Correlation Structure Research Articles

How to achieve model-robust inference in stepped wedge trials with model-based methods?

A stepped wedge design is an unidirectional crossover design where clusters are randomized to distinct treatment sequences. While model-based analysis of stepped wedge designs is a standard practice to evaluate treatment effects accounting for clustering and adjusting for covariates, their properties under misspecification have not been systematically explored. In this article, we focus on model-based methods, including linear mixed models and generalized estimating equationswith an independence, simple exchangeable, or nested exchangeable working correlation structure. We study when a potentially misspecified working model can offer consistent estimation of the marginal treatment effect estimands, which are defined nonparametrically with potential outcomes and may be functions of calendar time and/or exposure time. We prove a central result that consistency for nonparametric estimands usually requires a correctly specified treatment effect structure, but generally not the remaining aspects of the working model (functional form of covariates, random effects, and error distribution), and valid inference is obtained via the sandwich variance estimator. Furthermore, an additional g-computation step is required to achieve model-robust inference under non-identity link functions or for ratio estimands. The theoretical results are illustrated via several simulation experiments and re-analysis of a completed stepped wedge cluster randomized trial.

Structural and functional correlates of olfactory reward processing in behavioral variant frontotemporal dementia

The behavioral variant of frontotemporal dementia (bvFTD) includes symptoms that reflect altered pursuit of rewards, including food, alcohol, and money. Little is known, however, about how these reward changes relate to atrophy and functional connectivity within reward-related regions. The goal of this study was to examine the structural and functional correlates of valence perception for olfactory rewards in 24 patients with bvFTD. Regression analysis of resting-state brain functional connectivity indicated that more positive valence ratings of olfactory stimuli were predicted by ventral pallidum connectivity to other reward circuit regions, particularly functional connectivity between ventral pallidum and bilateral anterior cingulate cortex/ventromedial prefrontal cortex. Structural analysis showed that atrophy of the anterior cingulate cortex was also significantly associated with perceiving stimuli as more rewarding. Finally, there was a significant interaction between ventral pallidum connectivity and atrophy of the anterior cingulate cortex. More specifically, the ventral pallidum connectivity had a greater effect on the positive perception of olfactory stimuli in the setting of low anterior cingulate cortex volume. These findings indicate that atrophy and functional connectivity within reward-relevant regions exert independent and interacting effects on the perception of pleasantness in bvFTD, potentially due to changes in hedonic “liking” signals.

Bivariate iterated Farlie–Gumbel–Morgenstern stress–strength reliability model for Rayleigh margins: Properties and estimation

In this paper, we propose bivariate iterated Farlie–Gumbel–Morgenstern (FGM) due to [Huang and Kotz (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions. Biometrika 71(3), 633–636. https://doi.org/10.2307/2336577] with Rayleigh marginals. The dependence stress–strength reliability function is derived with its important reliability characteristics. Estimates of dependence reliability parameters are obtained. We analyse the effects of dependence parameters on the reliability function. We found that the upper bound of the positive correlation coefficient is attaining to 0.41 under a single iteration with Rayleigh marginals. A comprehensive comparison between classical FGM with iterated FGM copulas is graphically examined to assess the over or under estimation of reliability with respect to α and β. We propose a two-phase estimation procedure for estimating the reliability parameters. A Monte-Carlo simulation study is conducted to assess the finite sample behaviour of the proposed reliability estimators. Finally, the proposed estimators are examined and validated with real data sets.

A Simulation Study of the Effects of Additive, Multiplicative, Correlated, and Uncorrelated Errors on Principal Component Analysis

ABSTRACTMeasurement errors are ubiquitous in all experimental sciences. Depending on the particular experimental platform used to acquire data, different types of errors are introduced, amounting to an admixture of additive and multiplicative error components that can be uncorrelated or correlated. In this paper, we investigate the effect of different types of experimental error on the recovery of the subspace with principal component analysis (PCA) using numerical simulations. Specifically, we assessed how different error characteristics (variance, correlation, and correlation structure), loading structures, and data distributions influence the accuracy to estimate an error‐free (true) subspace from sampled data with PCA. Quality was assessed in terms of the mean squared reconstruction error and the congruence to the error‐free loadings, using the pseudorank and adjusting for rotational ambiguity. Analysis of variance reveals that the error variance, error correlation structure, and their interaction with the loading structure are the factors mostly affecting quality of loading estimation from sampled data. We advocate for the need to characterize and assess the nature of measurement error and the need to adapt formulations of PCA that can explicitly take into account error structures in the model fitting.

