Concordance in global office market cycles

A large proportion of international real estate investment is concentrated in the office markets of the world’s largest cities. However, many of these global cities are also key financial services centres, highlighting the possibility of reduced economic diversification from an investor’s perspective. This paper assesses the degree of synchronization in cycles across twenty of the world’s largest office markets, finding evidence of significant concordance across a large number of markets. The results highlight the problems associated with commonalities in the underlying economic bases of the markets. The concentration of investment also raises the possibility of common flow of funds effects that may further reduce diversification opportunities.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Chaos in high performance digital robot controllers

Active robot force control requires some form of dynamic inner loop control for stability. The author considers the implementation of position-based inner loop control on an industrial robot fitted with encoders only. It is shown that high gain velocity feedback for such a robot, which is effectively stationary when in contact with a stiff environment, involves problems beyond the usual caveats on the effects of unknown environment stiffness. It is shown that it is possible for the controlled joint to become chaotic at very low velocities if encoder edge timing data are used for velocity measurement. The results obtained indicate that there is a lower limit on controlled velocity when encoders are the only means of joint measurement. This lower limit to speed is determined by the desired amount of loop gain, which is itself determined by the severity of the nonlinearities present in the drive system.

Probabilistic evaluation of time series models : a comparison of several approaches

Several methods are examined which allow to produce forecasts for time series in the form of probability assignments. The necessary concepts are presented, addressing questions such as how to assess the performance of a probabilistic forecast. A particular class of models, cluster weighted models (CWMs), is given particular attention. CWMs, originally proposed for deterministic forecasts, can be employed for probabilistic forecasting with little modification. Two examples are presented. The first involves estimating the state of (numerically simulated) dynamical systems from noise corrupted measurements, a problem also known as filtering. There is an optimal solution to this problem, called the optimal filter, to which the considered time series models are compared. (The optimal filter requires the dynamical equations to be known.) In the second example, we aim at forecasting the chaotic oscillations of an experimental bronze spring system. Both examples demonstrate that the considered time series models, and especially the CWMs, provide useful probabilistic information about the underlying dynamical relations. In particular, they provide more than just an approximation to the conditional mean.

Sensitivity and out-of-sample error in continuous time data assimilation

Data assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time–tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade-off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade-off. A relation between the sensitivity and the out-of-sample error is established, which allows the latter to be calculated under operational conditions. A minimum out-of-sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade-off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, also known as synchronization. Numerical examples demonstrate the feasibility of the approach.

Numerical simulation of the Filchner overflow

The plume of Ice Shelf Water (ISW) flowing into the Weddell Sea over the Filchner sill contributes to the formation of Antarctic Bottom Water. The Filchner overflow is simulated using a hydrostatic, primitive equation three-dimensional ocean model with a 0.5–2 Sv ISW influx above the Filchner sill. The best fit to mooring temperature observations is found with influxes of 0.5 and 1 Sv, below a previous estimate of 1.6 ± 0.5 Sv based on sparse mooring velocities. The plume first moves north over the continental shelf, and then turns west, along slope of the continental shelf break where it breaks up into subplumes and domes, some of which then move downslope. Other subplumes run into the eastern submarine ridge and propagate along the ridge downslope in a chaotic manner. The next, western ridge is crossed by the plume through several paths. Despite a number of discrepancies with observational data, the model reproduces many attributes of the flow. In particular, we argue that the temporal variability shown by the observations can largely be attributed to the unstable structure of the flow, where the temperature fluctuations are determined by the motion of the domes past the moorings. Our sensitivity studies show that while thermobaricity plays a role, its effect is small for the flows considered. Smoothing the ridges out demonstrate that their presence strongly affects the plume shape around the ridges. An increase in the bottom drag or viscosity leads to slowing down, and hence thickening and widening of the plume

Influence of the quasi-biennial oscillation on the extratropical winter stratosphere in an atmospheric general circulation model and in reanalysis data

