ariaHidden":1907},[1917],[1841,4468,4470,4474,4515,4518,4521],{"className":4469},[1921],[1841,4471],{"className":4472,"style":4473},[1925],"height:0.7167em;vertical-align:-0.2861em;",[1841,4475,4477,4480],{"className":4476},[1930],[1841,4478,1893],{"className":4479,"style":2022},[1930,1934],[1841,4481,4483],{"className":4482},[1938],[1841,4484,4486,4507],{"className":4485},[1942,1943],[1841,4487,4489,4504],{"className":4488},[1947],[1841,4490,4492],{"className":4491,"style":3828},[1951],[1841,4493,4495,4498],{"style":4494},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1841,4496],{"className":4497,"style":1960},[1959],[1841,4499,4501],{"className":4500},[1964,1965,1966,1967],[1841,4502,3678],{"className":4503,"style":3843},[1930,1934,1967],[1841,4505,1975],{"className":4506},[1974],[1841,4508,4510],{"className":4509},[1947],[1841,4511,4513],{"className":4512,"style":3856},[1951],[1841,4514],{},[1841,4516],{"className":4517,"style":1989},[1988],[1841,4519,1877],{"className":4520},[1993],[1841,4522],{"className":4523,"style":1989},[1988],[1841,4525,4527,4530,4533,4536,4539,4542],{"className":4526},[1921],[1841,4528],{"className":4529,"style":2003},[1925],[1841,4531,2116],{"className":4532},[1930],[1841,4534,2119],{"className":4535,"style":2022},[1930,1934],[1841,4537,3678],{"className":4538,"style":3843},[1930,1934],[1841,4540,2110],{"className":4541},[1930],[1841,4543,3456],{"className":4544},[1930,1934],[1841,4546,4548],{"className":4547},[1844],[1841,4549,4551,4604],{"className":4550},[1848],[1841,4552,4554],{"className":4553},[1852],[1854,4555,4556],{"xmlns":1856,"display":1857},[1859,4557,4558,4601],{},[1862,4559,4560,4562,4564,4570,4572,4574,4599],{},[1868,4561,4255],{},[1875,4563,1890],{"stretchy":1889},[1865,4565,4566,4568],{},[1868,4567,1893],{},[1868,4569,3678],{},[1875,4571,1904],{"stretchy":1889},[1875,4573,1877],{},[2639,4575,4576,4593],{},[1862,4577,4578,4581,4587],{},[1868,4579,4580],{"mathvariant":2109},"∣",[1865,4582,4583,4585],{},[1868,4584,3022],{},[1868,4586,3678],{},[2680,4588,4589,4591],{},[1868,4590,4580],{"mathvariant":2109},[2114,4592,2116],{},[1862,4594,4595,4597],{},[2114,4596,2116],{},[1868,4598,2119],{},[1868,4600,3027],{"mathvariant":2109},[1910,4602,4603],{"encoding":1912},"I(\\omega_j)=\\frac{|d_j|^2}{2\\pi}.",[1841,4605,4607,4671],{"className":4606,"ariaHidden":1907},[1917],[1841,4608,4610,4613,4616,4619,4659,4662,4665,4668],{"className":4609},[1921],[1841,4611],{"className":4612,"style":3764},[1925],[1841,4614,4255],{"className":4615,"style":4389},[1930,1934],[1841,4617,1890],{"className":4618},[2018],[1841,4620,4622,4625],{"className":4621},[1930],[1841,4623,1893],{"className":4624,"style":2022},[1930,1934],[1841,4626,4628],{"className":4627},[1938],[1841,4629,4631,4651],{"className":4630},[1942,1943],[1841,4632,4634,4648],{"className":4633},[1947],[1841,4635,4637],{"className":4636,"style":3828},[1951],[1841,4638,4639,4642],{"style":4494},[1841,4640],{"className":4641,"style":1960},[1959],[1841,4643,4645],{"className":4644},[1964,1965,1966,1967],[1841,4646,3678],{"className":4647,"style":3843},[1930,1934,1967],[1841,4649,1975],{"className":4650},[1974],[1841,4652,4654],{"className":4653},[1947],[1841,4655,4657],{"className":4656,"style":3856},[1951],[1841,4658],{},[1841,4660,1904],{"className":4661},[2049],[1841,4663],{"className":4664,"style":1989},[1988],[1841,4666,1877],{"className":4667},[1993],[1841,4669],{"className":4670,"style":1989},[1988],[1841,4672,4674,4678,4814],{"className":4673},[1921],[1841,4675],{"className":4676,"style":4677},[1925],"height:2.1771em;vertical-align:-0.686em;",[1841,4679,4681,4684,4811],{"className":4680},[1930],[1841,4682],{"className":4683},[2018,2744],[1841,4685,4687],{"className":4686},[2639],[1841,4688,4690,4803],{"className":4689},[1942,1943],[1841,4691,4693,4800],{"className":4692},[1947],[1841,4694,4697,4711,4719],{"className":4695,"style":4696},[1951],"height:1.4911em;",[1841,4698,4699,4702],{"style":2760},[1841,4700],{"className":4701,"style":2764},[1959],[1841,4703,4705,4708],{"className":4704},[1930],[1841,4706,2116],{"className":4707},[1930],[1841,4709,2119],{"className":4710,"style":2022},[1930,1934],[1841,4712,4713,4716],{"style":2776},[1841,4714],{"className":4715,"style":2764},[1959],[1841,4717],{"className":4718,"style":2784},[2783],[1841,4720,4721,4724],{"style":2787},[1841,4722],{"className":4723,"style":2764},[1959],[1841,4725,4727,4730,4770],{"className":4726},[1930],[1841,4728,4580],{"className":4729},[1930],[1841,4731,4733,4736],{"className":4732},[1930],[1841,4734,3022],{"className":4735},[1930,1934],[1841,4737,4739],{"className":4738},[1938],[1841,4740,4742,4762],{"className":4741},[1942,1943],[1841,4743,4745,4759],{"className":4744},[1947],[1841,4746,4748],{"className":4747,"style":3828},[1951],[1841,4749,4750,4753],{"style":1955},[1841,4751],{"className":4752,"style":1960},[1959],[1841,4754,4756],{"className":4755},[1964,1965,1966,1967],[1841,4757,3678],{"className":4758,"style":3843},[1930,1934,1967],[1841,4760,1975],{"className":4761},[1974],[1841,4763,4765],{"className":4764},[1947],[1841,4766,4768],{"className":4767,"style":3856},[1951],[1841,4769],{},[1841,4771,4773,4776],{"className":4772},[1930],[1841,4774,4580],{"className":4775},[1930],[1841,4777,4779],{"className":4778},[1938],[1841,4780,4782],{"className":4781},[1942],[1841,4783,4785],{"className":4784},[1947],[1841,4786,4789],{"className":4787,"style":4788},[1951],"height:0.8141em;",[1841,4790,4791,4794],{"style":3635},[1841,4792],{"className":4793,"style":1960},[1959],[1841,4795,4797],{"className":4796},[1964,1965,1966,1967],[1841,4798,2116],{"className":4799},[1930,1967],[1841,4801,1975],{"className":4802},[1974],[1841,4804,4806],{"className":4805},[1947],[1841,4807,4809],{"className":4808,"style":2806},[1951],[1841,4810],{},[1841,4812],{"className":4813},[2049,2744],[1841,4815,3027],{"className":4816},[1930],[1793,4818,4820],{"id":4819},"_3-familiar-models-have-readable-spectra","3. Familiar