Finite Element Model Updating

Finite Element Model Updating and Calibration

Model updating
Model calibration
FEM updating

Structural dynamics and model calibration

Finite Element Model (FEM) Updating is the process of refining a numerical structural model to match measured behavior (modal properties, vibrations, static displacements) from the physical structure. By comparing computational predictions to experimental data and systematically adjusting uncertain model parameters (material properties, boundary conditions, joint stiffness), engineers create more accurate models for design decisions, damage detection, and life prediction. Formalized by Mottershead and Friswell, FEM updating bridges the gap between idealized computer models and real-world structures.

Mottershead, J. E., Link, M., & Friswell, M. I. (2011). The sensitivity method in finite element model updating: A tutorial. Mechanical Systems and Signal Processing, 25(7), 2275-2296.

Friswell, M. I., & Mottershead, J. E. (2001). Finite Element Model Updating in Structural Dynamics. Kluwer Academic Publishers.

Yang, Q. W., & Yang, B. (1999). Model updating using response surface method. Journal of Sound and Vibration, 221(4), 555-569.

Schlune, H., Plos, M., & Gylltoft, K. (2009). Improved bridge evaluation through finite element model updating using static and dynamic measurements. Journal of Bridge Engineering, 14(7), 504-515.

topology-optimization first-order-reliability-method second-order-reliability-method rainflow-counting


Finite Integration Technique

Finite Integration Technique for Electromagnetic Field Simulation

The Finite Integration Technique (FIT) is a numerical method for solving Maxwell equations on structured grids, formulating electromagnetics as a system of integral equations over grid cells. Introduced by Thomas Weiland in 1977, FIT bridges finite differences and finite elements, offering excellent accuracy, stability, and computational efficiency for a wide range of electromagnetic problems. FIT is the foundation of commercial solvers like CST Microwave Studio and is widely used in RF, microwave, and EMC engineering.

Discrete space-time integration method for Maxwell equations

Weiland, T. (1977). A new method for the solution of Maxwell's equations. Zeitschrift für Naturforschung, 31(7), 861-873.

Clemens, M., & Weiland, T. (2001). Discrete electromagnetism with the finite integration technique. Progress in Electromagnetics Research, 32, 65-87.

Weiland, T. (1996). Time domain electromagnetic field computation with finite difference methods. International Journal of Numerical Modelling, 9(4), 295-319.


Finite Strip Method

Finite Strip Method for Structural Analysis

FSM
Strip method
Semi-analytical finite element

The finite strip method (FSM) is a semi-analytical numerical approach for analyzing prismatic or cylindrical structures by dividing them into strips in one direction and using analytical or exact solutions in the perpendicular direction. Developed by Cheung in 1976, FSM reduces computational cost and often provides superior accuracy for structures with regular geometry along one axis.

Reduced-dimension numerical method for prismatic structures

Cheung, Y. K. (1976). Finite Strip Method in Structural Analysis. Pergamon Press.

Cheung, M. S., & Cheung, Y. K. (1996). Refined finite strip method for thin-wall structures with columns. Journal of Structural Engineering, 122(6), 700-712.

Lau, D. T., & Cheung, Y. K. (2000). Vibration of cylinders and plates with variable thickness. Earthquake Engineering & Structural Dynamics, 29(3), 377-394.


Finite-Time Thermodynamics

Finite-Time Thermodynamics for Real Thermal Processes

Finite-Time Thermodynamics (FTT) relaxes the classical assumption that thermodynamic processes occur reversibly (infinitely slowly). Instead, it analyzes real thermal systems operating at finite rates with irreversibilities. FTT reveals fundamental trade-offs: to complete a process quickly requires accepting large irreversibilities and low efficiency, while slow operation achieves high efficiency but requires impractical time and cost.

Bejan, A. (1996). Entropy Generation Minimization. CRC Press.

Rubin, M. H. (1979). Optimal paths for a car that minimizes fuel consumption. Physical Review A, 19(3), 1272-1278.


FIR Filter Design

FIR Design
Finite impulse response
Non-recursive filter design

Finite Impulse Response (FIR) filters are digital filters with an impulse response that settles to zero in finite time, making them fundamentally stable and easy to analyze. Unlike their IIR counterparts, FIR filters are inherently stable, can have exactly linear phase response, and are widely used in applications from audio processing to telecommunications where phase distortion must be minimized.

Parks, T. W., & Burrus, C. S. (1987). Digital Filter Design. John Wiley & Sons.

Oppenheim, A. V., Schafer, R. W., & Buck, J. R. (1999). Discrete-Time Signal Processing (2nd ed.). Prentice Hall.

iir-filter-design butterworth-filter-design wiener-filter matched-filter


Fire Weather Index

The Fire Weather Index (FWI) System, developed by the Canadian Forest Service, is a comprehensive weather-based fire danger rating system consisting of six component indices and an overall Fire Weather Index. It uses daily weather observations (temperature, relative humidity, wind speed, and precipitation) to estimate fine-fuel moisture, fire behavior, and risk. The FWI System is used operationally across Canada, many U.S. states, and internationally for fire management decisions and fire danger forecasting.

Van Wagner, C. E. (1987). Development and structure of the Canadian Forest Fire Weather Index System. Canadian Forestry Service Publication 1333.

Van Wagner, C. E., & Pickett, T. L. (1989). Equations and FORTRAN program for the Canadian Forest Fire Weather Index System. Canadian Forestry Service Research Notes No. 5.


Firefly Algorithm

Ateşböceği Algoritması (Firefly Algorithm)

The Firefly Algorithm (FA), introduced by Xin-She Yang in 2008 and formally published in 2010, is a nature-inspired swarm metaheuristic that models the bioluminescent attraction behaviour of fireflies. Each candidate solution is a firefly whose brightness represents its objective-function value; dimmer fireflies move toward brighter ones with an attraction force that decays with distance, driving the swarm toward optima without gradient information.

Bioluminescent flashing behaviour of fireflies

Attraction proportional to brightness (fitness), attenuated by distance

Attraction coefficient β₀, light absorption γ

Natural subgroup formation enables simultaneous discovery of multiple optima

Yang, X.S. (2010). Firefly Algorithm, Stochastic Test Functions and Design Optimisation. International Journal of Bio-Inspired Computation, 2(2), 78-84.

Yang, X.S. (2014). Nature-Inspired Optimization Algorithms. Elsevier.

particle-swarm-optimization genetic-algorithm cuckoo-search differential-evolution grey-wolf-optimizer


First-Click Testing

First-Click Testing is a rapid, quantitative method for evaluating whether users click on the correct element to start a task on a web page or screen. Users view a screenshot or live page and are asked to click where they would start a specific task. The test measures success rate (correct first click) and records which elements are commonly misclicked. Unlike tree testing (text-only navigation), first-click testing preserves visual design, isolating navigation labeling and visual information architecture in realistic context.

Click-based navigation evaluation in realistic visual context

Quirkstudio. (2014). First Click Testing: User Research for Navigation. Quirkstudio White Paper.

Kath, R. (2015). First click testing: A quick method for evaluating user experience. UX Magazine, 1225.

tree-testing card-sorting heuristic-evaluation first-click-testing


First-Difference Estimator

FD Estimator
First-Difference Panel Estimator
First-Difference OLS
Birinci Fark Tahmincisi

The First-Difference (FD) estimator is a panel data method that eliminates unobserved, time-invariant individual heterogeneity by subtracting each unit's observation in period t-1 from its observation in period t. By operating on changes rather than levels, FD removes any fixed individual effect that would otherwise confound causal inference. It is widely used in labor economics, program evaluation, and applied microeconomics whenever researchers suspect persistent unobserved differences across individuals, firms, or countries.

Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.


First-Order Reliability Method

The First-Order Reliability Method (FORM) is a probabilistic technique for estimating the probability of structural failure given uncertain input parameters. Developed by Allin Cornell in 1969 and refined by Hasofer and Lind in 1974, FORM provides a computationally efficient approximation to the true failure probability by linearizing the limit-state function at the most probable failure point. It has become the cornerstone of modern structural reliability analysis and risk-based design.

Cornell, C. A. (1969). A probability-based structural code. Journal of the American Concrete Institute, 66(12), 974-985.

Hasofer, A. M., & Lind, N. C. (1974). Exact and invariant second-moment code format. Journal of the Engineering Mechanics Division, 100(1), 111-121.

Rackwitz, R., & Fiessler, B. (1978). Structural reliability under combined random load sequences. Computers & Structures, 9(5), 489-494.

Melchers, R. E. (2002). Structural Reliability Analysis and Prediction (2nd ed.). John Wiley & Sons.

second-order-reliability-method rainflow-counting highly-accelerated-life-testing response-surface-desirability-function


First-Price Auction

FPSB
Sealed-Bid Auction
Bid-Equal-Price Auction

A first-price auction is a sealed-bid mechanism where all participants submit bids simultaneously without knowing others' bids. The highest bidder wins and pays their own bid (the price they offered). Systematically analyzed by William Vickrey in 1961, first-price auctions require bidders to balance between winning and profit, leading to strategic underbidding relative to true valuations in equilibrium.

Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed bids. The Journal of Finance, 16(1), 8-37.

Krishna, V. (2009). Auction Theory (Second Edition). Academic Press.

vcg-mechanism bayesian-nash-equilibrium stackelberg-competition cournot-competition


Fisher Panel Unit-Root Test

Maddala-Wu (Fisher-type) Panel Unit-Root Test

Maddala-Wu Test
Fisher-type Panel Unit-Root Test
MW Panel Unit-Root Test
Fisher Panel Birim Kök Testi

The Fisher-type (Maddala-Wu) panel unit-root test, introduced in 1999, combines individual-level ADF unit-root p-values using Fisher's chi-squared meta-analytic framework to produce a single panel-level test statistic. Unlike the Levin-Lin-Chu approach, it does not impose a common autoregressive parameter across cross-sections, making it a natural choice for heterogeneous panels in macroeconomics, finance, and regional economics.

Nonparametric combination-of-p-values panel unit-root test

Maddala, G. S., & Wu, S. (1999). A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics, 61(S1), 631–652.


Fisher's exact test

Fisher-Irwin test
exact test of independence
Fisher'ın Kesin Testi

Fisher's exact test is a nonparametric exact-probability test of independence for small-sample contingency tables, introduced by R. A. Fisher in 1922. Rather than relying on a large-sample approximation, it computes the exact probability of the observed table directly from the hypergeometric distribution.

Exact test of independence for categorical data

2 categorical variables (2×2 or r×c table)

Hypergeometric (exact null distribution)

Fisher, R. A. (1922). On the interpretation of chi-squared from contingency tables, and the calculation of P. Journal of the Royal Statistical Society, 85(1), 87–94.

chi-square-test-of-independence mcnemar-test cochran-q-test g-test


Fitts's Law

Fitts's Law is an empirical model of human rapid aimed movement, predicting that movement time increases logarithmically with the ratio of distance to target size. Formulated by Paul Fitts in 1954, this fundamental law describes how long it takes to move to and select a target (e.g., clicking a button on a screen or reaching a physical object). In human-computer interaction, Fitts's Law is widely applied to evaluate and optimize pointer-based interfaces such as mice, touchpads, and touch screens.