Correlation of Valence electron structure and properties of monolayer graphene and MX2 (M=Mo, W; X=S, Se, Te): Empirical Electron Theory (EET) investigation

The atomically thin layers of transition-metal dichalcogenide (TMDC) materials have garnered considerable attention due to their exceptional electrical, optical, mechanical, and thermal properties. Hence, it is important to investigate the mechanism of their excellent properties. In this paper, the study is focused on the correlation between valence electron structures (VESs) and mechanical as well as thermal properties of graphene and MX2 (M = Mo, W; X = S, Se, Te) for revealing their essential mechanisms of properties with an empirical electron theory (EET). A model of Young's modulus is built for the monolayer graphene and MX2 (M = Mo, W; X = S, Se, Te) based on the VES, which has been verified by the observed ones of elements in the 4th to 6th periods in the periodic table of elements. The calculated bond lengths and mechanical and thermal properties of graphene and MX2 are in good agreement with experimental ones. The study reveals that the thermal and mechanical properties of MX2 strongly depend on their valence electron structures. It shows that the melting point, cohesive energy, thermal conductivity, and Young's modulus are modulated by covalence electron pair nA, the averaged covalence electron per atom nc/atom, covalence electron pair nA and linear density of covalent electron on the strongest bond ρl, respectively. The study helps explain the thermal and mechanical properties of two-dimensional (2D) materials and also supplies a reference for their design with high performance by modulating their valence electron structures.

Research and application of joint-constrained inversion of transient electromagnetic multivariate parameter

Due to the phenomena of stratigraphic inclination, complex structure, and lateral discontinuity of resistivity or layer thickness in most of the coal seams, the traditional one-dimensional transient electromagnetic inversion method has limitations in interpretation accuracy. In addition, two- and three-dimensional inversion and artificial intelligence inversion have problems of large computation and large sample size, respectively, which limit their application in small- and medium-sized engineering exploration. To improve the inversion effect, this study proposes a method of joint-constrained inversion of transient electromagnetic multivariate parameters. This method achieves the joint constraint inversion of the transient electromagnetic multi-parameter by making full use of the geological data and a priori information to construct the initial model and adding the constraints such as the resistivity, the thickness, and the layer interface of each layer in the inversion objective function, and at the same time, taking into account the spatial correlation of the stratigraphic structure between the neighboring measurement points, as well as the transverse and vertical constraints between the measurement points along the direction of the survey line and perpendicular to the survey line. First, a series of typical geoelectric models are established and numerically simulated, and the results are compared with those of the traditional inversion method to verify the applicability and effectiveness of the method. Then, the constrained inversion is carried out on the physical simulation and measured data, and the results are in good agreement with the actual geological conditions. The numerical simulation, physical simulation and measured data inversion results consistently prove that this method can effectively reduce the uncertainty of the inversion at the isolated measuring points, improve the spatial continuity of the formation boundary, and better reflect the actual geoelectric characteristics of the formation.

Structural and functional neural correlates of sensorimotor deficits in progression of hepatic encephalopathy

Hepatic encephalopathy (HE) is a neurological condition that occurs as a complication of liver dysfunction that involves sensorimotor symptoms in addition to cognitive and behavioral changes, particularly in cases of severe liver disease or cirrhosis. Previous studies have reported spatially distributed structural and functional abnormalities related to HE, but the exact relationship between the structural and functional alterations with respect to disease progression remains unclear. In this study, we performed surface-based cortical thickness comparisons and functional connectivity (FC) analyses between three cross-sectional groups: healthy controls (HC, N=51), patients with minimal hepatic encephalopathy (MHE, N=50), patients with overt hepatic encephalopathy (OHE, N=51). In addition to the distributed cortical thinning that is extensively thought to be associated with cognitive decline in HE, we found significant cortical thickening in the left parahippocampal gyrus cortex in the OHE group (p<0.001, p=0.009) as compared to the HC and MHE group respectively, which is further corroborated by the significant correlation between the cortical thickness and digit symbol test (DST) scores. Furthermore, the decreased FC between the right postcentral gyrus and several sensory regions (bilateral somatosensory and visual cortices) was found to be significant in MHE patients as compared to the HC group. Our results revealed cross-sectional structural and functional variations concerning disease progression across different subsystems (e.g., visual, motor and sensory), providing evidence that can potentially explain the mechanisms underlying the sensorimotor and cognitive deficits related to HE.

