Before addressing these issues, it is worth mentioning that important strides have been taken in the field of detection and attribution (D&A) of individual events to climate change . To start, changes in global quantities that involve averaging over large spans of time and large areas of the globe have been both detected in and attributed to, with considerable confidence, anthropogenic changes in the atmospheric concentration of aerosols and greenhouse gases . The D&A of regional changes e.g., and, a fortiori, of individual events e.g., is considerably more difficult and much less incontrovertible.

Given the substantial impact of extreme events on human life and socioeconomic well-being e.g.,, an important step in achieving greater rigor in this field is a greater reliance on the counterfactual theory of necessary and sufficient causation, formulated by Judea Pearl , in the attribution of such events.

The counterfactual definition of causality goes back to the Scottish Enlightenment philosopher, historian, economist, and essayist David Hume (1711–1776), widely remembered for his empiricism and skepticism. It can be stated simply as follows: Y is caused by X if, and only if, Y would not have occurred were it not for X.

The usual identification of Pearl's causal theory as “counterfactual” appears to be, at first glance, rather counterintuitive. We take, therefore, a little detour here to explain briefly the theory and outline how it differs from the usual approach taken so far in D&A studies . In doing so, we follow .

An individual event is characterized by a binary variable, Y∈{0,1}, and, say, the threshold exceedance of surface air temperatures for a time interval, τ, and over an area, A. For brevity, we will use the “event Y” as a stand-in for the event defined by {Y=1}. The idea of causation of Y by a difference f∈F in the forcing – with F representing a set of values of insolation, atmospheric composition, etc. – is to distinguish between a situation in which f has the value measured during Y in the real, or factual, world, and the value f=0 that it would have had in an alternative, or counterfactual, world. The presence or absence of the extra forcing, f, is captured by another binary variable, Xf.

Obviously, the distinction between the two situations requires one to estimate the probability, p1=P(Y=1|Xf=1), of the event occurring in the factual world and the probability, p0=P(Y=1|Xf=0), of it occurring in the counterfactual world. The prevailing approach is to, given estimates p1 and p0, t compute the so-called fraction of attributable risk FAR as follows: 1FAR=1-p0/p1. We skip here several important steps in causality theory that involve comparing directed dependency graphs when one is interested in more than one possible effect – e.g., a dust devil and a hailstorm – and more than one cause may be at play, such as the values of the temperature field and those of the wind field in some neighborhood of the observed event. Please see Fig. 1, and the discussion thereof, and Sect. 2.

The key mathematical novelty in Pearl's counterfactual theory of causation is the realization that, following Hume, a cause should be both necessary and sufficient in order to unambiguously attribute an observed event to it. Instead of merely computing the fraction of attributable risk, FAR, as in Eq. (1), one needs to define and compute the probabilities PN, PS, and PNS of necessary, sufficient, and necessary and sufficient causation.

Thus, the probability, PN, of necessary causation is defined as the probability that the event, Y, would not have occurred in the absence of the event, X, given that both events, Y and X, did in fact occur. Sufficient causation, on the other hand, means that X always triggers Y but that Y may also occur for other reasons without requiring X. Finally, PNS is the probability that a cause is both necessary and sufficient. These three definitions are formally expressed as follows: 2aPN≡P(Y0=0|Y=1,X=1),2bPS≡P(Y1=1|Y=0,X=0),2cPNS≡P(Y0=0,Y1=1). Recall that the subscript 1 refers to the factual world, while the subscript 0 refers to the counterfactual one; see Eq. ().

The definitions in Eq. (2) are precise and unambiguously implementable, as long as a fully specified probabilistic model of the world is formulated. Under certain assumptions, spelled out by , the probabilities PN, PS, and PNS can be calculated as follows: 3PN=1-p0p1,PN=1-1-p11-p0,PNS=p1-p0. One can easily see that PN is more sensitive to p0 than to p1, and, conversely, that PS is more sensitive to p1 than to p0; necessary causation is enhanced further by an event being rare in the counterfactual world, whereas sufficient causation is enhanced further by it being frequent in the factual one; see, for instance, Fig. 2.