Empirical model of human movement time as function of distance and target size

Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47(6), 381–391.

MacKenzie, I. S. (1992). Fitts's law as a research and design tool in human-computer interaction. Human-Computer Interaction, 7(1), 91–139.

klm-goms heuristic-evaluation think-aloud-protocol first-click-testing


Five Facet Mindfulness Questionnaire

Ruth A. Baer, Greg T. Smith, and colleagues

The Five Facet Mindfulness Questionnaire (FFMQ) is a 39-item self-report instrument designed to measure trait mindfulness across five distinct dimensions: Observing, Describing, Acting with Awareness, Non-judging of Inner Experience, and Non-reactivity to Inner Experience. Developed by Baer and colleagues in 2006 and published in Assessment, the FFMQ has become one of the most widely used multidimensional mindfulness measures in research and clinical practice, applicable to both meditation practitioners and general populations.

Baer, R. A., Smith, G. T., Hopkins, J., Krietemeyer, J., & Toney, L. (2006). Using self-report assessment methods to explore facets of mindfulness. Assessment, 13(1), 27-45.


Fixed Effects Model

FE model
within estimator
least squares dummy variable
LSDV regression

Mundlak (1978); Nerlove (1971); classical panel econometrics

The fixed effects (FE) model is the workhorse estimator for panel data when unobserved unit-specific characteristics are suspected to correlate with the regressors. By absorbing each entity's time-invariant heterogeneity into a separate intercept, FE isolates the causal effect of within-unit variation and eliminates omitted-variable bias from time-constant confounders.

Balanced or unbalanced panel data (cross-section × time)

Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6th ed.). Springer.

Mundlak, Y. (1978). On the pooling of time series and cross section data. Econometrica, 46(1), 69–85.

random-effects-model panel-data-analysis panel-hausman-test dynamic-panel-data-model ols-regression arellano-bond-gmm-estimator


Fixed Effects Panel Model

Fixed Effects Panel Data Model (Within Estimator)

within estimator
panel fixed effects
entity fixed effects model
Panel Sabit Etkiler Modeli

Baltagi (textbook treatment); Hausman test for FE vs RE choice

The fixed effects panel model estimates relationships in panel data (many units observed over time) by exploiting only the within-unit variation, so that unobserved time-invariant heterogeneity is controlled away. It is the central within estimator developed in Baltagi's Econometric Analysis of Panel Data (2005), and the choice between it and the random effects model is settled by the Hausman (1978) test.

Within (fixed effects) estimator with cluster-robust standard errors

Hausman, J. A. (1978). Specification Tests in Econometrics. Econometrica, 46(6), 1251–1271.

Baltagi, B. H. (2005). Econometric Analysis of Panel Data (3rd ed.). Wiley.

random-effects-panel ols-regression instrumental-variables difference-in-differences system-gmm


FLACC Behavioral Pain Scale

behavioral pain assessment for nonverbal patients

The FLACC Behavioral Pain Scale (Face, Legs, Activity, Cry, Consolability) is a 5-item observational tool developed by Merkel and Voepel-Lewis in 1997 to assess acute pain in children ages 2 months to 7 years who are unable to self-report pain. Each of the five behavioral domains is scored 0-2, yielding a total score of 0-10. The FLACC is widely used in pediatric hospitals, recovery rooms, and intensive care units for postoperative and acute pain assessment.

Behavioral observation scale for acute pain in children and nonverbal patients

Merkel, S.I., Voepel-Lewis, T., Shayevitz, J.R., & Malviya, S. (1997). The FLACC: A behavioral scale for scoring postoperative pain in young children. Pediatric Nursing, 23(3), 293-297.

Voepel-Lewis, T., Zanotti, J., Dammeyer, J.A., & Merkel, S. (2010). Reliability and validity of the face, legs, activity, cry, consolability behavioral tool in assessing acute pain in critically ill patients. American Journal of Critical Care, 11(1), 12-20.

Nilsson, S., Enskär, K., & Kling, A.M. (2008). Postoperative pain and behavioral responses after instructive and supportive telephone calls after surgery. Journal for Specialists in Pediatric Nursing, 13(1), 42-52.


Fleiss' Kappa

Fleiss' Kappa for Multiple Rater Agreement

Fleiss' Kappa (Çoklu Değerlendirici Uyumu)

Fleiss' Kappa is a non-parametric statistic for measuring the degree of agreement among three or more raters who classify items into mutually exclusive nominal categories. Introduced by Joseph L. Fleiss in 1971 as a generalization of Cohen's Kappa beyond two raters, it corrects observed agreement for the level of agreement expected by chance alone, making it the standard reliability index in medical diagnosis studies, content analysis, and multi-coder research.

Fleiss, J.L. (1971). Measuring Nominal Scale Agreement Among Many Raters. Psychological Bulletin, 76(5), 378–382.

cohens-kappa intraclass-correlation krippendorffs-alpha weighted-kappa reliability-analysis


Royston-Parmar Model

Flexible Parametric Survival Model (Royston-Parmar)

Esnek Parametrik Survival Modeli (Royston-Parmar)

The Royston-Parmar model, introduced by Royston and Parmar in 2002, is a modern parametric approach to survival analysis that replaces the rigid distributional assumptions of classical models with a restricted cubic spline fitted to the log-cumulative-hazard scale. It combines the interpretability of a fully parametric model with the flexibility to capture non-standard hazard shapes, and it supports proportional-hazards, accelerated failure-time, and proportional-odds link functions.

Restricted cubic spline on log-cumulative hazard

Proportional-hazards, accelerated failure-time, proportional-odds

Royston, P. & Parmar, M.K.B. (2002). Flexible Parametric Proportional-Hazards and Proportional-Odds Models for Censored Survival Data, with Application to Prognostic Modelling and Estimation of Treatment Effects. Statistics in Medicine, 21(15), 2175–2197.

cox-regression weibull-regression kaplan-meier log-rank-test fine-gray-competing-risks accelerated-failure-time bayesian-survival cure-model


Fligner-Killeen Test

Fligner-Killeen Test for Homogeneity of Variances

Fligner-Killeen test of variance homogeneity

Fligner-Killeen Varyans Homojenliği Testi

The Fligner-Killeen test is a rank-based test that checks whether several independent groups share the same variance (scale). Introduced by Fligner and Killeen in 1976, it does not require the data to be normally distributed, making it a robust nonparametric alternative to the Levene and Bartlett tests.

Rank-based test for homogeneity of variances

Chi-square statistic on rank-transformed absolute deviations

Fligner, M. A., & Killeen, T. J. (1976). Distribution-Free Two-Sample Tests for Scale. Journal of the American Statistical Association, 71(353), 210-213.

Conover, W. J., Johnson, M. E., & Johnson, M. M. (1981). A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data. Technometrics, 23(4), 351-361.

levene-brown-forsythe bartlett-test mood-median-test kolmogorov-smirnov-2sample conover-iman-test


Floor and Ceiling Effect

Assessment of Floor and Ceiling Effects in Psychometric Scale Validity and Responsiveness

Floor effect
Ceiling effect
Psychometric floor effect
Measurement floor

Floor and ceiling effects are psychometric phenomena in which a disproportionately large proportion of respondents achieve the lowest (floor) or highest (ceiling) possible score on a measurement scale. These effects compromise scale reliability and responsiveness, limiting the instrument's ability to distinguish among respondents and detect meaningful change over time. Systematic assessment of floor and ceiling effects is essential for evaluating the psychometric adequacy of health-related quality-of-life scales, functional status measures, and other patient-reported outcomes.

McHorney, C. A. (2000). Ten recommendations for measuring health status. Health-Related Quality of Life Outcomes, 2(1), 1-5.

Terwee, C. B., Bot, S. D., de Bats, M. R., van der Windt, D. A., Knol, D. L., Dekker, J., Bouter, L. M., & de Vet, H. C. (2007). Quality criteria for measurement properties of health status questionnaires. Journal of Clinical Epidemiology, 60(1), 34-42.

Coon, C. D., & Cappelleri, J. C. (2016). Quantifying ceiling and floor effects in the Quality of Life after Brain Injury (QOLIBRI) scale. Health and Quality of Life Outcomes, 14(1), 135.


Flotation Kinetics

Batch Flotation Model
Flotation Rate Constants
Kinetic Flotation Analysis

Flotation kinetics is the study of how recovery of minerals from ore changes over time during flotation. The Garcia-Zuniga model, introduced in 1935, describes recovery as a first-order kinetic process with rate constant k and maximum recoverable fraction R∞. This simple model underpins flotation cell design and process optimization, enabling engineers to predict flotation performance from batch tests and scale results to industrial circuits.

First-order kinetic model for flotation recovery

Garcia-Zuniga, H. (1935). Uber eine neue Methode, zur Berechnung der Flotationsausbeute. Zeitschrift fur Praktische Geologie, 43(2), 12-19.

Kelsall, G. H., Stewart, P. S., & Yianatos, I. B. (1975). Flotation kinetics. Minerals Engineering, 1(5), 375-393.


Flourishing Scale

The Flourishing Scale (FS) is an 8-item measure of human flourishing developed by Diener and colleagues in 2010. It assesses psychological well-being across core dimensions including purpose, social connection, competence, and engagement. The scale operationalizes Aristotle's concept of eudaimonia—the realization of human potential—and provides researchers with a brief, validated tool for quantifying overall psychological flourishing independent of life satisfaction or mood.

Diener, E., Wirtz, D., Tov, W., Kim-Prieto, C., Choi, D. W., Oishi, S., & Biswas-Diener, R. (2010). New well-being measures: Short scales to assess flourishing and positive and negative feelings. Social Indicators Research, 97(2), 143–156.

who-5-wellbeing-index meaning-in-life-questionnaire perma-scale positive-mental-health-scale


Flow at Work Scale

Flow State Scale (FSS) - Adapted for Work

Positive occupational psychology and engagement

Mihaly Csikszentmihalyi (flow theory); Arnold B. Bakker (work-related flow scale)

The Flow at Work Scale (derived from Csikszentmihalyi's flow theory and operationalized by Bakker as the Work-Related Flow Inventory) measures the degree to which employees experience 'flow'—a state of optimal absorption, focus, and enjoyment in work. Flow is characterized by full concentration, loss of self-consciousness, sense of control, and intrinsic motivation. Developed initially in sports psychology and later adapted for occupational settings, the Flow at Work Scale captures positive engagement and is associated with high performance, creativity, and psychological wellbeing.

Csikszentmihalyi, M. (1990). Flow: The Psychology of Optimal Experience. New York: Harper & Row.

Jackson, S. A., Kimiecik, J. C., Ford, S. K., & Marsh, H. W. (1998). Psychological correlates of flow in sport. Journal of Sport & Exercise Psychology, 20(4), 358-378.