models have readable spectra",[2275,4822,4823,4833],{},[2278,4824,4825],{},[2281,4826,4827,4830],{},[2284,4828,4829],{},"Process",[2284,4831,4832],{},"Spectral shape",[2354,4834,4835,4843,4904,4992],{},[2281,4836,4837,4840],{},[2359,4838,4839],{},"white noise",[2359,4841,4842],{},"flat: no frequency dominates",[2281,4844,4845,4901],{},[2359,4846,4847,4848],{},"AR(1), ",[1841,4849,4851,4871],{"className":4850},[1848],[1841,4852,4854],{"className":4853},[1852],[1854,4855,4856],{"xmlns":1856},[1859,4857,4858,4868],{},[1862,4859,4860,4863,4866],{},[1868,4861,4862],{},"ϕ",[1875,4864,4865],{},">",[2114,4867,3277],{},[1910,4869,4870],{"encoding":1912},"\\phi>0",[1841,4872,4874,4892],{"className":4873,"ariaHidden":1907},[1917],[1841,4875,4877,4880,4883,4886,4889],{"className":4876},[1921],[1841,4878],{"className":4879,"style":2134},[1925],[1841,4881,4862],{"className":4882},[1930,1934],[1841,4884],{"className":4885,"style":1989},[1988],[1841,4887,4865],{"className":4888},[1993],[1841,4890],{"className":4891,"style":1989},[1988],[1841,4893,4895,4898],{"className":4894},[1921],[1841,4896],{"className":4897,"style":4033},[1925],[1841,4899,3277],{"className":4900},[1930],[2359,4902,4903],{},"high power near zero: persistent slow movement",[2281,4905,4906,4960],{},[2359,4907,4847,4908],{},[1841,4909,4911,4930],{"className":4910},[1848],[1841,4912,4914],{"className":4913},[1852],[1854,4915,4916],{"xmlns":1856},[1859,4917,4918,4927],{},[1862,4919,4920,4922,4925],{},[1868,4921,4862],{},[1875,4923,4924],{},"\u003C",[2114,4926,3277],{},[1910,4928,4929],{"encoding":1912},"\\phi\u003C0",[1841,4931,4933,4951],{"className":4932,"ariaHidden":1907},[1917],[1841,4934,4936,4939,4942,4945,4948],{"className":4935},[1921],[1841,4937],{"className":4938,"style":2134},[1925],[1841,4940,4862],{"className":4941},[1930,1934],[1841,4943],{"className":4944,"style":1989},[1988],[1841,4946,4924],{"className":4947},[1993],[1841,4949],{"className":4950,"style":1989},[1988],[1841,4952,4954,4957],{"className":4953},[1921],[1841,4955],{"className":4956,"style":4033},[1925],[1841,4958,3277],{"className":4959},[1930],[2359,4961,4962,4963,4991],{},"high power near ",[1841,4964,4966,4979],{"className":4965},[1848],[1841,4967,4969],{"className":4968},[1852],[1854,4970,4971],{"xmlns":1856},[1859,4972,4973,4977],{},[1862,4974,4975],{},[1868,4976,2119],{},[1910,4978,2587],{"encoding":1912},[1841,4980,4982],{"className":4981,"ariaHidden":1907},[1917],[1841,4983,4985,4988],{"className":4984},[1921],[1841,4986],{"className":4987,"style":2083},[1925],[1841,4989,2119],{"className":4990,"style":2022},[1930,1934],": alternating movement",[2281,4993,4994,4997],{},[2359,4995,4996],{},"seasonal process",[2359,4998,4999],{},"peak near seasonal frequency and possibly harmonics",[1798,5001,5002],{},"For AR(1),",[1841,5004,5006],{"className":5005},[1844],[1841,5007,5009,5079],{"className":5008},[1848],[1841,5010,5012],{"className":5011},[1852],[1854,5013,5014],{"xmlns":1856,"display":1857},[1859,5015,5016,5076],{},[1862,5017,5018,5020,5022,5024,5026,5028,5046,5074],{},[1868,5019,2102],{},[1875,5021,1890],{"stretchy":1889},[1868,5023,1893],{},[1875,5025,1904],{"stretchy":1889},[1875,5027,1877],{},[2639,5029,5030,5040],{},[2983,5031,5032,5035,5038],{},[1868,5033,5034],{},"σ",[1868,5036,5037],{},"ε",[2114,5039,2116],{},[1862,5041,5042,5044],{},[2114,5043,2116],{},[1868,5045,2119],{},[2639,5047,5048,5050],{},[2114,5049,2192],{},[1862,5051,5052,5054,5056,5058,5060,5062,5064,5066,5068],{},[2114,5053,2192],{},[1875,5055,2664],{},[2114,5057,2116],{},[1868,5059,4862],{},[1868,5061,1883],{},[1875,5063,1886],{},[1868,5065,1893],{},[1875,5067,1898],{},[2680,5069,5070,5072],{},[1868,5071,4862],{},[2114,5073,2116],{},[1868,5075,3027],{"mathvariant":2109},[1910,5077,5078],{"encoding":1912},"f(\\omega)=\\frac{\\sigma_\\varepsilon^2}{2\\pi}\n\\frac{1}{1-2\\phi\\cos\\omega+\\phi^2}.",[1841,5080,5082,5109],{"className":5081,"ariaHidden":1907},[1917],[1841,5083,5085,5088,5091,5094,5097,5100,5103,5106],{"className":5084},[1921],[1841,5086],{"className":5087,"style":2003},[1925],[1841,5089,2102],{"className":5090,"style":2138},[1930,1934],[1841,5092,1890],{"className":5093},[2018],[1841,5095,1893],{"className":5096,"style":2022},[1930,1934],[1841,5098,1904],{"className":5099},[2049],[1841,5101],{"className":5102,"style":1989},[1988],[1841,5104,1877],{"className":5105},[1993],[1841,5107],{"className":5108,"style":1989},[1988],[1841,5110,5112,5116,5230,5360],{"className":5111},[1921],[1841,5113],{"className":5114,"style":5115},[1925],"height:2.3715em;vertical-align:-0.8804em;",[1841,5117,5119,5122,5227],{"className":5118},[1930],[1841,5120],{"className":5121},[2018,2744],[1841,5123,5125],{"className":5124},[2639],[1841,5126,5128,5219],{"className":5127},[1942,1943],[1841,5129,5131,5216],{"className":5130},[1947],[1841,5132,5134,5148,5156],{"className":5133,"style":4696},[1951],[1841,5135,5136,5139],{"style":2760},[1841,5137],{"className":5138,"style":2764},[1959],[1841,5140,5142,5145],{"className":5141},[1930],[1841,5143,2116],{"className":5144},[1930],[1841,5146,2119],{"className":5147,"style":2022},[1930,1934],[1841,5149,5150,5153],{"style":2776},[1841,5151],{"className":5152,"style":2764},[1959],[1841,5154],{"className":5155,"style":2784},[2783],[1841,5157,5158,5161],{"style":2787},[1841,5159],{"className":5160,"style":2764},[1959],[1841,5162,5164],{"className":5163},[1930],[1841,5165,5167,5170],{"className":5166},[1930],[1841,5168,5034],{"className":5169,"style":2022},[1930,1934],[1841,5171,5173],{"className":5172},[1938],[1841,5174,5176,5208],{"className":5175},[1942,1943],[1841,5177,5179,5205],{"className":