Forecasting wind speeds at tall tower heights within Missouri

Forecasting of wind speeds is necessary for the planning and operations of the wind power generating plants. This research investigates the short term forecasting of wind speeds at tall tower heights for stations within Missouri: Columbia, Neosho and Blanchard. The first objective was to characterize the chaotic nature of this parameter using mono and multi fractal analysis using the Rescale Range Analysis (R/S Analysis) and the Multifractal Detrended Fluctuation Analysis respectively (MF-DFA). It was determined that the system was fractal and there were no trends indicative of increasing fractality and complexity with increasing height. The second objective was the qualitative and quantitative chaotic characterization of the wind speeds using phase-space portraits and the Largest Lyapunov Exponent (LLE) respectively. The methods confirm the results of the fractal analyses. A simple non-linear prediction algorithm, Empirical Dynamical Modeling (EDM) was then used to forecast the wind speeds using a moving window. It was determined that the EDM was comparable to persistence. It beats this benchmark model in the very short term range of one time step or 10 minutes. The third objective was to cluster the data using Self-Organizing Maps (SOMs), having identified the optimum number of clusters as 4 using the Elbow and Silhouette Methods, among others. Three continuous intervals belonging to a particular cluster, which represented approximately 50 percent and over of the input vectors or rows from the data frame were identified. These intervals were then used as inputs into a Long Short-Term Memory Network (LSTM) with variables, pressure and wind speeds, as well as a lagged series LSTM with embedding dimension, d, and time delay (tau). These were compared to the Moving window Auto Regressive Integrated Moving Average (ARIMA) and to persistence. It was determined that the lagged series LSTM improved on the LSTM with wind speed and pressure series inputs, and all models beat persistence. The lagged LSTM beats the Moving ARIMA for at least 2 of the forecasting times of 60 and 120 minutes for all intervals.Includes bibliographical references

Identification and prediction of nonlinear dynamics

Chaos theory and associated analyses are being applied to a growing number of disciplines. Studies of biological and ecological systems have shown the widest application of chaotic analyses thus far. When studying these systems, it is often only possible to measure a subset of the system\u27s many variables. To effectively perform a number of the analyses required to study a chaotic system, it is necessary to identify a complete strange attractor for the system. Consequently, it is necessary to reconstruct the system\u27s strange attractor from the available data. Many different methods exist for reconstructing strange attractors, but the effectiveness of each of these methods has not been studied and compared. This investigation examines the effectiveness of various reconstruction methods used to preserve the fractal structure of the attractor and the exponential divergence of nearby trajectories in an effort to determine the optimal method for reconstructing strange attractors. With an optimal method to reconstruct strange attractors for chaotic physical systems, engineers and scientists can more successfully characterize a nonlinear system and apply methods to predict its future behavior

Causes and Effects of Chaos

Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical tools used in the literature are explained and demonstrated, the rudiments of the current theory are outlined and that theory is discussed in the context of several examples: an electronic circuit, a chemical reaction and a system of satellites in the solar system

Bifurcation scenarios, dynamical integrity and control of noncontact atomic force microscopes

The research focuses on the description of the global dynamical behavior of a reduced-order model of noncontact Atomic Force Microscope. Different numerical analyses and continuation techniques are carried out to investigate the evolution of the main system periodic solutions and relevant basins of attraction under variations of the most significant system parameters. Local bifurcations, stability boundaries and basin erosion processes around primary and subharmonic resonance regions are studied in presence of both the parametrical horizontal excitation and the external one, and the obtained behavior charts are used not only to compare the results with the literature ones, but also as practical instruments to characterize the operation ranges in terms of the selected parameters. With the same perspective, dynamical integrity concepts, such as detection of basins of attraction, and quantification of their erosion process via integrity measures, are applied to determine acceptable frequency-dependent thresholds associated with a priori safe design targets. Furthermore, an external feedback control is introduced with the aim to take the system response to a selected reference one, thus providing a simple and efficient method to avoid possible unstable motions. Upon checking the effectiveness of the procedure in the weakly nonlinear regime via a perturbation approach, several numerical analyses in the strongly nonlinear regime are accomplished to achieve a description of its dynamical behavior as a function of the newly inserted parameters, and to critically evaluate the effectiveness of the control actuation on the system dynamics, with also a view to the overall response scenario