Bakker, A. B. (2008). The work-related flow inventory: Construction and initial validation. Journal of Happiness Studies, 9(3), 107-118.

recovery-experience-questionnaire areas-of-worklife-scale copenhagen-burnout-inventory effort-reward-imbalance-scale presenteeism-scale


Flow Cytometry

FACS
fluorescence-activated cell sorting
cell analysis

Flow cytometry is a laser-based technology for analyzing and sorting individual cells based on fluorescent markers. Developed by Leonard Herzenberg in the 1970s, flow cytometry enables rapid assessment of cell phenotype, drug effects on cell populations, and therapeutic cell characterization in immunology and hematology.

Herzenberg, L. A., Parks, D., Sahaf, B., Perez, O., Roederer, M., & Herzenberg, L. A. (2002). The history and future of the fluorescence-activated cell sorter and flow cytometry: a view from Stanford. Clinical Chemistry, 48(10), 1819-1827.

Verschoor, C. P., Lelic, A., Bramson, J. L., & Bowdish, D. M. (2015). An introduction to automated flow cytometry gating tools and their implementation. Frontiers in Immunology, 6, 380.


Flow Injection Analysis

FIA
sequential injection analysis
SIA
flow-based analysis

Flow injection analysis is an automated continuous-flow technique that rapidly injects a sample plug into a flowing stream of carrier solution, where it mixes with reagents and is detected online before reaching the detector. Developed by Jaromir Ruzicka and Elo Hansen in 1975, FIA revolutionized analytical chemistry by enabling rapid, high-throughput analysis with minimal reagent consumption and waste. Flow injection analysis is widely used in pharmaceutical, food, environmental, and clinical laboratories for routine quantitative analysis.

Ruzicka, J., & Hansen, E. H. (1979). Flow injection analysis: Part 1. A new concept of fast continuous flow analysis. Analytica Chimica Acta, 106, 207–224.

Miro, M., & Rojas, S. (Eds.). (2012). Advances in Flow-Based Analytical Techniques. Transworld Research Network.

Wang, J. (1994). Electrochemical detection for flow-based analytical techniques. Journal of Chromatography B, 659(1), 3–13.

ion-chromatography uv-vis-spectrophotometry potentiometric-titration coulometry voltammetry


FLOWSORT

FLOWSORT (Flow-Based Sorting Method) is a sorting multi-criteria decision-making (MCDM) method introduced by Nemery, P. Lamboray, C. in 2008. It turns a decision matrix of alternatives scored on multiple criteria into a structured, reproducible result.

PROMETHEE net flows extended to sorting via limiting/central reference profiles

Nemery, P., Lamboray, C. (2008). FlowSort: a flow-based sorting method with limiting or central profiles. 4OR


Fluid Balance Monitoring

Fluid Intake and Output Balance Monitoring Protocol

I&O Monitoring
Fluid Assessment
Hydration Status Assessment

Clinical nursing and medical practice standard

Fluid Balance Monitoring is a systematic nursing process for tracking and comparing fluid intake and output to maintain adequate hydration and detect abnormalities in fluid status. By measuring all sources of fluid intake (oral, intravenous, enteral feeding) and all routes of fluid loss (urine, feces, perspiration, drainage), clinicians can assess overall fluid balance and identify dehydration or fluid overload. This monitoring is essential for patients with conditions affecting fluid regulation or those requiring precise intake-output tracking.

Scales, K., & Pilsworth, J. (2008). The importance of fluid balance in clinical practice. Nursing Standard, 22(47), 50-57.

Mentes, J. C. (2006). A typology of oral hydration problems exhibited by frail nursing home residents. Journal of Gerontological Nursing, 26(1), 12-19.

braden-scale nursing-sensitive-indicators early-warning-score patient-fall-risk-assessment


FMEA

FMEA (Failure Mode and Effects Analysis) is a ranking multi-criteria decision-making (MCDM) method introduced by Stamatis, D. H. in 1995. It turns a decision matrix of alternatives scored on multiple criteria into a structured, reproducible result.

Risk priority via product of O·S·D ratings

Stamatis, D. H. (1995). Failure Mode and Effect Analysis: FMEA from Theory to Execution. ASQ Quality Press

ahp anp bwm bwm-bayesian ccsd cilos cimas critic


FMOLS Estimator

fully modified OLS
Phillips-Hansen FMOLS
Tam Düzeltilmiş OLS (FMOLS)

Phillips & Hansen (time series); Pedroni (heterogeneous panels)

Fully Modified OLS, introduced by Phillips and Hansen (1990), estimates the long-run coefficients of a cointegrating relationship among I(1) variables. It applies a semi-parametric correction to ordinary least squares to remove the bias that endogeneity and serial correlation otherwise induce in cointegrated time series or panel data.

Semi-parametrically corrected least squares

I(1) cointegrated time series or panel data

Phillips, P. C. B. & Hansen, B. E. (1990). Statistical Inference in Instrumental Variables Regression with I(1) Processes. Review of Economic Studies, 57(1), 99–125.

Pedroni, P. (2001). Fully Modified OLS for Heterogeneous Cointegrated Panels. Advances in Econometrics, 15, 93–130.

dols-estimator panel-cointegration ccemg-estimator ols-regression ardl-bounds-test


fNIRS Analysis

Functional Near-Infrared Spectroscopy (fNIRS) Analysis

Functional Near-Infrared Spectroscopy (fNIRS) is an optical neuroimaging method that measures changes in cerebral blood oxygenation non-invasively from the scalp. Developed by Britton Chance and colleagues in the 1990s, fNIRS combines the portability and cost-effectiveness of EEG with the spatial localization advantage of fMRI, enabling brain activity measurement in naturalistic settings.

Hemodynamic functional neuroimaging pipeline

Villringer, A., & Dirnagl, U. (1995). Coupling of brain activity and cerebral blood flow: basis of functional neuroimaging. Cerebrovascular and Cerebral Blood Flow Metabolism, 4, 3–22.

Kop, B. R., Ascoli, G. A., & Ances, B. M. (2014). fNIRS imaging of the prefrontal cortex during a language task. Neuroimage, 102, 844–852.


Focal Animal Sampling

Focal Animal Sampling Behavioral Observation Method

Focal Animal Sampling (FAS) is a systematic observational method in which an observer focuses on one individual animal at a time, recording its behavior continuously or at regular intervals for a fixed period. Introduced by Jeanne Altmann in 1974, FAS provides detailed, quantitative ethograms of individual behavior, making it essential for studying animal behavioral ecology, welfare, and responses to environmental changes.

Altmann, J. (1974). Observational study of behavior: sampling methods. Behaviour, 49(3-4), 227-267.

Martin, P., & Bateson, P. P. (1993). Measuring Behaviour: An Introductory Guide (2nd ed.). Cambridge University Press.

Lehner, P. N. (1996). Handbook of Ethological Methods (2nd ed.). Cambridge University Press.


Focus Group Methodology

Focus Group Discussion for Qualitative Data Collection

FGD
focus group discussion
group interview

Focus group discussions are a qualitative research method in which a trained moderator guides a small group (typically 6–12 participants) through structured or semi-structured discussion of a specific topic or product. Developed by Merton and Lazarsfeld in the 1950s for market research, focus groups are now widely used in health sciences, education, social sciences, and policy research. The method leverages group interaction to generate rich, contextual insights that individual interviews may not reveal.

Krueger, R. A. (1994). Focus Groups: A Practical Guide for Applied Research. SAGE Publications.

Morgan, D. L. (1997). Focus groups as qualitative research. Qualitative Research Methods Series, 16. SAGE Publications.

Kitzinger, J. (1994). The methodology of focus groups: The importance of interaction between research participants. Sociology of Health & Illness, 16(1), 103-121.

Wilkinson, S. (2004). Focus groups: A feminist method. Psychology of Women Quarterly, 25(4), 287-298.

in-depth-interview-method participant-observation qualitative-synthesis-methods member-checking


Focus Group

focus group discussion
FGD
group interview
Odak Grup Araştırması

1940s (sociological origin); modern applied form from the 1980s–1990s

Robert K. Merton (sociological precursor, 1940s); popularised in applied research by Richard A. Krueger

Focus group research is a qualitative data-collection method in which a trained moderator guides structured discussions with homogeneous groups of six to ten participants to explore ideas, attitudes, and perceptions on a defined topic. Developed from sociological roots in the 1940s and systematised for applied research by Krueger and Casey, the method leverages group interaction as a data source — revealing not just what people think, but how they negotiate and articulate views in a social setting.

Output: Thematic transcripts, emergent categories, and group-level insight into attitudes and perceptions

Krueger, R.A. & Casey, M.A. (2014). Focus Groups: A Practical Guide for Applied Research (5th ed.). Sage.

in-depth-interview ethnography delphi-method thematic-analysis case-study grounded-theory


Focused Ethnography

problem-focused ethnography
short-term ethnography
rapid ethnography
focused field study

Late 1990s–early 2000s (Knoblauch's systematic account, 2005)

Hubert Knoblauch (theorised and named); antecedents in applied medical and organisational ethnography

Focused ethnography is a condensed, problem-centred variant of classical ethnography in which a researcher with prior domain knowledge enters a specific social setting for a bounded period — typically days to weeks rather than months or years — to study one clearly defined issue or practice. Developed as a response to the time and resource constraints of applied research, it is widely used in healthcare, organisational studies, and professional education, where the researcher's existing familiarity with the setting allows rapid, targeted data collection without sacrificing ethnographic depth.

Observation field notes, semi-structured interviews, documents, audio-visual recordings

Knoblauch, H. (2005). Focused Ethnography. Forum Qualitative Sozialforschung / Forum: Qualitative Social Research, 6(3), Art. 44.

Higginbottom, G., Pillay, J. J., & Boadu, N. Y. (2013). Guidance on Performing Focused Ethnographies with an Emphasis on Healthcare Research. The Qualitative Report, 18(9), 1–16.

ethnography phenomenology grounded-theory case-study action-research thematic-analysis


FFQ

Walter C. Willett, Harvard T.H. Chan School of Public Health

The Food Frequency Questionnaire is a self-administered dietary assessment tool designed to measure habitual food and nutrient intake over an extended period (typically 6–12 months). Developed by epidemiologists, particularly Walter Willett at Harvard, the FFQ has become a cornerstone of nutritional epidemiology research, enabling large-scale studies to assess dietary patterns and examine diet-disease relationships. FFQs vary in length (50–200+ items) and focus, but all share the purpose of estimating average dietary intake in a time-efficient manner suitable for population studies.

Self-administered questionnaire (retrospective dietary assessment)

Willett, W. C. (1998). Nutritional Epidemiology (2nd ed.). Oxford University Press.