The interannual variability of the stratospheric polar vortex during winter in both hemispheres is observed to correlate strongly with the phase of the quasi-biennial oscillation (QBO) in tropical stratospheric winds. It follows that the lack of a spontaneously generated QBO in most atmospheric general circulation models (AGCMs) adversely affects the nature of polar variability in such models. This study examines QBO–vortex coupling in an AGCM in which a QBO is spontaneously induced by resolved and parameterized waves. The QBO–vortex coupling in the AGCM compares favorably to that seen in reanalysis data [from the 40-yr ECMWF Re-Analysis (ERA-40)], provided that careful attention is given to the definition of QBO phase. A phase angle representation of the QBO is employed that is based on the two leading empirical orthogonal functions of equatorial zonal wind vertical profiles. This yields a QBO phase that serves as a proxy for the vertical structure of equatorial winds over the whole depth of the stratosphere and thus provides a means of subsampling the data to select QBO phases with similar vertical profiles of equatorial zonal wind. Using this subsampling, it is found that the QBO phase that induces the strongest polar vortex response in early winter differs from that which induces the strongest late-winter vortex response. This is true in both hemispheres and for both the AGCM and ERA-40. It follows that the strength and timing of QBO influence on the vortex may be affected by the partial seasonal synchronization of QBO phase transitions that occurs both in observations and in the model. This provides a mechanism by which changes in the strength of QBO–vortex correlations may exhibit variability on decadal time scales. In the model, such behavior occurs in the absence of external forcings or interannual variations in sea surface temperatures.

What is the climate system able to do ‘on its own’?

The climate of the Earth, like planetary climates in general, is broadly controlled by solar irradiation, planetary albedo and emissivity as well as its rotation rate and distribution of land (with its orography) and oceans. However, the majority of climate fluctuations that affect mankind are internal modes of the general circulation of the atmosphere and the oceans. Some of these modes, such as El Nino-Southern Oscillation (ENSO), are quasi-regular and have some longer-term predictive skill; others like the Arctic and Antarctic Oscillation are chaotic and generally unpredictable beyond a few weeks. Studies using general circulation models indicate that internal processes dominate the regional climate and that some like ENSO events have even distinct global signatures. This is one of the reasons why it is so difficult to separate internal climate processes from external ones caused, for example, by changes in greenhouse gases and solar irradiation. However, the accumulation of the warmest seasons during the latest two decades is lending strong support to the forcing of the greenhouse gases. As models are getting more comprehensive, they show a gradually broader range of internal processes including those on longer time scales, challenging the interpretation of the causes of past and present climate events further.

Climate predictability experiments with a general circulation model

The atmospheric response to the evolution of the global sea surface temperatures from 1979 to 1992 is studied using the Max-Planck-Institut 19 level atmospheric general circulation model, ECHAM3 at T 42 resolution. Five separate 14-year integrations are performed and results are presented for each individual realization and for the ensemble-averaged response. The results are compared to a 30-year control integration using a climate monthly mean state of the sea surface temperatures and to analysis data. It is found that the ECHAM3 model, by and large, does reproduce the observed response pattern to El Nin˜o and La Nin˜a. During the El Nin˜ o events, the subtropical jet streams in both hemispheres are intensified and displaced equatorward, and there is a tendency towards weak upper easterlies over the equator. The Southern Oscillation is a very stable feature of the integrations and is accurately reproduced in all experiments. The inter-annual variability at middle- and high-latitudes, on the other hand, is strongly dominated by chaotic dynamics, and the tropical SST forcing only modulates the atmospheric circulation. The potential predictability of the model is investigated for six different regions. Signal to noise ratio is large in most parts of the tropical belt, of medium strength in the western hemisphere and generally small over the European area. The ENSO signal is most pronounced during the boreal spring. A particularly strong signal in the precipitation field in the extratropics during spring can be found over the southern United States. Western Canada is normally warmer during the warm ENSO phase, while northern Europe is warmer than normal during the ENSO cold phase. The reason is advection of warm air due to a more intense Pacific low than normal during the warm ENSO phase and a more intense Icelandic low than normal during the cold ENSO phase, respectively.