Informatics for EEG biomarker discovery in clinical neuroscience

Neurological and developmental disorders (NDDs) impose an enormous burden of disease on children throughout the world. Two of the most common are autism spectrum disorder (ASD) and epilepsy. ASD has recently been estimated to affect 1 in 68 children, making it the most common neurodevelopmental disorder in children. Epilepsy is also a spectrum disorder that follows a developmental trajectory, with an estimated prevalence of 1%, nearly as common as autism. ASD and epilepsy co-occur in approximately 30% of individuals with a primary diagnosis of either disorder. Although considered to be different disorders, the relatively high comorbidity suggests the possibility of common neuropathological mechanisms. Early interventions for NDDs lead to better long-term outcomes. But early intervention is predicated on early detection. Behavioral measures have thus far proven ineffective in detecting autism before about 18 months of age, in part because the behavioral repertoire of infants is so limited. Similarly, no methods for detecting emerging epilepsy before seizures begin are currently known. Because atypical brain development is likely to precede overt behavioral manifestations by months or even years, a critical developmental window for early intervention may be opened by the discovery of brain based biomarkers. Analysis of brain activity with EEG may be under-utilized for clinical applications, especially for neurodevelopment. The hypothesis investigated in this dissertation is that new methods of nonlinear signal analysis, together with methods from biomedical informatics, can extract information from EEG data that enables detection of atypical neurodevelopment. This is tested using data collected at Boston Children’s Hospital. Several results are presented. First, infants with a family history of ASD were found to have EEG features that may enable autism to be detected as early as 9 months. Second, significant EEG-based differences were found between children with absence epilepsy, ASD and control groups using short 30-second EEG segments. Comparison of control groups using different EEG equipment supported the claim that EEG features could be computed that were independent of equipment and lab conditions. Finally, the potential for this technology to help meet the clinical need for neurodevelopmental screening and monitoring in low-income regions of the world is discussed

Towards intelligent aquaculture. Development of an early Biological Warning System to monitor exposure to contaminants and fish welfare: from artificial vision to systems modelling