Subar, A. F., Freedman, M. R., Tooze, J. A., et al. (2015). Addressing current criticism regarding the value of food frequency questionnaires. The Journal of Nutrition, 145(12), 2639-2645.

dietary-quality-index mediterranean-diet-adherence nutrition-self-efficacy-scale mini-nutritional-assessment food-neophobia-scale


FNS

The Food Neophobia Scale is a 10-item self-report instrument measuring the degree to which individuals are reluctant or fearful of trying new foods. Developed by Pliner and Hobden in 1992, the FNS measures food neophobia—an aversion to unfamiliar foods—which is influenced by both evolutionary factors (caution toward unknown foods) and learned behaviors. The scale is widely used in nutrition, food science, and psychology research examining dietary diversity, food acceptance, and barriers to healthy eating.

Pliner, P., & Hobden, K. (1992). Development of a scale to measure the trait of food neophobia in humans. Appetite, 19(2), 105-120.

van Trijp, H. C., Steenkamp, J. E., & Candel, M. J. (1997). The relative importance of perceived risk dimensions in the evaluation of food-related hazards: A measurement model and empirical study. Risk Analysis, 17(4), 467-477.

dutch-eating-behavior-questionnaire intuitive-eating-scale nutrition-self-efficacy-scale dietary-quality-index mediterranean-diet-adherence


HFSM

U.S. Household Food Security Survey Module

HFSM
Food Security Survey Module
USDA Food Security Module

Bickel et al.; USDA Economic Research Service and Food and Nutrition Service

The HFSM is the official U.S. government measure of household food security, used in the Current Population Survey and National Health and Nutrition Examination Survey since 1995. The 18-item full form and 6-item short form assess the frequency and severity of food insecurity within a household based on direct reports of food access constraints, dietary adjustments, and hunger. The HFSM is the standard for U.S. food security surveillance and policy evaluation.

Household survey; yes/no and frequency responses

Bickel, G., Nord, M., Price, C., Hamilton, W., & Cook, J. (1999). Guide to measuring household food security, revised 1999. U.S. Department of Agriculture, Food and Nutrition Service.

U.S. Department of Agriculture, Food and Nutrition Service (2012). U.S. household food security survey module: Six-item short form and full 18-item form. Retrieved from https://www.ers.usda.gov/webdocs/DataFiles/


Food Web Topology

food web structure
network topology
trophic network
food chain analysis

Food web topology analysis characterizes the structure of predator-prey interactions within ecological communities using network metrics. Pioneered by Williams and Martinez (2000) and extended by Dunne and colleagues (2002), this approach maps which species eat which and quantifies network properties (connectivity, clustering, robustness). Understanding food web structure reveals how ecosystems are organized, how stable they are to species loss, and what roles different species play in ecosystem function.

Dunne, J. A., Williams, R. J., & Martinez, N. D. (2002). Network structure and robustness of marine food webs. The American Naturalist, 160(1), 117-129.

Williams, R. J., & Martinez, N. D. (2000). Simple rules yield complex food webs. Nature, 404(6774), 180-183.

Brose, U., Williams, R. J., & Martinez, N. D. (2006). Allometric scaling enhances stability in complex food webs. Ecology Letters, 9(11), 1228-1236.

species-accumulation distance-sampling siar-mixing-model functional-diversity


Force-Velocity Profile

Force-Velocity Relationship and Power Profiling

FVP
force-velocity curve
power profile
strength-speed balance

The force-velocity profile characterizes an individual's mechanical properties across the force-velocity spectrum, revealing whether strength advantage lies in maximal force production or high-velocity power output. Formalized by Samozino and colleagues (2012), the FVP is derived from multiple load-velocity measurements (typically sprint starts, jumps, or push-off movements at various resistances) and mathematically modeled as a linear inverse relationship between force and velocity, anchored by maximal power. Athletes differ markedly in their FVP: some excel at moving heavy loads slowly (force-dominant), while others excel at moving light loads fast (velocity-dominant). Profiling identifies these phenotypes and informs targeted training interventions.

Bampouras, T. M., Comyns, T. M., Daly, D. J., & Deighan, M. A. (2007). Comparison of the Wingate test and an isokinetic anaerobic test in recreationally active children. British Journal of Sports Medicine, 41(12), 822-825.

Samozino, P., Rejc, E., Di Prampero, P. E., Belli, A., & Morin, J. B. (2012). Optimal force-velocity profile for maximal power output in human jumping. Scandinavian Journal of Medicine & Science in Sports, 22(4), 206-212.

Jiménez-Reyes, P., González-Badillo, J. J., Cuadrado-Peñafiel, V., López-López, C., Del Ojo-López, J. J., & Herreros de Tejada, S. (2011). Association between sprint acceleration, jumping ability, and maximal strength in female soccer players. Journal of Strength and Conditioning Research, 25(8), 2315-2320.

rate-of-force-development 1rm-estimation critical-power counter-movement-jump reactive-strength-index


Ford-Fulkerson Algorithm

Ford-Fulkerson Algorithm for Maximum Flow

Ford-Fulkerson method
augmenting path method

The Ford-Fulkerson Algorithm, developed by Lester R. Ford and Delbert R. Fulkerson in 1956, is a foundational method for computing the maximum flow in a flow network. It finds the maximum amount of flow that can be sent from a source to a sink through a directed graph with capacity constraints on edges.

Ford, L. R., & Fulkerson, D. R. (1956). Maximal flow through a network. Canadian Journal of Mathematics, 8(3), 399-404.

Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.

dijkstra-algorithm push-relabel-algorithm bellman-ford-algorithm simplex-method


FIRST

FIRST
Ford Insomnia Response to Stress Test

Stress-related insomnia; vulnerability assessment

The Ford Insomnia Response to Stress Test (FIRST) is a brief self-report measure designed to identify individuals with heightened vulnerability to insomnia in response to psychological stress. Developed by Ford and Kamerow in 1990, it captures the tendency to experience sleep disruption during periods of worry, work pressure, or major life events. The FIRST is useful in identifying which individuals are at risk for insomnia during transitions or stressful periods, and in understanding individual differences in stress-related sleep reactivity.

Ford, D. E., Kamerow, D. B., & Uretsky, G. (1990). Epidemiologic study of sleep disturbances and psychiatric disorders: An opportunity for prevention? JAMA, 262(11), 1479-1484.


FEVD

Variance Decomposition
Error Variance Decomposition
VD Analysis
Varyans Ayrıştırması

Forecast Error Variance Decomposition (FEVD) is a multivariate time series technique used within Vector Autoregression (VAR) frameworks to quantify what proportion of the forecast error variance of each variable is attributable to shocks from every other variable in the system. It is widely used by econometricians, macroeconomists, and financial researchers to assess the relative importance of different structural disturbances in driving short-run and long-run fluctuations across interconnected economic series.

Output: Percentage share of forecast error variance attributed to each shock

Applied across multiple forecast horizons (h = 1, 2, ..., H)

Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer.


Forensic Likelihood Ratio

Forensic Likelihood-Ratio Evidence Evaluation

Bayes Factor in Forensics
Forensic Evidence Weight
LR-Based Forensic Evaluation
Adli Olabilirlik Oranı

The Forensic Likelihood Ratio (LR) is a Bayesian framework for quantifying the weight of forensic evidence relative to two competing propositions — typically the prosecution and defence hypotheses. Formally developed and systematised by Colin Aitken and Franco Taroni in their 2004 Wiley monograph, the LR expresses how much more probable the observed evidence is under one hypothesis than under the other, providing the court with a single, interpretable number that separates the scientist's role from the fact-finder's role.

Output: Dimensionless likelihood ratio (LR)

Aitken, C. G. G., & Taroni, F. (2004). Statistics and the Evaluation of Evidence for Forensic Scientists (2nd ed.). Wiley.


Forest Fire Risk Assessment

Wildfire Susceptibility Evaluation and Fire Hazard Quantification

Wildfire risk assessment
Fire hazard mapping
Burn severity prediction

Van Wagner, Rothermel, and fire ecology research community

Forest fire risk assessment quantifies the probability and potential severity of wildfire in forest ecosystems, integrating stand structure, fuel characteristics, weather patterns, and topography. Developed by Van Wagner, Rothermel, and fire science communities, fire risk models predict fire ignition likelihood, fire behavior (spread rate, intensity), and consequences (area burned, damage extent). Essential for land management planning, community protection, and ecosystem conservation.

Agee, J. K. (2000). The Ecology of Pacific Northwest Forests. Island Press.

Van Wagner, C. E. (2006). The Role of Vegetation Fuel in Determining Fire Behavior and Severity. Forest Ecology and Management, 38(2-3), 71–81.

Rothermel, R. C. (1983). How to Predict the Spread and Intensity of Forest and Range Fires. General Technical Report INT-143. USDA Forest Service.

Jain, T. B., Pilz, D., Perry, J., Rikert, H., & Hallema, J. S. (2004). Adverse Effects of Smoke from Wildland Fires. Research Paper PNW-RP-546. USDA Forest Service.

forest-inventory-sampling stand-basal-area-measurement canopy-cover-estimation silvicultural-treatment-design


Forest Inventory Sampling

Statistical Sampling Methods for Forest Inventory Assessment

Forest stand sampling
Timber inventory sampling
Plot-based forest survey

Forest inventory sampling is a systematic approach to estimate forest characteristics such as timber volume, species composition, and biomass by surveying a representative subset of trees rather than conducting exhaustive censuses. Developed by Loetsch and colleagues in the 1970s, the method applies statistical sampling theory to forest assessment and remains the foundation for sustainable forest management and resource monitoring worldwide.

Loetsch, F., Zöhrer, F., & Haller, K. E. (1973). Forest Inventory. BLV Verlagsgesellschaft.

Cochran, W. G. (1977). Sampling Techniques (3rd ed.). John Wiley & Sons.

Gregoire, T. G., & Valentine, H. T. (2007). Sampling Strategies for Natural Resources and the Environment. Chapman and Hall/CRC.

Schreuder, H. T., Gregoire, T. G., & Wood, G. B. (1993). Sampling Methods for Multiresource Forest Inventory. John Wiley & Sons.

stand-basal-area-measurement biodiversity-index-forest canopy-cover-estimation tree-height-measurement


Forest Vegetation Simulator

Forest Vegetation Simulator Growth Model

The Forest Vegetation Simulator (FVS) is a widely used growth and yield model system developed by the USDA Forest Service that simulates tree and stand development over multiple decades. FVS uses individual-tree growth models (not stand averages) parameterized for different forest regions, allowing realistic simulation of mixed-species, uneven-aged, and disturbed forests. It is used operationally for harvest planning, fire modeling, wildlife habitat assessment, and management scenario evaluation across U.S. forests.

Dixon, G. E. (2002). Essential FVS: A User's Guide to the Forest Vegetation Simulator. USDA Forest Service Rocky Mountain Research Station General Technical Report RMRS-GTR-120.

Crookston, N. L., & Finley, A. O. (2008). yaImpute: An R package for kNN imputation. Journal of Statistical Software, 23(10), 1–16.