5178},[1947],[1841,5180,5182,5194],{"className":5181,"style":4788},[1951],[1841,5183,5185,5188],{"style":5184},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[1841,5186],{"className":5187,"style":1960},[1959],[1841,5189,5191],{"className":5190},[1964,1965,1966,1967],[1841,5192,5037],{"className":5193},[1930,1934,1967],[1841,5195,5196,5199],{"style":3635},[1841,5197],{"className":5198,"style":1960},[1959],[1841,5200,5202],{"className":5201},[1964,1965,1966,1967],[1841,5203,2116],{"className":5204},[1930,1967],[1841,5206,1975],{"className":5207},[1974],[1841,5209,5211],{"className":5210},[1947],[1841,5212,5214],{"className":5213,"style":4364},[1951],[1841,5215],{},[1841,5217,1975],{"className":5218},[1974],[1841,5220,5222],{"className":5221},[1947],[1841,5223,5225],{"className":5224,"style":2806},[1951],[1841,5226],{},[1841,5228],{"className":5229},[2049,2744],[1841,5231,5233,5236,5357],{"className":5232},[1930],[1841,5234],{"className":5235},[2018,2744],[1841,5237,5239],{"className":5238},[2639],[1841,5240,5242,5348],{"className":5241},[1942,1943],[1841,5243,5245,5345],{"className":5244},[1947],[1841,5246,5248,5326,5334],{"className":5247,"style":2757},[1951],[1841,5249,5250,5253],{"style":2760},[1841,5251],{"className":5252,"style":2764},[1959],[1841,5254,5256,5259,5262,5265,5268,5271,5274,5277,5280,5283,5286,5289,5292,5295],{"className":5255},[1930],[1841,5257,2192],{"className":5258},[1930],[1841,5260],{"className":5261,"style":2029},[1988],[1841,5263,2664],{"className":5264},[2033],[1841,5266],{"className":5267,"style":2029},[1988],[1841,5269,2116],{"className":5270},[1930],[1841,5272,4862],{"className":5273},[1930,1934],[1841,5275],{"className":5276,"style":2010},[1988],[1841,5278,1883],{"className":5279},[2014],[1841,5281],{"className":5282,"style":2010},[1988],[1841,5284,1893],{"className":5285,"style":2022},[1930,1934],[1841,5287],{"className":5288,"style":2029},[1988],[1841,5290,1898],{"className":5291},[2033],[1841,5293],{"className":5294,"style":2029},[1988],[1841,5296,5298,5301],{"className":5297},[1930],[1841,5299,4862],{"className":5300},[1930,1934],[1841,5302,5304],{"className":5303},[1938],[1841,5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",[1841,5367,5369,5383],{"className":5368},[1848],[1841,5370,5372],{"className":5371},[1852],[1854,5373,5374],{"xmlns":1856},[1859,5375,5376,5380],{},[1862,5377,5378],{},[1868,5379,4862],{},[1910,5381,5382],{"encoding":1912},"\\phi",[1841,5384,5386],{"className":5385,"ariaHidden":1907},[1917],[1841,5387,5389,5392],{"className":5388},[1921],[1841,5390],{"className":5391,"style":2134},[1925],[1841,5393,4862],{"className":5394},[1930,1934]," approaches one, power concentrates near zero. That resemblance to trend is why low-frequency peaks should not be interpreted before stationarity and breaks are considered.",[1793,5397,5399],{"id":5398},"_4-the-periodogram-is-informative-and-noisy","4. 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high ordinate indicates that a sinusoid at that frequency explains substantial sample variation. But the raw periodogram is not a consistent pointwise estimator: as ",[1841,6135,6137,6150],{"className":6136},[1848],[1841,6138,6140],{"className":6139},[1852],[1854,6141,6142],{"xmlns":1856},[1859,6143,6144,6148],{},[1862,6145,6146],{},[1868,6147,3456],{},[1910,6149,3456],{"encoding":1912},[1841,6151,6153],{"className":6152,"ariaHidden":1907},[1917],[1841,6154,6156,6159],{"className":6155},[1921],[1841,6157],{"className":6158,"style":2083},[1925],[1841,6160,3456],{"className":6161},[1930,1934]," grows, its variance does not vanish. Smoothing trades resolution for stability.",[2275,6164,6165,6178],{},[2278,6166,6167],{},[2281,6168,6169,6172,6175],{},[2284,6170,6171],{},"Choice",[2284,6173,6174],{},"Gain",[2284,6176,6177],{},"Cost",[2354,6179,6180,6191,6202],{},[2281,6181,6182,6185,6188],{},[2359,6183,6184],{},"narrow smoothing window",[2359,6186,6187],{},"resolves close peaks",[2359,6189,6190],{},"noisy estimate",[2281,6192,6193,6196,6199],{},[2359,6194,6195],{},"wide smoothing window",[2359,6197,6198],{},"stable broad shape",[2359,6200,6201],{},"merges nearby cycles",[2281,6203,6204,6207,6210],{},[2359,6205,6206],{},"taper endpoints",[2359,6208,6209],{},"reduces leakage from boundary mismatch",[2359,6211,6212],{},"widens peaks and loses some information",[1793,6214,6216],{"id":6215},"_5-recover-two-known-cycles","5. Recover two known cycles",[1798,6218,6219,6220,6223],{},"The cell generates periods 12 and 4. Change ",[6221,6222,3456],"code",{}," from 120 to 118: incomplete cycles spread power across neighbouring frequencies, demonstrating spectral leakage.",[6225,6226],"web-r",{"code64":6227,"layout":6228,"locale":7,"title":6229},"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","vertical","Build a Fourier matrix and recover two periods",[1798,6231,6232],{},"The periodogram detects repetitive structure; it does not tell whether the cycle is stable, caused by a calendar mechanism, or useful for future prediction.",[1793,6234,6236],{"id":6235},"_6-aliasing-sampling-can-create-the-wrong-frequency","6. Aliasing: sampling can create the wrong frequency",[1798,6238,6239],{},"With one observation per hour, the Nyquist frequency is 0.5 cycles\u002Fhour. A true signal at 0.6 cycles\u002Fhour is indistinguishable from an alias at",[1841,6241,6243],{"className":6242},[1844],[1841,6244,6246,6279],{"className":6245},[1848],[1841,6247,6249],{"className":6248},[1852],[1854,6250,6251],{"xmlns":1856,"display":1857},[1859,6252,6253,6276],{},[1862,6254,6255,6257,6260,6262,6264,6266,6268,6271,6274],{},[1868,6256,4580],{"mathvariant":2109},[2114,6258,6259],{},"0.6",[1875,6261,2664],{},[2114,6263,2192],{},[1868,6265,4580],{"mathvariant":2109},[1875,6267,1877],{},[2114,6269,6270],{},"0.4",[3017,6272,6273],{}," cycles\u002Fhour",[1868,6275,3027],{"mathvariant":2109},[1910,6277,6278],{"encoding":1912},"|0.6-1|=0.4\\ \\text{cycles\u002Fhour}.",[1841,6280,6282,6301,6320],{"className":6281,"ariaHidden":1907},[1917],[1841,6283,6285,6288,6292,6295,6298],{"className":6284},[1921],[1841,6286],{"className":6287,"style":2003},[1925],[1841,6289,6291],{"className":6290},[1930],"∣0.6",[1841,6293],{"className":6294,"style":2029},[1988],[1841,6296,2664],{"className":6297},[2033],[1841,6299],{"className":6300,"style":2029},[1988],[1841,6302,6304,6307,6311,6314,6317],{"className":6303},[1921],[1841,6305],{"className":6306,"style":2003},[1925],[1841,6308,6310],{"className":6309},[1930],"1∣",[1841,6312],{"className":6313,"style":1989},[1988],[1841,6315,1877],{"className":6316},[1993],[1841,6318],{"className":6319,"style":1989},[1988],[1841,6321,6323,6326,6329,6333,6341],{"className":6322},[1921],[1841,6324],{"className":6325,"style":2003},[1925],[1841,6327,6270],{"className":6328},[1930],[1841,6330,6332],{"className":6331},[1988]," ",[1841,6334,6337],{"className":6335},[1930,6336],"text",[1841,6338,6340],{"className":6339},[1930],"cycles\u002Fhour",[1841,6342,3027],{"className":6343},[1930],[1798,6345,6346],{},"No post-processing can recover frequencies that the sampling design made indistinguishable. Sample faster, use an anti-alias filter before sampling, or restrict claims to the observable band.",[1793,6348,6350],{"id":6349},"_7-cross-spectra-describe-frequency-specific-association","7. Cross-spectra describe frequency-specific association",[1798,6352,6353],{},"For two stationary series, the cross-spectrum is the Fourier transform of cross-covariances. Squared coherence,",[1841,6355,6357],{"className":6356},[1844],[1841,6358,6360,6450],{"className":6359},[1848],[1841,6361,6363],{"className":6362},[1852],[1854,6364,6365],{"xmlns":1856,"display":1857},[1859,6366,6367,6447],{},[1862,6368,6369,6383,6385,6387,6389,6391,6445],{},[2983,6370,6371,6374,6381],{},[1868,6372,6373],{},"K",[1862,6375,6376,6378],{},[1868,6377,1870],{},[1868,6379,6380],{},"y",[2114,6382,2116],{},[1875,6384,1890],{"stretchy":1889},[1868,6386,1893],{},[1875,6388,1904],{"stretchy":1889},[1875,6390,1877],{},[2639,6392,6393,6419],{},[1862,6394,6395,6397,6407,6409,6411,6413],{},[1868,6396,4580],{"mathvariant":2109},[1865,6398,6399,6401],{},[1868,6400,2102],{},[1862,6402,6403,6405],{},[1868,6404,1870],{},[1868,6406,6380],{},[1875,6408,1890],{"stretchy":1889},[1868,6410,1893],{},[1875,6412,1904],{"stretchy":1889},[2680,6414,6415,6417],{},[1868,6416,4580],{"mathvariant":2109},[2114,6418,2116],{},[1862,6420,6421,6427,6429,6431,6433,6439,6441,6443],{},[1865,6422,6423,6425],{},[1868,6424,2102],{},[1868,6426,1870],{},[1875,6428,1890],{"stretchy":1889},[1868,6430,1893],{},[1875,6432,1904],{"stretchy":1889},[1865,6434,6435,6437],{},[1868,6436,2102],{},[1868,6438,6380],{},[1875,6440,1890],{"stretchy":1889},[1868,6442,1893],{},[1875,6444,1904],{"stretchy":1889},[1875,6446,1908],{"separator":1907},[1910,6448,6449],{"encoding":1912},"K^2_{xy}(\\omega)=\n\\frac{|f_{xy}(\\omega)|^2}{f_x(\\omega)f_y(\\omega)},",[1841,6451,6453,6539],{"className":6452,"ariaHidden":1907},[1917],[1841,6454,6456,6460,6521,6524,6527,6530,6533,6536],{"className":6455},[1921],[1841,6457],{"className":6458,"style":6459},[1925],"height:1.2472em;vertical-align:-0.3831em;",[1841,6461,6463,6467],{"className":6462},[1930],[1841,6464,6373],{"className":6465,"style":6466},[1930,1934],"margin-right:0.0715em;",[1841,6468,6470],{"className":6469},[1938],[1841,6471,6473,6512],{"className":6472},[1942,1943],[1841,6474,6476,6509],{"className":6475},[1947],[1841,6477,6480,6498],{"className":6478,"style":6479},[1951],"height:0.8641em;",[1841,6481,6483,6486],{"style":6482},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[1841,6484],{"className":6485,"style":1960},[1959],[1841,6487,6489],{"className":6488},[1964,1965,1966,1967],[1841,6490,6492,6495],{"className":6491},[1930,1967],[1841,6493,1870],{"className":6494},[1930,1934,1967],[1841,6496,6380],{"className":6497,"style":2022},[1930,1934,1967],[1841,6499,6500,6503],{"style":2931},[1841,6501],{"className":6502,"style":1960},[1959],[1841,6504,6506],{"className":6505},[1964,1965,1966,1967],[1841,6507,2116],{"className":6508},[1930,1967],[1841,6510,1975],{"className":6511},[1974],[1841,6513,6515],{"className":6514},[1947],[1841,6516,6519],{"className":6517,"style":6518},[1951],"height:0.3831em;",[1841,6520],{},[1841,6522,1890],{"className":6523},[2018],[1841,6525,1893],{"className":6526,"style":2022},[1930,1934],[1841,6528,1904],{"className":6529},[2049],[1841,6531],{"className":6532,"style":1989},[1988],[1841,6534,1877],{"className":6535},[1993],[1841,6537],{"className":6538,"style":1989},[1988],[1841,6540,6542,6546,6789],{"className":6541},[1921],[1841,6543],{"className":6544,"style":6545},[1925],"height:2.4632em;vertical-align:-0.9721em;",[1841,6547,6549,6552,6786],{"className":6548},[1930],[1841,6550],{"className":6551},[2018,2744],[1841,6553,6555],{"className":6554},[2639],[1841,6556,6558,6777],{"className":6557},[1942,1943],[1841,6559,6561,6774],{"className":6560},[1947],[1841,6562,6564,6671,6679],{"className":6563,"style":4696},[1951],[1841,6565,6566,6569],{"style":2760},[1841,6567],{"className":6568,"style":2764},[1959],[1841,6570,6572,6613,6616,6619,6622,6662,6665,6668],{"className":6571},[1930],[1841,6573,6575,6578],{"className":6574},[1930],[1841,6576,2102],{"className":6577,"style":2138},[1930,1934],[1841,6579,6581],{"className":6580},[1938],[1841,6582,6584,6605],{"className":6583},[1942,1943],[1841,6585,6587,6602],{"className":6586},[1947],[1841,6588,6590],{"className":6589,"style":3786},[1951],[1841,6591,6593,6596],{"style":6592},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[1841,6594],{"className":6595,"style":1960},[1959],[1841,6597