An interesting idea – first articulated by and further implemented by – is to apply data assimilation methodology e.g., for the computation of these three probabilities, using observations from the factual world and a model that encapsulates the knowledge of the system's evolution. One uses two versions of the latter model, namely the factual one with Xf=1 and the other one with Xf=0, and the data are supposed to tell one whether PNS is sufficiently close to unity or not.

Returning now to the issues of near-optimal control of climate change, it is important to realize that what needs to be controlled is not just the global and annual mean surface air temperature, T‾, as originally studied in the report. To outline the progress made in the 4 decades since the Charney report in thinking about anthropogenic effects on climate, please consider Fig.  herein.

The figure is a highly simplified conceptual diagram of the way that anthropogenic changes in radiative forcing would change the behavior of a climate system with increasingly complex characteristics, as one proceeds from Fig. a through Fig. b and on to Fig. c. Therefore, neither the time, t, on the abscissa nor the CO2 concentration and temperature, T‾, on the ordinate are labeled quantitatively in the three panels. The time we think of is years to decades, and the ranges of the CO2 concentration and T‾ correspond roughly to those expected for the difference in values between the end of the 21st century and the beginning of the 19th century. To keep things as simple as possible – but definitely not any simpler – anthropogenic changes in radiative forcing have been represented by a sudden jump in CO2 concentration, as in the Charney report.

The climate model represented in the Fig. a can be as simple as a forced linear, scalar, ordinary differential equation representing an energy balance model, as follows: 4x˙=-λx+H(t)(x-x1),x(t)≡T‾(t)-T‾0, with λ>0, and H(t) a Heaviside function that jumps from H=0, for t≤0, to H=1, for t>0. Here T‾0 and T‾1 are the model's equilibrium climates for the radiative forcings before and after the jump, respectively, while λ gives the rate of exponentially approaching the new equilibrium T‾1.

The case of Fig. b can be seen as an idealized climate system in which the El Niño–Southern Oscillation (ENSO) would be perfectly periodic, rather than having an irregular, 2–7 year periodicity with additional periodicities and chaotic components present, as in Fig. c. There are no serious doubts as to the long-term mean, T‾1, after the jump being larger than the preindustrial or current mean, T‾0. But, figuring out the higher moments of the long-term probability density function (pdf) after the jump is another matter entirely.

Recently, increasing attention has been paid by high-end modelers to the difficulties posed by the presence of internal variability in the climate system. For instance, and references therein point to this variability's imperfect simulation and to the consequences of attempting to use models with this marked deficiency to predict future climates on multidecadal timescales.

Concerning the ENSO's distribution of extreme events, investigated its dependence, in an idealized delay differential equation (DDE) model, on several model parameters. They also found that plotting the model's PBA, with respect to the seasonally periodic forcing, provided a much better understanding of the role of the seasonal cycle in the model.

found that parameter dependence in such a DDE model can lead to a critical transition between two types of chaotic behavior which differ substantially in their distribution of extreme events. This contrast is clearly apparent in Fig. , and it illustrates the types of nonequilibrium climate changes suggested by Fig. c.

The changes in the invariant, time-dependent measure, μt, supported on this ENSO model's PBA are plotted in Fig. a–c as a function of the control parameter a. The change in the PBA is clearly associated with the population lying towards the ends of the elongated filaments apparent in the figure. This population represents strong, warm El Niño and cold La Niña events.

The PBA experiences a critical transition at a value, a∗; here, h(t) is the thermocline depth anomaly from seasonal depth values at the domain's eastern boundary, with t in years; a=(1.12+δ))/180 and 0.015700<δ∗<0.015707. Thus, μt(a) faithfully encrypts the disappearance of such extreme events as a↗a∗. Adding stochastic perturbations to the model can smooth out the transition, which might make it less drastic in high-end models and in observations. Again, the study of the model's PBA greatly facilitates the understanding of the processes involved.