161 p.Etorkizun hurbilera begira gizakion kopurua ez ezik (Gerland et al., 2014; United Nations, 2014), itsas elikagaien ekoizpenaren eta kontsumoaren hazkundea espero da. Azken honen hazkundeak, batipat, akuikulturak bideratuko du (German Advisory Council on Global Change - WBGU, 2013). Akuikultura, ekoizpena handiagotzeko helburuagaz, itsas organismoen hazkunde kontrolatua da. Bertatik lortzen diren produktuak era askotakoak izan daitezke: arrainak, moluskuak, krustazeoak, algak eta itsas landareak dira ezagunenak. Baina kokodriloak, dortokak eta beste motatako zenbait animalia urlehortarrak ere ekoizten dira. Era berean, akuikultura instalazioek kokapen desberdinak euki ditzakete: itsas kostaldean zein itsasadarretan baina lur barnean zein ibai edo lakuen ondoan ere aurkitu daitezke.Beste edozein ekoizpen prozesu bezala, sortzen diren produktuen kalitatea erronka garrantzitsuenetariko bat da. Eta kalitatea, kontuan izanda sortzen diren produktuak gizakion kontsumorako direla, ekoiztutako animalien osasungarritasunarekin bat doa. Kezkarik garrantzitsuenetarikoa itsas elikagaien ekoizpenean ur ingurugiroan dauden kutsatzatzaileek ekoiztutako produktuetan duten efektua da. Kutsatzaileak gero eta kantitate haundiagoetan azaltzen dira eta gainera, gero eta kutsatzaile berri gehiago agertzen dira (Bevan et al., 2012; Roose et al., 2011). Kutsatzaileak ez dira bakarrik uretara izurtzen, animaliei emoten zaien elikagaiak ere kutsatuta egon daitezke (Dahle et al., 2010; Dobson et al., 2008; Sharma and Paradakar, 2010). Hoierariko kutsatzaile askok ere, animalien ongizatean efektu negatiboak izaten dituzte eta galera ekonomiko haundiak sortarazten dituzte. Gaur egun, kutsatzaileak antzemateko metodo ez-inbasiboen galera dago. Hare eta gehiago, kutsatzaile berriak detektatzeko metodorik ez dago, antzemate metodoak kutsatzaile bakoitzeko ezpezifikoki diseinatsen direlako eta gero eta kutsatzaile berri gehiago detektatzen direlako, adibidez gizakion kontsumorako uran (Dahle et al., 2010; Roose et al., 2011).Tesi honek, monitorizazio metodologia ez inbasibo bat garatzen du non arrainak Alerta Sistema Biologikoa (Biological Warning System - BWS) giza erabiltzen dira. Arrainak, euren igeriketa jokabidea aztertuz, sensore bat balira kontzideratzen dira. Metodologia honen abantailarik esanguratsugarrienak dira: i) teknologia eskuragarria bideratzea, ii) monitorizazio on-line-a uzten duela eta iii) mota ezberdinetako kutsatzaileekin, ezagun ala ezezagun, lan egiteko ahalmena izango duela.Azkenik, proposatutako Alerta Sistema Biologikoa beste hierarkia handiagoko sistemekin egon beharko luke harremanetan eta sistema guzti hauen emaitzak ekoiztutako produktuen trazabilitate agiriekin batera joan beharko lukete azken produktuak merkatuara heltzerakoan. Hierarkia handiagoko sistema hauek Arrizku Analisis eta Punto Kritikoen Kontrol (Hazard Analysis and Critical Control Point - HACCP) eta Arrain Ongizate Segurtasun Sistemak (Fish Welfare Assurance System - FWAS) dira hain zuzen ere (van de Vis et al., 2012).HipotesiaTesi hau ondoko hipotesian oinarrituta dago:Arrain multzo batek sentsore biologiko bat izango balitz bezala jokatuko du; arrainen jokabidea, arrainen erantzun biologiko eta fisiologikoak batzen dituena, kanpo estimuluen aurrean modu ez inbasiboan neurtu daitekelako. Honek, arrain-sistemaren jokabidearen aldaketa on-line monitorizaziorako tresna gisa erabiltzea bideratzen du.Helburuak eta ekarpenakTesi honen helburu garrantzitsuena arrainak sentsore giza erabiliz, arrainen ongizatea on-line monitorizatzeko eta akuikultura industrian aplikatzeko tresna baten disenua eta eraikuntza da. Honetarako, jakintza alor anitzak jorratzen dira, hala nola biologia, etologia, ingurugiroaren ikuskaketa, arrainen ongizatea, elikagain kalitatearen fidagarritasuna, sistemen ingenieritza edota seinaleen prozesamendu ez-lineala.Lupia (Dicentrarchus labrax) izan da Tesi hau garatzeko aukeratutako arrain mota bi arrazoi nagusirengatik. Batetik, mediterraneo itsasoko herrialdeetako akuikultura produkzioan oso hedatuta dagoelako eta, aurrekoarekin loturik, lupiaren produkzioan arrainen ongizatearen eta azken produktuaren kalitatea oso garrantzitsua delako, gehienbat gizakion kontsumora bideraturiko produktua delako hain zuzen.Bigarren mailako helburuak, proposatutako tresnaren ebaluazioarekin lotuta egoteaz gain, diseinatutako