,6599],{"className":6598},[1964,1965,1966,1967],[1841,6600,1870],{"className":6601},[1930,1934,1967],[1841,6603,1975],{"className":6604},[1974],[1841,6606,6608],{"className":6607},[1947],[1841,6609,6611],{"className":6610,"style":1982},[1951],[1841,6612],{},[1841,6614,1890],{"className":6615},[2018],[1841,6617,1893],{"className":6618,"style":2022},[1930,1934],[1841,6620,1904],{"className":6621},[2049],[1841,6623,6625,6628],{"className":6624},[1930],[1841,6626,2102],{"className":6627,"style":2138},[1930,1934],[1841,6629,6631],{"className":6630},[1938],[1841,6632,6634,6654],{"className":6633},[1942,1943],[1841,6635,6637,6651],{"className":6636},[1947],[1841,6638,6640],{"className":6639,"style":3786},[1951],[1841,6641,6642,6645],{"style":6592},[1841,6643],{"className":6644,"style":1960},[1959],[1841,6646,6648],{"className":6647},[1964,1965,1966,1967],[1841,6649,6380],{"className":6650,"style":2022},[1930,1934,1967],[1841,6652,1975],{"className":6653},[1974],[1841,6655,6657],{"className":6656},[1947],[1841,6658,6660],{"className":6659,"style":3856},[1951],[1841,6661],{},[1841,6663,1890],{"className":6664},[2018],[1841,6666,1893],{"className":6667,"style":2022},[1930,1934],[1841,6669,1904],{"className":6670},[2049],[1841,6672,6673,6676],{"style":2776},[1841,6674],{"className":6675,"style":2764},[1959],[1841,6677],{"className":6678,"style":2784},[2783],[1841,6680,6681,6684],{"style":2787},[1841,6682],{"className":6683,"style":2764},[1959],[1841,6685,6687,6690,6736,6739,6742,6745],{"className":6686},[1930],[1841,6688,4580],{"className":6689},[1930],[1841,6691,6693,6696],{"className":6692},[1930],[1841,6694,2102],{"className":6695,"style":2138},[1930,1934],[1841,6697,6699],{"className":6698},[1938],[1841,6700,6702,6728],{"className":6701},[1942,1943],[1841,6703,6705,6725],{"className":6704},[1947],[1841,6706,6708],{"className":6707,"style":3786},[1951],[1841,6709,6710,6713],{"style":6592},[1841,6711],{"className":6712,"style":1960},[1959],[1841,6714,6716],{"className":6715},[1964,1965,1966,1967],[1841,6717,6719,6722],{"className":6718},[1930,1967],[1841,6720,1870],{"className":6721},[1930,1934,1967],[1841,6723,6380],{"className":6724,"style":2022},[1930,1934,1967],[1841,6726,1975],{"className":6727},[1974],[1841,6729,6731],{"className":6730},[1947],[1841,6732,6734],{"className":6733,"style":3856},[1951],[1841,6735],{},[1841,6737,1890],{"className":6738},[2018],[1841,6740,1893],{"className":6741,"style":2022},[1930,1934],[1841,6743,1904],{"className":6744},[2049],[1841,6746,6748,6751],{"className":6747},[1930],[1841,6749,4580],{"className":6750},[1930],[1841,6752,6754],{"className":6753},[1938],[1841,6755,6757],{"className":6756},[1942],[1841,6758,6760],{"className":6759},[1947],[1841,6761,6763],{"className":6762,"style":4788},[1951],[1841,6764,6765,6768],{"style":3635},[1841,6766],{"className":6767,"style":1960},[1959],[1841,6769,6771],{"className":6770},[1964,1965,1966,1967],[1841,6772,2116],{"className":6773},[1930,1967],[1841,6775,1975],{"className":6776},[1974],[1841,6778,6780],{"className":6779},[1947],[1841,6781,6784],{"className":6782,"style":6783},[1951],"height:0.9721em;",[1841,6785],{},[1841,6787],{"className":6788},[2049,2744],[1841,6790,1908],{"className":6791},[2053],[1798,6793,6794,6795,6823],{},"lies between 0 and 1 and measures linear association at frequency ",[1841,6796,6798,6811],{"className":6797},[1848],[1841,6799,6801],{"className":6800},[1852],[1854,6802,6803],{"xmlns":1856},[1859,6804,6805,6809],{},[1862,6806,6807],{},[1868,6808,1893],{},[1910,6810,2073],{"encoding":1912},[1841,6812,6814],{"className":6813,"ariaHidden":1907},[1917],[1841,6815,6817,6820],{"className":6816},[1921],[1841,6818],{"className":6819,"style":2083},[1925],[1841,6821,1893],{"className":6822,"style":2022},[1930,1934],". Phase describes a frequency-specific lead or lag.",[1798,6825,6826],{},"High coherence at the annual frequency may simply reflect a shared calendar. It is not proof that one series causes the other; common forcing and filtering can create the same pattern.",[1798,6828,6829,6830,6860],{},"For a ",[1841,6831,6833,6847],{"className":6832},[1848],[1841,6834,6836],{"className":6835},[1852],[1854,6837,6838],{"xmlns":1856},[1859,6839,6840,6845],{},[1862,6841,6842],{},[1868,6843,6844],{},"k",[1910,6846,6844],{"encoding":1912},[1841,6848,6850],{"className":6849,"ariaHidden":1907},[1917],[1841,6851,6853,6856],{"className":6852},[1921],[1841,6854],{"className":6855,"style":3223},[1925],[1841,6857,6844],{"className":6858,"style":6859},[1930,1934],"margin-right:0.0315em;","-variate process, collect all auto- and cross-spectra in",[1841,6862,6864],{"className":6863},[1844],[1841,6865,6867,7048],{"className":6866},[1848],[1841,6868,6870],{"className":6869},[1852],[1854,6871,6872],{"xmlns":1856,"display":1857},[1859,6873,6874,7045],{},[1862,6875,6876,6878,6880,6882,6884,6886,7043],{},[1868,6877,2102],{},[1875,6879,1890],{"stretchy":1889},[1868,6881,1893],{},[1875,6883,1904],{"stretchy":1889},[1875,6885,1877],{},[1862,6887,6888,6891,7040],{},[1875,6889,6890],{"fence":1907},"[",[6892,6893,6897,6951,6988],"mtable",{"rowspacing":6894,"columnalign":6895,"columnspacing":6896},"0.16em","center center 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This is the frequency-domain analogue of a covariance matrix: for any complex vector ",[1841,7646,7648,7663],{"className":7647},[1848],[1841,7649,7651],{"className":7650},[1852],[1854,7652,7653],{"xmlns":1856},[1859,7654,7655,7660],{},[1862,7656,7657],{},[1868,7658,7659],{"mathvariant":3430},"a",[1910,7661,7662],{"encoding":1912},"\\mathbf a",[1841,7664,7666],{"className":7665,"ariaHidden":1907},[1917],[1841,7667,7669,7672],{"className":7668},[1921],[1841,7670],{"className":7671,"style":3480},[1925],[1841,7673,7659],{"className":7674},[1930,3484],", the scalar