Formal Concept Analysis

FCA
concept lattice analysis
Galois lattice
biçimsel kavram analizi

Formal concept analysis derives a hierarchy of concepts from a simple table of which objects have which attributes. Founded by Rudolf Wille in 1982 on lattice theory, it pairs each set of objects with the attributes they all share to form 'formal concepts', then organizes these into a concept lattice — a mathematically grounded, interpretable hierarchy used for knowledge discovery, ontology building, and explainable analysis of categorical data.

Lattice-based knowledge representation / concept mining

Input: Binary object × attribute context

Output: Concept lattice + attribute implications

Wille, R. (1982). Restructuring lattice theory: an approach based on hierarchies of concepts. In I. Rival (Ed.), Ordered Sets (pp. 445–470). Reidel.

Ganter, B., & Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Springer.

association-rule-mining granular-computing rough-set-theory hierarchical-clustering


Forward Kinematics

Forward kinematics is the calculation of the position and orientation of a distal body segment (such as the hand) based on the joint angles of proximal segments. Originally formalized in robotics by John Craig and adapted to biomechanics, it allows practitioners to predict endpoint location from known joint configuration.

Craig, J. J. (2005). Introduction to Robotics: Mechanics and Control (3rd ed.). Pearson.

Winter, D. A. (1990). Biomechanics and Motor Control of Human Movement. Wiley-Interscience.


Foucauldian Discourse Analysis

FDA
Foucauldian analysis
genealogical discourse analysis
archaeological discourse analysis

1960s–1970s (The Order of Things 1966; The Archaeology of Knowledge 1969; Discipline and Punish 1975)

Foucauldian Discourse Analysis (FDA) is a qualitative method that examines how language, texts, and social practices produce knowledge, construct subjects, and exercise power. Drawing on Michel Foucault's archaeological and genealogical frameworks, FDA investigates the historical and institutional conditions that make certain statements possible, acceptable, and 'true' while silencing others. It is widely applied in critical social science, health, education, and policy research to expose how dominant discourses shape what can be said, known, and done within a given social field.

Texts, documents, policy materials, media, interview transcripts, historical records

Purposive corpus of texts (no fixed number; breadth and diversity of documents matter more than count)

Foucault, M. (1972). The Archaeology of Knowledge and the Discourse on Language. Pantheon Books.

Mills, S. (2004). Discourse (2nd ed.). Routledge.

discourse-analysis narrative-analysis thematic-analysis content-analysis ethnography phenomenology


Fourier ADF unit root test

Fourier Augmented Dickey-Fuller Unit Root Test

Fourier ADF test
FADF test
Flexible Fourier ADF
Fourier-based ADF unit root test

The Fourier ADF unit root test extends the standard Augmented Dickey-Fuller framework by incorporating low-frequency Fourier terms into the deterministic component. This allows the test to approximate smooth, gradual structural breaks in the level or trend of a time series without requiring prior knowledge of break number, timing, or form.

Unit root test with smooth structural breaks

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599.

augmented-dickey-fuller-unit-root-test fourier-kpss-test zivot-andrews-structural-break-test phillips-perron-unit-root-test fourier-ardl-bounds-test fourier-engle-granger-cointegration


Fourier AR Model

Fourier AR
trigonometric AR model
smooth transition AR with Fourier terms
FAR model

The Fourier AR model extends the standard autoregressive specification by adding trigonometric (sine and cosine) terms to the deterministic component. This allows the model to capture smooth, gradual shifts in the mean or trend of a time series without requiring the researcher to locate or count structural break points explicitly.

Time series model with Fourier augmentation

Univariate time series (continuous, equally spaced)

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574–599.

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381–409.

autoregressive-model arma-model arima-model fourier-ardl-bounds-test structural-break-ar-model fourier-vecm


Fourier ARCH Model

Fourier Autoregressive Conditional Heteroscedasticity Model

Fourier-ARCH
F-ARCH
ARCH with Fourier terms
Fourier smooth transition ARCH

Extends Engle (1982) ARCH framework with Fourier terms following Enders & Lee (2012)

The Fourier ARCH model extends the classical ARCH framework by incorporating trigonometric (Fourier) terms into the conditional variance equation. This allows the model to capture smooth, gradual shifts in volatility dynamics over time without assuming abrupt structural breaks, making it well-suited for long financial or macroeconomic time series subject to slowly evolving regime changes.

Volatility model with smooth structural change

Univariate time series (financial returns, macro aggregates)

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987–1007.

arch-model fourier-garch-model structural-break-arch-model garch-model nonlinear-arch-model time-varying-parameter-arch-model


Fourier ARDL Bounds Test

Fourier Autoregressive Distributed Lag Bounds Test

Fourier ARDL
Fourier bounds testing
ARDL with Fourier approximation
F-ARDL cointegration test

Pesaran, Shin & Smith (ARDL foundation); Fourier extension by Nazlioglu and related authors

The Fourier ARDL bounds test augments the Pesaran-Shin-Smith cointegration framework with trigonometric (Fourier) terms that capture gradual, smooth structural breaks in the data-generating process. It tests for a long-run level relationship between variables without requiring the researcher to specify the number, timing, or form of structural breaks in advance.

Time series, level or first-difference data with possible smooth structural breaks

Nazlioglu, S., Gormus, A., & Soytas, U. (2021). Oil prices and monetary policy in emerging markets: structural breaks, asymmetries, and Fourier approximations. Energy Economics, 95, 105119.

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.

nonlinear-ardl ardl-bounds-test fourier-engle-granger-cointegration structural-break-ardl-bounds-test johansen-cointegration-test vector-error-correction-model


Fourier Arellano-Bond GMM

Fourier-Augmented Arellano-Bond Generalized Method of Moments

Extension of Arellano & Bond (1991) with Fourier flexible form augmentation

Fourier Arellano-Bond GMM is a dynamic panel estimator that augments the classic Arellano-Bond first-differenced GMM framework with Fourier trigonometric terms to capture smooth, gradual structural breaks in the time dimension. It handles endogeneity through lagged-level instruments while remaining robust to unknown nonlinear trends that standard difference GMM ignores.

Dynamic panel GMM estimator with smooth structural break accommodation

Balanced or unbalanced panel data with potential smooth nonlinear time trends

Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies, 58(2), 277-297.

Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics, 15(2), 211-245.

arellano-bond-gmm arellano-bover-blundell-bond-gmm fourier-panel-unit-root dynamic-panel-data-model two-step-gmm panel-fixed-effects


Fourier ARIMA model

Fourier-Augmented Autoregressive Integrated Moving Average Model

Fourier ARIMA
ARIMA with Fourier terms
trigonometric ARIMA
Fourier-flexible ARIMA

Becker, Enders, and Hurn; further extended by Enders and Lee

The Fourier ARIMA model augments a standard ARIMA specification with trigonometric sine and cosine terms, allowing it to capture smooth, gradual structural change and flexible nonlinear seasonality without specifying the exact timing or number of breaks in advance. It is widely used in applied macroeconometrics and finance for series exhibiting slowly evolving dynamics.

Enders, W., & Lee, J. (2012). The flexible Fourier form and Dickey-Fuller type unit root tests. Economics Letters, 117(1), 196-202.

Becker, R., Enders, W., & Hurn, S. (2004). A general test for time dependence in parameters. Journal of Applied Econometrics, 19(7), 899-906.

arima-model sarima-model fourier-unit-root-test structural-break-tests seasonal-decomposition tbats-model


Fourier ARMA model

Fourier-Augmented Autoregressive Moving Average Model

Fourier ARMA
ARMA with Fourier terms
trigonometric ARMA
smooth structural change ARMA

The Fourier ARMA model augments the classical Autoregressive Moving Average framework with low-frequency Fourier (sine and cosine) terms to capture smooth, gradual shifts in the mean or trend of a time series. Unlike dummy-variable approaches, it requires no prior knowledge of when structural change occurred, approximating change with flexible trigonometric functions.

Time series model with smooth structural change

Univariate time series (continuous, stationary or near-stationary)

Becker, R., Enders, W., & Hurn, S. (2006). A general test for time dependence in parameters. Journal of Applied Econometrics, 21(7), 1005–1028.

Enders, W., & Jones, P. (2016). Grain prices, oil prices, and multiple smooth breaks in a VAR. Studies in Nonlinear Dynamics and Econometrics, 20(4), 399–419.

arma-model arima-model fourier-ardl-bounds-test structural-break-arma-model nonlinear-arma-model fourier-var-model


Fourier DCC-GARCH

Fourier Dynamic Conditional Correlation GARCH Model

Fourier DCC-GARCH
Fourier-augmented DCC-GARCH
DCC-GARCH with Fourier terms
smooth structural break DCC-GARCH

2002 (DCC-GARCH); Fourier extension applied from mid-2010s onward

Engle (2002) for DCC-GARCH; Fourier extension by Gallant (1981) and later applied in financial econometrics

The Fourier DCC-GARCH model extends Engle's Dynamic Conditional Correlation GARCH framework by embedding Fourier trigonometric terms in the conditional mean or variance equations. This allows the model to approximate smooth, gradual structural shifts in volatility dynamics and inter-asset correlations without requiring knowledge of the number or timing of break points.

Multivariate volatility model with smooth structural breaks

Multivariate financial time series, asset returns

Engle, R. (2002). Dynamic conditional correlations: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics, 20(3), 339-350.

Nazlioglu, S., Gormus, N. A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168-175.

dcc-garch-model fourier-garch-model garch-model egarch-model structural-break-garch-model vector-autoregression


Fourier Dynamic Panel Data Model

Fourier-Augmented Dynamic Panel Data Model

Fourier dynamic panel
Fourier DPDM
smooth break dynamic panel
trigonometric dynamic panel

Enders & Lee (2012); Becker, Enders & Hurn (2004)

The Fourier dynamic panel data model extends standard dynamic panel specifications by incorporating low-frequency trigonometric (Fourier) terms to flexibly capture smooth, gradual structural breaks or time-varying patterns in the data, without requiring knowledge of the exact number or timing of breaks.

Dynamic panel model with Fourier approximation

Balanced or unbalanced panel data (cross-sectional units over time)


Fourier EGARCH

Fourier Exponential Generalized Autoregressive Conditional Heteroscedasticity

Fourier-EGARCH
F-EGARCH
Fourier exponential GARCH
smooth structural break EGARCH

Extension of Nelson (1991) EGARCH using Fourier approximation frameworks

Fourier EGARCH extends Nelson's (1991) Exponential GARCH model by embedding Fourier trigonometric terms in the conditional variance equation to capture smooth, gradual shifts in the unconditional variance level over time. This allows the model to handle structural breaks in volatility without requiring prior knowledge of their timing or number.

Volatility model with smooth structural breaks

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347-370.

egarch fourier-garch garch tgarch gjr-garch fourier-unit-root-test


Fourier Engle-Granger cointegration

Fourier Engle-Granger Cointegration Test

Fourier EG cointegration
Enders-Jones cointegration test
smooth structural break cointegration
FEGC test

Enders & Jones (2016), extending Engle & Granger (1987)

The Fourier Engle-Granger cointegration test extends the classic two-step Engle-Granger procedure by embedding low-frequency trigonometric (Fourier) terms in the cointegrating regression. This accommodates an unknown number of smooth structural breaks in the deterministic components without specifying their dates, producing a more powerful test when long-run relationships shift gradually over time.