The brain topological alterations in the structural connectome and correlations with clinical characteristics in type 1 narcolepsy

The brain topological alterations in the structural connectome and correlations with clinical characteristics in type 1 narcolepsy

Dynamic DeFi-G7 stock markets interactions and their potential role in diversifying and hedging strategies

AbstractThis study examines the link between stocks and decentralized finance (DeFi) in terms of returns and volatility. Major G7 exchange-traded funds (ETFs) and various highly traded DeFi assets are considered to ensure the robustness of the empirical experiment. Specifically, this study applies the vector autoregression generalized autoregressive conditional heteroskedasticity (VAR-GARCH) model to examine the information transmission of these two markets on a two-way basis and the dynamic conditional correlation (DCC)-GARCH model to assess the bivariate correlation structure between each DeFi and ETF pair. The volatility spillover analysis proves a contagion effect occurred between different geographic markets, and even between markets of different natures and typologies, during the most turbulent moments of the COVID-19 crisis and the war in the Ukraine. Our results also reveal a weak positive correlation between most DeFi and ETF pairs and positive hedge ratios that approach unity during turbulent times. In addition, DeFi assets, except for the Bazaar (BZR) Protocol, can offer diversification gains when included in financial investment portfolios. These results are particularly relevant for portfolio managers and policy-makers when designing investment strategies, especially during periods of financial crisis.

Geostatistical Simulation of Daily Rainfall Fields—Performance Assessment for Extremes in West Africa

Abstract The spatial description of high-resolution extreme daily rainfall fields is challenging because of the high spatial and temporal variability of rainfall, particularly in tropical regions due to the stochastic nature of convective rainfall. Geostatistical simulations offer a solution to this problem. In this study, a stochastic geostatistical simulation technique based on the spectral turning bands method is presented for modeling daily rainfall extremes in the data-scarce tropical Ouémé River basin (Benin). This technique uses meta-Gaussian frameworks built on Gaussian random fields, which are transformed into realistic rainfall fields using statistical transfer functions. The simulation framework can be conditioned on point observations and is computationally efficient in generating multiple ensembles of extreme rainfall fields. The results of tests and evaluations for multiple extremes demonstrate the effectiveness of the simulation framework in modeling more realistic rainfall fields and capturing their variability. It successfully reproduces the empirical cumulative distribution function of the observation samples and outperforms classical interpolation techniques like ordinary kriging in terms of spatial continuity and rainfall variability. The study also addresses the challenge of dealing with uncertainty in data-poor areas and proposes a novel approach for determining the spatial correlation structure even with low station density, resulting in a performance boost of 9.5% compared to traditional techniques. Additionally, we present a low-skill reference simulation method to facilitate a comprehensive comparison of the geostatistical simulation approaches. The simulations generated have the potential to provide valuable inputs for hydrological modeling.

Compact Eddy Structure of Turbulence Aft of an Inclined Slender Body

The compact eddy structure and the correlation structure of three distinct turbulent boundary layers were investigated, with a view to understanding the turbulent inflow seen by a stern-mounted propulsion tail rotor. The first is of a zero-pressure-gradient, equilibrium, flat plate boundary layer, and the second and third are on the lee and port sides of a body of revolution (BOR) inclined at a 5° angle of attack. The BOR is an axially symmetric body with an ellipsoidal nose, straight midbody, and 1.17-diam-long stern ramp with a stern-mounted tail rotor. Measurements on the BOR were taken at a body-diameter-based Reynolds number of 600,000 and a rotor advance ratio of J=1.27. Particle image velocimetry was taken at two principal locations at the stern of the BOR. Distortion of the correlation function was evident in the compact eddy structure of the BOR boundary layer on both the lee and port sides. On the port side, the turbulent modes were dominated by the embedded shear layer, whereas the lee side showed more significant pressure gradient effects in the eddy structure.

Discriminative Deep Canonical Correlation Analysis for Multi-View Data.