sistemak aldagai ezberdinen menpean duen portaera aztertzen dute, hala nola:Arrain-sistemaren erantzuna arrain kopurua aldatuz.Arrain-sistemaren erantzuna arrainen dietari selenioa gehitzerakoan.Arrain-sistemaren erantzuna urari kutsatzaile neurotoxikoa den metilmekurioa gehitzerakoan.Hirugarren eta azken helburua, aurreko bi helburuen bitartez garatutako informazioa, ezagutza-eredu batetan isladatzearekin dator bat.Helburu hauekin loturik, Tesi honek ondoko ekarpenak ditu:Sistemen ingenieritzan oinarritutako lan metodologia berri bat garatu da.Ez-inbasiboa, moldakorra, merkea eta Alerta Sistema Biologiko sistema batetan egokitzeko gai den irudi analisian oinarritutako tresna bat garatu da.Perturbazio bati erantzunez arrain multzoaren mugimendu ereduan oinarrituriko sentsore bat garatu da.Arrainen multzoaren igeriketa ereduari aplikatu ahal zaizkion zenbait seinale prozesaketa metodo ez-lineala garatu dira.Arrainen erantzuna kanpotiko perturbazio baten aurrean modelatzen duen eredu matematikoa garatu da.Garatutako metodologia eta monitorizazio tresna zenbait kasu experimentaletan aztertu da.Ikerketa MetodologiaIkerketa metodologiaren aldetik, Glass-ek (1995) proposatutako pausu berdinak jarraitu dira. Honela, Tesia lau faseetan banaturik dago: jakinarazte fasea, fase proposizionala, fase analitikoa eta ebaluaketa fasea. Era berean, erabilitako metodologia zati zientifiko eta zati enpiriko baten batura da. Ikerketa metodologia honek erdiesten dituen onurak ondokoak dira:Erabilitako ikerketa metodologia arrainen multzoak eta bere ingurugiroak osatzen duten sistemaren dinamika ulertzeko ezin bestekoa izan da.Tesian garatuta tresna zenbait kasu partikularretan aztertu da. Honela, tresnaren onurak ez ezik bere mugak ere aztertu ahal izan dira.Garatutako tresna benetako akuikultura instalazio batetan inplementatzeko bideragarritasuna aztertu da Norbegiako ¿Centre for Autonomous Marine Operations and Systems (AMOS)¿ Ikerketa Zentruan hiru hilabeteko ikerketa egonaldiari esker. Egonaldia Europa Batasuneko ikerlari mugikortasun beka batekin finantziatu da ¿European Economic Area (EEA) Researcher Mobility and Co-operation Grant, NILS Science and Sustainability Programme.Publikatutako lanen maila, kalitatea eta irismena handiagotu da.Etorkizunean, ikerketa proiektu eta gertaeretan parte hartzeko ahalmenaren handiagotzea, ikerlariaren kontaktu sarearen handiagotzearekin batera.Azkenik, Tesiaren zati experimentalak Yin-en (1993) kasu-ikerketaren oinarriak jarraitzen ditu. Kasu-ikerketa metodoa ezagutza induktiboa sortzeko erabiltzen da eta eguneroko gertakari erreal eta komplexuei aplikatzerakoan baliagarritasun handia erakusten du.Lanaren GarapenaTesi honen lehenengo zatian, irudi analisian oinarritutako metodologia ez-inbasibo bat garatzen da. Metedologia hau, video grabaketa, irudi prozesaketa eta seinale ez-linealen analisi eta prozesaketan datza. Arrain multzoaren erantzuna gertaera estokastiko edo aleatorio baten aurrean aztertzen eta neurtzen da. Irudietan agertzen diren objetuen antzematea, fluxu optikoko algoritmo baten bitartez burutzen da. Honela arrainak detektatu ez ezik, irudien atzealdea eta bestelako efektu ez desiratuak ezabatzen dira. Azkenik, arrain multzoaren zentruak jarraitzen duen ibilbidea, Shannon (Shannon, 1951, 1948) eta permutazio entropia (Bandt and Pompe, 2002) batetik; eta Higuchi (1988), Katz (1988) eta Castiglioni-k (2010) proposatutako Katz algoritmoaren aldaketaren bitartezko dimentsio fraktal algoritmoez aztertzen da.Garatutako monitorizazio tresna hiru kasu partikularretan aztertu zen. Lehenengoan, hiru lupia multzo aztertu ziren. Horietariko bi antzekoak ziren (kontrolak eta elastomero batekin markatutakoak), eta hirugarrena aldiz, 9 egunez metilmerkuriodun (4 µg MeHg/L) uretan murgildutako arrainez osoturik zegoen. Shannon-en entropia eta Katz-Castiglioni-ren algoritmoek izan ziren emaitza onenak erakutzi zutenak. Esan daiteke biek, arrainen erantzunak modu ez inbasibo batean kuantifikatzeko garaian, ahalmen nahikoa erakutsi zutela. Hala ere, Katz-Castiglioni-ren algoritmoak Shannon-en entropia baino askoz ere pisu konputazional handiagoa erakutsi zuen.Hasiera baten, arrain multzoa osotzen zuten animalien kopurua inolako eraginik euki zitzakela animalia sistema osoaren jokabidean garatutako aztertze metodologiari dagokionez, ez zen uste. Honetan