We present here, concisely, one particular EnBC model, and the role that active economic dynamics may have in modifying the effect of natural hazards on such an economy . The nonequilibrium dynamical model (NEDyM) of is a neoclassical model based on the model, in which equilibrium constraints associated with the clearing of goods and labor markets are replaced by dynamic relationships that involve adjustment delays. The model has eight state variables – which include production, capital, number of workers employed, wages, and prices – and the evolution of these variables is modeled by a set of ordinary differential equations. For a brief summary of the model equations, please see Appendix A; the parameters and their values are listed in Table 3.

NEDyM's main control parameter is the investment flexibility αinv, which measures the adjustment speed of investments in response to profitability signals. This parameter describes how rapidly investment can react to a profitability signal. If αinv is very large, investment soars when profits are high and collapses when profits are small, while a small αinv entails a much slower adjustment of the investment to the size of the profits. Introducing this parameter is equivalent to allocating an investment adjustment cost, as proposed by and by ; these authors found that introducing adjustment costs and delays helps to match the key features of macroeconomic models to the data.

In NEDyM, for small αinv, i.e., slow adjustment, the model has a stable equilibrium, which was calibrated to the economic state of the European Union (EU-15) in 2001 . As the adjustment flexibility increases, this equilibrium loses its stability and undergoes a Hopf bifurcation, after which the model exhibits a stable periodic solution .

Business cycles in NEDyM originate from the instability of the profit–investment feedback, which is quite similar to the Keynesian accelerator–multiplier effect. Furthermore, the cycles are constrained and limited in amplitude by the interplay of the following three processes: (i) a reserve army of labor effect, namely labor costs increasing when the employment rate is high, (ii) the inertia of production capacity, and (iii) the consequent inflation in goods prices when demand increases too rapidly. The model's bifurcation diagram is shown in Fig. .

For somewhat greater investment flexibility, the model exhibits chaotic behavior because a new constraint intervenes, namely limited investment capacity. In this chaotic regime, the cycles become quite irregular, with sharper recessions and recoveries of variable duration. In the present paper, we concentrate, for the sake of simplicity, on model behavior in the purely periodic regime, i.e., we have regular EnBCs but no chaos. Such periodic behavior is illustrated in Fig. .

The NEDyM business cycle is consistent with many stylized facts described in macroeconomic literature, such as the phasing of the distinct economic variables along the cycle, with the distinct phrases being referred to in this literature as comovements. The model also reproduces the observed asymmetry of the cycle, with recessions that are much shorter than expansions. This typical sawtooth shape of a business cycle is not well captured by RBC models, whose linear, auto-regressive character gives intrinsically symmetric behavior around the equilibrium. The amplitude of the price–wage oscillation, however, is too large in NEDyM, calling for a better calibration of the parameters and further refinements of the model.

In the setting of the 2008 economic and financial crisis, the banks' and other financial institutions' large losses clearly reduced access to credit; such a reduction very strongly affects investment flexibility. The EnBC model can thus help explain how changes in αinv can seriously perturb the behavior of the entire economic system, either by increasing or decreasing the variability in macroeconomic variables; see Fig. . Moreover, these losses also lead to a reduction in aggregated demand that, in turn, can lead to a reduction in economic production and a full-scale recession.

The immediate damage caused by a natural disaster is typically augmented by the cost of reconstruction, which is a major concern when considering the disaster's socioeconomic consequences. Reconstruction may also lead, though, to an increase in productivity by allowing for technical changes to be included in the reconstructed capital; technical changes can also sustain the demand and help economic recovery. Economic productivity may be reduced, however, during reconstruction because some vital sectors are not functional, and reconstruction investments crowd out investment into new production capacity e.g.,and references therein.

In particular, among others have suggested that the overall cost of a natural disaster might depend on the preexisting economic situation. For instance, the Marmara earthquake in 1999 caused destruction that amounted to 1.5 %–3 % of Turkey's GDP; its cost in terms of production loss, however, is believed to have been fairly modest due to the fact that the country was experiencing a strong recession of -7 % of the GDP in the year before the disaster .

To study how the state of the economy may influence the consequences of natural disasters, introduced into NEDyM the disaster-modeling scheme of , in which natural disasters destroy the productive capital through a modified production function. Furthermore, to account for market frictions and constraints in the reconstruction process, the reconstruction expenditures are limited.