Over the past few years, multimodal data analysis has emerged as an inevitable method for identifying sample categories. In the multi-view data classification problem, it is expected that the joint representation should include the supervised information of sample categories so that the similarity in the latent space implies the similarity in the corresponding concepts. Since each view has different statistical properties, the joint representation should be able to encapsulate the underlying nonlinear data distribution of the given observations. Another important aspect is the coherent knowledge of the multiple views. It is required that the learning objective of the multi-view model efficiently captures the nonlinear correlated structures across different modalities. In this context, this article introduces a novel architecture, termed discriminative deep canonical correlation analysis (D2CCA), for classifying given observations into multiple categories. The learning objective of the proposed architecture includes the merits of generative models to identify the underlying probability distribution of the given observations. In order to improve the discriminative ability of the proposed architecture, the supervised information is incorporated into the learning objective of the proposed model. It also enables the architecture to serve as both a feature extractor as well as a classifier. The theory of CCA is integrated with the objective function so that the joint representation of the multi-view data is learned from maximally correlated subspaces. The proposed framework is consolidated with corresponding convergence analysis. The efficacy of the proposed architecture is studied on different domains of applications, namely, object recognition, document classification, multilingual categorization, face recognition, and cancer subtype identification with reference to several state-of-the-art methods.

Rumor Detection With Hierarchical Representation on Bipartite Ad Hoc Event Trees.

The rapid growth of social media has caused tremendous effects on information propagation, raising extreme challenges in detecting rumors. Existing rumor detection methods typically exploit the reposting propagation of a rumor candidate for detection by regarding all reposts to a rumor candidate as a temporal sequence and learning semantics representations of the repost sequence. However, extracting informative support from the topological structure of propagation and the influence of reposting authors for debunking rumors is crucial, which generally has not been well addressed by existing methods. In this article, we organize a claim post in circulation as an ad hoc event tree, extract event elements, and convert it into bipartite ad hoc event trees in terms of both posts and authors, i.e., author tree and post tree. Accordingly, we propose a novel rumor detection model with hierarchical representation on the bipartite ad hoc event trees called BAET. Specifically, we introduce word embedding and feature encoder for the author and post tree, respectively, and design a root-aware attention module to perform node representation. Then we adopt the tree-like RNN model to capture the structural correlations and propose a tree-aware attention module to learn tree representation for the author tree and post tree, respectively. Extensive experimental results on two public Twitter datasets demonstrate the effectiveness of BAET in exploring and exploiting the rumor propagation structure and the superior detection performance of BAET over state-of-the-art baseline methods.

Molecular engineering on tyrian puprle natural dye as TiO2 based fined tuned photovoltaic dye material: DFT molecular analysis

In this research, molecular modification is employed to see the enhancement in the efficiency of Tyrian Purple (TP), a natural dye, for organic photovoltaic materials.By using Density Functional Theory (DFT) based molecular modeling, seven new structures are designed with pi spacer to extend electron donor moieties. Teheir Frontier Molecular Orbital (FMO) analysis demonstartes their charges with a similar pattern of distributions over their Highest Occupied and Lowed Unocuupied Molecular Orbitals (HOMO/lUMO). This analysls also show their energy gaps (Egaps) to range around 2.97-3.02 eV. Their maximum absorption wavelength (λmax) demosntartes 486-490 nm range to indicate their tendency of absorbing light efficiently. Their Transition Density Matrix (TDM) analysis also reveals their facile electronic transitions without a significant charges over spacers. From calculating their photovoltaic paramters, their Light Harvesting Efficiency (LHE) reaches to 72.4-95.5 %. Also their Open Circuit Voltage (Voc) varies across 1.16-1.34 V. It is found that dyes actively adsorb onto TiO2 clusters to demonstrate their promise for tuning their Conduction Band (CB). This research is an effort for to evaluate the structural correlations to the developm photovoltaic materials through molecular-level design and optimization.