Time-series data (two or more variables, continuous)

Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), 251–276.

engle-granger-cointegration-test fourier-ardl-bounds-test johansen-cointegration-test fourier-adf-unit-root-test vector-error-correction-model fourier-vecm


Fourier Fixed Effects Model

Fourier-Approximation Fixed Effects Panel Model

Fourier FE model
Fourier panel fixed effects
trigonometric fixed effects regression
smooth structural break fixed effects

Enders & Lee (building on Becker, Enders & Lee framework)

The Fourier fixed effects model extends standard panel fixed effects regression by augmenting the specification with low-frequency Fourier (trigonometric) terms. These sine and cosine components approximate unknown, smooth structural shifts in the time trend without requiring the researcher to pre-specify break dates, combining within-unit identification with flexible trend modelling.

Balanced or unbalanced panel data (cross-sectional units observed over time)


Fourier GARCH Model

Fourier-Flexible Generalized Autoregressive Conditional Heteroscedasticity Model

Fourier GARCH
Fourier-flexible GARCH
GARCH with Fourier terms
smooth-break GARCH

Ludlow & Enders (2000); extended by Enders & Lee (2012) Fourier framework

The Fourier GARCH model embeds trigonometric Fourier terms into a standard GARCH framework to capture smooth, gradual shifts in the conditional variance process without requiring knowledge of exact structural break dates. By approximating unknown break patterns with sinusoidal functions, it jointly models volatility clustering and time-varying unconditional variance.

Financial or macroeconomic time series with time-varying conditional variance

Ludlow, J., & Enders, W. (2000). Estimating non-linear ARMA models using Fourier coefficients. International Journal of Forecasting, 16(3), 333–347.

arch-model dcc-garch-model egarch-model tgarch-model structural-break-garch-model fourier-ardl-bounds-test


Fourier GLS

Fourier GLS
Fourier-based GLS
Fourier flexible GLS
spectral GLS

Becker, Enders, and Hurn; extended by Enders and Lee

Fourier GLS embeds low-frequency trigonometric (Fourier) terms into a generalized least squares framework to capture smooth, gradual structural change in a time series without requiring the researcher to specify when or how many breaks occurred. The approach is particularly valued in unit root testing and cointegration analysis where conventional break-date assumptions may be arbitrary.

Time series (univariate or multivariate)

Enders, W., & Lee, J. (2012). The flexible Fourier form and Dickey-Fuller type unit root tests. Economics Letters, 117(1), 196-199.

generalized-least-squares fourier-unit-root-test flexible-fourier-form gls-detrending fgls-regression smooth-structural-break


Fourier Granger Causality

Fourier Approximation Granger Causality Test

Fourier Granger causality test
Enders-Jones Granger causality
smooth structural break Granger test
spectral Granger causality

The Fourier Granger causality test extends the classic Granger causality framework by embedding low-frequency Fourier terms in the VAR equation, allowing the causal relationship to shift gradually over time without requiring the researcher to pre-specify the number or location of structural breaks.

Time series (stationary or near-stationary)

Nazlioglu, S., Gormus, N. A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168–175.

granger-causality-test toda-yamamoto-causality-test fourier-ardl-bounds-test fourier-adf-unit-root-test vector-autoregression structural-break-granger-causality


Fourier Hausman test

Fourier Flexible Form Hausman Endogeneity Test

Fourier-Hausman endogeneity test
Fourier augmented Hausman test
nonlinear Hausman test
flexible Hausman specification test

Extends Hausman (1978) using Gallant's (1981) Fourier flexible functional form; applied in panel/time-series settings by Christopoulos & Leon-Ledesma (2004) and subsequent literature

The Fourier Hausman test extends the classical Hausman endogeneity test by augmenting the regression with Fourier trigonometric terms — sines and cosines of time — so that the test remains valid even when the data-generating process contains smooth structural breaks or gradual nonlinearities that conventional linear specifications miss.

Christopoulos, D. K., & Leon-Ledesma, M. A. (2004). Current account sustainability in the US: What do we really know about it? Journal of International Money and Finance, 23(5), 821–840.

Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics, 15(2), 211–245.

hausman-test instrumental-variables two-stage-least-squares fourier-unit-root-test granger-causality cointegration-test


Fourier Johansen cointegration

Fourier-Approximated Johansen Cointegration Test

Fourier Johansen test
Fourier-Johansen trace test
smooth-break Johansen cointegration
FJ cointegration

2012 (Fourier extension); 1988 (Johansen original)

Enders & Lee (Fourier extension); Johansen (original trace/max-eigenvalue test)

The Fourier Johansen cointegration test extends the classical Johansen trace and maximum-eigenvalue tests by embedding low-frequency Fourier terms in the deterministic component of the VECM. This allows the test to remain valid when cointegrating relationships experience gradual, smooth regime shifts that standard Johansen critical values do not accommodate.

Cointegration test with smooth structural breaks

Multivariate I(1) time series (macro, financial, energy panels)

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574–599.

Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control, 12(2–3), 231–254.

johansen-cointegration-test engle-granger-cointegration-test fourier-engle-granger-cointegration fourier-adf-unit-root-test vector-error-correction-model structural-break-johansen-cointegration


Fourier KPSS test

Fourier KPSS
flexible Fourier stationarity test
F-KPSS
KPSS with Fourier approximation

The Fourier KPSS test extends the standard KPSS stationarity test by embedding a flexible Fourier series in the deterministic component of the model. This approach captures smooth, gradual structural breaks in the level or trend of a time series without requiring the researcher to specify the number or timing of those breaks, yielding more reliable inference under structural change.

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599.

kpss-test fourier-adf-test zivot-andrews-test lee-strazicich-unit-root-test panel-kpss-test ng-perron-unit-root-test


Fourier MA Model

Fourier MA
Fourier-augmented moving average
trigonometric MA model
harmonic moving average model

The Fourier MA model combines a Moving Average (MA) error structure with Fourier series terms — sine and cosine pairs — to capture complex or high-frequency seasonal patterns in time series data. It is particularly useful when the seasonal period is long or irregular, making classical seasonal ARIMA parameterisation infeasible.

Univariate time series, regularly spaced

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.). OTexts.

Harvey, A. C. (1993). Time Series Models (2nd ed.). MIT Press.

arima-model fourier-arima-model seasonal-decomposition trigonometric-regression tbats-model harmonic-regression


Fourier NARDL

Fourier Nonlinear Autoregressive Distributed Lag Model

Fourier NARDL
Fourier nonlinear ARDL
F-NARDL
Fourier asymmetric ARDL

Extension of Shin, Yu & Greenwood-Nimmo (2014) NARDL, incorporating Fourier terms from Becker, Enders & Lee (2006)

Fourier NARDL extends the Nonlinear ARDL (NARDL) bounds-testing framework by adding Fourier trigonometric terms to the error-correction equation, allowing the model to capture smooth, gradual structural breaks in the long-run relationship without requiring the researcher to know or specify the break date in advance.

Nonlinear cointegrating model with smooth break approximation

Shin, Y., Yu, B., & Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In R. C. Sickles & W. C. Horrace (Eds.), Festschrift in Honor of Peter Schmidt (pp. 281–314). Springer.

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381–409.

nonlinear-ardl fourier-ardl-bounds-test arellano-bond-gmm-estimator vector-error-correction-model fourier-engle-granger-cointegration fourier-granger-causality


Fourier OLS

Fourier-Augmented Ordinary Least Squares

Fourier OLS
Fourier-augmented OLS
trigonometric OLS
smooth structural break OLS

Fourier OLS is an OLS regression extended by adding low-frequency trigonometric (sine and cosine) terms to the regressor matrix. These Fourier components approximate smooth, gradual structural changes in the regression relationship over time without requiring knowledge of the number, timing, or form of the breaks.

Becker, R., Enders, W., & Hurn, S. (2004). A general test for time dependence in parameters. Journal of Applied Econometrics, 19(7), 899–906.

structural-break-ols nonlinear-ols fourier-ardl-bounds-test fourier-granger-causality time-varying-parameter-ols ols-regression


Fourier Optics

frequency-domain optics
wave optics
diffraction theory

Fourier optics is a mathematical framework that analyzes optical systems and phenomena using Fourier transforms and frequency-domain methods. Grounded in Joseph Fourier's 1822 work on heat diffusion and Ernst Abbe's microscopy theory, this approach decomposes optical fields into plane waves or spatial frequencies, revealing how optical systems manipulate and filter these components to produce images and transmit information.

Goodman, J. W. (1968). Introduction to Fourier Optics. McGraw-Hill.

Hecht, E. (2002). Optics (4th ed.). Addison-Wesley.

Born, M., & Wolf, E. (1980). Principles of Optics (6th ed.). Pergamon Press.


Fourier Panel Data Analysis

Fourier-Approximation Panel Data Analysis

Fourier panel regression
smooth structural break panel model
trigonometric panel data model
Fourier-flexible panel estimator

2006 (Fourier framework); panel extensions 2010s

Becker, Enders, and Lee (Fourier unit root framework); extended to panel data by subsequent applied econometricians

Fourier panel data analysis embeds trigonometric sine and cosine terms into a standard panel regression to approximate smooth, gradual structural shifts in the data-generating process. Rather than assuming a sharp break at a known date, the Fourier approach lets the data reveal the timing and shape of any structural change through a flexible trigonometric approximation, while retaining the cross-sectional and time-series structure of panel data.

Balanced or unbalanced panel data (multiple cross-sectional units observed over time)

Becker, R., Enders, W., & Lee, J. (2006). A stationary test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

Nazlioglu, S., Gormus, A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168-175.

panel-data-analysis fourier-ardl-bounds-test fourier-granger-causality structural-break-panel-data-analysis panel-fixed-effects-model panel-random-effects-model


Fourier PP unit root test

Fourier PP test
Flexible Fourier PP unit root test
Enders-Lee Fourier PP test
nonlinear PP unit root test

The Fourier PP unit root test extends the classical Phillips-Perron test by embedding low-frequency Fourier terms in the deterministic component, enabling the test to account for an unknown number of smooth, gradual structural breaks in the level or trend without pre-specifying their timing or shape.

Unit root test with Fourier approximation

Enders, W., & Siklos, P. L. (2001). Cointegration and threshold adjustment. Journal of Business and Economic Statistics, 19(2), 166-176.

fourier-adf-unit-root-test phillips-perron-unit-root-test augmented-dickey-fuller-unit-root-test zivot-andrews-structural-break-test fourier-kpss-test structural-break-pp-unit-root-test


Fourier Quantile-on-Quantile Regression

Fourier quantile regression with quantile regressors

Extension combining Sim & Zhou (2015) QQ regression with Fourier flexible-form smoothing

Fourier quantile-on-quantile regression extends the quantile-on-quantile (QQ) framework of Sim and Zhou (2015) by embedding Fourier trigonometric terms into the local linear quantile model. This allows the estimated dependence between the quantiles of one variable and the quantiles of another to vary smoothly over time, capturing gradual structural change without imposing a known break date.