These authors showed that the transition from an equilibrium regime to a nonequilibrium regime can radically change the long-term response to exogenous shocks in an EnBC model. Idealized as it may be, NEDyM shows that the long-term effects of a sequence of extreme events depend upon the economy's behavior; an economy in stable equilibrium with very little, or no, flexibility (αinv≲0.5, see Fig. ) is more vulnerable than a more flexible economy, albeit still at or near equilibrium (e.g., αinv≃1.0). Clearly, if investment flexibility is nil or very low, the economy is incapable of responding to the natural disasters through investment increases aimed at reconstruction; total production losses, therefore, are quite large. Such an economy behaves according to a pure growth model, where the savings, and therefore the investment, ratio is constant; see Table 1.

When investment can respond to profitability signals without destabilizing the economy, i.e., when αinv is nonzero but still lower than the critical bifurcation value of αinv≃1.39, the economy has greater freedom to improve its overall state and, thus, respond to productive capital influx. Such an economy is much more resilient to disasters because it can adjust its level of investment in the disaster's aftermath.

If investment flexibility, αinv, is larger than its Hopf bifurcation value, the economy undergoes periodic EnBCs, and along such a cycle, NEDyM passes through phases that differ in their stability. This, in turn, leads to a phase-dependent response to exogenous shocks and, consequently, to a phase-dependent vulnerability of the economic system, as illustrated in Fig. .

A key point we wish to make in our excursion into the economical aspects of problem 10 is precisely this phase dependency of the economy's response to natural hazards.

In fact, found an interesting vulnerability paradox. The indirect costs caused by extreme events during a growth phase of the economy are much higher than those that occur during a deep recession. Figure  illustrates this paradox by showing, in Figure a, a typical business cycle and, in Figure b, the corresponding losses for disasters hitting the economy in different phases of this cycle. The vertical lines in both panels, with blue at the end of the recession and red in the expansion phase, highlight the paradox. Sect. 2.2 discussed further aspects of this paradox and analogous considerations found in the much earlier work of .

Once noted in NEDyM behavior, this apparent paradox can be easily explained as disasters during high-growth episodes enhance preexisting disequilibria. Inventories are low and cannot compensate for the reduced production; employment is high, and hiring more employees induces wage inflation, while the producer lacks financial resources to increase investment. The opposite holds true during recessions as mobilizing investment and labor is much easier e.g.,.

As a consequence, production losses due to disasters that occur during expansion phases are strongly amplified, while they are reduced when the shocks occur during the recession phase. On average, however, (i) expansions last much longer than recessions in our NEDyM model as well as in reality, and (ii) amplification effects are larger than damping effects. It follows that the net effect of the cycle is strongly unfavorable to the economy, with an average production loss that is almost as large, for αinv=2.5, as for αinv=0; see again Table 1.

Beyond the obvious implications for disaster assessment, insurance, and other practical issues treated by and references therein, the findings shown schematically in Fig.  suggest a theoretically intriguing connection with the fluctuation–dissipation theory (FDT) in statistical mechanics. The FDT has its roots in the classical theory of many-particle systems in thermodynamic equilibrium. The idea goes back to , and it is very simple; the system's return to equilibrium will be the same whether the perturbation that modified its state is due to a small external impulse or to an internal, random fluctuation. The FDT thus establishes a useful relationship between the natural and the forced fluctuations of a system e.g.,; it is a cornerstone of statistical physics and has applications in many other areas and references therein. and have recently reviewed FDT applications in the climate sciences, in both the classical form used for systems in equilibrium , and in its more recent extensions to systems out of equilibrium, based on the Ruelle response theory .

The results in Sect.  above strongly suggest that the response to exogenous shocks of an economic system might differ from one phase of a business cycle to another. Hence, it is quite possible that the system's endogenous variability might also vary with the phase of a cycle that the system is in. More explicitly, the system's internal endogenous fluctuations may change in variance as the phase of the business cycle evolves in the same way as the exogenously driven ones do; i.e., larger “economic volatility” can be expected during expansions than during contractions of the economy. And, if this is the case, out-of-equilibrium response theory may apply to economic systems in the same way that it has been found to apply to the climate system, with both the local-in-time sensitivity and volatility being phase dependent.