Joint Deconvolution of Astronomical Images in the Presence of Poisson Noise

We present a new framework for joint likelihood deconvolution (Jolideco) of a set of astronomical observations of the same sky region in the presence of Poisson noise. The observations may be obtained from different instruments with different resolution, and different point-spread functions (PSFs). Jolideco reconstructs a single flux image by optimizing the posterior distribution based on the joint Poisson likelihood of all observations under one of several prior distributions. Most notably, we employ a patch-based image prior that is parameterized via a Gaussian mixture model, which we train on high-signal-to-noise astronomical images, including data from the JWST and the GLEAM radio survey. This prior favors correlation structures among the reconstructed pixel intensities that are characteristic of those observed in the training images. It is, however, not informative for the mean or scale of the reconstruction. By applying the method to simulated data, we show that the combination of multiple observations and the patch-based prior leads to much improved reconstruction quality in many different source scenarios and signal-to-noise regimes. We demonstrate that with the patch prior Jolideco yields superior reconstruction quality relative to alternative standard methods such as the Richardson–Lucy method. We illustrate the results of Jolideco applied to example data from the Chandra X-ray Observatory and the Fermi Gamma-ray Space Telescope. By comparing the measured width of a counts-based and the corresponding Jolideco flux profile of an X-ray filament in SNR 1E 0102.2–7219, we find the deconvolved width of 0.″58 ± 0.″02 to be consistent with the theoretical expectation derived from the known width of the PSF.

Spatial distribution characteristics of Tibetan Buddhism principal-subordinate monastery systems in the Hehuang region

Tibetan Buddhist monasteries constitute significant cultural heritages of ethnic minorities, evolving into the Principal-Subordinate Monastery System (PSMS) with profound historical and cultural significance. This system exhibits an integrated hierarchical structure in terms of religious dominance, political administration, and cultural diffusion. Existing research primarily focuses on internal humanistic factors such as governance patterns, while there is not any research focused on its patterns and manifestations in spatial distribution. This study aims to elucidate the spatial distribution characteristics of the PSMS in Hehuang region, including the Tibetan Buddhist monasteries from all sects in the Hehuang region since the Song Dynasty It establishes a hierarchical PSMS database based on the affiliation relationships, aimed at storing spatial and property information related to PSMS. The database standardizes the naming and coding of monasteries, and classifies them through hierarchical relationships to ensure data consistency and usability. Finally, the classified and coded monastery data were analyzed by GIS tools to form the PSMS spatial distribution characterisation framework. Results reveal that (1) Monasteries demonstrate notable spatial clustering patterns from both holistic and sectarian perspectives, with density being influenced by the principal monastery’s position. (2) Almost every PSMS exhibits a spatial pattern centered around the principal monastery, with subordinate monasteries clustering within a defined range correlated with the administrative region of the principal monastery. (3) PSMS scale variations are controlled by principal monasteries, exhibiting positive correlations in hierarchical structure, control quantity, and distribution range. (4) The spatial orientation of PSMS correlates with river distribution, while the relationships among mountain ranges require further investigation. These findings provide initial insights into the spatial distribution characteristics of PSMS, confirming the spatial influence of the principal monastery. Besides, this study established an innovative spatial research framework for heritage clusters with multiple types and hierarchies. Thus, this study offers new insights into the spatial distribution of Tibetan Buddhist monastery heritage and presents a framework for further examining the spatial distribution of the Tibetan Buddhist monastery heritage in other regions, as well as other heritage clusters where connected and clustered through religious and cultural ties.

Superconducting transition of GdBa2Cu3O7-x/La0.7A0.3MnO3 (A=Sr and Ca) bilayers governed by tunable interlayer structural coupling

Bilayers composed of high-Tc cuprate superconductors (HTS) with manganites have garnered significant attention due to the complex interaction between two layers, leading to competition among various ordering phenomena. When HTS and manganties are interfaced, new functionalities can emerge that are absent in the individual components. Notably, epitaxial strain can alter crystal structures and electronic states, significantly impacting superconductivity in HTS/manganite heterostructures. In this study, we analyze the superconducting and ferromagnetic behaviors of GdBa2Cu3O7-x(GdBCO)/La0.7A0.3MnO3(LAMO, where A=Sr and Ca) bilayers, maintain a constant GdBCO thickness of 500 nm while varying the LAMO thickness from 25 to 100 nm. We focus on the local structural correlations between the two layers. Temperature-dependent EXAFS spectra were systematically measured at the Cu and Mn K-edges to investigate the local atomic structures of both layers across the superconducting transition temperature (Tc). Our spectroscopic measurements reveal distinct conformations of the MnO6 octahedra near Tc, which are absent in single-layer LSMO, indicating structural coupling between the layers. Additionally, our magnetization studies show a strong relationship between the structural coupling, magnetic anisotropy (MA), and the superconducting transition of the GdBCO/LAMO bilayers. The additional distortion of the MnO6 octahedra in the GdBCO/LAMO bilayer, likely stemming from structural coupling, may account for the changes in the superconducting transition associated with magnetic anisotropy. This modification of structural coupling can be controlled by adjusting the LAMO thickness, yet it remains highly dependent on the bond geometry of LAMO. The Mn-O bond in LCMO is stiffer than in LSMO, resulting in a smaller change in structural coupling with varying thickness in LCMO compared to LSMO. The disparity in structural coupling directly impacts the magnetic properties of the LAMO layer and subsequently influences the superconducting property ΔTc. Our findings highlight the significance of structural coupling as a critical factor affecting the superconductivity of GdBCO/LAMO bilayers. Furthermore, this study provide foundational insight for designing various applications, such as magnetic sensors and spintronic devices, by enhancing our understanding of local dynamics at the interface between ferromagnets and superconductors.