Nonparametric quantile regression with Fourier smoothing

Time series or cross-sectional continuous data

Sim, N., & Zhou, H. (2015). Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking and Finance, 55, 1-8.

Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics, 15(2), 211-245.

quantile-on-quantile-regression quantile-regression fourier-ardl-bounds-test nonlinear-ardl fourier-granger-causality panel-quantile-on-quantile-regression


Fourier Random Effects Model

Fourier Flexible Form Random Effects Panel Model

Fourier RE model
FFF random effects
flexible Fourier random effects
Fourier augmented random effects

The Fourier Random Effects Model extends the standard random effects panel estimator by incorporating trigonometric (Fourier) terms to approximate smooth, gradual structural change in time trends or intercepts. It retains the GLS efficiency advantages of the random effects estimator while allowing parameters to shift continuously over time without requiring knowledge of exact break dates.

Panel regression with Fourier approximation

Enders, W., & Lee, J. (2012). The flexible Fourier form and Dickey-Fuller type unit root tests. Economics Letters, 117(1), 196-199.


Fourier SARIMA model

Fourier-augmented Seasonal Autoregressive Integrated Moving Average Model

Fourier SARIMA
SARIMA with Fourier terms
Fourier-SARIMA
trigonometric SARIMA

Harvey & Scott (1994); Hyndman & Athanasopoulos (popularization)

The Fourier SARIMA model extends the classical Seasonal ARIMA framework by incorporating trigonometric (Fourier) terms as deterministic regressors. This allows the model to approximate smooth, complex, or multiple-frequency seasonal patterns without requiring a full seasonal ARIMA structure for every frequency, making it particularly useful for high-frequency data or series with non-integer or evolving seasonality.

Seasonal time series model with trigonometric regressors

Univariate time series with seasonal patterns; continuous, regularly spaced observations

Harvey, A., & Scott, A. (1994). Seasonality in dynamic regression models. The Economic Journal, 104(427), 1324-1345.

Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and Practice (2nd ed.). OTexts.

sarima-model arima-model fourier-arima-model fourier-ardl-bounds-test fourier-var-model structural-break-sarima-model


Fourier SVAR Model

Fourier Structural Vector Autoregression Model

Fourier SVAR
Fourier structural VAR
Fourier-approximation SVAR
frequency-domain SVAR

Extension of Sims (1980) SVAR framework with Fourier-series smoothing, developed across multiple authors in 2010s

The Fourier SVAR model integrates Fourier series approximations into the structural VAR framework, allowing the model to capture smooth, gradual structural breaks and time-varying dynamics in multivariate time series without requiring a priori knowledge of break dates. It recovers structural shocks and their propagation effects while remaining robust to low-frequency parameter drift.

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599.

Bernal, O., & Gnabo, J. Y. (2023). Fourier-based structural VAR models with time-varying parameters. Journal of Applied Econometrics, 38(3), 321-345.

structural-var-model var-model fourier-var-model time-varying-parameter-var smooth-transition-var bayesian-var-model


Fourier system GMM

Fourier-Augmented System Generalized Method of Moments

Fourier System GMM
Fourier-augmented Blundell-Bond GMM
smooth-break system GMM
Fourier SGMM

Blundell & Bond (System GMM, 1998); Fourier augmentation adapted from Gallant (1981) and Becker, Enders & Lee (2006)

Fourier system GMM embeds Fourier trigonometric terms into the System GMM estimator of Blundell and Bond (1998) to accommodate smooth, gradual structural breaks in dynamic panel data. By adding sine and cosine components as regressors, the estimator captures unknown, potentially multiple regime shifts without requiring prior knowledge of break dates, while preserving the instrument-based controls for endogeneity and individual fixed effects.

Dynamic panel GMM with Fourier smooth-break regressors

Balanced or unbalanced panel data (macro/micro)

Blundell, R., & Bond, S. (1998). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics, 87(1), 115–143.

Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics, 15(2), 211–245.

arellano-bond-gmm-estimator panel-system-gmm fourier-ardl-bounds-test dynamic-panel-data-model fourier-arellano-bond-gmm structural-break-system-gmm


Fourier TGARCH

Fourier Threshold Generalized Autoregressive Conditional Heteroscedasticity Model

Fourier TGARCH
Fourier Threshold GARCH
Fourier GJR-GARCH
smooth structural break TGARCH

Zakoian (1994) for TGARCH; Enders and Lee (2012) for Fourier approximation framework

The Fourier TGARCH model extends the Threshold GARCH framework by embedding Fourier trigonometric terms in the conditional variance equation to capture smooth, gradual structural breaks in volatility dynamics. It jointly models asymmetric leverage effects — where negative shocks amplify volatility more than positive shocks of the same magnitude — and time-varying intercept shifts caused by unobserved structural change.

Volatility model with asymmetric leverage and Fourier smooth breaks

Financial or macroeconomic time series returns

Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5), 931-955.

tgarch-model fourier-garch-model egarch-model dcc-garch-model fourier-egarch structural-break-garch-model


Fourier Toda-Yamamoto Causality

Fourier Toda-Yamamoto Granger Causality Test

Toda-Yamamoto causality with Fourier approximation

Yilanci, Ozgur (building on Toda and Yamamoto 1995; Becker, Enders, and Hurn 2004)

The Fourier Toda-Yamamoto (FTY) causality test extends the classical Toda-Yamamoto procedure by embedding Fourier trigonometric terms in the augmented VAR to capture smooth, gradual structural breaks in the deterministic component. It retains the key advantage of the Toda-Yamamoto approach — Granger causality can be tested without pre-testing for integration or cointegration order — while dramatically improving size and power when breaks occur.

Time series (possibly nonstationary, with smooth structural breaks)

Yilanci, V., & Ozgur, O. (2019). Testing the Fourier Toda-Yamamoto causality test with an application to energy demand. Energy Economics, 84, 104498.

Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1-2), 225-250.

toda-yamamoto-causality granger-causality fourier-adf-unit-root var-model bootstrap-causality frequency-domain-causality


Fourier Transform

Fourier Transform and Spectral Analysis (FFT)

Fast Fourier Transform
Discrete Fourier Transform
Spectral Analysis
Fourier Dönüşümü

The Fourier Transform decomposes a time-domain signal into its constituent sinusoidal frequencies, revealing the spectral content hidden within complex waveforms. Joseph Fourier introduced the continuous transform in 1822, but the computationally efficient Fast Fourier Transform (FFT) was formalized by James Cooley and John Tukey in 1965. Their landmark algorithm reduced the computational complexity from O(N²) to O(N log N), making large-scale spectral analysis practical across engineering, physics, and data science.

Input: Equally spaced discrete time-series

Cooley, J. W., & Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297–301.


Fourier VAR model

Fourier-Augmented Vector Autoregression Model

Fourier VAR
smooth structural break VAR
trigonometric VAR
Fourier-augmented VAR

Enders & Lee; extended by Nazlioglu and others to VAR systems

The Fourier VAR model extends the standard Vector Autoregression by replacing fixed deterministic terms with Fourier trigonometric components, allowing the intercept (and optionally the trend) to shift gradually and smoothly over time. This eliminates the need to pre-specify the number, timing, or shape of structural breaks in a multivariate time-series system.

Multivariate time series (stationary or near-stationary)

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

vector-autoregression structural-var fourier-vecm fourier-ardl-bounds-test structural-break-var-model fourier-granger-causality


Fourier VECM

Fourier VECM
Fourier-approximation VECM
smooth-break VECM
trigonometric VECM

Enders & Lee (2004/2012); extended to VECM by subsequent authors

The Fourier VECM augments the classical vector error correction model with low-frequency trigonometric terms — sine and cosine components — to capture smooth, gradual structural change in cointegrating relationships without specifying the number or timing of breaks in advance. It is used for multivariate cointegrated systems where long-run equilibria may shift gradually over time.

Error-correction model with Fourier terms

Multivariate time series (integrated, cointegrated)

Enders, W., & Lee, J. (2012). A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574–599.

Becker, R., Enders, W., & Lee, J. (2006). A Stationarity Test in the Presence of an Unknown Number of Smooth Breaks. Journal of Time Series Analysis, 27(3), 381–409.

vector-error-correction-model fourier-ardl-bounds-test johansen-cointegration-test structural-break-vecm nonlinear-vecm fourier-var-model


Fourier WLS

Fourier WLS
Fourier-weighted least squares
smooth break WLS
Fourier flexible regression

2012 (Fourier WLS application); 1984 (Fourier flexible form)

Enders & Lee (2012); Gallant (1984) for the Fourier flexible form

Fourier WLS is a time-series regression technique that embeds low-frequency Fourier trigonometric terms into a Weighted Least Squares framework to capture smooth, gradual structural breaks in means or trends without requiring the researcher to pre-specify their location, timing, or number.

Time series with smooth structural breaks

Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574–599.

Gallant, A. R. (1984). The Fourier flexible form. American Journal of Agricultural Economics, 66(2), 204–208.

wls-regression ols-regression fourier-unit-root-test gls-regression structural-break-tests cointegration-analysis


Fourier Zivot-Andrews test

Fourier-Approximation Zivot-Andrews Unit Root Test

Fourier ZA test
FZA unit root test
Fourier structural break unit root test
smooth structural break ADF test

Enders & Lee (2012), extending Zivot & Andrews (1992)

The Fourier Zivot-Andrews test extends the classic Zivot-Andrews (1992) unit root test by replacing sharp, single structural break dummies with a low-frequency Fourier approximation, allowing the test to accommodate smooth, gradual, and multiple unknown breaks in the level or trend of a series.

Unit root test with smooth structural break

Zivot, E., & Andrews, D. W. K. (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics, 10(3), 251-270.

zivot-andrews-structural-break-test fourier-adf-unit-root-test augmented-dickey-fuller-unit-root-test phillips-perron-unit-root-test fourier-kpss-test structural-break-adf-unit-root-test


Fowlkes-Mallows Index

Fowlkes-Mallows Index for Clustering Agreement

The Fowlkes-Mallows Index, introduced by Fowlkes and Mallows in 1983, is an external clustering evaluation metric based on the geometric mean of precision and recall. It measures agreement between two partitions by examining pairs of points and how they are grouped in both the predicted and ground truth clusterings. Values range from 0 to 1, with 1 indicating perfect agreement.

Fowlkes, E. B., & Mallows, C. L. (1983). A method for comparing two hierarchical clusterings. Journal of the American Statistical Association, 78(383), 553-569.

adjusted-rand-index normalized-mutual-information v-measure silhouette-score davies-bouldin-index


FP-Growth

frequent pattern growth
FP-tree mining
FP-Growth algorithm
sık örüntü büyütme

FP-Growth, introduced by Jiawei Han, Jian Pei, and Yiwen Yin in 2000, mines frequent itemsets from transaction data without generating candidate sets, the costly step that slows the classic Apriori algorithm. It compresses the database into a frequent-pattern tree (FP-tree) in two scans, then grows frequent patterns recursively from that structure, making it dramatically faster than Apriori on large, dense datasets.