spectrum of ",[1841,7677,7679,7704],{"className":7678},[1848],[1841,7680,7682],{"className":7681},[1852],[1854,7683,7684],{"xmlns":1856},[1859,7685,7686,7701],{},[1862,7687,7688,7694],{},[2680,7689,7690,7692],{},[1868,7691,7659],{"mathvariant":3430},[1875,7693,4248],{},[1865,7695,7696,7699],{},[1868,7697,7698],{"mathvariant":3430},"X",[1868,7700,1873],{},[1910,7702,7703],{"encoding":1912},"\\mathbf a^*\\mathbf X_t",[1841,7705,7707],{"className":7706,"ariaHidden":1907},[1917],[1841,7708,7710,7714,7744],{"className":7709},[1921],[1841,7711],{"className":7712,"style":7713},[1925],"height:0.8387em;vertical-align:-0.15em;",[1841,7715,7717,7720],{"className":7716},[1930],[1841,7718,7659],{"className":7719},[1930,3484],[1841,7721,7723],{"className":7722},[1938],[1841,7724,7726],{"className":7725},[1942],[1841,7727,7729],{"className":7728},[1947],[1841,7730,7733],{"className":7731,"style":7732},[1951],"height:0.6887em;",[1841,7734,7735,7738],{"style":3635},[1841,7736],{"className":7737,"style":1960},[1959],[1841,7739,7741],{"className":7740},[1964,1965,1966,1967],[1841,7742,4248],{"className":7743},[2033,1967],[1841,7745,7747,7750],{"className":7746},[1930],[1841,7748,7698],{"className":7749},[1930,3484],[1841,7751,7753],{"className":7752},[1938],[1841,7754,7756,7776],{"className":7755},[1942,1943],[1841,7757,7759,7773],{"className":7758},[1947],[1841,7760,7762],{"className":7761,"style":1952},[1951],[1841,7763,7764,7767],{"style":1955},[1841,7765],{"className":7766,"style":1960},[1959],[1841,7768,7770],{"className":7769},[1964,1965,1966,1967],[1841,7771,1873],{"className":7772},[1930,1934,1967],[1841,7774,1975],{"className":7775},[1974],[1841,7777,7779],{"className":7778},[1947],[1841,7780,7782],{"className":7781,"style":1982},[1951],[1841,7783],{}," is ",[1841,7786,7788,7821],{"className":7787},[1848],[1841,7789,7791],{"className":7790},[1852],[1854,7792,7793],{"xmlns":1856},[1859,7794,7795,7818],{},[1862,7796,7797,7803,7805,7807,7809,7811,7813,7816],{},[2680,7798,7799,7801],{},[1868,7800,7659],{"mathvariant":3430},[1875,7802,4248],{},[1868,7804,2102],{},[1875,7806,1890],{"stretchy":1889},[1868,7808,1893],{},[1875,7810,1904],{"stretchy":1889},[1868,7812,7659],{"mathvariant":3430},[1875,7814,7815],{},"≥",[2114,7817,3277],{},[1910,7819,7820],{"encoding":1912},"\\mathbf a^*f(\\omega)\\mathbf a\\ge0",[1841,7822,7824,7883],{"className":7823,"ariaHidden":1907},[1917],[1841,7825,7827,7830,7859,7862,7865,7868,7871,7874,7877,7880],{"className":7826},[1921],[1841,7828],{"className":7829,"style":2003},[1925],[1841,7831,7833,7836],{"className":7832},[1930],[1841,7834,7659],{"className":7835},[1930,3484],[1841,7837,7839],{"className":7838},[1938],[1841,7840,7842],{"className":7841},[1942],[1841,7843,7845],{"className":7844},[1947],[1841,7846,7848],{"className":7847,"style":7732},[1951],[1841,7849,7850,7853],{"style":3635},[1841,7851],{"className":7852,"style":1960},[1959],[1841,7854,7856],{"className":7855},[1964,1965,1966,1967],[1841,7857,4248],{"className":7858},[2033,1967],[1841,7860,2102],{"className":7861,"style":2138},[1930,1934],[1841,7863,1890],{"className":7864},[2018],[1841,7866,1893],{"className":7867,"style":2022},[1930,1934],[1841,7869,1904],{"className":7870},[2049],[1841,7872,7659],{"className":7873},[1930,3484],[1841,7875],{"className":7876,"style":1989},[1988],[1841,7878,7815],{"className":7879},[1993],[1841,7881],{"className":7882,"style":1989},[1988],[1841,7884,7886,7889],{"className":7885},[1921],[1841,7887],{"className":7888,"style":4033},[1925],[1841,7890,3277],{"className":7891},[1930],[1793,7893,7895],{"id":7894},"_8-filters-can-leak-the-future","8. Filters can leak the future",[1798,7897,7898],{},"A centred moving average",[1841,7900,7902],{"className":7901},[1844],[1841,7903,7905,7967],{"className":7904},[1848],[1841,7906,7908],{"className":7907},[1852],[1854,7909,7910],{"xmlns":1856,"display":1857},[1859,7911,7912,7964],{},[1862,7913,7914,7921,7923],{},[1865,7915,7916,7919],{},[1868,7917,7918],{},"m",[1868,7920,1873],{},[1875,7922,1877],{},[2639,7924,7925,7961],{},[1862,7926,7927,7939,7941,7947,7949],{},[1865,7928,7929,7931],{},[1868,7930,1870],{},[1862,7932,7933,7935,7937],{},[1868,7934,1873],{},[1875,7936,2664],{},[2114,7938,2192],{},[1875,7940,1898],{},[1865,7942,7943,7945],{},[1868,7944,1870],{},[1868,7946,1873],{},[1875,7948,1898],{},[1865,7950,7951,7953],{},[1868,7952,1870],{},[1862,7954,7955,7957,7959],{},[1868,7956,1873],{},[1875,7958,1898],{},[2114,7960,2192],{},[2114,7962,7963],{},"3",[1910,7965,7966],{"encoding":1912},"m_t=\\frac{x_{t-1}+x_t+x_{t+1}}{3}",[1841,7968,7970,8025],{"className":7969,"ariaHidden":1907},[1917],[1841,7971,7973,7976,8016,8019,8022],{"className":7972},[1921],[1841,7974],{"className":7975,"style":1926},[1925],[1841,7977,7979,7982],{"className":7978},[1930],[1841,7980,7918],{"className":7981},[1930,1934],[1841,7983,7985],{"className":7984},[1938],[1841,7986,7988,8008],{"className":7987},[1942,1943],[1841,7989,7991,8005],{"className":7990},[1947],[1841,7992,7994],{"className":7993,"style":1952},[1951],[1841,7995,7996,7999],{"style":1955},[1841,7997],{"className":7998,"style":1960},[1959],[1841,8000,8002],{"className":8001},[1964,1965,1966,1967],[1841,8003,1873],{"className":8004},[1930,1934,1967],[1841,8006,1975],{"className":8007},[1974],[1841,8009,8011],{"className":8010},[1947],[1841,8012,8014],{"className":8013,"style":1982},[1951],[1841,8015],{},[1841,8017],{"className":8018,"style":1989},[1988],[1841,8020,1877],{"className":8021},[1993],[1841,8023],{"className":8024,"style":1989},[1988],[1841,8026,8028,8032],{"className":8027},[1921],[1841,8029],{"className":8030,"style":8031},[1925],"height:1.9463em;vertical-align:-0.686em;",[1841,8033,8035,8038,8245],{"className":8034},[1930],[1841,8036],{"className":8037},[2018,2744],[1841,8039,8041],{"className