sakontzeko, bigarren kasu partikular bat diseinatu zen, non sistemako aldagai bakarra arrain kopurua zen. Bigarren kasu partikular honen helbururik garrantzitsuena sistemarentzako Lan Puntuaren zehazpena zen, hau da, arrainek inolako perturbazio barik daudeneko egoera dinamikoa. Monitorizazio tresna doitzeko eta perturbazioak egoera ¿normaletik¿ bereiztu ahal izateko, Lan Puntuaren zehazpena ezin bestekoa da. Honetarako bi experimentu disenatu ziren:Arrain kopuruaren murrizketa. Arrain kopurua 50etik 1era pasatu zen 4 asteren buruan (50, 25, 13 eta arrain 1 aste bakoitzeko).Arrain kopururaren hazkuntza. Sistema osotzen duten arrainen kopurua 1etik 5era hazi zen egunero arrain berri bat sartuz (1, 2, 3, 4 eta 5 arrain egun bakoitzeko).Hirugarren kasu partikularrean, aldiz, arrainen urari bi sustantzia ezberdin gehitu zitzaizkion. Batetik, sodio selenitoa (Na2SeO3, 10 µg/L) gehitu zitzaion 7 egunen zehar eta bestetik metilmerkurioa (4 µg MeHg/L) 14 egunen zehar.Tesiaren azken atalean, aurreko kasuetan garatutako jakituriarekin ezagutza eredu bat eraiki da. Eredua, 3 azpi-ereduez osatuta dago, zeintzuk sistemaren eguneroko erantzuna eta Shannon-en entropiaren emaitzak, egoera basalean, gertaera aleatorio bateri erantzunez eta azken bien arteko erlazioari begira, integratzen dituzte.OndorioakOndoren Tesiaren ondoriorik esanguratsuenak adierazten dira:Arrainen monitorizaziorako tresna garatu da. Tresnaren oinarrian irudi eskuraketa eta prozesaketa egoteaz gain, arrain multzoak osatzen duen zentruaren ibilbidearen analizi ez-lineala ere badago.Arrainen sistemaren dinamika aztertzeko garaian Shannon-en entropia parametrorik onena izan da.1 eta 5 arrainen artean, lubia arrain sistemaren Shannon-en entropia eta sistema osotzen duten arrain kopuruaren erlazioa oso estua da.Lubia arrain sistemaren eta Shannon-en entropiaren arteko erlazioa exponentziala da.Garatutako tresna kasu partikularrei aplikatzerakoan era egokian lan egin du. Honek, ikerketa lanaren hasierako Tesia berretsi du.Espero zen bezala eta garatutako tresnak konfirmatuta, urari gehitutako sodio selenito kontsentrazioak ez du inolako efekturik izan lubia sistemaren Shannon-en entropian.Era berean, garatutako tresnak konfirmatu du urari gehituriko metilmerkurioak efektu ezezkorra izan duela lubia sistemaren Shannon-en entropian.Tresnaren aplikazioak kasu partikularretan lubia sistemaren Shannon-en entropiaren eguneroko balioa ez ezik bere egunean-eguneko bilakaera kontutan hartu behar dela erakutsi du.Garatutako ezagutza eredua lubia arrain sistemaren jokabide erantsunaren eguneroko bilakaeran oinarrituriko 3 azpi-ereduz osoturik dago: Azpi-eredu basala, gertaera aleatorioaren erantzuneko azpi eredua eta aurreko biak erlazionatzen dituen azpi-eredua.Garatutako tresna, zenbait hobekuntzekin, on-line monitorizazio arkitektura batetan inplementatzeko gaitasuna duela esan daiteke. Eta beraz, era basatian nahiz akuikulturan, arrain multzoak monitorizatzeko edo/eta ur kutsadura antzemateko ingurumen-programetan tresna baliagarria izan daiteke.Beraz, ikerketa lan honen Tesia arrainak sentsore biologiko moduan erabil daitezkela da; kanpo perturbazioen eraginez euren portaeran agertzen diren aldaketak modu ez-inbasiboan neurtu daitezkelako.Gerorako LanaGaur egun Errezirkulazio Akuikultura Sistemek (Recirculating Aquaculture Systems - RAS) gero eta aplikazio handiagoa dute munduan zehar. Tesi honetan oinarrituriko teknologia baten garapenak zekulako erabilgarritasuna eukiko luke mota honetako sistemetan, batez ere, produkzio unitate osoa monitorizatu daitekelako. Era berean, ur kutsadura antzemateko ingurumen-programetan tresna baliagarria izan daiteke. Azken honetarako, teknologia honetan oinarrituriko ikusketa guneak garatu beharko lirateke.Urpean lan egiteak desabantail ugari ditu. Hau ekiditeko datu eskuraketa teknika ezberdinak aztertu beharko lirateke, hala nola izpi-infragorriak, sonar teknologia edota irudi hiperespektralak zenbait esatearren.Azkenik, teknologia honetatik lortutako datuak bestelako tekniken bidez lorturiko datuekin bat egin beharko lirateke. Honela, arrainen datu biokimiko, histologiko eta fisiologikoez gain, teknika ez-inbasiboen bidez neurtutako igeriketa jokabideak ere kontutan hartu beharreko parametroa izan beharko luke.Research Center for Experimental Marine Biology and Biotechnology - Plentzia Marine Station (PiE