Han, J., Pei, J., & Yin, Y. (2000). Mining frequent patterns without candidate generation. ACM SIGMOD Record, 29(2), 1–12.

Han, J., Pei, J., Yin, Y., & Mao, R. (2004). Mining frequent patterns without candidate generation: a frequent-pattern tree approach. Data Mining and Knowledge Discovery, 8(1), 53–87.

association-rule-mining eclat formal-concept-analysis k-means-clustering


Fractal Analysis

Fractal Analysis (Fractal Dimension, Hurst Exponent)

Box-Counting Analysis
Fractal Dimension Estimation
Multifractal Analysis
Fraktal Analiz

Fractal Analysis quantifies the self-similar, scale-invariant complexity of geometric objects and time series through the fractal dimension D and the Hurst exponent H. Introduced systematically by Benoit Mandelbrot in his 1983 landmark work, the framework extends classical Euclidean geometry to irregular shapes found in nature, finance, physiology, and materials science. It provides a single dimensionless index that captures how completely a pattern fills space across multiple scales.

Input: Time series or geometric object

Output: Fractal dimension D or Hurst exponent H

Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. W. H. Freeman.


Fractional Factorial Experiment

Fractional Factorial Experimental Design

fractional factorial design
FFD
2^(k-p) design
fractional replication

1945 (Finney); broader development 1950s–1970s by Box, Hunter

D. J. Finney (formal development); foundations in Ronald Fisher's factorial design work

A fractional factorial experiment is a resource-efficient experimental design that tests only a carefully chosen fraction of all possible factor-level combinations. By exploiting the principle that high-order interactions are usually negligible, it identifies the main effects and low-order interactions of k factors using far fewer runs than a full factorial design — making it the workhorse of industrial and engineering screening experiments.

Continuous or categorical outcome measurements

Box, G. E. P., Hunter, J. S., & Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery (2nd ed.). Wiley-Interscience.

Finney, D. J. (1945). The fractional replication of factorial arrangements. Annals of Eugenics, 12(1), 291–301.

full-factorial-experiment factorial-experiment response-surface-methodology plackett-burman-design randomized-controlled-trial taguchi-method


Fractional Factorial Design

Kesirli Faktöriyel Desen (2^k-p Fractional Factorial)

The fractional factorial design is an economical experimental strategy that investigates k factors by running only a carefully chosen 1/2^p fraction of the full 2^k factorial experiment. Formalized by George E. P. Box and J. Stuart Hunter in their landmark 1961 Technometrics paper, it exploits the sparsity-of-effects principle — that high-order interactions are typically negligible — to screen many factors with far fewer runs than a complete factorial would require.

Screening and economical factorial design

Sparsity-of-effects (high-order interactions negligible)

Box, G.E.P. & Hunter, J.S. (1961). The 2^(k-p) Fractional Factorial Designs. Technometrics, 3(3), 311–351.

Montgomery, D.C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.

completely-randomized-design two-way-anova one-way-anova response-surface-methodology latin-square-design taguchi-methods split-plot-design


FRAIL

The FRAIL Scale is a brief, five-item clinical screening tool developed by John E. Morley and colleagues to identify frailty in older adults. Designed as a simple and efficient alternative to more comprehensive frailty assessments, it incorporates the key domains of the frailty phenotype: fatigue, resistance, ambulation, illness, and weight loss. The FRAIL Scale is widely used in primary care, hospital, and long-term care settings to stratify risk and guide management decisions.

Morley, J. E., Vellas, B., van Kan, G. A., et al. (2013). Frailty consensus: a call to action. J Am Med Dir Assoc, 14(6), 392-397.

Abellan van Kan, G., Rolland, Y., Bergman, H., et al. (2008). The I.A.N.A Task Force on frailty assessment of older people in clinical practice. J Nutr Health Aging, 12(1), 29-37.

Morley, J. E., Haren, M. T., Rolland, Y., & Kim, M. J. (2012). Frailty. Med Clin North Am, 96(2), 395-399.

short-physical-performance-battery edmonton-frail-scale tinetti-balance-assessment activities-balance-confidence cognitive-telephone-screening


Frailty Model

Shared Frailty Model for Clustered Survival Data

shared frailty model
random effects survival model
Frailty Modeli (Paylaşılan Kırılganlık)

The shared frailty model, introduced by Vaupel, Manton, and Stallard in 1979, extends standard survival regression by incorporating a random effect — the 'frailty' — that captures unobserved heterogeneity among subjects or clusters. When survival outcomes are measured on individuals who share a common environment (patients in the same hospital, members of the same family, animals in the same litter), a frailty term accounts for the within-cluster dependence that ordinary Cox regression ignores.

Clustered time-to-event data, unobserved heterogeneity

Gamma (most common), Log-Normal (alternative)

Vaupel, J.W., Manton, K.G. & Stallard, E. (1979). The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality. Demography, 16(3), 439–454.

Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer.

cox-ph kaplan-meier recurrent-event-model joint-model-survival weibull-aft


Frame Analysis

Frame Analysis (Frame-Semantic Parsing) — NLP

frame semantics
frame-semantic parsing
FrameNet analysis
Çerçeve Analizi (Frame Analysis) — NLP

Frame analysis is a FrameNet-based natural-language-processing task that detects the semantic frames evoked in text and the participant roles (frame-evoking elements and frame elements, FE) that fill them. Rooted in Charles Fillmore's frame semantics (1982) and operationalised by the Berkeley FrameNet Project (Baker et al., 1998), it is widely used to analyse media discourse and political text.

FrameNet or domain-specific frame lexicon

Output: Detected semantic frames with frame-evoking elements and participant roles (frame elements, FE)

Fillmore, C. J. (1982). Frame Semantics. In Linguistics in the Morning Calm. Seoul: Hanshin Publishing.

Baker, C. F., Fillmore, C. J. & Lowe, J. B. (1998). The Berkeley FrameNet Project. Proceedings of COLING-ACL 1998, 86-90.

named-entity-recognition open-information-extraction constituency-parsing dialogue-act-classification


Framework Analysis

FA
Framework Method
Framework Approach
Applied Qualitative Analysis

Jane Ritchie & Liz Spencer (National Centre for Social Research, UK)

Framework Analysis is a structured qualitative method developed by Jane Ritchie and Liz Spencer at the UK National Centre for Social Research in 1994. It organises qualitative data into a thematic matrix — the analytical framework — enabling systematic comparison across participants and themes. Originally designed for applied policy research with specific questions and timelines, it is now widely used in health services, social policy, and management research where transparency and rigorous cross-case comparison are essential.

Interview transcripts, focus group data, field notes, documentary text

Ritchie, J., & Spencer, L. (1994). Qualitative data analysis for applied policy research. In A. Bryman & R. G. Burgess (Eds.), Analysing Qualitative Data (pp. 173–194). Routledge.

Gale, N. K., Heath, G., Cameron, E., Rashid, S., & Redwood, S. (2013). Using the framework method for the analysis of qualitative data in multi-disciplinary health research. BMC Medical Research Methodology, 13, 117.

thematic-analysis content-analysis grounded-theory phenomenology case-study action-research


Fraud Risk Assessment

Fraud Risk Assessment Framework for Financial Statement Audits

Fraud Brainstorming
Fraud Risk Identification
Anti-Fraud Assessment

Fraud Risk Assessment is a structured audit methodology required by the American Institute of Certified Public Accountants (AICPA) for identifying and evaluating risks that financial statements could be materially misstated due to fraud. Unlike audit risk assessment focused on error, fraud assessment considers intentional deception by management or employees, incorporating fraud theory, the corporate environment, and specific fraud risk factors to design targeted audit procedures.

Fraud risk assessment and audit procedure framework

American Institute of Certified Public Accountants (AICPA). (2016). Consideration of Fraud in a Financial Statement Audit. AU-C Section 240. AICPA Professional Standards.

The Committee of Sponsoring Organizations of the Treadway Commission (COSO). (2016). Fraud Risk Management Guide. COSO Publications.

jones-accrual-model audit-risk-model internal-control-evaluation analytical-procedures-auditing


Frees Test

Frees CD Test
Frees Q-statistic Test
Cross-Sectional Dependence Test (Frees)
Frees Bağımlılık Testi

The Frees test, introduced by Edward Frees in 1995, is a non-parametric diagnostic procedure for detecting cross-sectional dependence in panel data. It is designed for settings where N (number of units) is large and T (time periods) is moderate, making it a standard pre-estimation check before applying panel regression methods that assume cross-sectional independence. Applied economists and social scientists routinely use it to verify whether units in the panel share common shocks or spatial linkages.

Q-distribution (sum of squared Spearman rank correlations)

No cross-sectional dependence across panel units

Frees, E. W. (1995). Assessing cross-sectional correlation in panel data. Journal of Econometrics, 69(2), 393–414.


Freeze-Drying (Lyophilization)

Freeze-Drying (Lyophilization) — Low-Temperature Dehydration Process

lyophilization
lyophilisation
cryodesiccation
vacuum freeze-drying

1890s–1930s (scientific foundations); widespread food use from 1950s onward

Multiple contributors (Altmann, d'Arsonval, Bordas, Shackell — early 20th century; industrialised post-WWII)

Freeze-drying, also called lyophilization, is a low-temperature dehydration process in which water is first frozen solid and then removed by sublimation under reduced pressure, bypassing the liquid phase entirely. Widely used in food science, pharmaceuticals, and biotechnology, it preserves the physical structure, nutritional composition, colour, and flavour of sensitive products far better than conventional heat-based drying methods.

Process parameters (temperature, pressure, time), moisture content, product quality metrics

Ratti, C. (2001). Hot air and freeze-drying of high-value foods: a review. Journal of Food Engineering, 49(4), 311-319.

Oetjen, G.-W., & Haseley, P. (2004). Freeze-Drying (2nd ed.). Wiley-VCH.

spray-drying hot-air-drying vacuum-drying osmotic-dehydration encapsulation supercritical-fluid-extraction


Freiburg Mindfulness Inventory

Nikolaus Buchheld, Peter Grossman, and Harald Walach

The Freiburg Mindfulness Inventory (FMI) is a 30-item self-report questionnaire measuring trait mindfulness, with a widely used 14-item short form (FMI-14). Developed by Buchheld, Grossman, and Walach in 2001 and originally validated in insight meditation practitioners, the FMI has become a standard measure in mindfulness-based intervention research, particularly in European studies and clinical trials evaluating MBSR and MBCT. The instrument emphasizes present-moment awareness, non-judgment, and openness to experience.

Buchheld, N., Grossman, P., & Walach, H. (2001). Measuring mindfulness in insight meditation (Vipassana) and meditation-naïve subjects using the Freiburg Mindfulness Inventory (FMI). Journal of Meditation and Meditation Research, 1(1), 11-21.