Antiferromagnetic Frustration Behavior with Face-Sharing CuAs4 Tetrahedrons in Conducting ACu6As3 (A = Li and Na).

New mixed-valence copper pnictides ACu6As3 (A = Li and Na) adopt a quasi-2D structure type, featuring alkalis and [Cu6As3]- slabs along the c-axis alternatively. The face-sharing connection between CuAs4 polyhedra leads to a higher valence state for the inside Cu ions than that of Cu ions with the other connectivity way. This is confirmed by X-ray photoemission spectroscopy results. The Cu2+ with near spin 1/2 located on bilayer triangular lattice is found to exhibit a peculiar hump in magnetic susceptibility along the c-axis and, most strikingly, nearly a constant at low temperatures from 1.8 K down to 0.4 K. Besides, high hole mobilities, 68.58 and 645.16 cm2 V-1 S-1, are observed in LiCu6As3 and NaCu6As3, respectively. These compounds provide a novel material system for researching the relationship among structure, valence state, and spin correlation in frustrated lattice.

Structure of Multi-State Correlation in Electronic Systems.

Beyond the Hohenberg-Kohn density functional theory for the ground state, it has been established that the Hamiltonian matrix for a finite number (N) of lowest eigenstates is a matrix density functional. Its fundamental variable─the matrix density D(r)─can be represented by, or mapped to, a set of auxiliary, multiconfigurational wave functions expressed as a linear combination of no more than N2 determinant configurations. The latter defines a minimal active space (MAS), which naturally leads to the introduction of the correlation matrix functional, responsible for the electronic correlation effects outside the MAS. In this study, we report a set of rigorous conditions in the Hamiltonian matrix functional, derived by enforcing the symmetry of a Hilbert subspace, namely the subspace invariance property. We further establish a fundamental theorem on the correlation matrix functional. That is, given the correlation functional for a single state in the N-dimensional subspace, all elements of the correlation matrix functional for the entire subspace are uniquely determined. These findings reveal the intricate structure of electronic correlation within the Hilbert subspace of lowest eigenstates and suggest a promising direction for efficient simulation of excited states.

Revisiting elastic string models of forward interest rates

Twenty five years ago, several authors proposed to describe the forward interest rate curve (FRC) as an elastic string along which idiosyncratic shocks propagate, accounting for the peculiar structure of the return correlation across different maturities. In this paper, we revisit the specific ‘stiff’ elastic string field theory of Baaquie and Bouchaud [Stiff field theory of interest rates and psychological future time. Wilmott Mag., 2004, 2–6] in a way that makes its micro-foundation more transparent. Our model can be interpreted as capturing the effect of market forces that set the rates of nearby tenors in a self-referential fashion. The model is parsimonious and accurately reproduces the whole correlation structure of the FRC over the time period 1994–2023, with an error around 1% and with only one adjustable parameter, the value of which being very stable across the last three decades. The dependence of correlation on time resolution (also called the Epps effect) is also faithfully reproduced within the model and leads to a cross-tenor information propagation time on the order of 30 minutes. Finally, we confirm that the perceived time in interest rate markets is a strongly sub-linear function of real time, as surmised by Baaquie and Bouchaud [Stiff field theory of interest rates and psychological future time. Wilmott Mag., 2004, 2–6]. In fact, our results are fully compatible with hyperbolic discounting, in line with the recent behavioral Finance literature Farmer and Geanakoplos [Hyperbolic Discounting is Rational: Valuing the Far Future with Uncertain Discount Rates, Cowles Foundation Discussion Papers, 2009 (Cowles Foundation for Research in Economics, Yale University)].