There is a long tradition of systematically analyzing cyclic behavior in economic data . Yet there is no trace, as far as we could tell, of an investigation along the lines proposed herein. Hence, set out to study the USA's macroeconomic data provided by the Bureau of Economic Analysis (BEA) for 1954–2005 to evaluate the evidence for the FDT conjecture suggested by the results reviewed in Sect.  above. The nine macroeconomic indicators these authors used were GDP, investment, consumption, employment rate (in %), total wage, change in private inventories, price, exports, and imports; see http://www.bea.gov/ (last access: 11 September 2020).

The nine indicators were each separately detrended by a filter, normalized by the trend values, and then collectively analyzed by using a data-adaptive multichannel singular spectrum analysis (M-SSA) filter; see , chap. 12 and Appendix B for details. The statistical significance of the results was carefully tested against an AR(1) null hypothesis , and they are illustrated in Fig. .

The nine detrended and normalized time series are shown in Fig. a, with the leading-mode pair of the joint M-SSA analysis in Fig. b. A simple counting of maxima and minima in Fig. b gives 10.5 cycles in 52 years, which agrees rather well with the NEDyM model's 5–6 year period and with the National Bureau of Economic Research's (NBER's) count of 11 cycles for the 65 year interval of 1945–2009, which yields an average period of 69 months = 5.75 years

https://www.nber.org/cycles/cyclesmain.html, last access: 6 April 2020; pdf version dated 23 April 2012.

In Fig. c the evolution in time of the local variance associated with all nine indicators is plotted, as measured over a sliding window of M=24 quarters = 6 years. It is clear that the local variance of the fluctuations, as defined in Appendix A, is consistent with the FDT hypothesis, especially over the latter part of the BEA data set; e.g., the local variance, VK(t), of the fluctuations during the NBER-defined recessions of July 1981 (16 months), July 1990 (8 months), and March 2001 (8 months) is at or very near to a minimum, while substantial local maxima of VK(t) are attained during the expansions in between.

Synchronization, known in the 1970s and 1980s as entrainment , is a key feature of nonlinear oscillators that has been known since Christiaan Huygens' experiment of 1665 in which two pendulum clocks with slightly different lengths synchronized. More recently, the synchronization of chaotic oscillators has become a topic of growing interest in the physical and biological sciences e.g.,.

Still, while the emergence of business cycle synchronization across countries has been widely acknowledged – especially in view of the ongoing globalization of economic activity – no agreement has emerged so far on basic issues like the quantification of comovements. Given the relative shortness of macroeconomic time series, efforts to apply advanced univariate analysis methods to them e.g., have provided interesting but not quite conclusive results.

To overcome the difficulties posed by the simultaneous analysis of a large number of time series, applied a suitable modification of M-SSA to macroeconomic data from the World Development Indicators (WDI) database of the World Bank at http://databank.worldbank.org (last access: 11 September 2020). The data set extracted from the WDI database comprises five macroeconomic indicators for 104 countries, namely GDP, gross fixed capital formation (GDI, formerly gross domestic fixed investment), final consumption expenditure (CON), exports (EXP), and imports (IMP) of goods and services, with all variables in constant 2010 USD. The data were only analyzed for the 46 year interval (1970–2015) for which at least one of the indicators chosen was available for each of the 104 economies selected. The main result of this analysis is shown in Fig. .

The leading mode in the figure has a rough periodicity of 7–11 years, captures 73 % of the trend residual's variance, and is statistically significant according to the stringent tests of . A key ingredient in the tests applied is the much larger number of data points used by the improved M-SSA methodology in examining not just GDP but a more complete set of indicators, with nine time series for Fig.  and five for Fig. .

The latter figure clearly illustrates the dominance of the USA economy over the 1970–2015 time interval, with the UK perfectly aligned on the USA indicators, while other European countries, including even the Russian Federation, lag somewhat behind. Japan is also in very good alignment with the USA, while China is in almost perfect phase opposition with it, whereas India and Indonesia follow the Chinese lead. More complicated lead-and-lag patterns are present in the much smaller economies of South America and Africa.