Essays on the nonlinear and nonstochastic nature of stock market data

The nature and structure of stock-market price dynamics is an area of ongoing and rigourous scientific debate. For almost three decades, most emphasis has been given on upholding the concepts of Market Efficiency and rational investment behaviour. Such an approach has favoured the development of numerous linear and nonlinear models mainly of stochastic foundations. Advances in mathematics have shown that nonlinear deterministic processes i.e. "chaos" can produce sequences that appear random to linear statistical techniques. Till recently, investment finance has been a science based on linearity and stochasticity. Hence it is important that studies of Market Efficiency include investigations of chaotic determinism and power laws. As far as chaos is concerned, there are rather mixed or inconclusive research results, prone with controversy. This inconclusiveness is attributed to two things: the nature of stock market time series, which are highly volatile and contaminated with a substantial amount of noise of largely unknown structure, and the lack of appropriate robust statistical testing procedures. In order to overcome such difficulties, within this thesis it is shown empirically and for the first time how one can combine novel techniques from recent chaotic and signal analysis literature, under a univariate time series analysis framework. Three basic methodologies are investigated: Recurrence analysis, Surrogate Data and Wavelet transforms. Recurrence Analysis is used to reveal qualitative and quantitative evidence of nonlinearity and nonstochasticity for a number of stock markets. It is then demonstrated how Surrogate Data, under a statistical hypothesis testing framework, can be simulated to provide similar evidence. Finally, it is shown how wavelet transforms can be applied in order to reveal various salient features of the market data and provide a platform for nonparametric regression and denoising. The results indicate that without the invocation of any parametric model-based assumptions, one can easily deduce that there is more to linearity and stochastic randomness in the data. Moreover, substantial evidence of recurrent patterns and aperiodicities is discovered which can be attributed to chaotic dynamics. These results are therefore very consistent with existing research indicating some types of nonlinear dependence in financial data. Concluding, the value of this thesis lies in its contribution to the overall evidence on Market Efficiency and chaotic determinism in financial markets. The main implication here is that the theory of equilibrium pricing in financial markets may need reconsideration in order to accommodate for the structures revealed

Intriguing pendula : founding metaphors in the analysis of economic fluctuations

The paper is an inquiry into the definition of the early econometric programme, namely into the discussions which Frisch and Schumpeter held in the early 1930s about the most suitable model for representing innovations, change and equilibrium in economics. The argument and its framework are briefly presented in the first section. The 1931 correspondence between the two founders of the Econometric Society is discussed in the second section. It provides a magnificent example of the importance of rhetorics in economics, of the heuristic role of constitutive metaphors in a research programme and of the difficulties in defining the most suitable mathematical formal ism for dealing with cycles and structural change. The third section presents the conclusion of the story: the bifurcation between the resulting contributions made by Frisch (Propagation problems and impulse problems in dynamic economics, pp. 171-205 in Koch, K. (ed.), Economic Essays in Honour of Gustav Cassel, London, Frank Cass, 1933) and Schumpeter {Business Cycles, New York, McGraw, 1939; and the posthumous volume, History of Economic Analysis, London, Routledge, 1954). Finally, the fourth section presents an alternative epilogue, highlighting some of the hidden implications of these verbal accounts of pendula as the founding metaphor for business cycles. The paper is based upon as yet unpublished papers that were found in Frisch's Collections (Oslo University Library and Frisch's Rommet at the Institute of Eco nomics) and Schumpeter's Collection (Harvard Universit)..info:eu-repo/semantics/publishedVersio