A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

Four-dimensional data assimilation and balanced dynamics

The problem of spurious excitation of gravity waves in the context of four-dimensional data assimilation is investigated using a simple model of balanced dynamics. The model admits a chaotic vortical mode coupled to a comparatively fast gravity wave mode, and can be initialized such that the model evolves on a so-called slow manifold, where the fast motion is suppressed. Identical twin assimilation experiments are performed, comparing the extended and ensemble Kalman filters (EKF and EnKF, respectively). The EKF uses a tangent linear model (TLM) to estimate the evolution of forecast error statistics in time, whereas the EnKF uses the statistics of an ensemble of nonlinear model integrations. Specifically, the case is examined where the true state is balanced, but observation errors project onto all degrees of freedom, including the fast modes. It is shown that the EKF and EnKF will assimilate observations in a balanced way only if certain assumptions hold, and that, outside of ideal cases (i.e., with very frequent observations), dynamical balance can easily be lost in the assimilation. For the EKF, the repeated adjustment of the covariances by the assimilation of observations can easily unbalance the TLM, and destroy the assumptions on which balanced assimilation rests. It is shown that an important factor is the choice of initial forecast error covariance matrix. A balance-constrained EKF is described and compared to the standard EKF, and shown to offer significant improvement for observation frequencies where balance in the standard EKF is lost. The EnKF is advantageous in that balance in the error covariances relies only on a balanced forecast ensemble, and that the analysis step is an ensemble-mean operation. Numerical experiments show that the EnKF may be preferable to the EKF in terms of balance, though its validity is limited by ensemble size. It is also found that overobserving can lead to a more unbalanced forecast ensemble and thus to an unbalanced analysis.

Stirring and mixing in two-dimensional divergent flow

While stirring and mixing properties in the stratosphere are reasonably well understood in the context of balanced (slow) dynamics, as is evidenced in numerous studies of chaotic advection, the strongly enhanced presence of high-frequency gravity waves in the mesosphere gives rise to a significant unbalanced (fast) component to the flow. The present investigation analyses result from two idealized shallow-water numerical simulations representative of stratospheric and mesospheric dynamics on a quasi-horizontal isentropic surface. A generalization of the Hua–Klein Eulerian diagnostic to divergent flow reveals that velocity gradients are strongly influenced by the unbalanced component of the flow. The Lagrangian diagnostic of patchiness nevertheless demonstrates the persistence of coherent features in the zonal component of the flow, in contrast to the destruction of coherent features in the meridional component. Single-particle statistics demonstrate t2 scaling for both the stratospheric and mesospheric regimes in the case of zonal dispersion, and distinctive scaling laws for the two regimes in the case of meridional dispersion. This is in contrast to two-particle statistics, which in the mesospheric (unbalanced) regime demonstrate a more rapid approach to Richardson’s t3 law in the case of zonal dispersion and is evidence of enhanced meridional dispersion.

Planning autonomous vehicles in the absence of speed lanes using lateral potentials

Chaotic traffic, prevalent in many countries, is marked by a large number of vehicles driving with different speeds without following any predefined speed lanes. Such traffic rules out using any planning algorithm for these vehicles which is based upon the maintenance of speed lanes and lane changes. The absence of speed lanes may imply more bandwidth and easier overtaking in cases where vehicles vary considerably in both their size and speed. Inspired by the performance of artificial potential fields in the planning of mobile robots, we propose here lateral potentials as measures to enable vehicles to decide about their lateral positions on the road. Each vehicle is subjected to a potential from obstacles and vehicles in front, road boundaries, obstacles and vehicles to the side and higher speed vehicles to the rear. All these potentials are lateral and only govern steering the vehicle. A speed control mechanism is also used for longitudinal control of vehicle. The proposed system is shown to perform well for obstacle avoidance, vehicle following and overtaking behaviors.