":8040},[2639],[1841,8042,8044,8237],{"className":8043},[1942,1943],[1841,8045,8047,8234],{"className":8046},[1947],[1841,8048,8051,8062,8070],{"className":8049,"style":8050},[1951],"height:1.2603em;",[1841,8052,8053,8056],{"style":2760},[1841,8054],{"className":8055,"style":2764},[1959],[1841,8057,8059],{"className":8058},[1930],[1841,8060,7963],{"className":8061},[1930],[1841,8063,8064,8067],{"style":2776},[1841,8065],{"className":8066,"style":2764},[1959],[1841,8068],{"className":8069,"style":2784},[2783],[1841,8071,8072,8075],{"style":2787},[1841,8073],{"className":8074,"style":2764},[1959],[1841,8076,8078,8127,8130,8133,8136,8176,8179,8182,8185],{"className":8077},[1930],[1841,8079,8081,8084],{"className":8080},[1930],[1841,8082,1870],{"className":8083},[1930,1934],[1841,8085,8087],{"className":8086},[1938],[1841,8088,8090,8119],{"className":8089},[1942,1943],[1841,8091,8093,8116],{"className":8092},[1947],[1841,8094,8096],{"className":8095,"style":3522},[1951],[1841,8097,8098,8101],{"style":1955},[1841,8099],{"className":8100,"style":1960},[1959],[1841,8102,8104],{"className":8103},[1964,1965,1966,1967],[1841,8105,8107,8110,8113],{"className":8106},[1930,1967],[1841,8108,1873],{"className":8109},[1930,1934,1967],[1841,8111,2664],{"className":8112},[2033,1967],[1841,8114,2192],{"className":8115},[1930,1967],[1841,8117,1975],{"className":8118},[1974],[1841,8120,8122],{"className":8121},[1947],[1841,8123,8125],{"className":8124,"style":3611},[1951],[1841,8126],{},[1841,8128],{"className":8129,"style":2029},[1988],[1841,8131,1898],{"className":8132},[2033],[1841,8134],{"className":8135,"style":2029},[1988],[1841,8137,8139,8142],{"className":8138},[1930],[1841,8140,1870],{"className":8141},[1930,1934],[1841,8143,8145],{"className":8144},[1938],[1841,8146,8148,8168],{"className":8147},[1942,1943],[1841,8149,8151,8165],{"className":8150},[1947],[1841,8152,8154],{"className":8153,"style":1952},[1951],[1841,8155,8156,8159],{"style":1955},[1841,8157],{"className":8158,"style":1960},[1959],[1841,8160,8162],{"className":8161},[1964,1965,1966,1967],[1841,8163,1873],{"className":8164},[1930,1934,1967],[1841,8166,1975],{"className":8167},[1974],[1841,8169,8171],{"className":8170},[1947],[1841,8172,8174],{"className":8173,"style":1982},[1951],[1841,8175],{},[1841,8177],{"className":8178,"style":2029},[1988],[1841,8180,1898],{"className":8181},[2033],[1841,8183],{"className":8184,"style":2029},[1988],[1841,8186,8188,8191],{"className":8187},[1930],[1841,8189,1870],{"className":8190},[1930,1934],[1841,8192,8194],{"className":8193},[1938],[1841,8195,8197,8226],{"className":8196},[1942,1943],[1841,8198,8200,8223],{"className":8199},[1947],[1841,8201,8203],{"className":8202,"style":3522},[1951],[1841,8204,8205,8208],{"style":1955},[1841,8206],{"className":8207,"style":1960},[1959],[1841,8209,8211],{"className":8210},[1964,1965,1966,1967],[1841,8212,8214,8217,8220],{"className":8213},[1930,1967],[1841,8215,1873],{"className":8216},[1930,1934,1967],[1841,8218,1898],{"className":8219},[2033,1967],[1841,8221,2192],{"className":8222},[1930,1967],[1841,8224,1975],{"className":8225},[1974],[1841,8227,8229],{"className":8228},[1947],[1841,8230,8232],{"className":8231,"style":3611},[1951],[1841,8233],{},[1841,8235,1975],{"className":8236},[1974],[1841,8238,8240],{"className":8239},[1947],[1841,8241,8243],{"className":8242,"style":2806},[1951],[1841,8244],{},[1841,8246],{"className":8247},[2049,2744],[1798,8249,8250,8251,8337,8338,8367],{},"uses ",[1841,8252,8254,8278],{"className":8253},[1848],[1841,8255,8257],{"className":8256},[1852],[1854,8258,8259],{"xmlns":1856},[1859,8260,8261,8275],{},[1862,8262,8263],{},[1865,8264,8265,8267],{},[1868,8266,1870],{},[1862,8268,8269,8271,8273],{},[1868,8270,1873],{},[1875,8272,1898],{},[2114,8274,2192],{},[1910,8276,8277],{"encoding":1912},"x_{t+1}",[1841,8279,8281],{"className":8280,"ariaHidden":1907},[1917],[1841,8282,8284,8288],{"className":8283},[1921],[1841,8285],{"className":8286,"style":8287},[1925],"height:0.6389em;vertical-align:-0.2083em;",[1841,8289,8291,8294],{"className":8290},[1930],[1841,8292,1870],{"className":8293},[1930,1934],[1841,8295,8297],{"className":8296},[1938],[1841,8298,8300,8329],{"className":8299},[1942,1943],[1841,8301,8303,8326],{"className":8302},[1947],[1841,8304,8306],{"className":8305,"style":3522},[1951],[1841,8307,8308,8311],{"style":1955},[1841,8309],{"className":8310,"style":1960},[1959],[1841,8312,8314],{"className":8313},[1964,1965,1966,1967],[1841,8315,8317,8320,8323],{"className":8316},[1930,1967],[1841,8318,1873],{"className":8319},[1930,1934,1967],[1841,8321,1898],{"className":8322},[2033,1967],[1841,8324,2192],{"className":8325},[1930,1967],[1841,8327,1975],{"className":8328},[1974],[1841,8330,8332],{"className":8331},[1947],[1841,8333,8335],{"className":8334,"style":3611},[1951],[1841,8336],{},". It is legitimate for retrospective description but unavailable for a real-time forecast made at ",[1841,8339,8341,8354],{"className":8340},[1848],[1841,8342,8344],{"className":8343},[1852],[1854,8345,8346],{"xmlns":1856},[1859,8347,8348,8352],{},[1862,8349,8350],{},[1868,8351,1873],{},[1910,8353,1873],{"encoding":1912},[1841,8355,8357],{"className":8356,"ariaHidden":1907},[1917],[1841,8358,8360,8364],{"className":8359},[1921],[1841,8361],{"className":8362,"style":8363},[1925],"height:0.6151em;",[1841,8365,1873],{"className":8366},[1930,1934],". Every filter should be labelled:",[2275,8369,8370,8380],{},[2278,8371,8372],{},[2281,8373,8374,8377],{},[2284,8375,8376],{},"Filter use",[2284,8378,8379],{},"Allowed data",[2354,8381,8382,8390,8398],{},[2281,8383,8384,8387],{},[2359,8385,8386],{},"retrospective decomposition",[2359,8388,8389],{},"past and future observations, if stated",[2281,8391,8392,8395],{},[2359,8393,8394],{},"real-time monitoring",[2359,8396,8397],{},"current and past only",[2281,8399,8400,8403],{},[2359,8401,8402],{},"forecasting backtest",[2359,8404,8405],{},"exactly what was available at each historical origin",[1798,8407,8408],{},"Frequency response describes what a filter attenuates or passes; it does not excuse information leakage.",[1793,8410,8412],{"id":8411},"_9-spectral-audit","9. Spectral audit",[8414,8415,8416,8419,8422,8425,8428,8431,8434,8437],"ol",{},[1812,8417,8418],{},"State sampling interval, units, and missing-data treatment.",[1812,8420,8421],{},"Remove or model mean, trend, seasonality, and breaks deliberately.",[1812,8423,8424],{},"Name candidate periods before interpreting peaks when domain knowledge exists.",[1812,8426,8427],{},"Report taper and smoothing choices.",[1812,8429,8430],{},"Check peak stability across windows and subsamples.",[1812,8432,8433],{},"Compare the frequency-domain story with ACF and a time-domain model.",[1812,8435,8436],{},"For cross-spectra, state alignment, common seasonal drivers, and phase uncertainty.",[1812,8438,8439],{},"For forecasts, ensure all filters are one-sided at each origin.",[1793,8441,8443],{"id":8442},"practice","Practice",[8414,8445,8446,8449,8452,8455],{},[1812,8447,8448],{},"Convert a frequency of 0.2 cycles\u002Fday to period and angular frequency.",[1812,8450,8451],{},"Where does an hourly signal at 0.75 cycles\u002Fhour alias when sampled hourly?",[1812,8453,8454],{},"Explain why a periodogram peak near zero may reflect a break rather than a genuine long cycle.",[1812,8456,8457],{},"Why does smoothing reduce variance but blur nearby frequencies?",[8459,8460,8462],"legacy-details",{"title":8461},"Answers",[8414,8463,8464,8502,8585,8588],{},[1812,8465,8466,8467,8501],{},"Period 5 days; angular frequency ",[1841,8468,8470,8486],{"className":8469},[1848],[1841,8471,8473],{"className":8472},[1852],[1854,8474,8475],{"xmlns":1856},[1859,8476,8477,8483],{},[1862,8478,8479,8481],{},[2114,8480,6270],{},[1868,8482,2119],{},[1910,8484,8485],{"encoding":1912},"0.4\\pi",[1841,8487,8489],{"className":8488,"ariaHidden":1907},[1917],[1841,8490,8492,8495,8498],{"className":8491},[1921],[1841,8493],{"className":8494,"style":4033},[1925],[1841,8496,6270],{"className":8497},[1930],[1841,8499,2119],{"className":8500,"style":2022},[1930,1934]," radians\u002Fday.",[1812,8503,8504,8584],{},[1841,8505,8507,8535],{"className":8506},[1848],[1841,8508,8510],{"className":8509},[1852],[1854,8511,8512],{"xmlns":1856},[1859,8513,8514,8532],{},[1862,8515,8516,8518,8521,8523,8525,8527,8529],{},[1868,8517,4580],{"mathvariant":2109},[2114,8519,8520],{},"0.75",[1875,8522,2664],{},[2114,8524,2192],{},[1868,8526,4580],{"mathvariant":2109},[1875,8528,1877],{},[2114,8530,8531],{},"0.25",[1910,8533,8534],{"encoding":1912},"|0.75-1|=0.25",[1841,8536,8538,8557,8575],{"className":8537,"ariaHidden":1907},[1917],[1841,8539,8541,8544,8548,8551,8554],{"className":8540},[1921],[1841,8542],{"className":8543,"style":2003},[1925],[1841,8545,8547],{"className":8546},[1930],"∣0.75",[1841,8549],{"className":8550,"style":2029},[1988],[1841,8552,2664],{"className":8553},[2033],[1841,8555],{"className":8556,"style":2029},[1988],[1841,8558,8560,8563,8566,8569,8572],{"className":8559},[1921],[1841,8561],{"className":8562,"style":2003},[1925],[1841,8564,6310],{"className":8565},[1930],[1841,8567],{"className":8568,"style":1989},[1988],[1841,8570,1877],{"className":8571},[1993],[1841,8573],{"className":8574,"style":1989},[1988],[1841,8576,8578,8581],{"className":8577},[1921],[1841,8579],{"className":8580,"style":4033},[1925],[1841,8582,8531],{"className":8583},[1930]," cycles\u002Fhour.",[1812,8586,8587],{},"A sudden level shift contributes strong low-frequency variation; stationarity is violated.",[1812,8589,8590],{},"Averaging neighbouring ordinates stabilises random variation but combines signals that are close in frequency.",[1793,8592,8594],{"id":8593},"takeaway","Takeaway",[1798,8596,8597,8598,8602,8603,8607],{},"Spectral analysis is a unitary change of coordinates for second-order structure. Sampling, stationarity, smoothing, and phase assumptions determine whether that change of coordinates is meaningful. Consolidate the matrix computations in the ",[7659,8599,8601],{"href":8600},"..\u002F07-r-implementation\u002F","R laboratory","; the ",[7659,8604,8606],{"href":8605},"..\u002F08-python-implementation\u002F","Python laboratory"," is an optional implementation contrast.",{"title":10,"searchDepth":8609,"depth":8609,"links":8610},2,[8611,8612,8613,8614,8618,8619,8620,8621,8622,8623,8624,8625,8626],{"id":1795,"depth":8609,"text":1796},{"id":1803,"depth":8609,"text":1804},{"id":1835,"depth":8609,"text":1836},{"id":2608,"depth":8609,"text":2609,"children":8615},[8616],{"id":3411,"depth":8617,"text":3412},3,{"id":4819,"depth":8609,"text":4820},{"id":5398,"depth":8609,"text":5399},{"id":6215,"depth":8609,"text":6216},{"id":6235,"depth":8609,"text":6236},{"id":6349,"depth":8609,"text":6350},{"id":7894,"depth":8609,"text":7895},{"id":8411,"depth":8609,"text":8412},{"id":8442,"depth":8609,"text":8443},{"id":8593,"depth":8609,"text":8594},"Translate autocovariance into frequency, estimate spectra, diagnose cycles, and avoid aliasing and filtering leakage.","md",{"sidebar":8630},{"order":8631},6,true,{"title":989,"description":8627},"JgL-WiSyYC_2uNjKit7wtNSonmwCXOMlhzWhN-xP48E",[8636,8638],{"title":983,"path":984,"stem":985,"description":8637,"children":-1},"Express vector autoregressions, seasonal operators, cointegration, and forecast constraints with block matrices and stable linear systems.",{"title":995,"path":996,"stem":997,"description":8639,"children":-1},"Four browser-based base R laboratories for covariance matrices, AR recursions, linear prediction, likelihood, and state space.",1785754738393]